#! /bin/sh # This is a shell archive. Remove anything before this line, then unpack # it by saving it into a file and typing "sh file". To overwrite existing # files, type "sh file -c". You can also feed this as standard input via # unshar, or by typing "sh 'AREADME.1ST' <<'END_OF_FILE' X *************************************************************************** X * All the software contained in this library is protected by copyright. * X * Permission to use, copy, modify, and distribute this software for any * X * purpose without fee is hereby granted, provided that this entire notice * X * is included in all copies of any software which is or includes a copy * X * or modification of this software and in all copies of the supporting * X * documentation for such software. * X *************************************************************************** X * THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED * X * WARRANTY. IN NO EVENT, NEITHER THE AUTHORS, NOR THE PUBLISHER, NOR ANY * X * MEMBER OF THE EDITORIAL BOARD OF THE JOURNAL "NUMERICAL ALGORITHMS", * X * NOR ITS EDITOR-IN-CHIEF, BE LIABLE FOR ANY ERROR IN THE SOFTWARE, ANY * X * MISUSE OF IT OR ANY DAMAGE ARISING OUT OF ITS USE. THE ENTIRE RISK OF * X * USING THE SOFTWARE LIES WITH THE PARTY DOING SO. * X *************************************************************************** X * ANY USE OF THE SOFTWARE CONSTITUTES ACCEPTANCE OF THE TERMS OF THE * X * ABOVE STATEMENT. * X *************************************************************************** X X AUTHOR: X X P. C. HANSEN X DEPT. OF MATHEMATICAL MODELLLING X TECHNICAL UNIVERSITY OF DENMARK X X REFERENCE: X X REGULARIZATION TOOLS: A MATLAB PACKAGE FOR ANALYSIS AND SOLUTION OF X DISCRETE ILL-POSED PROBLEMS, X NUMERICAL ALGORITHMS, 6 (1994), PP. 1-35 X X SOFTWARE REVISION: X X Ver 2.1 MARCH 31, 1998 X X SOFTWARE LANGUAGE: X X MATLAB 4 X X************************************************************************** X XRegularization Tools. XVersion 2.1 31-March-98. XCopyright (c) 1998 by Per Christian Hansen and UNI-C. X XThe installation of Regularization Tools is very simple: X X 1. Unpack the shell archive na4-matlab4 by executing the command X /bin/sh na4-matlab4 X X 2. Remove the file na4-matlab4 X X 3. If the Spline Toolbox is installed on your computer, then remove X the following 10 m-files X fnder.m sp2pp.m X ppbrk.m sorted.m X ppcut.m spbrk.m X ppmak.m spmak.m X ppual.m sprpp.m X X 4. The file Manual.ps contains the related manual in PostScript form X (not revised) X X*************************************************************** X* This is Version 2.1 of Regularization Tools for Matlab 4.2c * X*-------------------------------------------------------------* X* Per Christian Hansen, IMM * X*************************************************************** X XRevisions in Version 2.1 X X02/01/94: XFixed bug in cgls (s -> s2). X X08/03/94: XExpanded stopping criterion in newton. X X11/01/94: XModified get_l slightly such that the sign of L*x is correct. X X02/09/95: XRenamed csd to csdecomp (csd is now a function in the Signal Proc. Toolbox). XRevised gsvd to call csdecomp. X X07/02/97: XFixed bug in pcgls when computing filter factors. X X11/11/97: XModified gen_hh to compensate for Matlab's signum function. X X12/29/97: XCorrected bugs in discrep and lsqi. 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Fo(|except)p eop X%%Page: 9 10 X9 9 bop 59 166 a Fm(2.3.)34 b(SVD)15 b(AND)g(GENERALIZED)h(SVD)969 Xb Fo(9)p 59 178 1772 2 v 59 306 a(for)16 b(singular)h(v)o(ectors)e X(asso)q(ciated)i(with)f(m)o(ultiple)i(singular)f(v)m(alues.)24 Xb(In)17 b(connection)g(with)g(discrete)59 362 y(ill-p)q(osed)h X(problems,)d(t)o(w)o(o)f(c)o(haracteristic)i(features)f(of)f(the)i(SVD) Xf(of)g Fn(A)g Fo(are)g(v)o(ery)g(often)f(found.)127 455 Xy Fk(\017)23 b Fo(The)10 b(singular)i(v)m(alues)f Fn(\033)585 X462 y Fl(i)610 455 y Fo(deca)o(y)f(gradually)h(to)f(zero)g(with)h(no)g X(particular)g(gap)f(in)h(the)g(sp)q(ectrum.)173 512 y(An)i(increase)h X(of)f(the)g(dimensions)h(of)f Fn(A)g Fo(will)i(increase)f(the)f(n)o(um) Xo(b)q(er)g(of)g(small)h(singular)g(v)m(alues.)127 605 Xy Fk(\017)23 b Fo(The)14 b(left)g(and)g(righ)o(t)g(singular)h(v)o X(ectors)e Fn(u)889 612 y Fl(i)917 605 y Fo(and)h Fn(v)1026 X612 y Fl(i)1053 605 y Fo(tend)h(to)e(ha)o(v)o(e)h(more)f(sign)h(c)o X(hanges)g(in)h(their)173 661 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y(n)o(umerical)k(n)o(ull-v)o(ectors)f(of)f Fn(A)p XFo(.)28 b(F)l(rom)16 b(this)i(and)g(the)g(c)o(haracteristic)g(features) Xf(of)g Fn(A)h Fo(w)o(e)f(conclude)59 1529 y(that)i(the)g(matrix)g(in)h X(a)f(discrete)i(ill-p)q(osed)h(problem)e(is)g(alw)o(a)o(ys)e(highly)j X(ill-conditione)q(d,)i(and)c(its)59 1586 y(n)o(umerical)e(n)o X(ull-space)g(is)e(spanned)h(b)o(y)f(v)o(ectors)g(with)g(man)o(y)g(sign) Xh(c)o(hanges.)130 1642 y(The)10 b(SVD)h(also)f(giv)o(es)g(imp)q(ortan)o X(t)g(insigh)o(t)h(in)o(to)g(another)f(asp)q(ect)g(of)g(discrete)h X(ill-p)q(osed)i(problems,)59 1699 y(namely)k(the)g(smo)q(othing)f X(e\013ect)h(t)o(ypically)h(asso)q(ciated)f(with)f(a)h(square)f(in)o X(tegrable)h(k)o(ernel.)25 b(Notice)59 1755 y(that)19 Xb(as)g Fn(\033)248 1762 y Fl(i)281 1755 y Fo(decreases,)i(the)f X(singular)g(v)o(ectors)f Fn(u)941 1762 y Fl(i)974 1755 Xy Fo(and)h Fn(v)1089 1762 y Fl(i)1122 1755 y Fo(b)q(ecome)g(more)f(and) Xh(more)f(oscillatory)l(.)59 1812 y(Consider)i(no)o(w)e(the)h(mapping)h 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Fl(n)1501 1694 y Fo(\))14 Xb(are)g(iden)o(tical)i(to)59 1751 y(the)f(singular)h(v)m(alues)h(of)e XFn(A)p Fo(|except)h(for)e(the)i(ordering)f(of)g(the)g(singular)h(v)m X(alues)h(and)e(v)o(ectors.)130 1807 y(In)g(general,)f(there)h(is)g(no)f X(connection)h(b)q(et)o(w)o(een)g(the)g(generalized)h(singular)f(v)m X(alues/v)o(ectors)f(and)59 1863 y(the)e(ordinary)g(singular)h(v)m X(alues/v)o(ectors.)19 b(F)l(or)11 b(discrete)i(ill-p)q(osed)i X(problems,)e(though,)f(w)o(e)g(can)g(actu-)59 1920 y(ally)17 Xb(sa)o(y)d(something)i(ab)q(out)g(the)f(SVD-GSVD)h(connection)g(b)q X(ecause)h Fn(L)e Fo(is)h(t)o(ypically)h(a)e(reasonably)59 X1976 y(w)o(ell-conditioned)23 b(matrix.)33 b(When)20 Xb(this)g(is)g(the)g(case,)h(then)f(it)g(can)f(b)q(e)i(sho)o(wn)e(that)g X(the)h(matrix)59 2033 y Fn(X)h Fo(in)d(\(2.11\))e(is)i(also)f(w)o X(ell-conditioned.)30 b(Hence,)18 b(the)g(diagonal)g(matrix)f(\006)h(m)o X(ust)f(displa)o(y)h(the)g(ill-)59 2089 y(conditioning)g(of)d XFn(A)p Fo(,)h(and)g(since)h Fn(\015)660 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X(example)f(of)f(the)h(use)g(of)f(GSVD)h(in)g(the)g(analysis)g(of)f X(discrete)i(regulariza-)59 2428 y(tion)d(problems,)g(w)o(e)g(men)o X(tion)g(the)f(follo)o(wing)i(p)q(erturbation)f(b)q(ound)h(for)e X(Tikhono)o(v)h(regularization)59 2484 y(deriv)o(ed)j(in)g([40].)28 Xb(Let)19 b Fn(E)h Fo(and)f Fn(e)f Fo(denote)g(the)h(p)q(erturbations)f X(of)g Fn(A)h Fo(and)f Fn(b)p Fo(,)g(resp)q(ectiv)o(ely)l(,)j(and)d(let) X62 2541 y(\026)-26 b Fn(x)85 2548 y Fl(\025)124 2541 Xy Fo(denote)18 b(the)g(exact)f(solution)i(to)e(the)g(unp)q(erturb)q(ed) Xi(problem;)h(then)d(the)h(relativ)o(e)g(error)f(in)i(the)59 X2597 y(p)q(erturb)q(ed)d(solution)g Fn(x)469 2604 y Fl(\025)506 X2597 y Fo(satis\014es)207 2678 y Fk(k)p Fn(x)256 2685 Xy Fl(\025)288 2678 y Fk(\000)d Fo(\026)-26 b Fn(x)359 X2685 y Fl(\025)381 2678 y Fk(k)404 2685 y Fj(2)p 207 X2698 216 2 v 259 2740 a Fk(k)s Fo(\026)g Fn(x)308 2747 Xy Fl(\025)329 2740 y Fk(k)352 2747 y Fj(2)469 2709 y XFk(\024)591 2678 y(k)p Fn(A)p Fk(k)671 2685 y Fj(2)697 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Fi(\023)1715 2833 y Fo(\(2.14\))p eop X%%Page: 11 12 X11 11 bop 59 166 a Fm(2.4.)34 b(THE)15 b(DISCRETE)h(PICARD)f(CONDITION) Xh(AND)g(FIL)l(TER)g(F)-5 b(A)o(CTORS)251 b Fo(11)p 59 X178 1772 2 v 59 306 a(where)14 b(w)o(e)g(ha)o(v)o(e)g(de\014ned)h XFn(b)534 313 y Fl(\025)568 306 y Fo(=)e Fn(A)8 b(x)684 X313 y Fl(\025)720 306 y Fo(and)14 b Fn(r)828 313 y Fl(\025)862 X306 y Fo(=)f Fn(b)7 b Fk(\000)i Fn(b)1001 313 y Fl(\025)1022 X306 y Fo(.)19 b(The)c(imp)q(ortan)o(t)e(conclusion)j(w)o(e)e(can)g(mak) Xo(e)59 362 y(from)i(this)h(relation)g(is)g(that)e(for)h(all)h X(reasonable)g Fn(\025)f Fo(the)h(p)q(erturbation)g(b)q(ound)g(for)f X(the)g(regularized)59 419 y(solution)h Fn(x)258 426 y XFl(\025)295 419 y Fo(is)f(prop)q(ortional)h(to)e Fn(\025)687 X402 y Ff(\000)p Fj(1)747 419 y Fo(and)h(to)f(the)h(norm)f(of)g(the)h X(matrix)g Fn(X)t Fo(.)21 b(The)16 b(latter)f(quan)o(tit)o(y)59 X475 y(is)i(analyzed)g(in)g([41)o(])f(where)g(it)h(is)f(sho)o(wn)g(that) Xg Fk(k)p Fn(X)t Fk(k)988 482 y Fj(2)1021 475 y Fo(is)h(appro)o X(ximately)f(b)q(ounded)i(b)o(y)e Fk(k)p Fn(L)1673 459 Xy Ff(y)1690 475 y Fk(k)1713 482 y Fj(2)1731 475 y Fo(,)g(i.e.,)59 X532 y(b)o(y)i(the)g(in)o(v)o(erse)g(of)f(the)h(smallest)h(singular)f(v) Xm(alue)h(of)f Fn(L)p Fo(.)27 b(Hence,)19 b(in)g(addition)g(to)e(con)o X(trolling)i(the)59 588 y(smo)q(othness)12 b(of)f(the)h(regularized)h X(solution,)g Fn(\025)f Fo(and)g Fn(L)f Fo(also)h(con)o(trol)g(its)g X(sensitivit)o(y)h(to)e(p)q(erturbations)59 645 y(of)k XFn(A)g Fo(and)g Fn(b)p Fo(.)130 702 y(The)k(SVD)g(and)h(the)f(GSVD)g X(are)g(computed)h(b)o(y)f(means)g(of)g(routines)h Fe(csvd)g XFo(and)f Fe(gsvd)h Fo(in)g(this)59 759 y(pac)o(k)m(age.)g(The)15 Xb(routine)h Fe(bsvd)h Fo(computes)e(the)g(SVD)g(of)g(a)g(bidiagonal)i X(matrix.)59 894 y Fs(2.4.)h(The)g(Discrete)f(Picard)i(Condition)g(and)g X(Filter)e(F)-5 b(actors)59 998 y Fo(As)13 b(w)o(e)g(ha)o(v)o(e)g(seen)h X(in)g(Section)g(2.3,)e(the)h(in)o(tegration)g(in)h(Eq.)f(\(2.1\))f 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y(\026)637 1889 y Fn(b)f Fo(is)h(the)f(unp)q X(erturb)q(ed)i(righ)o(t-hand)f(side.)33 b(Both)1540 1877 Xy(\026)1541 1889 y Fn(b)19 b Fo(and)g(the)h(cor-)59 1946 Xy(resp)q(onding)g(unp)q(erturb)q(ed)h(solution)h(\026)-26 Xb Fn(x)19 b Fo(represen)o(t)g(the)g(underlying)i(unp)q(erturb)q(ed)f X(and)f(unkno)o(wn)59 2002 y(problem.)h(No)o(w,)12 b(if)h(w)o(e)g(w)o X(an)o(t)f(to)g(b)q(e)h(able)h(to)e(compute)h(a)f(regularized)j X(solution)e Fn(x)1482 2009 y Fl(r)q(eg)1546 2002 y Fo(from)f(the)h(giv) Xo(en)59 2059 y(righ)o(t-hand)h(side)g Fn(b)e Fo(suc)o(h)i(that)e XFn(x)625 2066 y Fl(r)q(eg)690 2059 y Fo(appro)o(ximates)g(the)i(exact)f X(solution)j(\026)-26 b Fn(x)p Fo(,)14 b(then)f(it)h(is)f(sho)o(wn)g(in) Xh([44)o(])59 2115 y(that)g(the)h(corresp)q(onding)g(exact)f(righ)o X(t-hand)h(side)957 2103 y(\026)959 2115 y Fn(b)f Fo(m)o(ust)g(satisfy)g X(a)g(criterion)i(v)o(ery)e(similar)i(to)e(the)59 2172 Xy(Picard)i(condition:)59 2289 y Fp(The)21 b(discrete)g(Picard)g X(condition)p Fo(.)31 b(The)18 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X(v)o(e)f(this)i(curv)o(e.)59 2407 y(The)16 b(solution)h(computed)f(b)o X(y)g(Tikhono)o(v)g(regularization)g(is)h(therefore)e(optimal)i(in)f X(the)g(sense)g(that)59 2463 y(for)f(a)g(giv)o(en)h(residual)h(norm)e X(there)g(do)q(es)h(not)f(exist)h(a)f(solution)h(with)g(smaller)g X(seminorm)f(than)h(the)59 2520 y(Tikhono)o(v)k(solution|and)h(the)f X(same)g(is)g(true)g(with)g(the)g(roles)g(of)f(the)h(norms)f(in)o(terc)o X(hanged.)35 b(A)59 2576 y(consequence)17 b(of)d(this)i(is)g(that)e(one) Xi(can)f(compare)g(other)g(regularization)h(metho)q(ds)f(with)h(Tikhono) Xo(v)59 2633 y(regularization)k(b)o(y)f(insp)q(ecting)j(ho)o(w)c(close)i X(the)g(L-curv)o(e)f(for)g(the)g(alternativ)o(e)h(metho)q(d)f(is)h(to)f X(the)59 2689 y(Tikhono)o(v)e(L-curv)o(e.)25 b(If)497 X2677 y(\026)499 2689 y Fn(b)16 b Fo(satis\014es)h(the)g(discrete)g X(Picard)h(condition,)g(then)f(the)g(t)o(w)o(o)f(L-curv)o(es)h(are)59 X2746 y(close)k(to)e(eac)o(h)h(other)f(and)h(the)g(solutions)h(computed) 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X(corner)f(and)h(iden)o(ti\014es)i(the)e(corresp)q(onding)59 X929 y(regularization)e(parameter.)29 b(Giv)o(en)19 b(a)f(discrete)h X(set)f(of)h(v)m(alues)g(of)f Fk(k)p Fn(A)8 b(x)1356 936 Xy Fl(r)q(eg)1419 929 y Fk(\000)13 b Fn(b)p Fk(k)1510 X936 y Fj(2)1546 929 y Fo(and)19 b Fk(k)p Fn(L)8 b(x)1726 X936 y Fl(r)q(eg)1777 929 y Fk(k)1800 936 y Fj(2)1818 X929 y Fo(,)59 985 y(routine)14 b Fe(plot)p 290 985 V X16 w(lc)f Fo(plots)h(the)f(corresp)q(onding)h(L-curv)o(e,)f(while)i X(routine)e Fe(l)p 1283 985 V 17 w(co)o(rner)f Fo(lo)q(cates)h(the)g X(L-curv)o(e's)59 1042 y(corner.)59 1178 y Fs(2.6.)18 Xb(T)-5 b(ransformation)18 b(to)g(Standard)i(F)-5 b(orm)59 X1283 y Fo(A)16 b(regularization)h(problem)f(with)h(side)f(constrain)o X(t)g(\012\()p Fn(x)p Fo(\))d(=)h Fk(k)p Fn(L)8 b Fo(\()p XFn(x)h Fk(\000)i Fn(x)1330 1267 y Ff(\003)1349 1283 y XFo(\))p Fk(k)1390 1290 y Fj(2)1424 1283 y Fo(\(2.5\))j(is)j(said)f(to)f X(b)q(e)i(in)59 1340 y Fg(standar)n(d)f(form)f Fo(if)g(the)g(matrix)f XFn(L)h Fo(is)g(the)g(iden)o(tit)o(y)g(matrix)f Fn(I)1121 X1347 y Fl(n)1144 1340 y Fo(.)19 b(In)d(man)o(y)e(applications,)i X(regulariza-)59 1396 y(tion)g(in)h(standard)f(form)f(is)h(not)g(the)g X(b)q(est)g(c)o(hoice,)h(i.e.,)e(one)i(should)f(use)h(some)e XFn(L)f Fk(6)p Fo(=)g Fn(I)1583 1403 y Fl(n)1622 1396 Xy Fo(in)j(the)f(side)59 1453 y(constrain)o(t)j(\012\()p XFn(x)p Fo(\).)30 b(The)19 b(prop)q(er)g(c)o(hoice)h(of)e(matrix)h XFn(L)g Fo(dep)q(ends)h(on)f(the)g(particular)g(application,)59 X1509 y(but)c(often)g(an)g(appro)o(ximation)h(to)e(the)h(\014rst)g(or)g X(second)h(deriv)m(ativ)o(e)g(op)q(erator)e(giv)o(es)i(go)q(o)q(d)f X(results.)130 1567 y(Ho)o(w)o(ev)o(er,)23 b(from)e(a)i(n)o(umerical)h X(p)q(oin)o(t)f(of)f(view)h(it)g(is)g(m)o(uc)o(h)f(simpler)i(to)e(treat) Xg(problems)h(in)59 1624 y(standard)10 b(form,)h(basically)h(b)q(ecause) Xg(only)f(one)g(matrix,)f Fn(A)p Fo(,)i(is)f(in)o(v)o(olv)o(ed)g X(instead)g(of)g(the)f(t)o(w)o(o)g(matrices)59 1680 y XFn(A)17 b Fo(and)f Fn(L)p Fo(.)24 b(Hence,)17 b(it)g(is)g(con)o(v)o X(enien)o(t)g(to)f(b)q(e)h(able)g(to)f(transform)f(a)i(giv)o(en)g X(regularization)g(problem)59 1737 y(in)i(general)f(form)f(in)o(to)g(an) Xh(equiv)m(alen)o(t)h(one)f(in)g(standard)g(form)e(b)o(y)i(means)g(of)f X(n)o(umerically)i(stable)59 1793 y(metho)q(ds.)h(F)l(or)14 Xb(example,)h(for)f(Tikhono)o(v)h(regularization)h(w)o(e)e(w)o(an)o(t)g X(a)g(n)o(umerically)i(stable)f(metho)q(d)59 1850 y(for)e(transforming)g X(the)g(general-form)h(problem)g(\(2.6\))e(in)o(to)i(the)f(follo)o(wing) Xh(standard-form)f(problem)608 1957 y(min)699 1923 y Fi(\010)724 X1957 y Fk(k)758 1946 y Fo(\026)747 1957 y Fn(A)d Fo(\026)-26 Xb Fn(x)10 b Fk(\000)868 1945 y Fo(\026)870 1957 y Fn(b)o XFk(k)912 1938 y Fj(2)912 1968 y(2)940 1957 y Fo(+)h Fn(\025)1013 X1938 y Fj(2)1039 1957 y Fk(k)s Fo(\026)-26 b Fn(x)9 b XFk(\000)14 b Fo(\026)-26 b Fn(x)1169 1938 y Ff(\003)1188 X1957 y Fk(k)1211 1938 y Fj(2)1211 1968 y(2)1229 1923 Xy Fi(\011)1276 1957 y Fn(;)426 b Fo(\(2.21\))59 2065 Xy(where)13 b(the)f(new)h(matrix)511 2053 y(\026)499 2065 Xy Fn(A)q Fo(,)f(the)h(new)g(righ)o(t-hand)g(side)1033 X2053 y(\026)1035 2065 y Fn(b)p Fo(,)f(and)h(the)g(v)o(ector)h(\026)-26 Xb Fn(x)1401 2048 y Ff(\003)1433 2065 y Fo(are)12 b(deriv)o(ed)i(from)e X(the)59 2121 y(original)18 b(quan)o(tities)f Fn(A)p Fo(,)g XFn(L)p Fo(,)g Fn(b)p Fo(,)f(and)h Fn(x)726 2105 y Ff(\003)745 X2121 y Fo(.)25 b(Moreo)o(v)o(er,)15 b(w)o(e)i(also)f(w)o(an)o(t)g(a)g X(n)o(umerically)j(stable)e(sc)o(heme)59 2178 y(for)h(transforming)f X(the)h(solution)k(\026)-26 b Fn(x)686 2185 y Fl(\025)726 X2178 y Fo(to)18 b(\(2.21\))e(bac)o(k)i(to)g(the)g(general-form)h X(setting.)29 b(Finally)l(,)20 b(w)o(e)59 2234 y(prefer)14 Xb(a)g(transformation)f(that)g(leads)i(to)e(a)h(simple)h(relationship)h X(b)q(et)o(w)o(een)e(the)h(SVD)f(of)1644 2223 y(\026)1632 X2234 y Fn(A)g Fo(and)h(the)59 2291 y(GSVD)e(of)g(\()p XFn(A;)8 b(L)p Fo(\),)k(for)h(then)h(w)o(e)g(ha)o(v)o(e)f(a)g(p)q 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b(turns)g(out)f(that)g(it)h(is)59 2633 y(a)f(go)q(o)q(d)g(idea) Xg(to)g(distinguish)i(b)q(et)o(w)o(een)e(direct)h(and)f(iterativ)o(e)g X(regularization)h(metho)q(ds|cf.)g(the)59 2689 y(next)d(t)o(w)o(o)f X(sections.)27 b(F)l(or)17 b(the)g(direct)h(metho)q(ds)f(w)o(e)g(need)i X(to)d(b)q(e)i(able)g(to)f(compute)g(the)h(matrix)1809 X2678 y(\026)1797 2689 y Fn(A)59 2746 y Fo(explicitly)h(b)o(y)d X(standard)g(metho)q(ds)h(suc)o(h)g(as)e(the)i(QR)g(factorization.)23 Xb(F)l(or)16 b(the)g(iterativ)o(e)h(metho)q(ds,)59 2802 Xy(on)f(the)g(other)f(hand,)h(w)o(e)g(merely)g(need)h(to)e(b)q(e)i(able) Xg(to)e(compute)h(the)g(matrix-v)o(ector)f(pro)q(duct)1775 X2791 y(\026)1763 2802 y Fn(A)c Fo(\026)-26 b Fn(x)59 X2859 y Fo(e\016cien)o(tly)l(.)32 b(Belo)o(w,)20 b(w)o(e)e(describ)q(e)j X(t)o(w)o(o)c(metho)q(ds)i(for)f(transformation)f(to)h(standard)h(form)f X(whic)o(h)p eop X%%Page: 16 17 X16 16 bop 59 166 a Fo(16)956 b Fm(DISCRETE)15 b(ILL-POSED)i(PR)o X(OBLEMS)p 59 178 1772 2 v 59 306 a Fo(are)k(suited)i(for)e(direct)i X(and)e(iterativ)o(e)h(metho)q(ds,)i(resp)q(ectiv)o(ely)l(.)41 Xb(W)l(e)22 b(assume)f(that)g(the)h(matrix)59 362 y Fn(L)13 Xb Fk(2)g Fn(I)l(R)196 342 y Fl(p)p Ff(\002)p Fl(n)277 X362 y Fo(has)i(full)i(ro)o(w)d(rank,)h(i.e.,)f(the)i(rank)f(of)f XFn(L)i Fo(is)f Fn(p)p Fo(.)59 490 y Fp(2.6.1.)h(T)l(ransformation)i X(for)f(Direct)h(Metho)q(ds)59 579 y Fo(The)i(standard-form)e X(transformation)h(for)f(direct)j(metho)q(ds)e(describ)q(ed)j(here)e(w)o X(as)f(dev)o(elop)q(ed)i(b)o(y)59 635 y(Eld)o(\023)-21 Xb(en)11 b([22)o(],)g(and)g(it)h(is)f(based)g(on)g(t)o(w)o(o)f(QR)h X(factorizations.)19 b(The)11 b(description)h(of)f(this)g X(transformation)59 692 y(is)18 b(quite)g(algorithmic,)h(and)e(it)h(is)g X(summarized)g(b)q(elo)o(w)g(\(where,)f(for)g(con)o(v)o(enience,)i(the)f X(subscripts)59 748 y Fn(p)p Fo(,)d Fn(o)p Fo(,)h(and)f XFn(q)j Fo(denote)e(matrices)f(with)h Fn(p)p Fo(,)f Fn(n)c XFk(\000)g Fn(p)p Fo(,)k(and)h Fn(m)10 b Fk(\000)h Fo(\()p XFn(n)f Fk(\000)h 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y(and)16 b(w)o(e)g(stress)g(that)f(the)h(most)f X(e\016cien)o(t)i(w)o(a)o(y)e(to)g(compute)1148 1630 y(\026)1136 X1642 y Fn(A)h Fo(and)1273 1630 y(\026)1275 1642 y Fn(b)f XFo(is)i(to)e(apply)i(the)f(orthogonal)59 1698 y(transformations)e(that) Xg(mak)o(e)g(up)i Fn(K)h Fo(and)e Fn(H)k Fo(\\on)14 b(the)h(\015y")g(to) Xf Fn(A)h Fo(and)g Fn(b)g Fo(as)f(the)h(QR)h(factorizations)59 X1755 y(in)f(\(2.22\))c(and)j(\(2.23\))e(are)h(computed.)20 Xb(When)14 b(\(2.21\))e(has)h(b)q(een)i(solv)o(ed)f(for)i(\026)-26 Xb Fn(x)p Fo(,)13 b(the)h(transformation)59 1811 y(bac)o(k)h(to)g(the)g X(general-form)g(setting)g(then)h(tak)o(es)e(the)i(form)605 X1917 y Fn(x)c Fo(=)h Fn(L)722 1899 y Ff(y)743 1917 y XFo(\026)-26 b Fn(x)10 b Fo(+)g Fn(K)860 1929 y Fl(o)886 X1917 y Fn(T)919 1899 y Ff(\000)p Fj(1)913 1929 y Fl(o)963 X1917 y Fn(H)1005 1899 y Fl(T)1001 1929 y(o)1031 1917 Xy Fo(\()p Fn(b)f Fk(\000)i Fn(A)d(L)1197 1899 y Ff(y)1216 X1917 y Fo(\026)-25 b Fn(x)o Fo(\))15 b Fn(:)430 b Fo(\(2.25\))130 X2025 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b(REGULARIZA)l(TION)i X(METHODS)673 b Fo(25)p 59 178 1772 2 v 130 306 a(Regarding)21 Xb(the)h(\014lter)f(factors)f(for)h(CG)f(and)h(LSQR,)h(w)o(e)f(ha)o(v)o X(e)g(found)g(that)g(the)g(expression)59 362 y(\(2.55\))c(using)j(the)e X(Ritz)i(p)q(olynomial)h(is)e(extremely)g(sensitiv)o(e)h(to)f(rounding)g X(errors.)30 b(Instead,)20 b(w)o(e)59 419 y(compute)15 Xb(the)h(\014lter)f(factors)f Fn(f)605 396 y Fj(\()p Fl(k)q XFj(\))600 429 y Fl(i)667 419 y Fo(b)o(y)h(means)g(of)g(n)o(umerically)i X(more)d(robust)h(recursions)h(deriv)o(ed)g(in)59 475 Xy([78)o(])h(and)g([49)o(].)25 b(Notice)18 b(that)e(the)h(exact)g X(singular)h(v)m(alues)g Fn(\033)1130 482 y Fl(i)1161 X475 y Fo(of)f Fn(A)g Fo(are)g(required)h(to)e(compute)i(the)59 X532 y(\014lter)e(factors;)d(hence)k(this)e(option)h(is)g(mainly)g(of)f X(p)q(edagogical)h(in)o(terest.)59 668 y Fp(2.8.2.)g(Bidiagonali)q(za)q X(tio)q(n)k(with)e(Regularization)59 759 y Fo(It)11 b(is)h(p)q(ossible)h 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6194 3739 L X 898 3520 mt 925 3520 L X6221 3520 mt 6194 3520 L X 898 3302 mt 925 3302 L X6221 3302 mt 6194 3302 L X 898 3083 mt 925 3083 L X6221 3083 mt 6194 3083 L X 898 2864 mt 925 2864 L X6221 2864 mt 6194 2864 L X 898 3739 mt 951 3739 L X6221 3739 mt 6168 3739 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 512 3801 mt (10) s X X/Helvetica 10 FMS X X 698 3690 mt (0) s X X/Helvetica 14 FMS X X 898 3520 mt 925 3520 L X6221 3520 mt 6194 3520 L X 898 3302 mt 925 3302 L X6221 3302 mt 6194 3302 L X 898 3083 mt 925 3083 L X6221 3083 mt 6194 3083 L X 898 2864 mt 925 2864 L X6221 2864 mt 6194 2864 L X 898 3302 mt 951 3302 L X6221 3302 mt 6168 3302 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 512 3364 mt (10) s X X/Helvetica 10 FMS X X 698 3253 mt (10) s X X/Helvetica 14 FMS X X 898 3083 mt 925 3083 L X6221 3083 mt 6194 3083 L X 898 2864 mt 925 2864 L X6221 2864 mt 6194 2864 L X 898 2864 mt 951 2864 L X6221 2864 mt 6168 2864 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 512 2926 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3918 L Xgs 898 2864 5324 1751 MR c np Xgr X2231 3848 mt 2303 3920 L X2303 3848 mt 2231 3920 L Xgs 898 2864 5324 1751 MR c np Xgr X2383 3835 mt 2455 3907 L X2455 3835 mt 2383 3907 L Xgs 898 2864 5324 1751 MR c np Xgr X2535 3829 mt 2607 3901 L X2607 3829 mt 2535 3901 L Xgs 898 2864 5324 1751 MR c np Xgr X2687 3846 mt 2759 3918 L X2759 3846 mt 2687 3918 L Xgs 898 2864 5324 1751 MR c np Xgr X2839 3841 mt 2911 3913 L X2911 3841 mt 2839 3913 L Xgs 898 2864 5324 1751 MR c np Xgr X2991 3841 mt 3063 3913 L X3063 3841 mt 2991 3913 L Xgs 898 2864 5324 1751 MR c np Xgr X3143 3836 mt 3215 3908 L X3215 3836 mt 3143 3908 L Xgs 898 2864 5324 1751 MR c np Xgr X3295 3838 mt 3367 3910 L X3367 3838 mt 3295 3910 L Xgs 898 2864 5324 1751 MR c np Xgr X3447 3826 mt 3519 3898 L X3519 3826 mt 3447 3898 L Xgs 898 2864 5324 1751 MR c np Xgr X3600 3834 mt 3672 3906 L X3672 3834 mt 3600 3906 L Xgs 898 2864 5324 1751 MR c np Xgr X3752 3823 mt 3824 3895 L X3824 3823 mt 3752 3895 L Xgs 898 2864 5324 1751 MR c np Xgr X3904 3826 mt 3976 3898 L X3976 3826 mt 3904 3898 L Xgs 898 2864 5324 1751 MR c np Xgr X4056 3842 mt 4128 3914 L X4128 3842 mt 4056 3914 L Xgs 898 2864 5324 1751 MR c np Xgr X4208 3855 mt 4280 3927 L X4280 3855 mt 4208 3927 L Xgs 898 2864 5324 1751 MR c np Xgr X4360 3836 mt 4432 3908 L X4432 3836 mt 4360 3908 L Xgs 898 2864 5324 1751 MR c np Xgr X4512 3846 mt 4584 3918 L X4584 3846 mt 4512 3918 L Xgs 898 2864 5324 1751 MR c np Xgr X4664 3835 mt 4736 3907 L X4736 3835 mt 4664 3907 L Xgs 898 2864 5324 1751 MR c np Xgr X4816 3841 mt 4888 3913 L X4888 3841 mt 4816 3913 L Xgs 898 2864 5324 1751 MR c np Xgr X4968 3839 mt 5040 3911 L X5040 3839 mt 4968 3911 L Xgs 898 2864 5324 1751 MR c np Xgr X5120 3856 mt 5192 3928 L X5192 3856 mt 5120 3928 L Xgs 898 2864 5324 1751 MR c np Xgr X5272 3823 mt 5344 3895 L X5344 3823 mt 5272 3895 L Xgs 898 2864 5324 1751 MR c np Xgr X5425 3851 mt 5497 3923 L X5497 3851 mt 5425 3923 L Xgs 898 2864 5324 1751 MR c np Xgr X5577 3837 mt 5649 3909 L X5649 3837 mt 5577 3909 L Xgs 898 2864 5324 1751 MR c np Xgr X5729 3862 mt 5801 3934 L X5801 3862 mt 5729 3934 L Xgs 898 2864 5324 1751 MR c np Xgr X1014 3676 1086 3748 FO Xgs 898 2864 5324 1751 MR c np Xgr X1166 3701 1238 3773 FO Xgs 898 2864 5324 1751 MR c np Xgr X1318 3681 1390 3753 FO Xgs 898 2864 5324 1751 MR c np Xgr X1470 3699 1542 3771 FO Xgs 898 2864 5324 1751 MR c np Xgr X1622 3717 1694 3789 FO Xgs 898 2864 5324 1751 MR c np Xgr X1775 3715 1847 3787 FO Xgs 898 2864 5324 1751 MR c np Xgr X1927 3714 1999 3786 FO Xgs 898 2864 5324 1751 MR c np Xgr X2079 3742 2151 3814 FO Xgs 898 2864 5324 1751 MR c np Xgr X2231 3721 2303 3793 FO Xgs 898 2864 5324 1751 MR c np Xgr X2383 3653 2455 3725 FO Xgs 898 2864 5324 1751 MR c np Xgr X2535 3606 2607 3678 FO Xgs 898 2864 5324 1751 MR c np Xgr X2687 3598 2759 3670 FO Xgs 898 2864 5324 1751 MR c np Xgr X2839 3561 2911 3633 FO Xgs 898 2864 5324 1751 MR c np Xgr X2991 3520 3063 3592 FO Xgs 898 2864 5324 1751 MR c np Xgr X3143 3466 3215 3538 FO Xgs 898 2864 5324 1751 MR c np Xgr X3295 3417 3367 3489 FO Xgs 898 2864 5324 1751 MR c np Xgr X3447 3363 3519 3435 FO Xgs 898 2864 5324 1751 MR c np Xgr X3600 3301 3672 3373 FO Xgs 898 2864 5324 1751 MR c np Xgr X3752 3266 3824 3338 FO Xgs 898 2864 5324 1751 MR c np Xgr X3904 3246 3976 3318 FO Xgs 898 2864 5324 1751 MR c np Xgr X4056 3171 4128 3243 FO Xgs 898 2864 5324 1751 MR c np Xgr X4208 3171 4280 3243 FO Xgs 898 2864 5324 1751 MR c np Xgr X4360 3149 4432 3221 FO Xgs 898 2864 5324 1751 MR c np Xgr X4512 3153 4584 3225 FO Xgs 898 2864 5324 1751 MR c np Xgr X4664 3139 4736 3211 FO Xgs 898 2864 5324 1751 MR c np Xgr X4816 3142 4888 3214 FO Xgs 898 2864 5324 1751 MR c np Xgr X4968 3130 5040 3202 FO Xgs 898 2864 5324 1751 MR c np Xgr X5120 3144 5192 3216 FO Xgs 898 2864 5324 1751 MR c np Xgr X5272 3104 5344 3176 FO Xgs 898 2864 5324 1751 MR c np Xgr X5425 3128 5497 3200 FO Xgs 898 2864 5324 1751 MR c np Xgr X5577 3100 5649 3172 FO Xgs 898 2864 5324 1751 MR c np Xgr X5729 3111 5801 3183 FO Xgs 898 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restore} bdef X/bplot /gsave ldef X/eplot {stroke grestore} bdef X X/llx 0 def X/lly 0 def X/urx 0 def X/ury 0 def X/bbox {/ury xdef /urx xdef /lly xdef /llx xdef} bdef X X/portraitMode (op) def X/landscapeMode (ol) def X/Orientation portraitMode def X/portrait {/Orientation portraitMode def} bdef X/landscape {/Orientation landscapeMode def} bdef X X/dpi2point 0 def X X/FontSize 0 def X/FMS { X /FontSize xstore %save size off stack X findfont X [FontSize dpi2point mul 0 0 FontSize dpi2point mul neg 0 0] X makefont X setfont X }bdef X/setPortrait { X 1 dpi2point div -1 dpi2point div scale X llx ury neg translate X } bdef X/setLandscape { X 1 dpi2point div -1 dpi2point div scale X urx ury neg translate X 90 rotate X } bdef X X/csm {Orientation portraitMode eq {setPortrait} {setLandscape} ifelse} bdef X/SO { [] 0 setdash } bdef X/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef X/DA { [6 dpi2point mul] 0 setdash } bdef X/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef X X/L { % LineTo X lineto X stroke X } bdef X/MP { % MakePoly X 3 1 roll moveto X 1 sub {rlineto} repeat X } bdef X/AP { % AddPoly X {rlineto} repeat X } bdef X/PP { % PaintPoly X closepath fill X } bdef X/DP { % DrawPoly X closepath stroke X } bdef X/MR { % MakeRect X 4 -2 roll moveto X dup 0 exch rlineto X exch 0 rlineto X neg 0 exch rlineto X closepath X } bdef X/FR { % FrameRect X MR stroke X } bdef X/PR { % PaintRect X MR fill X } bdef X/L1i { % Level 1 Image X { currentfile picstr readhexstring pop } image X } bdef X X/half_width 0 def X/half_height 0 def X/MakeOval { X newpath X /ury xstore /urx xstore /lly xstore /llx xstore X /half_width urx llx sub 2 div store X /half_height ury lly sub 2 div store X llx half_width add lly half_height add translate X half_width half_height scale X .5 half_width div setlinewidth X 0 0 1 0 360 arc X } bdef X/FO { X gsave X MakeOval stroke X grestore X } bdef X/PO { X gsave X MakeOval fill X grestore X } bdef X X/PD { X 2 copy moveto lineto stroke X } bdef X X Xcurrentdict end def %dictionary X XMathWorks begin X X X1 setlinecap 1 setlinejoin X Xend X XMathWorks begin Xbpage X Xbplot X X/dpi2point 12 def X0216 2160 7128 7344 bbox portrait csm X X0 0 6912 5184 MR c np X6.00 setlinewidth X/colortable 76 dict begin X/c0 { 0 0 0 sc} bdef X/c1 { 1 1 1 sc} bdef X/c2 { 1 0 0 sc} bdef X/c3 { 0 1 0 sc} bdef X/c4 { 0 0 1 sc} bdef X/c5 { 1 1 0 sc} bdef X/c6 { 1 0 1 sc} bdef X/c7 { 0 1 1 sc} bdef Xcurrentdict end def % Colortable X Xcolortable begin X X X/Helvetica 14 FMS X Xc1 X 0 0 6912 5184 PR XDO XSO Xc0 X1851 2165 mt 3094 1825 L X 898 1722 mt 1851 2165 L X 898 1722 mt 898 754 L X1851 2165 mt 1881 2179 L X1905 2346 mt (0) s X2473 1995 mt 2502 2009 L X2526 2176 mt (5) s X3094 1825 mt 3123 1838 L X3148 2006 mt (10) s X1175 1850 mt 1144 1859 L X 932 2022 mt (10) s X1482 1993 mt 1451 2002 L X1239 2165 mt (20) s X1790 2136 mt 1759 2145 L X1547 2308 mt (30) s X 898 1542 mt 869 1528 L X 695 1579 mt (-5) s X 898 1201 mt 869 1188 L X 751 1238 mt (0) s X 898 860 mt 869 847 L X 751 898 mt (5) s X1302 281 mt (Tikhonov solutions) s Xgs 898 414 2197 1752 MR c np Xcolortable /c8 { 0.000000 1.000000 0.476190 sc} put Xc8 X X-30 910 124 -351 31 -185 2016 1008 4 MP PP Xc0 X X-30 910 124 -351 31 -185 2016 1008 4 MP DP Xcolortable /c9 { 0.000000 0.761905 1.000000 sc} put Xc9 X X-30 -135 124 214 31 50 2139 948 4 MP PP Xc0 X X-30 -135 124 214 31 50 2139 948 4 MP DP Xcolortable /c10 { 0.000000 0.952381 1.000000 sc} put Xc10 X X-31 72 124 135 31 7 2170 998 4 MP PP Xc0 X X-31 72 124 135 31 7 2170 998 4 MP DP Xc9 X X-31 185 124 -97 31 -8 1892 928 4 MP PP Xc0 X X-31 185 124 -97 31 -8 1892 928 4 MP DP Xcolortable /c11 { 0.000000 0.380952 1.000000 sc} put Xc11 X X-31 -316 125 129 30 78 2109 870 4 MP PP Xc0 X X-31 -316 125 129 30 78 2109 870 4 MP DP Xcolortable /c12 { 0.000000 0.571429 1.000000 sc} put Xc12 X X-31 16 124 -123 31 10 1923 920 4 MP PP Xc0 X X-31 16 124 -123 31 10 1923 920 4 MP DP Xc10 X X-31 196 124 -29 31 -32 2201 1005 4 MP PP Xc0 X X-31 196 124 -29 31 -32 2201 1005 4 MP DP Xcolortable /c13 { 0.000000 0.666667 1.000000 sc} put Xc13 X X-31 8 124 -44 31 16 1768 948 4 MP PP Xc0 X X-31 8 124 -44 31 16 1768 948 4 MP DP Xc12 X X-31 -63 125 -78 30 18 1954 930 4 MP PP Xc0 X X-31 -63 125 -78 30 18 1954 930 4 MP DP Xc12 X X-30 -78 124 -15 31 15 1984 948 4 MP PP Xc0 X X-30 -78 124 -15 31 15 1984 948 4 MP DP Xc13 X X-31 -10 125 -47 30 13 1799 964 4 MP PP Xc0 X X-31 -10 125 -47 30 13 1799 964 4 MP DP Xc13 X X-31 -50 124 24 31 11 2015 963 4 MP PP Xc0 X X-31 -50 124 24 31 11 2015 963 4 MP DP Xc4 X X-31 -318 124 -109 31 63 2078 807 4 MP PP Xc0 X X-31 -318 124 -109 31 63 2078 807 4 MP DP Xcolortable /c14 { 0.000000 0.190476 1.000000 sc} put Xc14 X X-31 29 124 -364 31 -16 2047 823 4 MP PP Xc0 X X-31 29 124 -364 31 -16 2047 823 4 MP DP Xc13 X X-30 -18 124 -39 31 10 1829 977 4 MP PP Xc0 X X-30 -18 124 -39 31 10 1829 977 4 MP DP Xc12 X X-31 -7 124 28 31 3 2046 974 4 MP PP Xc0 X X-31 -7 124 28 31 3 2046 974 4 MP DP Xc13 X X-31 -16 125 -35 30 12 1644 987 4 MP PP Xc0 X X-31 -16 125 -35 30 12 1644 987 4 MP DP Xc13 X X-31 -15 124 -30 31 6 1860 987 4 MP PP Xc0 X X-31 -15 124 -30 31 6 1860 987 4 MP DP Xc13 X X-30 -13 124 -33 31 11 1674 999 4 MP PP Xc0 X X-30 -13 124 -33 31 11 1674 999 4 MP DP Xc12 X X-31 32 125 -3 30 -1 2077 977 4 MP PP Xc0 X X-31 32 125 -3 30 -1 2077 977 4 MP DP Xc13 X X-31 191 125 -175 30 -45 2232 973 4 MP PP Xc0 X X-31 191 125 -175 30 -45 2232 973 4 MP DP Xc12 X X-31 -11 124 -24 31 5 1891 993 4 MP PP Xc0 X X-31 -11 124 -24 31 5 1891 993 4 MP DP Xc13 X X-31 -10 124 -31 31 8 1705 1010 4 MP PP Xc0 X X-31 -10 124 -31 31 8 1705 1010 4 MP DP Xc9 X X-30 -12 124 -35 31 11 1519 1023 4 MP PP Xc0 X X-30 -12 124 -35 31 11 1519 1023 4 MP DP Xc12 X X-31 -3 125 -25 30 4 1922 998 4 MP PP Xc0 X X-31 -3 125 -25 30 4 1922 998 4 MP DP Xcolortable /c15 { 0.000000 0.476190 1.000000 sc} put Xc15 X X-30 45 124 -48 31 0 2107 976 4 MP PP Xc0 X X-30 45 124 -48 31 0 2107 976 4 MP DP Xc13 X X-31 -6 124 -31 31 6 1736 1018 4 MP PP Xc0 X X-31 -6 124 -31 31 6 1736 1018 4 MP DP Xc13 X X-31 -11 124 -32 31 8 1550 1034 4 MP PP Xc0 X X-31 -11 124 -32 31 8 1550 1034 4 MP DP Xc15 X X-30 -62 124 106 31 53 2385 1026 4 MP PP Xc0 X X-30 -62 124 106 31 53 2385 1026 4 MP DP Xc14 X X-31 -173 125 97 30 67 2355 959 4 MP PP Xc0 X X-31 -173 125 97 30 67 2355 959 4 MP DP Xc15 X X-30 1 124 -30 31 4 1952 1002 4 MP PP Xc0 X X-30 1 124 -30 31 4 1952 1002 4 MP DP Xc12 X X-31 -5 125 -32 30 6 1767 1024 4 MP PP Xc0 X X-31 -5 125 -32 30 6 1767 1024 4 MP DP Xc9 X X-31 72 124 19 31 15 2416 1079 4 MP PP Xc0 X X-31 72 124 19 31 15 2416 1079 4 MP DP Xc13 X X-31 -8 124 -32 31 8 1581 1042 4 MP PP Xc0 X X-31 -8 124 -32 31 8 1581 1042 4 MP DP Xc13 X X-31 -11 124 -32 31 10 1395 1056 4 MP PP Xc0 X X-31 -11 124 -32 31 10 1395 1056 4 MP DP Xcolortable /c16 { 0.000000 0.285714 1.000000 sc} put Xc16 X X-30 71 124 -217 31 -29 2262 928 4 MP PP Xc0 X X-30 71 124 -217 31 -29 2262 928 4 MP DP Xc11 X X-31 29 124 -83 31 6 2138 976 4 MP PP Xc0 X X-31 29 124 -83 31 6 2138 976 4 MP DP Xc4 X X-31 -185 124 -9 31 50 2324 909 4 MP PP Xc0 X X-31 -185 124 -9 31 50 2324 909 4 MP DP Xc12 X X-30 -4 124 -33 31 5 1797 1030 4 MP PP Xc0 X X-30 -4 124 -33 31 5 1797 1030 4 MP DP Xc12 X X-31 -6 125 -33 30 7 1612 1050 4 MP PP Xc0 X X-31 -6 125 -33 30 7 1612 1050 4 MP DP Xc11 X X-31 0 124 -37 31 7 1983 1006 4 MP PP Xc0 X X-31 0 124 -37 31 7 1983 1006 4 MP DP Xc13 X X-31 -8 124 -34 31 10 1426 1066 4 MP PP Xc0 X X-31 -8 124 -34 31 10 1426 1066 4 MP DP Xc4 X X-31 -83 124 -144 31 10 2293 899 4 MP PP Xc0 X X-31 -83 124 -144 31 10 2293 899 4 MP DP Xc16 X X-31 -10 124 -87 31 14 2169 982 4 MP PP Xc0 X X-31 -10 124 -87 31 14 2169 982 4 MP DP Xc12 X X-30 -6 124 -33 31 6 1642 1057 4 MP PP Xc0 X X-30 -6 124 -33 31 6 1642 1057 4 MP DP Xc15 X X-31 -4 124 -36 31 7 1828 1035 4 MP PP Xc0 X X-31 -4 124 -36 31 7 1828 1035 4 MP DP Xc11 X X-31 -6 124 -42 31 11 2014 1013 4 MP PP Xc0 X X-31 -6 124 -42 31 11 2014 1013 4 MP DP Xc13 X X-31 -8 125 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MP DP Xc15 X X-31 -15 124 11 31 18 2292 1065 4 MP PP Xc0 X X-31 -15 124 11 31 18 2292 1065 4 MP DP Xc11 X X-30 -21 124 -38 31 17 2075 1038 4 MP PP Xc0 X X-30 -21 124 -38 31 17 2075 1038 4 MP DP Xc11 X X-31 -11 125 -36 30 10 1890 1050 4 MP PP Xc0 X X-31 -11 125 -36 30 10 1890 1050 4 MP DP Xc12 X X-31 -6 124 -36 31 7 1518 1092 4 MP PP Xc0 X X-31 -6 124 -36 31 7 1518 1092 4 MP DP Xc15 X X-31 -7 124 -35 31 7 1704 1070 4 MP PP Xc0 X X-31 -7 124 -35 31 7 1704 1070 4 MP DP Xc13 X X-30 -8 124 -38 31 9 1332 1113 4 MP PP Xc0 X X-30 -8 124 -38 31 9 1332 1113 4 MP DP Xc13 X X-31 -11 125 -33 30 11 1147 1124 4 MP PP Xc0 X X-31 -11 125 -33 30 11 1147 1124 4 MP DP Xc15 X X-31 20 125 -22 30 13 2323 1083 4 MP PP Xc0 X X-31 20 125 -22 30 13 2323 1083 4 MP DP Xc11 X X-31 -25 124 -30 31 17 2106 1055 4 MP PP Xc0 X X-31 -25 124 -30 31 17 2106 1055 4 MP DP Xc12 X X-31 93 125 -163 30 -28 2478 1074 4 MP PP Xc0 X X-31 93 125 -163 30 -28 2478 1074 4 MP DP Xc11 X X-30 -14 124 -35 31 13 1920 1060 4 MP PP Xc0 X X-30 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1394 1131 4 MP PP Xc0 X X-31 -7 124 -38 31 6 1394 1131 4 MP DP Xc15 X X-31 -7 125 -36 30 7 1580 1106 4 MP PP Xc0 X X-31 -7 125 -36 30 7 1580 1106 4 MP DP Xc14 X X-31 -142 124 59 31 47 2570 1068 4 MP PP Xc0 X X-31 -142 124 59 31 47 2570 1068 4 MP DP Xc13 X X-31 -9 124 -33 31 9 1208 1146 4 MP PP Xc0 X X-31 -9 124 -33 31 9 1208 1146 4 MP DP Xc9 X X-30 -11 124 -36 31 12 1022 1159 4 MP PP Xc0 X X-30 -11 124 -36 31 12 1022 1159 4 MP DP Xc16 X X-30 -24 124 -133 31 -6 2508 1046 4 MP PP Xc0 X X-30 -24 124 -133 31 -6 2508 1046 4 MP DP Xc11 X X-31 -17 124 -32 31 17 1982 1087 4 MP PP Xc0 X X-31 -17 124 -32 31 17 1982 1087 4 MP DP Xc14 X X-31 -125 124 -36 31 28 2539 1040 4 MP PP Xc0 X X-31 -125 124 -36 31 28 2539 1040 4 MP DP Xc12 X X-30 54 124 28 31 3 2631 1151 4 MP PP Xc0 X X-30 54 124 28 31 3 2631 1151 4 MP DP Xc11 X X-30 -13 124 -33 31 20 2198 1109 4 MP PP Xc0 X X-30 -13 124 -33 31 20 2198 1109 4 MP DP Xc15 X X-31 6 124 -84 31 15 2384 1109 4 MP PP Xc0 X X-31 6 124 -84 31 15 2384 1109 4 MP DP Xc11 X X-31 -13 124 -34 31 11 1796 1096 4 MP PP Xc0 X X-31 -13 124 -34 31 11 1796 1096 4 MP DP Xc12 X X-31 -7 125 -37 30 6 1425 1137 4 MP PP Xc0 X X-31 -7 125 -37 30 6 1425 1137 4 MP DP Xc13 X X-31 -9 124 -32 31 8 1239 1155 4 MP PP Xc0 X X-31 -9 124 -32 31 8 1239 1155 4 MP DP Xc15 X X-30 -9 124 -34 31 7 1610 1113 4 MP PP Xc0 X X-30 -9 124 -34 31 7 1610 1113 4 MP DP Xc13 X X-31 -11 124 -37 31 12 1053 1171 4 MP PP Xc0 X X-31 -11 124 -37 31 12 1053 1171 4 MP DP Xc11 X X-31 -18 125 -32 30 18 2013 1104 4 MP PP Xc0 X X-31 -18 125 -32 30 18 2013 1104 4 MP DP Xc15 X X-31 -13 124 -40 31 20 2229 1129 4 MP PP Xc0 X X-31 -13 124 -40 31 20 2229 1129 4 MP DP Xc15 X X-31 -28 124 -69 31 13 2415 1124 4 MP PP Xc0 X X-31 -28 124 -69 31 13 2415 1124 4 MP DP Xc11 X X-31 -14 124 -33 31 13 1827 1107 4 MP PP Xc0 X X-31 -14 124 -33 31 13 1827 1107 4 MP DP Xc11 X X-31 -10 124 -32 31 8 1641 1120 4 MP PP Xc0 X X-31 -10 124 -32 31 8 1641 1120 4 MP DP Xc12 X X-31 -6 125 -33 30 7 1270 1163 4 MP PP Xc0 X X-31 -6 125 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1170 4 MP PP Xc0 X X-30 -6 124 -32 31 5 1300 1170 4 MP DP Xc11 X X-30 -36 124 4 31 1 2476 1146 4 MP PP Xc0 X X-30 -36 124 4 31 1 2476 1146 4 MP DP Xc15 X X-31 -7 124 -32 31 4 1486 1148 4 MP PP Xc0 X X-31 -7 124 -32 31 4 1486 1148 4 MP DP Xc15 X X-31 -13 125 -40 30 11 2291 1166 4 MP PP Xc0 X X-31 -13 125 -40 30 11 2291 1166 4 MP DP Xc11 X X-30 -18 124 -33 31 19 1888 1136 4 MP PP Xc0 X X-30 -18 124 -33 31 19 1888 1136 4 MP DP Xc15 X X-31 -20 124 -37 31 21 2105 1165 4 MP PP Xc0 X X-31 -20 124 -37 31 21 2105 1165 4 MP DP Xc16 X X-31 -13 125 -32 30 13 1703 1139 4 MP PP Xc0 X X-31 -13 125 -32 30 13 1703 1139 4 MP DP Xc13 X X-30 -7 124 -41 31 9 1145 1202 4 MP PP Xc0 X X-30 -7 124 -41 31 9 1145 1202 4 MP DP Xc11 X X-31 -3 124 15 31 -8 2507 1147 4 MP PP Xc0 X X-31 -3 124 15 31 -8 2507 1147 4 MP DP Xc15 X X-31 -5 124 -32 31 5 1331 1175 4 MP PP Xc0 X X-31 -5 124 -32 31 5 1331 1175 4 MP DP Xc11 X X-31 -8 124 -30 31 6 1517 1152 4 MP PP Xc0 X X-31 -8 124 -30 31 6 1517 1152 4 MP DP Xc11 X X-31 -21 124 -34 31 22 1919 1155 4 MP PP Xc0 X X-31 -21 124 -34 31 22 1919 1155 4 MP DP Xc15 X X-30 -9 124 -33 31 2 2321 1177 4 MP PP Xc0 X X-30 -9 124 -33 31 2 2321 1177 4 MP DP Xc15 X X-31 -17 125 -36 30 16 2136 1186 4 MP PP Xc0 X X-31 -17 125 -36 30 16 2136 1186 4 MP DP Xc16 X X-30 -16 124 -33 31 17 1733 1152 4 MP PP Xc0 X X-30 -16 124 -33 31 17 1733 1152 4 MP DP Xc11 X X-31 -22 124 -35 31 23 1950 1177 4 MP PP Xc0 X X-31 -22 124 -35 31 23 1950 1177 4 MP DP Xc12 X X-31 -5 124 -44 31 8 1176 1211 4 MP PP Xc0 X X-31 -5 124 -44 31 8 1176 1211 4 MP DP Xc16 X X-31 -11 125 -29 30 10 1548 1158 4 MP PP Xc0 X X-31 -11 125 -29 30 10 1548 1158 4 MP DP Xc15 X X-31 -4 124 -33 31 5 1362 1180 4 MP PP Xc0 X X-31 -4 124 -33 31 5 1362 1180 4 MP DP Xc12 X X-30 -11 124 -34 31 9 2166 1202 4 MP PP Xc0 X X-30 -11 124 -34 31 9 2166 1202 4 MP DP Xc11 X X-31 -1 124 -26 31 -6 2352 1179 4 MP PP Xc0 X X-31 -1 124 -26 31 -6 2352 1179 4 MP DP Xc14 X X-31 32 124 -6 31 -11 2538 1139 4 MP PP Xc0 X X-31 32 124 -6 31 -11 2538 1139 4 MP DP Xc11 X X-31 -19 124 -35 31 21 1764 1169 4 MP PP Xc0 X X-31 -19 124 -35 31 21 1764 1169 4 MP DP Xc15 X X-31 -21 125 -35 30 21 1981 1200 4 MP PP Xc0 X X-31 -21 125 -35 30 21 1981 1200 4 MP DP Xc14 X X-31 114 124 -140 31 -43 2693 1122 4 MP PP Xc0 X X-31 114 124 -140 31 -43 2693 1122 4 MP DP Xc16 X X-30 -13 124 -30 31 14 1578 1168 4 MP PP Xc0 X X-30 -13 124 -30 31 14 1578 1168 4 MP DP Xc12 X X-31 -5 124 -47 31 8 1207 1219 4 MP PP Xc0 X X-31 -5 124 -47 31 8 1207 1219 4 MP DP Xc11 X X-31 -6 125 -35 30 8 1393 1185 4 MP PP Xc0 X X-31 -6 125 -35 30 8 1393 1185 4 MP DP Xc11 X X-31 -22 124 -36 31 23 1795 1190 4 MP PP Xc0 X X-31 -22 124 -36 31 23 1795 1190 4 MP DP Xc15 X X-31 -2 124 -32 31 0 2197 1211 4 MP PP Xc0 X X-31 -2 124 -32 31 0 2197 1211 4 MP DP Xc12 X X-30 -16 124 -35 31 16 2011 1221 4 MP PP Xc0 X X-30 -16 124 -35 31 16 2011 1221 4 MP DP Xc11 X X-31 -47 125 43 30 84 2847 1216 4 MP PP Xc0 X X-31 -47 125 43 30 84 2847 1216 4 MP DP Xc16 X X-31 8 124 -25 31 -9 2383 1173 4 MP PP Xc0 X X-31 8 124 -25 31 -9 2383 1173 4 MP DP Xc16 X X-31 -17 124 -32 31 19 1609 1182 4 MP PP Xc0 X X-31 -17 124 -32 31 19 1609 1182 4 MP DP Xc15 X X-31 -23 125 -37 30 24 1826 1213 4 MP PP Xc0 X X-31 -23 125 -37 30 24 1826 1213 4 MP DP Xc4 X X-31 43 125 -43 30 -6 2569 1128 4 MP PP Xc0 X X-31 43 125 -43 30 -6 2569 1128 4 MP DP Xc15 X X-31 -5 125 -50 30 8 1238 1227 4 MP PP Xc0 X X-31 -5 125 -50 30 8 1238 1227 4 MP DP Xc9 X X-30 62 124 -69 31 50 2877 1300 4 MP PP Xc0 X X-30 62 124 -69 31 50 2877 1300 4 MP DP Xc16 X X-30 -10 124 -36 31 11 1423 1193 4 MP PP Xc0 X X-30 -10 124 -36 31 11 1423 1193 4 MP DP Xcolortable /c17 { 0.095238 0.000000 1.000000 sc} put Xc17 X X-31 -138 124 80 31 85 2816 1131 4 MP PP Xc0 X X-31 -138 124 80 31 85 2816 1131 4 MP DP Xc12 X X-31 -9 124 -34 31 8 2042 1237 4 MP PP Xc0 X X-31 -9 124 -34 31 8 2042 1237 4 MP DP Xc11 X X-31 6 124 -32 31 -6 2228 1211 4 MP PP Xc0 X X-31 6 124 -32 31 -6 2228 1211 4 MP DP Xc16 X X-31 -21 124 -36 31 25 1640 1201 4 MP PP Xc0 X X-31 -21 124 -36 31 25 1640 1201 4 MP DP Xc15 X X-30 -21 124 -36 31 20 1856 1237 4 MP PP Xc0 X X-30 -21 124 -36 31 20 1856 1237 4 MP DP Xcolortable /c18 { 0.190476 0.000000 1.000000 sc} put Xc18 X X-31 23 125 -139 30 -24 2724 1079 4 MP PP Xc0 X X-31 23 125 -139 30 -24 2724 1079 4 MP DP Xcolortable /c19 { 0.000000 0.095238 1.000000 sc} put Xc19 X X-31 11 125 -29 30 -7 2414 1164 4 MP PP Xc0 X X-31 11 125 -29 30 -7 2414 1164 4 MP DP Xc16 X X-31 -14 124 -38 31 16 1454 1204 4 MP PP Xc0 X X-31 -14 124 -38 31 16 1454 1204 4 MP DP Xc15 X X-30 -8 124 -52 31 10 1268 1235 4 MP PP Xc0 X X-30 -8 124 -52 31 10 1268 1235 4 MP DP Xcolortable /c20 { 0.380952 0.000000 1.000000 sc} put Xc20 X X-31 -154 124 27 31 60 2785 1071 4 MP PP Xc0 X X-31 -154 124 27 31 60 2785 1071 4 MP DP Xc11 X X-31 -23 125 -39 30 26 1671 1226 4 MP PP Xc0 X X-31 -23 125 -39 30 26 1671 1226 4 MP DP Xc15 X X-31 0 124 -33 31 -1 2073 1245 4 MP PP Xc0 X X-31 0 124 -33 31 -1 2073 1245 4 MP DP Xc12 X X-31 -16 124 -34 31 14 1887 1257 4 MP PP Xc0 X X-31 -16 124 -34 31 14 1887 1257 4 MP DP Xc17 X X-30 24 124 -75 31 8 2599 1122 4 MP PP Xc0 X X-30 24 124 -75 31 8 2599 1122 4 MP DP Xc20 X X-30 -88 124 -67 31 16 2754 1055 4 MP PP Xc0 X X-30 -88 124 -67 31 16 2754 1055 4 MP DP Xc16 X X-31 9 125 -32 30 -9 2259 1205 4 MP PP Xc0 X X-31 9 125 -32 30 -9 2259 1205 4 MP DP Xcolortable /c21 { 0.000000 1.000000 0.952381 sc} put Xc21 X X-31 97 124 -154 31 -12 2908 1350 4 MP PP Xc0 X X-31 97 124 -154 31 -12 2908 1350 4 MP DP Xc16 X X-31 -19 124 -39 31 20 1485 1220 4 MP PP Xc0 X X-31 -19 124 -39 31 20 1485 1220 4 MP DP Xc15 X X-30 -24 124 -40 31 25 1701 1252 4 MP PP Xc0 X X-30 -24 124 -40 31 25 1701 1252 4 MP DP Xc15 X X-31 -11 124 -52 31 11 1299 1245 4 MP PP Xc0 X X-31 -11 124 -52 31 11 1299 1245 4 MP DP Xc4 X X-30 6 124 -37 31 2 2444 1157 4 MP PP Xc0 X X-30 6 124 -37 31 2 2444 1157 4 MP DP Xc12 X X-31 -8 124 -32 31 6 1918 1271 4 MP PP Xc0 X X-31 -8 124 -32 31 6 1918 1271 4 MP DP Xc11 X X-31 -25 125 -39 30 25 1516 1240 4 MP PP Xc0 X X-31 -25 125 -39 30 25 1516 1240 4 MP DP Xc17 X X-31 -16 124 -84 31 25 2630 1130 4 MP PP Xc0 X X-31 -16 124 -84 31 25 2630 1130 4 MP DP Xc11 X X-31 6 125 -32 30 -7 2104 1244 4 MP PP Xc0 X X-31 6 125 -32 30 -7 2104 1244 4 MP DP Xc12 X X-31 -20 124 -38 31 18 1732 1277 4 MP PP Xc0 X X-31 -20 124 -38 31 18 1732 1277 4 MP DP Xc16 X X-30 -84 124 11 31 47 2722 1242 4 MP PP Xc0 X X-30 -84 124 11 31 47 2722 1242 4 MP DP Xc15 X X-31 -50 124 22 31 39 2753 1289 4 MP PP Xc0 X X-31 -50 124 22 31 39 2753 1289 4 MP DP Xc15 X X-31 -16 124 -50 31 14 1330 1256 4 MP PP Xc0 X X-31 -16 124 -50 31 14 1330 1256 4 MP DP Xc19 X X-31 -85 125 -26 30 48 2692 1194 4 MP PP Xc0 X X-31 -85 125 -26 30 48 2692 1194 4 MP DP Xc17 X X-31 -60 124 -63 31 39 2661 1155 4 MP PP Xc0 X X-31 -60 124 -63 31 39 2661 1155 4 MP DP Xc19 X X-30 7 124 -34 31 -5 2289 1196 4 MP PP Xc0 X X-30 7 124 -34 31 -5 2289 1196 4 MP DP Xc11 X X-30 -26 124 -38 31 25 1546 1265 4 MP PP Xc0 X X-30 -26 124 -38 31 25 1546 1265 4 MP DP Xc17 X X-31 -8 124 -42 31 13 2475 1159 4 MP PP Xc0 X X-31 -8 124 -42 31 13 2475 1159 4 MP DP Xc12 X X-31 -14 124 -34 31 10 1763 1295 4 MP PP Xc0 X X-31 -14 124 -34 31 10 1763 1295 4 MP DP Xc15 X X-31 1 125 -30 30 -3 1949 1277 4 MP PP Xc0 X X-31 1 125 -30 30 -3 1949 1277 4 MP DP Xc13 X X-31 12 124 -13 31 23 2784 1328 4 MP PP Xc0 X X-31 12 124 -13 31 23 2784 1328 4 MP DP Xc11 X X-31 -20 125 -46 30 16 1361 1270 4 MP PP Xc0 X X-31 -20 125 -46 30 16 1361 1270 4 MP DP Xcolortable /c22 { 0.000000 0.857143 1.000000 sc} put Xc22 X X-31 -48 124 -10 31 -96 2939 1338 4 MP PP Xc0 X X-31 -48 124 -10 31 -96 2939 1338 4 MP DP Xc15 X X-31 -25 124 -34 31 21 1577 1290 4 MP PP Xc0 X X-31 -25 124 -34 31 21 1577 1290 4 MP DP Xc16 X X-30 9 124 -33 31 -8 2134 1237 4 MP PP Xc0 X X-30 9 124 -33 31 -8 2134 1237 4 MP DP Xc17 X X-31 -25 124 -43 31 26 2506 1172 4 MP PP Xc0 X X-31 -25 124 -43 31 26 2506 1172 4 MP DP Xc4 X X-31 -2 124 -36 31 4 2320 1191 4 MP PP Xc0 X X-31 -2 124 -36 31 4 2320 1191 4 MP DP Xc12 X X-31 -6 125 -29 30 1 1794 1305 4 MP PP Xc0 X X-31 -6 125 -29 30 1 1794 1305 4 MP DP Xc15 X X-30 -25 124 -38 31 17 1391 1286 4 MP PP Xc0 X X-30 -25 124 -38 31 17 1391 1286 4 MP DP Xc4 X X-31 -39 125 -39 30 35 2537 1198 4 MP PP Xc0 X X-31 -39 125 -39 30 35 2537 1198 4 MP DP Xc12 X X-31 -18 124 -31 31 15 1608 1311 4 MP PP Xc0 X X-31 -18 124 -31 31 15 1608 1311 4 MP DP Xc11 X X-30 7 124 -31 31 -6 1979 1274 4 MP PP Xc0 X X-30 7 124 -31 31 -6 1979 1274 4 MP DP Xc19 X X-30 -48 124 -31 31 40 2567 1233 4 MP PP Xc0 X X-30 -48 124 -31 31 40 2567 1233 4 MP DP Xc9 X X-31 96 125 -109 30 0 2815 1351 4 MP PP Xc0 X X-31 96 125 -109 30 0 2815 1351 4 MP DP Xc19 X X-31 5 124 -34 31 -4 2165 1229 4 MP PP Xc0 X X-31 5 124 -34 31 -4 2165 1229 4 MP DP Xc16 X X-31 -47 124 -25 31 41 2598 1273 4 MP PP Xc0 X X-31 -47 124 -25 31 41 2598 1273 4 MP DP Xc15 X X-31 -25 124 -28 31 15 1422 1303 4 MP PP Xc0 X X-31 -25 124 -28 31 15 1422 1303 4 MP DP Xc17 X X-31 -13 124 -37 31 14 2351 1195 4 MP PP Xc0 X X-31 -13 124 -37 31 14 2351 1195 4 MP DP Xc12 X X-31 -10 125 -28 30 7 1639 1326 4 MP PP Xc0 X X-31 -10 125 -28 30 7 1639 1326 4 MP DP Xc15 X X-31 -39 124 -23 31 37 2629 1314 4 MP PP Xc0 X X-31 -39 124 -23 31 37 2629 1314 4 MP DP Xc15 X X-30 3 124 -28 31 -4 1824 1306 4 MP PP Xc0 X X-30 3 124 -28 31 -4 1824 1306 4 MP DP Xc16 X X-31 8 124 -33 31 -6 2010 1268 4 MP PP Xc0 X X-31 8 124 -33 31 -6 2010 1268 4 MP DP Xc15 X X-31 -21 124 -21 31 14 1453 1318 4 MP PP Xc0 X X-31 -21 124 -21 31 14 1453 1318 4 MP DP Xc17 X X-31 -26 125 -36 30 25 2382 1209 4 MP PP Xc0 X X-31 -26 125 -36 30 25 2382 1209 4 MP DP Xc12 X X-31 -23 125 -30 30 30 2660 1351 4 MP PP Xc0 X X-31 -23 125 -30 30 30 2660 1351 4 MP DP Xc4 X X-31 -4 124 -35 31 5 2196 1225 4 MP PP Xc0 X X-31 -4 124 -35 31 5 2196 1225 4 MP DP Xc4 X X-30 -35 124 -35 31 34 2412 1234 4 MP PP Xc0 X X-30 -35 124 -35 31 34 2412 1234 4 MP DP Xc15 X X-30 -1 124 -26 31 -1 1669 1333 4 MP PP Xc0 X X-30 -1 124 -26 31 -1 1669 1333 4 MP DP Xc13 X X-30 0 124 -52 31 22 2690 1381 4 MP PP Xc0 X X-30 0 124 -52 31 22 2690 1381 4 MP DP Xc16 X X-31 6 124 -27 31 -7 1855 1302 4 MP PP Xc0 X X-31 6 124 -27 31 -7 1855 1302 4 MP DP Xc15 X X-31 -15 125 -16 30 10 1484 1332 4 MP PP Xc0 X X-31 -15 125 -16 30 10 1484 1332 4 MP DP Xc19 X X-31 -40 124 -32 31 37 2443 1268 4 MP PP Xc0 X X-31 -40 124 -32 31 37 2443 1268 4 MP DP Xc19 X X-31 4 124 -36 31 -1 2041 1262 4 MP PP Xc0 X X-31 4 124 -36 31 -1 2041 1262 4 MP DP Xc17 X X-31 -14 125 -36 30 15 2227 1230 4 MP PP Xc0 X X-31 -14 125 -36 30 15 2227 1230 4 MP DP Xc16 X X-31 -41 124 -31 31 40 2474 1305 4 MP PP Xc0 X X-31 -41 124 -31 31 40 2474 1305 4 MP DP Xc15 X X-31 -37 125 -31 30 37 2505 1345 4 MP PP Xc0 X X-31 -37 125 -31 30 37 2505 1345 4 MP DP Xc11 X X-31 4 124 -27 31 -3 1700 1332 4 MP PP Xc0 X X-31 4 124 -27 31 -3 1700 1332 4 MP DP Xc15 X X-30 -7 124 -17 31 8 1514 1342 4 MP PP Xc0 X X-30 -7 124 -17 31 8 1514 1342 4 MP DP Xc4 X X-30 -25 124 -36 31 25 2257 1245 4 MP PP Xc0 X X-30 -25 124 -36 31 25 2257 1245 4 MP DP Xc14 X X-31 6 124 -30 31 -3 1886 1295 4 MP PP Xc0 X X-31 6 124 -30 31 -3 1886 1295 4 MP DP Xc12 X X-30 -30 124 -34 31 33 2535 1382 4 MP PP Xc0 X X-30 -30 124 -34 31 33 2535 1382 4 MP DP Xc4 X X-31 -5 125 -38 30 7 2072 1261 4 MP PP Xc0 X X-31 -5 125 -38 30 7 2072 1261 4 MP DP Xc4 X X-31 -34 124 -35 31 33 2288 1270 4 MP PP Xc0 X X-31 -34 124 -35 31 33 2288 1270 4 MP DP Xc13 X X-31 -22 124 -39 31 27 2566 1415 4 MP PP Xc0 X X-31 -22 124 -39 31 27 2566 1415 4 MP DP Xc11 X X-31 1 124 -23 31 5 1545 1350 4 MP PP Xc0 X X-31 1 124 -23 31 5 1545 1350 4 MP DP Xc16 X X-31 7 124 -30 31 -4 1731 1329 4 MP PP Xc0 X X-31 7 124 -30 31 -4 1731 1329 4 MP DP Xc19 X X-31 -37 124 -35 31 37 2319 1303 4 MP PP Xc0 X X-31 -37 124 -35 31 37 2319 1303 4 MP DP Xc4 X X-30 -15 124 -40 31 17 2102 1268 4 MP PP Xc0 X X-30 -15 124 -40 31 17 2102 1268 4 MP DP Xc19 X X-31 1 125 -35 30 4 1917 1292 4 MP PP Xc0 X X-31 1 125 -35 30 4 1917 1292 4 MP DP Xc16 X X-31 -40 125 -32 30 37 2350 1340 4 MP PP Xc0 X X-31 -40 125 -32 30 37 2350 1340 4 MP DP Xc11 X X-30 -37 124 -31 31 36 2380 1377 4 MP PP Xc0 X X-30 -37 124 -31 31 36 2380 1377 4 MP DP Xc16 X X-31 3 124 -33 31 7 1576 1355 4 MP PP Xc0 X X-31 3 124 -33 31 7 1576 1355 4 MP DP Xc4 X X-31 -25 124 -39 31 24 2133 1285 4 MP PP Xc0 X X-31 -25 124 -39 31 24 2133 1285 4 MP DP Xc14 X X-31 3 125 -35 30 2 1762 1325 4 MP PP Xc0 X X-31 3 125 -35 30 2 1762 1325 4 MP DP Xc12 X X-31 -33 124 -31 31 33 2411 1413 4 MP PP Xc0 X X-31 -33 124 -31 31 33 2411 1413 4 MP DP Xc4 X X-30 -7 124 -39 31 11 1947 1296 4 MP PP Xc0 X X-30 -7 124 -39 31 11 1947 1296 4 MP DP Xc19 X X-31 -33 124 -37 31 31 2164 1309 4 MP PP Xc0 X X-31 -33 124 -37 31 31 2164 1309 4 MP DP Xc13 X X-31 -27 124 -32 31 28 2442 1446 4 MP PP Xc0 X X-31 -27 124 -32 31 28 2442 1446 4 MP DP Xc14 X X-31 -37 125 -33 30 33 2195 1340 4 MP PP Xc0 X X-31 -37 125 -33 30 33 2195 1340 4 MP DP Xc16 X X-31 4 125 -45 30 8 1607 1362 4 MP PP Xc0 X X-31 4 125 -45 30 8 1607 1362 4 MP DP Xc4 X X-31 -17 124 -42 31 20 1978 1307 4 MP PP Xc0 X X-31 -17 124 -42 31 20 1978 1307 4 MP DP Xc19 X X-30 -4 124 -39 31 8 1792 1327 4 MP PP Xc0 X X-30 -4 124 -39 31 8 1792 1327 4 MP DP Xc16 X X-30 -37 124 -30 31 34 2225 1373 4 MP PP Xc0 X X-30 -37 124 -30 31 34 2225 1373 4 MP DP Xc4 X X-31 -24 124 -43 31 25 2009 1327 4 MP PP Xc0 X X-31 -24 124 -43 31 25 2009 1327 4 MP DP Xc11 X X-31 -36 124 -27 31 33 2256 1407 4 MP PP Xc0 X X-31 -36 124 -27 31 33 2256 1407 4 MP DP Xc16 X X-30 -2 124 -56 31 13 1637 1370 4 MP PP Xc0 X X-30 -2 124 -56 31 13 1637 1370 4 MP DP Xc19 X X-31 -11 124 -43 31 15 1823 1335 4 MP PP Xc0 X X-31 -11 124 -43 31 15 1823 1335 4 MP DP Xc15 X X-31 -33 124 -24 31 30 2287 1440 4 MP PP Xc0 X X-31 -33 124 -24 31 30 2287 1440 4 MP DP Xc19 X X-31 -31 125 -41 30 29 2040 1352 4 MP PP Xc0 X X-31 -31 125 -41 30 29 2040 1352 4 MP DP Xc12 X X-31 -28 125 -22 30 26 2318 1470 4 MP PP Xc0 X X-31 -28 125 -22 30 26 2318 1470 4 MP DP Xc14 X X-30 -33 124 -38 31 30 2070 1381 4 MP PP Xc0 X X-30 -33 124 -38 31 30 2070 1381 4 MP DP Xc14 X X-31 -8 124 -65 31 17 1668 1383 4 MP PP Xc0 X X-31 -8 124 -65 31 17 1668 1383 4 MP DP Xc19 X X-31 -20 124 -45 31 22 1854 1350 4 MP PP Xc0 X X-31 -20 124 -45 31 22 1854 1350 4 MP DP Xc16 X X-31 -34 124 -34 31 30 2101 1411 4 MP PP Xc0 X X-31 -34 124 -34 31 30 2101 1411 4 MP DP Xc19 X X-31 -25 125 -46 30 26 1885 1372 4 MP PP Xc0 X X-31 -25 125 -46 30 26 1885 1372 4 MP DP Xc16 X X-31 -15 124 -70 31 20 1699 1400 4 MP PP Xc0 X X-31 -15 124 -70 31 20 1699 1400 4 MP DP Xc11 X X-31 -33 124 -29 31 28 2132 1441 4 MP PP Xc0 X X-31 -33 124 -29 31 28 2132 1441 4 MP DP Xc14 X X-30 -29 124 -45 31 28 1915 1398 4 MP PP Xc0 X X-30 -29 124 -45 31 28 1915 1398 4 MP DP Xc15 X X-31 -30 125 -25 30 26 2163 1469 4 MP PP Xc0 X X-31 -30 125 -25 30 26 2163 1469 4 MP DP Xc16 X X-31 -22 125 -71 30 23 1730 1420 4 MP PP Xc0 X X-31 -22 125 -71 30 23 1730 1420 4 MP DP Xc16 X X-31 -30 124 -44 31 29 1946 1426 4 MP PP Xc0 X X-31 -30 124 -44 31 29 1946 1426 4 MP DP Xc12 X X-30 -26 124 -23 31 24 2193 1495 4 MP PP Xc0 X X-30 -26 124 -23 31 24 2193 1495 4 MP DP Xc11 X X-30 -26 124 -69 31 24 1760 1443 4 MP PP Xc0 X X-30 -26 124 -69 31 24 1760 1443 4 MP DP Xc11 X X-31 -30 124 -41 31 27 1977 1455 4 MP PP Xc0 X X-31 -30 124 -41 31 27 1977 1455 4 MP DP Xc11 X X-31 -28 124 -66 31 25 1791 1467 4 MP PP Xc0 X X-31 -28 124 -66 31 25 1791 1467 4 MP DP Xc15 X X-31 -28 125 -39 30 26 2008 1482 4 MP PP Xc0 X X-31 -28 125 -39 30 26 2008 1482 4 MP DP Xc15 X X-31 -29 124 -60 31 23 1822 1492 4 MP PP Xc0 X X-31 -29 124 -60 31 23 1822 1492 4 MP DP Xc15 X X-30 -26 124 -37 31 24 2038 1508 4 MP PP Xc0 X X-30 -26 124 -37 31 24 2038 1508 4 MP DP Xc12 X X-31 -27 125 -56 30 23 1853 1515 4 MP PP Xc0 X X-31 -27 125 -56 30 23 1853 1515 4 MP DP Xc12 X X-31 -24 124 -35 31 22 2069 1532 4 MP PP Xc0 X X-31 -24 124 -35 31 22 2069 1532 4 MP DP Xc12 X X-30 -26 124 -51 31 21 1883 1538 4 MP PP Xc0 X X-30 -26 124 -51 31 21 1883 1538 4 MP DP Xc13 X X-31 -24 124 -47 31 20 1914 1559 4 MP PP Xc0 X X-31 -24 124 -47 31 20 1914 1559 4 MP DP Xc13 X X-31 -22 124 -43 31 18 1945 1579 4 MP PP Xc0 X X-31 -22 124 -43 31 18 1945 1579 4 MP DP Xgr XDO XSO X4979 2165 mt 6221 1825 L X4026 1722 mt 4979 2165 L X4026 1722 mt 4026 754 L X4979 2165 mt 5008 2179 L X5033 2346 mt (0) s X5600 1995 mt 5629 2009 L X5654 2176 mt (5) s X6221 1825 mt 6250 1838 L X6275 2006 mt (10) s X4303 1850 mt 4272 1859 L X4060 2022 mt (10) s X4610 1993 mt 4579 2002 L X4367 2165 mt (20) s X4918 2136 mt 4886 2145 L X4674 2308 mt (30) s X4026 1600 mt 3997 1586 L X3730 1637 mt (-30) s X4026 1318 mt 3997 1304 L X3730 1355 mt (-20) s X4026 1036 mt 3997 1022 L X3730 1073 mt (-10) s X4133 281 mt (Tikh filter factors, log scale) s Xgs 4026 414 2196 1752 MR c np Xc2 X X-31 -14 124 -34 31 14 5144 448 4 MP PP Xc0 X X-31 -14 124 -34 31 14 5144 448 4 MP DP Xc2 X X-30 -15 124 -34 30 15 5175 462 4 MP PP Xc0 X X-30 -15 124 -34 30 15 5175 462 4 MP DP Xc2 X X-31 -14 124 -34 31 14 5205 477 4 MP PP Xc0 X X-31 -14 124 -34 31 14 5205 477 4 MP DP Xc2 X X-31 -14 125 -34 30 14 5020 482 4 MP PP Xc0 X X-31 -14 125 -34 30 14 5020 482 4 MP DP Xc2 X X-31 -14 124 -34 31 14 5236 491 4 MP PP Xc0 X X-31 -14 124 -34 31 14 5236 491 4 MP DP Xc2 X X-30 -15 124 -34 31 15 5050 496 4 MP PP Xc0 X X-30 -15 124 -34 31 15 5050 496 4 MP DP Xc2 X X-31 -14 124 -34 31 14 5267 505 4 MP PP Xc0 X X-31 -14 124 -34 31 14 5267 505 4 MP DP Xc2 X X-31 -14 124 -34 31 14 5081 511 4 MP PP Xc0 X X-31 -14 124 -34 31 14 5081 511 4 MP DP Xc2 X X-30 -14 124 -34 31 14 4895 516 4 MP PP Xc0 X X-30 -14 124 -34 31 14 4895 516 4 MP DP Xc2 X X-30 -15 124 -34 30 15 5298 519 4 MP PP Xc0 X X-30 -15 124 -34 30 15 5298 519 4 MP DP Xc3 X X-31 -32 125 -63 30 31 5759 1373 4 MP PP Xc0 X X-31 -32 125 -63 30 31 5759 1373 4 MP DP Xc2 X X-31 -14 124 -34 31 14 5112 525 4 MP PP Xc0 X X-31 -14 124 -34 31 14 5112 525 4 MP DP Xc2 X X-31 -15 124 -34 31 15 4926 530 4 MP PP Xc0 X X-31 -15 124 -34 31 15 4926 530 4 MP DP Xcolortable /c23 { 0.000000 1.000000 0.761905 sc} put Xc23 X X-31 -131 124 -64 31 131 5728 1242 4 MP PP Xc0 X X-31 -131 124 -64 31 131 5728 1242 4 MP DP Xc2 X X-31 -14 124 -34 31 14 5328 534 4 MP PP Xc0 X X-31 -14 124 -34 31 14 5328 534 4 MP DP Xcolortable /c24 { 0.095238 1.000000 0.000000 sc} put Xc24 X X-30 -18 124 -64 31 19 5789 1404 4 MP PP Xc0 X X-30 -18 124 -64 31 19 5789 1404 4 MP DP Xc22 X X-30 -45 124 -63 31 44 5666 1153 4 MP PP Xc0 X X-30 -45 124 -63 31 44 5666 1153 4 MP DP Xc2 X X-31 -14 125 -35 30 15 5143 539 4 MP PP Xc0 X X-31 -14 125 -35 30 15 5143 539 4 MP DP Xc21 X X-31 -44 124 -64 31 45 5697 1197 4 MP PP Xc0 X X-31 -44 124 -64 31 45 5697 1197 4 MP DP Xc16 X X-31 -105 125 -64 30 105 5636 1048 4 MP PP Xc0 X X-31 -105 125 -64 30 105 5636 1048 4 MP DP Xc2 X X-31 -14 124 -34 31 14 4957 545 4 MP PP Xc0 X X-31 -14 124 -34 31 14 4957 545 4 MP DP Xc24 X X-31 -22 124 -63 31 21 5820 1423 4 MP PP Xc0 X X-31 -22 124 -63 31 21 5820 1423 4 MP DP Xcolortable /c25 { 0.190476 1.000000 0.000000 sc} put Xc25 X X-30 -31 124 -61 31 31 5634 1434 4 MP PP Xc0 X X-30 -31 124 -61 31 31 5634 1434 4 MP DP Xc2 X X-31 -14 124 -34 31 14 5359 548 4 MP PP Xc0 X X-31 -14 124 -34 31 14 5359 548 4 MP DP Xc2 X X-31 -14 124 -34 31 14 4771 550 4 MP PP Xc0 X X-31 -14 124 -34 31 14 4771 550 4 MP DP Xc4 X X-31 -67 124 -64 31 68 5605 980 4 MP PP Xc0 X X-31 -67 124 -64 31 68 5605 980 4 MP DP Xc2 X X-30 -15 124 -34 31 14 5173 554 4 MP PP Xc0 X X-30 -15 124 -34 31 14 5173 554 4 MP DP Xcolortable /c26 { 0.000000 1.000000 0.571429 sc} put Xc26 X X-31 -131 125 -61 30 131 5604 1303 4 MP PP Xc0 X X-31 -131 125 -61 30 131 5604 1303 4 MP DP Xcolortable /c27 { 0.476190 0.000000 1.000000 sc} put Xc27 X X-31 -81 124 -63 31 80 5574 900 4 MP PP Xc0 X X-31 -81 124 -63 31 80 5574 900 4 MP DP Xcolortable /c28 { 0.285714 1.000000 0.000000 sc} put Xc28 X X-31 -19 124 -61 31 19 5665 1465 4 MP PP Xc0 X X-31 -19 124 -61 31 19 5665 1465 4 MP DP Xc25 X X-31 -18 124 -63 31 18 5851 1444 4 MP PP Xc0 X X-31 -18 124 -63 31 18 5851 1444 4 MP DP Xc2 X X-31 -14 125 -34 30 14 4988 559 4 MP PP Xc0 X X-31 -14 125 -34 30 14 4988 559 4 MP DP Xc2 X X-31 -17 125 -45 30 28 5390 562 4 MP PP Xc0 X X-31 -17 125 -45 30 28 5390 562 4 MP DP Xc21 X X-31 -44 124 -61 31 44 5542 1214 4 MP PP Xc0 X X-31 -44 124 -61 31 44 5542 1214 4 MP DP Xc2 X X-31 -15 124 -34 31 15 4802 564 4 MP PP Xc0 X X-31 -15 124 -34 31 15 4802 564 4 MP DP Xcolortable /c29 { 0.857143 0.000000 1.000000 sc} put Xc29 X X-30 -77 124 -64 31 77 5543 823 4 MP PP Xc0 X X-30 -77 124 -64 31 77 5543 823 4 MP DP Xc23 X X-31 -45 124 -61 31 45 5573 1258 4 MP PP Xc0 X X-31 -45 124 -61 31 45 5573 1258 4 MP DP Xc2 X X-30 -45 124 -63 31 63 5420 590 4 MP PP Xc0 X X-30 -45 124 -63 31 63 5420 590 4 MP DP Xc15 X X-30 -105 124 -61 31 105 5511 1109 4 MP PP Xc0 X X-30 -105 124 -61 31 105 5511 1109 4 MP DP Xc2 X X-31 -14 124 -34 31 14 5204 568 4 MP PP Xc0 X X-31 -14 124 -34 31 14 5204 568 4 MP DP Xcolortable /c30 { 1.000000 0.000000 0.380952 sc} put Xc30 X X-31 -47 124 -63 31 47 5451 653 4 MP PP Xc0 X X-31 -47 124 -63 31 47 5451 653 4 MP DP Xcolortable /c31 { 1.000000 0.000000 0.761905 sc} put Xc31 X X-31 -68 125 -64 30 69 5513 754 4 MP PP Xc0 X X-31 -68 125 -64 30 69 5513 754 4 MP DP Xc28 X X-31 -21 124 -61 31 21 5696 1484 4 MP PP Xc0 X X-31 -21 124 -61 31 21 5696 1484 4 MP DP Xc25 X X-31 -18 125 -64 30 19 5882 1462 4 MP PP Xc0 X X-31 -18 125 -64 30 19 5882 1462 4 MP DP Xcolortable /c32 { 1.000000 0.000000 0.571429 sc} put Xc32 X X-31 -54 124 -63 31 54 5482 700 4 MP PP Xc0 X X-31 -54 124 -63 31 54 5482 700 4 MP DP Xc28 X X-31 -31 124 -64 31 32 5510 1497 4 MP PP Xc0 X X-31 -31 124 -64 31 32 5510 1497 4 MP DP Xc2 X X-30 -15 124 -34 31 15 5018 573 4 MP PP Xc0 X X-30 -15 124 -34 31 15 5018 573 4 MP DP Xc19 X X-31 -68 125 -61 30 68 5481 1041 4 MP PP Xc0 X X-31 -68 125 -61 30 68 5481 1041 4 MP DP Xc2 X X-31 -14 125 -34 30 14 4833 579 4 MP PP Xc0 X X-31 -14 125 -34 30 14 4833 579 4 MP DP Xcolortable /c33 { 0.000000 1.000000 0.380952 sc} put Xc33 X X-30 -131 124 -63 31 131 5479 1366 4 MP PP Xc0 X X-30 -131 124 -63 31 131 5479 1366 4 MP DP Xc2 X X-31 -14 124 -35 31 15 5235 582 4 MP PP Xc0 X X-31 -14 124 -35 31 15 5235 582 4 MP DP Xcolortable /c34 { 0.285714 0.000000 1.000000 sc} put Xc34 X X-31 -80 124 -61 31 80 5450 961 4 MP PP Xc0 X X-31 -80 124 -61 31 80 5450 961 4 MP DP Xc25 X X-30 -28 124 -64 31 28 5912 1481 4 MP PP Xc0 X X-30 -28 124 -64 31 28 5912 1481 4 MP DP Xcolortable /c35 { 0.476190 1.000000 0.000000 sc} put Xc35 X X-31 -19 124 -63 31 18 5541 1529 4 MP PP Xc0 X X-31 -19 124 -63 31 18 5541 1529 4 MP DP Xc28 X X-31 -18 125 -61 30 18 5727 1505 4 MP PP Xc0 X X-31 -18 125 -61 30 18 5727 1505 4 MP DP Xc2 X X-31 -14 124 -34 31 14 4647 584 4 MP PP Xc0 X X-31 -14 124 -34 31 14 4647 584 4 MP DP Xc23 X X-31 -44 124 -64 31 45 5418 1277 4 MP PP Xc0 X X-31 -44 124 -64 31 45 5418 1277 4 MP DP Xc2 X X-31 -14 124 -34 31 14 5049 588 4 MP PP Xc0 X X-31 -14 124 -34 31 14 5049 588 4 MP DP Xcolortable /c36 { 0.666667 0.000000 1.000000 sc} put Xc36 X X-31 -77 124 -61 31 77 5419 884 4 MP PP Xc0 X X-31 -77 124 -61 31 77 5419 884 4 MP DP Xc26 X X-31 -45 125 -63 30 44 5449 1322 4 MP PP Xc0 X X-31 -45 125 -63 30 44 5449 1322 4 MP DP Xc2 X X-30 -28 124 -57 30 50 5266 597 4 MP PP Xc0 X X-30 -28 124 -57 30 50 5266 597 4 MP DP Xc13 X X-31 -105 124 -63 31 105 5387 1172 4 MP PP Xc0 X X-31 -105 124 -63 31 105 5387 1172 4 MP DP Xc2 X X-30 -14 124 -34 31 14 4863 593 4 MP PP Xc0 X X-30 -14 124 -34 31 14 4863 593 4 MP DP Xcolortable /c37 { 1.000000 0.000000 0.190476 sc} put Xc37 X X-31 -63 124 -61 31 67 5296 647 4 MP PP Xc0 X X-31 -63 124 -61 31 67 5296 647 4 MP DP Xcolortable /c38 { 1.000000 0.000000 0.476190 sc} put Xc38 X X-31 -47 124 -61 31 47 5327 714 4 MP PP Xc0 X X-31 -47 124 -61 31 47 5327 714 4 MP DP Xc28 X X-31 -18 124 -63 31 17 5943 1509 4 MP PP Xc0 X X-31 -18 124 -63 31 17 5943 1509 4 MP DP Xcolortable /c39 { 1.000000 0.000000 0.952381 sc} put Xc39 X X-30 -69 124 -61 31 69 5388 815 4 MP PP Xc0 X X-30 -69 124 -61 31 69 5388 815 4 MP DP Xc35 X X-31 -21 125 -64 30 22 5572 1547 4 MP PP Xc0 X X-31 -21 125 -64 30 22 5572 1547 4 MP DP Xcolortable /c40 { 0.380952 1.000000 0.000000 sc} put Xc40 X X-30 -19 124 -61 31 19 5757 1523 4 MP PP Xc0 X X-30 -19 124 -61 31 19 5757 1523 4 MP DP Xc31 X X-31 -54 125 -61 30 54 5358 761 4 MP PP Xc0 X X-31 -54 125 -61 30 54 5358 761 4 MP DP Xc35 X X-31 -32 124 -61 31 32 5386 1558 4 MP PP Xc0 X X-31 -32 124 -61 31 32 5386 1558 4 MP DP Xc2 X X-31 -15 125 -34 30 15 4678 598 4 MP PP Xc0 X X-31 -15 125 -34 30 15 4678 598 4 MP DP Xc2 X X-31 -14 124 -35 31 15 5080 602 4 MP PP Xc0 X X-31 -14 124 -35 31 15 5080 602 4 MP DP Xc16 X X-30 -68 124 -63 30 67 5357 1105 4 MP PP Xc0 X X-30 -68 124 -63 30 67 5357 1105 4 MP DP Xcolortable /c41 { 0.000000 1.000000 0.190476 sc} put Xc41 X X-31 -131 124 -61 31 131 5355 1427 4 MP PP Xc0 X X-31 -131 124 -61 31 131 5355 1427 4 MP DP Xc28 X X-31 -23 124 -64 31 24 5974 1526 4 MP PP Xc0 X X-31 -23 124 -64 31 24 5974 1526 4 MP DP Xc2 X X-31 -15 124 -34 31 15 4894 607 4 MP PP Xc0 X X-31 -15 124 -34 31 15 4894 607 4 MP DP Xc17 X X-31 -80 124 -64 31 81 5326 1024 4 MP PP Xc0 X X-31 -80 124 -64 31 81 5326 1024 4 MP DP Xc40 X X-31 -28 124 -61 31 28 5788 1542 4 MP PP Xc0 X X-31 -28 124 -61 31 28 5788 1542 4 MP DP Xcolortable /c42 { 0.571429 1.000000 0.000000 sc} put Xc42 X X-31 -18 124 -61 31 18 5417 1590 4 MP PP Xc0 X X-31 -18 124 -61 31 18 5417 1590 4 MP DP Xc35 X X-30 -18 124 -63 31 17 5602 1569 4 MP PP Xc0 X X-30 -18 124 -63 31 17 5602 1569 4 MP DP Xc26 X X-31 -45 124 -60 31 44 5294 1338 4 MP PP Xc0 X X-31 -45 124 -60 31 44 5294 1338 4 MP DP Xc2 X X-30 -14 124 -34 31 14 4708 613 4 MP PP Xc0 X X-30 -14 124 -34 31 14 4708 613 4 MP DP Xc27 X X-31 -77 124 -63 31 77 5295 947 4 MP PP Xc0 X X-31 -77 124 -63 31 77 5295 947 4 MP DP Xc33 X X-30 -44 124 -61 30 45 5325 1382 4 MP PP Xc0 X X-30 -44 124 -61 30 45 5325 1382 4 MP DP Xc2 X X-31 -15 125 -40 30 20 5111 617 4 MP PP Xc0 X X-31 -15 125 -40 30 20 5111 617 4 MP DP Xc22 X X-31 -105 124 -61 31 105 5263 1233 4 MP PP Xc0 X X-31 -105 124 -61 31 105 5263 1233 4 MP DP Xc2 X X-31 -14 124 -34 31 14 4523 618 4 MP PP Xc0 X X-31 -14 124 -34 31 14 4523 618 4 MP DP Xc40 X X-31 -19 125 -64 30 19 6005 1550 4 MP PP Xc0 X X-31 -19 125 -64 30 19 6005 1550 4 MP DP Xc30 X X-31 -67 124 -63 31 67 5172 710 4 MP PP Xc0 X X-31 -67 124 -63 31 67 5172 710 4 MP DP Xcolortable /c43 { 1.000000 0.000000 0.666667 sc} put Xc43 X X-31 -47 124 -64 31 48 5203 777 4 MP PP Xc0 X X-31 -47 124 -64 31 48 5203 777 4 MP DP Xc35 X X-31 -17 124 -61 31 17 5819 1570 4 MP PP Xc0 X X-31 -17 124 -61 31 17 5819 1570 4 MP DP Xc29 X X-31 -69 124 -63 31 68 5264 879 4 MP PP Xc0 X X-31 -69 124 -63 31 68 5264 879 4 MP DP Xcolortable /c44 { 0.666667 1.000000 0.000000 sc} put Xc44 X X-30 -22 124 -61 30 22 5448 1608 4 MP PP Xc0 X X-30 -22 124 -61 30 22 5448 1608 4 MP DP Xc42 X X-31 -19 124 -63 31 19 5633 1586 4 MP PP Xc0 X X-31 -19 124 -63 31 19 5633 1586 4 MP DP Xc39 X X-30 -54 124 -64 30 54 5234 825 4 MP PP Xc0 X X-30 -54 124 -64 30 54 5234 825 4 MP DP Xc2 X X-30 -50 124 -63 31 73 5141 637 4 MP PP Xc0 X X-30 -50 124 -63 31 73 5141 637 4 MP DP Xc2 X X-31 -14 124 -34 31 14 4925 622 4 MP PP Xc0 X X-31 -14 124 -34 31 14 4925 622 4 MP DP Xc44 X X-31 -32 124 -63 31 31 5262 1622 4 MP PP Xc0 X X-31 -32 124 -63 31 31 5262 1622 4 MP DP Xc15 X X-30 -67 124 -61 31 67 5232 1166 4 MP PP Xc0 X X-30 -67 124 -61 31 67 5232 1166 4 MP DP Xc2 X X-31 -14 124 -35 31 15 4739 627 4 MP PP Xc0 X X-31 -14 124 -35 31 15 4739 627 4 MP DP Xc40 X X-30 -32 124 -63 31 31 6035 1569 4 MP PP Xc0 X X-30 -32 124 -63 31 31 6035 1569 4 MP DP Xcolortable /c45 { 0.000000 1.000000 0.095238 sc} put Xc45 X X-31 -131 124 -64 31 131 5231 1491 4 MP PP Xc0 X X-31 -131 124 -64 31 131 5231 1491 4 MP DP Xc35 X X-31 -24 125 -61 30 24 5850 1587 4 MP PP Xc0 X X-31 -24 125 -61 30 24 5850 1587 4 MP DP Xc19 X X-31 -81 125 -61 30 81 5202 1085 4 MP PP Xc0 X X-31 -81 125 -61 30 81 5202 1085 4 MP DP Xc42 X X-31 -28 124 -63 31 28 5664 1605 4 MP PP Xc0 X X-31 -28 124 -63 31 28 5664 1605 4 MP DP Xcolortable /c46 { 0.761905 1.000000 0.000000 sc} put Xc46 X X-31 -18 125 -64 30 19 5293 1653 4 MP PP Xc0 X X-31 -18 125 -64 30 19 5293 1653 4 MP DP Xc2 X X-30 -15 124 -34 30 15 4554 632 4 MP PP Xc0 X X-30 -15 124 -34 30 15 4554 632 4 MP DP Xc44 X X-31 -17 124 -61 31 17 5478 1630 4 MP PP Xc0 X X-31 -17 124 -61 31 17 5478 1630 4 MP DP Xc2 X X-31 -15 125 -38 30 19 4956 636 4 MP PP Xc0 X X-31 -15 125 -38 30 19 4956 636 4 MP DP Xc8 X X-31 -44 125 -64 30 45 5170 1401 4 MP PP Xc0 X X-31 -44 125 -64 30 45 5170 1401 4 MP DP Xc35 X X-31 -33 124 -63 31 33 6066 1600 4 MP PP Xc0 X X-31 -33 124 -63 31 33 6066 1600 4 MP DP Xc20 X X-31 -77 124 -61 31 77 5171 1008 4 MP PP Xc0 X X-31 -77 124 -61 31 77 5171 1008 4 MP DP Xcolortable /c47 { 0.000000 1.000000 0.285714 sc} put Xc47 X X-30 -45 124 -64 31 45 5200 1446 4 MP PP Xc0 X X-30 -45 124 -64 31 45 5200 1446 4 MP DP Xc2 X X-31 -15 124 -35 31 15 4770 642 4 MP PP Xc0 X X-31 -15 124 -35 31 15 4770 642 4 MP DP Xc10 X X-31 -105 124 -63 31 104 5139 1297 4 MP PP Xc0 X X-31 -105 124 -63 31 104 5139 1297 4 MP DP Xc42 X X-30 -19 124 -61 31 19 5880 1611 4 MP PP Xc0 X X-30 -19 124 -61 31 19 5880 1611 4 MP DP Xcolortable /c48 { 1.000000 0.000000 0.857143 sc} put Xc48 X X-31 -48 125 -61 30 48 5079 838 4 MP PP Xc0 X X-31 -48 125 -61 30 48 5079 838 4 MP DP Xc32 X X-31 -67 124 -61 31 67 5048 771 4 MP PP Xc0 X X-31 -67 124 -61 31 67 5048 771 4 MP DP Xc44 X X-31 -17 125 -64 30 18 5695 1633 4 MP PP Xc0 X X-31 -17 125 -64 30 18 5695 1633 4 MP DP Xc36 X X-31 -68 124 -61 31 68 5140 940 4 MP PP Xc0 X X-31 -68 124 -61 31 68 5140 940 4 MP DP Xcolortable /c49 { 0.857143 1.000000 0.000000 sc} put Xc49 X X-30 -22 124 -63 31 21 5323 1672 4 MP PP Xc0 X X-30 -22 124 -63 31 21 5323 1672 4 MP DP Xc44 X X-31 -19 124 -61 31 19 5509 1647 4 MP PP Xc0 X X-31 -19 124 -61 31 19 5509 1647 4 MP DP Xcolortable /c50 { 0.952381 0.000000 1.000000 sc} put Xc50 X X-30 -54 124 -61 31 54 5109 886 4 MP PP Xc0 X X-30 -54 124 -61 31 54 5109 886 4 MP DP Xc49 X X-31 -31 125 -61 30 32 5138 1682 4 MP PP Xc0 X X-31 -31 125 -61 30 32 5138 1682 4 MP DP Xc2 X X-31 -14 124 -34 31 14 4584 647 4 MP PP Xc0 X X-31 -14 124 -34 31 14 4584 647 4 MP DP Xc2 X X-30 -20 124 -51 31 33 4986 655 4 MP PP Xc0 X X-30 -20 124 -51 31 33 4986 655 4 MP DP Xcolortable /c51 { 1.000000 0.000000 0.095238 sc} put Xc51 X X-31 -73 124 -61 31 83 5017 688 4 MP PP Xc0 X X-31 -73 124 -61 31 83 5017 688 4 MP DP Xc13 X X-31 -67 124 -64 31 68 5108 1229 4 MP PP Xc0 X X-31 -67 124 -64 31 68 5108 1229 4 MP DP Xc2 X X-31 -14 125 -34 30 14 4399 652 4 MP PP Xc0 X X-31 -14 125 -34 30 14 4399 652 4 MP DP Xc42 X X-31 -31 124 -61 31 31 5911 1630 4 MP PP Xc0 X X-31 -31 124 -61 31 31 5911 1630 4 MP DP Xc24 X X-31 -131 124 -60 31 130 5107 1552 4 MP PP Xc0 X X-31 -131 124 -60 31 130 5107 1552 4 MP DP Xc2 X X-31 -14 125 -36 30 15 4801 657 4 MP PP Xc0 X X-31 -14 125 -36 30 15 4801 657 4 MP DP Xc44 X X-30 -24 124 -63 31 23 5725 1651 4 MP PP Xc0 X X-30 -24 124 -63 31 23 5725 1651 4 MP DP Xc16 X X-30 -81 124 -63 31 80 5077 1149 4 MP PP Xc0 X X-30 -81 124 -63 31 80 5077 1149 4 MP DP Xc46 X X-31 -28 125 -61 30 28 5540 1666 4 MP PP Xc0 X X-31 -28 125 -61 30 28 5540 1666 4 MP DP Xcolortable /c52 { 0.952381 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124 -61 31 28 5291 1791 4 MP DP Xcolortable /c61 { 1.000000 0.666667 0.000000 sc} put Xc61 X X-31 -18 124 -61 31 19 4920 1838 4 MP PP Xc0 X X-31 -18 124 -61 31 19 4920 1838 4 MP DP Xc60 X X-31 -18 125 -63 30 18 5106 1817 4 MP PP Xc0 X X-31 -18 125 -63 30 18 5106 1817 4 MP DP Xc24 X X-31 -44 124 -61 31 44 4797 1587 4 MP PP Xc0 X X-31 -44 124 -61 31 44 4797 1587 4 MP DP Xc55 X X-30 -33 124 -61 31 34 5693 1785 4 MP PP Xc0 X X-30 -33 124 -61 31 34 5693 1785 4 MP DP Xc14 X X-31 -77 124 -64 31 77 4798 1196 4 MP PP Xc0 X X-31 -77 124 -64 31 77 4798 1196 4 MP DP Xc28 X X-31 -45 124 -61 31 45 4828 1631 4 MP PP Xc0 X X-31 -45 124 -61 31 45 4828 1631 4 MP DP Xc2 X X-31 -15 124 -39 31 19 4336 716 4 MP PP Xc0 X X-31 -15 124 -39 31 19 4336 716 4 MP DP Xc8 X X-31 -105 124 -61 31 105 4766 1482 4 MP PP Xc0 X X-31 -105 124 -61 31 105 4766 1482 4 MP DP Xc51 X X-30 -19 124 -58 31 22 4552 751 4 MP PP Xc0 X X-30 -19 124 -58 31 22 4552 751 4 MP DP Xc55 X X-31 -19 124 -64 31 19 5508 1799 4 MP PP Xc0 X X-31 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124 -61 31 67 4551 1020 4 MP PP Xc0 X X-31 -67 124 -61 31 67 4551 1020 4 MP DP Xc61 X X-31 -17 124 -63 31 17 5198 1882 4 MP PP Xc0 X X-31 -17 124 -63 31 17 5198 1882 4 MP DP Xc56 X X-31 -22 124 -60 31 22 4428 811 4 MP PP Xc0 X X-31 -22 124 -60 31 22 4428 811 4 MP DP Xc4 X X-30 -68 124 -61 31 69 4643 1188 4 MP PP Xc0 X X-30 -68 124 -61 31 69 4643 1188 4 MP DP Xc63 X X-30 -21 124 -64 30 22 4827 1920 4 MP PP Xc0 X X-30 -21 124 -64 30 22 4827 1920 4 MP DP Xc62 X X-31 -19 124 -61 31 19 5012 1896 4 MP PP Xc0 X X-31 -19 124 -61 31 19 5012 1896 4 MP DP Xc18 X X-31 -55 125 -60 30 54 4613 1134 4 MP PP Xc0 X X-31 -55 125 -60 30 54 4613 1134 4 MP DP Xc2 X X-31 -19 125 -53 30 31 4212 757 4 MP PP Xc0 X X-31 -19 125 -53 30 31 4212 757 4 MP DP Xc31 X X-30 -85 124 -61 31 85 4520 935 4 MP PP Xc0 X X-30 -85 124 -61 31 85 4520 935 4 MP DP Xc30 X X-31 -57 124 -61 31 58 4459 833 4 MP PP Xc0 X X-31 -57 124 -61 31 58 4459 833 4 MP DP Xc58 X X-30 -67 124 -63 31 67 4611 1478 4 MP PP Xc0 X X-30 -67 124 -63 31 67 4611 1478 4 MP DP Xc32 X X-31 -44 125 -61 30 44 4490 891 4 MP PP Xc0 X X-31 -44 125 -61 30 44 4490 891 4 MP DP Xc51 X X-30 -49 124 -63 31 59 4242 788 4 MP PP Xc0 X X-30 -49 124 -63 31 59 4242 788 4 MP DP Xc60 X X-30 -31 124 -61 31 31 5414 1879 4 MP PP Xc0 X X-30 -31 124 -61 31 31 5414 1879 4 MP DP Xc61 X X-31 -24 125 -63 30 24 5229 1899 4 MP PP Xc0 X X-31 -24 125 -63 30 24 5229 1899 4 MP DP Xc10 X X-31 -81 125 -63 30 81 4581 1397 4 MP PP Xc0 X X-31 -81 125 -63 30 81 4581 1397 4 MP DP Xc62 X X-31 -28 124 -61 31 28 5043 1915 4 MP PP Xc0 X X-31 -28 124 -61 31 28 5043 1915 4 MP DP Xcolortable /c64 { 1.000000 0.380952 0.000000 sc} put Xc64 X X-31 -18 124 -64 31 18 4857 1942 4 MP PP Xc0 X X-31 -18 124 -64 31 18 4857 1942 4 MP DP Xc30 X X-31 -27 124 -63 31 27 4273 847 4 MP PP Xc0 X X-31 -27 124 -63 31 27 4273 847 4 MP DP Xc62 X X-31 -33 124 -61 31 33 5445 1910 4 MP PP Xc0 X X-31 -33 124 -61 31 33 5445 1910 4 MP DP Xc12 X X-31 -77 124 -63 31 77 4550 1320 4 MP PP Xc0 X X-31 -77 124 -63 31 77 4550 1320 4 MP DP Xc62 X X-30 -19 124 -63 31 19 5259 1923 4 MP PP Xc0 X X-30 -19 124 -63 31 19 5259 1923 4 MP DP Xc18 X X-31 -47 125 -64 30 48 4458 1150 4 MP PP Xc0 X X-31 -47 125 -64 30 48 4458 1150 4 MP DP Xc59 X X-31 -67 124 -63 31 67 4427 1083 4 MP PP Xc0 X X-31 -67 124 -63 31 67 4427 1083 4 MP DP Xc63 X X-31 -17 125 -61 30 17 5074 1943 4 MP PP Xc0 X X-31 -17 125 -61 30 17 5074 1943 4 MP DP Xc14 X X-31 -69 124 -63 31 68 4519 1252 4 MP PP Xc0 X X-31 -69 124 -63 31 68 4519 1252 4 MP DP Xc38 X X-31 -22 124 -64 31 23 4304 874 4 MP PP Xc0 X X-31 -22 124 -64 31 23 4304 874 4 MP DP Xc64 X X-31 -19 124 -63 31 18 4888 1960 4 MP PP Xc0 X X-31 -19 124 -63 31 18 4888 1960 4 MP DP Xc4 X X-30 -54 124 -64 31 54 4488 1198 4 MP PP Xc0 X X-30 -54 124 -64 31 54 4488 1198 4 MP DP Xc39 X X-31 -85 124 -63 31 85 4396 998 4 MP PP Xc0 X X-31 -85 124 -63 31 85 4396 998 4 MP DP Xc32 X X-31 -58 125 -63 30 57 4335 897 4 MP PP Xc0 X X-31 -58 125 -63 30 57 4335 897 4 MP DP Xc31 X X-30 -44 124 -63 31 44 4365 954 4 MP PP Xc0 X X-30 -44 124 -63 31 44 4365 954 4 MP DP Xc62 X X-31 -31 124 -63 31 31 5290 1942 4 MP PP Xc0 X X-31 -31 124 -63 31 31 5290 1942 4 MP DP Xc63 X X-30 -24 124 -61 31 24 5104 1960 4 MP PP Xc0 X X-30 -24 124 -61 31 24 5104 1960 4 MP DP Xc64 X X-31 -28 125 -64 30 29 4919 1978 4 MP PP Xc0 X X-31 -28 125 -64 30 29 4919 1978 4 MP DP Xc64 X X-31 -33 124 -64 31 34 5321 1973 4 MP PP Xc0 X X-31 -33 124 -64 31 34 5321 1973 4 MP DP Xc64 X X-31 -19 124 -61 31 19 5135 1984 4 MP PP Xc0 X X-31 -19 124 -61 31 19 5135 1984 4 MP DP Xcolortable /c65 { 1.000000 0.285714 0.000000 sc} put Xc65 X X-30 -17 124 -64 31 17 4949 2007 4 MP PP Xc0 X X-30 -17 124 -64 31 17 4949 2007 4 MP DP Xc64 X X-31 -31 124 -61 31 31 5166 2003 4 MP PP Xc0 X X-31 -31 124 -61 31 31 5166 2003 4 MP DP Xc65 X X-31 -24 124 -63 31 23 4980 2024 4 MP PP Xc0 X X-31 -24 124 -63 31 23 4980 2024 4 MP DP Xc65 X X-31 -34 125 -60 30 33 5197 2034 4 MP PP Xc0 X X-31 -34 125 -60 30 33 5197 2034 4 MP DP Xcolortable /c66 { 1.000000 0.190476 0.000000 sc} put Xc66 X X-31 -19 124 -63 31 19 5011 2047 4 MP PP Xc0 X X-31 -19 124 -63 31 19 5011 2047 4 MP DP Xc66 X X-31 -31 125 -64 30 32 5042 2066 4 MP PP Xc0 X X-31 -31 125 -64 30 32 5042 2066 4 MP DP Xcolortable /c67 { 1.000000 0.095238 0.000000 sc} put Xc67 X X-30 -33 124 -64 31 33 5072 2098 4 MP PP Xc0 X X-30 -33 124 -64 31 33 5072 2098 4 MP DP Xgr XDO XSO X1851 4614 mt 3094 4274 L X 898 4171 mt 1851 4614 L X 898 4171 mt 898 3204 L X2162 4529 mt 2191 4543 L X2216 4710 mt (5) s X2550 4423 mt 2580 4436 L X2604 4604 mt (10) s X2939 4317 mt 2968 4330 L X2992 4498 mt (15) s X1175 4300 mt 1144 4308 L X 932 4471 mt (10) s X1482 4443 mt 1451 4451 L X1239 4614 mt (20) s X1790 4585 mt 1759 4594 L X1547 4757 mt (30) s X 898 4023 mt 869 4010 L X 695 4060 mt (-4) s X 898 3836 mt 869 3823 L X 695 3873 mt (-2) s X 898 3649 mt 869 3635 L X 751 3686 mt (0) s X 898 3462 mt 869 3448 L X 751 3499 mt (2) s X 898 3274 mt 869 3261 L X 751 3311 mt (4) s X1413 2731 mt (LSQR solutions) s Xgs 898 2864 2197 1751 MR c np Xc13 X X-30 808 77 -295 31 4 2063 3314 4 MP PP Xc0 X X-30 808 77 -295 31 4 2063 3314 4 MP DP Xc12 X X-31 -4 78 -28 31 23 1985 3323 4 MP PP Xc0 X X-31 -4 78 -28 31 23 1985 3323 4 MP DP Xc15 X X-31 -32 78 279 30 -2 2217 3358 4 MP PP Xc0 X X-31 -32 78 279 30 -2 2217 3358 4 MP DP Xc12 X X-30 -206 77 245 31 3 2186 3355 4 MP PP Xc0 X X-30 -206 77 245 31 3 2186 3355 4 MP DP Xc13 X X-30 -13 77 -31 31 16 2016 3346 4 MP PP Xc0 X X-30 -13 77 -31 31 16 2016 3346 4 MP DP Xc12 X X-31 -23 78 -21 30 23 1908 3344 4 MP PP Xc0 X X-31 -23 78 -21 30 23 1908 3344 4 MP DP Xc13 X X-31 -14 77 -27 31 10 2047 3362 4 MP PP Xc0 X X-31 -14 77 -27 31 10 2047 3362 4 MP DP Xc13 X X-31 -16 78 -22 31 17 1938 3367 4 MP PP Xc0 X X-31 -16 78 -22 31 17 1938 3367 4 MP DP Xc12 X X-31 -294 78 42 31 10 2155 3345 4 MP PP Xc0 X X-31 -294 78 42 31 10 2155 3345 4 MP DP Xc11 X X-31 136 78 146 31 -3 2247 3356 4 MP PP Xc0 X X-31 136 78 146 31 -3 2247 3356 4 MP DP Xc13 X X-30 -23 77 -24 31 12 1830 3379 4 MP PP Xc0 X X-30 -23 77 -24 31 12 1830 3379 4 MP DP Xc12 X X-31 -10 78 -20 30 3 2078 3372 4 MP PP Xc0 X X-31 -10 78 -20 30 3 2078 3372 4 MP DP Xc13 X X-31 -10 78 -21 31 9 1969 3384 4 MP PP Xc0 X X-31 -10 78 -21 31 9 1969 3384 4 MP DP Xc12 X X-31 130 78 -438 30 13 2094 3318 4 MP PP Xc0 X X-31 130 78 -438 30 13 2094 3318 4 MP DP Xc13 X X-31 -17 78 -17 30 10 1861 3391 4 MP PP Xc0 X X-31 -17 78 -17 30 10 1861 3391 4 MP DP Xc13 X X-31 -12 78 -22 31 13 1752 3400 4 MP PP Xc0 X X-31 -12 78 -22 31 13 1752 3400 4 MP DP Xc12 X X-31 -210 78 -242 31 14 2124 3331 4 MP PP Xc0 X X-31 -210 78 -242 31 14 2124 3331 4 MP DP Xc12 X X-31 -3 78 -15 31 -2 2108 3375 4 MP PP Xc0 X X-31 -3 78 -15 31 -2 2108 3375 4 MP DP Xc12 X X-30 -3 77 -21 31 3 2000 3393 4 MP PP Xc0 X X-30 -3 77 -21 31 3 2000 3393 4 MP DP Xc12 X X-31 -9 78 -14 31 6 1891 3401 4 MP PP Xc0 X X-31 -9 78 -14 31 6 1891 3401 4 MP DP Xc13 X X-30 -10 77 -21 31 9 1783 3413 4 MP PP Xc0 X X-30 -10 77 -21 31 9 1783 3413 4 MP DP Xc15 X X-30 2 77 -15 31 -2 2139 3373 4 MP PP Xc0 X X-30 2 77 -15 31 -2 2139 3373 4 MP DP Xc13 X X-31 -13 78 -21 30 13 1675 3421 4 MP PP Xc0 X X-31 -13 78 -21 30 13 1675 3421 4 MP DP Xc12 X X-31 2 78 -21 30 -2 2031 3396 4 MP PP Xc0 X X-31 2 78 -21 30 -2 2031 3396 4 MP DP Xc12 X X-31 -3 78 -15 31 4 1922 3407 4 MP PP Xc0 X X-31 -3 78 -15 31 4 1922 3407 4 MP DP Xc16 X X-31 216 78 -70 31 0 2278 3353 4 MP PP Xc0 X X-31 216 78 -70 31 0 2278 3353 4 MP DP Xc12 X X-31 -6 77 -21 31 6 1814 3422 4 MP PP Xc0 X X-31 -6 77 -21 31 6 1814 3422 4 MP DP Xc13 X X-31 -9 78 -21 31 9 1705 3434 4 MP PP Xc0 X X-31 -9 78 -21 31 9 1705 3434 4 MP DP Xc11 X X-31 3 77 -17 31 -1 2170 3371 4 MP PP Xc0 X X-31 3 77 -17 31 -1 2170 3371 4 MP DP Xc13 X X-30 -13 77 -23 31 8 1597 3449 4 MP PP Xc0 X X-30 -13 77 -23 31 8 1597 3449 4 MP DP Xc15 X X-31 2 78 -21 31 -2 2061 3394 4 MP PP Xc0 X X-31 2 78 -21 31 -2 2061 3394 4 MP DP Xc15 X X-30 2 77 -19 31 2 1953 3411 4 MP PP Xc0 X X-30 2 77 -19 31 2 1953 3411 4 MP DP Xc12 X X-31 -4 78 -21 30 4 1845 3428 4 MP PP Xc0 X X-31 -4 78 -21 30 4 1845 3428 4 MP DP Xc12 X X-31 -6 78 -21 31 6 1736 3443 4 MP PP Xc0 X X-31 -6 78 -21 31 6 1736 3443 4 MP DP Xc13 X X-31 -9 78 -20 30 6 1628 3457 4 MP PP Xc0 X X-31 -9 78 -20 30 6 1628 3457 4 MP DP Xc16 X X-31 0 78 -22 30 5 2201 3370 4 MP PP Xc0 X X-31 0 78 -22 30 5 2201 3370 4 MP DP Xc13 X X-31 -8 78 -21 31 8 1519 3470 4 MP PP Xc0 X X-31 -8 78 -21 31 8 1519 3470 4 MP DP Xc11 X X-31 1 78 -22 31 0 2092 3392 4 MP PP Xc0 X X-31 1 78 -22 31 0 2092 3392 4 MP DP Xc11 X X-31 2 78 -23 30 2 1984 3413 4 MP PP Xc0 X X-31 2 78 -23 30 2 1984 3413 4 MP DP Xc15 X X-31 -2 78 -21 31 2 1875 3432 4 MP PP Xc0 X X-31 -2 78 -21 31 2 1875 3432 4 MP DP Xc12 X X-30 -4 77 -21 31 4 1767 3449 4 MP PP Xc0 X X-30 -4 77 -21 31 4 1767 3449 4 MP DP Xc12 X X-31 -6 78 -18 31 4 1658 3463 4 MP PP Xc0 X X-31 -6 78 -18 31 4 1658 3463 4 MP DP Xc14 X X-31 -6 78 -27 31 11 2231 3375 4 MP PP Xc0 X X-31 -6 78 -27 31 11 2231 3375 4 MP DP Xc14 X X-30 172 77 -248 31 6 2309 3353 4 MP PP Xc0 X X-30 172 77 -248 31 6 2309 3353 4 MP DP Xc13 X X-30 -6 77 -21 31 6 1550 3478 4 MP PP Xc0 X X-30 -6 77 -21 31 6 1550 3478 4 MP DP Xc16 X X-31 -20 78 212 30 19 2463 3441 4 MP PP Xc0 X X-31 -20 78 212 30 19 2463 3441 4 MP DP Xc16 X X-30 -5 77 -21 31 4 2123 3392 4 MP PP Xc0 X X-30 -5 77 -21 31 4 2123 3392 4 MP DP Xc16 X X-30 -182 77 211 31 21 2432 3420 4 MP PP Xc0 X X-30 -182 77 211 31 21 2432 3420 4 MP DP Xc11 X X-31 0 78 -26 31 3 2014 3415 4 MP PP Xc0 X X-31 0 78 -26 31 3 2014 3415 4 MP DP Xc13 X X-31 -8 78 -21 30 7 1442 3492 4 MP PP Xc0 X X-31 -8 78 -21 30 7 1442 3492 4 MP DP Xc11 X X-30 -2 77 -21 31 2 1906 3434 4 MP PP Xc0 X X-30 -2 77 -21 31 2 1906 3434 4 MP DP Xc15 X X-31 -2 78 -21 30 2 1798 3453 4 MP PP Xc0 X X-31 -2 78 -21 30 2 1798 3453 4 MP DP Xc14 X X-30 -16 77 -27 31 16 2262 3386 4 MP PP Xc0 X X-30 -16 77 -27 31 16 2262 3386 4 MP DP Xc12 X X-31 -4 78 -18 31 4 1689 3467 4 MP PP Xc0 X X-31 -4 78 -18 31 4 1689 3467 4 MP DP Xc12 X X-31 -4 77 -21 31 4 1581 3484 4 MP PP Xc0 X X-31 -4 77 -21 31 4 1581 3484 4 MP DP Xc14 X X-31 -11 78 -21 30 11 2154 3396 4 MP PP Xc0 X X-31 -11 78 -21 30 11 2154 3396 4 MP DP Xc13 X X-31 -6 78 -21 31 6 1472 3499 4 MP PP Xc0 X X-31 -6 78 -21 31 6 1472 3499 4 MP DP Xc16 X X-31 -4 78 -27 31 5 2045 3418 4 MP PP Xc0 X X-31 -4 78 -27 31 5 2045 3418 4 MP DP Xc14 X X-31 -234 78 50 31 24 2401 3396 4 MP PP Xc0 X X-31 -234 78 50 31 24 2401 3396 4 MP DP Xc11 X X-31 -3 77 -21 31 3 1937 3436 4 MP PP Xc0 X X-31 -3 77 -21 31 3 1937 3436 4 MP DP Xc14 X X-31 -21 77 -25 31 19 2293 3402 4 MP PP Xc0 X X-31 -21 77 -25 31 19 2293 3402 4 MP DP Xc13 X X-30 -7 77 -14 31 9 1364 3504 4 MP PP Xc0 X X-30 -7 77 -14 31 9 1364 3504 4 MP DP Xc16 X X-31 141 78 54 31 17 2493 3460 4 MP PP Xc0 X X-31 141 78 54 31 17 2493 3460 4 MP DP Xc11 X X-31 -2 78 -21 31 2 1828 3455 4 MP PP Xc0 X X-31 -2 78 -21 31 2 1828 3455 4 MP DP Xc15 X X-30 -2 77 -19 31 3 1720 3471 4 MP PP Xc0 X X-30 -2 77 -19 31 3 1720 3471 4 MP DP Xc14 X X-31 -16 78 -21 31 16 2184 3407 4 MP PP Xc0 X X-31 -16 78 -21 31 16 2184 3407 4 MP DP Xc19 X X-31 23 78 -287 30 16 2340 3359 4 MP PP Xc0 X X-31 23 78 -287 30 16 2340 3359 4 MP DP Xc12 X X-31 -4 78 -21 30 4 1612 3488 4 MP PP Xc0 X X-31 -4 78 -21 30 4 1612 3488 4 MP DP Xc14 X X-31 -24 78 -19 30 18 2324 3421 4 MP PP Xc0 X X-31 -24 78 -19 30 18 2324 3421 4 MP DP Xc12 X X-31 -4 78 -22 31 5 1503 3505 4 MP PP Xc0 X X-31 -4 78 -22 31 5 1503 3505 4 MP DP Xc14 X X-30 -11 77 -25 31 9 2076 3423 4 MP PP Xc0 X X-30 -11 77 -25 31 9 2076 3423 4 MP DP Xc19 X X-31 -148 78 -160 31 21 2370 3375 4 MP PP Xc0 X X-31 -148 78 -160 31 21 2370 3375 4 MP DP Xc16 X X-31 -5 78 -22 30 6 1968 3439 4 MP PP Xc0 X X-31 -5 78 -22 30 6 1968 3439 4 MP DP Xc12 X X-31 -6 78 -15 30 7 1395 3513 4 MP PP Xc0 X X-31 -6 78 -15 30 7 1395 3513 4 MP DP Xc14 X X-31 -19 78 -21 31 19 2215 3423 4 MP PP Xc0 X X-31 -19 78 -21 31 19 2215 3423 4 MP DP Xc11 X X-31 -3 78 -21 31 3 1859 3457 4 MP PP Xc0 X X-31 -3 78 -21 31 3 1859 3457 4 MP DP Xc11 X X-31 -2 78 -21 30 4 1751 3474 4 MP PP Xc0 X X-31 -2 78 -21 30 4 1751 3474 4 MP DP Xc13 X X-31 -9 78 -21 31 9 1286 3525 4 MP PP Xc0 X X-31 -9 78 -21 31 9 1286 3525 4 MP DP Xc14 X X-31 -21 78 -16 31 18 2354 3439 4 MP PP Xc0 X X-31 -21 78 -16 31 18 2354 3439 4 MP DP Xc15 X X-31 -3 78 -21 31 3 1642 3492 4 MP PP Xc0 X X-31 -3 78 -21 31 3 1642 3492 4 MP DP Xc14 X X-31 -16 78 -21 30 12 2107 3432 4 MP PP Xc0 X X-31 -16 78 -21 30 12 2107 3432 4 MP DP Xc12 X X-30 -4 77 -21 31 3 1534 3510 4 MP PP Xc0 X X-30 -4 77 -21 31 3 1534 3510 4 MP DP Xc14 X X-30 -18 77 -21 31 18 2246 3442 4 MP PP Xc0 X X-30 -18 77 -21 31 18 2246 3442 4 MP DP Xc14 X X-31 -9 78 -21 31 8 1998 3445 4 MP PP Xc0 X X-31 -9 78 -21 31 8 1998 3445 4 MP DP Xc12 X X-31 -5 78 -17 31 7 1425 3520 4 MP PP Xc0 X X-31 -5 78 -17 31 7 1425 3520 4 MP DP Xc16 X X-30 -19 77 -16 31 19 2385 3457 4 MP PP Xc0 X X-30 -19 77 -16 31 19 2385 3457 4 MP DP Xc16 X X-30 -6 77 -21 31 6 1890 3460 4 MP PP Xc0 X X-30 -6 77 -21 31 6 1890 3460 4 MP DP Xc12 X X-30 -7 77 -21 31 7 1317 3534 4 MP PP Xc0 X X-30 -7 77 -21 31 7 1317 3534 4 MP DP Xc11 X X-31 -3 78 -23 31 5 1781 3478 4 MP PP Xc0 X X-31 -3 78 -23 31 5 1781 3478 4 MP DP Xc14 X X-31 -19 78 -16 31 14 2137 3444 4 MP PP Xc0 X X-31 -19 78 -16 31 14 2137 3444 4 MP DP Xc13 X X-31 -9 78 -24 30 8 1209 3550 4 MP PP Xc0 X X-31 -9 78 -24 30 8 1209 3550 4 MP DP Xc11 X X-30 -4 77 -21 31 4 1673 3495 4 MP PP Xc0 X X-30 -4 77 -21 31 4 1673 3495 4 MP DP Xc11 X X-31 191 78 -155 31 18 2524 3477 4 MP PP Xc0 X X-31 191 78 -155 31 18 2524 3477 4 MP DP Xc14 X X-31 -18 78 -22 30 19 2277 3460 4 MP PP Xc0 X X-31 -18 78 -22 30 19 2277 3460 4 MP DP Xc15 X X-31 -3 78 -21 30 3 1565 3513 4 MP PP Xc0 X X-31 -3 78 -21 30 3 1565 3513 4 MP DP Xc14 X X-30 -12 77 -21 31 12 2029 3453 4 MP PP Xc0 X X-30 -12 77 -21 31 12 2029 3453 4 MP DP Xc16 X X-31 -17 77 -20 31 21 2416 3476 4 MP PP Xc0 X X-31 -17 77 -20 31 21 2416 3476 4 MP DP Xc15 X X-31 -3 78 -20 31 6 1456 3527 4 MP PP Xc0 X X-31 -3 78 -20 31 6 1456 3527 4 MP DP Xc14 X X-31 -8 78 -21 30 8 1921 3466 4 MP PP Xc0 X X-31 -8 78 -21 30 8 1921 3466 4 MP DP Xc14 X X-31 -18 78 -16 31 18 2168 3458 4 MP PP Xc0 X X-31 -18 78 -16 31 18 2168 3458 4 MP DP Xc12 X X-31 -7 77 -21 31 7 1348 3541 4 MP PP Xc0 X X-31 -7 77 -21 31 7 1348 3541 4 MP DP Xc16 X X-31 -6 78 -24 31 7 1812 3483 4 MP PP Xc0 X X-31 -6 78 -24 31 7 1812 3483 4 MP DP Xc16 X X-31 -19 78 -21 31 18 2307 3479 4 MP PP Xc0 X X-31 -19 78 -21 31 18 2307 3479 4 MP DP Xc13 X X-31 -7 78 -24 31 7 1239 3558 4 MP PP Xc0 X X-31 -7 78 -24 31 7 1239 3558 4 MP DP Xc16 X X-31 -18 78 -24 30 22 2447 3497 4 MP PP Xc0 X X-31 -18 78 -24 30 22 2447 3497 4 MP DP Xc11 X X-31 -5 77 -21 31 5 1704 3499 4 MP PP Xc0 X X-31 -5 77 -21 31 5 1704 3499 4 MP DP Xc14 X X-31 -14 77 -22 31 15 2060 3465 4 MP PP Xc0 X X-31 -14 77 -22 31 15 2060 3465 4 MP DP Xc13 X X-30 -8 77 -28 31 12 1131 3574 4 MP PP Xc0 X X-30 -8 77 -28 31 12 1131 3574 4 MP DP Xc11 X X-31 -4 78 -21 31 4 1595 3516 4 MP PP Xc0 X X-31 -4 78 -21 31 4 1595 3516 4 MP DP Xc14 X X-30 -19 77 -18 31 21 2199 3476 4 MP PP Xc0 X X-30 -19 77 -18 31 21 2199 3476 4 MP DP Xc15 X X-30 -3 77 -23 31 6 1487 3533 4 MP PP Xc0 X X-30 -3 77 -23 31 6 1487 3533 4 MP DP Xc14 X X-31 -12 78 -21 31 12 1951 3474 4 MP PP Xc0 X X-31 -12 78 -21 31 12 1951 3474 4 MP DP Xc16 X X-31 -21 78 -21 31 21 2338 3497 4 MP PP Xc0 X X-31 -21 78 -21 31 21 2338 3497 4 MP DP Xc15 X X-31 -6 78 -21 30 6 1379 3548 4 MP PP Xc0 X X-31 -6 78 -21 30 6 1379 3548 4 MP DP Xc11 X X-31 -17 78 -27 31 20 2477 3519 4 MP PP Xc0 X X-31 -17 78 -27 31 20 2477 3519 4 MP DP Xc14 X X-30 -8 77 -25 31 9 1843 3490 4 MP PP Xc0 X X-30 -8 77 -25 31 9 1843 3490 4 MP DP Xc14 X X-31 -18 78 -21 30 17 2091 3480 4 MP PP Xc0 X X-31 -18 78 -21 30 17 2091 3480 4 MP DP Xc12 X X-31 -7 78 -23 31 6 1270 3565 4 MP PP Xc0 X X-31 -7 78 -23 31 6 1270 3565 4 MP DP Xc16 X X-31 -7 78 -21 30 7 1735 3504 4 MP PP Xc0 X X-31 -7 78 -21 30 7 1735 3504 4 MP DP Xc14 X X-31 -18 77 -23 31 23 2230 3497 4 MP PP Xc0 X X-31 -18 77 -23 31 23 2230 3497 4 MP DP Xc13 X X-31 -7 78 -31 30 10 1162 3586 4 MP PP Xc0 X X-31 -7 78 -31 30 10 1162 3586 4 MP DP 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3658 4 MP DP Xc13 X X-31 540 78 -427 30 6 2986 3746 4 MP PP Xc0 X X-31 540 78 -427 30 6 2986 3746 4 MP DP Xc16 X X-31 -28 78 -23 31 26 1639 3691 4 MP PP Xc0 X X-31 -28 78 -23 31 26 1639 3691 4 MP DP Xc34 X X-30 1 77 -21 31 -1 2429 3565 4 MP PP Xc0 X X-30 1 77 -21 31 -1 2429 3565 4 MP DP Xc17 X X-31 12 78 -21 31 -12 2320 3598 4 MP PP Xc0 X X-31 12 78 -21 31 -12 2320 3598 4 MP DP Xc11 X X-31 -17 78 -21 31 17 1778 3719 4 MP PP Xc0 X X-31 -17 78 -21 31 17 1778 3719 4 MP DP Xc11 X X-31 -44 77 -21 31 44 2769 3708 4 MP PP Xc0 X X-31 -44 77 -21 31 44 2769 3708 4 MP DP Xc14 X X-31 -23 78 -43 30 29 1392 3672 4 MP PP Xc0 X X-31 -23 78 -43 30 29 1392 3672 4 MP DP Xc20 X X-30 -32 77 -24 31 28 2568 3557 4 MP PP Xc0 X X-30 -32 77 -24 31 28 2568 3557 4 MP DP Xc12 X X-31 -33 78 -21 30 33 2800 3752 4 MP PP Xc0 X X-31 -33 78 -21 30 33 2800 3752 4 MP DP Xc14 X X-31 -25 77 -30 31 33 1531 3688 4 MP PP Xc0 X X-31 -25 77 -30 31 33 1531 3688 4 MP DP Xc11 X X-30 -25 77 -20 31 22 1670 3717 4 MP PP Xc0 X X-30 -25 77 -20 31 22 1670 3717 4 MP DP Xc11 X X-31 -24 78 -48 31 15 1283 3705 4 MP PP Xc0 X X-31 -24 78 -48 31 15 1283 3705 4 MP DP Xc11 X X-31 -13 77 -15 31 3 1175 3717 4 MP PP Xc0 X X-31 -13 77 -15 31 3 1175 3717 4 MP DP Xc11 X X-31 7 78 -21 30 -7 1918 3720 4 MP PP Xc0 X X-31 7 78 -21 30 -7 1918 3720 4 MP DP Xc34 X X-31 -45 77 -19 31 40 2599 3585 4 MP PP Xc0 X X-31 -45 77 -19 31 40 2599 3585 4 MP DP Xc16 X X-30 15 77 -21 31 -15 2026 3692 4 MP PP Xc0 X X-30 15 77 -21 31 -15 2026 3692 4 MP DP Xc15 X X-31 -5 78 -21 31 5 1809 3736 4 MP PP Xc0 X X-31 -5 78 -21 31 5 1809 3736 4 MP DP Xc20 X X-31 -14 78 -21 30 14 2460 3564 4 MP PP Xc0 X X-31 -14 78 -21 30 14 2460 3564 4 MP DP Xc16 X X-31 -30 78 -45 31 32 1422 3701 4 MP PP Xc0 X X-31 -30 78 -45 31 32 1422 3701 4 MP DP Xc11 X X-31 -26 78 -35 30 31 1562 3721 4 MP PP Xc0 X X-31 -26 78 -35 30 31 1562 3721 4 MP DP Xc13 X X-31 -19 78 -21 31 19 2830 3785 4 MP PP Xc0 X X-31 -19 78 -21 31 19 2830 3785 4 MP DP Xc17 X X-31 -51 78 -14 30 46 2630 3625 4 MP PP Xc0 X X-31 -51 78 -14 30 46 2630 3625 4 MP DP Xc19 X X-31 18 78 -21 31 -18 2134 3656 4 MP PP Xc0 X X-31 18 78 -21 31 -18 2134 3656 4 MP DP Xc34 X X-31 1 78 -21 31 -1 2351 3586 4 MP PP Xc0 X X-31 1 78 -21 31 -1 2351 3586 4 MP DP Xc17 X X-31 12 77 -19 31 -12 2243 3617 4 MP PP Xc0 X X-31 12 77 -19 31 -12 2243 3617 4 MP DP Xc19 X X-31 -51 78 -12 31 49 2660 3671 4 MP PP Xc0 X X-31 -51 78 -12 31 49 2660 3671 4 MP DP Xc11 X X-30 -29 77 -36 31 17 1314 3720 4 MP PP Xc0 X X-30 -29 77 -36 31 17 1314 3720 4 MP DP Xc11 X X-31 -17 78 -17 30 14 1701 3739 4 MP PP Xc0 X X-31 -17 78 -17 30 14 1701 3739 4 MP DP Xc16 X X-31 -44 78 -14 31 46 2691 3720 4 MP PP Xc0 X X-31 -44 78 -14 31 46 2691 3720 4 MP DP Xc11 X X-31 -33 78 -40 31 28 1453 3733 4 MP PP Xc0 X X-31 -33 78 -40 31 28 1453 3733 4 MP DP Xc16 X X-31 -15 78 -4 30 4 1206 3720 4 MP PP Xc0 X X-31 -15 78 -4 30 4 1206 3720 4 MP DP Xc20 X X-31 -28 78 -22 31 29 2490 3578 4 MP PP Xc0 X X-31 -28 78 -22 31 29 2490 3578 4 MP DP Xc15 X X-31 -22 78 -33 31 20 1592 3752 4 MP PP Xc0 X X-31 -22 78 -33 31 20 1592 3752 4 MP DP Xc15 X X-30 -33 77 -23 31 42 2722 3766 4 MP PP Xc0 X X-30 -33 77 -23 31 42 2722 3766 4 MP DP Xc11 X X-31 -32 78 -19 30 15 1345 3737 4 MP PP Xc0 X X-31 -32 78 -19 30 15 1345 3737 4 MP DP Xc15 X X-30 -31 77 -28 31 19 1484 3761 4 MP PP Xc0 X X-30 -31 77 -28 31 19 1484 3761 4 MP DP Xc11 X X-30 7 77 -21 31 -7 1840 3741 4 MP PP Xc0 X X-30 7 77 -21 31 -7 1840 3741 4 MP DP Xc34 X X-31 -40 78 -21 31 39 2521 3607 4 MP PP Xc0 X X-31 -40 78 -21 31 39 2521 3607 4 MP DP Xc16 X X-31 15 78 -20 31 -16 1948 3713 4 MP PP Xc0 X X-31 15 78 -20 31 -16 1948 3713 4 MP DP Xc20 X X-30 -14 77 -21 31 14 2382 3585 4 MP PP Xc0 X X-30 -14 77 -21 31 14 2382 3585 4 MP DP Xc13 X X-31 -19 78 -39 30 35 2753 3808 4 MP PP Xc0 X X-31 -19 78 -39 30 35 2753 3808 4 MP DP Xc11 X X-31 -5 78 -16 31 4 1731 3753 4 MP PP Xc0 X X-31 -5 78 -16 31 4 1731 3753 4 MP DP Xc19 X X-31 18 77 -21 31 -18 2057 3677 4 MP PP Xc0 X X-31 18 77 -21 31 -18 2057 3677 4 MP DP Xc17 X X-30 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X-31 -39 78 -22 30 40 2444 3628 4 MP PP Xc0 X X-31 -39 78 -22 30 40 2444 3628 4 MP DP Xc11 X X-31 7 78 -17 31 -6 1762 3757 4 MP PP Xc0 X X-31 7 78 -17 31 -6 1762 3757 4 MP DP Xc13 X X-30 -35 77 -21 31 35 2675 3829 4 MP PP Xc0 X X-30 -35 77 -21 31 35 2675 3829 4 MP DP Xc16 X X-31 16 78 -21 30 -16 1871 3734 4 MP PP Xc0 X X-31 16 78 -21 30 -16 1871 3734 4 MP DP Xc11 X X-31 -19 78 8 31 7 1406 3765 4 MP PP Xc0 X X-31 -19 78 8 31 7 1406 3765 4 MP DP Xc20 X X-31 -14 78 -22 31 16 2304 3605 4 MP PP Xc0 X X-31 -14 78 -22 31 16 2304 3605 4 MP DP Xc14 X X-30 -15 77 20 31 4 1267 3728 4 MP PP Xc0 X X-30 -15 77 20 31 4 1267 3728 4 MP DP Xc15 X X-31 -4 77 -17 31 -6 1654 3780 4 MP PP Xc0 X X-31 -4 77 -17 31 -6 1654 3780 4 MP DP Xc19 X X-31 18 78 -21 31 -17 1979 3697 4 MP PP Xc0 X X-31 18 78 -21 31 -17 1979 3697 4 MP DP Xc17 X X-31 -47 78 -21 31 46 2474 3668 4 MP PP Xc0 X X-31 -47 78 -21 31 46 2474 3668 4 MP DP Xc34 X X-30 0 77 -22 31 1 2196 3626 4 MP PP Xc0 X X-30 0 77 -22 31 1 2196 3626 4 MP DP Xc17 X X-31 12 78 -22 30 -11 2088 3659 4 MP PP Xc0 X X-31 12 78 -22 30 -11 2088 3659 4 MP DP Xc15 X X-31 -8 78 -4 31 -4 1545 3788 4 MP PP Xc0 X X-31 -8 78 -4 31 -4 1545 3788 4 MP DP Xc19 X X-30 -48 77 -21 31 48 2505 3714 4 MP PP Xc0 X X-30 -48 77 -21 31 48 2505 3714 4 MP DP Xc16 X X-31 -47 77 -21 31 47 2536 3762 4 MP PP Xc0 X X-31 -47 77 -21 31 47 2536 3762 4 MP DP Xc20 X X-31 -29 78 -23 31 30 2335 3621 4 MP PP Xc0 X X-31 -29 78 -23 31 30 2335 3621 4 MP DP Xc15 X X-31 -41 78 -22 30 42 2567 3809 4 MP PP Xc0 X X-31 -41 78 -22 30 42 2567 3809 4 MP DP Xc16 X X-30 -8 77 15 31 1 1437 3772 4 MP PP Xc0 X X-30 -8 77 15 31 1 1437 3772 4 MP DP Xc19 X X-31 -13 77 26 31 7 1298 3732 4 MP PP Xc0 X X-31 -13 77 26 31 7 1298 3732 4 MP DP Xc34 X X-30 -40 77 -23 31 40 2366 3651 4 MP PP Xc0 X X-30 -40 77 -23 31 40 2366 3651 4 MP DP Xc13 X X-31 -35 78 -22 31 35 2597 3851 4 MP PP Xc0 X X-31 -35 78 -22 31 35 2597 3851 4 MP DP Xc14 X X-30 16 77 -20 31 -13 1793 3751 4 MP PP Xc0 X X-30 16 77 -20 31 -13 1793 3751 4 MP DP Xc20 X X-31 -16 78 -21 30 15 2227 3627 4 MP PP Xc0 X X-31 -16 78 -21 30 15 2227 3627 4 MP DP Xc16 X X-31 6 78 -10 30 -13 1685 3774 4 MP PP Xc0 X X-31 6 78 -10 30 -13 1685 3774 4 MP DP Xc17 X X-31 -46 78 -24 30 47 2397 3691 4 MP PP Xc0 X X-31 -46 78 -24 30 47 2397 3691 4 MP DP Xc19 X X-31 17 78 -21 31 -17 1901 3718 4 MP PP Xc0 X X-31 17 78 -21 31 -17 1901 3718 4 MP DP Xc11 X X-31 6 78 2 31 -12 1576 3784 4 MP PP Xc0 X X-31 6 78 2 31 -12 1576 3784 4 MP DP Xc34 X X-31 -1 78 -21 31 0 2118 3648 4 MP PP Xc0 X X-31 -1 78 -21 31 0 2118 3648 4 MP DP Xc17 X X-30 11 77 -21 31 -11 2010 3680 4 MP PP Xc0 X X-30 11 77 -21 31 -11 2010 3680 4 MP DP Xc19 X X-31 -48 78 -23 31 47 2427 3738 4 MP PP Xc0 X X-31 -48 78 -23 31 47 2427 3738 4 MP DP Xc16 X X-31 -47 78 -21 31 45 2458 3785 4 MP PP Xc0 X X-31 -47 78 -21 31 45 2458 3785 4 MP DP Xc19 X X-31 -7 78 25 30 8 1329 3739 4 MP PP Xc0 X X-31 -7 78 25 30 8 1329 3739 4 MP DP Xc20 X X-31 -30 78 -21 31 30 2257 3642 4 MP PP Xc0 X X-31 -30 78 -21 31 30 2257 3642 4 MP DP Xc14 X X-31 4 77 14 31 -3 1468 3773 4 MP PP Xc0 X X-31 4 77 14 31 -3 1468 3773 4 MP DP Xc15 X X-30 -42 77 -19 31 40 2489 3830 4 MP PP Xc0 X X-30 -42 77 -19 31 40 2489 3830 4 MP DP Xc34 X X-31 -40 78 -21 31 40 2288 3672 4 MP PP Xc0 X X-31 -40 78 -21 31 40 2288 3672 4 MP DP Xc13 X X-31 -35 78 -17 30 33 2520 3870 4 MP PP Xc0 X X-31 -35 78 -17 30 33 2520 3870 4 MP DP Xc20 X X-30 -15 77 -21 31 15 2149 3648 4 MP PP Xc0 X X-30 -15 77 -21 31 15 2149 3648 4 MP DP Xc17 X X-30 -47 77 -21 31 47 2319 3712 4 MP PP Xc0 X X-30 -47 77 -21 31 47 2319 3712 4 MP DP Xc14 X X-31 13 78 -7 31 -16 1715 3761 4 MP PP Xc0 X X-31 13 78 -7 31 -16 1715 3761 4 MP DP Xc19 X X-31 17 77 -24 31 -13 1824 3738 4 MP PP Xc0 X X-31 17 77 -24 31 -13 1824 3738 4 MP DP Xc34 X X-31 0 78 -22 30 1 2041 3669 4 MP PP Xc0 X X-31 0 78 -22 30 1 2041 3669 4 MP DP Xc17 X X-31 11 78 -21 31 -11 1932 3701 4 MP PP Xc0 X X-31 11 78 -21 31 -11 1932 3701 4 MP DP Xc19 X X-31 -47 78 -22 30 48 2350 3759 4 MP PP Xc0 X X-31 -47 78 -22 30 48 2350 3759 4 MP DP Xc14 X X-30 13 77 2 31 -13 1607 3772 4 MP PP Xc0 X X-30 13 77 2 31 -13 1607 3772 4 MP DP Xc4 X X-31 -1 78 15 31 11 1359 3747 4 MP PP Xc0 X X-31 -1 78 15 31 11 1359 3747 4 MP DP Xc16 X X-31 -45 78 -22 31 45 2380 3807 4 MP PP Xc0 X X-31 -45 78 -22 31 45 2380 3807 4 MP DP Xc20 X X-31 -30 77 -21 31 30 2180 3663 4 MP PP Xc0 X X-31 -30 77 -21 31 30 2180 3663 4 MP DP Xc19 X X-31 12 78 7 30 -5 1499 3770 4 MP PP Xc0 X X-31 12 78 7 30 -5 1499 3770 4 MP DP Xc15 X X-31 -40 78 -22 31 40 2411 3852 4 MP PP Xc0 X X-31 -40 78 -22 31 40 2411 3852 4 MP DP Xc34 X X-31 -40 78 -22 30 41 2211 3693 4 MP PP Xc0 X X-31 -40 78 -22 30 41 2211 3693 4 MP DP Xc13 X X-30 -33 77 -22 31 33 2442 3892 4 MP PP Xc0 X X-30 -33 77 -22 31 33 2442 3892 4 MP DP Xc20 X X-31 -15 78 -24 31 17 2071 3670 4 MP PP Xc0 X X-31 -15 78 -24 31 17 2071 3670 4 MP DP Xc17 X X-31 -47 78 -21 31 46 2241 3734 4 MP PP Xc0 X X-31 -47 78 -21 31 46 2241 3734 4 MP DP Xc4 X X-31 3 78 -1 31 13 1390 3758 4 MP PP Xc0 X X-31 3 78 -1 31 13 1390 3758 4 MP DP Xc17 X X-31 11 78 -29 30 -6 1855 3725 4 MP PP Xc0 X X-31 11 78 -29 30 -6 1855 3725 4 MP DP Xc4 X X-31 13 78 -10 31 -10 1746 3745 4 MP PP Xc0 X X-31 13 78 -10 31 -10 1746 3745 4 MP DP Xc34 X X-30 -1 77 -22 31 2 1963 3690 4 MP PP Xc0 X X-30 -1 77 -22 31 2 1963 3690 4 MP DP Xc19 X X-30 -48 77 -21 31 48 2272 3780 4 MP PP Xc0 X X-30 -48 77 -21 31 48 2272 3780 4 MP DP Xc4 X X-31 16 78 -6 30 -8 1638 3759 4 MP PP Xc0 X X-31 16 78 -6 30 -8 1638 3759 4 MP DP Xc16 X X-31 -45 77 -21 31 45 2303 3828 4 MP PP Xc0 X X-31 -45 77 -21 31 45 2303 3828 4 MP DP Xc20 X X-31 -30 78 -23 31 29 2102 3687 4 MP PP Xc0 X X-31 -30 78 -23 31 29 2102 3687 4 MP DP Xc4 X X-31 13 78 -4 31 -2 1529 3765 4 MP PP Xc0 X X-31 13 78 -4 31 -2 1529 3765 4 MP DP Xc15 X X-31 -40 78 -21 30 40 2334 3873 4 MP PP Xc0 X X-31 -40 78 -21 30 40 2334 3873 4 MP DP Xc34 X X-30 -41 77 -22 31 40 2133 3716 4 MP PP Xc0 X X-30 -41 77 -22 31 40 2133 3716 4 MP DP Xc4 X X-30 5 77 -23 31 17 1421 3771 4 MP PP Xc0 X X-30 5 77 -23 31 17 1421 3771 4 MP DP Xc20 X X-31 -17 78 -21 30 16 1994 3692 4 MP PP Xc0 X X-31 -17 78 -21 30 16 1994 3692 4 MP DP Xc13 X X-31 -33 78 -21 31 33 2364 3913 4 MP PP Xc0 X X-31 -33 78 -21 31 33 2364 3913 4 MP DP Xc17 X X-31 -46 78 -21 30 45 2164 3756 4 MP PP Xc0 X X-31 -46 78 -21 30 45 2164 3756 4 MP DP Xc34 X X-31 -2 78 -31 31 4 1885 3719 4 MP PP Xc0 X X-31 -2 78 -31 31 4 1885 3719 4 MP DP Xc18 X X-30 6 77 -16 31 0 1777 3735 4 MP PP Xc0 X X-30 6 77 -16 31 0 1777 3735 4 MP DP Xc19 X X-31 -48 78 -19 31 46 2194 3801 4 MP PP Xc0 X X-31 -48 78 -19 31 46 2194 3801 4 MP DP Xc17 X X-31 10 78 -16 31 0 1668 3751 4 MP PP Xc0 X X-31 10 78 -16 31 0 1668 3751 4 MP DP Xc16 X X-31 -45 78 -18 31 44 2225 3847 4 MP PP Xc0 X X-31 -45 78 -18 31 44 2225 3847 4 MP DP Xc20 X X-31 -29 78 -22 31 30 2024 3708 4 MP PP Xc0 X X-31 -29 78 -22 31 30 2024 3708 4 MP DP Xc17 X X-30 8 77 -16 31 4 1560 3763 4 MP PP Xc0 X X-30 8 77 -16 31 4 1560 3763 4 MP DP Xc15 X X-30 -40 77 -17 31 39 2256 3891 4 MP PP Xc0 X X-30 -40 77 -17 31 39 2256 3891 4 MP DP Xc4 X X-31 2 78 -45 30 20 1452 3788 4 MP PP Xc0 X X-31 2 78 -45 30 20 1452 3788 4 MP DP Xc34 X X-31 -40 78 -21 31 39 2055 3738 4 MP PP Xc0 X X-31 -40 78 -21 31 39 2055 3738 4 MP DP Xc34 X X-30 -16 77 -30 31 15 1916 3723 4 MP PP Xc0 X X-30 -16 77 -30 31 15 1916 3723 4 MP DP Xc13 X X-31 -33 78 -16 30 32 2287 3930 4 MP PP Xc0 X X-31 -33 78 -16 30 32 2287 3930 4 MP DP Xc34 X X-31 -4 78 -24 30 12 1808 3735 4 MP PP Xc0 X X-31 -4 78 -24 30 12 1808 3735 4 MP DP Xc17 X X-30 -45 77 -21 31 45 2086 3777 4 MP PP Xc0 X X-30 -45 77 -21 31 45 2086 3777 4 MP DP Xc19 X X-31 -46 78 -21 30 46 2117 3822 4 MP PP Xc0 X X-31 -46 78 -21 30 46 2117 3822 4 MP DP Xc18 X X-31 0 78 -26 31 10 1699 3751 4 MP PP Xc0 X X-31 0 78 -26 31 10 1699 3751 4 MP DP Xc4 X X-31 -4 78 -63 31 22 1482 3808 4 MP PP Xc0 X X-31 -4 78 -63 31 22 1482 3808 4 MP DP Xc34 X X-31 -30 77 -27 31 27 1947 3738 4 MP PP Xc0 X X-31 -30 77 -27 31 27 1947 3738 4 MP DP Xc16 X X-31 -44 78 -21 31 44 2147 3868 4 MP PP Xc0 X X-31 -44 78 -21 31 44 2147 3868 4 MP DP Xc18 X X-31 0 77 -28 31 12 1591 3767 4 MP PP Xc0 X X-31 0 77 -28 31 12 1591 3767 4 MP DP Xc15 X X-31 -39 78 -21 31 39 2178 3912 4 MP PP Xc0 X X-31 -39 78 -21 31 39 2178 3912 4 MP DP Xc34 X X-31 -15 78 -31 31 22 1838 3747 4 MP PP Xc0 X X-31 -15 78 -31 31 22 1838 3747 4 MP DP Xc18 X X-31 -39 78 -22 30 34 1978 3765 4 MP PP Xc0 X X-31 -39 78 -22 30 34 1978 3765 4 MP DP Xc34 X X-30 -12 77 -35 31 21 1730 3761 4 MP PP Xc0 X X-30 -12 77 -35 31 21 1730 3761 4 MP DP Xc13 X X-30 -32 77 -22 31 33 2209 3951 4 MP PP Xc0 X X-30 -32 77 -22 31 33 2209 3951 4 MP DP Xc19 X X-31 -12 78 -76 31 25 1513 3830 4 MP PP Xc0 X X-31 -12 78 -76 31 25 1513 3830 4 MP DP Xc17 X X-31 -45 78 -16 31 39 2008 3799 4 MP PP Xc0 X X-31 -45 78 -16 31 39 2008 3799 4 MP DP Xc18 X X-31 -10 78 -38 30 20 1622 3779 4 MP PP Xc0 X X-31 -10 78 -38 30 20 1622 3779 4 MP DP Xc34 X X-31 -27 78 -34 31 30 1869 3769 4 MP PP Xc0 X X-31 -27 78 -34 31 30 1869 3769 4 MP DP Xc19 X X-30 -46 77 -9 31 39 2039 3838 4 MP PP Xc0 X X-30 -46 77 -9 31 39 2039 3838 4 MP DP Xc14 X X-31 -44 77 -4 31 39 2070 3877 4 MP PP Xc0 X X-31 -44 77 -4 31 39 2070 3877 4 MP DP Xc18 X X-31 -22 78 -41 30 28 1761 3782 4 MP PP Xc0 X X-31 -22 78 -41 30 28 1761 3782 4 MP DP Xc14 X X-30 -20 77 -83 31 27 1544 3855 4 MP PP Xc0 X X-30 -20 77 -83 31 27 1544 3855 4 MP DP Xc18 X X-30 -34 77 -34 31 34 1900 3799 4 MP PP Xc0 X X-30 -34 77 -34 31 34 1900 3799 4 MP DP Xc18 X X-31 -21 78 -43 31 26 1652 3799 4 MP PP Xc0 X X-31 -21 78 -43 31 26 1652 3799 4 MP DP Xc11 X X-31 -39 78 0 30 35 2101 3916 4 MP PP Xc0 X X-31 -39 78 0 30 35 2101 3916 4 MP DP Xc17 X X-31 -30 78 -44 31 33 1791 3810 4 MP PP Xc0 X X-31 -30 78 -44 31 33 1791 3810 4 MP DP Xc4 X X-31 -39 78 -31 30 36 1931 3833 4 MP PP Xc0 X X-31 -39 78 -31 30 36 1931 3833 4 MP DP Xc16 X X-31 -26 78 -84 30 27 1575 3882 4 MP PP Xc0 X X-31 -26 78 -84 30 27 1575 3882 4 MP DP Xc15 X X-31 -33 78 2 31 31 2131 3951 4 MP PP Xc0 X X-31 -33 78 2 31 31 2131 3951 4 MP DP Xc17 X X-30 -28 77 -45 31 30 1683 3825 4 MP PP Xc0 X X-30 -28 77 -45 31 30 1683 3825 4 MP DP Xc19 X X-31 -39 78 -26 31 34 1961 3869 4 MP PP Xc0 X X-31 -39 78 -26 31 34 1961 3869 4 MP DP Xc4 X X-31 -34 78 -44 31 34 1822 3843 4 MP PP Xc0 X X-31 -34 78 -44 31 34 1822 3843 4 MP DP Xc16 X X-31 -30 78 -80 31 26 1605 3909 4 MP PP Xc0 X X-31 -30 78 -80 31 26 1605 3909 4 MP DP Xc16 X X-31 -39 78 -20 31 33 1992 3903 4 MP PP Xc0 X X-31 -39 78 -20 31 33 1992 3903 4 MP DP Xc4 X X-31 -33 77 -44 31 32 1714 3855 4 MP PP Xc0 X X-31 -33 77 -44 31 32 1714 3855 4 MP DP Xc19 X X-30 -36 77 -42 31 34 1853 3877 4 MP PP Xc0 X X-30 -36 77 -42 31 34 1853 3877 4 MP DP Xc11 X X-30 -35 77 -15 31 30 2023 3936 4 MP PP Xc0 X X-30 -35 77 -15 31 30 2023 3936 4 MP DP Xc19 X X-31 -34 78 -42 30 32 1745 3887 4 MP PP Xc0 X X-31 -34 78 -42 30 32 1745 3887 4 MP DP Xc11 X X-31 -32 78 -73 31 25 1636 3935 4 MP PP Xc0 X X-31 -32 78 -73 31 25 1636 3935 4 MP DP Xc16 X X-31 -34 78 -39 30 31 1884 3911 4 MP PP Xc0 X X-31 -34 78 -39 30 31 1884 3911 4 MP DP Xc15 X X-31 -31 78 -10 30 26 2054 3966 4 MP PP Xc0 X X-31 -31 78 -10 30 26 2054 3966 4 MP DP Xc16 X X-31 -34 78 -38 31 30 1775 3919 4 MP PP Xc0 X X-31 -34 78 -38 31 30 1775 3919 4 MP DP Xc15 X X-30 -32 77 -65 31 24 1667 3960 4 MP PP Xc0 X X-30 -32 77 -65 31 24 1667 3960 4 MP DP Xc11 X X-31 -33 78 -35 31 29 1914 3942 4 MP PP Xc0 X X-31 -33 78 -35 31 29 1914 3942 4 MP DP Xc11 X X-30 -31 77 -35 31 28 1806 3949 4 MP PP Xc0 X X-30 -31 77 -35 31 28 1806 3949 4 MP DP Xc15 X X-31 -30 78 -31 31 26 1945 3971 4 MP PP Xc0 X X-31 -30 78 -31 31 26 1945 3971 4 MP DP Xc12 X X-31 -30 78 -57 30 22 1698 3984 4 MP PP Xc0 X X-31 -30 78 -57 30 22 1698 3984 4 MP DP Xc15 X X-31 -29 77 -32 31 26 1837 3977 4 MP PP Xc0 X X-31 -29 77 -32 31 26 1837 3977 4 MP DP Xc15 X X-30 -26 77 -29 31 24 1976 3997 4 MP PP Xc0 X X-30 -26 77 -29 31 24 1976 3997 4 MP DP Xc12 X X-31 -28 78 -50 31 21 1728 4006 4 MP PP Xc0 X X-31 -28 78 -50 31 21 1728 4006 4 MP DP Xc15 X X-31 -26 78 -30 30 24 1868 4003 4 MP PP Xc0 X X-31 -26 78 -30 30 24 1868 4003 4 MP DP Xc12 X X-31 -26 78 -43 31 19 1759 4027 4 MP PP Xc0 X X-31 -26 78 -43 31 19 1759 4027 4 MP DP Xc12 X X-31 -24 78 -27 31 21 1898 4027 4 MP PP Xc0 X X-31 -24 78 -27 31 21 1898 4027 4 MP DP Xc13 X X-30 -24 77 -38 31 19 1790 4046 4 MP PP Xc0 X X-30 -24 77 -38 31 19 1790 4046 4 MP DP Xc13 X X-31 -21 78 -34 30 17 1821 4065 4 MP PP Xc0 X X-31 -21 78 -34 30 17 1821 4065 4 MP DP Xgr XDO XSO X4979 4614 mt 6221 4274 L X4026 4171 mt 4979 4614 L X4026 4171 mt 4026 3204 L X5290 4529 mt 5319 4543 L X5343 4710 mt (5) s X5678 4423 mt 5707 4436 L X5731 4604 mt (10) s X6066 4317 mt 6095 4330 L X6119 4498 mt (15) s X4303 4300 mt 4272 4308 L X4060 4471 mt (10) s X4610 4443 mt 4579 4451 L X4367 4614 mt (20) s X4918 4585 mt 4886 4594 L X4674 4757 mt (30) s X4026 4027 mt 3997 4014 L X3730 4064 mt (-30) s X4026 3754 mt 3997 3741 L X3730 3791 mt (-20) s X4026 3481 mt 3997 3468 L X3730 3518 mt (-10) s X4026 3209 mt 3997 3195 L X3879 3246 mt (0) s X4063 2731 mt (LSQR filter factors, log scale) s Xgs 4026 2864 2196 1751 MR c np Xc2 X X-31 -14 78 -21 31 14 5190 2890 4 MP PP Xc0 X X-31 -14 78 -21 31 14 5190 2890 4 MP DP Xc2 X X-30 -14 77 -21 31 14 5221 2904 4 MP PP Xc0 X X-30 -14 77 -21 31 14 5221 2904 4 MP DP Xc2 X X-31 -14 78 -21 30 14 5113 2911 4 MP PP Xc0 X X-31 -14 78 -21 30 14 5113 2911 4 MP DP Xc2 X X-31 -14 77 -22 31 15 5252 2918 4 MP PP Xc0 X X-31 -14 77 -22 31 15 5252 2918 4 MP DP Xc2 X X-31 -14 78 -22 31 15 5143 2925 4 MP PP Xc0 X X-31 -14 78 -22 31 15 5143 2925 4 MP DP Xc2 X X-30 -14 77 -22 31 15 5035 2932 4 MP PP Xc0 X X-30 -14 77 -22 31 15 5035 2932 4 MP DP Xc2 X X-31 -15 78 -21 30 14 5283 2933 4 MP PP Xc0 X X-31 -15 78 -21 30 14 5283 2933 4 MP DP Xc2 X X-31 -15 78 -21 31 14 5174 2940 4 MP PP Xc0 X X-31 -15 78 -21 31 14 5174 2940 4 MP DP Xc2 X X-31 -15 77 -21 31 14 5066 2947 4 MP PP Xc0 X X-31 -15 77 -21 31 14 5066 2947 4 MP DP Xc2 X X-31 -14 78 -21 31 14 5313 2947 4 MP PP Xc0 X X-31 -14 78 -21 31 14 5313 2947 4 MP DP Xc2 X X-31 -15 78 -21 31 14 4957 2954 4 MP PP Xc0 X X-31 -15 78 -21 31 14 4957 2954 4 MP DP Xc2 X X-30 -14 77 -21 31 14 5205 2954 4 MP PP Xc0 X X-30 -14 77 -21 31 14 5205 2954 4 MP DP Xc2 X X-31 -14 78 -21 30 14 5097 2961 4 MP PP Xc0 X X-31 -14 78 -21 30 14 5097 2961 4 MP DP Xc2 X X-30 -14 77 -22 31 15 5344 2961 4 MP PP Xc0 X X-30 -14 77 -22 31 15 5344 2961 4 MP DP Xc24 X X-31 -31 78 -64 31 31 5805 3828 4 MP PP Xc0 X X-31 -31 78 -64 31 31 5805 3828 4 MP DP Xc2 X X-31 -14 78 -21 31 14 4988 2968 4 MP PP Xc0 X X-31 -14 78 -21 31 14 4988 2968 4 MP DP Xc2 X X-31 -14 78 -22 30 15 5236 2968 4 MP PP Xc0 X X-31 -14 78 -22 30 15 5236 2968 4 MP DP Xc26 X X-31 -127 78 -64 31 127 5774 3701 4 MP PP Xc0 X X-31 -127 78 -64 31 127 5774 3701 4 MP DP Xc28 X X-31 -31 78 -46 30 31 5728 3874 4 MP PP Xc0 X X-31 -31 78 -46 30 31 5728 3874 4 MP DP Xc25 X X-30 -18 77 -65 31 19 5836 3859 4 MP PP Xc0 X X-30 -18 77 -65 31 19 5836 3859 4 MP DP Xc21 X X-30 -44 77 -64 31 43 5713 3614 4 MP PP Xc0 X X-30 -44 77 -64 31 43 5713 3614 4 MP DP Xc2 X X-31 -14 77 -21 31 14 4880 2975 4 MP PP Xc0 X X-31 -14 77 -21 31 14 4880 2975 4 MP DP Xc2 X X-31 -14 78 -22 31 15 5127 2975 4 MP PP Xc0 X X-31 -14 78 -22 31 15 5127 2975 4 MP DP Xc2 X X-31 -15 77 -21 31 14 5375 2976 4 MP PP Xc0 X X-31 -15 77 -21 31 14 5375 2976 4 MP DP Xc23 X X-31 -44 78 -64 30 44 5744 3657 4 MP PP Xc0 X X-31 -44 78 -64 30 44 5744 3657 4 MP DP Xc8 X X-31 -127 77 -46 31 127 5697 3747 4 MP PP Xc0 X X-31 -127 77 -46 31 127 5697 3747 4 MP DP Xc15 X X-31 -102 78 -65 31 102 5682 3512 4 MP PP Xc0 X X-31 -102 78 -65 31 102 5682 3512 4 MP DP Xc2 X X-30 -14 77 -21 31 14 5019 2982 4 MP PP Xc0 X X-30 -14 77 -21 31 14 5019 2982 4 MP DP Xc2 X X-31 -15 78 -21 31 14 5266 2983 4 MP PP Xc0 X X-31 -15 78 -21 31 14 5266 2983 4 MP DP Xc40 X X-31 -19 78 -45 31 18 5758 3905 4 MP PP Xc0 X X-31 -19 78 -45 31 18 5758 3905 4 MP DP Xc53 X X-31 -43 78 -46 31 44 5635 3659 4 MP PP Xc0 X X-31 -43 78 -46 31 44 5635 3659 4 MP DP Xc25 X X-31 -21 78 -65 30 21 5867 3878 4 MP PP Xc0 X X-31 -21 78 -65 30 21 5867 3878 4 MP DP Xc19 X X-31 -65 78 -65 31 66 5651 3446 4 MP PP Xc0 X X-31 -65 78 -65 31 66 5651 3446 4 MP DP Xc2 X X-31 -14 78 -21 30 14 4911 2989 4 MP PP Xc0 X X-31 -14 78 -21 30 14 4911 2989 4 MP DP Xc58 X X-30 -44 77 -46 31 44 5666 3703 4 MP PP Xc0 X X-30 -44 77 -46 31 44 5666 3703 4 MP DP Xc2 X X-30 -15 77 -21 31 14 5158 2990 4 MP PP Xc0 X X-30 -15 77 -21 31 14 5158 2990 4 MP DP Xc28 X X-30 -31 77 -21 31 31 5650 3895 4 MP PP Xc0 X X-30 -31 77 -21 31 31 5650 3895 4 MP DP Xc2 X X-31 -14 78 -21 30 14 5406 2990 4 MP PP Xc0 X X-31 -14 78 -21 30 14 5406 2990 4 MP DP Xc12 X X-31 -102 78 -45 30 102 5605 3557 4 MP PP Xc0 X X-31 -102 78 -45 30 102 5605 3557 4 MP DP Xc34 X X-31 -79 78 -64 30 78 5621 3368 4 MP PP Xc0 X X-31 -79 78 -64 30 78 5621 3368 4 MP DP Xc40 X X-30 -21 77 -45 31 21 5789 3923 4 MP PP Xc0 X X-30 -21 77 -45 31 21 5789 3923 4 MP DP Xc2 X X-31 -14 78 -21 31 14 4802 2996 4 MP PP Xc0 X X-31 -14 78 -21 31 14 4802 2996 4 MP DP Xc2 X X-31 -15 78 -21 30 15 5050 2996 4 MP PP Xc0 X X-31 -15 78 -21 30 15 5050 2996 4 MP DP Xc2 X X-31 -14 78 -21 31 14 5297 2997 4 MP PP Xc0 X X-31 -14 78 -21 31 14 5297 2997 4 MP DP Xc8 X X-31 -127 78 -21 31 127 5619 3768 4 MP PP Xc0 X X-31 -127 78 -21 31 127 5619 3768 4 MP DP Xc28 X X-31 -18 78 -64 31 17 5897 3899 4 MP PP Xc0 X X-31 -18 78 -64 31 17 5897 3899 4 MP DP Xc16 X X-31 -66 77 -45 31 65 5574 3492 4 MP PP Xc0 X X-31 -66 77 -45 31 65 5574 3492 4 MP DP Xc36 X X-30 -75 77 -65 31 75 5590 3293 4 MP PP Xc0 X X-30 -75 77 -65 31 75 5590 3293 4 MP DP Xc40 X X-31 -18 78 -21 30 18 5681 3926 4 MP PP Xc0 X X-31 -18 78 -21 30 18 5681 3926 4 MP DP Xc2 X X-31 -14 78 -64 31 57 5436 3004 4 MP PP Xc0 X X-31 -14 78 -64 31 57 5436 3004 4 MP DP Xc53 X X-31 -44 78 -21 30 43 5558 3681 4 MP PP Xc0 X X-31 -44 78 -21 30 43 5558 3681 4 MP DP Xc56 X X-30 -65 77 -65 31 66 5467 3061 4 MP PP Xc0 X X-30 -65 77 -65 31 66 5467 3061 4 MP DP Xc32 X X-31 -47 77 -64 31 46 5498 3127 4 MP PP Xc0 X X-31 -47 77 -64 31 46 5498 3127 4 MP DP Xc2 X X-31 -14 78 -22 31 15 4941 3003 4 MP PP Xc0 X X-31 -14 78 -22 31 15 4941 3003 4 MP DP Xc2 X X-31 -14 77 -21 31 14 5189 3004 4 MP PP Xc0 X X-31 -14 77 -21 31 14 5189 3004 4 MP DP Xc17 X X-30 -78 77 -46 31 79 5543 3413 4 MP PP Xc0 X X-30 -78 77 -46 31 79 5543 3413 4 MP DP Xc50 X X-31 -66 78 -65 31 67 5559 3226 4 MP PP Xc0 X X-31 -66 78 -65 31 67 5559 3226 4 MP DP Xc31 X X-31 -53 78 -64 30 53 5529 3173 4 MP PP Xc0 X X-31 -53 78 -64 30 53 5529 3173 4 MP DP Xc35 X X-31 -31 78 -50 31 31 5572 3945 4 MP PP Xc0 X X-31 -31 78 -50 31 31 5572 3945 4 MP DP Xc58 X X-31 -44 78 -21 31 44 5588 3724 4 MP PP Xc0 X X-31 -44 78 -21 31 44 5588 3724 4 MP DP Xc12 X X-30 -102 77 -22 31 102 5527 3579 4 MP PP Xc0 X X-30 -102 77 -22 31 102 5527 3579 4 MP DP Xc40 X X-31 -17 77 -46 31 18 5820 3944 4 MP PP Xc0 X X-31 -17 77 -46 31 18 5820 3944 4 MP DP Xc2 X X-30 -14 77 -22 31 15 4833 3010 4 MP PP Xc0 X X-30 -14 77 -22 31 15 4833 3010 4 MP DP Xc27 X X-31 -75 78 -45 31 75 5512 3338 4 MP PP Xc0 X X-31 -75 78 -45 31 75 5512 3338 4 MP DP Xc2 X X-31 -14 78 -21 31 14 5080 3011 4 MP PP Xc0 X X-31 -14 78 -21 31 14 5080 3011 4 MP DP Xc2 X X-30 -14 77 -21 31 14 5328 3011 4 MP PP Xc0 X X-30 -14 77 -21 31 14 5328 3011 4 MP DP Xc28 X X-31 -19 78 -64 31 19 5928 3916 4 MP PP Xc0 X X-31 -19 78 -64 31 19 5928 3916 4 MP DP Xc30 X X-31 -66 78 -45 31 65 5389 3107 4 MP PP Xc0 X X-31 -66 78 -45 31 65 5389 3107 4 MP DP Xc40 X X-31 -21 78 -22 31 22 5711 3944 4 MP PP Xc0 X X-31 -21 78 -22 31 22 5711 3944 4 MP DP Xc31 X X-31 -46 78 -46 31 47 5420 3172 4 MP PP Xc0 X X-31 -46 78 -46 31 47 5420 3172 4 MP DP Xc47 X X-31 -127 78 -50 30 127 5542 3818 4 MP PP Xc0 X X-31 -127 78 -50 30 127 5542 3818 4 MP DP Xc29 X X-31 -67 78 -45 30 66 5482 3272 4 MP PP Xc0 X X-31 -67 78 -45 30 66 5482 3272 4 MP DP Xc39 X X-30 -53 77 -46 31 53 5451 3219 4 MP PP Xc0 X X-30 -53 77 -46 31 53 5451 3219 4 MP DP Xc16 X X-31 -65 78 -22 31 66 5496 3513 4 MP PP Xc0 X X-31 -65 78 -22 31 66 5496 3513 4 MP DP Xc2 X X-31 -57 78 -46 30 82 5359 3025 4 MP PP Xc0 X X-31 -57 78 -46 30 82 5359 3025 4 MP DP Xc42 X X-30 -18 77 -50 31 18 5603 3976 4 MP PP Xc0 X X-30 -18 77 -50 31 18 5603 3976 4 MP DP Xc2 X X-31 -14 78 -22 30 15 4725 3017 4 MP PP Xc0 X X-31 -14 78 -22 30 15 4725 3017 4 MP DP Xc2 X X-30 -15 77 -21 31 14 4972 3018 4 MP PP Xc0 X X-30 -15 77 -21 31 14 4972 3018 4 MP DP Xc2 X X-31 -14 78 -21 30 14 5220 3018 4 MP PP Xc0 X X-31 -14 78 -21 30 14 5220 3018 4 MP DP Xc58 X X-30 -43 77 -50 31 43 5480 3731 4 MP PP Xc0 X X-30 -43 77 -50 31 43 5480 3731 4 MP DP Xc17 X X-31 -79 78 -21 30 79 5466 3434 4 MP PP Xc0 X X-31 -79 78 -21 30 79 5466 3434 4 MP DP Xc35 X X-31 -19 78 -46 30 19 5851 3962 4 MP PP Xc0 X X-31 -19 78 -46 30 19 5851 3962 4 MP DP Xc35 X X-31 -31 78 -21 30 31 5495 3966 4 MP PP Xc0 X X-31 -31 78 -21 30 31 5495 3966 4 MP DP Xc28 X X-30 -27 77 -65 31 28 5959 3935 4 MP PP Xc0 X X-30 -27 77 -65 31 28 5959 3935 4 MP DP Xc8 X X-31 -44 77 -50 31 44 5511 3774 4 MP PP Xc0 X X-31 -44 77 -50 31 44 5511 3774 4 MP DP Xc9 X X-31 -102 78 -50 31 102 5449 3629 4 MP PP Xc0 X X-31 -102 78 -50 31 102 5449 3629 4 MP DP Xc2 X X-31 -15 78 -21 30 14 4864 3025 4 MP PP Xc0 X X-31 -15 78 -21 30 14 4864 3025 4 MP DP Xc2 X X-31 -14 78 -21 31 14 5111 3025 4 MP PP Xc0 X X-31 -14 78 -21 31 14 5111 3025 4 MP DP Xc40 X X-31 -18 78 -21 31 17 5742 3966 4 MP PP Xc0 X X-31 -18 78 -21 31 17 5742 3966 4 MP DP Xc27 X X-31 -75 77 -21 31 75 5435 3359 4 MP PP Xc0 X X-31 -75 77 -21 31 75 5435 3359 4 MP DP Xc42 X X-31 -22 77 -50 31 22 5634 3994 4 MP PP Xc0 X X-31 -22 77 -50 31 22 5634 3994 4 MP DP Xc47 X X-30 -127 77 -21 31 127 5464 3839 4 MP PP Xc0 X X-30 -127 77 -21 31 127 5464 3839 4 MP DP Xc30 X X-31 -65 77 -22 31 66 5312 3128 4 MP PP Xc0 X X-31 -65 77 -22 31 66 5312 3128 4 MP DP Xc31 X X-31 -47 78 -21 30 46 5343 3194 4 MP PP Xc0 X X-31 -47 78 -21 30 46 5343 3194 4 MP DP Xc2 X X-31 -15 78 -21 31 14 4755 3032 4 MP PP Xc0 X X-31 -15 78 -21 31 14 4755 3032 4 MP DP Xc2 X X-31 -14 77 -21 31 14 5003 3032 4 MP PP Xc0 X X-31 -14 77 -21 31 14 5003 3032 4 MP DP Xc29 X X-30 -66 77 -21 31 66 5404 3293 4 MP PP Xc0 X X-30 -66 77 -21 31 66 5404 3293 4 MP DP Xc2 X X-31 -14 78 -22 31 15 5250 3032 4 MP PP Xc0 X X-31 -14 78 -22 31 15 5250 3032 4 MP DP Xc15 X X-31 -66 78 -50 30 66 5419 3563 4 MP PP Xc0 X X-31 -66 78 -50 30 66 5419 3563 4 MP DP Xc35 X X-31 -28 78 -45 31 27 5881 3981 4 MP PP Xc0 X X-31 -28 78 -45 31 27 5881 3981 4 MP DP Xc39 X X-31 -53 78 -21 31 53 5373 3240 4 MP PP Xc0 X X-31 -53 78 -21 31 53 5373 3240 4 MP DP Xc42 X X-31 -18 78 -22 31 19 5525 3997 4 MP PP Xc0 X X-31 -18 78 -22 31 19 5525 3997 4 MP DP Xc40 X X-31 -17 78 -65 30 17 5990 3963 4 MP PP Xc0 X X-31 -17 78 -65 30 17 5990 3963 4 MP DP Xc58 X X-31 -43 78 -21 31 43 5402 3752 4 MP PP Xc0 X X-31 -43 78 -21 31 43 5402 3752 4 MP DP Xc2 X X-30 -82 77 -21 31 81 5281 3047 4 MP PP Xc0 X X-30 -82 77 -21 31 81 5281 3047 4 MP DP Xc19 X X-30 -79 77 -50 31 78 5388 3485 4 MP PP Xc0 X X-30 -79 77 -50 31 78 5388 3485 4 MP DP Xc2 X X-30 -15 77 -21 31 14 4647 3039 4 MP PP Xc0 X X-30 -15 77 -21 31 14 4647 3039 4 MP DP Xc35 X X-30 -19 77 -21 31 19 5773 3983 4 MP PP Xc0 X X-30 -19 77 -21 31 19 5773 3983 4 MP DP Xc2 X X-31 -14 78 -21 31 14 4894 3039 4 MP PP Xc0 X X-31 -14 78 -21 31 14 4894 3039 4 MP DP Xc8 X X-31 -44 78 -21 31 44 5433 3795 4 MP PP Xc0 X X-31 -44 78 -21 31 44 5433 3795 4 MP DP Xc2 X X-30 -14 77 -21 31 14 5142 3039 4 MP PP Xc0 X X-30 -14 77 -21 31 14 5142 3039 4 MP DP Xc9 X X-31 -102 78 -21 30 102 5372 3650 4 MP PP Xc0 X X-31 -102 78 -21 30 102 5372 3650 4 MP DP Xc42 X X-31 -17 78 -50 30 17 5665 4016 4 MP PP Xc0 X X-31 -17 78 -50 30 17 5665 4016 4 MP DP Xc35 X X-30 -31 77 -22 31 31 5417 3988 4 MP PP Xc0 X X-30 -31 77 -22 31 31 5417 3988 4 MP DP Xc34 X X-31 -75 78 -51 31 76 5357 3409 4 MP PP Xc0 X X-31 -75 78 -51 31 76 5357 3409 4 MP DP Xc42 X X-30 -17 77 -45 31 17 5912 4008 4 MP PP Xc0 X X-30 -17 77 -45 31 17 5912 4008 4 MP DP Xc2 X X-30 -14 77 -21 31 14 4786 3046 4 MP PP Xc0 X X-30 -14 77 -21 31 14 4786 3046 4 MP DP Xc42 X X-31 -22 78 -21 31 21 5556 4016 4 MP PP Xc0 X X-31 -22 78 -21 31 21 5556 4016 4 MP DP Xc2 X X-31 -14 78 -22 30 15 5034 3046 4 MP PP Xc0 X X-31 -14 78 -22 30 15 5034 3046 4 MP DP Xc32 X X-31 -66 78 -50 31 66 5234 3178 4 MP PP Xc0 X X-31 -66 78 -50 31 66 5234 3178 4 MP DP Xc40 X X-31 -24 78 -64 31 23 6020 3980 4 MP PP Xc0 X X-31 -24 78 -64 31 23 6020 3980 4 MP DP Xc39 X X-30 -46 77 -50 31 46 5265 3244 4 MP PP Xc0 X X-30 -46 77 -50 31 46 5265 3244 4 MP DP Xc36 X X-31 -66 78 -50 31 66 5326 3343 4 MP PP Xc0 X X-31 -66 78 -50 31 66 5326 3343 4 MP DP Xc15 X X-30 -66 77 -21 31 66 5341 3584 4 MP PP Xc0 X X-30 -66 77 -21 31 66 5341 3584 4 MP DP Xc29 X X-31 -53 78 -50 30 53 5296 3290 4 MP PP Xc0 X X-31 -53 78 -50 30 53 5296 3290 4 MP DP Xc35 X X-31 -27 78 -22 30 28 5804 4002 4 MP PP Xc0 X X-31 -27 78 -22 30 28 5804 4002 4 MP DP Xc47 X X-31 -127 78 -22 31 128 5386 3860 4 MP PP Xc0 X X-31 -127 78 -22 31 128 5386 3860 4 MP DP Xc2 X X-31 -15 78 -49 30 43 5173 3053 4 MP PP Xc0 X X-31 -15 78 -49 30 43 5173 3053 4 MP DP Xc37 X X-31 -81 78 -50 31 82 5203 3096 4 MP PP Xc0 X X-31 -81 78 -50 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Xc0 X X-31 -14 78 -21 31 14 4569 3060 4 MP DP Xc2 X X-31 -14 77 -21 31 14 4817 3060 4 MP PP Xc0 X X-31 -14 77 -21 31 14 4817 3060 4 MP DP Xc34 X X-31 -76 78 -21 30 75 5280 3431 4 MP PP Xc0 X X-31 -76 78 -21 30 75 5280 3431 4 MP DP Xc8 X X-31 -44 78 -21 30 43 5356 3817 4 MP PP Xc0 X X-31 -44 78 -21 30 43 5356 3817 4 MP DP Xc2 X X-31 -14 78 -21 31 13 5064 3061 4 MP PP Xc0 X X-31 -14 78 -21 31 13 5064 3061 4 MP DP Xc9 X X-30 -102 77 -21 31 102 5294 3671 4 MP PP Xc0 X X-30 -102 77 -21 31 102 5294 3671 4 MP DP Xc42 X X-31 -17 78 -22 31 17 5834 4030 4 MP PP Xc0 X X-31 -17 78 -22 31 17 5834 4030 4 MP DP Xc32 X X-31 -66 78 -21 30 65 5157 3200 4 MP PP Xc0 X X-31 -66 78 -21 30 65 5157 3200 4 MP DP Xc39 X X-31 -46 78 -21 31 46 5187 3265 4 MP PP Xc0 X X-31 -46 78 -21 31 46 5187 3265 4 MP DP Xc36 X X-31 -66 77 -22 31 67 5249 3364 4 MP PP Xc0 X X-31 -66 77 -22 31 67 5249 3364 4 MP DP Xc29 X X-30 -53 77 -21 31 53 5218 3311 4 MP PP Xc0 X X-30 -53 77 -21 31 53 5218 3311 4 MP DP Xc45 X X-31 -128 78 -58 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PP Xc0 X X-31 -46 77 -21 31 46 4877 3387 4 MP DP Xc20 X X-31 -66 78 -22 31 67 4938 3486 4 MP PP Xc0 X X-31 -66 78 -22 31 67 4938 3486 4 MP DP Xc2 X X-31 -14 78 -21 31 14 4367 3138 4 MP PP Xc0 X X-31 -14 78 -21 31 14 4367 3138 4 MP DP Xc52 X X-31 -32 78 -22 31 33 5802 4191 4 MP PP Xc0 X X-31 -32 78 -22 31 33 5802 4191 4 MP DP Xc49 X X-30 -21 77 -30 31 21 5168 4168 4 MP PP Xc0 X X-30 -21 77 -30 31 21 5168 4168 4 MP DP Xc2 X X-31 -14 78 -20 30 13 4615 3138 4 MP PP Xc0 X X-31 -14 78 -20 30 13 4615 3138 4 MP DP Xc36 X X-31 -53 78 -21 30 53 4908 3433 4 MP PP Xc0 X X-31 -53 78 -21 30 53 4908 3433 4 MP DP Xc2 X X-31 -50 77 -22 31 50 4754 3146 4 MP PP Xc0 X X-31 -50 77 -22 31 50 4754 3146 4 MP DP Xc49 X X-30 -19 77 -21 31 18 5663 4163 4 MP PP Xc0 X X-30 -19 77 -21 31 18 5663 4163 4 MP DP Xc30 X X-31 -83 78 -21 31 82 4815 3239 4 MP PP Xc0 X X-31 -83 78 -21 31 82 4815 3239 4 MP DP Xc9 X X-30 -66 77 -30 31 65 4953 3737 4 MP PP Xc0 X X-30 -66 77 -30 31 65 4953 3737 4 MP DP Xc3 X X-31 -127 78 -21 31 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414 mt 5293 441 L X5515 4614 mt 5515 4587 L X5515 414 mt 5515 441 L X5687 4614 mt 5687 4587 L X5687 414 mt 5687 441 L X5827 4614 mt 5827 4587 L X5827 414 mt 5827 441 L X5946 4614 mt 5946 4587 L X5946 414 mt 5946 441 L X6049 4614 mt 6049 4587 L X6049 414 mt 6049 441 L X6140 4614 mt 6140 4587 L X6140 414 mt 6140 441 L X6221 4614 mt 6221 4587 L X6221 414 mt 6221 441 L X6221 4614 mt 6221 4561 L X6221 414 mt 6221 467 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X6095 4863 mt (10) s X X/Helvetica 10 FMS X X6281 4752 mt (0) s X X/Helvetica 14 FMS X X 898 4614 mt 925 4614 L X6221 4614 mt 6194 4614 L X 898 4614 mt 951 4614 L X6221 4614 mt 6168 4614 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 619 4676 mt (10) s X X/Helvetica 10 FMS X X 805 4565 mt (0) s X X/Helvetica 14 FMS X X 898 4193 mt 925 4193 L X6221 4193 mt 6194 4193 L X 898 3946 mt 925 3946 L X6221 3946 mt 6194 3946 L X 898 3771 mt 925 3771 L X6221 3771 mt 6194 3771 L X 898 3635 mt 925 3635 L X6221 3635 mt 6194 3635 L X 898 3525 mt 925 3525 L X6221 3525 mt 6194 3525 L X 898 3431 mt 925 3431 L X6221 3431 mt 6194 3431 L X 898 3350 mt 925 3350 L X6221 3350 mt 6194 3350 L X 898 3278 mt 925 3278 L X6221 3278 mt 6194 3278 L X 898 3214 mt 925 3214 L X6221 3214 mt 6194 3214 L X 898 3214 mt 951 3214 L X6221 3214 mt 6168 3214 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 619 3276 mt (10) s X X/Helvetica 10 FMS X X 805 3165 mt (1) s X X/Helvetica 14 FMS X X 898 2793 mt 925 2793 L X6221 2793 mt 6194 2793 L X 898 2546 mt 925 2546 L X6221 2546 mt 6194 2546 L X 898 2371 mt 925 2371 L X6221 2371 mt 6194 2371 L X 898 2235 mt 925 2235 L X6221 2235 mt 6194 2235 L X 898 2125 mt 925 2125 L X6221 2125 mt 6194 2125 L X 898 2031 mt 925 2031 L X6221 2031 mt 6194 2031 L X 898 1950 mt 925 1950 L X6221 1950 mt 6194 1950 L X 898 1878 mt 925 1878 L X6221 1878 mt 6194 1878 L X 898 1814 mt 925 1814 L X6221 1814 mt 6194 1814 L X 898 1814 mt 951 1814 L X6221 1814 mt 6168 1814 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 619 1876 mt (10) s X X/Helvetica 10 FMS X X 805 1765 mt (2) s X X/Helvetica 14 FMS X X 898 1393 mt 925 1393 L X6221 1393 mt 6194 1393 L X 898 1146 mt 925 1146 L X6221 1146 mt 6194 1146 L X 898 971 mt 925 971 L X6221 971 mt 6194 971 L X 898 835 mt 925 835 L X6221 835 mt 6194 835 L X 898 725 mt 925 725 L X6221 725 mt 6194 725 L X 898 631 mt 925 631 L X6221 631 mt 6194 631 L X 898 550 mt 925 550 L X6221 550 mt 6194 550 L X 898 478 mt 925 478 L X6221 478 mt 6194 478 L X 898 414 mt 925 414 L X6221 414 mt 6194 414 L X 898 414 mt 951 414 L X6221 414 mt 6168 414 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 619 476 mt (10) s X X/Helvetica 10 FMS X X 805 365 mt (3) s X X/Helvetica 14 FMS X Xgs 898 414 5324 4201 MR c np Xgr X5642 3550 5714 3622 FO Xgs 898 414 5324 4201 MR c np Xgr X3641 3535 3713 3607 FO Xgs 898 414 5324 4201 MR c np Xgr X3285 3532 3357 3604 FO Xgs 898 414 5324 4201 MR c np Xgr X2912 3530 2984 3602 FO Xgs 898 414 5324 4201 MR c np Xgr X2073 3527 2145 3599 FO Xgs 898 414 5324 4201 MR c np Xgr X2068 3527 2140 3599 FO Xgs 898 414 5324 4201 MR c np Xgr X2064 3525 2136 3597 FO Xgs 898 414 5324 4201 MR c np Xgr X2048 2929 2120 3001 FO Xgs 898 414 5324 4201 MR c np Xgr X2014 1463 2086 1535 FO Xgs 898 414 5324 4201 MR c np Xgr X2009 1100 2081 1172 FO Xgs 898 414 5324 4201 MR c np Xgr X5642 3550 mt 5714 3622 L X5714 3550 mt 5642 3622 L Xgs 898 414 5324 4201 MR c np Xgr X2912 3530 mt 2984 3602 L X2984 3530 mt 2912 3602 L Xgs 898 414 5324 4201 MR c np Xgr X2064 3525 mt 2136 3597 L X2136 3525 mt 2064 3597 L Xgs 898 414 5324 4201 MR c np Xgr X2009 1100 mt 2081 1172 L X2081 1100 mt 2009 1172 L Xgs 898 414 5324 4201 MR c np Xgr X5678 3647 mt (3) s X2948 3627 mt (6) s X2100 3622 mt (9) s X2045 1197 mt (12) s X2649 5040 mt (residual norm || A x - b ||) s X 565 3219 mt -90 rotate(solution norm || x ||) s X90 rotate X2587 281 mt (L-curve, TSVD corner at 9) s Xgs 898 414 5324 4201 MR c np XDA X2100 0 0 3561 2 MP stroke X0 -1623 2100 5184 2 MP stroke Xgr X 898 4614 mt 6221 4614 L X 898 414 mt 6221 414 L X 898 4614 mt 898 414 L X6221 4614 mt 6221 414 L X 898 414 mt 898 414 L X6221 414 mt 6221 414 L X Xend % pop colortable dictionary X Xeplot X Xepage Xend X Xshowpage X X%%EndDocument X X endTexFig X 84 1045 a X 16120153 12120418 1184071 11840716 39074365 40258437 startTexFig X 84 1045 a X%%BeginDocument: tutorial/fig4c.eps X X X/MathWorks 120 dict begin X X/bdef {bind def} bind def X/ldef {load def} bind def X/xdef {exch def} bdef X/xstore {exch store} bdef X X/c /clip ldef X/cc /concat ldef X/cp /closepath ldef X/gr /grestore ldef X/gs /gsave ldef X/mt /moveto ldef X/np /newpath ldef X/rc {rectclip} bdef X/rf {rectfill} bdef X/rm /rmoveto ldef X/rl /rlineto ldef X/s /show ldef X/sc /setrgbcolor ldef X X/pgsv () def X/bpage {/pgsv save def} bdef X/epage {pgsv restore} bdef X/bplot /gsave ldef X/eplot {stroke grestore} bdef X X/llx 0 def X/lly 0 def X/urx 0 def X/ury 0 def X/bbox {/ury xdef /urx xdef /lly xdef /llx xdef} bdef X X/portraitMode (op) def X/landscapeMode (ol) def X/Orientation portraitMode def X/portrait {/Orientation portraitMode def} bdef X/landscape {/Orientation landscapeMode def} bdef X X/dpi2point 0 def X X/FontSize 0 def X/FMS { X /FontSize xstore %save size off stack X findfont X [FontSize dpi2point mul 0 0 FontSize dpi2point mul neg 0 0] X makefont X setfont X }bdef X/setPortrait { X 1 dpi2point div -1 dpi2point div scale X llx ury neg translate X } bdef X/setLandscape { X 1 dpi2point div -1 dpi2point div scale X urx ury neg translate X 90 rotate X } bdef X X/csm {Orientation portraitMode eq {setPortrait} {setLandscape} ifelse} bdef X/SO { [] 0 setdash } bdef X/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef X/DA { [6 dpi2point mul] 0 setdash } bdef X/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef X X/L { % LineTo X lineto X stroke X } bdef X/MP { % MakePoly X 3 1 roll moveto X 1 sub {rlineto} repeat X } bdef X/AP { % AddPoly X {rlineto} repeat X } bdef X/PP { % PaintPoly X closepath fill X } bdef X/DP { % DrawPoly X closepath stroke X } bdef X/MR { % MakeRect X 4 -2 roll moveto X dup 0 exch rlineto X exch 0 rlineto X neg 0 exch rlineto X closepath X } bdef X/FR { % FrameRect X MR stroke X } bdef X/PR { % PaintRect X MR fill X } bdef X/L1i { % Level 1 Image X { currentfile picstr readhexstring pop } image X } bdef X X/half_width 0 def X/half_height 0 def X/MakeOval { X newpath X /ury xstore /urx xstore /lly xstore /llx xstore X /half_width urx llx sub 2 div store X /half_height ury lly sub 2 div store X llx half_width add lly half_height add translate X half_width half_height scale X .5 half_width div setlinewidth X 0 0 1 0 360 arc X } bdef X/FO { X gsave X MakeOval stroke X grestore X } bdef X/PO { X gsave X MakeOval fill X grestore X } bdef X X/PD { X 2 copy moveto lineto stroke X } bdef X X Xcurrentdict end def %dictionary X XMathWorks begin X X X1 setlinecap 1 setlinejoin X Xend X XMathWorks begin Xbpage X Xbplot X X/dpi2point 12 def X0216 2160 7128 7344 bbox portrait csm X X0 0 6912 5184 MR c np X6.00 setlinewidth X/colortable 76 dict begin X/c0 { 0 0 0 sc} bdef X/c1 { 1 1 1 sc} bdef X/c2 { 1 0 0 sc} bdef X/c3 { 0 1 0 sc} bdef X/c4 { 0 0 1 sc} bdef X/c5 { 1 1 0 sc} bdef X/c6 { 1 0 1 sc} bdef X/c7 { 0 1 1 sc} bdef Xcurrentdict end def % Colortable X Xcolortable begin X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X Xc1 X 0 0 6912 5184 PR XDO XSO Xc0 X 898 4614 mt 6221 4614 L X 898 414 mt 6221 414 L X 898 4614 mt 898 414 L X6221 4614 mt 6221 414 L X 898 4614 mt 898 4614 L X6221 4614 mt 6221 4614 L X 898 4614 mt 6221 4614 L X 898 4614 mt 898 414 L X 898 4614 mt 898 4614 L X 898 4614 mt 898 4587 L X 898 414 mt 898 441 L X 898 4614 mt 898 4561 L X 898 414 mt 898 467 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 752 4863 mt (10) s X X/Helvetica 10 FMS X X 938 4752 mt (-6) s X X/Helvetica 14 FMS X X1165 4614 mt 1165 4587 L X1165 414 mt 1165 441 L X1321 4614 mt 1321 4587 L X1321 414 mt 1321 441 L X1432 4614 mt 1432 4587 L X1432 414 mt 1432 441 L X1518 4614 mt 1518 4587 L X1518 414 mt 1518 441 L X1588 4614 mt 1588 4587 L X1588 414 mt 1588 441 L X1648 4614 mt 1648 4587 L X1648 414 mt 1648 441 L X1699 4614 mt 1699 4587 L X1699 414 mt 1699 441 L X1745 4614 mt 1745 4587 L X1745 414 mt 1745 441 L X1785 4614 mt 1785 4587 L X1785 414 mt 1785 441 L X1785 4614 mt 1785 4561 L X1785 414 mt 1785 467 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X1639 4863 mt (10) s X X/Helvetica 10 FMS X X1825 4752 mt (-5) s X X/Helvetica 14 FMS X X2052 4614 mt 2052 4587 L X2052 414 mt 2052 441 L X2208 4614 mt 2208 4587 L X2208 414 mt 2208 441 L X2319 4614 mt 2319 4587 L X2319 414 mt 2319 441 L X2405 4614 mt 2405 4587 L X2405 414 mt 2405 441 L X2476 4614 mt 2476 4587 L X2476 414 mt 2476 441 L X2535 4614 mt 2535 4587 L X2535 414 mt 2535 441 L X2586 4614 mt 2586 4587 L X2586 414 mt 2586 441 L X2632 4614 mt 2632 4587 L X2632 414 mt 2632 441 L X2672 4614 mt 2672 4587 L X2672 414 mt 2672 441 L X2672 4614 mt 2672 4561 L X2672 414 mt 2672 467 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X2526 4863 mt (10) s X X/Helvetica 10 FMS X X2712 4752 mt (-4) s X X/Helvetica 14 FMS X X2939 4614 mt 2939 4587 L X2939 414 mt 2939 441 L X3096 4614 mt 3096 4587 L X3096 414 mt 3096 441 L X3206 4614 mt 3206 4587 L X3206 414 mt 3206 441 L X3292 4614 mt 3292 4587 L X3292 414 mt 3292 441 L X3363 4614 mt 3363 4587 L X3363 414 mt 3363 441 L X3422 4614 mt 3422 4587 L X3422 414 mt 3422 441 L X3474 4614 mt 3474 4587 L X3474 414 mt 3474 441 L X3519 4614 mt 3519 4587 L X3519 414 mt 3519 441 L X3560 4614 mt 3560 4587 L X3560 414 mt 3560 441 L X3560 4614 mt 3560 4561 L X3560 414 mt 3560 467 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X3414 4863 mt (10) s X X/Helvetica 10 FMS X X3600 4752 mt (-3) s X X/Helvetica 14 FMS X X3827 4614 mt 3827 4587 L X3827 414 mt 3827 441 L X3983 4614 mt 3983 4587 L X3983 414 mt 3983 441 L X4094 4614 mt 4094 4587 L X4094 414 mt 4094 441 L X4180 4614 mt 4180 4587 L X4180 414 mt 4180 441 L X4250 4614 mt 4250 4587 L X4250 414 mt 4250 441 L X4309 4614 mt 4309 4587 L X4309 414 mt 4309 441 L X4361 4614 mt 4361 4587 L X4361 414 mt 4361 441 L X4406 4614 mt 4406 4587 L X4406 414 mt 4406 441 L X4447 4614 mt 4447 4587 L X4447 414 mt 4447 441 L X4447 4614 mt 4447 4561 L X4447 414 mt 4447 467 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X4301 4863 mt (10) s X X/Helvetica 10 FMS X X4487 4752 mt (-2) s X X/Helvetica 14 FMS X X4714 4614 mt 4714 4587 L X4714 414 mt 4714 441 L X4870 4614 mt 4870 4587 L X4870 414 mt 4870 441 L X4981 4614 mt 4981 4587 L X4981 414 mt 4981 441 L X5067 4614 mt 5067 4587 L X5067 414 mt 5067 441 L X5137 4614 mt 5137 4587 L X5137 414 mt 5137 441 L X5196 4614 mt 5196 4587 L X5196 414 mt 5196 441 L X5248 4614 mt 5248 4587 L X5248 414 mt 5248 441 L X5293 4614 mt 5293 4587 L X5293 414 mt 5293 441 L X5334 4614 mt 5334 4587 L X5334 414 mt 5334 441 L X5334 4614 mt 5334 4561 L X5334 414 mt 5334 467 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X5188 4863 mt (10) s X X/Helvetica 10 FMS X X5374 4752 mt (-1) s X X/Helvetica 14 FMS X X5601 4614 mt 5601 4587 L X5601 414 mt 5601 441 L X5757 4614 mt 5757 4587 L X5757 414 mt 5757 441 L X5868 4614 mt 5868 4587 L X5868 414 mt 5868 441 L X5954 4614 mt 5954 4587 L X5954 414 mt 5954 441 L X6024 4614 mt 6024 4587 L X6024 414 mt 6024 441 L X6084 4614 mt 6084 4587 L X6084 414 mt 6084 441 L X6135 4614 mt 6135 4587 L X6135 414 mt 6135 441 L X6180 4614 mt 6180 4587 L X6180 414 mt 6180 441 L X6221 4614 mt 6221 4587 L X6221 414 mt 6221 441 L X6221 4614 mt 6221 4561 L X6221 414 mt 6221 467 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X6095 4863 mt (10) s X X/Helvetica 10 FMS X X6281 4752 mt (0) s X X/Helvetica 14 FMS X X 898 4614 mt 925 4614 L X6221 4614 mt 6194 4614 L X 898 4614 mt 951 4614 L X6221 4614 mt 6168 4614 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 579 4676 mt (10) s X X/Helvetica 10 FMS X X 765 4565 mt (-9) s X X/Helvetica 14 FMS X X 898 4456 mt 925 4456 L X6221 4456 mt 6194 4456 L X 898 4364 mt 925 4364 L X6221 4364 mt 6194 4364 L X 898 4298 mt 925 4298 L X6221 4298 mt 6194 4298 L X 898 4247 mt 925 4247 L X6221 4247 mt 6194 4247 L X 898 4205 mt 925 4205 L X6221 4205 mt 6194 4205 L X 898 4170 mt 925 4170 L X6221 4170 mt 6194 4170 L X 898 4140 mt 925 4140 L X6221 4140 mt 6194 4140 L X 898 4113 mt 925 4113 L X6221 4113 mt 6194 4113 L X 898 4089 mt 925 4089 L X6221 4089 mt 6194 4089 L X 898 4089 mt 951 4089 L X6221 4089 mt 6168 4089 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 579 4151 mt (10) s X X/Helvetica 10 FMS X X 765 4040 mt (-8) s X X/Helvetica 14 FMS X X 898 3931 mt 925 3931 L X6221 3931 mt 6194 3931 L X 898 3839 mt 925 3839 L X6221 3839 mt 6194 3839 L X 898 3773 mt 925 3773 L X6221 3773 mt 6194 3773 L X 898 3722 mt 925 3722 L X6221 3722 mt 6194 3722 L X 898 3680 mt 925 3680 L X6221 3680 mt 6194 3680 L X 898 3645 mt 925 3645 L X6221 3645 mt 6194 3645 L X 898 3615 mt 925 3615 L X6221 3615 mt 6194 3615 L X 898 3588 mt 925 3588 L X6221 3588 mt 6194 3588 L X 898 3564 mt 925 3564 L X6221 3564 mt 6194 3564 L X 898 3564 mt 951 3564 L X6221 3564 mt 6168 3564 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 579 3626 mt (10) s X X/Helvetica 10 FMS X X 765 3515 mt (-7) s X X/Helvetica 14 FMS X X 898 3406 mt 925 3406 L X6221 3406 mt 6194 3406 L X 898 3314 mt 925 3314 L X6221 3314 mt 6194 3314 L X 898 3248 mt 925 3248 L X6221 3248 mt 6194 3248 L X 898 3197 mt 925 3197 L X6221 3197 mt 6194 3197 L X 898 3155 mt 925 3155 L X6221 3155 mt 6194 3155 L X 898 3120 mt 925 3120 L X6221 3120 mt 6194 3120 L X 898 3090 mt 925 3090 L X6221 3090 mt 6194 3090 L X 898 3063 mt 925 3063 L X6221 3063 mt 6194 3063 L X 898 3039 mt 925 3039 L X6221 3039 mt 6194 3039 L X 898 3039 mt 951 3039 L X6221 3039 mt 6168 3039 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 579 3101 mt (10) s X X/Helvetica 10 FMS X X 765 2990 mt (-6) s X X/Helvetica 14 FMS X X 898 2881 mt 925 2881 L X6221 2881 mt 6194 2881 L X 898 2789 mt 925 2789 L X6221 2789 mt 6194 2789 L X 898 2723 mt 925 2723 L X6221 2723 mt 6194 2723 L X 898 2672 mt 925 2672 L X6221 2672 mt 6194 2672 L X 898 2630 mt 925 2630 L X6221 2630 mt 6194 2630 L X 898 2595 mt 925 2595 L X6221 2595 mt 6194 2595 L X 898 2565 mt 925 2565 L X6221 2565 mt 6194 2565 L X 898 2538 mt 925 2538 L X6221 2538 mt 6194 2538 L X 898 2514 mt 925 2514 L X6221 2514 mt 6194 2514 L X 898 2514 mt 951 2514 L X6221 2514 mt 6168 2514 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 579 2576 mt (10) s X X/Helvetica 10 FMS X X 765 2465 mt (-5) s X X/Helvetica 14 FMS X X 898 2356 mt 925 2356 L X6221 2356 mt 6194 2356 L X 898 2264 mt 925 2264 L X6221 2264 mt 6194 2264 L X 898 2198 mt 925 2198 L X6221 2198 mt 6194 2198 L X 898 2147 mt 925 2147 L X6221 2147 mt 6194 2147 L X 898 2105 mt 925 2105 L X6221 2105 mt 6194 2105 L X 898 2070 mt 925 2070 L X6221 2070 mt 6194 2070 L X 898 2040 mt 925 2040 L X6221 2040 mt 6194 2040 L X 898 2013 mt 925 2013 L X6221 2013 mt 6194 2013 L X 898 1989 mt 925 1989 L X6221 1989 mt 6194 1989 L X 898 1989 mt 951 1989 L X6221 1989 mt 6168 1989 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 579 2051 mt (10) s X X/Helvetica 10 FMS X X 765 1940 mt (-4) s X X/Helvetica 14 FMS X X 898 1831 mt 925 1831 L X6221 1831 mt 6194 1831 L X 898 1739 mt 925 1739 L X6221 1739 mt 6194 1739 L X 898 1673 mt 925 1673 L X6221 1673 mt 6194 1673 L X 898 1622 mt 925 1622 L X6221 1622 mt 6194 1622 L X 898 1580 mt 925 1580 L X6221 1580 mt 6194 1580 L X 898 1545 mt 925 1545 L X6221 1545 mt 6194 1545 L X 898 1515 mt 925 1515 L X6221 1515 mt 6194 1515 L X 898 1488 mt 925 1488 L X6221 1488 mt 6194 1488 L X 898 1464 mt 925 1464 L X6221 1464 mt 6194 1464 L X 898 1464 mt 951 1464 L X6221 1464 mt 6168 1464 L X X/Helvetica 10 FMS X X X/Helvetica 14 FMS X X 579 1526 mt (10) s X X/Helvetica 10 FMS X X 765 1415 mt (-3) s X X/Helvetica 14 FMS X X 898 1306 mt 925 1306 L X6221 1306 mt 6194 1306 L X 898 1214 mt 925 1214 L X6221 1214 mt 6194 1214 L X 898 1148 mt 925 1148 L X6221 1148 mt 6194 1148 L X 898 1097 mt 925 1097 L X6221 1097 mt 6194 1097 L X 898 1055 mt 925 1055 L X6221 1055 mt 6194 1055 L X 898 1020 mt 925 1020 L X6221 1020 mt 6194 1020 L X 898 990 mt 925 990 L X6221 990 mt 6194 990 L X 898 963 mt 925 963 L X6221 963 mt 6194 963 L X 898 939 mt 925 939 L X6221 939 mt 6194 939 L X 898 939 mt 951 939 L X6221 939 mt 6168 939 L X X/Helvetica 10 FMS X X 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X%%Page: 57 58 X57 57 bop 59 157 a Fb(gcv)1662 b Fo(57)p 59 178 1772 X2 v 59 306 a Fp(Diagnostics:)130 375 y Fo(The)14 b(n)o(um)o(b)q(er)g X(of)f(p)q(oin)o(ts)h(on)g(the)g(GCV)f(curv)o(e)h(for)f(Tikhono)o(v)h X(regularization)g(is)h(determined)g(b)o(y)59 431 y(the)g(parameter)g XFe(np)q(oints)i Fo(whic)o(h)f(can)f(easily)i(b)q(e)f(c)o(hanged)f(b)o X(y)g(the)g(user)h(to)e(giv)o(e)i(a)f(\014ner)g(resolution.)59 X538 y Fp(See)i(also:)130 606 y Fe(discrep)p Fo(,)f Fe(l)p X303 606 14 2 v 16 w(curve)p Fo(,)f Fe(quasiopt)59 713 Xy Fp(References:)115 784 y Fo(1.)22 b(G.)14 b(W)l(ah)o(ba,)g XFg(Spline)h(Mo)n(dels)g(for)i(Observational)e(Data)p XFo(,)h(CBMS-NSF)f(Regional)h(Conference)173 840 y(Series)g(in)g X(Applied)h(Mathematics,)e(V)l(ol.)g(59,)f(SIAM,)i(Philadelphia,)h X(1990.)p eop X%%Page: 58 59 X58 58 bop 59 157 a Fo(58)1556 b Fb(gen)p 1730 157 14 X2 v 17 w(fo)o(rm)p 59 178 1772 2 v 59 306 a Fa(gen)p X161 306 20 2 v 23 w(fo)n(rm)59 407 y Fp(Purp)q(ose:)130 X476 y Fo(T)l(ransformation)14 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Xf(R.)g(Pik)o(e,)g Fg(Line)n(ar)g(inverse)h(pr)n(oblems)g(with)g(discr)n X(ete)g(data:)22 b(II.)152 1760 y(Stability)16 b(and)g(r)n(e)n X(gularization)p Fo(,)f(In)o(v)o(erse)g(Problems)h Fp(4)f XFo(\(1988\),)e(573{594.)82 1852 y([8])152 1844 y(\027)152 X1852 y(A.)f(Bj\177)-23 b(orc)o(k,)12 b Fg(A)h(bidiagonalization)g X(algorithm)h(for)g(solving)e(lar)n(ge)h(and)g(sp)n(arse)g(il)r(l-p)n X(ose)n(d)g(systems)152 1909 y(of)k(line)n(ar)e(e)n(quations)p XFo(,)g(BIT)g Fp(28)h Fo(\(1988\),)d(659{670.)82 2001 Xy([9])152 1993 y(\027)152 2001 y(A.)18 b(Bj\177)-23 b(orc)o(k,)17 Xb Fg(L)n(e)n(ast)g(Squar)n(es)g(Metho)n(ds)p Fo(;)h(in)g(P)l(.)g(G.)e X(Ciarlet)i(&)g(J.)f(L.)h(Lions,)g Fg(Handb)n(o)n(ok)g(of)g(Nu-)152 X2058 y(meric)n(al)e(A)o(nalysis)f(V)m(ol.)g(I)p Fo(,)g(Elsevier,)h X(Amsterdam,)e(1990.)59 2150 y([10])152 2142 y(\027)152 X2150 y(A.)k(Bj\177)-23 b(orc)o(k)18 b(&)g(L.)h(Eld)o(\023)-21 Xb(en,)18 b Fg(Metho)n(ds)h(in)g(numeric)n(al)f(algebr)n(a)h(for)g(il)r X(l-p)n(ose)n(d)f(pr)n(oblems)p Fo(,)g(Rep)q(ort)152 2207 Xy(LiTH-MA)l(T-R33-1979,)d(Dept.)g(of)g(Mathematics,)f(Link\177)-23 Xb(oping)17 b(Univ)o(ersit)o(y)l(,)f(1979.)59 2299 y([11])21 Xb(H.)28 b(Brakhage,)i Fg(On)d(il)r(l-p)n(ose)n(d)g(pr)n(oblems)g(and)h X(the)g(metho)n(d)g(of)g(c)n(onjugate)g(gr)n(adients)p XFo(;)33 b(in)152 2355 y(H.)21 b(W.)f(Engl)h(&)g(C.)f(W.)g(Gro)q(etsc)o X(h)g(\(Eds.\),)h Fg(Inverse)f(and)i(Il)r(l-Pose)n(d)e(Pr)n(oblems)p XFo(,)h(Academic)152 2412 y(Press,)15 b(Boston,)f(1987.)59 X2504 y([12])21 b(T.)12 b(F.)g(Chan)g(&)g(P)l(.)g(C.)g(Hansen,)h XFg(Some)g(applic)n(ations)g(of)h(the)f(r)n(ank)g(r)n(eve)n(aling)f(QR)i X(factorization)p Fo(,)152 2561 y(SIAM)i(J.)f(Sci.)h(Stat.)e(Comput.)h XFp(13)g Fo(\(1992\),)e(727{741.)59 2653 y([13])21 b(J.)c(A.)f(Co)q(c)o X(hran,)g Fg(The)h(A)o(nalysis)e(of)j(Line)n(ar)e(Inte)n(gr)n(al)g X(Equations)p Fo(,)g(McGra)o(w-Hill,)h(New)f(Y)l(ork,)152 X2710 y(1972.)59 2802 y([14])21 b(D.)14 b(Colton)g(&)h(R.)f(Kress,)g XFg(Inte)n(gr)n(al)g(Equation)i(Metho)n(ds)f(for)h(Sc)n(attering)e(The)n X(ory)p Fo(,)g(John)h(Wiley)l(,)152 2859 y(New)h(Y)l(ork,)e(1983.)911 X2973 y(105)p eop X%%Page: 106 107 X106 106 bop 59 166 a Fo(106)1339 b Fm(BIBLIOGRAPHY)p X59 178 1772 2 v 59 306 a Fo([15])21 b(I.)15 b(J.)g(D.)f(Craig)g(&)h(J.) Xf(C.)g(Bro)o(wn,)g Fg(Inverse)h(Pr)n(oblems)f(in)i(Astr)n(onomy)p XFo(,)e(Adam)g(Hilger,)h(Bristol,)152 362 y(1986.)59 461 Xy([16])21 b(J.)12 b(J.)g(M.)f(Cupp)q(en,)i Fg(A)g(Numeric)n(al)g X(Solution)g(of)g(the)g(Inverse)f(Pr)n(oblem)h(of)g(Ele)n(ctr)n(o)n(c)n X(ar)n(dio)n(gr)n(aphy)p Fo(,)152 517 y(Ph.)i(D.)g(Thesis,)g(Dept.)g(of) Xg(Mathematics,)f(Univ.)i(of)f(Amsterdam,)f(1983.)59 616 Xy([17])21 b(L.)13 b(M.)f(Delv)o(es)h(&)g(J.)f(L.)h(Mohamed,)f XFg(Computational)j(Metho)n(ds)e(for)h(Inte)n(gr)n(al)f(Equations)p XFo(,)f(Cam-)152 673 y(bridge)17 b(Univ)o(ersit)o(y)e(Press,)g(Cam)o X(bridge,)g(1985.)59 771 y([18])21 b(L.)14 b(M.)e(Delv)o(es)h(&)h(J.)f X(W)l(alsh)g(\(Eds.\),)f Fg(Numeric)n(al)i(Solution)g(of)h(Inte)n(gr)n X(al)d(Equations)p Fo(,)h(Clarendon)152 828 y(Press,)i(Oxford,)g(1974.) X59 926 y([19])21 b(J.)16 b(B.)f(Drak)o(e,)f Fg(ARIES:)i(a)h(c)n X(omputer)g(pr)n(o)n(gr)n(am)f(for)h(the)g(solution)f(of)h(\014rst)f X(kind)g(inte)n(gr)n(al)f(e)n(qua-)152 983 y(tions)e(with)h(noisy)f X(data)p Fo(,)g(Rep)q(ort)g(K/CSD/TM-43,)e(Dept.)g(of)h(Computer)g X(Science,)i(Oak)e(Ridge)152 1039 y(National)k(Lab)q(oratory)l(,)e X(Octob)q(er)i(1983.)59 1138 y([20])21 b(M.)h(P)l(.)g(Ekstrom)f(&)h(R.)g X(L.)h(Rho)q(des,)h Fg(On)e(the)h(applic)n(ation)g(of)f(eigenve)n(ctor)g X(exp)n(ansions)f(to)152 1194 y(numeric)n(al)16 b(de)n(c)n(onvolution)p XFo(,)e(J.)h(Comp.)g(Ph)o(ys.)f Fp(14)i Fo(\(1974\),)d(319-340.)59 X1293 y([21])21 b(L.)12 b(Eld)o(\023)-21 b(en,)13 b Fg(A)g(pr)n(o)n(gr)n X(am)g(for)g(inter)n(active)g(r)n(e)n(gularization)p Fo(,)e(Rep)q(ort)h X(LiTH-MA)l(T-R-79-25,)h(Dept.)152 1350 y(of)i(Mathematics,)g(Link\177) X-23 b(oping)17 b(Univ)o(ersit)o(y)l(,)e(Sw)o(eden,)h(1979.)59 X1448 y([22])21 b(L.)15 b(Eld)o(\023)-21 b(en,)15 b Fg(A)o(lgorithms)g X(for)i(r)n(e)n(gularization)e(of)h(il)r(l-c)n(onditione)n(d)f(le)n X(ast-squar)n(es)f(pr)n(oblems)p Fo(,)g(BIT)152 1505 y XFp(17)i Fo(\(1977\),)d(134{145.)59 1603 y([23])21 b(L.)13 Xb(Eld)o(\023)-21 b(en,)13 b Fg(A)g(weighte)n(d)h(pseudoinverse,)g X(gener)n(alize)n(d)e(singular)h(values,)h(and)f(c)n(onstr)n(aine)n(d)g X(le)n(ast)152 1660 y(squar)n(es)j(pr)n(oblems)p Fo(,)e(BIT)i XFp(22)g Fo(\(1982\),)d(487{501.)59 1758 y([24])21 b(L.)c(Eld)o(\023)-21 Xb(en,)16 b Fg(A)h(note)f(on)h(the)g(c)n(omputation)h(of)g(the)f(gener)n X(alize)n(d)e(cr)n(oss-validation)i(function)f(for)152 X1815 y(il)r(l-c)n(onditione)n(d)g(le)n(ast)f(squar)n(es)h(pr)n(oblems)p XFo(,)e(BIT)i Fp(24)f Fo(\(1984\),)e(467{472.)59 1914 Xy([25])21 b(H.)f(W.)g(Engl)h(&)f(J.)h(Gfrerer,)f Fg(A)h(p)n(osteriori)g X(p)n(ar)n(ameter)g(choic)n(e)g(for)g(gener)n(al)f(r)n(e)n(gularization) X152 1970 y(metho)n(ds)f(for)f(solving)f(line)n(ar)g(il)r(l-p)n(ose)n(d) Xg(pr)n(oblems)p Fo(,)g(App.)g(Numerical)i(Math.)d Fp(4)h XFo(\(1988\),)f(395{)152 2027 y(417.)59 2125 y([26])21 Xb(R.)f(D.)e(Fierro)h(&)g(J.)g(R.)g(Bunc)o(h,)i Fg(Col)r(line)n(arity)e X(and)g(total)i(le)n(ast)d(squar)n(es)p Fo(,)i(Rep)q(ort,)g(Dept.)e(of) X152 2182 y(Mathematics,)j(Univ.)g(of)f(California,)i(San)e(Diego,)i X(1991;)f(to)f(app)q(ear)g(in)h(SIAM)g(J.)f(Matrix)152 X2238 y(Anal.)c(Appl.)59 2337 y([27])21 b(R.)c(D.)g(Fierro,)f(G.)h(H.)f X(Golub,)i(P)l(.)e(C.)h(Hansen)g(&)g(D.)f(P)l(.)h(O'Leary)l(,)g XFg(R)n(e)n(gularization)h(and)f(total)152 2393 y(le)n(ast)f(squar)n(es) Xp Fo(,)e(man)o(uscript)i(in)g(preparation)f(for)f(SIAM)i(J.)f(Sci.)h X(Comput.)59 2492 y([28])21 b(R.)h(Fletc)o(her,)h Fg(Pr)n(actic)n(al)f X(Optimization)g(Metho)n(ds.)g(V)m(ol.)f(1,)j(Unc)n(onstr)n(aine)n(d)c X(Optimization)p Fo(,)152 2548 y(Wiley)l(,)d(Chic)o(hester,)e(1980.)59 X2647 y([29])21 b(G.)15 b(H.)f(Golub)h(&)h(C.)e(F.)g(V)l(an)h(Loan,)g XFg(Matrix)h(Computations)p Fo(,)f(2.)f(Ed.,)g(Johns)h(Hopkins,)h X(Balti-)152 2704 y(more,)f(1989.)59 2802 y([30])21 b(G.)d(H.)g(Golub)h X(&)g(U.)f(v)o(on)g(Matt,)g Fg(Quadr)n(atic)n(al)r(ly)h(c)n(onstr)n X(aine)n(d)f(le)n(ast)g(squar)n(es)h(and)g(quadr)n(atic)152 X2859 y(pr)n(oblems)p Fo(,)c(Numerisc)o(he)h(Mathematik)f XFp(59)g Fo(\(1991\),)e(561{580.)p eop X%%Page: 107 108 X107 107 bop 59 166 a Fm(BIBLIOGRAPHY)1343 b Fo(107)p X59 178 1772 2 v 59 306 a([31])21 b(C.)c(W.)f(Gro)q(etsc)o(h,)h XFg(The)h(The)n(ory)f(of)i(Tikhonov)e(R)n(e)n(gularization)g(for)i(F)m X(r)n(e)n(dholm)e(Equations)h(of)152 362 y(the)f(First)f(Kind)p XFo(,)e(Pitman,)h(Boston,)f(1984.)59 458 y([32])21 b(C.)10 Xb(W.)g(Gro)q(etsc)o(h,)g Fg(Inverse)h(Pr)n(oblems)g(in)g(the)h X(Mathematic)n(al)g(Scienc)n(es)p Fo(,)d(View)o(eg)i(V)l(erlag,)g(Wies-) X152 515 y(baden,)16 b(1993.)59 611 y([33])21 b(C.)g(W.)g(Gro)q(etsc)o X(h)g(&)g(C.)g(R.)h(V)l(ogel,)h Fg(Asymptotic)f(the)n(ory)g(of)g X(\014ltering)f(for)h(line)n(ar)f(op)n(er)n(ator)152 667 Xy(e)n(quations)16 b(with)h(discr)n(ete)f(noisy)f(data)p XFo(,)h(Math.)e(Comp.)h Fp(49)g Fo(\(1987\),)e(499{506.)59 X763 y([34])21 b(J.)13 b(Hadamard,)e Fg(L)n(e)n(ctur)n(es)h(on)h X(Cauchy's)h(Pr)n(oblem)f(in)g(Line)n(ar)f(Partial)i(Di\013er)n(ential)e X(Equations)p Fo(,)152 820 y(Y)l(ale)k(Univ)o(ersit)o(y)g(Press,)f(New)g X(Ha)o(v)o(en,)g(1923.)59 916 y([35])21 b(M.)11 b(Hank)o(e,)h XFg(R)n(e)n(gularization)g(with)h(di\013er)n(ential)f(op)n(er)n(ators.)h X(A)o(n)f(iter)n(ative)g(appr)n(o)n(ach)p Fo(,)h(J.)e(Numer.)152 X972 y(F)l(unct.)16 b(Anal.)f(Optim.)h Fp(13)f Fo(\(1992\),)e(523{540.) X59 1068 y([36])21 b(M.)15 b(Hank)o(e,)g Fg(Iter)n(ative)h(solution)g X(of)h(under)n(determine)n(d)f(line)n(ar)f(systems)h(by)g(tr)n X(ansformation)g(to)152 1125 y(standar)n(d)h(form)p Fo(,)e(submitted)h X(to)f(Num.)g(Lin.)h(Alg.)f(Appl.)59 1221 y([37])21 b(M.)14 Xb(Hank)o(e)h(&)f(P)l(.)h(C.)e(Hansen,)i Fg(R)n(e)n(gularization)g X(Metho)n(ds)g(for)h(L)n(ar)n(ge-Sc)n(ale)e(Pr)n(oblems)p XFo(,)f(Rep)q(ort)152 1277 y(UNIC-92-04,)f(UNI)p Fk(\017)p XFo(C,)e(August)h(1992;)g(to)g(app)q(ear)g(in)h(Surv)o(eys)f(on)g X(Mathematics)g(for)f(Industry)l(.)59 1373 y([38])21 b(P)l(.)f(C.)g X(Hansen,)h Fg(The)f(trunc)n(ate)n(d)h(SVD)f(as)g(a)h(metho)n(d)g(for)g X(r)n(e)n(gularization)p Fo(,)f(BIT)h Fp(27)f Fo(\(1987\),)152 X1430 y(543{553.)59 1526 y([39])h(P)l(.)d(C.)g(Hansen,)h XFg(Computation)g(of)g(the)h(singular)e(value)h(exp)n(ansion)p XFo(,)e(Computing)i Fp(40)f Fo(\(1988\),)152 1582 y(185{199.)59 X1678 y([40])j(P)l(.)12 b(C.)g(Hansen,)h Fg(Perturb)n(ation)h(b)n(ounds) Xf(for)h(discr)n(ete)f(Tikhonov)g(r)n(e)n(gularization)p XFo(,)e(In)o(v)o(erse)i(Prob-)152 1735 y(lems)j Fp(5)g XFo(\(1989\),)d(L41{L44.)59 1831 y([41])21 b(P)l(.)15 Xb(C.)g(Hansen,)g Fg(R)n(e)n(gularization,)g(GSVD)h(and)g(trunc)n(ate)n X(d)g(GSVD)p Fo(,)e(BIT)i Fp(29)f Fo(\(1989\),)e(491{504.)59 X1927 y([42])21 b(P)l(.)28 b(C.)f(Hansen,)32 b Fg(T)m(runc)n(ate)n(d)26 Xb(SVD)i(solutions)f(to)h(discr)n(ete)g(il)r(l-p)n(ose)n(d)f(pr)n X(oblems)h(with)g(il)r(l-)152 1983 y(determine)n(d)17 Xb(numeric)n(al)e(r)n(ank)p Fo(,)g(SIAM)g(J.)g(Sci.)h(Stat.)e(Comput.)h XFp(11)g Fo(\(1990\),)e(503{518.)59 2079 y([43])21 b(P)l(.)16 Xb(C.)f(Hansen,)g Fg(R)n(elations)h(b)n(etwe)n(en)f(SVD)i(and)f(GSVD)h X(of)f(discr)n(ete)g(r)n(e)n(gularization)g(pr)n(oblems)152 X2136 y(in)g(standar)n(d)h(and)f(gener)n(al)f(form)p Fo(,)g(Lin.)h(Alg.) Xg(Appl.)g Fp(141)f Fo(\(1990\),)e(165{176.)59 2232 y([44])21 Xb(P)l(.)c(C.)g(Hansen,)h Fg(The)f(discr)n(ete)h(Pic)n(ar)n(d)g(c)n X(ondition)f(for)h(discr)n(ete)g(il)r(l-p)n(ose)n(d)f(pr)n(oblems)p XFo(,)g(BIT)g Fp(30)152 2288 y Fo(\(1990\),)d(658{672.)59 X2384 y([45])21 b(P)l(.)15 b(C.)g(Hansen,)g Fg(A)o(nalysis)f(of)i(discr) Xn(ete)g(il)r(l-p)n(ose)n(d)f(pr)n(oblems)h(by)g(me)n(ans)f(of)h(the)h X(L-curve)p Fo(,)d(SIAM)152 2441 y(Review)j Fp(34)e Fo(\(1992\),)e X(561{580.)59 2537 y([46])21 b(P)l(.)12 b(C.)f(Hansen,)i XFg(Numeric)n(al)g(to)n(ols)g(for)g(analysis)f(and)i(solution)e(of)i(F)m X(r)n(e)n(dholm)e(inte)n(gr)n(al)g(e)n(quations)152 2593 Xy(of)17 b(the)f(\014rst)g(kind)p Fo(,)f(In)o(v)o(erse)g(Problems)g XFp(8)h Fo(\(1992\),)d(849{872.)59 2689 y([47])21 b(P)l(.)f(C.)g X(Hansen,)h Fg(R)n(e)n(gularization)f(T)m(o)n(ols,)g(a)h(Matlab)f(p)n X(ackage)h(for)g(analysis)f(and)g(solution)h(of)152 2746 Xy(discr)n(ete)e(il)r(l-p)n(ose)n(d)f(pr)n(oblems;)i(V)m(ersion)d(2.0)i X(for)h(Matlab)e(4.0)p Fo(,)h(Rep)q(ort)f(UNIC-92-03,)g(Marc)o(h)152 X2802 y(1993.)c(Av)m(ailable)j(in)e(P)o(ostScript)g(form)f(via)h(Netlib) Xh(\()p Fc(netlib@research.att.co)o(m)p Fo(\))c(from)i(the)152 X2859 y(library)i(NUMERALGO.)p eop X%%Page: 108 109 X108 108 bop 59 166 a Fo(108)1339 b Fm(BIBLIOGRAPHY)p X59 178 1772 2 v 59 306 a Fo([48])21 b(P)l(.)d(C.)f(Hansen)h(&)g(D.)f(P) Xl(.)g(O'Leary)l(,)i Fg(The)f(use)g(of)h(the)g(L-curve)f(in)g(the)h(r)n X(e)n(gularization)f(of)h(dis-)152 362 y(cr)n(ete)h(il)r(l-p)n(ose)n(d)f X(pr)n(oblems)p Fo(,)g(Rep)q(ort)g(CS-TR-2781,)g(Dept.)f(of)h(Computer)f X(Science,)k(Univ.)d(of)152 419 y(Maryland,)c(1991;)f(to)h(app)q(ear)g X(in)h(SIAM)g(J.)f(Sci.)h(Comput.)59 512 y([49])21 b(S.)12 Xb(C.)g(Eisenstat,)g(P)l(.)g(C.)f(Hansen,)i(D.)f(P)l(.)f(O'Leary)i(&)f X(G.)f(W.)h(Stew)o(art,)f Fg(R)n(e)n(gularizing)h(c)n(onjugate)152 X569 y(gr)n(adient)k(iter)n(ations)g(for)h(solving)e(discr)n(ete)g(il)r X(l-p)n(ose)n(d)h(pr)n(oblems)p Fo(,)e(w)o(ork)h(in)h(progress.)59 X663 y([50])21 b(P)l(.)g(C.)f(Hansen,)i(T.)e(Sekii)i(&)f(H.)f X(Shibahashi,)j Fg(The)e(mo)n(di\014e)n(d)g(trunc)n(ate)n(d)g(SVD)g X(metho)n(d)g(for)152 719 y(r)n(e)n(gularization)16 b(is)g(gener)n(al)f X(form)p Fo(,)g(SIAM)h(J.)f(Sci.)h(Stat.)e(Comput.)g Fp(13)i XFo(\(1992\),)d(1142-1150.)59 813 y([51])21 b(B.)g(Hofmann,)h XFg(R)n(e)n(gularization)f(for)h(Applie)n(d)f(Inverse)f(and)i(Il)r X(l-Pose)n(d)e(Pr)n(oblems)p Fo(,)h(T)l(eubner,)152 869 Xy(Leipzig,)c(1986.)59 963 y([52])k(T.)d(Kitaga)o(w)o(a,)f XFg(A)i(deterministic)f(appr)n(o)n(ach)h(to)h(optimal)f(r)n(e)n X(gularization|the)g(\014nite)f(dimen-)152 1019 y(sional)e(c)n(ase)p XFo(,)e(Japan)h(J.)h(Appl.)g(Math.)e Fp(4)h Fo(\(1987\),)e(371{391.)59 X1113 y([53])21 b(R.)16 b(Kress,)f Fg(Line)n(ar)g(Inte)n(gr)n(al)g X(Equations)p Fo(,)f(Springer,)i(Berlin,)g(1989.)59 1207 Xy([54])21 b(C.)14 b(L.)h(La)o(wson)f(&)h(R.)g(J.)f(Hanson,)g XFg(Solving)h(L)n(e)n(ast)f(Squar)n(es)h(Pr)n(oblems)p XFo(,)f(Pren)o(tice-Hall,)i(Engle-)152 1263 y(w)o(o)q(o)q(d)f(Cli\013s,) Xh(1974.)59 1357 y([55])21 b(P)l(.)h(Linz,)j Fg(Unc)n(ertainty)d(in)g X(the)g(solution)h(of)f(line)n(ar)g(op)n(er)n(ator)h(e)n(quations)p XFo(,)g(BIT)g Fp(24)f Fo(\(1984\),)152 1413 y(92{101.)59 X1507 y([56])f Fg(Matlab)c(R)n(efer)n(enc)n(e)d(Guide)p XFo(,)i(The)f(MathW)l(orks,)f(Mass.,)f(1992.)59 1601 y([57])21 Xb(K.)13 b(Miller,)h Fg(L)n(e)n(ast)e(squar)n(es)i(metho)n(ds)f(for)i X(il)r(l-p)n(ose)n(d)e(pr)n(oblems)g(with)h(a)g(pr)n(escrib)n(e)n(d)f(b) Xn(ound)p Fo(,)f(SIAM)152 1657 y(J.)k(Math.)e(Anal.)h XFp(1)g Fo(\(1970\),)f(52{74.)59 1751 y([58])21 b(V.)g(A.)g(Morozo)o(v,) Xg Fg(Metho)n(ds)h(for)g(Solving)f(Inc)n(orr)n(e)n(ctly)f(Pose)n(d)h(Pr) Xn(oblems)p Fo(,)h(Springer)g(V)l(erlag,)152 1807 y(New)16 Xb(Y)l(ork,)e(1984.)59 1901 y([59])21 b(F.)14 b(Natterer,)g XFg(The)h(Mathematics)h(of)g(Computerize)n(d)g(T)m(omo)n(gr)n(aphy)p XFo(,)e(John)h(Wiley)l(,)h(New)e(Y)l(ork,)152 1958 y(1986.)59 X2051 y([60])21 b(F.)15 b(Natterer,)f Fg(Numeric)n(al)i(tr)n(e)n(atment) Xg(of)g(il)r(l-p)n(ose)n(d)g(pr)n(oblems)p Fo(,)e(in)i([67)o(].)59 X2145 y([61])21 b(D.)12 b(P)l(.)f(O'Leary)h(&)g(J.)f(A.)h(Simmons,)g XFg(A)h(bidiagonalization-r)n(e)n(gularization)f(pr)n(o)n(c)n(e)n(dur)n X(e)h(for)g(lar)n(ge)152 2202 y(sc)n(ale)h(discr)n(etizations)g(of)h(il) Xr(l-p)n(ose)n(d)g(pr)n(oblems)p Fo(,)e(SIAM)h(J.)g(Sci.)g(Stat.)f X(Comput.)g Fp(2)h Fo(\(1981\),)e(474{)152 2258 y(489.)59 X2352 y([62])21 b(C.)16 b(C.)g(P)o(aige)g(&)g(M.)g(A.)f(Saunders,)i XFg(LSQR:)f(an)h(algorithm)h(for)f(sp)n(arse)g(line)n(ar)f(e)n(quations) Xh(and)152 2408 y(sp)n(arse)f(le)n(ast)g(squar)n(es)p XFo(,)e(A)o(CM)g(T)l(rans.)h(Math.)f(Soft)o(w)o(are)g XFp(8)h Fo(\(1982\),)e(43{71.)59 2502 y([63])21 b(R.)d(L.)g(P)o(ark)o X(er,)g Fg(Understanding)f(inverse)h(the)n(ory)p Fo(,)g(Ann.)g(Rev.)h X(Earth)e(Planet)h(Sci.)h Fp(5)f Fo(\(1977\),)152 2558 Xy(35{64.)59 2652 y([64])j(D.)15 b(L.)g(Phillips,)i Fg(A)f(te)n(chnique) Xf(for)h(the)h(numeric)n(al)e(solution)h(of)g(c)n(ertain)f(inte)n(gr)n X(al)g(e)n(quations)h(of)152 2708 y(the)h(\014rst)f(kind)p XFo(,)e(J.)h(A)o(CM)g Fp(9)g Fo(\(1962\),)e(84{97.)59 X2802 y([65])21 b(F.)h(San)o(tosa,)h(Y.-H.)f(P)o(ao,)h(W.)f(W.)g(Symes)g X(&)h(C.)f(Holland)i(\(Eds.\),)f Fg(Inverse)e(Pr)n(oblems)i(of)152 X2859 y(A)n(c)n(oustic)16 b(and)g(Elastic)g(Waves)p Fo(,)f(SIAM,)g X(Philadelphia,)j(1984.)p eop X%%Page: 109 110 X109 109 bop 59 166 a Fm(BIBLIOGRAPHY)1343 b Fo(109)p X59 178 1772 2 v 59 306 a([66])21 b(C.)13 b(Ra)o(y)g(Smith)h(&)f(W.)g X(T.)f(Grandy)l(,)h(Jr.)g(\(Eds.\),)f Fg(Maximum-Entr)n(opy)k(and)e X(Bayesian)g(Metho)n(ds)152 362 y(in)i(Inverse)f(Pr)n(oblems)p XFo(,)f(Reidel,)j(Boston,)d(1985.)59 457 y([67])21 b(G.)c(T)l(alen)o X(ti,)h Fg(Inverse)f(Pr)n(oblems)p Fo(,)g(Lecture)h(Notes)f(in)h X(Mathematics)f(1225,)f(Springer)j(V)l(erlag,)152 514 Xy(Berlin,)e(1986.)59 609 y([68])k(H.)12 b(J.)f(J.)h(te)f(Riele,)j XFg(A)f(pr)n(o)n(gr)n(am)g(for)h(solving)d(\014rst)i(kind)f(F)m(r)n(e)n X(dholm)g(inte)n(gr)n(al)g(e)n(quations)h(by)g(me)n(ans)152 X665 y(of)k(r)n(e)n(gularization)p Fo(,)d(Computer)h(Ph)o(ysics)h(Comm.) Xe Fp(36)h Fo(\(1985\),)e(423{432.)59 760 y([69])21 b(A.)e(N.)g(Tikhono) Xo(v,)h Fg(Solution)f(of)h(inc)n(orr)n(e)n(ctly)f(formulate)n(d)h(pr)n X(oblems)f(and)h(the)g(r)n(e)n(gularization)152 816 y(metho)n(d)p XFo(,)13 b(Dokl.)e(Ak)m(ad.)g(Nauk.)h(SSSR)g Fp(151)g XFo(\(1963\),)e(501{504)f(=)j(So)o(viet)g(Math.)e(Dokl.)h XFp(4)h Fo(\(1963\),)152 873 y(1035{1038.)59 968 y([70])21 Xb(A.)16 b(N.)g(Tikhono)o(v)g(&)g(V.)f(Y.)h(Arsenin,)h XFg(Solutions)f(of)h(Il)r(l-Pose)n(d)f(Pr)n(oblems)p Fo(,)f(Winston)h(&) Xg(Sons,)152 1024 y(W)l(ashington,)f(D.C.,)f(1977.)59 X1119 y([71])21 b(A.)f(N.)g(Tikhono)o(v)g(&)h(A.)f(V.)g(Gonc)o(harsky)l X(,)g Fg(Il)r(l-Pose)n(d)g(Pr)n(oblems)g(in)g(the)h(Natur)n(al)g(Scienc) Xn(es)p Fo(,)152 1176 y(MIR)16 b(Publishers,)h(Mosco)o(w,)c(1987.)59 X1271 y([72])21 b(A.)15 b(v)m(an)h(der)g(Sluis,)g Fg(The)g(c)n(onver)n X(genc)n(e)e(b)n(ehavior)j(of)f(c)n(onjugate)g(gr)n(adients)g(and)g(R)o X(itz)g(values)g(in)152 1327 y(various)k(cir)n(cumstanc)n(es)p XFo(;)e(in)h(R.)g(Beau)o(w)o(ens)f(&)h(P)l(.)f(de)h(Gro)q(en)f X(\(Eds.\),)g Fg(Iter)n(ative)g(Metho)n(ds)h(in)152 1383 Xy(Line)n(ar)d(A)o(lgebr)n(a)p Fo(,)e(North-Holland,)h(Amsterdam,)g X(1992.)59 1478 y([73])21 b(A.)e(v)m(an)h(der)f(Sluis)i(&)f(H.)e(A.)h(v) Xm(an)h(der)f(V)l(orst,)g Fg(SIR)m(T-)g(and)h(CG-typ)n(e)g(metho)n(ds)g X(for)g(iter)n(ative)152 1535 y(solution)15 b(of)h(sp)n(arse)e(line)n X(ar)g(le)n(ast-squar)n(es)g(pr)n(oblems)p Fo(,)g(Lin.)h(Alg.)f(Appl.)h XFp(130)f Fo(\(1990\),)e(257{302.)59 1630 y([74])21 b(J.)c(M.)g(V)l X(arah,)f Fg(On)i(the)g(numeric)n(al)f(solution)h(of)g(il)r(l-c)n X(onditione)n(d)f(line)n(ar)g(systems)g(with)h(appli-)152 X1686 y(c)n(ations)e(to)h(il)r(l-p)n(ose)n(d)e(pr)n(oblems)p XFo(,)f(SIAM)i(J.)f(Numer.)g(Anal.)h Fp(10)f Fo(\(1973\),)e(257{267.)59 X1781 y([75])21 b(J.)c(M.)e(V)l(arah,)h Fg(A)h(pr)n(actic)n(al)g X(examination)g(of)g(some)g(numeric)n(al)g(metho)n(ds)g(for)h(line)n(ar) Xe(discr)n(ete)152 1838 y(il)r(l-p)n(ose)n(d)g(pr)n(oblems)p XFo(,)e(SIAM)i(Rev.)f Fp(21)h Fo(\(1979\),)d(100{111.)59 X1932 y([76])21 b(J.)c(M.)e(V)l(arah,)h Fg(Pitfal)r(ls)g(in)h(the)g X(numeric)n(al)g(solution)g(of)g(line)n(ar)f(il)r(l-p)n(ose)n(d)h(pr)n X(oblems)p Fo(,)e(SIAM)i(J.)152 1989 y(Sci.)f(Stat.)e(Comput.)h XFp(4)g Fo(\(1983\),)e(164{176.)59 2084 y([77])21 b(C.)c(R.)g(V)l(ogel,) Xh Fg(Optimal)g(choic)n(e)g(of)g(a)h(trunc)n(ation)e(level)g(for)i(the)f X(trunc)n(ate)n(d)g(SVD)g(solution)g(of)152 2140 y(line)n(ar)h(\014rst)g X(kind)h(inte)n(gr)n(al)e(e)n(quations)h(when)h(data)g(ar)n(e)g(noisy)p XFo(,)f(SIAM)g(J.)g(Numer.)f(Anal.)h Fp(23)152 2197 y XFo(\(1986\),)14 b(109{117.)59 2292 y([78])21 b(C.)27 Xb(R.)g(V)l(ogel,)j Fg(Solving)25 b(il)r(l-c)n(onditione)n(d)h(line)n X(ar)g(systems)g(using)h(the)g(c)n(onjugate)g(gr)n(adient)152 X2348 y(metho)n(d)p Fo(,)16 b(Rep)q(ort,)f(Dept.)g(of)f(Mathematical)i X(Sciences,)g(Mon)o(tana)e(State)h(Univ)o(ersit)o(y)l(,)g(1987.)59 X2443 y([79])21 b(G.)c(W)l(ah)o(ba,)g Fg(Spline)g(Mo)n(dels)g(for)i X(Observational)e(Data)p Fo(,)h(CBMS-NSF)g(Regional)g(Conference)152 X2499 y(Series)f(in)f(Applied)h(Mathematics,)d(V)l(ol.)i(59,)e(SIAM,)h X(Philadelphi)q(a,)i(1990.)59 2594 y([80])k(G.)c(M.)g(Wing,)h XFg(Condition)g(numb)n(ers)f(of)i(matric)n(es)f(arising)f(fr)n(om)i(the) Xg(numeric)n(al)f(solution)g(of)152 2651 y(line)n(ar)j(inte)n(gr)n(al)e X(e)n(quations)i(of)g(the)h(\014rst)e(kind)p Fo(,)h(J.)f(In)o(tegral)h X(Equations)f Fp(9)h Fo(\(Suppl.\))g(\(1985\),)152 2707 Xy(191{204.)59 2802 y([81])g(G.)13 b(M.)g(Wing)g(&)h(J.)f(D.)g(Zahrt,)f XFg(A)j(Primer)g(on)f(Inte)n(gr)n(al)f(Equations)h(of)h(the)g(First)f X(Kind)p Fo(,)f(SIAM,)152 2859 y(Philadelphi)q(a,)k(1991.)p Xeop X%%Page: 110 111 X110 110 bop 59 166 a Fo(110)1339 b Fm(BIBLIOGRAPHY)p X59 178 1772 2 v 59 306 a Fo([82])21 b(H.)16 b(Zha)g(&)h(P)l(.)f(C.)f X(Hansen,)i Fg(R)n(e)n(gularization)f(and)i(the)f(gener)n(al)f X(Gauss-Markov)i(line)n(ar)e(mo)n(del)p Fo(,)152 362 y(Math.)f(Comp.)f XFp(55)h Fo(\(1990\),)f(613{624.)p eop X%%Trailer Xend Xuserdict /end-hook known{end-hook}if X%%EOF END_OF_FILE if test 893990 -ne `wc -c <'Manual.ps'`; then echo shar: \"'Manual.ps'\" unpacked with wrong size! fi # end of 'Manual.ps' fi if test -f 'Contents.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'Contents.m'\" else echo shar: Extracting \"'Contents.m'\" \(4792 characters\) sed "s/^X//" >'Contents.m' <<'END_OF_FILE' X% Regularization Tools. X% Version 2.1 16-March-93, Revised 31-March-98. X% Copyright (c) 1993 by Per Christian Hansen and UNI-C. X% X% Demonstration. X% regudemo - Tutorial introduction to Regularization Tools. X% X% Test examples. X% baart - Fredholm integral equation of the first kind. X% deriv2 - Computation of the second derivative. X% foxgood - Severely ill-posed problem. X% heat - Inverse heat equation. X% ilaplace - Inverse Laplace transformation. X% parallax - Stellar parallax problem with 28 fixed observations. X% phillips - Philips' "famous" test problem. X% shaw - One-dimensional image restoration problem. X% spikes - Test problem with a "spiky" solution. X% ursell - Integral equation with no square integrable solution. X% wing - Test problem with a discontinuous solution. X% X% Regularization routines. X% cgls - Computes the least squares solution based on k steps X% of the conjugate gradient algorithm. X% discrep - Minimizes the solution (semi-)norm subject to an upper X% bound on the residual norm (discrepancy principle). X% dsvd - Computes a damped SVD/GSVD solution. X% lsqi - Minimizes the residual norm subject to an upper bound X% on the (semi-)norm of the solution. X% lsqr - Computes the least squares solution based on k steps X% of the LSQR algorithm (Lanczos bidiagonalization). X% maxent - Computes the maximum entropy regularized solution. X% mtsvd - Computes the modified TSVD solution. X% nu - Computes the solution based on k steps of Brakhage's X% iterative nu-method. X% pcgls - Same as cgls, but for general-form regularization. X% plsqr - Same as lsqr, but for general-form regularization. X% pnu - Same as nu, but for general-form regularization. X% tgsvd - Computes the truncated GSVD solution. X% tikhonov - Computes the Tikhonov regularized solution. X% tsvd - Computes the truncated SVD solution. X% ttls - Computes the truncated TLS solution. X% X% Analysis routines. X% fil_fac - Computes filter factors for some regularization methods. X% gcv - Plots the GCV function and computes its minimum. X% l_corner - Locates the L-shaped corner of the L-curve. X% l_curve - Computes the L-curve, plots it, and computes its corner. X% lagrange - Plots the Lagrange function ||Ax-b||^2 + lambda^2*||Lx||^, X% and its derivative. X% picard - Plots the (generalized) singular values, X% the Fourier coefficient for the right-hand side, and a X% (smoothed curve of) the solution Fourier-coefficients. X% plot_lc - Plots an L-curve. X% quasiopt - Plots the quasi-optimality function and computes its minimum. X% X% Routines for transforming a problem in general form into one in X% standard form, and back again. X% gen_form - Transforms a standard-form solution back into the X% general-form setting. X% std_form - Transforms a general-form problem into one in X% standard form. X% X% Utility routines. X% bidiag - Bidiagonalization of a matrix by Householder transformations. X% bsvd - Computes the singular values, or the compact SVD, X% of a bidiagonal matrix stored in compact form. X% csdecomp - Computes the CS decomposition. X% csvd - Computes the compact SVD of an m-by-n matrix. X% get_l - Produces a p-by-n matrix which is the discrete X% approximation to the d'th order derivative operator. X% gsvd - Computes the generalized SVD of a matrix pair. X% lanc_b - Performs k steps of the Lanczos bidiagonalization X% process with/without reorthogonalization. X% X% Auxiliary routines required by some of the above routines. X% app_hh_l - Applies a Householder transformation from the left. X% gen_hh - Generates a Householder transformation. X% heb_new - Newton-Raphson iteration with Hebden's rational X% approximation, used in lsqi. X% heb_new2 - Ditto, used in discrep. X% lsolve - Inversion with A-weighted generalized inverse of L. X% ltsolve - Inversion with transposed A-weighted inverse of L. X% mgs - Modified Gram-Schmidt orthonormalization. X% newton - Newton-Raphson iteration, used in discrep. X% pinit - Initialization for treating general-form problems. X% pythag - Computes sqrt(a^2 + b^2). X% spleval - Computes points on a spline or spline curve. X% X% The routine l_corner requires the following routines from the Spline X% Toolbox: X% fnder, ppbrk, ppcut, ppmak, sp2pp, spbrk, spmak. X% If the Spline Toolbox is not available, then dummy functions with these X% names should reside in the same directory as Regularization Tools. END_OF_FILE if test 4792 -ne `wc -c <'Contents.m'`; then echo shar: \"'Contents.m'\" unpacked with wrong size! fi # end of 'Contents.m' fi if test -f 'app_hh.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'app_hh.m'\" else echo shar: Extracting \"'app_hh.m'\" \(304 characters\) sed "s/^X//" >'app_hh.m' <<'END_OF_FILE' Xfunction A = app_hh(A,beta,v) X%APP_HH Apply a Householder transformation. X% X% A = app_hh(A,beta,v) X% X% Applies the Householder transformation, defined by X% vector v and scaler beta, to the matrix A; i.e. X% A = (eye - beta*v*v')*A . X X% Per Christian Hansen, UNI-C, 03/11/92. X XA = A - (beta*v)*(v'*A); END_OF_FILE if test 304 -ne `wc -c <'app_hh.m'`; then echo shar: \"'app_hh.m'\" unpacked with wrong size! fi # end of 'app_hh.m' fi if test -f 'baart.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'baart.m'\" else echo shar: Extracting \"'baart.m'\" \(1577 characters\) sed "s/^X//" >'baart.m' <<'END_OF_FILE' Xfunction [A,b,x] = baart(n) X%BAART Test problem: Fredholm integral equation of the first kind. X% X% [A,b,x] = baart(n) X% X% Discretization of a first kind Fredholm integral equation with X% kernel K and right-hand side g given by X% K(s,t) = exp(s*cos(t)) , g(s) = 2*sinh(s)/s , X% and with integration intervals s in [0,pi/2] , t in [0,pi] . X% The solution is given by X% f(t) = sin(t) . X% X% The order n must be even. X X% Reference: M. L. Baart, "The use of auto-correlation for pseudo- X% rank determination in noisy ill-conditioned linear least-squares X% problems", IMA J. Numer. Anal. 2 (1982), 241-247. X X% Discretized by Galerkin method with orthonormal box functions; X% one integration is exact, the other is done by Simpson's rule. X X% Per Christian Hansen, UNI-C, 09/16/92. X X% Check input. Xif (rem(n,2)~=0), error('The order n must be even'), end X X% Generate the matrix. Xhs = pi/(2*n); ht = pi/n; c = 1/(3*sqrt(2)); XA = zeros(n,n); ihs = [0:n]'*hs; n1 = n+1; nh = n/2; Xf3 = exp(ihs(2:n1)) - exp(ihs(1:n)); Xfor j=1:n X f1 = f3; co2 = cos((j-.5)*ht); co3 = cos(j*ht); X f2 = (exp(ihs(2:n1)*co2) - exp(ihs(1:n)*co2))/co2; X if (j==nh) X f3 = hs*ones(n,1); X else X f3 = (exp(ihs(2:n1)*co3) - exp(ihs(1:n)*co3))/co3; X end X A(:,j) = c*(f1 + 4*f2 + f3); Xend X X% Generate the right-hand side. Xif (nargout>1) X si(1:2*n) = [.5:.5:n]'*hs; si = sinh(si)./si; X b = zeros(n,1); X b(1) = 1 + 4*si(1) + si(2); X b(2:n) = si(2:2:2*n-2) + 4*si(3:2:2*n-1) + si(4:2:2*n); X b = b*sqrt(hs)/3; Xend X X% Generate the solution. Xif (nargout==3) X x = -diff(cos([0:n]'*ht))/sqrt(ht); Xend END_OF_FILE if test 1577 -ne `wc -c <'baart.m'`; then echo shar: \"'baart.m'\" unpacked with wrong size! fi # end of 'baart.m' fi if test -f 'bidiag.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'bidiag.m'\" else echo shar: Extracting \"'bidiag.m'\" \(2059 characters\) sed "s/^X//" >'bidiag.m' <<'END_OF_FILE' Xfunction [U,B_n,V] = bidiag(A) X%BIDIAG Bidiagonalization of an m-times-n matrix with m >= n. X% X% B_n = bidiag(A) X% [U,B_n,V] = bidiag(A) X% X% Computes the bidiagonalization of the m-times-n matrix A X% with m >= n: X% A = U*B*V' , X% where B is an upper bidiagonal n-times-n matrix, and U and X% V have orthogonal columns. X% X% The matrix B is stored in compact form in B_n as follows: X% [b_11 b_12 ] [b_11 b_12] X% B = [ b_22 b_23 ] B_n = [b_22 b_23] . X% [ . . ] [ . . ] X% [ b_nn ] [b_nn NaN ] X% The NaN in B_n(n,2) is used to distinguish a "compact" upper X% bidiagonal matrix from a "compact" lower bidiagonal one. X X% Reference: L. Elden, "Algorithms for regularization of ill- X% conditioned least-squares problems", BIT 17 (1977), 134-145. X X% Per Christian Hansen, UNI-C, 03/11/92. X X% Initialization. X[m,n] = size(A); Xif (m 1), U = [eye(n);zeros(m-n,n)]; betaU = zeros(n,1); end Xif (nargout==3), V = eye(n); betaV = zeros(n,1); end X X% Bidiagonalization; save Householder quantities. Xif (m > n), k_last = n; else k_last = n-1; end Xfor k=1:k_last X X [B_n(k,1),beta,A(k:m,k)] = gen_hh(A(k:m,k)); X if (k < n), A(k:m,k+1:n) = app_hh(A(k:m,k+1:n),beta,A(k:m,k)); end X if (nargout>1), betaU(k) = beta; end X X if (k < n-1) X [B_n(k,2),beta,v] = gen_hh(A(k,k+1:n)'); A(k,k+1:n) = v'; X A(k+1:m,k+1:n) = app_hh(A(k+1:m,k+1:n)',beta,A(k,k+1:n)')'; X if (nargout==3), betaV(k) = beta;, end X elseif (k == n-1) X B_n(n-1,2) = A(n-1,n); X end X Xend X X% Save bottom element if A is square. Xif (k_last < n), B_n(n,1) = A(n,n); end X X% Put a NaN in bottom element of B_n. XB_n(n,2) = NaN; X X% Compute U if wanted. Xif (nargout>1) X for k=k_last:-1:1 X U(k:m,k:n) = app_hh(U(k:m,k:n),betaU(k),A(k:m,k)); X end Xend X X% Compute V if wanted. Xif (nargout==3) X for k=n-2:-1:1 X V(k+1:n,k:n) = app_hh(V(k+1:n,k:n),betaV(k),A(k,k+1:n)'); X end Xend X Xif (nargout==1), U = B_n; end END_OF_FILE if test 2059 -ne `wc -c <'bidiag.m'`; then echo shar: \"'bidiag.m'\" unpacked with wrong size! fi # end of 'bidiag.m' fi if test -f 'bsvd.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'bsvd.m'\" else echo shar: Extracting \"'bsvd.m'\" \(1091 characters\) sed "s/^X//" >'bsvd.m' <<'END_OF_FILE' Xfunction [U,s,V] = bsvd(B_k) X%BSVD SVD of a bidiagonal matrix stored in "compact form". X% X% s = bsvd(B_k) X% [U,s,V] = bsvd(B_k) X% X% Computes the singular values, or the compact SVD, of the X% bidiagonal matrix B stored in compact form in B_k. X% X% If the bottom right element of B_k is a NAN, then B_k repre- X% sents an upper bidiagonal matrix (such as produced by bidiag), X% stored with its diagonal and upper bidiagonal in the first and X% second columns of B_k, repsectively. X% X% Otherwise, B_k represents a lower bidiagonal matrix (such as X% produced by lanc_b), stored with its lower bidiagonal and its X% diagonal in the first and second columns of B_k, respectively. X X% Per Christian Hansen, UNI-C, 03/11/92. X X% Initialization. X[k,l] = size(B_k); Xif (l~=2), error('B_k does not represent a bidiagonal matrix'), end X X% Determine which bidiagonal form. Xif (B_k(k,2)==NaN) X B = diag(B_k(:,1)) + diag(B_k(1:k-1,2),1); Xelse X B = diag(B_k(:,1),-1) + diag([B_k(:,2);0]); X [k1,k1] = size(B); B = B(:,1:k1-1); Xend X X% Compute the SVD. Xif (nargout<=1) X U = svd(B); Xelse X [U,s,V] = csvd(B); Xend END_OF_FILE if test 1091 -ne `wc -c <'bsvd.m'`; then echo shar: \"'bsvd.m'\" unpacked with wrong size! fi # end of 'bsvd.m' fi if test -f 'cgls.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'cgls.m'\" else echo shar: Extracting \"'cgls.m'\" \(2084 characters\) sed "s/^X//" >'cgls.m' <<'END_OF_FILE' Xfunction [X,rho,eta,F] = cgls(A,b,k,s) X%CGLS Conjugate gradient algorithm applied implicitly to the normal equations. X% X% [X,rho,eta,F] = cgls(A,b,k,s) X% X% Performs k steps of the conjugate gradient algorithm applied X% implicitly to the normal equations A'*A*x = A'*b. X% X% The routine returns all k solutions, stored as columns of X% the matrix X. The solution norm and residual norm are returned X% in eta and rho, respectively. X% X% If the singular values s are also provided, cgls computes the X% filter factors associated with each step and stores them X% columnwise in the matrix F. X X% References: A. Bjorck, "Least Squares Methods", in P. G. X% Ciarlet & J. L Lions (Eds.), "Handbook of Numerical Analysis, X% Vol. I", Elsevier, Amsterdam, 1990; p. 560. X% C. R. Vogel, "Solving ill-conditioned linear systems using the X% conjugate gradient method", Report, Dept. of Mathematical X% Sciences, Montana State University, 1987. X X% Per Christian Hansen, UNI-C, 03/31/98. X X% The fudge threshold is used to prevent filter factors from exploding. Xfudge_thr = 1e-4; X X% Initialization. Xif (k < 1), error('Number of steps k must be positive'), end X[m,n] = size(A); X = zeros(n,k); Xif (nargout > 1) X eta = zeros(k,1); rho = eta; Xend Xif (nargout==4 & nargin==3), error('Too few imput arguments'), end Xif (nargin==4) X F = zeros(n,k); Fd = zeros(n,1); s2 = s.^2; Xend X X% Prepare for CG iteration. Xx = zeros(n,1); Xd = A'*b; Xr = b; Xnormr2 = d'*d; X X% Iterate. Xfor j=1:k X X Ad = A*d; alpha = normr2/(Ad'*Ad); X x = x + alpha*d; X r = r - alpha*Ad; X s = A'*r; X normr2_new = s'*s; X beta = normr2_new/normr2; X normr2 = normr2_new; X d = s + beta*d; X X(:,j) = x; X if (nargout>1), rho(j) = norm(r); end X if (nargout>2), eta(j) = norm(x); end X X if (nargin==4) X if (j==1) X F(:,1) = alpha*s2; X Fd = s2 - s2.*F(:,1) + beta*s2; X else X F(:,j) = F(:,j-1) + alpha*Fd; X Fd = s2 - s2.*F(:,j) + beta*Fd; X end X if (j > 2) X f = find(abs(F(:,j-1)-1) < fudge_thr & abs(F(:,j-2)-1) < fudge_thr); X if (length(f) > 0), F(f,j) = ones(length(f),1); end X end X end X Xend END_OF_FILE if test 2084 -ne `wc -c <'cgls.m'`; then echo shar: \"'cgls.m'\" unpacked with wrong size! fi # end of 'cgls.m' fi if test -f 'csdecomp.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'csdecomp.m'\" else echo shar: Extracting \"'csdecomp.m'\" \(1794 characters\) sed "s/^X//" >'csdecomp.m' <<'END_OF_FILE' Xfunction [U1,U2,cs,V] = csdecomp(Q1,Q2) X%CSD CS decomposition. X% cs = csdecomp(Q1,Q2) X% [U1,U2,cs,V] = csdecomp(Q1,Q2) , cs = [c,s] X% X% Computes the CS decomposition X% [ Q1 ] = [ U1 0 ]*[ diag(c) 0 ]*V' X% [ Q2 ] [ 0 U2 ] [ 0 eye(n-p) ] X% [ diag(s) 0 ] X% of a matrix with orthonormal columns. The number of rows in Q1 X% must be greater than or equal to the number of rows in Q2. X% U1 is m-by-n , c is p-by-1 X% U2 is p-by-p , s is p-by-1 X% V is n-by-n . X X% Reference: C. F. Van Loan, "Computing the CS and the generalized X% singular value decomposition", Numer. Math. 46 (1985), 479-491. X X% Per Christian Hansen, IMM, 04/17/97. X X% Initialization. X[m,n1] = size(Q1); [p,n] = size(Q2); Xif (m n-p+k), R(i,:) = -R(i,:); end X U1(:,n+1-i) = -U1(:,n+1-i); X end Xend Xc(p:-1:pk+1) = abs(diag(R(n-p+1:n-pk,n-p+1:n-pk))); XR = R(n:-1:n-pk+1,n:-1:n-pk+1); X X% Compute c, U1 and V. X[U1t,gamma,Vt] = svd(R); Xc(pk:-1:1) = diag(gamma); clear gamma; XU1(:,1:pk) = U1(:,1:pk)*U1t(:,pk:-1:1); XV(:,1:pk) = V(:,1:pk)*Vt(:,pk:-1:1); XR = Vt; for i=1:pk, R(i,:) = s(i)*R(i,pk:-1:1); end X X% Compute s and U2. XU2t = zeros(pk,pk); Xfor i=1:pk, X s(i) = norm(R(:,i)); X U2t(:,i) = R(:,i)/s(i); Xend XU2(:,1:pk) = U2(:,1:pk)*U2t; X X% Make sure that c and s do not exceed one. Xix = find(c>1); c(ix) = ones(length(ix),1); Xix = find(s>1); s(ix) = ones(length(ix),1); X X% Return the desired quantities. Xcs = [c,s]; Xif (nargout < 2), U1 = cs; end END_OF_FILE if test 1794 -ne `wc -c <'csdecomp.m'`; then echo shar: \"'csdecomp.m'\" unpacked with wrong size! fi # end of 'csdecomp.m' fi if test -f 'csvd.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'csvd.m'\" else echo shar: Extracting \"'csvd.m'\" \(724 characters\) sed "s/^X//" >'csvd.m' <<'END_OF_FILE' Xfunction [U,s,V] = csvd(A,tst) X%CSVD Compact singular value decomposition. X% X% s = csvd(A) X% [U,s,V] = csvd(A) X% [U,s,V] = csvd(A,'full') X% X% Computes the compact form of the SVD of A: X% A = U*diag(s)*V', X% where X% U is m-by-min(m,n) X% s is min(m,n)-by-1 X% V is n-by-min(m,n). X% X% If a second argument is present, the full U and V are returned. X X% Per Christian Hansen, UNI-C, 06/22/93. X Xif (nargin==1) X if (nargout > 1) X [m,n] = size(A); X if (m >= n) X [U,s,V] = svd(full(A),0); s = diag(s); X else X [V,s,U] = svd(full(A)',0); s = diag(s); X end X else X U = svd(full(A)); X end Xelse X if (nargout > 1) X [U,s,V] = svd(full(A)); s = diag(s); X else X U = svd(full(A)); X end Xend END_OF_FILE if test 724 -ne `wc -c <'csvd.m'`; then echo shar: \"'csvd.m'\" unpacked with wrong size! fi # end of 'csvd.m' fi if test -f 'deriv2.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'deriv2.m'\" else echo shar: Extracting \"'deriv2.m'\" \(2671 characters\) sed "s/^X//" >'deriv2.m' <<'END_OF_FILE' Xfunction [A,b,x] = deriv2(n,case) X%DERIV2 Test problem: computation of the second derivative. X% X% [A,b,x] = deriv2(n,case) X% X% This is a mildly ill-posed problem. It is a discretization of a X% first kind Fredholm integral equation whose kernel K is the X% Green's function for the second derivative: X% K(s,t) = | s(t-1) , s < t . X% | t(s-1) , s >= t X% Both integration intervals are [0,1], and as right-hand side g X% and correspond solution f one can choose between the following: X% case = 1 : g(s) = (s^3 - s)/6 , f(t) = t X% case = 2 : g(s) = exp(s) + (1-e)s - 1 , f(t) = exp(t) X% case = 3 : g(s) = | (4s^3 - 3s)/24 , s < 0.5 X% | (-4s^3 + 12s^2 - 9s + 1)/24 , s >= 0.5 X% f(t) = | t , t < 0.5 X% | 1-t , t >= 0.5 X X% References. The first two examples are from L. M. Delves & J. L. X% Mohamed, "Computational Methods for Integral Equations", Cambridge X% University Press, 1985; p. 310. The third example is from A. K. X% Louis & P. Maass, "A mollifier method for linear operator equations X% of the first kind", Inverse Problems 6 (1990), 427-440. X X% Discretized by Galerkin method with orthonormal box functions. X X% Per Christian Hansen, UNI-C, 05/28/93. X X% Initialization. Xif (nargin==1), case = 1; end Xh = 1/n; sqh = sqrt(h); h32 = h*sqh; h2 = h^2; sqhi = 1/sqh; Xt = 2/3; A = zeros(n,n); X X% Compute the matrix A. Xfor i=1:n X A(i,i) = h2*((i^2 - i + 0.25)*h - (i - t)); X for j=1:i-1 X A(i,j) = h2*(j-0.5)*((i-0.5)*h-1); X end Xend XA = A + tril(A,-1)'; X X% Compute the right-hand side vector b. Xif (nargout>1) X b = zeros(n,1); X if (case==1) X for i=1:n X b(i) = h32*(i-0.5)*((i^2 + (i-1)^2)*h2/2 - 1)/6; X end X elseif (case==2) X ee = 1 - exp(1); X for i=1:n X b(i) = sqhi*(exp(i*h) - exp((i-1)*h) + ee*(i-0.5)*h2 - h); X end X elseif (case==3) X if (rem(n,2)~=0), error('Order n must be even'), else X for i=1:n/2 X s12 = (i*h)^2; s22 = ((i-1)*h)^2; X b(i) = sqhi*(s12 + s22 - 1.5)*(s12 - s22)/24; X end X for i=n/2+1:n X s1 = i*h; s12 = s1^2; s2 = (i-1)*h; s22 = s2^2; X b(i) = sqhi*(-(s12+s22)*(s12-s22) + 4*(s1^3 - s2^3) - ... X 4.5*(s12 - s22) + h)/24; X end X end X else X error('Illegal value of case') X end Xend X X% Compute the solution vector x. Xif (nargout==3) X x = zeros(n,1); X if (case==1) X for i=1:n, x(i) = h32*(i-0.5); end X elseif(case==2) X for i=1:n, x(i) = sqhi*(exp(i*h) - exp((i-1)*h)); end X else X for i=1:n/2, x(i) = sqhi*((i*h)^2 - ((i-1)*h)^2)/2; end X for i=n/2+1:n, x(i) = sqhi*(h - ((i*h)^2 - ((i-1)*h)^2)/2); end X end Xend END_OF_FILE if test 2671 -ne `wc -c <'deriv2.m'`; then echo shar: \"'deriv2.m'\" unpacked with wrong size! fi # end of 'deriv2.m' fi if test -f 'discrep.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'discrep.m'\" else echo shar: Extracting \"'discrep.m'\" \(2756 characters\) sed "s/^X//" >'discrep.m' <<'END_OF_FILE' Xfunction [x_delta,lambda] = discrep(U,s,V,b,delta,x_0) X%DISCREP Discrepancy principle criterion for choosing the reg. parameter. X% X% [x_delta,lambda] = discrep(U,s,V,b,delta,x_0) X% [x_delta,lambda] = discrep(U,sm,X,b,delta,x_0) , sm = [sigma,mu] X% X% Least squares minimization with a quadratic inequality constraint: X% min || x - x_0 || subject to || A x - b || <= delta X% min || L (x - x_0) || subject to || A x - b || <= delta X% where x_0 is an initial guess of the solution, and delta is a X% positive constant. Requires either the compact SVD of A saved as X% U, s, and V, or part of the GSVD of (A,L) saved as U, sm, and X. X% The regularization parameter lambda is also returned. X% X% If delta is a vector, then x_delta is a matrix such that X% x_delta = [ x_delta(1), x_delta(2), ... ] . X% X% If x_0 is not specified, x_0 = 0 is used. X X% Reference: V. A. Morozov, "Methods for Solving Incorrectly Posed X% Problems", Springer, 1984; Chapter 26. X X% Per Christian Hansen, UNI-C, 02/20/92. X X% Initialization. X[n,p] = size(V); [p,ps] = size(s); ld = length(delta); Xx_k = zeros(n,ld); lambda = zeros(ld,1); rho = zeros(p,1); Xif (min(delta)<0) X error('Illegal inequality constraint delta') Xend Xif (nargin==5), x_0 = zeros(n,1); end Xif (ps == 1), omega = V'*x_0; else, omega = V\x_0; end X X% Compute residual norms corresponding to TSVD/TGSVD. Xbeta = U'*b; Xnb = norm(b); Xsnz = length(find(s(:,1)>0)); Xif (ps == 1) X delta_0 = norm(b - U*beta); X rho(n) = delta_0^2; X for i=n:-1:2 X rho(i-1) = rho(i) + (beta(i) - s(i)*omega(i))^2; X end Xelse X delta_0 = norm(b - U*beta); X rho(1) = delta_0^2; X for i=1:p-1 X rho(i+1) = rho(i) + (beta(i) - s(i,1)*omega(i))^2; X end Xend X X% Check input. Xif (min(delta) < delta_0) X error('Irrelevant delta < || (I - U*U'')*b ||') Xend X Xif (ps == 1) X s2 = s.^2; X for k=1:ld X if (delta(k)^2 >= norm(beta - s.*omega)^2 + delta_0^2) X x_delta(:,k) = x_0; X else X [dummy,kmin] = min(abs(rho - delta(k)^2)); X lambda_0 = s(kmin); X lambda(k) = newton(lambda_0,delta(k),s,beta,omega,delta_0); X e = s./(s2 + lambda(k)^2); f = s.*e; X x_delta(:,k) = V(:,1:p)*(e.*beta + (1-f).*omega); X end X end Xelse X omega = omega(1:p); gamma = s(:,1)./s(:,2); X x_u = V(:,p+1:n)*beta(p+1:n); X for k=1:ld X if (delta(k)^2 >= norm(beta(1:p) - s(:,1).*omega)^2 + delta_0^2) X x_delta(:,k) = V*[omega;U(:,p+1:n)'*b]; X else X [dummy,kmin] = min(abs(rho - delta(k)^2)); X lambda_0 = gamma(kmin); X lambda(k) = newton(lambda_0,delta(k),s,beta(1:p),omega,delta_0); X e = gamma./(gamma.^2 + lambda(k)^2); f = gamma.*e; X x_delta(:,k) = V(:,1:p)*(e.*beta(1:p)./s(:,2) + ... X (1-f).*s(:,2).*omega) + x_u; X end X end Xend END_OF_FILE if test 2756 -ne `wc -c <'discrep.m'`; then echo shar: \"'discrep.m'\" unpacked with wrong size! fi # end of 'discrep.m' fi if test -f 'dsvd.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'dsvd.m'\" else echo shar: Extracting \"'dsvd.m'\" \(1192 characters\) sed "s/^X//" >'dsvd.m' <<'END_OF_FILE' Xfunction x_lambda = dsvd(U,s,V,b,lambda) X%DSVD Damped SVD regularization. X% X% x_lambda = dsvd(U,s,V,b,lambda) X% x_lambda = dsvd(U,sm,X,b,lambda) , sm = [sigma,mu] X% X% Computes the damped SVD solution defined as X% x_lambda = V*inv(diag(s + lambda))*U'*b . X% If lambda is a vector, then x_lambda is a matrix such that X% x_lambda = [ x_lambda(1), x_lambda(2), ... ] . X% X% If sm and X are specified, then the damped GSVD solution: X% x_lambda = X*[ inv(diag(sigma + lambda*mu)) 0 ]*U'*b X% [ 0 I ] X% is computed. X X% Reference: M. P. Ekstrom & R. L. Rhoads, "On the application of X% eigenvector expansions to numerical deconvolution", J. Comp. X% Phys. 14 (1974), 319-340. X X% Per Christian Hansen, UNI-C, 07/21/90. X X% Initialization. Xif (min(lambda)<0) X error('Illegal regularization parameter lambda') Xend X[n,pv] = size(V); [p,ps] = size(s); X X% Compute x_lambda. Xbeta = U(:,1:p)'*b; Xll = length(lambda); x_lambda = zeros(n,ll); Xif (ps==1) X for i=1:ll X x_lambda(:,i) = V(:,1:p)*(beta./(s + lambda(i))); X end Xelse X x0 = V(:,p+1:n)*U(:,p+1:n)'*b; X for i=1:ll X x_lambda(:,i) = V(:,1:p)*(beta./(s(:,1) + lambda(i)*s(:,2))) + x0; X end Xend END_OF_FILE if test 1192 -ne `wc -c <'dsvd.m'`; then echo shar: \"'dsvd.m'\" unpacked with wrong size! fi # end of 'dsvd.m' fi if test -f 'fil_fac.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'fil_fac.m'\" else echo shar: Extracting \"'fil_fac.m'\" \(2527 characters\) sed "s/^X//" >'fil_fac.m' <<'END_OF_FILE' Xfunction f = fil_fac(s,reg_param,method,s1,V1) X%FIL_FAC Filter factors for some regularization methods. X% X% f = fil_fac(s,reg_param,method) X% f = fil_fac(sm,reg_param,method) , sm = [sigma,mu] X% f = fil_fac(s1,k,'ttls',s1,V1) X% X% Computes all the filter factors corresponding to the X% singular values in s and the regularization parameter X% reg_param, for the following methods: X% method = 'dsvd' : damped SVD X% method = 'tsvd' : truncated SVD X% method = 'Tikh' : Tikhonov regularization X% method = 'ttls' : truncated TLS. X% If sm = [sigma,mu] is specified, then the filter factors X% for the corresponding generalized methods are computed. X% X% If method = 'ttls' then the singular values s1 and the X% right singular matrix V1 of [A,b] must also be supplied. X% X% If method is not specified, 'Tikh' is default. X X% Per Christian Hansen, UNI-C, 06/23/93. X X% Initialization. X[p,ps] = size(s); lr = length(reg_param); Xif (nargin==2), method = 'Tikh'; end Xf = zeros(p,lr); X X% Check input data. Xif (min(reg_param) <= 0) X error('Regularization parameter must be positive') Xend Xif (method ~= 'Tikh' & min(reg_param) > p) X error('Truncation parameter too large') Xend X X% Compute the filter factors. Xfor j=1:lr X if (method(1:2)=='cg' | method(1:2)=='nu' | method(1:2)=='ls') X error('Filter factors for iterative methods are not supported') X elseif (method(1:4)=='dsvd') X if (ps==1) X f(:,j) = s./(s + reg_param(j)); X else X f(:,j) = s(:,1)./(s(:,1) + reg_param(j)*s(:,2)); X end X elseif (method(1:4)=='Tikh' | method(1:4)=='tikh') X if (ps==1) X f(:,j) = (s.^2)./(s.^2 + reg_param(j)^2); X else X f(:,j) = (s(:,1).^2)./(s(:,1).^2 + reg_param(j)^2*s(:,2).^2); X end X elseif (method(1:4)=='tsvd' | method(1:4)=='tgsv') X if (ps==1) X f(:,j) = [ones(reg_param(j),1);zeros(p-reg_param(j),1)]; X else X f(:,j) = [zeros(p-reg_param(j),1);ones(reg_param(j),1)]; X end X elseif (method(1:4)=='ttls') X if (ps==1) X coef = ((V1(p+1,:).^2)')/norm(V1(p+1,reg_param(j)+1:p+1))^2; X for i=1:p X k = reg_param(j); X f(i,j) = s(i)^2*... X sum( coef(k+1:p+1)./(s(i)+s1(k+1:p+1))./(s(i)-s1(k+1:p+1)) ); X if (f(i,j) < 0), f(i,j) = eps; end X if (i > 1) X if (f(i-1,j) <= eps & f(i,j) > f(i-1,j)), f(i,j) = f(i-1,j); end X end X end X else X error('The SVD of [A,b] must be supplied') X end X elseif (method(1:4)=='mtsv') X error('Filter factors for MTSVD are not supported') X else X error('Illegal method') X end Xend END_OF_FILE if test 2527 -ne `wc -c <'fil_fac.m'`; then echo shar: \"'fil_fac.m'\" unpacked with wrong size! fi # end of 'fil_fac.m' fi if test -f 'fnder.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'fnder.m'\" else echo shar: Extracting \"'fnder.m'\" \(79 characters\) sed "s/^X//" >'fnder.m' <<'END_OF_FILE' Xfunction fprime=fnder(f,dorder) X%FNDER Dummy function for Regularization Tools END_OF_FILE if test 79 -ne `wc -c <'fnder.m'`; then echo shar: \"'fnder.m'\" unpacked with wrong size! fi # end of 'fnder.m' fi if test -f 'foxgood.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'foxgood.m'\" else echo shar: Extracting \"'foxgood.m'\" \(610 characters\) sed "s/^X//" >'foxgood.m' <<'END_OF_FILE' Xfunction [A,b,x] = foxgood(n) X%FOXGOOD Severely ill-posed testproblem. X% X% [A,b,x] = foxgood(n) X% X% This is a model problem which does not satisfy the X% discrete Picard condition for the small singular values. X% The problem was first used by Fox & Goodwin. X X% Reference: C. T. H. Baker, "The Numerical Treatment of X% Integral Equations", Clarendon Press, Oxford, 1977; p. 665. X X% Discretized by simple quadrature (midpoint rule). X X% Per Christian Hansen, UNI-C, 03/16/93. X X% Initialization. Xh = 1/n; t = h*([1:n]' - 0.5); X XA = h*sqrt((t.^2)*ones(1,n) + ones(n,1)*(t.^2)'); Xx = t; b = ((1+t.^2).^1.5 - t.^3)/3; END_OF_FILE if test 610 -ne `wc -c <'foxgood.m'`; then echo shar: \"'foxgood.m'\" unpacked with wrong size! fi # end of 'foxgood.m' fi if test -f 'gcv.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'gcv.m'\" else echo shar: Extracting \"'gcv.m'\" \(3891 characters\) sed "s/^X//" >'gcv.m' <<'END_OF_FILE' Xfunction [reg_min,G,reg_param] = gcv(U,s,b,method) X%GCV Plot the GCV function and find its minimum. X% X% [reg_min,G,reg_param] = gcv(U,s,b,method) X% [reg_min,G,reg_param] = gcv(U,sm,b,method) , sm = [sigma,mu] X% X% Plots the GCV-function X% || A*x - b ||^2 X% G = ------------------- X% (trace(I - A*A_I)^2 X% as a function of the regularization parameter reg_param. X% Here, A_I is a matrix which produces the regularized solution. X% X% The following methods are allowed: X% method = 'Tikh' : Tikhonov regularization (solid line ) X% method = 'tsvd' : truncated SVD or GSVD (o markers ) X% method = 'dsvd' : damped SVD or GSVD (dotted line) X% If method is not specified, 'Tikh' is default. X% X% If any output arguments are specified, then the minimum of G is X% identified and the corresponding reg. parameter reg_min is returned. X X% Per Christian Hansen, UNI-C, 03/16/93. X X% Reference: G. Wahba, "Spline Models for Observational Data", X% SIAM, 1990. X X% Set defaults. Xif (nargin==3), method='Tikh'; end % Default method. Xnpoints = 100; % Number of points on the curve. Xsmin_ratio = 16*eps; % Smallest regularization parameter. X X% Initialization. X[m,n] = size(U); [p,ps] = size(s); Xbeta = U'*b; beta2 = b'*b - beta'*beta; Xif (ps==2) X s = s(p:-1:1,1)./s(p:-1:1,2); beta = beta(p:-1:1); Xend Xif (nargout > 0), find_min = 1; else find_min = 0; end X Xif (method(1:4)=='Tikh' | method(1:4)=='tikh') X X reg_param = zeros(npoints,1); G = reg_param; s2 = s.^2; X reg_param(npoints) = max([s(p),s(1)*smin_ratio]); X ratio = (s(1)/reg_param(npoints))^(1/(npoints-1)); X ratio = 1.2*(s(1)/reg_param(npoints))^(1/(npoints-1)); X for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end X delta0 = 0; X if (m > n & beta2 > 0), delta0 = beta2; end X for i=1:npoints X f1 = (reg_param(i)^2)./(s2 + reg_param(i)^2); X fb = f1.*beta(1:p); rho2 = fb'*fb + delta0; X G(i) = rho2/(m - n + sum(f1))^2; X end X loglog(reg_param,G,'-'), xlabel('lambda'), ylabel('G(lambda)') X title('GCV function') X if (find_min) X [minG,minGi] = min(G); reg_min = reg_param(minGi); X HoldState = ishold; hold on; X loglog(reg_min,minG,'*',[reg_min,reg_min],[minG/1000,minG],':') X title(['GCV function, minimum at ',num2str(reg_min)]) X if (~HoldState), hold off; end X end X Xelseif (method(1:4)=='tsvd' | method(1:4)=='tgsv') X X rho2(p-1) = beta(p)^2; X if (m > n & beta2 > 0), rho2(p-1) = rho2(p-1) + beta2; end X for k=p-2:-1:1, rho2(k) = rho2(k+1) + beta(k+1)^2; end X for k=1:p-1 X G(k) = rho2(k)/(m - k + (n - p))^2; X end X reg_param = [1:p-1]'; X semilogy(reg_param,G,'o'), xlabel('k'), ylabel('G(k)') X title('GCV function') X if (find_min) X [minG,reg_min] = min(G); X HoldState = ishold; hold on; X semilogy(reg_min,minG,'*',[reg_min,reg_min],[minG/1000,minG],'--') X title(['GCV function, minimum at ',num2str(reg_min)]) X if (~HoldState), hold off; end X end X Xelseif (method(1:4)=='dsvd' | method(1:4)=='dgsv') X X reg_param = zeros(npoints,1); G = reg_param; X reg_param(npoints) = max([s(p),s(1)*smin_ratio]); X ratio = (s(1)/reg_param(npoints))^(1/(npoints-1)); X for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end X delta0 = 0; X if (m > n & beta2 > 0), delta0 = beta2; end X for i=1:npoints X f1 = reg_param(i)./(s + reg_param(i)); X fb = f1.*beta(1:p); rho2 = fb'*fb + delta0; X G(i) = rho2/(m - n + sum(f1))^2; X end X loglog(reg_param,G,':'), xlabel('lambda'), ylabel('G(lambda)') X title('GCV function') X if (find_min) X [minG,minGi] = min(G); reg_min = reg_param(minGi); X HoldState = ishold; hold on; X loglog(reg_min,minG,'*',[reg_min,reg_min],[minG/1000,minG],'--') X tiel(['GCV function, minimum at ',num2str(reg_min)]) X if (~HoldState), hold off; end X end X Xelseif (method(1:4)=='mtsv') X X error('The MTSVD method is not supported') X Xelse, error('Illegal method'), end END_OF_FILE if test 3891 -ne `wc -c <'gcv.m'`; then echo shar: \"'gcv.m'\" unpacked with wrong size! fi # end of 'gcv.m' fi if test -f 'gen_form.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'gen_form.m'\" else echo shar: Extracting \"'gen_form.m'\" \(1350 characters\) sed "s/^X//" >'gen_form.m' <<'END_OF_FILE' Xfunction x = gen_form(L_p,x_s,A,b,K,M) X%GEN_FORM Transform a standard-form problem back to the general-form setting. X% X% x = gen_form(L_p,x_s,A,b,K,M) (method 1) X% x = gen_form(L_p,x_s,x_0) (method 2) X% X% Transforms the standard-form solution x_s back to the required X% solution to the general-form problem: X% x = L_p*x_s + d , X% where L_p and d depend on the method as follows: X% method = 1: L_p = pseudoinverse of L, d = K*(b - A*L_p*x_s) X% method = 2: L_p = A-weighted pseudoinverse of L, d = x_0. X% X% Usually, the standard-form problem is generated by means of X% function std_form. X% X% Note that x_s may have more that one column. X X% References: L. Elden, "Algorithms for regularization of ill- X% conditioned least-squares problems", BIT 17 (1977), 134-145. X% L. Elden, "A weighted pseudoinverse, generalized singular values, X% and constrained lest squares problems", BIT 22 (1982), 487-502. X% M. Hanke, "Regularization with differential operators. An itera- X% tive approach", J. Numer. Funct. Anal. Optim. 13 (1992), 523-540. X X% Per Christian Hansen, UNI-C, 06/12/93. X X% Nargin determines which method. Xif (nargin==6) X [p,q] = size(x_s); [Km,Kn] = size(K); X if (Km==0) X x = L_p*x_s; X else X x = L_p*x_s + K*(M*(b*ones(1,q) - A*(L_p*x_s))); X end Xelse X x_0 = A; [p,q] = size(x_s); X x = L_p*x_s + x_0*ones(1,q); Xend END_OF_FILE if test 1350 -ne `wc -c <'gen_form.m'`; then echo shar: \"'gen_form.m'\" unpacked with wrong size! fi # end of 'gen_form.m' fi if test -f 'gen_hh.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'gen_hh.m'\" else echo shar: Extracting \"'gen_hh.m'\" \(554 characters\) sed "s/^X//" >'gen_hh.m' <<'END_OF_FILE' Xfunction [x1,beta,v] = gen_hh(x) X%GEN_HH Generate a Householder transformation. X% X% [x1,beta,v] = gen_hh(x) X% X% Given a vector x, gen_hh computes the scalar beta and the vector v X% determining a Householder transformation X% H = (I - beta*v*v'), X% such that H*x = +-norm(x)*e_1. x1 is the first element of H*x. X X% Per Christian Hansen, UNI-C, 11/11/1997. X Xv = x; alpha = norm(v); Xif (alpha==0), X beta = 0; Xelse X beta = 1/(alpha*(alpha + abs(v(1)))); Xend Xif (v(1) >= 0) X v(1) = v(1) + alpha; x1 = -alpha; Xelse X v(1) = v(1) - alpha; x1 = +alpha; Xend END_OF_FILE if test 554 -ne `wc -c <'gen_hh.m'`; then echo shar: \"'gen_hh.m'\" unpacked with wrong size! fi # end of 'gen_hh.m' fi if test -f 'get_l.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'get_l.m'\" else echo shar: Extracting \"'get_l.m'\" \(793 characters\) sed "s/^X//" >'get_l.m' <<'END_OF_FILE' Xfunction [L,W] = get_l(n,d) X%GET_L Compute discrete derivative operators. X% X% [L,W] = get_l(n,d) X% X% Computes the discrete approximation L to the derivative operator X% of order d on a regular grid with n points, i.e. L is (n-d)-by-n. X% X% L is stored as a sparse matrix. X% X% Also computes W, an orthonormal basis for the null space of L. X X% Per Christian Hansen, UNI-C, 05/26/93. X X% Initialization. Xif (d<1), error ('Order d must be positive'), end Xnd = n-d; X X% Compute L. Xc = [-1,1,zeros(1,d-1)]; Xfor i=2:d, c = [c(1:d),0] - [0,c(1:d)]; end XL = sparse(nd,n); Xfor i=1:d+1 X L = L + sparse(1:nd,[1:nd]+i-1,c(i)*ones(1,nd),nd,n); Xend X X% If required, compute the null vectors W. Xif (nargout==2) X W = zeros(n,d); X W(:,1) = ones(n,1); X for i=2:d, W(:,i) = W(:,i-1).*[1:n]'; end X W = mgs(W); Xend END_OF_FILE if test 793 -ne `wc -c <'get_l.m'`; then echo shar: \"'get_l.m'\" unpacked with wrong size! fi # end of 'get_l.m' fi if test -f 'gsvd.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'gsvd.m'\" else echo shar: Extracting \"'gsvd.m'\" \(1110 characters\) sed "s/^X//" >'gsvd.m' <<'END_OF_FILE' Xfunction [U,V,sm,X] = gsvd(A,L) X%GSVD Generalized SVD of a matrix pair. X% X% sm = gsvd(A,L) X% [U,V,sm,X] = gsvd(A,L) , sm = [sigma,mu] X% X% Computes the generalized SVD of the matrix pair (A,L): X% [ A ] = [ U 0 ]*[ diag(sigma) 0 ]*inv(X) X% [ L ] [ 0 V ] [ 0 eye(n-p) ] X% [ diag(mu) 0 ] X% where X% U is m-by-n , sigma is p-by-1 X% V is p-by-p , mu is p-by-1 X% X is n-by-n . X% X% It is assumed that m >= n >= p . X X% Reference: C. F. Van Loan, "Computing the CS and the generalized X% singular value decomposition", Numer. Math. 46 (1985), 479-491. X X% Per Christian Hansen, UNI-C, 06/22/93. X X% Initialization. X[m,n] = size(A); [p,n1] = size(L); Xif (n1 ~= n | m < n | n < p) X error('Incorrect dimensions of A and L') Xend X X% Compute the GSVD in compact form via the CS decomposition. X[Q,d,X] = svd([full(A);full(L)]); Q = Q(:,1:n); d = diag(d); Xif (nargout > 1) X [U,V,sm,Z] = csdecomp(Q(1:m,:),Q(m+1:m+p,:)); X if (nargout==4) X for j=1:n, X(:,j) = X(:,j)/d(j); end X X = X*Z; X end Xelse X U = csd(Q(1:m,:),Q(m+1:m+p,:)); Xend END_OF_FILE if test 1110 -ne `wc -c <'gsvd.m'`; then echo shar: \"'gsvd.m'\" unpacked with wrong size! fi # end of 'gsvd.m' fi if test -f 'heat.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'heat.m'\" else echo shar: Extracting \"'heat.m'\" \(1632 characters\) sed "s/^X//" >'heat.m' <<'END_OF_FILE' Xfunction [A,b,x] = heat(n,kappa) X%HEAT Test problem: inverse heat equation. X% X% [A,b,x] = heat(n,kappa) X% X% A first kind Volterra integral equation with [0,1] as X% integration interval. The kernel is K(s,t) = k(s-t) with X% k(t) = t^(-3/2)/(2*kappa*sqrt(pi))*exp(-1/(4*kappa^2*t^2)) . X% Here, kappa controls the ill-conditioning of the matrix: X% kappa = 5 gives a well-conditioned problem X% kappa = 1 gives an ill-conditioned problem. X% The default is kappa = 1. X% X% An exact soltuion is constructed, and then the right-hand side X% b is produced as b = A*x. X X% Reference: A. S. Carasso, "Determining surface temperatures X% from interior observations", SIAM J. Appl. Math. 42 (1982), X% 558-574. See also L. Elden, "The numerical solution of a X% non-characteristic Cauchy problem for a parabolic equation"; X% in P. Deuflhand & E. Hairer (Eds.), "Numerical Treatment of X% Inverse Problems in Differential and Integral Equations", X% Birkhauser, 1983. X X% Discretization by means of simple quadrature (midpoint rule). X X% Per Christian Hansen, UNI-C, 09/18/92. X X% Set default kappa. Xif (nargin==1), kappa = 1; end X X% Initialization. Xh = 1/n; t = h/2:h:1; e = ones(1,length(t)); Xc = h/(2*kappa*sqrt(pi)); d = 1/(4*kappa^2); X X% Compute the matrix A. Xk = c*t.^(-1.5).*exp(-d*e./t); Xr = zeros(1,length(t)); r(1) = k(1); A = toeplitz(k,r); X X% Compute the vectors x and b. Xif (nargout>1) X x = zeros(n,1); X for i=1:n/2 X ti = i*20/n; X if (ti < 2) X x(i) = 0.75*ti^2/4; X elseif (ti < 3) X x(i) = 0.75 + (ti-2)*(3-ti); X else X x(i) = 0.75*exp(-(ti-3)*2); X end X end X x(n/2+1:n) = zeros(1,n/2); X b = A*x; Xend END_OF_FILE if test 1632 -ne `wc -c <'heat.m'`; then echo shar: \"'heat.m'\" unpacked with wrong size! fi # end of 'heat.m' fi if test -f 'heb_new.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'heb_new.m'\" else echo shar: Extracting \"'heb_new.m'\" \(1781 characters\) sed "s/^X//" >'heb_new.m' <<'END_OF_FILE' Xfunction lambda = heb_new(lambda_0,alpha,s,beta,omega) X%HEB_NEW Newton iteration with Hebden model (utility routine for LSQI). X% X% lambda = heb_new(lambda_0,alpha,s,beta,omega) X% X% Uses Newton iteration with a Hebden (rational) model to find the X% solution lambda to the secular equation X% || L (x_lambda - x_0) || = alpha , X% where x_lambda is the solution defined by Tikhonov regularization. X% X% The initial guess is lambda_0. X% X% The norm || L (x_lambda - x_0) || is computed via s, beta and omega. X% Here, s holds either the singular values of A, if L = I, or the X% c,s-pairs of the GSVD of (A,L), if L ~= I. Moreover, beta = U'*b X% and omega is either V'*x_0 or the first p elements of inv(X)*x_0. X X% Reference: T. F. Chan, J. Olkin & D. W. Cooley, "Solving quadratically X% constrained least squares using block box unconstrained solvers", X% BIT 32 (1992), 481-495. X% Extension to the case x_0 ~= 0 by Per Chr. Hansen, UNI-C, 11/20/91. X X% Per Christian Hansen, UNI-C, 02/07/92. X X% Set defaults. Xthr = eps; % Relative stopping criterion. Xit_max = 50; % Max number of iterations. X X% Initialization. Xif (lambda_0 < 0) X error('Initial guess lambda_0 must be nonnegative') Xend X[p,ps] = size(s); Xif (ps==2), mu = s(:,2); s = s(:,1)./s(:,2); end Xs2 = s.^2; X X% Iterate. Xlambda = lambda_0^2; step = 1; it = 0; Xwhile (abs(step) > thr*lambda & it < it_max), it = it+1; X e = s./(s2 + lambda); f = 2.*e; X if (ps==1) X Lx = e.*beta - f.*omega; X else X Lx = e.*beta - f.*mu.*omega; X end X norm_Lx = norm(Lx); X Lv = Lx./(s2 + lambda); X step = (norm_Lx - alpha)*norm_Lx^2/((Lv'*Lx)*alpha); X lambda = lambda + step; Xend X X% Terminate with an error if too many iterations. Xif (abs(step) > thr*lambda), error('Max. number of iterations reached'), end X Xlambda = sqrt(lambda); END_OF_FILE if test 1781 -ne `wc -c <'heb_new.m'`; then echo shar: \"'heb_new.m'\" unpacked with wrong size! fi # end of 'heb_new.m' fi if test -f 'ilaplace.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'ilaplace.m'\" else echo shar: Extracting \"'ilaplace.m'\" \(1976 characters\) sed "s/^X//" >'ilaplace.m' <<'END_OF_FILE' Xfunction [A,b,x] = ilaplace(n,example) X%ILAPLACE Test problem: inverse Laplace transformation. X% X% [A,b,x] = ilaplace(n,example) X% X% Discretization of the inverse Laplace transformation by means of X% Gauss-Laguerre quadrature. The kernel K is given by X% K(s,t) = exp(-s*t) , X% and both integration intervals are [0,inf). X% The following examples are implemented, where f denotes X% the solution, and g denotes the right-hand side: X% 1: f(t) = exp(-t/2), g(s) = 1/(s + 0.5) X% 2: f(t) = 1 - exp(-t/2), g(s) = 1/s - 1/(s + 0.5) X% 3: f(t) = t^2*exp(-t/2), g(s) = 2/(s + 0.5)^3 X% 4: f(t) = | 0 , t <= 2, g(s) = exp(-2*s)/s. X% | 1 , t > 2 X X% Reference: J. M. Varah, "Pitfalls in the numerical solution of linear X% ill-posed problems", SIAM J. Sci. Stat. Comput. 4 (1983), 164-176. X X% Per Christian Hansen, UNI-C, 09/18/92. X X% Initialization. Xif (n <= 0), error('The order n must be positive'); end Xif (nargin == 1), example = 1; end X X% Compute equidistand collocation points s. Xs = (10/n)*[1:n]'; X X% Compute abscissas t and weights w from the eigensystem of the X% symmetric tridiagonal system derived from the recurrence X% relation for the Laguerre polynomials. Sorting of the X% eigenvalues and -vectors is necessary. Xt = diag(2*[1:n]-1) - diag([1:n-1],1) - diag([1:n-1],-1); X[Q,t] = eig(t); t = diag(t); [t,indx] = sort(t); Xw = Q(1,indx).^2; Q = []; X X% Set up the coefficient matrix A. XA = zeros(n,n); Xfor i=1:n X for j=1:n X A(i,j) = (1-s(i))*t(j); X end Xend XA = exp(A)*diag(w); X X% Compute the right-hand side b and the solution x by means of X% simple collocation. Xif (example==1) X b = ones(n,1)./(s + .5); X x = exp(-t/2); Xelseif (example==2) X b = ones(n,1)./s - ones(n,1)./(s + .5); X x = ones(n,1) - exp(-t/2); Xelseif (example==3) X b = 2*ones(n,1)./((s + .5).^3); X x = (t.^2).*exp(-t/2); Xelseif (example==4) X b = exp(-2*s)./s; X x = ones(n,1); f = find(t<=2); x(f) = zeros(length(f),1); Xelse X error('Illegal example') Xend END_OF_FILE if test 1976 -ne `wc -c <'ilaplace.m'`; then echo shar: \"'ilaplace.m'\" unpacked with wrong size! fi # end of 'ilaplace.m' fi if test -f 'l_corner.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'l_corner.m'\" else echo shar: Extracting \"'l_corner.m'\" \(8126 characters\) sed "s/^X//" >'l_corner.m' <<'END_OF_FILE' Xfunction [reg_c,rho_c,eta_c] = l_corner(rho,eta,reg_param,U,s,b,method,M) X%L_CORNER Locate the "corner" of the L-curve. X% X% [reg_c,rho_c,eta_c] = X% l_corner(rho,eta,reg_param) X% l_corner(rho,eta,reg_param,U,s,b,method,M) X% l_corner(rho,eta,reg_param,U,sm,b,method,M) , sm = [sigma,mu] X% X% Locates the "corner" of the L-curve in log-log scale. X% X% It is assumed that corresponding values of || A x - b ||, || L x ||, X% and the regularization parameter are stored in the arrays rho, eta, X% and reg_param, respectively (such as the output from routine l_curve). X% X% If nargin = 3, then no particular method is assumed, and if X% nargin = 2 then it is issumed that reg_param = 1:length(rho). X% X% If nargin >= 6, then the following methods are allowed: X% method = 'Tikh' : Tikhonov regularization X% method = 'tsvd' : truncated SVD or GSVD X% method = 'dsvd' : damped SVD or GSVD X% method = 'mtsvd' : modified TSVD, X% and if no method is specified, 'Tikh' is default. If the Spline Toolbox X% is not available, then only 'Tikh' and 'dsvd' can be used. X% X% An eighth argument M specifies an upper bound for eta, below which X% the corner should be found. X X% The following routines from the Spline Toolbox are needed if X% method differs from 'Tikh' or 'dsvd': X% fnder, ppbrk, ppcut, ppmak, sp2pp, spbrk, spmak. X X% Per Christian Hansen, UNI-C, 03/17/93. X X% Set default regularization method. Xif (nargin <= 3) X method = 'none'; X if (nargin==2), reg_param = [1:length(rho)]'; end Xelse X if (nargin==6), method = 'Tikh'; end Xend X X% Set threshold for skipping very small singular values in the X% L-curve analysis. Xs_thr = eps; % Neglect singular values less than s_thr. X X% Set default parameters for treatment of discrete L-curve. Xdeg = 2; % Degree of local smooting polynomial. Xq = 2; % Half-width of local smoothing interval. Xorder = 4; % Order of fitting 2-D spline curve. X X% Initialization. Xif (length(rho) < order) X error('Too few data points for L-curve analysis') Xend Xif (nargin > 3) X [p,ps] = size(s); [m,n] = size(U); X if (ps==2), s = s(p:-1:1,1)./s(p:-1:1,2); U = U(:,p:-1:1); end X beta = U'*b; xi = beta./s; Xend X X% Restrict the analysis of the L-curve according to M (if specified) X% and s_thr. Xif (nargin==8) X index = find(eta < M); X rho = rho(index); eta = eta(index); reg_param = reg_param(index); X s = s(index); beta = beta(index); xi = xi(index); Xend X Xif (method(1:4)=='Tikh' | method(1:4)=='tikh') X X % The L-curve is differentiable; computation of curvature in X % log-log scale is easy. X X % Initialization. X [reg_m,reg_n] = size(reg_param); X phi = zeros(reg_m,reg_n); dphi = phi; psi = phi; dpsi = phi; X s2 = s.^2; beta2 = beta.^2; xi2 = xi.^2; X X % Compute some intermediate quantities. X for i = 1:length(reg_param) X f = s2./(s2 + reg_param(i)^2); cf = 1 - f; X f1 = -2*f.*cf/reg_param(i); X f2 = -f1.*(3-4*f)/reg_param(i); X phi(i) = sum(f.*f1.*xi2); X psi(i) = sum(cf.*f1.*beta2); X dphi(i) = sum((f1.^2 + f.*f2).*xi2); X dpsi(i) = sum((-f1.^2 + cf.*f2).*beta2); X end X X % Now compute the first and second derivatives of eta and rho X % with respect to lambda; X deta = phi./eta; X drho = -psi./rho; X ddeta = dphi./eta - deta.*(deta./eta); X ddrho = -dpsi./rho - drho.*(drho./rho); X X % Convert to derivatives of log(eta) and log(rho). X dlogeta = deta./eta; X dlogrho = drho./rho; X ddlogeta = ddeta./eta - (dlogeta).^2; X ddlogrho = ddrho./rho - (dlogrho).^2; X X % Let g = curvature. X g = (dlogrho.*ddlogeta - ddlogrho.*dlogeta)./... X (dlogrho.^2 + dlogeta.^2).^(1.5); X X % Locate the corner. If the curvature is negative everywhere, X % then define the leftmost point of the L-curve as the corner. X [gmax,gi] = max(g); X if (gmax < 0) X lr = length(rho); X reg_c = reg_param(lr); rho_c = rho(lr); eta_c = eta(lr); X else X rho_c = rho(gi); eta_c = eta(gi); reg_c = reg_param(gi); X end X Xelseif (method(1:4)=='tsvd' | method(1:4)=='tgsv' | ... X method(1:4)=='mtsv' | method(1:4)=='none') X X % The L-curve is discrete and may include unwanted fine-grained X % corners. Use local smoothing, followed by fitting a 2-D spline X % curve to the smoothed discrete L-curve. X X % Check if the Spline Toolbox exists, otherwise return. X if (exist('spdemos')~=2) X error('The Spline Toolbox in not available so l_corner cannot be used') X end X X % For TSVD, TGSVD, and MTSVD, restrict the analysis of the L-curve X % according to s_thr. X if (nargin > 3) X index = find(s > s_thr); X rho = rho(index); eta = eta(index); reg_param = reg_param(index); X s = s(index); beta = beta(index); xi = xi(index); X end X X % Convert to logarithms. X lr = length(rho); X lrho = log(rho); leta = log(eta); slrho = lrho; sleta = leta; X X % For all interior points k = q+1:length(rho)-q-1 on the discrete X % L-curve, perform local smoothing with a polynomial of degree deg X % to the points k-q:k+q. X v = [-q:q]'; A = zeros(2*q+1,deg+1); A(:,1) = ones(length(v),1); X for j = 2:deg+1, A(:,j) = A(:,j-1).*v; end X for k = q+1:lr-q-1 X cr = A\lrho(k+v); slrho(k) = cr(1); X ce = A\leta(k+v); sleta(k) = ce(1); X end X X % Fit a 2-D spline curve to the smoothed discrete L-curve. X sp = spmak([1:lr+order],[slrho';sleta']); X pp = ppcut(sp2pp(sp),[4,lr+1]); X X % Extract abscissa and ordinate splines and differentiate them. X P = spleval(pp); dpp = fnder(pp); X D = spleval(dpp); ddpp = fnder(pp,2); X DD = spleval(ddpp); X ppx = P(1,:); ppy = P(2,:); X dppx = D(1,:); dppy = D(2,:); X ddppx = DD(1,:); ddppy = DD(2,:); X X % Compute the corner of the spline curve via max. curvature. X % Define curvature = 0 where both dppx and dppy are zero. X k1 = dppx.*ddppy - ddppx.*dppy; X k2 = (dppx.^2 + dppy.^2).^(1.5); X I_nz = find(k2 ~= 0); X kappa = zeros(1,length(dppx)); X kappa(I_nz) = -k1(I_nz)./k2(I_nz); X [kmax,ikmax] = max(kappa); X x_corner = ppx(ikmax); y_corner = ppy(ikmax); X X % Locate the point on the discrete L-curve which is closest to the X % corner of the spline curve. Prefer a point below and to the X % left of the corner. If the curvature is negative everywhere, X % then define the leftmost point of the L-curve as the corner. X if (kmax < 0) X reg_c = reg_param(lr); rho_c = rho(lr); eta_c = eta(lr); X else X index = find(lrho < x_corner & leta < y_corner); X if (length(index) > 0) X [dummy,rpi] = min((lrho(index)-x_corner).^2 + (leta(index)-y_corner).^2); X rpi = index(rpi); X else X [dummy,rpi] = min((lrho-x_corner).^2 + (leta-y_corner).^2); X end X reg_c = reg_param(rpi); rho_c = rho(rpi); eta_c = eta(rpi); X end X Xelseif (method(1:4)=='dsvd' | method(1:4)=='dgsv') X X % The L-curve is differentiable; computation of curvature in X % log-log scale is easy. X X % Initialization. X [reg_m,reg_n] = size(reg_param); X phi = zeros(reg_m,reg_n); dphi = phi; psi = phi; dpsi = phi; X beta2 = beta.^2; xi2 = xi.^2; X X % Compute some intermediate quantities. X for i = 1:length(reg_param) X f = s./(s + reg_param(i)); cf = 1 - f; X f1 = -f.*cf/reg_param(i); X f2 = -2*f1.*cf/reg_param(i); X phi(i) = sum(f.*f1.*xi2); X psi(i) = sum(cf.*f1.*beta2); X dphi(i) = sum((f1.^2 + f.*f2).*xi2); X dpsi(i) = sum((-f1.^2 + cf.*f2).*beta2); X end X X % Now compute the first and second derivatives of eta and rho X % with respect to lambda; X deta = phi./eta; X drho = -psi./rho; X ddeta = dphi./eta - deta.*(deta./eta); X ddrho = -dpsi./rho - drho.*(drho./rho); X X % Convert to derivatives of log(eta) and log(rho). X dlogeta = deta./eta; X dlogrho = drho./rho; X ddlogeta = ddeta./eta - (dlogeta).^2; X ddlogrho = ddrho./rho - (dlogrho).^2; X X % Let g = curvature. X g = (dlogrho.*ddlogeta - ddlogrho.*dlogeta)./... X (dlogrho.^2 + dlogeta.^2).^(1.5); X X % Locate the corner. If the curvature is negative everywhere, X % then define the leftmost point of the L-curve as the corner. X [gmax,gi] = max(g); X if (gmax < 0) X lr = length(rho); X reg_c = reg_param(lr); rho_c = rho(lr); eta_c = eta(lr); X else X rho_c = rho(gi); eta_c = eta(gi); reg_c = reg_param(gi); X end X Xelse, error('Illegal method'), end END_OF_FILE if test 8126 -ne `wc -c <'l_corner.m'`; then echo shar: \"'l_corner.m'\" unpacked with wrong size! fi # end of 'l_corner.m' fi if test -f 'l_curve.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'l_curve.m'\" else echo shar: Extracting \"'l_curve.m'\" \(4936 characters\) sed "s/^X//" >'l_curve.m' <<'END_OF_FILE' Xfunction [reg_corner,rho,eta,reg_param] = l_curve(U,sm,b,method,L,V) X%L_CURVE Plot the L-curve and find its "corner". X% X% [reg_corner,rho,eta,reg_param] = X% l_curve(U,s,b,method) X% l_curve(U,sm,b,method) , sm = [sigma,mu] X% l_curve(U,s,b,method,L,V) X% X% Plots the L-shaped curve of eta, the solution norm || x || or X% semi-norm || L x ||, as a function of rho, the residual norm X% || A x - b ||, for the following methods: X% method = 'Tikh' : Tikhonov regularization (solid line ) X% method = 'tsvd' : truncated SVD or GSVD (o markers ) X% method = 'dsvd' : damped SVD or GSVD (dotted line) X% method = 'mtsvd' : modified TSVD (x markers ) X% The corresponding reg. parameters are returned in reg_param. X% If no method is specified, 'Tikh' is default. X% X% If any output arguments are specified, then the corner of the L-curve X% is identified and the corresponding reg. parameter reg_corner is X% returned. Use routine l_corner if an upper bound on eta is required. X% X% If the Spline Toolbox is not available and reg_corner is requested, X% then the routine returns reg_corner = NaN for 'tsvd' and 'mtsvd'. X X% Reference: P. C. Hansen & D. P. O'Leary, "The use of the L-curve in X% the regularization of discrete ill-posed problems", Report UMIACS- X% TR-91-142, Dept. of Computer Science, Univ. of Maryland, 1991; X% to appear in SIAM J. Sci. Comp. X X% Per Christian Hansen, UNI-C, 03/17/93. X X% Set defaults. Xif (nargin==3), method='Tikh'; end % Tikhonov reg. is default. Xnpoints = 100; % Number of points on the L-curve for Tikh and dsvd. Xsmin_ratio = 16*eps; % Smallest regularization parameter. X X% Initialization. X[m,n] = size(U); [p,ps] = size(sm); Xif (nargout > 0), locate = 1; else locate = 0; end Xbeta = U'*b; beta2 = b'*b - beta'*beta; Xif (ps==1) X s = sm; beta = beta(1:p); Xelse X s = sm(p:-1:1,1)./sm(p:-1:1,2); beta = beta(p:-1:1); Xend Xxi = beta(1:p)./s; X Xif (method(1:4)=='Tikh' | method(1:4)=='tikh') X X eta = zeros(npoints,1); rho = eta; reg_param = eta; s2 = s.^2; X reg_param(npoints) = max([s(p),s(1)*smin_ratio]); X ratio = (s(1)/reg_param(npoints))^(1/(npoints-1)); X ratio = (s(1)/reg_param(npoints))^(1/(npoints-1)); X for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end X for i=1:npoints X f = s2./(s2 + reg_param(i)^2); X eta(i) = norm(f.*xi); X rho(i) = norm((1-f).*beta(1:p)); X end X if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end X marker = '-'; pos = .8; txt = 'Tikh.'; X Xelseif (method(1:4)=='tsvd' | method(1:4)=='tgsv') X X eta = zeros(p,1); rho = eta; X eta(1) = xi(1)^2; X for k=2:p, eta(k) = eta(k-1) + xi(k)^2; end X eta = sqrt(eta); X if (m > n) X if (beta2 > 0), rho(p) = beta2; else rho(p) = eps^2; end X else X rho(p) = eps^2; X end X for k=p-1:-1:1, rho(k) = rho(k+1) + beta(k+1)^2; end X rho = sqrt(rho); X reg_param = [1:p]'; marker = 'o'; pos = .75; X if (ps==1) X U = U(:,1:p); txt = 'TSVD'; X else X U = U(:,1:p); txt = 'TGSVD'; X end X Xelseif (method(1:4)=='dsvd' | method(1:4)=='dgsv') X X eta = zeros(npoints,1); rho = eta; reg_param = eta; X reg_param(npoints) = max([s(p),s(1)*smin_ratio]); X ratio = (s(1)/reg_param(npoints))^(1/(npoints-1)); X for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end X for i=1:npoints X f = s./(s + reg_param(i)); X eta(i) = norm(f.*xi); X rho(i) = norm((1-f).*beta(1:p)); X end X if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end X marker = ':'; pos = .85; X if (ps==1), txt = 'DSVD'; else txt = 'DGSVD'; end X Xelseif (method(1:4)=='mtsv') X X if (nargin~=6) X error('The matrices L and V must also be specified') X end X [p,n] = size(L); rho = zeros(p,1); eta = rho; X [Q,R] = qr(L*V(:,n:-1:n-p)); X for i=1:p X k = n-p+i; X Lxk = L*V(:,1:k)*xi(1:k); X zk = R(1:n-k,1:n-k)\(Q(:,1:n-k)'*Lxk); zk = zk(n-k:-1:1); X eta(i) = norm(Q(:,n-k+1:p)'*Lxk); X if (i < p) X rho(i) = norm(beta(k+1:n) + s(k+1:n).*zk); X else X rho(i) = eps; X end X end X if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end X reg_param = [n-p+1:n]'; txt = 'MTSVD'; X U = U(:,reg_param); sm = sm(reg_param); X marker = 'x'; pos = .7; ps = 2; % General form regularization. X Xelse, error('Illegal method'), end X X% Locate the "corner" of the L-curve, if required. If the Spline X% Toolbox is not available, return NaN for reg_corner. Xif (locate) X SkipCorner = ( (method(1:4)=='tsvd' | method(1:4)=='tgsv' | ... X method(1:4)=='mtsv') & exist('spdemos')~=2 ); X if (SkipCorner) X reg_corner = NaN; X else X [reg_corner,rho_c,eta_c] = l_corner(rho,eta,reg_param,U,sm,b,method); X end Xend X X% Make plot. Xplot_lc(rho,eta,marker,ps,reg_param); Xif (locate & ~SkipCorner) X HoldState = ishold; hold on; X loglog([min(rho)/100,rho_c],[eta_c,eta_c],'--',... X [rho_c,rho_c],[min(eta)/100,eta_c],'--') X title(['L-curve, ',txt,' corner at ',num2str(reg_corner)]); X if (~HoldState), hold off; end Xend END_OF_FILE if test 4936 -ne `wc -c <'l_curve.m'`; then echo shar: \"'l_curve.m'\" unpacked with wrong size! fi # end of 'l_curve.m' fi if test -f 'lagrange.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'lagrange.m'\" else echo shar: Extracting \"'lagrange.m'\" \(1654 characters\) sed "s/^X//" >'lagrange.m' <<'END_OF_FILE' Xfunction [La,dLa,lambda0] = lagrange(U,s,b,more) X%LAGRANGE Plot the Lagrange function for Tikhonov regularization. X% X% [La,dLa,lambda0] = lagrange(U,s,b,more) X% [La,dLa,lambda0] = lagrange(U,sm,b,more) , sm = [sigma,mu] X% X% Plots the Lagrange function X% La(lambda) = || A x - b ||^2 + lambda^2*|| L x ||^2 X% and its first derivative dLa = dLa/dlambda versus lambda. X% Here, x is the Tikhonov regularized solution. X% X% If nargin = 4, || A x - b || and || L x || are also plotted. X% X% Returns La, dLa, and the value lambda0 of lambda for which X% dLa has its minimum. X X% Per Christian Hansen, UNI-C, 07/14/92. X X% Initialization. X[m,n] = size(U); [p,ps] = size(s); npoints = 40; Xbeta = U'*b; beta2 = b'*b - beta'*beta; Xif (ps==2) X s = s(p:-1:1,1)./s(p:-1:1,2); beta = beta(p:-1:1); Xend Xxi = beta(1:p)./s; X X% Compute the L-curve. Xeta = zeros(npoints,1); rho = eta; Xlambda(npoints,1) = s(p); Xratio = (s(1)/s(p))^(1/(npoints-1)); Xfor i=npoints-1:-1:1, lambda(i) = ratio*lambda(i+1); end Xfor i=1:npoints X f = fil_fac(s,lambda(i)); X eta(i) = norm(f.*xi); X rho(i) = norm((1-f).*beta(1:p)); Xend Xif (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end X X% Compute the Lagrange function and its derivative. XLa = rho.^2 + (lambda.^2).*(eta.^2); XdLa = 2*lambda.*(eta.^2); X[mindLa,mindLi] = min(dLa); lambda0 = lambda(mindLi); X X% Plot the functions. Xif (nargin==3) X loglog(lambda,La,'-',lambda,dLa,'--',lambda0,mindLa,'o') X title('--- La - - - dLa/dlambda') Xelse X loglog(lambda,La,'-',lambda,dLa,'--',lambda,eta,':',lambda,rho,'-.',... X lambda0,mindLa,'o') X title('--- La - - - dLa/dlambda ... || Lx || -.-. || Ax-b ||') Xend Xxlabel('lambda') END_OF_FILE if test 1654 -ne `wc -c <'lagrange.m'`; then echo shar: \"'lagrange.m'\" unpacked with wrong size! fi # end of 'lagrange.m' fi if test -f 'lanc_b.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'lanc_b.m'\" else echo shar: Extracting \"'lanc_b.m'\" \(2826 characters\) sed "s/^X//" >'lanc_b.m' <<'END_OF_FILE' Xfunction [U,B_k,V] = lanc_b(A,p,k,reorth) X%LANC_B Lanczos bidiagonalization. X% X% B_k = lanc_b(A,p,k,reorth) X% [U,B_k,V] = lanc_b(A,p,k,reorth) X% X% Performs k steps of the Lanczos bidiagonalization process X% with starting vector p, producing a lower bidiagonal matrix X% [b_11 ] [b_21 b_11] X% [b_21 b_22 ] [b_32 b_22] X% B = [ b_32 . ] stored as B_k = [ . . ] . X% [ . b_kk ] [b_k+1,k b_kk] X% [ b_k+1,k] X% U and V consist of the left and right Lanczos vectors. X% X% Reorthogonalization is controlled by means of reorth: X% reorth = 0 : no reorthogonalization, X% reorth = 1 : reorthogonalization by means of MGS, X% reorth = 2 : Householder-reorthogonalization. X% No reorthogonalization is assumed if reorth is not specified. X X% Reference: G. H. Golub & C. F. Van Loan, "Matrix Computations", X% 2. Ed., Johns Hopkins, 1989. Section 9.3.4. X X% Per Christian Hansen, UNI-C, 05/25/93. X X% Initialization. Xif (k<1), error('Number of steps k must be positive'), end Xif (nargin < 4), reorth = 0; end Xif (reorth < 0 | reorth > 2), error('Illegal reorth'), end Xif (nargout==2), error('Not enough output arguments'), end X[m,n] = size(A); Xif (nargout>1 | reorth==1) X U = zeros(m,k); V = zeros(n,k); UV = 1; Xelse X UV = 0; Xend Xif (reorth==2) X if (k>=n), error('No. of iterations must satisfy k < n'), end X HHU = zeros(m,k); HHV = zeros(n,k); X HHalpha = zeros(1,k); HHbeta = HHalpha; Xend X X% Prepare for Lanczos iteration. Xv = zeros(n,1); Xbeta = norm(p); Xif (beta==0), error('Starting vector must be nonzero'), end Xif (reorth==2) X [beta,HHbeta(1),HHU(:,1)] = gen_hh(p); Xend Xu = p/beta; Xif (UV), U(:,1) = u; end X X% Perform Lanczos bidiagonalization with/without reorthogonalization. Xfor i=1:k X X r = A'*u - beta*v; X if (reorth==0) X alpha = norm(r); v = r/alpha; X elseif (reorth==1) X for j=1:i-1, r = r - (V(:,j)'*r)*V(:,j); end X alpha = norm(r); v = r/alpha; X else X for j=1:i-1 X r(j:n) = app_hh(r(j:n),HHalpha(j),HHV(j:n,j)); X end X [alpha,HHalpha(i),HHV(i:n,i)] = gen_hh(r(i:n)); X v = zeros(n,1); v(i) = 1; X for j=i:-1:1 X v(j:n) = app_hh(v(j:n),HHalpha(j),HHV(j:n,j)); X end X end X B_k(i,2) = alpha; if (UV), V(:,i) = v; end X X p = A*v - alpha*u; X if (reorth==0) X beta = norm(p); u = p/beta; X elseif (reorth==1) X for j=1:i, p = p - (U(:,j)'*p)*U(:,j); end X beta = norm(p); u = p/beta; X else X for j=1:i X p(j:m) = app_hh(p(j:m),HHbeta(j),HHU(j:m,j)); X end X [beta,HHbeta(i+1),HHU(i+1:m,i+1)] = gen_hh(p(i+1:m)); X u = zeros(m,1); u(i+1) = 1; X for j=i+1:-1:1 X u(j:m) = app_hh(u(j:m),HHbeta(j),HHU(j:m,j)); X end X end X B_k(i,1) = beta; if (UV), U(:,i+1) = u; end X Xend X Xif (nargout==1), U = B_k; end END_OF_FILE if test 2826 -ne `wc -c <'lanc_b.m'`; then echo shar: \"'lanc_b.m'\" unpacked with wrong size! fi # end of 'lanc_b.m' fi if test -f 'lsolve.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'lsolve.m'\" else echo shar: Extracting \"'lsolve.m'\" \(1089 characters\) sed "s/^X//" >'lsolve.m' <<'END_OF_FILE' Xfunction x = lsolve(L,y,W,NAA) X%LSOLVE Utility routine for "preconditioned" iterative methods. X% X% x = lsolve(L,y,W,NAA) X% X% Computes the vector X% x = L_p*y X% where L_p is the A-weighted generalized inverse of L. X% X% Typically, L is a p-by-n band matrix with bandwidth n-p+1, W holds X% a basis for the null space of L, and NAA is a utility matrix which X% should be computed by routine pinit. X% X% Alternatively, L is square and dense, and W and NAA are not needed. X% X% Notice that x and y may be matrices, in which case X% x(:,i) = L_p*y(:,i) . X X% Reference: M. Hanke, "Regularization with differential operators. X% An iterative approach", J. Numer. Funct. Anal. Optim. 13 (1992), X% 523-540. X X% Per Christian Hansen, UNI-C, and Martin Hanke, Institut fuer X% Praktische Mathematik, Universitaet Karlsruhe, 05/26/93. X X% Initialization. X[p,n] = size(L); nu = n-p; [py,ly] = size(y); X X% Special treatment of square L. Xif (nu==0), x = L\y; return; end X X% Compute a particular solution Xx = [[eye(nu),zeros(nu,p)];L]\[zeros(nu,ly);y]; X X% Perform the necessary projection. Xx = x - W*(NAA*x); END_OF_FILE if test 1089 -ne `wc -c <'lsolve.m'`; then echo shar: \"'lsolve.m'\" unpacked with wrong size! fi # end of 'lsolve.m' fi if test -f 'lsqi.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'lsqi.m'\" else echo shar: Extracting \"'lsqi.m'\" \(2636 characters\) sed "s/^X//" >'lsqi.m' <<'END_OF_FILE' Xfunction [x_alpha,lambda] = lsqi(U,s,V,b,alpha,x_0) X%LSQI Least squares minimizaiton with a quadratic inequality constraint. X% X% [x_alpha,lambda] = lsqi(U,s,V,b,alpha,x_0) X% [x_alpha,lambda] = lsqi(U,sm,X,b,alpha,x_0) , sm = [sigma,mu] X% X% Least squares minimization with a quadratic inequality constraint: X% min || A x - b || subject to || x - x_0 || <= alpha X% min || A x - b || subject to || L (x - x_0) || <= alpha X% where x_0 is an initial guess of the solution, and alpha is a X% positive constant. Requires either the compact SVD of A saved as X% U, s, and V, or part of the GSVD of (A,L) saved as U, sm, and X. X% The regularization parameter lambda is also returned. X% X% If alpha is a vector, then x_alpha is a matrix such that X% x_alpha = [ x_alpha(1), x_alpha(2), ... ] . X% X% If x_0 is not specified, x_0 = 0 is used. X X% Reference: T. F. Chan, J. Olkin & D. W. Cooley, "Solving quadratically X% constrained least squares using block box unconstrained solvers", X% BIT 32 (1992), 481-495. X% Extension to the case x_0 ~= 0 by Per Chr. Hansen, UNI-C, 11/20/91. X% Key point: the initial lambda is almost unaffected by x_0 because X% || x_unreg || >> || x_0 ||. X X% Per Christian Hansen, UNI-C, 11/22/91. X X% Initialization. X[n,p] = size(V); [p,ps] = size(s); Xif (min(alpha)<0) X error('Illegal inequality constraint alpha') Xend Xif (nargin==5), x_0 = zeros(n,1); end Xla = length(alpha); Xx_k = zeros(n,la); lambda = zeros(la,1); Xsnz = length(find(s(:,1)>0)); beta = U'*b; X Xif (ps == 1) X xi = beta(1:snz)./s(1:snz); omega = V'*x_0; s2 = s.^2; X x_unreg = V(:,1:snz)*xi; norm_x_unreg = norm(x_unreg - x_0); X for k=1:la X if (norm_x_unreg <= alpha(k)) X x_alpha(:,k) = x_unreg; lambda(k) = 0; X else X lambda_0 = s(snz)*(norm_x_unreg/alpha(k) - 1); X lambda(k) = heb_new(lambda_0,alpha(k),s,beta,omega); X e = s./(s2 + lambda(k)^2); f = s.*e; X x_alpha(:,k) = V(:,1:p)*(e.*beta + (1-f).*omega); X end X end Xelse X x_u = V(:,p+1:n)*beta(p+1:n); ps1 = p-snz+1; X xi = beta(ps1:p)./s(ps1:p,1); gamma = s(:,1)./s(:,2); X omega = V\x_0; omega = omega(1:p); X x_unreg = V(:,ps1:p)*xi + x_u; X norm_Lx_unreg = norm(s(ps1:p,2).*(xi - omega(ps1:p))); X for k=1:la X if (norm_Lx_unreg <= alpha(k)) X x_alpha(:,k) = x_unreg; lambda(k) = 0; X else X lambda_0 = (s(ps1,1)/s(ps1,2))*(norm_Lx_unreg/alpha(k) - 1); X lambda(k) = heb_new(lambda_0,alpha(k),s,beta(1:p),omega); X e = gamma./(gamma.^2 + lambda(k)^2); f = gamma.*e; X x_alpha(:,k) = V(:,1:p)*(e.*beta(1:p)./s(:,2) + ... X (1-f).*s(:,2).*omega) + x_u; X end X end Xend END_OF_FILE if test 2636 -ne `wc -c <'lsqi.m'`; then echo shar: \"'lsqi.m'\" unpacked with wrong size! fi # end of 'lsqi.m' fi if test -f 'lsqr.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'lsqr.m'\" else echo shar: Extracting \"'lsqr.m'\" \(4571 characters\) sed "s/^X//" >'lsqr.m' <<'END_OF_FILE' Xfunction [X,rho,eta,F] = lsqr(A,b,k,reorth,s) X%LSQR Solution of least squares problems by Lanczos bidiagonalization. X% X% [X,rho,eta,F] = lsqr(A,b,k,reorth,s) X% X% Performs k steps of the LSQR Lanczos bidiagonalization algorithm X% applied to the system X% min || A x - b || . X% The routine returns all k solutions, stored as columns of X% the matrix X. The solution norm and residual norm are returned X% in eta and rho, respectively. X% X% If the singular values s are also provided, lsqr computes the X% filter factors associated with each step and stores them columnwise X% in the matrix F. X% X% Reorthogonalization is controlled by means of reorth: X% reorth = 0 : no reorthogonalization (default), X% reorth = 1 : reorthogonalization by means of MGS, X% reorth = 2 : Householder-reorthogonalization. X X% Reference: C. C. Paige & M. A. Saunders, "LSQR: an algorithm for X% sparse linear equations and sparse least squares", ACM Trans. X% Math. Software 8 (1982), 43-71. X X% Per Christian Hansen, UNI-C, 05/25/93. X X% The fudge threshold is used to prevent filter factors from exploding. Xfudge_thr = 1e-4; X X% Initialization. Xif (k < 1), error('Number of steps k must be positive'), end Xif (nargin==3), reorth = 0; end Xif (nargout==4 & nargin<5), error('Too few input arguments'), end X[m,n] = size(A); X = zeros(n,k); Xif (reorth==0) X UV = 0; Xelseif (reorth==1) X U = zeros(m,k); V = zeros(n,k); UV = 1; Xelseif (reorth==2) X if (k>=n), error('No. of iterations must satisfy k < n'), end X UV = 0; HHU = zeros(m,k); HHV = zeros(n,k); X HHalpha = zeros(1,k); HHbeta = HHalpha; Xelse X error('Illegal reorth') Xend Xif (nargout > 1) X eta = zeros(k,1); rho = eta; X c2 = -1; s2 = 0; xnorm = 0; z = 0; Xend Xif (nargin==5) X ls = length(s); X F = zeros(ls,k); Fv = zeros(ls,1); Fw = Fv; X s = s.^2; Xend X X% Prepare for LSQR iteration. Xv = zeros(n,1); x = v; beta = norm(b); Xif (beta==0), error('Right-hand side must be nonzero'), end Xif (reorth==2) X [beta,HHbeta(1),HHU(:,1)] = gen_hh(b); Xend Xu = b/beta; if (UV), U(:,1) = u; end Xr = A'*u; alpha = norm(r); Xif (reorth==2) X [alpha,HHalpha(1),HHV(:,1)] = gen_hh(r); Xend Xv = r/alpha; if (UV), V(:,1) = v; end Xphi_bar = beta; rho_bar = alpha; w = v; Xif (nargin==5), Fv = s/(alpha*beta); Fw = Fv; end X X% Perform Lanczos bidiagonalization with/without reorthogonalization. Xfor i=2:k+1 X X alpha_old = alpha; beta_old = beta; X X % Compute A*v - alpha*u. X p = A*v - alpha*u; X if (reorth==0) X beta = norm(p); u = p/beta; X elseif (reorth==1) X for j=1:i-1, p = p - (U(:,j)'*p)*U(:,j); end X beta = norm(p); u = p/beta; X else X for j=1:i-1 X p(j:m) = app_hh(p(j:m),HHbeta(j),HHU(j:m,j)); X end X [beta,HHbeta(i),HHU(i:m,i)] = gen_hh(p(i:m)); X u = zeros(m,1); u(i) = 1; X for j=i:-1:1 X u(j:m) = app_hh(u(j:m),HHbeta(j),HHU(j:m,j)); X end X end X X % Compute A'*u - beta*v. X r = A'*u - beta*v; X if (reorth==0) X alpha = norm(r); v = r/alpha; X elseif (reorth==1) X for j=1:i-1, r = r - (V(:,j)'*r)*V(:,j); end X alpha = norm(r); v = r/alpha; X else X for j=1:i-1 X r(j:n) = app_hh(r(j:n),HHalpha(j),HHV(j:n,j)); X end X [alpha,HHalpha(i),HHV(i:n,i)] = gen_hh(r(i:n)); X v = zeros(n,1); v(i) = 1; X for j=i:-1:1 X v(j:n) = app_hh(v(j:n),HHalpha(j),HHV(j:n,j)); X end X end X X % Store U and V if necessary. X if (UV), U(:,i) = u; V(:,i) = v; end X X % Construct and apply orthogonal transformation. X rrho = pythag(rho_bar,beta); c1 = rho_bar/rrho; X s1 = beta/rrho; theta = s1*alpha; rho_bar = -c1*alpha; X phi = c1*phi_bar; phi_bar = s1*phi_bar; X X % Compute solution norm and residual norm if necessary; X if (nargout > 1) X delta = s2*rrho; gamma_bar = -c2*rrho; rhs = phi - delta*z; X z_bar = rhs/gamma_bar; eta(i-1) = pythag(xnorm,z_bar); X gamma = pythag(gamma_bar,theta); X c2 = gamma_bar/gamma; s2 = theta/gamma; X z = rhs/gamma; xnorm = pythag(xnorm,z); X rho(i-1) = abs(phi_bar); X end X X % If required, compute the filter factors. X if (nargin==5) X X if (i==2) X Fv_old = Fv; X Fv = Fv.*(s - beta^2 - alpha_old^2)/(alpha*beta); X F(:,i-1) = (phi/rrho)*Fw; X else X tmp = Fv; X Fv = (Fv.*(s - beta^2 - alpha_old^2) - ... X Fv_old*alpha_old*beta_old)/(alpha*beta); X Fv_old = tmp; X F(:,i-1) = F(:,i-2) + (phi/rrho)*Fw; X end X if (i > 3) X f = find(abs(F(:,i-2)-1) < fudge_thr & abs(F(:,i-3)-1) < fudge_thr); X if (length(f) > 0), F(f,i-1) = ones(length(f),1); end X end X Fw = Fv - (theta/rrho)*Fw; X X end X X % Update the solution. X x = x + (phi/rrho)*w; w = v - (theta/rrho)*w; X X(:,i-1) = x; X Xend END_OF_FILE if test 4571 -ne `wc -c <'lsqr.m'`; then echo shar: \"'lsqr.m'\" unpacked with wrong size! fi # end of 'lsqr.m' fi if test -f 'ltsolve.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'ltsolve.m'\" else echo shar: Extracting \"'ltsolve.m'\" \(1083 characters\) sed "s/^X//" >'ltsolve.m' <<'END_OF_FILE' Xfunction x = ltsolve(L,y,W,NAA) X%LTSOLVE Utility routine for "preconditioned" iterative methods. X% X% x = ltsolve(L,y,W,NAA) X% X% Computes the vector x from the relation X% [ x ] = inv([ L ]')*y . X% [ z ] ([ I 0 ] ) X% Typically, L is a p-by-n band matrix with bandwidth n-p+1. X% Alternatively, L is square and dense. X% X% If W and NAA are also specified, then x = L_p*y instead, where X% L_p is the A-weighted generalized inverse of L. X% X% Notice that x and y may be matrices, in which case x(:,i) X% corresponds to y(:,i). X X% Reference: M. Hanke, "Regularization with differential operators. X% An iterative approach", J. Numer. Funct. Anal. Optim. 13 (1992), X% 523-540. X X% Per Christian Hansen, UNI-C, and Martin Hanke, Institut fuer X% Praktische Mathematik, Universitaet Karlsruhe, 05/26/93. X X% Initialization. X[p,n] = size(L); nu = n-p; [ny,ly] = size(y); X X% Special treatment of square L. Xif (nu==0), x = (L')\y; return; end X X% Perform the projection, if necessary. Xif (nargin > 2), y = y - NAA'*(W'*y); end X X% Compute x. Xx = y'/[L;[zeros(nu,p),eye(nu)]]; Xx = x(:,1:p)'; END_OF_FILE if test 1083 -ne `wc -c <'ltsolve.m'`; then echo shar: \"'ltsolve.m'\" unpacked with wrong size! fi # end of 'ltsolve.m' fi if test -f 'maxent.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'maxent.m'\" else echo shar: Extracting \"'maxent.m'\" \(4639 characters\) sed "s/^X//" >'maxent.m' <<'END_OF_FILE' Xfunction [x_lambda,rho,eta,data,X] = maxent(A,b,lambda,w,x0) X%MAXENT Maximum entropy regularization. X% X% [x_lambda,rho,eta] = maxent(A,b,lambda) X% X% Maximum entropy regularization: X% min { || A x - b ||^2 + lambda^2*x'*log(diag(w)*x) } , X% where -x'*log(diag(w)*x) is the entropy of the solution x. X% If no weights w are specified, unit weights are used. X% X% If lambda is a vector, then x_lambda is a matrix such that X% x_lambda = [x_lambda(1), x_lambda(2), ... ] . X% X% This routine uses a nonlinear conjugate gradient algorithm with "soft" X% line search and a step-length control that ensures a positive solution. X% If the starting vector x0 is not specified, then the default is X% x0 = norm(b)/norm(A,1)*ones(n,1) . X X% Per Christian Hansen, UNI-C and Tommy Elfving, Dept. of Mathematics, X% Linkoping University, 06/10/92. X X% Reference: R. Fletcher, "Practical Methods for Optimization", X% Second Edition, Wiley, Chichester, 1987. X X% Set defaults. Xflat = 1e-3; % Measures a flat minimum. Xflatrange = 10; % How many iterations before a minimum is considered flat. Xmaxit = 150; % Maximum number of CG iterations; Xminstep = 1e-12; % Determines the accuracy of x_lambda. Xsigma = 0.5; % Threshold used in descent test. Xtau0 = 1e-3; % Initial threshold used in secant root finder. X X% Initialization. X[m,n] = size(A); x_lambda = zeros(n,length(lambda)); F = zeros(maxit,1); Xif (min(lambda) <= 0) X error('Regularization parameter lambda must be positive') Xend Xif (nargin ==3), w = ones(n,1); end Xif (nargin < 5), x0 = ones(n,1); end X X% Treat each lambda separately. Xfor j=1:length(lambda); X X % Prepare for nonlinear CG iteration. X l2 = lambda(j)^2; X x = x0; Ax = A*x; X g = 2*A'*(Ax - b) + l2*(1 + log(w.*x)); X p = -g; X r = Ax - b; X X % Start the nonlinear CG iteration here. X delta_x = x; dF = 1; it = 0; phi0 = p'*g; X while (norm(delta_x) > minstep*norm(x) & dF > flat & it < maxit & phi0 < 0) X it = it + 1; X X % Compute some CG quantities. X Ap = A*p; gamma = Ap'*Ap; v = A'*Ap; X X % Determine the steplength alpha by "soft" line search in which X % the minimum of phi(alpha) = p'*g(x + alpha*p) is determined to X % a certain "soft" tolerance. X % First compute initial parameters for the root finder. X alpha_left = 0; phi_left = phi0; X if (min(p) >= 0) X alpha_right = -phi0/(2*gamma); X h = 1 + alpha_right*p./x; X else X % Step-length control to ensure a positive x + alpha*p. X I = find(p < 0); X alpha_right = min(-x(I)./p(I)); X h = 1 + alpha_right*p./x; delta = eps; X while (min(h) <= 0) X alpha_right = alpha_right*(1 - delta); X h = 1 + alpha_right*p./x; X delta = delta*2; X end X end X z = log(h); X phi_right = phi0 + 2*alpha_right*gamma + l2*p'*z; X alpha = alpha_right; phi = phi_right; X X if (phi_right <= 0) X X % Special treatment of the case when phi(alpha_right) = 0. X z = log(1 + alpha*p./x); X g_new = g + l2*z + 2*alpha*v; t = g_new'*g_new; X beta = (t - g'*g_new)/(phi - phi0); X X else X X % The regular case: improve the steplength alpha iteratively X % until the new step is a descent step. X t = 1; u = 1; tau = tau0; X while (u > -sigma*t) X X % Use the secant method to improve the root of phi(alpha) = 0 X % to within an accuracy determined by tau. X while (abs(phi/phi0) > tau) X alpha = (alpha_left*phi_right - alpha_right*phi_left)/... X (phi_right - phi_left); X z = log(1 + alpha*p./x); X phi = phi0 + 2*alpha*gamma + l2*p'*z; X if (phi > 0) X alpha_right = alpha; phi_right = phi; X else X alpha_left = alpha; phi_left = phi; X end X end X X % To check the descent step, compute u = p'*g_new and X % t = norm(g_new)^2, where g_new is the gradient at x + alpha*p. X g_new = g + l2*z + 2*alpha*v; t = g_new'*g_new; X beta = (t - g'*g_new)/(phi - phi0); X u = -t + beta*phi; X tau = tau/10; X X end % End of improvement iteration. X X end % End of regular case. X X % Update the iteration vectors. X g = g_new; delta_x = alpha*p; X x = x + delta_x; X p = -g + beta*p; X r = r + alpha*Ap; X phi0 = p'*g; X X % Compute some norms and check for flat minimum. X rho(j,1) = norm(r); eta(j,1) = x'*log(w.*x); X F(it) = rho(j,1)^2 + l2*eta(j,1); X if (it <= flatrange) X dF = 1; X else X dF = abs(F(it) - F(it-flatrange))/abs(F(it)); X end X X data(it,:) = [F(it),norm(delta_x),norm(g)]; X X(:,it) = x; X X end % End of iteration for x_lambda(j). X X x_lambda(:,j) = x; X Xend END_OF_FILE if test 4639 -ne `wc -c <'maxent.m'`; then echo shar: \"'maxent.m'\" unpacked with wrong size! fi # end of 'maxent.m' fi if test -f 'mgs.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'mgs.m'\" else echo shar: Extracting \"'mgs.m'\" \(432 characters\) sed "s/^X//" >'mgs.m' <<'END_OF_FILE' Xfunction Q = mgs(A) X%MGS Modified Gram-Schmidt orthonormalization. X% X% Q = mgs(A) X% X% Applies the modified Gram-Schmidt process to the matrix A X% (assuming that A has full rank), and returns a matrix Q whose X% columns are orthonormal and span the range of A. X X% Per Christian Hansen, UNI-C, 04/11/90. X XQ = A; [m,n] = size(A); X Xfor k=1:n X Q(:,k) = Q(:,k)/norm(Q(:,k)); X Q(:,k+1:n) = Q(:,k+1:n) - Q(:,k)*(Q(:,k)'*Q(:,k+1:n)); Xend END_OF_FILE if test 432 -ne `wc -c <'mgs.m'`; then echo shar: \"'mgs.m'\" unpacked with wrong size! fi # end of 'mgs.m' fi if test -f 'mtsvd.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'mtsvd.m'\" else echo shar: Extracting \"'mtsvd.m'\" \(1441 characters\) sed "s/^X//" >'mtsvd.m' <<'END_OF_FILE' Xfunction x_k = mtsvd(U,s,V,b,k,L) X%MTSVD Modified truncated SVD regularization. X% X% x_k = mtsvd(U,s,V,b,k,L) X% X% Computes the modified TSVD solution: X% x_k = V*[ xi_k ] . X% [ xi_0 ] X% Here, xi_k defines the usual TSVD solution X% xi_k = inv(diag(s(1:k)))*U(:,1:k)'*b , X% and xi_0 is chosen so as to minimize the seminorm || L x_k ||. X% This leads to choosing xi_0 as follows: X% xi_0 = -pinv(L*V(:,k+1:n))*L*V(:,1:k)*xi_k . X% X% The truncation parameter must satisfy k > n-p. X% X% If k is a vector, then x_k is a matrix such that X% x_k = [ x_k(1), x_k(2), ... ] . X X% Reference: P. C. Hansen, T. Sekii & H. Shibahashi, "The modified X% truncated-SVD method for regularization in general form", SIAM J. X% Sci. Stat. Comput. 13 (1992), 1142-1150. X X% Per Christian Hansen, UNI-C, 05/26/93. X X% Initialization. X[m,n1] = size(U); [p,n] = size(L); Xlk = length(k); kmin = min(k); Xif (kminn) X error('Illegal truncation parameter k') Xend Xx_k = zeros(n,lk); xi = (U(:,1:n)'*b)./s; X X% Compute large enough QR factorization. X[Q,R] = qr(L*V(:,n:-1:kmin+1)); X X% Treat each k separately. Xfor j=1:lk X kj = k(j); tmp = V(:,1:kj)*xi(1:kj); X if (kj==n) X x_k(:,j) = tmp; X else X z = R(1:n-kj,1:n-kj)\(Q(:,1:n-kj)'*(L*tmp)); X z = z(n-kj:-1:1); X x_k(:,j) = tmp - V(:,kj+1:n)*z; X end X if (m > n) X beta = U(:,1:n)'*b; X beta2 = b'*b - beta'*beta; X if (beta2 > 0), rho = sqrt(rho.^2 + beta2); end X end Xend END_OF_FILE if test 1441 -ne `wc -c <'mtsvd.m'`; then echo shar: \"'mtsvd.m'\" unpacked with wrong size! fi # end of 'mtsvd.m' fi if test -f 'newton.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'newton.m'\" else echo shar: Extracting \"'newton.m'\" \(1866 characters\) sed "s/^X//" >'newton.m' <<'END_OF_FILE' Xfunction lambda = newton(lambda_0,delta,s,beta,omega,delta_0) X%NEWTON Newton iteration (utility routine for DISCREP). X% X% lambda = newton(lambda_0,delta,s,beta,omega,delta_0) X% X% Uses Newton iteration to find the solution lambda to the equation X% || A x_lambda - b || = delta , X% where x_lambda is the solution defined by Tikhonov regularization. X% X% The initial guess is lambda_0. X% X% The norm || A x_lambda - b || is computed via s, beta, omega and X% delta_0. Here, s holds either the singular values of A, if L = I, X% or the c,s-pairs of the GSVD of (A,L), if L ~= I. Moreover, X% beta = U'*b and omega is either V'*x_0 or the first p elements of X% inv(X)*x_0. Finally, delta_0 is the incompatibility measure. X X% Reference: V. A. Morozov, "Methods for Solving Incorrectly Posed X% Problems", Springer, 1984; Chapter 26. X X% Per Christian Hansen, UNI-C, 02/21/92. X X% Set defaults. Xthr = 100*sqrt(eps); % Relative stopping criterion. Xit_max = 50; % Max number of iterations. X% lambda_0 = eps; X X% Initialization. Xif (lambda_0 < 0) X error('Initial guess lambda_0 must be nonnegative') Xend X[p,ps] = size(s); Xif (ps==2), sigma = s(:,1); mu = s(:,2); s = s(:,1)./s(:,2); end Xs2 = s.^2; X X% Iterate; avoid negative values of lambda. Xlambda = lambda_0^2; step = 1; it = 0; Xwhile (abs(step) > thr*lambda & abs(step) > thr & it < it_max), it = it+1; X f = s2./(s2 + lambda); X if (ps==1) X r = (1-f).*(beta - s.*omega); X dr = f.*(beta - omega)./(s2 + lambda); X else X r = (1-f).*(beta - sigma.*omega); X dr = f.*(beta - sigma.*omega)./(s2 + lambda); X end X res = sqrt(r'*r + delta_0^2); X step = -(res - delta)*res/(dr'*r); X lambda = lambda + step; X if (lambda <= 0), lambda = eps*max(s2); end Xend X X% Terminate with an error if too many iterations. Xif (abs(step) > thr*lambda) X error('Max. number of iterations reached') Xend X Xlambda = sqrt(lambda); END_OF_FILE if test 1866 -ne `wc -c <'newton.m'`; then echo shar: \"'newton.m'\" unpacked with wrong size! fi # end of 'newton.m' fi if test -f 'nu.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'nu.m'\" else echo shar: Extracting \"'nu.m'\" \(2725 characters\) sed "s/^X//" >'nu.m' <<'END_OF_FILE' Xfunction [X,rho,eta,F] = nu(A,b,k,nu,s) X%NU Brakhage's nu-method. X% X% [X,rho,eta,F] = nu(A,b,k,nu,s) X% X% Performs k steps of Brakhage's nu-method for the problem X% min || A x - b || . X% The routine returns all k solutions, stored as columns of X% the matrix X. The solution norm and residual norm are returned X% in eta and rho, respectively. X% X% If nu is not specified, nu = .5 is the default value, which gives X% the Chebychev method of Nemirovskii and Polyak. X% X% If the singular values s are also provided, nu computes the X% filter factors associated with each step and stores them X% columnwise in the matrix F. X X% Reference: H. Brakhage, "On ill-posed problems and the method of X% conjugate gradients"; in H. W. Engl & G. W. Groetsch, "Inverse and X% Ill-Posed Problems", Academic Press, 1987. X X% Martin Hanke, Institut fuer Praktische Mathematik, Universitaet X% Karlsruhe and Per Christian Hansen, UNI-C, 03/21/92. X X% Set parameter. Xl_steps = 3; % Number of Lanczos steps for est. of || A ||. Xfudge = 0.99; % Scale A and b by fudge/|| A*L_p ||. Xfudge_thr = 1e-4; % Used to prevent filter factors from exploding. X X% Initialization. Xif (k < 1), error('Number of steps k must be positive'), end Xif (nargin==3), nu = .5; end X[m,n] = size(A); X = zeros(n,k); Xif (nargout > 1) X rho = zeros(k,1); eta = rho; Xend; Xif (nargin==5) X F = zeros(n,k); Fd = zeros(n,1); s = s.^2; Xend XV = zeros(n,l_steps); B = zeros(l_steps+1,l_steps); Xv = zeros(n,1); eta = zeros(l_steps+1,1); X X% Compute a rough estimate of the norm of A by means of a few X% steps of Lanczos bidiagonalization, and scale A and b such X% that || A || is slightly less than one. Xbeta = norm(b); u = b/beta; Xfor i=1:l_steps X r = A'*u - beta*v; X alpha = norm(r); v = r/alpha; X B(i,i) = alpha; V(:,i) = v; X p = A*v - alpha*u; X beta = norm(p); u = p/beta; X B(i+1,i) = beta; Xend Xscale = fudge/norm(B); A = scale*A; b = scale*b; Xif (nargin==5), s = scale^2*s; end X X% Prepare for iteration. Xx = zeros(n,1); Xd = A'*b; Xr = d; Xif (nargout>1), z = b; end X X% Iterate. Xfor j=0:k-1 X X alpha = 4*(j+nu)*(j+nu+0.5)/(j+2*nu)/(j+2*nu+0.5); X beta = (j+nu)*(j+1)*(j+0.5)/(j+2*nu)/(j+2*nu+0.5)/(j+nu+1); X Ad = A*d; AAd = A'*Ad; X x = x + alpha*d; X r = r - alpha*AAd; X d = r + beta*d; X X(:,j+1) = x; X if (nargout>1) X z = z - alpha*Ad; rho(j+1) = norm(z)/scale; X end; X if (nargout>2), eta(j+1) = norm(x); end; X X if (nargin==5) X if (j==0) X F(:,1) = alpha*s; X Fd = s - s.*F(:,1) + beta*s; X else X F(:,j+1) = F(:,j) + alpha*Fd; X Fd = s - s.*F(:,j+1) + beta*Fd; X end X if (j > 1) X f = find(abs(F(:,j)-1) < fudge_thr & abs(F(:,j-1)-1) < fudge_thr); X if (length(f) > 0), F(f,j+1) = ones(length(f),1); end X end X end X Xend END_OF_FILE if test 2725 -ne `wc -c <'nu.m'`; then echo shar: \"'nu.m'\" unpacked with wrong size! fi # end of 'nu.m' fi if test -f 'parallax.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'parallax.m'\" else echo shar: Extracting \"'parallax.m'\" \(1856 characters\) sed "s/^X//" >'parallax.m' <<'END_OF_FILE' Xfunction [A,b] = parallax(n) X%PARALLAX Stellar parallax problem with 28 fixed, real observations. X% X% [A,b] = parallax(n) X% X% Stellar parallax problem with 28 fixed, real observations. X% X% The underlying problem is a Fredholm integral equation of the X% first kind with kernel X% K(s,t) = (1/sigma*sqrt(2*pi))*esp(-0.5*((s-t)/sigma)^2) , X% and it is discretized by means of a Galerkin method with n X% orthonormal basis functions. The right-hand side consists of X% a measured distribution function of stellar parallaxes, and its X% length is fixed, m = 26. The exact solution, which represents X% the true distribution of stellar parallaxes, in not known. X X% Reference: W. M. Smart, "Stellar Dynamics", Cambridge X% University Press, 1938; p. 30. X X% Discretized by Galerkin method with orthonormal box functions; X% 2-D integration is done by means of the computational molecule: X% 1 4 1 X% 4 16 1 X% 1 4 1 X X% Per Christian Hansen, UNI-C, 09/16/92. X X% Initialization. Xa = 0; b = 0.1; m = 26; sigma = 0.014234; Xhs = 0.130/m; hx = (b-a)/n; hsh = hs/2; hxh = hx/2; Xss = (-0.03 + [0:m-1]'*hs)*ones(1,n); Xxx = ones(m,1)*(a + [0:n-1]*hx); X X% Set up the matrix. XA = 16*exp(-0.5*((ss+hsh - xx-hxh)/sigma).^2); XA = A + 4*(exp(-0.5*((ss+hsh - xx )/sigma).^2) + ... X exp(-0.5*((ss+hsh - xx-hx )/sigma).^2) + ... X exp(-0.5*((ss - xx-hxh)/sigma).^2) + ... X exp(-0.5*((ss+hs - xx-hxh)/sigma).^2)); XA = A + (exp(-0.5*((ss - xx )/sigma).^2) + ... X exp(-0.5*((ss+hs - xx )/sigma).^2) + ... X exp(-0.5*((ss - xx-hx )/sigma).^2) + ... X exp(-0.5*((ss+hs - xx-hx )/sigma).^2)); XA = sqrt(hs*hx)/(36*sigma*sqrt(2*pi))*A; X X% Set up the normalized right-hand side. Xb = [3;7;7;17;27;39;46;51;56;50;43;45;43;32;33;29;... X 21;12;17;13;15;12;6;6;5;5]/(sqrt(hs)*640); END_OF_FILE if test 1856 -ne `wc -c <'parallax.m'`; then echo shar: \"'parallax.m'\" unpacked with wrong size! fi # end of 'parallax.m' fi if test -f 'pcgls.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'pcgls.m'\" else echo shar: Extracting \"'pcgls.m'\" \(2810 characters\) sed "s/^X//" >'pcgls.m' <<'END_OF_FILE' Xfunction [X,rho,eta,F] = pcgls(A,L,W,b,k,sm) X%PCGLS "Preconditioned" conjugate gradients appl. implicitly to normal equations. X% [X,rho,eta,F] = pcgls(A,L,W,b,k,sm) X% X% Performs k steps of the `preconditioned' conjugate gradient X% algorithm applied implicitly to the normal equations X% (A*L_p)'*(A*L_p)*x = (A*L_p)'*b , X% where L_p is the A-weighted generalized inverse of L. Notice X% that the matrix W holding a basis for the null space of L must X% also be specified. X% X% The routine returns all k solutions, stored as columns of the X% matrix X. The solution seminorm and residual norm are returned in eta X% and rho, respectively. X% X% If the generalized singular values sm of (A,L) are also provided, X% pcgls computes the filter factors associated with each step and X% stores them columnwise in the matrix F. X X% References: A. Bjorck, "Least Squares Methods", in P. G. X% Ciarlet & J. L Lions (Eds.), "Handbook of Numerical Analysis, X% Vol. I", Elsevier, Amsterdam, 1990; p. 560. X% M. Hanke, "Regularization with differential operators. An itera- X% tive approach", J. Numer. Funct. Anal. Optim. 13 (1992), 523-540. X% C. R. Vogel, "Solving ill-conditioned linear systems using the X% conjugate gradient method", Report, Dept. of Mathematical X% Sciences, Montana State University, 1987. X X% Per Christian Hansen, UNI-C and Martin Hanke, Institut fuer X% Praktische Mathematik, Universitaet Karlsruhe, 11/05/92. X X% The fudge threshold is used to prevent filter factors from exploding. Xfudge_thr = 1e-4; X X% Initialization Xif (k < 1), error('Number of steps k must be positive'), end X[m,n] = size(A); [p,n1] = size(L); X = zeros(n,k); Xif (nargout > 1) X eta = zeros(k,1); rho = eta; Xend Xif (nargout>3 & nargin==5), error('Too few imput arguments'), end Xif (nargin==6) X F = zeros(p,k); Fd = zeros(p,1); gamma = (sm(:,1)./sm(:,2)).^2; Xend X X% Prepare for computations with L_p. X[NAA,x_0] = pinit(W,A,b); X X% Prepare for CG iteartion. Xx = x_0; Xr = b - A*x_0; s = A'*r; Xq1 = ltsolve(L,s); Xq = lsolve(L,q1,W,NAA); Xz = q; Xdq = s'*q; Xif (nargout>2), z1 = q1; x1 = zeros(p,1); end X X% Iterate. Xfor j=1:k X X Az = A*z; alpha = dq/(Az'*Az); X x = x + alpha*z; X r = r - alpha*Az; s = A'*r; X q1 = ltsolve(L,s); X q = lsolve(L,q1,W,NAA); X dq2 = s'*q; beta = dq2/dq; X dq = dq2; X z = q + beta*z; X X(:,j) = x; X if (nargout>1), rho(j) = norm(r); end X if (nargout>2) X x1 = x1 + alpha*z1; z1 = q1 + beta*z1; eta(j) = norm(x1); X end X X if (nargin==6) X if (j==1) X F(:,1) = alpha*gamma; X Fd = gamma - gamma.*F(:,1) + beta*gamma; X else X F(:,j) = F(:,j-1) + alpha*Fd; X Fd = gamma - gamma.*F(:,j) + beta*Fd; X end X if (j > 2) X f = find(abs(F(:,j-1)-1) < fudge_thr & abs(F(:,j-2)-1) < fudge_thr); X if (length(f) > 0), F(f,j) = ones(length(f),1); end X end X end X Xend END_OF_FILE if test 2810 -ne `wc -c <'pcgls.m'`; then echo shar: \"'pcgls.m'\" unpacked with wrong size! fi # end of 'pcgls.m' fi if test -f 'phillips.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'phillips.m'\" else echo shar: Extracting \"'phillips.m'\" \(1685 characters\) sed "s/^X//" >'phillips.m' <<'END_OF_FILE' Xfunction [A,b,x] = phillips(n) X%PHILLIPS Phillips' "famous" test problem. X% X% [A,b,x] = phillips(n) X% X% Discretization of the `famous' first-kind Fredholm integral X% equation deviced by D. L. Phillips. Define the function X% phi(x) = | 1 + cos(x*pi/3) , |x| < 3 . X% | 0 , |x| >= 3 X% Then the kernel K, the solution f, and the right-hand side X% g are given by: X% K(s,t) = phi(s-t) , X% f(t) = phi(t) , X% g(s) = (6-|s|)*(1+.5*cos(s*pi/3)) + 9/(2*pi)*sin(|s|*pi/3) . X% Both integration intervals are [-6,6]. X% X% The order n must be a multiple of 4. X X% Reference: D. L. Phillips, "A technique for the numerical solution X% of certain integral equations of the first kind", J. ACM 9 X% (1962), 84-97. X X% Discretized by Galerkin method with orthonormal box functions. X X% Per Christian Hansen, UNI-C, 09/17/92. X X% Check input. Xif (rem(n,4)~=0), error('The order n must be a multiple of 4'), end X X% Compute the matrix A. Xh = 12/n; n4 = n/4; r1 = zeros(1,n); Xc = cos([-1:n4]*4*pi/n); Xr1(1:n4) = h + 9/(h*pi^2)*(2*c(2:n4+1) - c(1:n4) - c(3:n4+2)); Xr1(n4+1) = h/2 + 9/(h*pi^2)*(cos(4*pi/n)-1); XA = toeplitz(r1); X X% Compute the right-hand side b. Xif (nargout>1), X b = zeros(n,1); c = pi/3; X for i=n/2+1:n X t1 = -6 + i*h; t2 = t1 - h; X b(i) = t1*(6-abs(t1)/2) ... X + ((3-abs(t1)/2)*sin(c*t1) - 2/c*(cos(c*t1) - 1))/c ... X - t2*(6-abs(t2)/2) ... X - ((3-abs(t2)/2)*sin(c*t2) - 2/c*(cos(c*t2) - 1))/c; X b(n-i+1) = b(i); X end X b = b/sqrt(h); Xend X X% Compute the solution x. Xif (nargout==3), X x = zeros(n,1); X x(2*n4+1:3*n4) = (h + diff(sin([0:h:(3+10*eps)]'*c))/c)/sqrt(h); X x(n4+1:2*n4) = x(3*n4:-1:2*n4+1); Xend END_OF_FILE if test 1685 -ne `wc -c <'phillips.m'`; then echo shar: \"'phillips.m'\" unpacked with wrong size! fi # end of 'phillips.m' fi if test -f 'picard.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'picard.m'\" else echo shar: Extracting \"'picard.m'\" \(1286 characters\) sed "s/^X//" >'picard.m' <<'END_OF_FILE' Xfunction xi = picard(U,s,b,d) X%PICARD Visual inspection of the Picard condition. X% X% xi = picard(U,s,b,d) X% xi = picard(U,sm,b,d) , sm = [sigma,mu] X% X% Plots the singular values, s(i), the abs. value of the Fourier X% coefficients, |U(:,i)'*b|, and a (possibly smoothed) curve of X% the solution coefficients xi(i) = |U(:,i)'*b|/s(i). X% X% If s = [sigma,mu], where gamma = sigma./mu are the generalized X% singular values, then this routine plots gamma(i), |U(:,i)'*b|, X% and (smoothed) xi(i) = |U(:,i)'*b|/gamma(i). X% X% The smoothing is a geometric mean over 2*d+1 points, centered X% at point # i. If nargin = 3, then d = 0 (i.e, no smothing). X X% Reference: P. C. Hansen, "The discrete Picard condition for X% discrete ill-posed problems", BIT 30 (1990), 658-672. X X% Per Christian Hansen, UNI-C, 04/11/90. X X% Initialization. X[n,ps] = size(s); beta = abs(U(:,1:n)'*b); eta = zeros(n,1); Xif (nargin==3), d = 0; end; Xif (ps==2), s = s(:,1)./s(:,2); end Xd21 = 2*d+1; keta = 1+d:n-d; Xfor i=keta X eta(i) = (prod(beta(i-d:i+d))^(1/d21))/s(i); Xend X X% Plot the data. Xsemilogy(1:n,s,'-',1:n,beta,'x',keta,eta(keta),'o') Xxlabel('i') Xif (ps==1) X title('--- = sing. values x = rhs. coef. o = solution coef.') Xelse X title('--- = gen. sing. values x = rhs. coef. o = sol. coef.') Xend END_OF_FILE if test 1286 -ne `wc -c <'picard.m'`; then echo shar: \"'picard.m'\" unpacked with wrong size! fi # end of 'picard.m' fi if test -f 'pinit.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'pinit.m'\" else echo shar: Extracting \"'pinit.m'\" \(1004 characters\) sed "s/^X//" >'pinit.m' <<'END_OF_FILE' Xfunction [NAA,x_0] = pinit(W,A,b) X%PINIT Utility init.-procedure for "preconditioned" iterative methods. X% X% NAA = pinit(W,A) X% [NAA,x_0] = pinit(W,A,b) X% X% Initialization for `preconditioning' of general-form problems. X% Here, W holds a basis for the null space of L. X% X% Determines the matrix NAA needed in the iterative routines for X% treating regularization problems in general form. X% X% If b is also specified then x_0, the component of the solution in X% the null space of L, is also computed. X X% Reference: M. Hanke, "Regularization with differential operators. X% An iterative approach", J. Numer. Funct. Anal. Optim. 13 (1992), X% 523-540. X X% Per Christian Hansen, UNI-C, and Martin Hanke, Institut fuer X% Praktische Mathematik, Universitaet Karlsruhe, 05/26/93. X X% Initialization. X[n,nu] = size(W); X X% Special treatment of square L. Xif (nu==0), NAA = []; x_0 = zeros(n,1); return, end X X% Compute NAA. XT = pinv(A*W); XNAA = T*A; X X% If required, also compute x_0. Xif (nargin==3), x_0 = W*(T*b); end END_OF_FILE if test 1004 -ne `wc -c <'pinit.m'`; then echo shar: \"'pinit.m'\" unpacked with wrong size! fi # end of 'pinit.m' fi if test -f 'plot_lc.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'plot_lc.m'\" else echo shar: Extracting \"'plot_lc.m'\" \(1739 characters\) sed "s/^X//" >'plot_lc.m' <<'END_OF_FILE' Xfunction plot_lc(rho,eta,marker,ps,reg_param) X%PLOT_LC Plot the L-curve. X% X% plot_lc(rho,eta,marker,ps,reg_param) X% X% Plots the L-shaped curve of the solution norm X% eta = || x || if ps = 1 X% eta = || L x || if ps = 2 X% as a function of the residual norm rho = || A x - b ||. If ps is X% not specified, the value ps = 1 is assumed. X% X% The text string marker is used as marker. If marker is not X% specified, the marker '-' is used. X% X% If a fifth argument reg_param is present, holding the regularization X% parameters corresponding to rho and eta, then some points on the X% L-curve are identified by their corresponding parameter. X X% Per Christian Hansen, UNI-C, 03/17/93. X X% Set defaults. Xif (nargin==2), marker = '-'; end % Default marker. Xif (nargin < 4), ps = 1; end % Std. form is default. Xnp = 10; % Number of identified points. X X% Initialization. Xif (ps < 1 | ps > 2), error('Illegal value of ps'), end Xn = length(rho); ni = round(n/np); X X% Make plot. Xif (max(eta)/min(eta) > 10 | max(rho)/min(rho) > 10) X if (nargin < 5) X loglog(rho,eta,marker) X else X loglog(rho,eta,marker,rho(ni:ni:n),eta(ni:ni:n),'x') X HoldState = ishold; hold on; X for k = ni:ni:n X text(rho(k),eta(k),num2str(reg_param(k))); X end X if (~HoldState), hold off; end X end Xelse X if (nargin < 5) X plot(rho,eta,marker) X else X plot(rho,eta,marker,rho(ni:ni:n),eta(ni:ni:n),'x') X HoldState = ishold; hold on; X for k = ni:ni:n X text(rho(k),eta(k),num2str(reg_param(k))); X end X if (~HoldState), hold off; end X end Xend Xxlabel('residual norm || A x - b ||') Xif (ps==1) X ylabel('solution norm || x ||') Xelse X ylabel('solution semi-norm || L x ||') Xend Xtitle('L-curve') END_OF_FILE if test 1739 -ne `wc -c <'plot_lc.m'`; then echo shar: \"'plot_lc.m'\" unpacked with wrong size! fi # end of 'plot_lc.m' fi if test -f 'plsqr.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'plsqr.m'\" else echo shar: Extracting \"'plsqr.m'\" \(5334 characters\) sed "s/^X//" >'plsqr.m' <<'END_OF_FILE' Xfunction [X,rho,eta,F] = plsqr(A,L,W,b,k,reorth,sm) X%PLSQR "Preconditioned" version of the LSQR Lanczos bidiagonalization algorithm. X% X% [X,rho,eta,F] = plsqr(A,L,W,b,k,reorth,sm) X% X% Performs k steps of the `preconditioned' LSQR Lanczos X% bidiagonalization algorithm applied to the system X% min || (A*L_p) x - b || , X% where L_p is the A-weighted generalized inverse of L. Notice X% that the matrix W holding a basis for the null space of L must X% also be specified. X% X% The routine returns all k solutions, stored as columns of X% the matrix X. The solution seminorm and the residual norm are X% returned in eta and rho, respectively. X% X% If the generalized singular values sm of (A,L) are also provided, X% then glsqr computes the filter factors associated with each step X% and stores them columnwise in the matrix F. X% X% Reorthogonalization is controlled by means of reorth: X% reorth = 0 : no reorthogonalization (default), X% reorth = 1 : reorthogonalization by means of MGS, X% reorth = 2 : Householder-reorthogonalization. X X% References: C. C. Paige & M. A. Saunders, "LSQR: an algorithm for X% sparse linear equations and sparse least squares", ACM Trans. X% Math. Software 8 (1982), 43-71. X% M. Hanke, "Regularization with differential operators. An itera- X% tive approach", J. Numer. Funct. Anal. Optim. 13 (1992), 523-540. X X% Per Christian Hansen, UNI-C, 05/26/93. X X% The fudge threshold is used to prevent filter factors from exploding. Xfudge_thr = 1e-4; X X% Initialization Xif (k < 1), error('Number of steps k must be positive'), end Xif (nargin==5), reorth = 0; end Xif (nargout==4 & nargin<7), error('Too few input arguments'), end X[m,n] = size(A); X = zeros(n,k); [pp,n1] = size(L); Xif (n1 ~= n | m < n | n < pp) X error('Incorrect dimensions of A and L') Xend Xif (reorth==0) X UV = 0; Xelseif (reorth==1) X U = zeros(m,k); V = zeros(pp,k); UV = 1; Xelseif (reorth==2) X if (k>=n), error('No. of iterations must satisfy k < n'), end X UV = 0; HHU = zeros(m,k); HHV = zeros(pp,k); X HHalpha = zeros(1,k); HHbeta = HHalpha; Xelse X error('Illegal reorth') Xend Xif (nargout > 1) X eta = zeros(k,1); rho = eta; X c2 = -1; s2 = 0; xnorm = 0; z = 0; Xend Xif (nargin==7) X [ls,ms] = size(sm); X F = zeros(ls,k); Fv = zeros(ls,1); Fw = Fv; X s = (sm(:,1)./sm(:,2)).^2; Xend X X% Prepare for computations with L_p. X[NAA,x_0] = pinit(W,A,b); X X% By subtracting the component A*x_0 from b we ensure that X% the corrent residual norms are computed. Xb = b - A*x_0; X X% Prepare for LSQR iteration. Xv = zeros(pp,1); x = v; beta = norm(b); Xif (beta==0), error('Right-hand side must be nonzero'), end Xif (reorth==2) X [beta,HHbeta(1),HHU(:,1)] = gen_hh(b); Xend Xu = b/beta; if (UV), U(:,1) = u; end Xr = ltsolve(L,A'*u,W,NAA); alpha = norm(r); Xif (reorth==2) X [alpha,HHalpha(1),HHV(:,1)] = gen_hh(r); Xend Xv = r/alpha; if (UV), V(:,1) = v; end Xphi_bar = beta; rho_bar = alpha; w = v; Xif (nargin==7), Fv = s/(alpha*beta); Fw = Fv; end X X% Perform Lanczos bidiagonalization with/without reorthogonalization. Xfor i=2:k+1 X X alpha_old = alpha; beta_old = beta; X X % Compute (A*L_p)*v - alpha*u. X p = A*lsolve(L,v,W,NAA) - alpha*u; X if (reorth==0) X beta = norm(p); u = p/beta; X elseif (reorth==1) X for j=1:i-1, p = p - (U(:,j)'*p)*U(:,j); end X beta = norm(p); u = p/beta; X else X for j=1:i-1 X p(j:m) = app_hh(p(j:m),HHbeta(j),HHU(j:m,j)); X end X [beta,HHbeta(i),HHU(i:m,i)] = gen_hh(p(i:m)); X u = zeros(m,1); u(i) = 1; X for j=i:-1:1 X u(j:m) = app_hh(u(j:m),HHbeta(j),HHU(j:m,j)); X end X end X X % Compute L_p'*A'*u - beta*v. X r = ltsolve(L,A'*u,W,NAA) - beta*v; X if (reorth==0) X alpha = norm(r); v = r/alpha; X elseif (reorth==1) X for j=1:i-1, r = r - (V(:,j)'*r)*V(:,j); end X alpha = norm(r); v = r/alpha; X else X for j=1:i-1 X r(j:pp) = app_hh(r(j:pp),HHalpha(j),HHV(j:pp,j)); X end X [alpha,HHalpha(i),HHV(i:pp,i)] = gen_hh(r(i:pp)); X v = zeros(pp,1); v(i) = 1; X for j=i:-1:1 X v(j:pp) = app_hh(v(j:pp),HHalpha(j),HHV(j:pp,j)); X end X end X X % Store U and V if necessary. X if (UV), U(:,i) = u; V(:,i) = v; end X X % Construct and apply orthogonal transformation. X rrho = pythag(rho_bar,beta); c1 = rho_bar/rrho; X s1 = beta/rrho; theta = s1*alpha; rho_bar = -c1*alpha; X phi = c1*phi_bar; phi_bar = s1*phi_bar; X X % Compute solution norm and residual norm if necessary; X if (nargout > 1) X delta = s2*rrho; gamma_bar = -c2*rrho; rhs = phi - delta*z; X z_bar = rhs/gamma_bar; eta(i-1) = pythag(xnorm,z_bar); X gamma = pythag(gamma_bar,theta); X c2 = gamma_bar/gamma; s2 = theta/gamma; X z = rhs/gamma; xnorm = pythag(xnorm,z); X rho(i-1) = abs(phi_bar); X end X X % If required, compute the filter factors. X if (nargin==7) X X if (i==2) X Fv_old = Fv; X Fv = Fv.*(s - beta^2 - alpha_old^2)/(alpha*beta); X F(:,i-1) = (phi/rrho)*Fw; X else X tmp = Fv; X Fv = (Fv.*(s - beta^2 - alpha_old^2) - ... X Fv_old*alpha_old*beta_old)/(alpha*beta); X Fv_old = tmp; X F(:,i-1) = F(:,i-2) + (phi/rrho)*Fw; X end X if (i > 3) X f = find(abs(F(:,i-2)-1) < fudge_thr & abs(F(:,i-3)-1) < fudge_thr); X if (length(f) > 0), F(f,i-1) = ones(length(f),1); end X end X Fw = Fv - (theta/rrho)*Fw; X X end X X % Update the solution. X x = x + (phi/rrho)*w; w = v - (theta/rrho)*w; X X(:,i-1) = lsolve(L,x,W,NAA) + x_0; X Xend END_OF_FILE if test 5334 -ne `wc -c <'plsqr.m'`; then echo shar: \"'plsqr.m'\" unpacked with wrong size! fi # end of 'plsqr.m' fi if test -f 'pnu.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'pnu.m'\" else echo shar: Extracting \"'pnu.m'\" \(3323 characters\) sed "s/^X//" >'pnu.m' <<'END_OF_FILE' Xfunction [X,rho,eta,F] = pnu(A,L,W,b,k,nu,sm) X%PNU "Preconditioned" version of Brakhage's nu-method. X% X% [X,rho,eta,F] = pnu(A,L,W,b,k,nu,sm) X% X% Performs k steps of a `preconditioned' version of Brakhage's X% nu-method for the problem X% min || (A*L_p) x - b || , X% where L_p is the A-weighted generalized inverse of L. Notice X% that the matrix W holding a basis for the null space of L must X% also be specified. X% X% The routine returns all k solutions, stored as columns of X% the matrix X. The solution seminorm and residual norm are returned X% in eta and rho, respectively. X% X% If nu is not specified, nu = .5 is the default value, which gives X% the Chebychev method of Nemirovskii and Polyak. X% X% If the generalized singular values sm of (A,L) are also provided, X% then pnu computes the filter factors associated with each step and X% stores them columnwise in the matrix F. X X% Reference: H. Brakhage, "On ill-posed problems and the method of X% conjugate gradients"; in H. W. Engl & G. W. Groetsch, "Inverse and X% Ill-Posed Problems", Academic Press, 1987. X X% Martin Hanke, Institut fuer Praktische Mathematik, Universitaet X% Karlsruhe and Per Christian Hansen, UNI-C, 06/25/92. X X% Set parameters. Xl_steps = 3; % Number of Lanczos steps for est. of || A*L_p ||. Xfudge = 0.99; % Scale A and b by fudge/|| A*L_p ||. Xfudge_thr = 1e-4; % Used to prevent filter factors from exploding. X X% Initialization. Xif (k < 1), error('Number of steps k must be positive'), end Xif (nargin==5), nu = .5; end X[m,n] = size(A); [p,n1] = size(L); X = zeros(n,k); Xif (nargout > 1) X rho = zeros(k,1); eta = rho; Xend; Xif (nargin==7) X F = zeros(n,k); Fd = zeros(n,1); s = (sm(:,1)./sm(:,2)).^2; Xend XV = zeros(p,l_steps); B = zeros(l_steps+1,l_steps); Xv = zeros(p,1); eta = zeros(l_steps+1,1); X X% Prepare for computations with L_p. X[NAA,x_0] = pinit(W,A,b); x1 = x_0; X X% Compute a rough estimate of || A*L_p || by means of a few X% steps of Lanczos bidiagonalization, and scale A and b such X% that || A*L_p || is slightly less than one. Xb_0 = b - A*x_0; beta = norm(b_0); u = b_0/beta; Xfor i=1:l_steps X r = ltsolve(L,A'*u,W,NAA) - beta*v; X alpha = norm(r); v = r/alpha; X B(i,i) = alpha; V(:,i) = v; X p = A*lsolve(L,v,W,NAA) - alpha*u; X beta = norm(p); u = p/beta; X B(i+1,i) = beta; Xend Xscale = fudge/norm(B); A = scale*A; b = scale*b; Xif (nargin==7), s = scale^2*s; end X X% Prepare for iteration. Xx = x_0; Xz = -scale*b_0; Xr = A'*z; Xd1 = ltsolve(L,r); Xd = lsolve(L,d1,W,NAA); Xif (nargout>2), x1 = L*x_0; end X X% Iterate. Xfor j=0:k-1 X X alpha = 4*(j+nu)*(j+nu+0.5)/(j+2*nu)/(j+2*nu+0.5); X beta = -(j+nu)*(j+1)*(j+0.5)/(j+2*nu)/(j+2*nu+0.5)/(j+nu+1); X Ad = A*d; AAd = A'*Ad; X x = x - alpha*d; X r = r - alpha*AAd; X rr1 = ltsolve(L,r); X rr = lsolve(L,rr1,W,NAA); X d = rr - beta*d; X X(:,j+1) = x; X if (nargout>1 ) X z = z - alpha*Ad; rho(j+1) = norm(z)/scale; X end; X if (nargout>2) X x1 = x1 - alpha*d1; d1 = rr1 - beta*d1; X eta(j+1) = norm(x1); X end; X X if (nargin==7) X if (j==0) X F(:,1) = alpha*s; X Fd = s - s.*F(:,1) + beta*s; X else X F(:,j+1) = F(:,j) + alpha*Fd; X Fd = s - s.*F(:,j+1) + beta*Fd; X end X if (j > 1) X f = find(abs(F(:,j)-1) < fudge_thr & abs(F(:,j-1)-1) < fudge_thr); X if (length(f) > 0), F(f,j+1) = ones(length(f),1); end X end X end X Xend END_OF_FILE if test 3323 -ne `wc -c <'pnu.m'`; then echo shar: \"'pnu.m'\" unpacked with wrong size! fi # end of 'pnu.m' fi if test -f 'ppbrk.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'ppbrk.m'\" else echo shar: Extracting \"'ppbrk.m'\" \(93 characters\) sed "s/^X//" >'ppbrk.m' <<'END_OF_FILE' Xfunction [breaks,coefs,l,k,d]=ppbrk(pp,print) X%PPBRK Dummy function for Regularization Tools END_OF_FILE if test 93 -ne `wc -c <'ppbrk.m'`; then echo shar: \"'ppbrk.m'\" unpacked with wrong size! fi # end of 'ppbrk.m' fi if test -f 'ppcut.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'ppcut.m'\" else echo shar: Extracting \"'ppcut.m'\" \(76 characters\) sed "s/^X//" >'ppcut.m' <<'END_OF_FILE' Xfunction pc=ppcut(pp,bounds) X%PPCUT Dummy function for Regularization Tools END_OF_FILE if test 76 -ne `wc -c <'ppcut.m'`; then echo shar: \"'ppcut.m'\" unpacked with wrong size! fi # end of 'ppcut.m' fi if test -f 'ppmak.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'ppmak.m'\" else echo shar: Extracting \"'ppmak.m'\" \(81 characters\) sed "s/^X//" >'ppmak.m' <<'END_OF_FILE' Xfunction pp=ppmak(breaks,coefs,d) X%PPMAK Dummy function for Regularization Tools END_OF_FILE if test 81 -ne `wc -c <'ppmak.m'`; then echo shar: \"'ppmak.m'\" unpacked with wrong size! fi # end of 'ppmak.m' fi if test -f 'ppual.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'ppual.m'\" else echo shar: Extracting \"'ppual.m'\" \(71 characters\) sed "s/^X//" >'ppual.m' <<'END_OF_FILE' Xfunction v=ppual(pp,xx) X%PPUAL Dummy function for Regularization Tools END_OF_FILE if test 71 -ne `wc -c <'ppual.m'`; then echo shar: \"'ppual.m'\" unpacked with wrong size! fi # end of 'ppual.m' fi if test -f 'pythag.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'pythag.m'\" else echo shar: Extracting \"'pythag.m'\" \(314 characters\) sed "s/^X//" >'pythag.m' <<'END_OF_FILE' Xfunction x = pythag(y,z) X%PYTHAG Computes sqrt( y^2 + z^2 ). X% X% x = pythag(y,z) X% X% Returns sqrt(y^2 + z^2) but is careful to scale to avoid overflow. X X% Christian H. Bischof, Argonne National Laboratory, 03/31/89. X Xrmax = max(abs([y;z])); Xif (rmax==0) X x = 0; Xelse X x = rmax*sqrt((y/rmax)^2 + (z/rmax)^2); Xend END_OF_FILE if test 314 -ne `wc -c <'pythag.m'`; then echo shar: \"'pythag.m'\" unpacked with wrong size! fi # end of 'pythag.m' fi if test -f 'quasiopt.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'quasiopt.m'\" else echo shar: Extracting \"'quasiopt.m'\" \(2949 characters\) sed "s/^X//" >'quasiopt.m' <<'END_OF_FILE' Xfunction [reg_min,Q,reg_param] = quasiopt(U,s,b,method) X%QUASIOPT Quasi-optimality criterion for choosing the regularization parameter. X% X% [reg_min,Q,reg_param] = quasiopt(U,s,b,method) X% [reg_min,Q,reg_param] = quasiopt(U,sm,b,method) , sm = [sigma,mu] X% X% Plots the quasi-optimality function Q for the following methods: X% method = 'Tikh' : Tikhonov regularization (solid line ) X% method = 'tsvd' : truncated SVD or GSVD (o markers ) X% method = 'dsvd' : damped SVD or GSVD (dotted line) X% If no method is specified, 'Tikh' is default. X% X% If any output arguments are specified, then the minimum of Q is X% identified and the corresponding reg. parameter reg_min is returned. X X% Reference: T. Kitagawa, "A deterministic approach to optimal X% regularization - the finite dimensional case", Japan J. Appl. X% Math. 4 (1987), 371-391. X X% Per Christian Hansen, UNI-C, 03/17/93. X X% Set defaults. Xnpoints = 80; % Number of points for 'Tikh' and 'dsvd'. Xif (nargin==3), method = 'Tikh'; end % Default method. X X% Initialization. X[m,n] = size(U); [p,ps] = size(s); Xif (ps==2), s = s(p:-1:1,1)./s(p:-1:1,2); U = U(:,p:-1:1); end Xxi = (U'*b)./s; Xif (nargout > 0), find_min = 1; else find_min = 0; end X X% Compute the quasioptimality function Q. Xif (method(1:4)=='Tikh' | method(1:4)=='tikh') X X Q = zeros(npoints,1); reg_param = Q; X reg_param(npoints) = s(p); X ratio = (s(1)/s(p))^(1/(npoints-1)); X for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end X for i=1:npoints X f = (s.^2)./(s.^2 + reg_param(i)^2); X Q(i) = norm((1 - f).*f.*xi); X end X Xelseif (method(1:4)=='dsvd' | method(1:4)=='dgsv') X X Q = zeros(npoints,1); reg_param = Q; X reg_param(npoints,1) = s(p); X ratio = (s(1)/s(p))^(1/(npoints-1)); X for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end X for i=1:npoints X f = s./(s + reg_param(i)); X Q(i) = norm((1 - f).*f.*xi); X end X Xelseif (method(1:4)=='tsvd' | method(1:4)=='tgsv') X X Q = abs(xi); reg_param = [1:p]'; X Xelse, error('Illegal method'), end X X% Plot the function. Xif (method=='tsvd' | method=='tgsv') X semilogy(reg_param,Q,'o'), xlabel('k') X title('Quasi-optimality function') X if (find_min) X [minQ,minQi] = min(Q); reg_min = reg_param(minQi); X HoldState = ishold; hold on; X semilogy([reg_min,reg_min],[minQ,minQ/1000],'--') X if (~HoldState), hold off; end X title(['Quasi-optimality function, minimum at ',num2str(reg_min)]) X end Xelse X if (method(1:4)=='Tikh' | method(1:4)=='tikh' | ... X method(1:4)=='dsvd' | method(1:4)=='dgsv' ) X loglog(reg_param,Q), xlabel('lambda') X else X loglog(reg_param,Q,':'), xlabel('lambda') X end X title('Quasi-optimality function') X if (find_min) X [minQ,minQi] = min(Q); reg_min = reg_param(minQi); X HoldState = ishold; hold on; X loglog([reg_min,reg_min],[minQ,minQ/1000],'--') X if (~HoldState), hold off; end X title(['Quasi-optimality function, minimum at ',num2str(reg_min)]) X end Xend END_OF_FILE if test 2949 -ne `wc -c <'quasiopt.m'`; then echo shar: \"'quasiopt.m'\" unpacked with wrong size! fi # end of 'quasiopt.m' fi if test -f 'regudemo.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'regudemo.m'\" else echo shar: Extracting \"'regudemo.m'\" \(5039 characters\) sed "s/^X//" >'regudemo.m' <<'END_OF_FILE' X%REGUDEMO Tutorial script for Regularization Tools. X X% Per Christian Hansen, UNI-C, 03/17/93. X Xecho on, clf X X% Part 1. The discrete Picard condition X% -------------------------------------- X% X% First generate a "pure" test problem where only rounding X% errors are present. Then generate another "noisy" test X% problem by adding white noise to the right-hand side. X% X% Next compute the SVD of the coefficient matrix A. X% X% Finally, check the Picard condition for both test problems X% graphically. Notice that for both problems the condition is X% indeed satisfied for the coefficients corresponding to the X% larger singular values, while the noise eventually starts to X% dominate. X X[A,b_bar,x] = shaw(32); Xrandn('seed',70957); Xe = 1e-3*randn(size(b_bar)); b = b_bar + e; X[U,s,V] = csvd(A); Xpause % Strike any key to continue Xsubplot(2,1,1); picard(U,s,b_bar); Xsubplot(2,1,2); picard(U,s,b); pause Xclf X X% Part 2. Filter factors X% ----------------------- X% X% Compute regularized solutions to the "noisy" problem from Part 1 X% by means of Tikhonov's method and LSQR without reorthogonalization. X% Also, compute the corresponding filter factors. X% X% A surface (or mesh) plot of the solutions clearly shows their dependence X% on the regularization parameter (lambda or the iteration number). X Xlambda = [1,3e-1,1e-1,3e-2,1e-2,3e-3,1e-3,3e-4,1e-4,3e-5]; XX_tikh = tikhonov(U,s,V,b,lambda); XF_tikh = fil_fac(s,lambda); Xiter = 30; reorth = 0; X[X_lsqr,rho,eta,F_lsqr] = lsqr(A,b,iter,reorth,s); Xpause % Strike any key to continue Xsubplot(2,2,1); surf(X_tikh), title('Tikhonov solutions'), axis('ij') Xsubplot(2,2,2); surf(log10(F_tikh)), axis('ij') X title('Tikh filter factors, log scale') Xsubplot(2,2,3); surf(X_lsqr(:,1:17)), title('LSQR solutions'), axis('ij') Xsubplot(2,2,4); surf(log10(F_lsqr(:,1:17))), axis('ij') X title('LSQR filter factors, log scale'), pause Xclf X X% Part 3. The L-curve X% -------------------- X% X% Plot the L-curves for Tikhonov regularization and for X% LSQR for the "noisy" test problem from Part 1. X% X% Notice the similarity between the two L-curves and thus, X% in turn, by the two methods. X Xpause % Strike any key to continue Xsubplot(1,2,1); l_curve(U,s,b); axis([1e-3,1,1,1e3]) Xsubplot(1,2,2); plot_lc(rho,eta,'o'); axis([1e-3,1,1,1e3]), pause Xclf X X% Part 4. Regularization parameters X% ---------------------------------- X% X% Use the L-curve criterion and GCV to determine the regularization X% parameters for Tikhonov regularization and truncated SVD. X% X% Then compute the relative errors for the four solutions. X Xpause % Strike any key to continue Xlambda_l = l_curve(U,s,b); axis([1e-3,1,1,1e3]), pause Xk_l = l_curve(U,s,b,'tsvd'); axis([1e-3,1,1,1e3]), pause Xlambda_gcv = gcv(U,s,b); axis([1e-6,1,1e-9,1e-1]), pause Xk_gcv = gcv(U,s,b,'tsvd'); axis([0,20,1e-9,1e-1]), pause X Xx_tikh_l = tikhonov(U,s,V,b,lambda_l); Xx_tikh_gcv = tikhonov(U,s,V,b,lambda_gcv); Xif isnan(k_l) X x_tsvd_l = zeros(32,1); % Spline Toolbox not available. Xelse X x_tsvd_l = tsvd(U,s,V,b,k_l); Xend Xx_tsvd_gcv = tsvd(U,s,V,b,k_gcv); X[norm(x-x_tikh_l),norm(x-x_tikh_gcv),... X norm(x-x_tsvd_l),norm(x-x_tsvd_gcv)]/norm(x) Xpause % Strike any key to continue X X% Part 5. Standard form versus general form X% ------------------------------------------ X% X% Generate a new test problem: inverse Laplace transformation X% with white noise in the right-hand side. X% X% For the general-form regularization, choose minimization of X% the first derivative. X% X% First display some left singular vectors of SVD and GSVD; then X% compare truncated SVD solutions with truncated GSVD solutions. X% Notice that TSVD cannot reproduce the asymptotic part of the X% solution in the right part of the figure. X Xn = 16; [A,b,x] = ilaplace(n,2); Xb = b + 1e-4*randn(size(b)); XL = get_l(n,1); X[U,s,V] = csvd(A); [UU,VV,sm,XX] = gsvd(A,L); Xpause % Strike any key to continue XI = 1; Xfor i=[3,6,9,12] X subplot(2,2,I); plot(1:n,V(:,i)); axis([1,n,-1,1]) X xlabel(['i = ',num2str(i)]), I = I + 1; Xend Xsubplot(2,2,1), text(12,1.2,'Right singular vectors V(:,i)'), pause Xclf XI = 1; Xfor i=[n-2,n-5,n-8,n-11] X subplot(2,2,I); plot(1:n,XX(:,i)), axis([1,n,-1,1]); X xlabel(['i = ',num2str(i)]), I = I + 1; Xend Xsubplot(2,2,1) Xtext(10,1.2,'Right generalized singular vectors XX(:,i)'), pause Xclf X Xk_tsvd = 7; k_tgsvd = 6; XX_I = tsvd(U,s,V,b,1:k_tsvd); XX_L = tgsvd(UU,sm,XX,b,1:k_tgsvd); Xpause % Strike any key to continue Xsubplot(2,1,1); X plot(1:n,X_I,1:n,x,'x'), axis([1,n,0,1.2]), xlabel('L = I') Xsubplot(2,1,2); X plot(1:n,X_L,1:n,x,'x'), axis([1,n,0,1.2]), xlabel('L ~= I'), pause Xclf X X% Part 6. No square integrable solution X% -------------------------------------- X% X% In the last example there is no square integrable solution to X% the underlying integral equation (NB: no noise is added). X% X% Notice that the discrete Picard condition does not seem to X% be satisfied, which indicates trouble! X X[A,b] = ursell(32); [U,s,V] = csvd(A); Xpause % Strike any key to continue Xpicard(U,s,b); pause X X% This concludes the demo. Xecho off END_OF_FILE if test 5039 -ne `wc -c <'regudemo.m'`; then echo shar: \"'regudemo.m'\" unpacked with wrong size! fi # end of 'regudemo.m' fi if test -f 'shaw.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'shaw.m'\" else echo shar: Extracting \"'shaw.m'\" \(1404 characters\) sed "s/^X//" >'shaw.m' <<'END_OF_FILE' Xfunction [A,b,x] = shaw(n) X%SHAW Test problem: one-dimensional iimage restoration model. X% X% [A,b,x] = shaw(n) X% X% Discretization of a first kind Fredholm integral equation with X% [-pi/2,pi/2] as both integration intervals. The kernel K and X% the solution f, which are given by X% K(s,t) = (cos(s) + cos(t))*(sin(u)/u)^2 X% u = pi*(sin(s) + sin(t)) X% f(t) = a1*exp(-c1*(t - t1)^2) + a2*exp(-c2*(t - t2)^2) , X% are discretized by simple quadrature to produce A and x. X% Then the discrete right-hand b side is produced as b = A*x. X% X% The order n must be even. X X% Reference: C. B. Shaw, Jr., "Improvements of the resolution of X% an instrument by numerical solution of an integral equation", X% J. Math. Anal. Appl. 37 (1972), 83-112. X X% Per Christian Hansen, UNI-C, 08/20/91. X X% Check input. Xif (rem(n,2)~=0), error('The order n must be even'), end X X% Initialization. Xh = pi/n; A = zeros(n,n); X X% Compute the matrix A. Xco = cos(-pi/2 + [.5:n-.5]*h); Xpsi = pi*sin(-pi/2 + [.5:n-.5]*h); Xfor i=1:n/2 X for j=i:n-i X ss = psi(i) + psi(j); X A(i,j) = ((co(i) + co(j))*sin(ss)/ss)^2; X A(n-j+1,n-i+1) = A(i,j); X end X A(i,n-i+1) = (2*co(i))^2; Xend XA = A + triu(A,1)'; A = A*h; X X% Compute the vectors x and b. Xa1 = 2; c1 = 6; t1 = .8; Xa2 = 1; c2 = 2; t2 = -.5; Xif (nargout>1) X x = a1*exp(-c1*(-pi/2 + [.5:n-.5]'*h - t1).^2) ... X + a2*exp(-c2*(-pi/2 + [.5:n-.5]'*h - t2).^2); X b = A*x; Xend END_OF_FILE if test 1404 -ne `wc -c <'shaw.m'`; then echo shar: \"'shaw.m'\" unpacked with wrong size! fi # end of 'shaw.m' fi if test -f 'sorted.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'sorted.m'\" else echo shar: Extracting \"'sorted.m'\" \(87 characters\) sed "s/^X//" >'sorted.m' <<'END_OF_FILE' Xfunction pointer=sorted(knots, points) X%SORTED Dummy function for Regularization Tools END_OF_FILE if test 87 -ne `wc -c <'sorted.m'`; then echo shar: \"'sorted.m'\" unpacked with wrong size! fi # end of 'sorted.m' fi if test -f 'sp2pp.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'sp2pp.m'\" else echo shar: Extracting \"'sp2pp.m'\" \(73 characters\) sed "s/^X//" >'sp2pp.m' <<'END_OF_FILE' Xfunction pp=sp2pp(spline) X%SP2PP Dummy function for Regularization Tools END_OF_FILE if test 73 -ne `wc -c <'sp2pp.m'`; then echo shar: \"'sp2pp.m'\" unpacked with wrong size! fi # end of 'sp2pp.m' fi if test -f 'spbrk.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'spbrk.m'\" else echo shar: Extracting \"'spbrk.m'\" \(96 characters\) sed "s/^X//" >'spbrk.m' <<'END_OF_FILE' Xfunction [knots,coefs,n,k,d]=spbrk(spline,print) X%SPBRK Dummy function for Regularization Tools END_OF_FILE if test 96 -ne `wc -c <'spbrk.m'`; then echo shar: \"'spbrk.m'\" unpacked with wrong size! fi # end of 'spbrk.m' fi if test -f 'spikes.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'spikes.m'\" else echo shar: Extracting \"'spikes.m'\" \(1214 characters\) sed "s/^X//" >'spikes.m' <<'END_OF_FILE' Xfunction [A,b,x] = spikes(n,t_max) X%SPIKES Test problem with a "spiky" solution. X% X% [A,b,x] = spikes(n,t_max) X% X% Artificially generated discrete ill-posed problem. X% X% The solution x consists of a unit step at t = .5, and a pulse train X% of spikes of decrasing magnitude at t = .5, 1.5, 2.5, ... X% X% The parameter t_max is optional; its default value is 5. X% It controls the length of the pulse train. X X% Per Christian Hansen, UNI-C, 06/24/91. X X% Initialization. Xif (nargin == 1), t_max = 5; end Xt = t_max*[1:n]/n; del = t_max/n; X X% Compute the matrix A. X[t,sigma] = meshdom(del:del:t_max,del:del:t_max); sigma = flipud(sigma); XA = sigma./(2*sqrt(pi*t.^3)).*exp(-(sigma.^2)./(4*t)); X X% Compute the right-hand side b and the solution x. Xif (nargout > 1) X heights = 2*ones(t_max,1); heights(1) = 25; X heights(2) = 9; heights(3) = 5; heights(4) = 4; heights(5) = 3; X x = zeros(n,1); n_h = 1; X peak = 0.5/t_max; peak_dist = 1/t_max; X if (peak < 1) X n_peak = round(peak*n); x(n_peak) = heights(n_h); X x(n_peak+1:n) = ones(n-n_peak,1); X peak = peak + peak_dist; n_h = n_h + 1; X end X while (peak < 1) X x(round(peak*n)) = heights(n_h); X peak = peak + peak_dist; n_h = n_h + 1; X end X b = A*x; Xend END_OF_FILE if test 1214 -ne `wc -c <'spikes.m'`; then echo shar: \"'spikes.m'\" unpacked with wrong size! fi # end of 'spikes.m' fi if test -f 'spleval.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'spleval.m'\" else echo shar: Extracting \"'spleval.m'\" \(521 characters\) sed "s/^X//" >'spleval.m' <<'END_OF_FILE' Xfunction points = spleval(f) X%SPLEVAL Evaluation of a spline or spline curve. X% X% points = spleval(f) X% X% Computes points on the given spline or spline curve f between X% its extreme breaks. X X% Original routine fnplt by C. de Boor / latest change: Feb.25, 1989 X% Simplified by Per Christian Hansen, UNI-C, 11/19/91. X Xif (f(1)==11), f = sp2pp(f); end X X[breaks,coefs,l,k,d] = ppbrk(f); Xnpoints=100; Xx = breaks(1) + [0:npoints]*((breaks(l+1)-breaks(1))/npoints); Xv=ppual(f,x); X Xif (d==1), points=[x;v]; else, points = v; end END_OF_FILE if test 521 -ne `wc -c <'spleval.m'`; then echo shar: \"'spleval.m'\" unpacked with wrong size! fi # end of 'spleval.m' fi if test -f 'spmak.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'spmak.m'\" else echo shar: Extracting \"'spmak.m'\" \(82 characters\) sed "s/^X//" >'spmak.m' <<'END_OF_FILE' Xfunction spline=spmak(knots,coefs) X%SPMAK Dummy function for Regularization Tools END_OF_FILE if test 82 -ne `wc -c <'spmak.m'`; then echo shar: \"'spmak.m'\" unpacked with wrong size! fi # end of 'spmak.m' fi if test -f 'sprpp.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'sprpp.m'\" else echo shar: Extracting \"'sprpp.m'\" \(76 characters\) sed "s/^X//" >'sprpp.m' <<'END_OF_FILE' Xfunction [v,b] = sprpp(tx,a) X%SPRPP Dummy function for Regularization Tools END_OF_FILE if test 76 -ne `wc -c <'sprpp.m'`; then echo shar: \"'sprpp.m'\" unpacked with wrong size! fi # end of 'sprpp.m' fi if test -f 'std_form.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'std_form.m'\" else echo shar: Extracting \"'std_form.m'\" \(2544 characters\) sed "s/^X//" >'std_form.m' <<'END_OF_FILE' Xfunction [A_s,b_s,L_p,K,M] = std_form(A,L,b,W) X%STD_FORM Transform a general-form reg. problem into one in standard form. X% X% [A_s,b_s,L_p,K,M] = std_form(A,L,b) (method 1) X% [A_s,b_s,L_p,x_0] = std_form(A,L,b,W) (method 2) X% X% Transforms a regularization problem in general form X% min { || A x - b ||^2 + lambda^2 || L x ||^2 } X% into one in standard form X% min { || A_s x_s - b_s ||^2 + lambda^2 || x_s ||^2 } . X% X% Two methods are available. In both methods, the regularized X% solution to the original problem can be written as X% x = L_p*x_s + d X% where L_p and d depend on the method as follows: X% method = 1: L_p = pseudoinverse of L, d = K*M*(b - A*L_p*x_s) X% method = 2: L_p = A-weighted pseudoinverse of L, d = x_0. X% X% The transformation from x_s back to x can be carried out by means X% of the subroutine gen_form. X X% References: L. Elden, "Algorithms for regularization of ill- X% conditioned least-squares problems", BIT 17 (1977), 134-145. X% L. Elden, "A weighted pseudoinverse, generalized singular values, X% and constrained lest squares problems", BIT 22 (1982), 487-502. X% M. Hanke, "Regularization with differential operators. An itera- X% tive approach", J. Numer. Funct. Anal. Optim. 13 (1992), 523-540. X X% Per Christian Hansen, UNI-C, 05/26/93. X X% Nargin determines which method. Xif (nargin==3) X X % Initialization for method 1. X [m,n] = size(A); [p,np] = size(L); X if (np~=n), error('A and L must have the same number of columns'), end X X % Special treatment of the case where L is square. X if (p==n) X L_p = inv(L); K = []; M = []; A_s = A/L; b_s = b; X return X end X X % Compute a QR factorization of L'. X [K,R] = qr(full(L')); R = R(1:p,:); X X % Compute a QR factorization of A*K(:,p+1:n)). X [H,T] = qr(A*K(:,p+1:n)); T = T(1:n-p,:); X X % Compute the transformed quantities. X L_p = (R\(K(:,1:p)'))'; X K = K(:,p+1:n); X M = T\(H(:,1:n-p)'); X A_s = H(:,n-p+1:m)'*A*L_p; X b_s = H(:,n-p+1:m)'*b; X Xelse X X % Initialization for method 2. X [m,n] = size(A); [p,nl] = size(L); nu = n-p; X if (nl~=n), error('A and L must have the same number of columns'), end X X % Special treatment of the case where L is square. X if (p==n) X L_p = inv(L); A_s = A/L; b_s = b; X x_0 = zeros(n,1); K = x_0; % Fix output name. X return X end X X % Compute NAA and x_0; X [NAA,x_0] = pinit(W,A,b); X b_s = b - A*x_0; X X % Compute the transformed quantities. X L1 = inv([[eye(nu),zeros(nu,p)];L]); L1 = full(L1(:,nu+1:n)); X L_p = L1 - W*(NAA*L1); X A_s = A*L_p; X X % Fix output name. X K = x_0; X Xend END_OF_FILE if test 2544 -ne `wc -c <'std_form.m'`; then echo shar: \"'std_form.m'\" unpacked with wrong size! fi # end of 'std_form.m' fi if test -f 'tgsvd.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'tgsvd.m'\" else echo shar: Extracting \"'tgsvd.m'\" \(891 characters\) sed "s/^X//" >'tgsvd.m' <<'END_OF_FILE' Xfunction x_k = tgsvd(U,sm,X,b,k) X%TGSVD Truncated GSVD regularization. X% X% x_k = tgsvd(U,sm,X,b,k) , sm = [sigma,mu] X% X% Computes the truncated GSVD solution X% [ 0 0 0 ] X% x_k = X*[ 0 inv(diag(sigma(p-k+1:p))) 0 ]*U'*b . X% [ 0 0 eye(n-p) ] X% If k is a vector, then x_k is a matrix such that X% x_k = [ x_k(1), x_k(2), ... ] . X X% Reference: P. C. Hansen, "Regularization, GSVD and truncated GSVD", X% BIT 29 (1989), 491-504. X X% Per Christian Hansen, UNI-C, 11/18/91. X X% Initialization. X[n,n] = size(X); p = length(sm(:,1)); lk = length(k); Xif (min(k)<1 | max(k)>p) X error('Illegal truncation parameter k') Xend X X% Treat each k separately. Xx_k = zeros(n,lk); xi = (U(:,1:p)'*b)./sm(:,1); Xx_0 = X(:,p+1:n)*U(:,p+1:n)'*b; Xfor j=1:lk X i = k(j); pi1 = p-i+1; X x_k(:,j) = X(:,pi1:p)*xi(pi1:p) + x_0; Xend END_OF_FILE if test 891 -ne `wc -c <'tgsvd.m'`; then echo shar: \"'tgsvd.m'\" unpacked with wrong size! fi # end of 'tgsvd.m' fi if test -f 'tikhonov.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'tikhonov.m'\" else echo shar: Extracting \"'tikhonov.m'\" \(1291 characters\) sed "s/^X//" >'tikhonov.m' <<'END_OF_FILE' Xfunction x_lambda = tikhonov(U,s,V,b,lambda) X%TIKHONOV Tikhonov regularization. X% X% x_lambda = tikhonov(U,s,V,b,lambda) X% x_lambda = tikhonov(U,sm,X,b,lambda) , sm = [sigma,mu] X% X% Computes the Tikhonov regularized solution x_lambda. If the X% SVD is used, i.e. if U, s, and V are specified, then standard- X% form regularization is applied: X% min { || A x - b ||^2 + lambda^2 || x ||^2 } . X% If, on the other hand, the GSVD is used, i.e. if U, sm, and X X% are specified, then general-form regularization is applied: X% min { || A x - b ||^2 + lambda^2 || L x ||^2 } . X% X% If lambda is a vector, then x_lambda is a matrix such that X% x_lambda = [ x_lambda(1), x_lambda(2), ... ] . X X% Per Christian Hansen, UNI-C, 03/10/90. X X% Reference: A. N. Tikhonov & V. Y. Arsenin, "Solutions of X% Ill-Posed Problems", Wiley, 1977. X X% Initialization. Xif (min(lambda)<0) X error('Illegal regularization parameter lambda') Xend X[n,pv] = size(V); [p,ps] = size(s); Xeta = s(:,1).*(U(:,1:p)'*b); Xll = length(lambda); x_lambda = zeros(n,ll); X X% Treat each lambda separately. Xif (ps==1) X for i=1:ll X x_lambda(:,i) = V(:,1:p)*(eta./(s.^2 + lambda(i)^2)); X end Xelse X x0 = V(:,p+1:n)*U(:,p+1:n)'*b; X for i=1:ll X x_lambda(:,i) = V(:,1:p)*(eta./(s(:,1).^2 + lambda(i)^2*s(:,2).^2)) + x0; X end Xend END_OF_FILE if test 1291 -ne `wc -c <'tikhonov.m'`; then echo shar: \"'tikhonov.m'\" unpacked with wrong size! fi # end of 'tikhonov.m' fi if test -f 'tsvd.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'tsvd.m'\" else echo shar: Extracting \"'tsvd.m'\" \(566 characters\) sed "s/^X//" >'tsvd.m' <<'END_OF_FILE' Xfunction x_k = tsvd(U,s,V,b,k) X%TSVD Truncated SVD regularization. X% X% x_k = tsvd(U,s,V,b,k) X% X% Computes the truncated SVD solution X% x_k = V(:,1:k)*inv(diag(s(1:k)))*U(:,1:k)'*b . X% If k is a vector, then x_k is a matrix such that X% x_k = [ x_k(1), x_k(2), ... ] . X X% Per Christian Hansen, UNI-C, 11/18/91. X X% Initialization. X[n,p] = size(V); lk = length(k); Xif (min(k)<1 | max(k)>p) X error('Illegal truncation parameter k') Xend Xx_k = zeros(n,lk); xi = (U(:,1:p)'*b)./s; X X% Treat each k separately. Xfor j=1:lk X i = k(j); X x_k(:,j) = V(:,1:i)*xi(1:i); Xend END_OF_FILE if test 566 -ne `wc -c <'tsvd.m'`; then echo shar: \"'tsvd.m'\" unpacked with wrong size! fi # end of 'tsvd.m' fi if test -f 'ttls.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'ttls.m'\" else echo shar: Extracting \"'ttls.m'\" \(1308 characters\) sed "s/^X//" >'ttls.m' <<'END_OF_FILE' Xfunction [x_k,rho,eta] = ttls(V1,k,s1) X%TTLS Truncated TLS regularization. X% X% [x_k,rho,eta] = ttls(V1,k,s1) X% X% Computes the truncated TLS solution X% x_k = - V1(1:n,k+1:n+1)*pinv(V1(n+1,k+1:n+1)) X% where V1 is the right singular matrix in the SVD of the matrix X% [A,b] = U1*diag(s1)*V1' . X% X% If k is a vector, then x_k is a matrix such that X% x_k = [ x_k(1), x_k(2), ... ] . X% If k is not specified, k = n is used. X% X% The solution norms and TLS residual norms corresponding to x_k are X% returned in eta and rho, respectively. Notice that the singular X% values s1 are required to compute rho. X X% Per Christian Hansen, UNI-C, 03/18/93. X X% Initialization. X[n1,m1] = size(V1); n = n1-1; Xif (m1 ~= n1), error('The matrix V1 must be square'), end Xif (nargin == 1), k = n; end Xlk = length(k); Xif (min(k) < 1 | max(k) > n) X error('Illegal truncation parameter k') Xend Xx_k = zeros(n,lk); Xif (nargout > 1) X if (nargin < 3) X error('The singular values must also be specified') X end X ns = length(s1); rho = zeros(lk,1); Xend Xif (nargout==3), eta = zeros(lk,1); end X X% Treat each k separately. Xfor j=1:lk X i = k(j); X v = V1(n1,i+1:n1); gamma = 1/(v*v'); X x_k(:,j) = - V1(1:n,i+1:n1)*v'*gamma; X if (nargout > 1), rho(j) = norm(s1(i+1:ns)); end X if (nargout == 3), eta(j) = sqrt(gamma - 1); end Xend END_OF_FILE if test 1308 -ne `wc -c <'ttls.m'`; then echo shar: \"'ttls.m'\" unpacked with wrong size! fi # end of 'ttls.m' fi if test -f 'ursell.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'ursell.m'\" else echo shar: Extracting \"'ursell.m'\" \(1026 characters\) sed "s/^X//" >'ursell.m' <<'END_OF_FILE' Xfunction [A,b] = ursell(n) X%URSELL Test problem: integral equation wiht no square integrable solution. X% X% [A,b] = ursell(n) X% X% Discretization of a first kind Fredholm integral equation with X% kernel K and right-hand side g given by X% K(s,t) = 1/(s+t+1) , g(s) = 1 , X% where both integration itervals are [0,1]. X% X% Note: this integral equation has NO square integrable solution. X X% Reference: F. Ursell, "Introduction to the theory of linear X% integral equations", Chapter 1 in L. M. Delves & J. Walsh (Eds.), X% "Numerical Solution of Integral Equations", Clarendon Press, 1974. X X% Discretized by Galerkin method with orthonormal box functions. X X% Per Christian Hansen, UNI-C, 09/16/92. X X% Compute the matrix A. Xfor k = 1:n X d1 = 1 + (1+k)/n; d2 = 1 + k/n; d3 = 1 + (k-1)/n; X c(k) = n*(d1*log(d1) + d3*log(d3) - 2*d2*log(d2)); X e1 = 1 + (n+k)/n; e2 = 1 + (n+k-1)/n; e3 = 1 + (n+k-2)/n; X r(k) = n*(e1*log(e1) + e3*log(e3) - 2*e2*log(e2)); Xend XA = hankel(c,r); X X% Compute the right-hand side b. Xb = ones(n,1)/sqrt(n); END_OF_FILE if test 1026 -ne `wc -c <'ursell.m'`; then echo shar: \"'ursell.m'\" unpacked with wrong size! fi # end of 'ursell.m' fi if test -f 'wing.m' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'wing.m'\" else echo shar: Extracting \"'wing.m'\" \(1306 characters\) sed "s/^X//" >'wing.m' <<'END_OF_FILE' Xfunction [A,b,x] = wing(n,t1,t2) X%WING Test problem with a discontinuous solution. X% X% [A,b,x] = wing(n,t1,t2) X% X% Discretization of a first kind Fredholm integral eqaution with X% kernel K and right-hand side g given by X% K(s,t) = t*exp(-s*t^2) 0 < s,t < 1 X% g(s) = (exp(-s*t1^2) - exp(-s*t2^2)/(2*s) 0 < s < 1 X% and with the solution f given by X% f(t) = | 1 for t1 < t < t2 X% | 0 elsewhere. X% X% Here, t1 and t2 are constants satisfying t1 < t2. If they are X% not speficied, the values t1 = 1/3 and t2 = 2/3 are used. X X% Reference: G. M. Wing, "A Primer on Integral Equations of the X% First Kind", SIAM, 1991. X X% Discretized by Galerkin method with orthonormal box functions; X% both integrations are done by the midpoint rule. X X% Per Christian Hansen, UNI-C, 09/17/92. X X% Initialization. Xif (nargin==1) X t1 = 1/3; t2 = 2/3; Xelse X if (t1 > t2), error('t1 must be smaller than t2'), end Xend XA = zeros(n,n); h = 1/n; sh = sqrt(h); X X% Set up matrix. Xsti = ([1:n]-0.5)*h; Xfor i=1:n X A(i,:) = h*sti.*exp(-sti(i)*sti.^2); Xend X X% Set up right-hand side. Xif (nargout > 1) X b = sqrt(h)*0.5*(exp(-sti*t1^2)' - exp(-sti*t2^2)')./sti'; Xend X X% Set up solution. Xif (nargout==3) X I = find(t1 < sti & sti < t2); X x = zeros(n,1); x(I) = sqrt(h)*ones(length(I),1); Xend END_OF_FILE if test 1306 -ne `wc -c <'wing.m'`; then echo shar: \"'wing.m'\" unpacked with wrong size! fi # end of 'wing.m' fi echo shar: End of shell archive. exit 0 Michela Redivo-Zaglia Universita` di Padova - Dipartimento di Elettronica e Informatica Via G. Gradenigo 6/A - 35131 Padova - Italy Phone ++39-49-8277625 e-mail: michela@dei.unipd.it Fax ++39-49-8277699