C********************************************************************** C this package solves boundary value problems for C ordinary differential equations with constraints, C as described below. for more see [1]. C C COLDAE is a modification of the package COLNEW by bader and C ascher [4], which in turn is a modification of COLSYS by ascher, C christiansen and russell [3]. it does what colnew does plus C optionally solving semi-explicit differential-algebraic equations C with index at most 2. C********************************************************************** C---------------------------------------------------------------------- C p a r t 1 C main storage allocation and program control subroutines C---------------------------------------------------------------------- C SUBROUTINE COLDAE (NCOMP, NY, M, ALEFT, ARIGHT, ZETA, IPAR, LTOL, 1 TOL, FIXPNT, ISPACE, FSPACE, IFLAG, 2 FSUB, DFSUB, GSUB, DGSUB, GUESS) C C C********************************************************************** C C written by C uri ascher and ray spiteri, C department of computer science, C university of british columbia, C vancouver, b. c., canada v6t 1z2 C C********************************************************************** C C purpose C C this package solves a multi-point boundary value C problem for a mixed order system of ode-s with constraints C given by C C (m(i)) C u = f ( x; z(u(x)), y(x) ) i = 1, ... ,ncomp C i i C C 0 = f ( x; z(u(x)), y(x) ) i = ncomp+1,...,ncomp+ny C i C C aleft .lt. x .lt. aright, C C C g ( zeta(j); z(u(zeta(j))) ) = 0 j = 1, ... ,mstar C j C mstar = m(1)+m(2)+...+m(ncomp), C C C where t t C u = (u , u , ... ,u ) , y = (y ,y , ... ,y ) C 1 2 ncomp 1 2 ny C C is the exact solution vector: u are the differential solution C components and y are the algebraic solution components. C C (mi) C u is the mi=m(i) th derivative of u C i i C C (1) (m1-1) (mncomp-1) C z(u(x)) = ( u (x),u (x),...,u (x),...,u (x) ) C 1 1 1 ncomp C C f (x,z(u),y) is a (generally) nonlinear function of C i C z(u)=z(u(x)) and y=y(x). C C g (zeta(j);z(u)) is a (generally) nonlinear function C j C used to represent a boundary condition. C C the boundary points satisfy C aleft .le. zeta(1) .le. .. .le. zeta(mstar) .le. aright C C the orders mi of the differential equations satisfy C 1 .le. m(i) .le. 4. C C regarding the dae, note that: C i) with ny=0, the code is essentially identical to the ode C code colnew C ii) no explicit checking of the index of the problem C is provided. if the index is > 2 then the code will C not work well. C iii) the constraints are treated like de-s of order 0 and C correspondingly the approximation to y is sought in C a piecewise discontinuous polynomial space. C iv) the number of boundary conditions required is independent C of the index. it is the user's responsibility to ensure C that these conditions are consistent with the constraints. C the conditions at the left end point aleft must include C a subset equivalent to specifying the index-2 C constraints there. C v) for an index-2 problem in hessenberg form, the projected C collocation method of ascher and petzold [2] is used. C vi) if the constraints are of a mixed type (and possibly C a mixed index) then coldae can transform and project C appropriately -- see description of ipar(12). C C********************************************************************** C C method C C the method used to approximate the solution u is C collocation at gaussian points, requiring m(i)-1 continuous C derivatives in the i-th component, i = 1, ..., ncomp. C here, k is the number of collocation points (stages) per C subinterval and is chosen such that k .ge. max m(i). C a runge-kutta-monomial solution representation is utilized. C for hessenberg index-2 daes, a projection on the constraint C manifold at each interval's end is used [2]. C C references C C [1] u. ascher and r. spiteri, C collocation software for boundary value differential- C algebraic equations, C siam j. scient. stat. comput., to appear. C C [2] u. ascher and l. petzold, C projected implicit runge-kutta methods for differential- C algebraic equations, C siam j. num. anal. 28 (1991), 1097-1120. C C [3] u. ascher, j. christiansen and r.d. russell, C collocation software for boundary-value odes, C acm trans. math software 7 (1981), 209-222. C C [4] g. bader and u. ascher, C a new basis implementation for a mixed order C boundary value ode solver, C siam j. scient. stat. comput. 8 (1987), 483-500. C C [5] u. ascher, j. christiansen and r.d. russell, C a collocation solver for mixed order C systems of boundary value problems, C math. comp. 33 (1979), 659-679. C C [6] u. ascher, j. christiansen and r.d. russell, C colsys - a collocation code for boundary C value problems, C lecture notes comp.sc. 76, springer verlag, C b. childs et. al. (eds.) (1979), 164-185. C C [7] c. deboor and r. weiss, C solveblok: a package for solving almost block diagonal C linear systems, C acm trans. math. software 6 (1980), 80-87. C C C********************************************************************** C C *************** input to coldae *************** C C variables C C ncomp - no. of differential equations (ncomp .le. 20) C C ny - no. of constraints (ny .le. 20) C C m(j) - order of the j-th differential equation C ( mstar = m(1) + ... + m(ncomp) .le. 40 ) C C aleft - left end of interval C C aright - right end of interval C C zeta(j) - j-th side condition point (boundary point). must C have zeta(j) .le. zeta(j+1). all side condition C points must be mesh points in all meshes used, C see description of ipar(11) and fixpnt below. C C ipar - an integer array dimensioned at least 11. C a list of the parameters in ipar and their meaning follows C some parameters are renamed in coldae; these new names are C given in parentheses. C C ipar(1) ( = nonlin ) C = 0 if the problem is linear C = 1 if the problem is nonlinear C C ipar(2) = no. of collocation points per subinterval (= k ) C where max m(i) .le. k .le. 7 . if ipar(2)=0 then C coldae sets k = max ( max m(i)+1, 5-max m(i) ) C C ipar(3) = no. of subintervals in the initial mesh ( = n ). C if ipar(3) = 0 then coldae arbitrarily sets n = 5. C C ipar(4) = no. of solution and derivative tolerances. ( = ntol ) C we require 0 .lt. ntol .le. mstar. C C ipar(5) = dimension of fspace. ( = ndimf ) C C ipar(6) = dimension of ispace. ( = ndimi ) C C ipar(7) - output control ( = iprint ) C = -1 for full diagnostic printout C = 0 for selected printout C = 1 for no printout C C ipar(8) ( = iread ) C = 0 causes coldae to generate a uniform initial mesh. C = 1 if the initial mesh is provided by the user. it C is defined in fspace as follows: the mesh C aleft=x(1).lt.x(2).lt. ... .lt.x(n).lt.x(n+1)=aright C will occupy fspace(1), ..., fspace(n+1). the C user needs to supply only the interior mesh C points fspace(j) = x(j), j = 2, ..., n. C = 2 if the initial mesh is supplied by the user C as with ipar(8)=1, and in addition no adaptive C mesh selection is to be done. C C ipar(9) ( = iguess ) C = 0 if no initial guess for the solution is C provided. C = 1 if an initial guess is provided by the user C in subroutine guess. C = 2 if an initial mesh and approximate solution C coefficients are provided by the user in fspace. C (the former and new mesh are the same). C = 3 if a former mesh and approximate solution C coefficients are provided by the user in fspace, C and the new mesh is to be taken twice as coarse; C i.e.,every second point from the former mesh. C = 4 if in addition to a former initial mesh and C approximate solution coefficients, a new mesh C is provided in fspace as well. C (see description of output for further details C on iguess = 2, 3, and 4.) C C ipar(10)= -1 if the first relax factor is RSTART C (use for an extra sensitive nonlinear problem only) C = 0 if the problem is regular C = 1 if the newton iterations are not to be damped C (use for initial value problems). C = 2 if we are to return immediately upon (a) two C successive nonconvergences, or (b) after obtaining C error estimate for the first time. C C ipar(11)= no. of fixed points in the mesh other than aleft C and aright. ( = nfxpnt , the dimension of fixpnt) C the code requires that all side condition points C other than aleft and aright (see description of C zeta ) be included as fixed points in fixpnt. C C ipar(12) ( = index ) C this parameter is ignored if ny=0. C = 0 if the index of the dae is not as per one of the C following cases C = 1 if the index of the dae is 1. in this case the C ny x ny jacobian matrix of the constraints with C respect to the algebraic unknowns, i.e. C df(i,j), i=ncomp+1,...,ncomp+ny, C j=mstar+1,...,mstar+ny C (see description of dfsub below) C is nonsingular wherever it is evaluated. this C allows usual collocation to be safely used. C = 2 if the index of the dae is 2 and it is in Hessenberg C form. in this case the C ny x ny jacobian matrix of the constraints with C respect to the algebraic unknowns is 0, and the C ny x ny matrix CB is nonsingular, where C C(i,j) = df(i+ncomp, m(j)), i=1,...,ny, C j=1,...,ncomp C B(i,j) = df(i,j+mstar), i=1,...,ncomp, C j=1,...,ny C the projected collocation method described in [2] C is then used. C in case of ipar(12)=0 and ny > 0, coldae determines the C appropriate projection needed at the right end of each C mesh subinterval using SVD. this is the most expensive C and most general option. C C ltol - an array of dimension ipar(4). ltol(j) = l specifies C that the j-th tolerance in tol controls the error C in the l-th component of z(u). also require that C 1.le.ltol(1).lt.ltol(2).lt. ... .lt.ltol(ntol).le.mstar C C tol - an array of dimension ipar(4). tol(j) is the C error tolerance on the ltol(j) -th component C of z(u). thus, the code attempts to satisfy C for j=1,...,ntol on each subinterval C abs(z(v)-z(u)) .le. tol(j)*abs(z(u)) +tol(j) C ltol(j) ltol(j) C C if v(x) is the approximate solution vector. C C fixpnt - an array of dimension ipar(11). it contains C the points, other than aleft and aright, which C are to be included in every mesh. C C ispace - an integer work array of dimension ipar(6). C its size provides a constraint on nmax, C the maximum number of subintervals. choose C ipar(6) according to the formula C ipar(6) .ge. nmax*nsizei C where C nsizei = 3 + kdm C with C kdm = kdy + mstar ; kdy = k * (ncomp+ny) ; C nrec = no. of right end boundary conditions. C C C fspace - a real work array of dimension ipar(5). C its size provides a constraint on nmax. C choose ipar(5) according to the formula C ipar(5) .ge. nmax*nsizef C where C nsizef = 4 + 3 * mstar + (5+kdy) * kdm + C (2*mstar-nrec) * 2*mstar + C ncomp*(mstar+ny+2) + kdy. C C C iflag - the mode of return from coldae. C = 1 for normal return C = 0 if the collocation matrix is singular. C =-1 if the expected no. of subintervals exceeds storage C specifications. C =-2 if the nonlinear iteration has not converged. C =-3 if there is an input data error. C C C********************************************************************** C C ************* user supplied subroutines ************* C C C the following subroutines must be declared external in the C main program which calls coldae. C C C fsub - name of subroutine for evaluating f(x,z(u(x)),y(x)) = C t C (f ,...,f ) at a point x in (aleft,aright). it C 1 ncomp+ny C should have the heading C C subroutine fsub (x , z , y, f) C C where f is the vector containing the value of fi(x,z(u),y) C in the i-th component and y(x) = (y(1),...y(ny)) and C z(u(x))=(z(1),...,z(mstar)) C are defined as above under purpose . C C C dfsub - name of subroutine for evaluating the jacobian of C f(x,z(u),y) at a point x. it should have the heading C C subroutine dfsub (x , z , y, df) C C where z(u(x)) and y(x) are defined as for fsub and the C (ncomp+ny) by (mstar+ny) C array df should be filled by the partial deriv- C atives of f, viz, for a particular call one calculates C df(i,j) = dfi / dzj, i=1,...,ncomp+ny C j=1,...,mstar, C df(i,mstar+j) = dfi / dyj, i=1,...,ncomp+ny C j=1,...,ny. C C C gsub - name of subroutine for evaluating the i-th component of C g(x,z(u(x))) = g (zeta(i),z(u(zeta(i)))) at a point x = C i C zeta(i) where 1.le.i.le.mstar. it should have the heading C C subroutine gsub (i , z , g) C C where z(u) is as for fsub, and i and g=g are as above. C i C note that in contrast to f in fsub , here C only one value per call is returned in g. C also, g is independent of y and, in case of a higher C index dae (index = 2), should include the constraints C sampled at aleft plus independent additional constraints, C or an equivalent set. C C C dgsub - name of subroutine for evaluating the i-th row of C the jacobian of g(x,z(u(x))). it should have the heading C C subroutine dgsub (i , z , dg) C C where z(u) is as for fsub, i as for gsub and the mstar- C vector dg should be filled with the partial derivatives C of g, viz, for a particular call one calculates C dg(i,j) = dgi / dzj j=1,...,mstar. C C C guess - name of subroutine to evaluate the initial C approximation for z(u(x)), y(x) and for dmval(u(x))= vector C of the mj-th derivatives of u(x). it should have the C heading C C subroutine guess (x , z , y, dmval) C C note that this subroutine is needed only if using C ipar(9) = 1, and then all mstar components of z, C ny components of y C and ncomp components of dmval should be specified C for any x, aleft .le. x .le. aright . C C C********************************************************************** C C ************ use of output from coldae ************ C C *** solution evaluation *** C C on return from coldae, the arrays fspace and ispace C contain information specifying the approximate solution. C the user can produce the solution vector z( u(x) ), y(x) at C any point x, aleft .le. x .le. aright, by the statement, C C call appsln (x, z, y, fspace, ispace) C C when saving the coefficients for later reference, only C ispace(1),...,ispace(8+ncomp) and C fspace(1),...,fspace(ispace(8)) need to be saved as C these are the quantities used by appsln. C C C *** simple continuation *** C C C a formerly obtained solution can easily be used as the C first approximation for the nonlinear iteration for a C new problem by setting (iguess =) ipar(9) = 2, 3 or 4. C C if the former solution has just been obtained then the C values needed to define the first approximation are C already in ispace and fspace. C alternatively, if the former solution was obtained in a C previous run and its coefficients were saved then those C coefficients must be put back into C ispace(1),..., ispace(8+ncomp) and C fspace(1),..., fspace(ispace(8)). C C for ipar(9) = 2 or 3 set ipar(3) = ispace(1) ( = the C size of the previous mesh ). C C for ipar(9) = 4 the user specifies a new mesh of n subintervals C as follows. C the values in fspace(1),...,fspace(ispace(8)) have to be C shifted by n+1 locations to fspace(n+2),..,fspace(ispace(8)+n+1) C and the new mesh is then specified in fspace(1),..., fspace(n+1). C also set ipar(3) = n. C C C********************************************************************** C C *************** package subroutines *************** C C the following description gives a brief overview of how the C procedure is broken down into the subroutines which make up C the package called coldae . for further details the C user should refer to documentation in the various subroutines C and to the references cited above. C C the subroutines fall into four groups: C C part 1 - the main storage allocation and program control subr C C coldae - tests input values, does initialization and breaks up C the work areas, fspace and ispace, into the arrays C used by the program. C C contrl - is the actual driver of the package. this routine C contains the strategy for nonlinear equation solving. C C skale - provides scaling for the control C of convergence in the nonlinear iteration. C C C part 2 - mesh selection and error estimation subroutines C C consts - is called once by coldae to initialize constants C which are used for error estimation and mesh selection. C C newmsh - generates meshes. it contains the test to decide C whether or not to redistribute a mesh. C C errchk - produces error estimates and checks against the C tolerances at each subinterval C C C part 3 - collocation system set-up subroutines C C lsyslv - controls the set-up and solution of the linear C algebraic systems of collocation equations which C arise at each newton iteration. C C gderiv - is used by lsyslv to set up the equation associated C with a side condition point. C C vwblok - is used by lsyslv to set up the equation(s) associated C with a collocation point. C C gblock - is used by lsyslv to construct a block of the global C collocation matrix or the corresponding right hand C side. C C C part 4 - service subroutines C C appsln - sets up a standard call to approx . C C approx - evaluates a piecewise polynomial solution. C C rkbas - evaluates the mesh independent runge-kutta basis C C vmonde - solves a vandermonde system for given right hand C side C C horder - evaluates the highest order derivatives of the C current collocation solution used for mesh refinement. C C C part 5 - linear algebra subroutines C C to solve the global linear systems of collocation equations C constructed in part 3, coldae uses a column oriented version C of the package solveblok originally due to de boor and weiss. C C to solve the linear systems for static parameter condensation C in each block of the collocation equations, the linpack C routines dgefa and dgesl are included. but these C may be replaced when solving problems on vector processors C or when solving large scale sparse jacobian problems. C C---------------------------------------------------------------------- IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION M(1), ZETA(1), IPAR(1), LTOL(1), TOL(1), DUMMY(1), 1 FIXPNT(1), ISPACE(1), FSPACE(1) DIMENSION DUMMY2(840) C COMMON /COLOUT/ PRECIS, IOUT, IPRINT COMMON /COLLOC/ RHO(7), COEF(49) COMMON /COLORD/ K, NC, NNY, NCY, MSTAR, KD, KDY, MMAX, MT(20) COMMON /COLAPR/ N, NOLD, NMAX, NZ, NDMZ COMMON /COLMSH/ MSHFLG, MSHNUM, MSHLMT, MSHALT COMMON /COLSID/ TZETA(40), TLEFT, TRIGHT, IZETA, IDUM COMMON /COLNLN/ NONLIN, ITER, LIMIT, ICARE, IGUESS, INDEX COMMON /COLEST/ TTL(40), WGTMSH(40), WGTERR(40), TOLIN(40), 1 ROOT(40), JTOL(40), LTTOL(40), NTOL C EXTERNAL FSUB, DFSUB, GSUB, DGSUB, GUESS C C********************************************************************* C C the actual subroutine coldae serves as an interface with C the package of subroutines referred to collectively as C coldae. the subroutine serves to test some of the input C parameters, rename some of the parameters (to make under- C standing of the coding easier), to do some initialization, C and to break the work areas fspace and ispace up into the C arrays needed by the program. C C********************************************************************** C C... specify machine dependent output unit iout and compute machine C... dependent constant precis = 100 * machine unit roundoff C IF ( IPAR(7) .LE. 0 ) WRITE(6,99) 99 FORMAT(//,' VERSION *1* OF COLDAE .' ,//) C IOUT = 6 PRECIS = 1.D0 10 PRECIS = PRECIS / 2.D0 PRECP1 = PRECIS + 1.D0 IF ( PRECP1 .GT. 1.D0 ) GO TO 10 PRECIS = PRECIS * 100.D0 C C... in case incorrect input data is detected, the program returns C... immediately with iflag=-3. C IFLAG = -3 NCY = NCOMP + NY IF ( NCOMP .LT. 0 .OR. NCOMP .GT. 20 ) RETURN IF ( NY .LT. 0 .OR. NY .GT. 20 ) RETURN IF ( NCY .LT. 1 .OR. NCY .GT. 40 ) RETURN DO 20 I=1,NCOMP IF ( M(I) .LT. 1 .OR. M(I) .GT. 4 ) RETURN 20 CONTINUE C C... rename some of the parameters and set default values. C NONLIN = IPAR(1) K = IPAR(2) N = IPAR(3) IF ( N .EQ. 0 ) N = 5 IREAD = IPAR(8) IGUESS = IPAR(9) IF ( NONLIN .EQ. 0 .AND. IGUESS .EQ. 1 ) IGUESS = 0 IF ( IGUESS .GE. 2 .AND. IREAD .EQ. 0 ) IREAD = 1 ICARE = IPAR(10) NTOL = IPAR(4) NDIMF = IPAR(5) NDIMI = IPAR(6) NFXPNT = IPAR(11) IPRINT = IPAR(7) INDEX = IPAR(12) IF (NY .EQ. 0) INDEX = 0 MSTAR = 0 MMAX = 0 DO 30 I = 1, NCOMP MMAX = MAX0 ( MMAX, M(I) ) MSTAR = MSTAR + M(I) MT(I) = M(I) 30 CONTINUE IF ( K .EQ. 0 ) K = MAX0( MMAX + 1 , 5 - MMAX ) DO 40 I = 1, MSTAR 40 TZETA(I) = ZETA(I) DO 50 I = 1, NTOL LTTOL(I) = LTOL(I) 50 TOLIN(I) = TOL(I) TLEFT = ALEFT TRIGHT = ARIGHT NC = NCOMP NNY = NY KD = K * NCOMP KDY = K * NCY C C... print the input data for checking. C IF ( IPRINT .GT. -1 ) GO TO 80 IF ( NONLIN .EQ. 0 ) THEN WRITE (IOUT,260) NCOMP, (M(IP), IP=1,NCOMP) ELSE WRITE (IOUT,270) NCOMP, (M(IP), IP=1,NCOMP) END IF WRITE (IOUT,275) NY IF (NY .GT. 0 .AND. INDEX .EQ. 0) THEN WRITE (IOUT,276) ELSE WRITE (IOUT,278) INDEX END IF WRITE (IOUT,279) WRITE (IOUT,280) (ZETA(IP), IP=1,MSTAR) IF ( NFXPNT .GT. 0 ) 1 WRITE (IOUT,340) NFXPNT, (FIXPNT(IP), IP=1,NFXPNT) WRITE (IOUT,290) K WRITE (IOUT,299) WRITE (IOUT,300) (LTOL(IP), IP=1,NTOL) WRITE (IOUT,309) WRITE (IOUT,310) (TOL(IP), IP=1,NTOL) IF (IGUESS .GE. 2) WRITE (IOUT,320) IF (IREAD .EQ. 2) WRITE (IOUT,330) 80 CONTINUE C C... check for correctness of data C IF ( K .LT. 0 .OR. K .GT. 7 ) RETURN IF ( N .LT. 0 ) RETURN IF ( IREAD .LT. 0 .OR. IREAD .GT. 2 ) RETURN IF ( IGUESS .LT. 0 .OR. IGUESS .GT. 4 ) RETURN IF ( ICARE .LT. -1 .OR. ICARE .GT. 2 ) RETURN IF ( INDEX .LT. 0 .OR. INDEX .GT. 2 ) RETURN IF ( NTOL .LT. 0 .OR. NTOL .GT. MSTAR ) RETURN IF ( NFXPNT .LT. 0 ) RETURN IF ( IPRINT .LT. (-1) .OR. IPRINT .GT. 1 ) RETURN IF ( MSTAR .LT. 0 .OR. MSTAR .GT. 40 ) RETURN IP = 1 DO 100 I = 1, MSTAR IF ( DABS(ZETA(I) - ALEFT) .LT. PRECIS .OR. 1 DABS(ZETA(I) - ARIGHT) .LT. PRECIS ) GO TO 100 90 IF ( IP .GT. NFXPNT ) RETURN IF ( ZETA(I) - PRECIS .LT. FIXPNT(IP) ) GO TO 95 IP = IP + 1 GO TO 90 95 IF ( ZETA(I) + PRECIS .LT. FIXPNT(IP) ) RETURN 100 CONTINUE C C... set limits on iterations and initialize counters. C... limit = maximum number of newton iterations per mesh. C... see subroutine newmsh for the roles of mshlmt , mshflg , C... mshnum , and mshalt . C MSHLMT = 3 MSHFLG = 0 MSHNUM = 1 MSHALT = 1 LIMIT = 40 C C... compute the maxium possible n for the given sizes of C... ispace and fspace. C NREC = 0 DO 110 I = 1, MSTAR IB = MSTAR + 1 - I IF ( ZETA(IB) .GE. ARIGHT ) NREC = I 110 CONTINUE NFIXI = MSTAR NSIZEI = 3 + KDY + MSTAR NFIXF = NREC * (2*MSTAR) + 5 * MSTAR + 3 NSIZEF = 4 + 3 * MSTAR + (KDY+5) * (KDY+MSTAR) + 1(2*MSTAR-NREC) * 2*MSTAR + (MSTAR+NY+2)*NCOMP + KDY NMAXF = (NDIMF - NFIXF) / NSIZEF NMAXI = (NDIMI - NFIXI) / NSIZEI IF ( IPRINT .LT. 1 ) THEN WRITE(IOUT,350) WRITE(IOUT,351) NMAXF, NMAXI ENDIF NMAX = MIN0( NMAXF, NMAXI ) IF ( NMAX .LT. N ) RETURN IF ( NMAX .LT. NFXPNT+1 ) RETURN IF (NMAX .LT. 2*NFXPNT+2 .AND. IPRINT .LT. 1) WRITE(IOUT,360) C C... generate pointers to break up fspace and ispace . C LXI = 1 LG = LXI + NMAX + 1 LXIOLD = LG + 2*MSTAR * (NMAX * (2*MSTAR-NREC) + NREC) LW = LXIOLD + NMAX + 1 LV = LW + KDY**2 * NMAX LFC = LV + MSTAR * KDY * NMAX LZ = LFC + (MSTAR+NY+2) * NCOMP * NMAX LDMZ = LZ + MSTAR * (NMAX + 1) LDMV = LDMZ + KDY* NMAX LDELZ = LDMV + KDY * NMAX LDELDZ = LDELZ + MSTAR * (NMAX + 1) LDQZ = LDELDZ + KDY * NMAX LDQDMZ = LDQZ + MSTAR * (NMAX + 1) LRHS = LDQDMZ + KDY * NMAX LVALST = LRHS + KDY * NMAX + MSTAR LSLOPE = LVALST + 4 * MSTAR * NMAX LACCUM = LSLOPE + NMAX LSCL = LACCUM + NMAX + 1 LDSCL = LSCL + MSTAR * (NMAX + 1) LPVTG = 1 LPVTW = LPVTG + MSTAR * (NMAX + 1) LINTEG = LPVTW + KDY * NMAX C C... if iguess .ge. 2, move xiold, z, and dmz to their proper C... locations in fspace. C IF ( IGUESS .LT. 2 ) GO TO 160 NOLD = N IF (IGUESS .EQ. 4) NOLD = ISPACE(1) NZ = MSTAR * (NOLD + 1) NDMZ = KDY * NOLD NP1 = N + 1 IF ( IGUESS .EQ. 4 ) NP1 = NP1 + NOLD + 1 DO 120 I=1,NZ 120 FSPACE( LZ+I-1 ) = FSPACE( NP1+I ) IDMZ = NP1 + NZ DO 125 I=1,NDMZ 125 FSPACE( LDMZ+I-1 ) = FSPACE( IDMZ+I ) NP1 = NOLD + 1 IF ( IGUESS .EQ. 4 ) GO TO 140 DO 130 I=1,NP1 130 FSPACE( LXIOLD+I-1 ) = FSPACE( LXI+I-1 ) GO TO 160 140 DO 150 I=1,NP1 150 FSPACE( LXIOLD+I-1 ) = FSPACE( N+1+I ) 160 CONTINUE C C... initialize collocation points, constants, mesh. C CALL CONSTS ( K, RHO, COEF ) IF (NY.EQ.0) THEN NYCB = 1 ELSE NYCB = NY ENDIF CALL NEWMSH (3+IREAD, FSPACE(LXI), FSPACE(LXIOLD), DUMMY, 1 DUMMY, DUMMY, DUMMY, DUMMY, DUMMY, NFXPNT, FIXPNT, 2 DUMMY2, DFSUB, DUMMY2, DUMMY2, NCOMP, NYCB) C C... determine first approximation, if the problem is nonlinear. C IF (IGUESS .GE. 2) GO TO 230 NP1 = N + 1 DO 210 I = 1, NP1 210 FSPACE( I + LXIOLD - 1 ) = FSPACE( I + LXI - 1 ) NOLD = N IF ( NONLIN .EQ. 0 .OR. IGUESS .EQ. 1 ) GO TO 230 C C... system provides first approximation of the solution. C... choose z(j) = 0 for j=1,...,mstar. C DO 220 I=1, NZ 220 FSPACE( LZ-1+I ) = 0.D0 DO 225 I=1, NDMZ 225 FSPACE( LDMZ-1+I ) = 0.D0 230 CONTINUE IF (IGUESS .GE. 2) IGUESS = 0 CALL CONTRL (FSPACE(LXI),FSPACE(LXIOLD),FSPACE(LZ),FSPACE(LDMZ), + FSPACE(LDMV), 1 FSPACE(LRHS),FSPACE(LDELZ),FSPACE(LDELDZ),FSPACE(LDQZ), 2 FSPACE(LDQDMZ),FSPACE(LG),FSPACE(LW),FSPACE(LV),FSPACE(LFC), 3 FSPACE(LVALST),FSPACE(LSLOPE),FSPACE(LSCL),FSPACE(LDSCL), 4 FSPACE(LACCUM),ISPACE(LPVTG),ISPACE(LINTEG),ISPACE(LPVTW), 5 NFXPNT,FIXPNT,IFLAG,FSUB,DFSUB,GSUB,DGSUB,GUESS ) C C... prepare output C ISPACE(1) = N ISPACE(2) = K ISPACE(3) = NCOMP ISPACE(4) = NY ISPACE(5) = MSTAR ISPACE(6) = MMAX ISPACE(7) = NZ + NDMZ + N + 2 K2 = K * K ISPACE(8) = ISPACE(7) + K2 - 1 DO 240 I = 1, NCOMP 240 ISPACE(8+I) = M(I) DO 250 I = 1, NZ 250 FSPACE( N+1+I ) = FSPACE( LZ-1+I ) IDMZ = N + 1 + NZ DO 255 I = 1, NDMZ 255 FSPACE( IDMZ+I ) = FSPACE( LDMZ-1+I ) IC = IDMZ + NDMZ DO 258 I = 1, K2 258 FSPACE( IC+I ) = COEF(I) RETURN C---------------------------------------------------------------------- 260 FORMAT(/// ' THE NUMBER OF (LINEAR) DIFF EQNS IS' , I3/ 1X, 1 16HTHEIR ORDERS ARE, 20I3) 270 FORMAT(/// 40H THE NUMBER OF (NONLINEAR) DIFF EQNS IS , I3/ 1X, 1 16HTHEIR ORDERS ARE, 20I3) 275 FORMAT( ' THERE ARE',I3,' ALGEBRAIC CONSTRAINTS') 276 FORMAT( ' THE PROBLEM HAS MIXED INDEX CONSTRAINTS') 278 FORMAT( ' THE INDEX IS', I2) 279 FORMAT(27H SIDE CONDITION POINTS ZETA ) 280 FORMAT(8F10.6, 4( / 8F10.6)) 290 FORMAT(37H NUMBER OF COLLOC PTS PER INTERVAL IS, I3) 299 FORMAT(40H COMPONENTS OF Z REQUIRING TOLERANCES - ) 300 FORMAT(100(/8I10)) 309 FORMAT(33H CORRESPONDING ERROR TOLERANCES -) 310 FORMAT(8D10.2) 320 FORMAT(44H INITIAL MESH(ES) AND Z,DMZ PROVIDED BY USER) 330 FORMAT(27H NO ADAPTIVE MESH SELECTION) 340 FORMAT(10H THERE ARE ,I5,27H FIXED POINTS IN THE MESH - , 1 10(6F10.6/)) 350 FORMAT(35H THE MAXIMUM NUMBER OF SUBINTERVALS) 351 FORMAT(9H IS MIN (, I4,23H (ALLOWED FROM FSPACE),,I4, 1 24H (ALLOWED FROM ISPACE) )) 360 FORMAT(/53H INSUFFICIENT SPACE TO DOUBLE MESH FOR ERROR ESTIMATE) END SUBROUTINE CONTRL (XI, XIOLD, Z, DMZ, DMV, RHS, DELZ, DELDMZ, 1 DQZ, DQDMZ, G, W, V, FC, VALSTR, SLOPE, SCALE, DSCALE, 2 ACCUM, IPVTG, INTEGS, IPVTW, NFXPNT, FIXPNT, IFLAG, 3 FSUB, DFSUB, GSUB, DGSUB, GUESS ) C C********************************************************************** C C purpose C this subroutine is the actual driver. the nonlinear iteration C strategy is controlled here ( see [6] ). upon convergence, errchk C is called to test for satisfaction of the requested tolerances. C C variables C C check - maximum tolerance value, used as part of criteria for C checking for nonlinear iteration convergence C relax - the relaxation factor for damped newton iteration C relmin - minimum allowable value for relax (otherwise the C jacobian is considered singular). C rlxold - previous relax C rstart - initial value for relax when problem is sensitive C ifrz - number of fixed jacobian iterations C lmtfrz - maximum value for ifrz before performing a reinversion C iter - number of iterations (counted only when jacobian C reinversions are performed). C xi - current mesh C xiold - previous mesh C ipred = 0 if relax is determined by a correction C = 1 if relax is determined by a prediction C ifreez = 0 if the jacobian is to be updated C = 1 if the jacobian is currently fixed (frozen) C iconv = 0 if no previous convergence has been obtained C = 1 if convergence on a previous mesh has been obtained C icare =-1 no convergence occurred (used for regular problems) C = 0 a regular problem C = 1 no damped newton C = 2 used for continuation (see description of ipar(10) C in coldae). C rnorm - norm of rhs (right hand side) for current iteration C rnold - norm of rhs for previous iteration C anscl - scaled norm of newton correction C anfix - scaled norm of newton correction at next step C anorm - scaled norm of a correction obtained with jacobian fixed C nz - number of components of z (see subroutine approx) C ndmz - number of components of dmz (see subroutine approx) C imesh - a control variable for subroutines newmsh and errchk C = 1 the current mesh resulted from mesh selection C or is the initial mesh. C = 2 the current mesh resulted from doubling the C previous mesh C C********************************************************************** C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION XI(1), XIOLD(1), Z(1), DMZ(1), RHS(1), DMV(1) DIMENSION G(1), W(1), V(1), VALSTR(1), SLOPE(1), ACCUM(1) DIMENSION DELZ(1), DELDMZ(1), DQZ(1), DQDMZ(1) , FIXPNT(1) DIMENSION DUMMY(1), SCALE(1), DSCALE(1), FC(1), DF(800) DIMENSION FCSP(40,60), CBSP(20,20) DIMENSION INTEGS(1), IPVTG(1), IPVTW(1) C COMMON /COLOUT/ PRECIS, IOUT, IPRINT COMMON /COLORD/ K, NCOMP, NY, NCY, MSTAR, KD, KDY, MMAX, M(20) COMMON /COLAPR/ N, NOLD, NMAX, NZ, NDMZ COMMON /COLMSH/ MSHFLG, MSHNUM, MSHLMT, MSHALT COMMON /COLSID/ ZETA(40), ALEFT, ARIGHT, IZETA, IDUM COMMON /COLNLN/ NONLIN, ITER, LIMIT, ICARE, IGUESS, INDEX COMMON /COLEST/ TOL(40), WGTMSH(40), WGTERR(40), TOLIN(40), 1 ROOT(40), JTOL(40), LTOL(40), NTOL C EXTERNAL FSUB, DFSUB, GSUB, DGSUB, GUESS C C... constants for control of nonlinear iteration C RELMIN = 1.D-3 RSTART = 1.D-2 LMTFRZ = 4 C C... compute the maximum tolerance C CHECK = 0.D0 DO 10 I = 1, NTOL 10 CHECK = DMAX1 ( TOLIN(I), CHECK ) IMESH = 1 ICONV = 0 IF ( NONLIN .EQ. 0 ) ICONV = 1 ICOR = 0 NOCONV = 0 MSING = 0 ISING = 0 C C... the main iteration begins here . C... loop 20 is executed until error tolerances are satisfied or C... the code fails (due to a singular matrix or storage limitations) C 20 CONTINUE C C... initialization for a new mesh C ITER = 0 IF ( NONLIN .GT. 0 ) GO TO 50 C C... the linear case. C... set up and solve equations C CALL LSYSLV (MSING, XI, XIOLD, DUMMY, DUMMY, Z, DMZ, G, 1 W, V, FC, RHS, DUMMY, INTEGS, IPVTG, IPVTW, RNORM, 0, 2 FSUB, DFSUB, GSUB, DGSUB, GUESS, ISING ) C C... check for a singular matrix C IF (ISING .NE. 0) THEN IF ( IPRINT .LT. 1 ) WRITE (IOUT,497) IFLAG = 0 RETURN END IF IF ( MSING .EQ. 0 ) GO TO 400 30 IF ( MSING .LT. 0 ) GO TO 40 IF ( IPRINT .LT. 1 ) WRITE (IOUT,495) GO TO 460 40 IF ( IPRINT .LT. 1 ) WRITE (IOUT,490) IFLAG = 0 RETURN C C... iteration loop for nonlinear case C... define the initial relaxation parameter (= relax) C 50 RELAX = 1.D0 C C... check for previous convergence and problem sensitivity C IF ( ICARE .EQ. (-1) ) RELAX = RSTART IF ( ICARE .EQ. 1 ) RELAX = 1.D0 IF ( ICONV .EQ. 0 ) GO TO 160 C C... convergence on a previous mesh has been obtained. thus C... we have a very good initial approximation for the newton C... process. proceed with one full newton and then iterate C... with a fixed jacobian. C IFREEZ = 0 C C... evaluate right hand side and its norm and C... find the first newton correction C CALL LSYSLV (MSING, XI, XIOLD, Z, DMZ, DELZ, DELDMZ, G, 1 W, V, FC, RHS, DQDMZ, INTEGS, IPVTG, IPVTW, RNOLD, 1, 2 FSUB, DFSUB, GSUB, DGSUB, GUESS, ISING ) C IF ( IPRINT .LT. 0 ) WRITE(IOUT,530) IF ( IPRINT .LT. 0 ) WRITE (IOUT,510) ITER, RNOLD GO TO 70 C C... solve for the next iterate . C... the value of ifreez determines whether this is a full C... newton step (=0) or a fixed jacobian iteration (=1). C 60 IF ( IPRINT .LT. 0 ) WRITE (IOUT,510) ITER, RNORM RNOLD = RNORM CALL LSYSLV (MSING, XI, XIOLD, Z, DMZ, DELZ, DELDMZ, G, 1 W, V, FC, RHS, DUMMY, INTEGS, IPVTG, IPVTW, RNORM, 2 3+IFREEZ, FSUB, DFSUB, GSUB, DGSUB, GUESS, ISING ) C C... check for a singular matrix C 70 IF ( MSING .NE. 0 ) GO TO 30 IF (ISING .NE. 0) THEN IF ( IPRINT .LT. 1 ) WRITE (IOUT,497) IFLAG = 0 RETURN END IF IF ( IFREEZ .EQ. 1 ) GO TO 80 C C... a full newton step C ITER = ITER + 1 IFRZ = 0 80 CONTINUE C C... update z and dmz , compute new rhs and its norm C DO 90 I = 1, NZ Z(I) = Z(I) + DELZ(I) 90 CONTINUE DO 100 I = 1, NDMZ DMZ(I) = DMZ(I) + DELDMZ(I) 100 CONTINUE CALL LSYSLV (MSING, XI, XIOLD, Z, DMZ, DELZ, DELDMZ, G, 1 W, V, FC, RHS, DUMMY, INTEGS, IPVTG, IPVTW, RNORM, 2, 2 FSUB, DFSUB, GSUB, DGSUB, GUESS, ISING ) C C... check monotonicity. if the norm of rhs gets smaller, C... proceed with a fixed jacobian; else proceed cautiously, C... as if convergence has not been obtained before (iconv=0). C IF ( RNORM .LT. PRECIS ) GO TO 390 IF ( RNORM .GT. RNOLD ) GO TO 130 IF ( IFREEZ .EQ. 1 ) GO TO 110 IFREEZ = 1 GO TO 60 C C... verify that the linear convergence with fixed jacobian C... is fast enough. C 110 IFRZ = IFRZ + 1 IF ( IFRZ .GE. LMTFRZ ) IFREEZ = 0 IF ( RNOLD .LT. 4.D0*RNORM ) IFREEZ = 0 C C... check convergence (iconv = 1). C DO 120 IT = 1, NTOL INZ = LTOL(IT) DO 120 IZ = INZ, NZ, MSTAR IF ( DABS(DELZ(IZ)) .GT. 1 TOLIN(IT) * (DABS(Z(IZ)) + 1.D0)) GO TO 60 120 CONTINUE C C... convergence obtained C IF ( IPRINT .LT. 1 ) WRITE (IOUT,560) ITER GO TO 400 C C... convergence of fixed jacobian iteration failed. C 130 IF ( IPRINT .LT. 0 ) WRITE (IOUT,510) ITER, RNORM IF ( IPRINT .LT. 0 ) WRITE (IOUT,540) ICONV = 0 IF ( ICARE .NE. 1 ) RELAX = RSTART DO 140 I = 1, NZ Z(I) = Z(I) - DELZ(I) 140 CONTINUE DO 150 I = 1, NDMZ DMZ(I) = DMZ(I) - DELDMZ(I) 150 CONTINUE C C... update old mesh C NP1 = N + 1 DO 155 I = 1, NP1 155 XIOLD(I) = XI(I) NOLD = N C ITER = 0 C C... no previous convergence has been obtained. proceed C... with the damped newton method. C... evaluate rhs and find the first newton correction. C 160 IF(IPRINT .LT. 0) WRITE (IOUT,500) CALL LSYSLV (MSING, XI, XIOLD, Z, DMZ, DELZ, DELDMZ, G, 1 W, V, FC, RHS, DQDMZ, INTEGS, IPVTG, IPVTW, RNOLD, 1, 2 FSUB, DFSUB, GSUB, DGSUB, GUESS, ISING ) C C... check for a singular matrix C IF ( MSING .NE. 0 ) GO TO 30 IF (ISING .NE. 0) THEN IF ( IPRINT .LT. 1 ) WRITE (IOUT,497) IFLAG = 0 RETURN END IF C C... bookkeeping for first mesh C IF ( IGUESS .EQ. 1 ) IGUESS = 0 C C... find initial scaling C CALL SKALE (N, MSTAR, KDY, Z, DMZ, XI, SCALE, DSCALE) RLXOLD = RELAX IPRED = 1 GO TO 220 C C... main iteration loop C 170 RNOLD = RNORM IF ( ITER .GE. LIMIT ) GO TO 430 C C... update scaling C CALL SKALE (N, MSTAR, KDY, Z, DMZ, XI, SCALE, DSCALE) C C... compute norm of newton correction with new scaling C ANSCL = 0.D0 DO 180 I = 1, NZ ANSCL = ANSCL + (DELZ(I) * SCALE(I))**2 180 CONTINUE DO 190 I = 1, NDMZ ANSCL = ANSCL + (DELDMZ(I) * DSCALE(I))**2 190 CONTINUE ANSCL = DSQRT(ANSCL / FLOAT(NZ+NDMZ)) C C... find a newton direction C CALL LSYSLV (MSING, XI, XIOLD, Z, DMZ, DELZ, DELDMZ, G, 1 W, V, FC, RHS, DUMMY, INTEGS, IPVTG, IPVTW, RNORM, 3, 2 FSUB, DFSUB, GSUB, DGSUB, GUESS, ISING ) C C... check for a singular matrix C IF ( MSING .NE. 0 ) GO TO 30 IF (ISING .NE. 0) THEN IF ( IPRINT .LT. 1 ) WRITE (IOUT,497) IFLAG = 0 RETURN END IF C C... predict relaxation factor for newton step. C IF (ICARE .NE. 1) THEN ANDIF = 0.D0 DO 200 I = 1, NZ ANDIF = ANDIF + ((DQZ(I) - DELZ(I)) * SCALE(I))**2 200 CONTINUE DO 210 I = 1, NDMZ ANDIF = ANDIF + ((DQDMZ(I) - DELDMZ(I)) * DSCALE(I))**2 210 CONTINUE ANDIF = DSQRT(ANDIF/FLOAT(NZ+NDMZ) + PRECIS) RELAX = RELAX * ANSCL / ANDIF IF ( RELAX .GT. 1.D0 ) RELAX = 1.D0 RLXOLD = RELAX IPRED = 1 END IF 220 ITER = ITER + 1 C C... determine a new z and dmz and find new rhs and its norm C DO 230 I = 1, NZ Z(I) = Z(I) + RELAX * DELZ(I) 230 CONTINUE DO 240 I = 1, NDMZ DMZ(I) = DMZ(I) + RELAX * DELDMZ(I) 240 CONTINUE 250 CALL LSYSLV (MSING, XI, XIOLD, Z, DMZ, DQZ, DQDMZ, G, 1 W, V, FC, RHS, DUMMY, INTEGS, IPVTG, IPVTW, RNORM, 2, 2 FSUB, DFSUB, GSUB, DGSUB, GUESS, ISING ) C C... compute a fixed jacobian iterate (used to control relax) C CALL LSYSLV (MSING, XI, XIOLD, Z, DMZ, DQZ, DQDMZ, G, 1 W, V, FC, RHS, DUMMY, INTEGS, IPVTG, IPVTW, RNORM, 4, 2 FSUB, DFSUB, GSUB, DGSUB, GUESS, ISING ) C C... find scaled norms of various terms used to correct relax C ANORM = 0.D0 ANFIX = 0.D0 DO 260 I = 1, NZ ANORM = ANORM + (DELZ(I) * SCALE(I))**2 ANFIX = ANFIX + (DQZ(I) * SCALE(I))**2 260 CONTINUE DO 270 I = 1, NDMZ ANORM = ANORM + (DELDMZ(I) * DSCALE(I))**2 ANFIX = ANFIX + (DQDMZ(I) * DSCALE(I))**2 270 CONTINUE ANORM = DSQRT(ANORM / FLOAT(NZ+NDMZ)) ANFIX = DSQRT(ANFIX / FLOAT(NZ+NDMZ)) IF ( ICOR .EQ. 1 ) GO TO 280 IF (IPRINT .LT. 0) WRITE (IOUT,520) ITER, RELAX, ANORM, 1 ANFIX, RNOLD, RNORM GO TO 290 280 IF (IPRINT .LT. 0) WRITE (IOUT,550) RELAX, ANORM, ANFIX, 1 RNOLD, RNORM 290 ICOR = 0 C C... check for monotonic decrease in delz and deldmz. C IF (ANFIX.LT.PRECIS .OR. RNORM.LT.PRECIS) GO TO 390 IF ( ANFIX .GT. ANORM .AND. ICARE .NE. 1) GO TO 300 C C... we have a decrease. C... if dqz and dqdmz small, check for convergence C IF ( ANFIX .LE. CHECK ) GO TO 350 C C... correct the predicted relax unless the corrected C... value is within 10 percent of the predicted one. C IF ( IPRED .NE. 1 ) GO TO 170 300 IF ( ITER .GE. LIMIT ) GO TO 430 IF ( ICARE .EQ. 1 ) GO TO 170 C C... correct the relaxation factor. C IPRED = 0 ARG = (ANFIX/ANORM - 1.D0) / RELAX + 1.D0 IF ( ARG .LT. 0.D0 ) GO TO 170 IF (ARG .LE. .25D0*RELAX+.125D0*RELAX**2) GO TO 310 FACTOR = -1.D0 + DSQRT (1.D0+8.D0 * ARG) IF ( DABS(FACTOR-1.D0) .LT. .1D0*FACTOR ) GO TO 170 IF ( FACTOR .LT. 0.5D0 ) FACTOR = 0.5D0 RELAX = RELAX / FACTOR GO TO 320 310 IF ( RELAX .GE. .9D0 ) GO TO 170 RELAX = 1.D0 320 ICOR = 1 IF ( RELAX .LT. RELMIN ) GO TO 440 FACT = RELAX - RLXOLD DO 330 I = 1, NZ Z(I) = Z(I) + FACT * DELZ(I) 330 CONTINUE DO 340 I = 1, NDMZ DMZ(I) = DMZ(I) + FACT * DELDMZ(I) 340 CONTINUE RLXOLD = RELAX GO TO 250 C C... check convergence (iconv = 0). C 350 CONTINUE DO 360 IT = 1, NTOL INZ = LTOL(IT) DO 360 IZ = INZ, NZ, MSTAR IF ( DABS(DQZ(IZ)) .GT. 1 TOLIN(IT) * (DABS(Z(IZ)) + 1.D0) ) GO TO 170 360 CONTINUE C C... convergence obtained C IF ( IPRINT .LT. 1 ) WRITE (IOUT,560) ITER C C... since convergence obtained, update z and dmz with term C... from the fixed jacobian iteration. C DO 370 I = 1, NZ Z(I) = Z(I) + DQZ(I) 370 CONTINUE DO 380 I = 1, NDMZ DMZ(I) = DMZ(I) + DQDMZ(I) 380 CONTINUE 390 IF ( (ANFIX .LT. PRECIS .OR. RNORM .LT. PRECIS) 1 .AND. IPRINT .LT. 1 ) WRITE (IOUT,560) ITER ICONV = 1 IF ( ICARE .EQ. (-1) ) ICARE = 0 C C... if full output has been requested, print values of the C... solution components z at the meshpoints and y at C... collocation points. C 400 IF ( IPRINT .GE. 0 ) GO TO 420 DO 410 J = 1, MSTAR WRITE(IOUT,610) J 410 WRITE(IOUT,620) (Z(LJ), LJ = J, NZ, MSTAR) DO 415 J = 1, NY WRITE(IOUT,630) J 415 WRITE(IOUT,620) (DMZ(LJ), LJ = J+NCOMP, NDMZ, KDY) C C... check for error tolerance satisfaction C 420 IFIN = 1 IF (IMESH .EQ. 2) CALL ERRCHK (XI, Z, DMZ, VALSTR, IFIN) IF ( IMESH .EQ. 1 .OR. 1 IFIN .EQ. 0 .AND. ICARE .NE. 2) GO TO 460 IFLAG = 1 RETURN C C... diagnostics for failure of nonlinear iteration. C 430 IF ( IPRINT .LT. 1 ) WRITE (IOUT,570) ITER GO TO 450 440 IF( IPRINT .LT. 1 ) THEN WRITE(IOUT,580) RELAX WRITE(IOUT,581) RELMIN ENDIF 450 IFLAG = -2 NOCONV = NOCONV + 1 IF ( ICARE .EQ. 2 .AND. NOCONV .GT. 1 ) RETURN IF ( ICARE .EQ. 0 ) ICARE = -1 C C... update old mesh C 460 NP1 = N + 1 DO 470 I = 1, NP1 470 XIOLD(I) = XI(I) NOLD = N C C... pick a new mesh C... check safeguards for mesh refinement C IMESH = 1 IF ( ICONV .EQ. 0 .OR. MSHNUM .GE. MSHLMT 1 .OR. MSHALT .GE. MSHLMT ) IMESH = 2 IF ( MSHALT .GE. MSHLMT .AND. 1 MSHNUM .LT. MSHLMT ) MSHALT = 1 IF (NY.EQ.0) THEN NYCB = 1 ELSE NYCB = NY ENDIF CALL NEWMSH (IMESH, XI, XIOLD, Z, DMZ, DMV, VALSTR, 1 SLOPE, ACCUM, NFXPNT, FIXPNT, DF, DFSUB, 2 FCSP, CBSP, NCOMP, NYCB) C C... exit if expected n is too large (but may try n=nmax once) C IF ( N .LE. NMAX ) GO TO 480 N = N / 2 IFLAG = -1 IF ( ICONV .EQ. 0 .AND. IPRINT .LT. 1 ) WRITE (IOUT,590) IF ( ICONV .EQ. 1 .AND. IPRINT .LT. 1 ) WRITE (IOUT,600) RETURN 480 IF ( ICONV .EQ. 0 ) IMESH = 1 C IF ( ICARE .EQ. 1 ) ICONV = 0 GO TO 20 C --------------------------------------------------------------- 490 FORMAT(//35H THE GLOBAL BVP-MATRIX IS SINGULAR ) 495 FORMAT(//40H A LOCAL ELIMINATION MATRIX IS SINGULAR ) 497 FORMAT(// 'SINGULAR PROJECTION MATRIX DUE TO INDEX > 2' ) 500 FORMAT(/30H FULL DAMPED NEWTON ITERATION,) 510 FORMAT(13H ITERATION = , I3, 15H NORM (RHS) = , D10.2) 520 FORMAT(13H ITERATION = ,I3,22H RELAXATION FACTOR = ,D10.2 1 /33H NORM OF SCALED RHS CHANGES FROM ,D10.2,3H TO,D10.2 2 /33H NORM OF RHS CHANGES FROM ,D10.2,3H TO,D10.2, 2 D10.2) 530 FORMAT(/27H FIXED JACOBIAN ITERATIONS,) 540 FORMAT(/35H SWITCH TO DAMPED NEWTON ITERATION,) 550 FORMAT(40H RELAXATION FACTOR CORRECTED TO RELAX = , D10.2 1 /33H NORM OF SCALED RHS CHANGES FROM ,D10.2,3H TO,D10.2 2 /33H NORM OF RHS CHANGES FROM ,D10.2,3H TO,D10.2 2 ,D10.2) 560 FORMAT(/18H CONVERGENCE AFTER , I3,11H ITERATIONS /) 570 FORMAT(/22H NO CONVERGENCE AFTER , I3, 11H ITERATIONS/) 580 FORMAT(/37H NO CONVERGENCE. RELAXATION FACTOR =,D10.3 1 ,13H IS TOO SMALL ) 581 FORMAT(10H(LESS THAN, D10.3, 1H)/) 590 FORMAT(18H (NO CONVERGENCE) ) 600 FORMAT(50H (PROBABLY TOLERANCES TOO STRINGENT, OR NMAX TOO 1 ,6HSMALL) ) 610 FORMAT( 19H MESH VALUES FOR Z(, I2, 2H), ) 620 FORMAT(1H , 5D15.7) 630 FORMAT( ' VALUES AT 1st COLLOCATION POINTS FOR Y(', I2, 2H), ) END SUBROUTINE SKALE (N, MSTAR, KDY, Z, DMZ, XI, SCALE, DSCALE) C C********************************************************************** C C purpose C provide a proper scaling of the state variables, used C to control the damping factor for a newton iteration [4]. C C variables C C n = number of mesh subintervals C mstar = number of unknomns in z(u(x)) C kdy = number of unknowns in dmz per mesh subinterval C z = the global current solution vector C dmz = the global current highest derivs vector C xi = the current mesh C scale = scaling vector for z C dscale = scaling vector for dmz C C********************************************************************** C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION Z(MSTAR,1), SCALE(MSTAR,1), DMZ(KDY,N), DSCALE(KDY,N) DIMENSION XI(1), BASM(5) C COMMON /COLORD/ K, NCOMP, NY, NCY, ID1, KD, ID3, MMAX, M(20) C BASM(1) = 1.D0 DO 50 J=1,N IZ = 1 H = XI(J+1) - XI(J) DO 10 L = 1, MMAX BASM(L+1) = BASM(L) * H / FLOAT(L) 10 CONTINUE DO 40 ICOMP = 1, NCOMP SCAL = (DABS(Z(IZ,J)) + DABS(Z(IZ,J+1))) * .5D0 + 1.D0 MJ = M(ICOMP) DO 20 L = 1, MJ SCALE(IZ,J) = BASM(L) / SCAL IZ = IZ + 1 20 CONTINUE SCAL = BASM(MJ+1) / SCAL DO 30 IDMZ = ICOMP, KDY, NCY DSCALE(IDMZ,J) = SCAL 30 CONTINUE 40 CONTINUE DO 45 ICOMP = 1+NCOMP,NCY SCAL = 1.D0 / (DABS(DMZ(ICOMP,J)) + 1.D0) DO 45 IDMZ = ICOMP, KDY, NCY DSCALE(IDMZ,J) = SCAL 45 CONTINUE 50 CONTINUE NP1 = N + 1 DO 60 IZ = 1, MSTAR SCALE(IZ,NP1) = SCALE(IZ,N) 60 CONTINUE RETURN END C---------------------------------------------------------------------- C p a r t 2 C mesh selection, error estimation, (and related C constant assignment) routines -- see [5], [6] C---------------------------------------------------------------------- C SUBROUTINE NEWMSH (MODE, XI, XIOLD, Z, DMZ, DMV, VALSTR, 1 SLOPE, ACCUM, NFXPNT, FIXPNT, DF, DFSUB, 2 FC, CB, NCOMP, NYCB) C C********************************************************************** C C purpose C select a mesh on which a collocation solution is to be C determined C C there are 5 possible modes of action: C mode = 5,4,3 - deal mainly with definition of an initial C mesh for the current boundary value problem C = 2,1 - deal with definition of a new mesh, either C by simple mesh halving or by mesh selection C more specifically, for C mode = 5 an initial (generally nonuniform) mesh is C defined by the user and no mesh selection is to C be performed C = 4 an initial (generally nonuniform) mesh is C defined by the user C = 3 a simple uniform mesh (except possibly for some C fixed points) is defined; n= no. of subintervals C = 1 the automatic mesh selection procedure is used C (see [5] for details) C = 2 a simple mesh halving is performed C C********************************************************************** C C variables C C n = number of mesh subintervals C nold = number of subintervals for former mesh C xi - mesh point array C xiold - former mesh point array C mshlmt - maximum no. of mesh selections which are permitted C for a given n before mesh halving C mshnum - no. of mesh selections which have actually been C performed for the given n C mshalt - no. of consecutive times ( plus 1 ) the mesh C selection has alternately halved and doubled n. C if mshalt .ge. mshlmt then contrl requires C that the current mesh be halved. C mshflg = 1 the mesh is a halving of its former mesh C (so an error estimate has been calculated) C = 0 otherwise C iguess - ipar(9) in subroutine coldae. it is used C here only for mode=5 and 4, where C = 2 the subroutine sets xi=xiold. this is C used e.g. if continuation is being per- C formed, and a mesh for the old differen- C tial equation is being used C = 3 same as for =2, except xi uses every other C point of xiold (so mesh xiold is mesh xi C halved) C = 4 xi has been defined by the user, and an old C mesh xiold is also available C otherwise, xi has been defined by the user C and we set xiold=xi in this subroutine C slope - an approximate quantity to be equidistributed for C mesh selection (see [5]), viz, C . (k+mj) C slope(i)= max (weight(l) *u (xi(i))) C 1.le.l.le.ntol j C C where j=jtol(l) C slphmx - maximum of slope(i)*(xiold(i+1)-xiold(i)) for C i = 1 ,..., nold. C accum - accum(i) is the integral of slope from aleft C to xiold(i). C valstr - is assigned values needed in errchk for the C error estimate. C fc - you know C********************************************************************** C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION D1(40), D2(40), SLOPE(1), ACCUM(1), VALSTR(1), DMV(1) DIMENSION XI(1), XIOLD(1), Z(1), DMZ(1), FIXPNT(1), DUMMY(1) DIMENSION FC(NCOMP,60), ZVAL(40), YVAL(40), A(28), DF(NCY,1) DIMENSION CB(NYCB,NYCB), IPVTCB(40), BCOL(40), U(400), V(400) EXTERNAL DFSUB C COMMON /COLLOC/ RHO(7), COEF(49) COMMON /COLOUT/ PRECIS, IOUT, IPRINT COMMON /COLORD/ K, NCDUM, NY, NCY, MSTAR, KD, KDY, MMAX, M(20) COMMON /COLAPR/ N, NOLD, NMAX, NZ, NDMZ COMMON /COLMSH/ MSHFLG, MSHNUM, MSHLMT, MSHALT COMMON /COLNLN/ NONLIN, ITER, LIMIT, ICARE, IGUESS, INDEX COMMON /COLSID/ ZETA(40), ALEFT, ARIGHT, IZETA, IDUM COMMON /COLBAS/ B(28), ACOL(28,7), ASAVE(28,4) COMMON /COLEST/ TOL(40), WGTMSH(40), WGTERR(40), TOLIN(40), 1 ROOT(40), JTOL(40), LTOL(40), NTOL C NFXP1 = NFXPNT +1 GO TO (180, 100, 50, 20, 10), MODE C C... mode=5 set mshlmt=1 so that no mesh selection is performed C 10 MSHLMT = 1 C C... mode=4 the user-specified initial mesh is already in place. C 20 IF ( IGUESS .LT. 2 ) GO TO 40 C C... iguess=2, 3 or 4. C NOLDP1 = NOLD + 1 IF (IPRINT .LT. 1) WRITE(IOUT,360) NOLD, (XIOLD(I), I=1,NOLDP1) IF ( IGUESS .NE. 3 ) GO TO 40 C C... if iread ( ipar(8) ) .ge. 1 and iguess ( ipar(9) ) .eq. 3 C... then the first mesh is every second point of the C... mesh in xiold . C N = NOLD /2 I = 0 DO 30 J = 1, NOLD, 2 I = I + 1 30 XI(I) = XIOLD(J) 40 CONTINUE NP1 = N + 1 XI(1) = ALEFT XI(NP1) = ARIGHT GO TO 320 C C... mode=3 generate a (piecewise) uniform mesh. if there are C... fixed points then ensure that the n being used is large enough. C 50 IF ( N .LT. NFXP1 ) N = NFXP1 NP1 = N + 1 XI(1) = ALEFT ILEFT = 1 XLEFT = ALEFT C C... loop over the subregions between fixed points. C DO 90 J = 1, NFXP1 XRIGHT = ARIGHT IRIGHT = NP1 IF ( J .EQ. NFXP1 ) GO TO 60 XRIGHT = FIXPNT(J) C C... determine where the j-th fixed point should fall in the C... new mesh - this is xi(iright) and the (j-1)st fixed C... point is in xi(ileft) C NMIN = (XRIGHT-ALEFT) / (ARIGHT-ALEFT) * FLOAT(N) + 1.5D0 IF (NMIN .GT. N-NFXPNT+J) NMIN = N - NFXPNT + J IRIGHT = MAX0 (ILEFT+1, NMIN) 60 XI(IRIGHT) = XRIGHT C C... generate equally spaced points between the j-1st and the C... j-th fixed points. C NREGN = IRIGHT - ILEFT - 1 IF ( NREGN .EQ. 0 ) GO TO 80 DX = (XRIGHT - XLEFT) / FLOAT(NREGN+1) DO 70 I = 1, NREGN 70 XI(ILEFT+I) = XLEFT + FLOAT(I) * DX 80 ILEFT = IRIGHT XLEFT = XRIGHT 90 CONTINUE GO TO 320 C C... mode=2 halve the current mesh (i.e. double its size) C 100 N2 = 2 * N C C... check that n does not exceed storage limitations C IF ( N2 .LE. NMAX ) GO TO 120 C C... if possible, try with n=nmax. redistribute first. C IF ( MODE .EQ. 2 ) GO TO 110 N = NMAX / 2 GO TO 220 110 IF ( IPRINT .LT. 1 ) WRITE(IOUT,370) N = N2 RETURN C C... calculate the old approximate solution values at C... points to be used in errchk for error estimates. C... if mshflg =1 an error estimate was obtained for C... for the old approximation so half the needed values C... will already be in valstr . C 120 IF ( MSHFLG .EQ. 0 ) GO TO 140 C C... save in valstr the values of the old solution C... at the relative positions 1/6 and 5/6 in each subinterval. C KSTORE = 1 DO 130 I = 1, NOLD HD6 = (XIOLD(I+1) - XIOLD(I)) / 6.D0 X = XIOLD(I) + HD6 CALL APPROX (I, X, VALSTR(KSTORE), DUMMY, ASAVE(1,1), + DUMMY, XIOLD, NOLD, Z, DMZ, 1 K, NCOMP, NY, MMAX, M, MSTAR, 4, DUMMY, 0) X = X + 4.D0 * HD6 KSTORE = KSTORE + 3 * MSTAR CALL APPROX (I, X, VALSTR(KSTORE), DUMMY, ASAVE(1,4), + DUMMY, XIOLD, NOLD, Z, DMZ, 1 K, NCOMP, NY, MMAX, M, MSTAR, 4, DUMMY, 0) KSTORE = KSTORE + MSTAR 130 CONTINUE GO TO 160 C C... save in valstr the values of the old solution C... at the relative positions 1/6, 2/6, 4/6 and 5/6 in C... each subinterval. C 140 KSTORE = 1 DO 150 I = 1, N X = XI(I) HD6 = (XI(I+1) - XI(I)) / 6.D0 DO 150 J = 1, 4 X = X + HD6 IF ( J.EQ.3 ) X = X + HD6 CALL APPROX (I, X, VALSTR(KSTORE), DUMMY, ASAVE(1,J), + DUMMY, XIOLD, NOLD, Z, DMZ, 1 K, NCOMP, NY, MMAX, M, MSTAR, 4, DUMMY, 0) KSTORE = KSTORE + MSTAR 150 CONTINUE 160 MSHFLG = 0 MSHNUM = 1 MODE = 2 C C... generate the halved mesh. C J = 2 DO 170 I = 1, N XI(J) = (XIOLD(I) + XIOLD(I+1)) / 2.D0 XI(J+1) = XIOLD(I+1) 170 J = J + 2 N = N2 GO TO 320 C C... mode=1 we do mesh selection if it is deemed worthwhile C 180 IF ( NOLD .EQ. 1 ) GO TO 100 IF ( NOLD .LE. 2*NFXPNT ) GO TO 100 C C... we now project DMZ for mesh selection strategy, if required C... but set DMV = DMZ in case it is not IDMZ = 1 DO 183 I = 1, NOLD DO 182 KK = 1, K DO 181 J = 1, NCY DMV(IDMZ) = DMZ(IDMZ) IDMZ = IDMZ + 1 181 CONTINUE 182 CONTINUE 183 CONTINUE IF (INDEX.NE.1 .AND. NY .GT. 0) THEN IDMZ = 1 DO 500 I=1, NOLD XI1 = XIOLD(I+1) CALL APPROX (I, XI1, ZVAL, YVAL, A, 1 COEF, XIOLD, NOLD, Z, DMZ, 1 K, NCOMP, NY, MMAX, M, MSTAR, 3, DUMMY, 1) CALL DFSUB (XI1, ZVAL, YVAL, DF) C... if index=2, form projection matrices directly C... otherwise use svd to define appropriate projection IF (INDEX.EQ.0) THEN CALL PRJSVD (FC,DF,CB,U,V,NCOMP,NCY,NY,IPVTCB,ISING,2) ELSE C... form cb DO 212 J = 1, NY DO 212 J1 = 1, NY FACT = 0.0D0 ML = 0 DO 211 L = 1, NCOMP ML = ML + M(L) FACT = FACT + DF(J+NCOMP,ML)*DF(L,MSTAR+J1) 211 CONTINUE CB(J,J1) = FACT 212 CONTINUE C... decompose cb CALL DGEFA (CB, NY, NY, IPVTCB, ISING) IF (ISING.NE.0) RETURN C... form columns of fc ML = 0 DO 215 L = 1, NCOMP ML = ML + M(L) DO 213 J1 = 1, NY BCOL(J1) = DF(J1+NCOMP,ML) 213 CONTINUE CALL DGESL(CB, NY, NY, IPVTCB, BCOL, 0) DO 215 J1 = 1, NCOMP FACT = 0.0D0 DO 214 J = 1, NY FACT = FACT + DF(J1, J+MSTAR)*BCOL(J) 214 CONTINUE FC(J1,L) = FACT C CONTINUE 215 CONTINUE ENDIF C.. finally, replace fc with the true projection SR = I - fc DO 217 J = 1, NCOMP DO 216 L = 1, NCOMP FC(J,L) = -FC(J,L) IF (J.EQ.L) FC(J,L) = FC(J,L) + 1.0D0 216 CONTINUE 217 CONTINUE C... project DMZ for the k collocation points, store in DMV DO 221 KK = 1, K DO 219 J = 1, NCOMP FACT = 0.0D0 DO 218 L = 1, NCOMP FACT = FACT + FC(J,L)*DMZ(IDMZ+L-1) 218 CONTINUE DMV(IDMZ+J-1) = FACT 219 CONTINUE IDMZ = IDMZ + NCY 221 CONTINUE 500 CONTINUE ENDIF C C... the first interval has to be treated separately from the C... other intervals (generally the solution on the (i-1)st and ith C... intervals will be used to approximate the needed derivative, but C... here the 1st and second intervals are used.) C I = 1 HIOLD = XIOLD(2) - XIOLD(1) CALL HORDER (1, D1, HIOLD, DMV, NCOMP, NCY, K) IDMZ = IDMZ + (NCOMP + NY) * K HIOLD = XIOLD(3) - XIOLD(2) CALL HORDER (2, D2, HIOLD, DMV, NCOMP, NCY, K) ACCUM(1) = 0.D0 SLOPE(1) = 0.D0 ONEOVH = 2.D0 / ( XIOLD(3) - XIOLD(1) ) DO 190 J = 1, NTOL JJ = JTOL(J) JZ = LTOL(J) 190 SLOPE(1) = DMAX1(SLOPE(1),(DABS(D2(JJ)-D1(JJ))*WGTMSH(J)* 1 ONEOVH / (1.D0 + DABS(Z(JZ)))) **ROOT(J)) SLPHMX = SLOPE(1) * (XIOLD(2) - XIOLD(1)) ACCUM(2) = SLPHMX IFLIP = 1 C C... go through the remaining intervals generating slope C... and accum . C DO 210 I = 2, NOLD HIOLD = XIOLD(I+1) - XIOLD(I) IF ( IFLIP .EQ. -1 ) 1 CALL HORDER ( I, D1, HIOLD, DMV, NCOMP, NCY, K) IF ( IFLIP .EQ. 1 ) 1 CALL HORDER ( I, D2, HIOLD, DMV, NCOMP, NCY, K) ONEOVH = 2.D0 / ( XIOLD(I+1) - XIOLD(I-1) ) SLOPE(I) = 0.D0 C C... evaluate function to be equidistributed C DO 200 J = 1, NTOL JJ = JTOL(J) JZ = LTOL(J) + (I-1) * MSTAR SLOPE(I) = DMAX1(SLOPE(I),(DABS(D2(JJ)-D1(JJ))*WGTMSH(J)* 1 ONEOVH / (1.D0 + DABS(Z(JZ)))) **ROOT(J)) 200 CONTINUE C C... accumulate approximate integral of function to be C... equidistributed C TEMP = SLOPE(I) * (XIOLD(I+1)-XIOLD(I)) SLPHMX = DMAX1(SLPHMX,TEMP) ACCUM(I+1) = ACCUM(I) + TEMP IFLIP = - IFLIP 210 CONTINUE AVRG = ACCUM(NOLD+1) / FLOAT(NOLD) DEGEQU = AVRG / DMAX1(SLPHMX,PRECIS) C C... naccum=expected n to achieve .1x user requested tolerances C NACCUM = ACCUM(NOLD+1) + 1.D0 IF ( IPRINT .LT. 0 ) WRITE(IOUT,350) DEGEQU, NACCUM C C... decide if mesh selection is worthwhile (otherwise, halve) C IF ( AVRG .LT. PRECIS ) GO TO 100 IF ( DEGEQU .GE. .5D0 ) GO TO 100 C C... nmx assures mesh has at least half as many subintervals as the C... previous mesh C NMX = MAX0 ( NOLD+1, NACCUM ) / 2 C C... this assures that halving will be possible later (for error est) C NMAX2 = NMAX / 2 C C... the mesh is at most halved C N = MIN0 ( NMAX2, NOLD, NMX ) 220 NOLDP1 = NOLD + 1 IF ( N .LT. NFXP1 ) N = NFXP1 MSHNUM = MSHNUM + 1 C C... if the new mesh is smaller than the old mesh set mshnum C... so that the next call to newmsh will produce a halved C... mesh. if n .eq. nold / 2 increment mshalt so there can not C... be an infinite loop alternating between n and n/2 points. C IF ( N .LT. NOLD ) MSHNUM = MSHLMT IF ( N .GT. NOLD/2 ) MSHALT = 1 IF ( N .EQ. NOLD/2 ) MSHALT = MSHALT + 1 MSHFLG = 0 C C... having decided to generate a new mesh with n subintervals we now C... do so, taking into account that the nfxpnt points in the array C... fixpnt must be included in the new mesh. C IN = 1 ACCL = 0.D0 LOLD = 2 XI(1) = ALEFT XI(N+1) = ARIGHT DO 310 I = 1, NFXP1 IF ( I .EQ. NFXP1 ) GO TO 250 DO 230 J = LOLD, NOLDP1 LNEW = J IF ( FIXPNT(I) .LE. XIOLD(J) ) GO TO 240 230 CONTINUE 240 CONTINUE ACCR = ACCUM(LNEW) + (FIXPNT(I)-XIOLD(LNEW))*SLOPE(LNEW-1) NREGN = (ACCR-ACCL) / ACCUM(NOLDP1) * FLOAT(N) - .5D0 NREGN = MIN0(NREGN, N - IN - NFXP1 + I) XI(IN + NREGN + 1) = FIXPNT(I) GO TO 260 250 ACCR = ACCUM(NOLDP1) LNEW = NOLDP1 NREGN = N - IN 260 IF ( NREGN .EQ. 0 ) GO TO 300 TEMP = ACCL TSUM = (ACCR - ACCL) / FLOAT(NREGN+1) DO 290 J = 1, NREGN IN = IN + 1 TEMP = TEMP + TSUM DO 270 L = LOLD, LNEW LCARRY = L IF ( TEMP .LE. ACCUM(L) ) GO TO 280 270 CONTINUE 280 CONTINUE LOLD = LCARRY 290 XI(IN) = XIOLD(LOLD-1) + (TEMP - ACCUM(LOLD-1)) / 1 SLOPE(LOLD-1) 300 IN = IN + 1 ACCL = ACCR LOLD = LNEW 310 CONTINUE MODE = 1 320 CONTINUE NP1 = N + 1 IF ( IPRINT .LT. 1 ) THEN WRITE(IOUT,340) N WRITE(IOUT,341) (XI(I),I=1,NP1) ENDIF NZ = MSTAR * (N + 1) NDMZ = KDY * N RETURN C---------------------------------------------------------------- 340 FORMAT(/17H THE NEW MESH (OF,I5,14H SUBINTERVALS) ) 341 FORMAT(100(/6F12.6)) 350 FORMAT(/21H MESH SELECTION INFO,/30H DEGREE OF EQUIDISTRIBUTION = 1 , F8.5, 28H PREDICTION FOR REQUIRED N = , I8) 360 FORMAT(/20H THE FORMER MESH (OF,I5,15H SUBINTERVALS),, 1 100(/6F12.6)) 370 FORMAT (/23H EXPECTED N TOO LARGE ) END SUBROUTINE CONSTS (K, RHO, COEF) C C********************************************************************** C C purpose C assign (once) values to various array constants. C C arrays assigned during compilation: C cnsts1 - weights for extrapolation error estimate C cnsts2 - weights for mesh selection C (the above weights come from the theoretical form for C the collocation error -- see [5]) C C arrays assigned during execution: C wgterr - the particular values of cnsts1 used for current run C (depending on k, m) C wgtmsh - gotten from the values of cnsts2 which in turn are C the constants in the theoretical expression for the C errors. the quantities in wgtmsh are 10x the values C in cnsts2 so that the mesh selection algorithm C is aiming for errors .1x as large as the user C requested tolerances. C jtol - components of differential system to which tolerances C refer (viz, if ltol(i) refers to a derivative of u(j), C then jtol(i)=j) C root - reciprocals of expected rates of convergence of compo- C nents of z(j) for which tolerances are specified C rho - the k collocation points on (0,1) C coef - C acol - the runge-kutta coefficients values at collocation C points C C********************************************************************** C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION RHO(7), COEF(K,1), CNSTS1(28), CNSTS2(28), DUMMY(1) C COMMON /COLORD/ KDUM, NCOMP, NY, NCY, MSTAR, KD, KDY, MMAX, M(20) COMMON /COLBAS/ B(28), ACOL(28,7), ASAVE(28,4) COMMON /COLEST/ TOL(40), WGTMSH(40), WGTERR(40), TOLIN(40), 1 ROOT(40), JTOL(40), LTOL(40), NTOL C DATA CNSTS1 / .25D0, .625D-1, 7.2169D-2, 1.8342D-2, 1 1.9065D-2, 5.8190D-2, 5.4658D-3, 5.3370D-3, 1.8890D-2, 2 2.7792D-2, 1.6095D-3, 1.4964D-3, 7.5938D-3, 5.7573D-3, 3 1.8342D-2, 4.673D-3, 4.150D-4, 1.919D-3, 1.468D-3, 4 6.371D-3, 4.610D-3, 1.342D-4, 1.138D-4, 4.889D-4, 5 4.177D-4, 1.374D-3, 1.654D-3, 2.863D-3 / DATA CNSTS2 / 1.25D-1, 2.604D-3, 8.019D-3, 2.170D-5, 1 7.453D-5, 5.208D-4, 9.689D-8, 3.689D-7, 3.100D-6, 2 2.451D-5, 2.691D-10, 1.120D-9, 1.076D-8, 9.405D-8, 3 1.033D-6, 5.097D-13, 2.290D-12, 2.446D-11, 2.331D-10, 4 2.936D-9, 3.593D-8, 7.001D-16, 3.363D-15, 3.921D-14, 5 4.028D-13, 5.646D-12, 7.531D-11, 1.129D-9 / C C... assign weights for error estimate C KOFF = K * ( K + 1 ) / 2 IZ = 1 DO 10 J = 1, NCOMP MJ = M(J) DO 10 L = 1, MJ WGTERR(IZ) = CNSTS1(KOFF - MJ + L) IZ = IZ + 1 10 CONTINUE C C... assign array values for mesh selection: wgtmsh, jtol, and root C JCOMP = 1 MTOT = M(1) DO 40 I = 1, NTOL LTOLI = LTOL(I) 20 CONTINUE IF ( LTOLI .LE. MTOT ) GO TO 30 JCOMP = JCOMP + 1 MTOT = MTOT + M(JCOMP) GO TO 20 30 CONTINUE JTOL(I) = JCOMP WGTMSH(I) = 1.D1 * CNSTS2(KOFF+LTOLI-MTOT) / TOLIN(I) ROOT(I) = 1.D0 / FLOAT(K+MTOT-LTOLI+1) 40 CONTINUE C C... specify collocation points C GO TO (50,60,70,80,90,100,110), K 50 RHO(1) = 0.D0 GO TO 120 60 RHO(2) = .57735026918962576451D0 RHO(1) = - RHO(2) GO TO 120 70 RHO(3) = .77459666924148337704D0 RHO(2) = .0D0 RHO(1) = - RHO(3) GO TO 120 80 RHO(4) = .86113631159405257523D0 RHO(3) = .33998104358485626480D0 RHO(2) = - RHO(3) RHO(1) = - RHO(4) GO TO 120 90 RHO(5) = .90617984593866399280D0 RHO(4) = .53846931010568309104D0 RHO(3) = .0D0 RHO(2) = - RHO(4) RHO(1) = - RHO(5) GO TO 120 100 RHO(6) = .93246951420315202781D0 RHO(5) = .66120938646626451366D0 RHO(4) = .23861918608319690863D0 RHO(3) = -RHO(4) RHO(2) = -RHO(5) RHO(1) = -RHO(6) GO TO 120 110 RHO(7) = .949107991234275852452D0 RHO(6) = .74153118559939443986D0 RHO(5) = .40584515137739716690D0 RHO(4) = 0.D0 RHO(3) = -RHO(5) RHO(2) = -RHO(6) RHO(1) = -RHO(7) 120 CONTINUE C C... map (-1,1) to (0,1) by t = .5 * (1. + x) C DO 130 J = 1, K RHO(J) = .5D0 * (1.D0 + RHO(J)) 130 CONTINUE C C... now find runge-kutta coeffitients b, acol and asave C... the values of asave are to be used in newmsh and errchk . C DO 140 J = 1, K DO 135 I = 1, K 135 COEF(I,J) = 0.D0 COEF(J,J) = 1.D0 CALL VMONDE (RHO, COEF(1,J), K) 140 CONTINUE CALL RKBAS ( 1.D0, COEF, K, MMAX, B, DUMMY, 0) DO 150 I = 1, K CALL RKBAS ( RHO(I), COEF, K, MMAX, ACOL(1,I), DUMMY, 0) 150 CONTINUE CALL RKBAS ( 1.D0/6.D0, COEF, K, MMAX, ASAVE(1,1), DUMMY, 0) CALL RKBAS ( 1.D0/3.D0, COEF, K, MMAX, ASAVE(1,2), DUMMY, 0) CALL RKBAS ( 2.D0/3.D0, COEF, K, MMAX, ASAVE(1,3), DUMMY, 0) CALL RKBAS ( 5.D0/6.D0, COEF, K, MMAX, ASAVE(1,4), DUMMY, 0) RETURN END SUBROUTINE ERRCHK (XI, Z, DMZ, VALSTR, IFIN) C C********************************************************************** C C purpose C determine the error estimates and test to see if the C error tolerances are satisfied. C C variables C xi - current mesh points C valstr - values of the previous solution which are needed C for the extrapolation- like error estimate. C wgterr - weights used in the extrapolation-like error C estimate. the array values are assigned in C subroutine consts. C errest - storage for error estimates C err - temporary storage used for error estimates C z - approximate solution on mesh xi C ifin - a 0-1 variable. on return it indicates whether C the error tolerances were satisfied C mshflg - is set by errchk to indicate to newmsh whether C any values of the current solution are stored in C the array valstr. (0 for no, 1 for yes) C C********************************************************************** C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION ERR(40), ERREST(40), DUMMY(1) DIMENSION XI(1), Z(1), DMZ(1), VALSTR(1) C COMMON /COLOUT/ PRECIS, IOUT, IPRINT COMMON /COLORD/ K, NCOMP, NY, NCY, MSTAR, KD, KDY, MMAX, M(20) COMMON /COLAPR/ N, NOLD, NMAX, NZ, NDMZ COMMON /COLMSH/ MSHFLG, MSHNUM, MSHLMT, MSHALT COMMON /COLBAS/ B(28), ACOL(28,7), ASAVE(28,4) COMMON /COLEST/ TOL(40), WGTMSH(40), WGTERR(40), TOLIN(40), 1 ROOT(40), JTOL(40), LTOL(40), NTOL C C... error estimates are to be generated and tested C... to see if the tolerance requirements are satisfied. C IFIN = 1 MSHFLG = 1 DO 10 J = 1, MSTAR 10 ERREST(J) = 0.D0 DO 60 IBACK = 1, N I = N + 1 - IBACK C C... the error estimates are obtained by combining values of C... the numerical solutions for two meshes. C... for each value of iback we will consider the two C... approximations at 2 points in each of C... the new subintervals. we work backwards through C... the subinterval so that new values can be stored C... in valstr in case they prove to be needed later C... for an error estimate. the routine newmsh C... filled in the needed values of the old solution C... in valstr. C KNEW = ( 4 * (I-1) + 2 ) * MSTAR + 1 KSTORE = ( 2 * (I-1) + 1 ) * MSTAR + 1 X = XI(I) + (XI(I+1)-XI(I)) * 2.D0 / 3.D0 CALL APPROX (I, X, VALSTR(KNEW), DUMMY, ASAVE(1,3), + DUMMY, XI, N, Z, DMZ, 1 K, NCOMP, NY, MMAX, M, MSTAR, 4, DUMMY, 0) DO 20 L = 1,MSTAR ERR(L) = WGTERR(L) * DABS(VALSTR(KNEW) - 1 VALSTR(KSTORE)) KNEW = KNEW + 1 KSTORE = KSTORE + 1 20 CONTINUE KNEW = ( 4 * (I-1) + 1 ) * MSTAR + 1 KSTORE = 2 * (I-1) * MSTAR + 1 X = XI(I) + (XI(I+1)-XI(I)) / 3.D0 CALL APPROX (I, X, VALSTR(KNEW), DUMMY, ASAVE(1,2), + DUMMY, XI, N, Z, DMZ, 1 K, NCOMP, NY, MMAX, M, MSTAR, 4, DUMMY, 0) DO 30 L = 1,MSTAR ERR(L) = ERR(L) + WGTERR(L) * DABS(VALSTR(KNEW) - 1 VALSTR(KSTORE)) KNEW = KNEW + 1 KSTORE = KSTORE + 1 30 CONTINUE C C... find component-wise maximum error estimate C DO 40 L = 1,MSTAR ERREST(L) = DMAX1(ERREST(L),ERR(L)) 40 CONTINUE C C... test whether the tolerance requirements are satisfied C... in the i-th interval. C IF ( IFIN .EQ. 0 ) GO TO 60 DO 50 J = 1, NTOL LTOLJ = LTOL(J) LTJZ = LTOLJ + (I-1) * MSTAR IF ( ERR(LTOLJ) .GT. 1 TOLIN(J) * (DABS(Z(LTJZ))+1.D0) ) IFIN = 0 50 CONTINUE 60 CONTINUE IF ( IPRINT .GE. 0 ) RETURN WRITE(IOUT,130) LJ = 1 DO 70 J = 1,NCOMP MJ = LJ - 1 + M(J) WRITE(IOUT,120) J, (ERREST(L), L= LJ, MJ) LJ = MJ + 1 70 CONTINUE RETURN C-------------------------------------------------------------- 120 FORMAT (3H U(, I2, 3H) -,4D12.4) 130 FORMAT (/26H THE ESTIMATED ERRORS ARE,) END C--------------------------------------------------------------------- C p a r t 3 C collocation system setup routines C--------------------------------------------------------------------- C SUBROUTINE LSYSLV (MSING, XI, XIOLD, Z, DMZ, DELZ, DELDMZ, 1 G, W, V, FC, RHS, DMZO, INTEGS, IPVTG, IPVTW, RNORM, 2 MODE, FSUB, DFSUB, GSUB, DGSUB, GUESS, ISING ) C********************************************************************* C C purpose C this routine controls the set up and solution of a linear C system of collocation equations. C the matrix g is cast into an almost block diagonal C form by an appropriate ordering of the columns and solved C using the package of de boor-weiss [7] modified. C the matrix is composed of n blocks. the i-th block has the size C integs(1,i) * integs(2,i). C it contains in its last rows the linearized collocation C equations, condensed as described in [4], C and the linearized side conditions corresponding to C the i-th subinterval. integs(3,i) steps of gaussian C elimination are applied to it to achieve a partial plu C decomposition. the right hand side vector is put into rhs C and the solution vector is returned in delz and deldmz. C note that the presence of algebraic solution components C does not affect the structure (size) of g -- only the contents C of the blocks (and the size of deldmz) changes. C C lsyslv operates according to one of 5 modes: C mode = 0 - set up the collocation matrices v , w , g C and the right hand side rhs , and solve. C (for linear problems only.) C mode = 1 - set up the collocation matrices v , w , g C and the right hand sides rhs and dmzo , C and solve. also set up integs . C (first iteration of nonlinear problems only). C mode = 2 - set up rhs only and compute its norm. C mode = 3 - set up v, w, g only and solve system. C mode = 4 - perform forward and backward substitution only C (do not set up the matrices nor form the rhs). C C variables C C ig,izeta - pointers to g,zeta respectively C (necessary to keep track of blocks of g C during matrix manipulations) C idmz,irhs,iv,iw - pointers to rhs,v,w rspectively C df - partial derivatives of f from dfsub C rnorm - euclidean norm of rhs C lside - number of side conditions in current and previous blocks C iguess = 1 when current soln is user specified via guess C = 0 otherwise C dmzo - an array used to project the initial solution into C the current pp-space, when mode=1. C in the case mode=1 the current solution iterate may not C be in the right space, being defined by an arbitrary C user's guess or as a pp on a different mesh. C when forming collocation equations we are using values C of z, y and dmval at collocation points and of z at C boundary points. at the end of lsyslv (with mode=1) C a similar projection used to obtain the corrections C delz and deldmz is used to obtain the projected initial C iterate z and dmz. C fc - an array used to store projection matrices for C the case of projected collocation C C C********************************************************************* IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION Z(1), DMZ(1), DELZ(1), DELDMZ(1), XI(1), XIOLD(1) DIMENSION G(1), W(1), V(1), RHS(1), DMZO(1), DUMMY(1), Y(1) DIMENSION INTEGS(3,1), IPVTG(1), IPVTW(1), YVAL(20) DIMENSION ZVAL(40), F(40), DGZ(40), DMVAL(20), DF(800), AT(28) DIMENSION FC(1), CB(400), IPVTCB(20) C COMMON /COLOUT/ PRECIS, IOUT, IPRINT COMMON /COLLOC/ RHO(7), COEF(49) COMMON /COLORD/ K, NCOMP, NY, NCY, MSTAR, KD, KDY, MMAX, M(20) COMMON /COLSID/ ZETA(40), ALEFT, ARIGHT, IZETA, IZSAVE COMMON /COLAPR/ N, NOLD, NMAX, NZ, NDMZ COMMON /COLNLN/ NONLIN, ITER, LIMIT, ICARE, IGUESS, INDEX COMMON /COLBAS/ B(28), ACOL(28,7), ASAVE(28,4) C EXTERNAL DFSUB, DGSUB C IF (NY.EQ.0) THEN NYCB = 1 ELSE NYCB = NY ENDIF INFC = (MSTAR+NY) * NCOMP M1 = MODE + 1 GO TO (10, 30, 30, 30, 310), M1 C C... linear problem initialization C 10 DO 20 I=1,MSTAR 20 ZVAL(I) = 0.D0 DO 25 I=1,NY 25 YVAL(I) = 0.D0 C C... initialization C 30 IDMZ = 1 IDMZO = 1 IRHS = 1 IG = 1 IW = 1 IV = 1 IFC = 1 IZETA = 1 LSIDE = 0 IOLD = 1 NCOL = 2 * MSTAR RNORM = 0.D0 IF ( MODE .GT. 1 ) GO TO 80 C C... build integs (describing block structure of matrix) C DO 70 I = 1,N INTEGS(2,I) = NCOL IF (I .LT. N) GO TO 40 INTEGS(3,N) = NCOL LSIDE = MSTAR GO TO 60 40 INTEGS(3,I) = MSTAR 50 IF( LSIDE .EQ. MSTAR ) GO TO 60 IF ( ZETA(LSIDE+1) .GE. XI(I)+PRECIS ) GO TO 60 LSIDE = LSIDE + 1 GO TO 50 60 NROW = MSTAR + LSIDE 70 INTEGS(1,I) = NROW 80 CONTINUE IF ( MODE .EQ. 2 ) GO TO 90 C C... zero the matrices to be computed C LW = KDY * KDY * N DO 84 L = 1, LW 84 W(L) = 0.D0 C C... the do loop 290 sets up the linear system of equations. C 90 CONTINUE DO 290 I=1, N C C... construct a block of a and a corresponding piece of rhs. C XII = XI(I) H = XI(I+1) - XI(I) NROW = INTEGS(1,I) C C... go thru the ncomp collocation equations and side conditions C... in the i-th subinterval C 100 IF ( IZETA .GT. MSTAR ) GO TO 140 IF ( ZETA(IZETA) .GT. XII + PRECIS ) GO TO 140 C C... build equation for a side condition. C IF ( MODE .EQ. 0 ) GO TO 110 IF ( IGUESS .NE. 1 ) GO TO 102 C C... case where user provided current approximation C CALL GUESS (XII, ZVAL, YVAL, DMVAL) GO TO 110 C C... other nonlinear case C 102 IF ( MODE .NE. 1 ) GO TO 106 CALL APPROX (IOLD, XII, ZVAL, Y, AT, COEF, XIOLD, NOLD, 1 Z, DMZ, K, NCOMP, NY, MMAX, M, MSTAR, 2, DUMMY, 0) GO TO 110 106 CALL APPROX (I, XII, ZVAL, Y, AT, DUMMY, XI, N, Z, DMZ, 1 K, NCOMP, NY, MMAX, M, MSTAR, 1, DUMMY, 0) 108 IF ( MODE .EQ. 3 ) GO TO 120 C C... find rhs boundary value. C 110 CALL GSUB (IZETA, ZVAL, GVAL) RHS(NDMZ+IZETA) = -GVAL RNORM = RNORM + GVAL**2 IF ( MODE .EQ. 2 ) GO TO 130 C C... build a row of a corresponding to a boundary point C 120 CALL GDERIV (G(IG), NROW, IZETA, ZVAL, DGZ, 1, DGSUB) 130 IZETA = IZETA + 1 GO TO 100 C C... assemble collocation equations C 140 DO 220 J = 1, K HRHO = H * RHO(J) XCOL = XII + HRHO C C... this value corresponds to a collocation (interior) C... point. build the corresponding ncy equations. C IF ( MODE .EQ. 0 ) GO TO 200 IF ( IGUESS .NE. 1 ) GO TO 160 C C... use initial approximation provided by the user. C CALL GUESS (XCOL, ZVAL, YVAL, DMZO(IRHS) ) GO TO 170 C C... find rhs values C 160 IF ( MODE .NE. 1 ) GO TO 190 CALL APPROX (IOLD, XCOL, ZVAL, YVAL, AT, COEF, + XIOLD, NOLD, Z, DMZ, 1 K, NCOMP, NY, MMAX, M, MSTAR, 2, DMZO(IRHS), 2) C 170 CALL FSUB (XCOL, ZVAL, YVAL, F) DO 175 JJ = NCOMP+1,NCY 175 DMZO(IRHS+JJ-1) = 0.D0 DO 180 JJ = 1, NCY VALUE = DMZO(IRHS) - F(JJ) RHS(IRHS) = - VALUE RNORM = RNORM + VALUE**2 IRHS = IRHS + 1 180 CONTINUE GO TO 210 C C... evaluate former collocation solution C 190 CALL APPROX (I, XCOL, ZVAL, Y, ACOL(1,J), COEF, XI, N, 1 Z, DMZ, K, NCOMP, NY, MMAX, M, MSTAR, 4, DUMMY, 0) IF ( MODE .EQ. 3 ) GO TO 210 C C... fill in rhs values (and accumulate its norm). C CALL FSUB (XCOL, ZVAL, DMZ(IRHS+NCOMP), F) DO 195 JJ = 1, NCY VALUE = F(JJ) IF (JJ .LE. NCOMP) VALUE = VALUE - DMZ(IRHS) RHS(IRHS) = VALUE RNORM = RNORM + VALUE**2 IRHS = IRHS + 1 195 CONTINUE GO TO 220 C C... the linear case C 200 CALL FSUB (XCOL, ZVAL, YVAL, RHS(IRHS)) IRHS = IRHS + NCY C C... fill in ncy rows of w and v C 210 CALL VWBLOK (XCOL, HRHO, J, W(IW), V(IV), IPVTW(IDMZ), 1 KDY, ZVAL, YVAL, DF, ACOL(1,J), DMZO(IDMZO), 2 NCY, DFSUB, MSING) IF ( MSING .NE. 0 ) RETURN 220 CONTINUE C C... build global bvp matrix g C IF (INDEX .NE. 1 .AND. NY .GT. 0) THEN C C... projected collocation: find solution at xi(i+1) C XI1 = XI(I+1) IF (MODE .NE. 0) THEN IF (IGUESS .EQ. 1) THEN CALL GUESS (XI1, ZVAL, YVAL, DMVAL) ELSE IF (MODE .EQ. 1) THEN CALL APPROX (IOLD, XI1, ZVAL, YVAL, AT,COEF, + XIOLD, NOLD, Z, DMZ, K, NCOMP, NY, MMAX, + M, MSTAR, 2, DUMMY, 1) IF (I .EQ. N) + CALL APPROX (NOLD+1, XI1, ZVAL, YVAL, AT,COEF, + XIOLD, NOLD, Z, DMZ, K, NCOMP, NY, MMAX, + M, MSTAR, 1, DUMMY, 0) ELSE CALL APPROX (I, XI1, ZVAL, YVAL, AT,COEF, + XI, N, Z, DMZ, K, NCOMP, NY, MMAX, + M, MSTAR, 3, DUMMY, 1) CALL APPROX (I+1, XI1, ZVAL, YVAL, AT,COEF, + XI, N, Z, DMZ, K, NCOMP, NY, MMAX, + M, MSTAR, 1, DUMMY, 0) END IF END IF END IF C C... find rhs at next mesh point (also for linear case) C CALL FSUB (XI1, ZVAL, YVAL, F) END IF C CALL GBLOCK (H, G(IG), NROW, IZETA, W(IW), V(IV), KDY, 2 DUMMY, DELDMZ(IDMZ), IPVTW(IDMZ), 1, MODE, + XI1, ZVAL, YVAL, F, DF, CB, IPVTCB, + FC(IFC), DFSUB, ISING, NCOMP, NYCB, NCY ) IF (ISING .NE. 0) RETURN IF ( I .LT. N ) GO TO 280 IZSAVE = IZETA 240 IF ( IZETA .GT. MSTAR ) GO TO 290 C C... build equation for a side condition. C IF ( MODE .EQ. 0 ) GO TO 250 IF ( IGUESS .NE. 1 ) GO TO 245 C C... case where user provided current approximation C CALL GUESS (ARIGHT, ZVAL, YVAL, DMVAL) GO TO 250 C C... other nonlinear case C 245 IF ( MODE .NE. 1 ) GO TO 246 CALL APPROX (NOLD+1, ARIGHT, ZVAL, Y, AT, COEF, + XIOLD, NOLD, Z, DMZ, 1 K, NCOMP, NY, MMAX, M, MSTAR, 1, DUMMY, 0) GO TO 250 246 CALL APPROX (N+1, ARIGHT, ZVAL, Y, AT, COEF, XI, N, 1 Z, DMZ, K, NCOMP, NY, MMAX, M, MSTAR, 1, DUMMY, 0) 248 IF ( MODE .EQ. 3 ) GO TO 260 C C... find rhs boundary value. C 250 CALL GSUB (IZETA, ZVAL, GVAL) RHS(NDMZ+IZETA) = - GVAL RNORM = RNORM + GVAL**2 IF ( MODE .EQ. 2 ) GO TO 270 C C... build a row of a corresponding to a boundary point C 260 CALL GDERIV (G(IG), NROW, IZETA+MSTAR, ZVAL, DGZ, 2, DGSUB) 270 IZETA = IZETA + 1 GO TO 240 C C... update counters -- i-th block completed C 280 IG = IG + NROW * NCOL IV = IV + KDY * MSTAR IW = IW + KDY * KDY IDMZ = IDMZ + KDY IF ( MODE .EQ. 1 ) IDMZO = IDMZO + KDY IFC = IFC + INFC+2*NCOMP 290 CONTINUE C C... assembly process completed C IF ( MODE .EQ. 0 .OR. MODE .EQ. 3 ) GO TO 300 RNORM = DSQRT(RNORM / FLOAT(NZ+NDMZ)) IF ( MODE .EQ. 2 ) RETURN C C... solve the linear system. C C... matrix decomposition C 300 CALL FCBLOK (G, INTEGS, N, IPVTG, DF, MSING) C C... check for singular matrix C MSING = - MSING IF( MSING .NE. 0 ) RETURN C C... perform forward and backward substitution . C 310 CONTINUE DO 311 L = 1, NDMZ DELDMZ(L) = RHS(L) 311 CONTINUE IZ = 1 IDMZ = 1 IW = 1 IFC = 1 IZET = 1 DO 320 I=1, N NROW = INTEGS(1,I) IZETA = NROW + 1 - MSTAR IF ( I .EQ. N ) IZETA = IZSAVE 322 IF ( IZET .EQ. IZETA ) GO TO 324 DELZ(IZ-1+IZET) = RHS(NDMZ+IZET) IZET = IZET + 1 GO TO 322 324 H = XI(I+1) - XI(I) CALL GBLOCK (H, G(1), NROW, IZETA, W(IW), V(1), KDY, 1 DELZ(IZ), DELDMZ(IDMZ), IPVTW(IDMZ), 2, MODE, + XI1, ZVAL, YVAL, FC(IFC+INFC), DF, CB, + IPVTCB, FC(IFC), DFSUB, ISING, NCOMP, NYCB, NCY ) IZ = IZ + MSTAR IDMZ = IDMZ + KDY IW = IW + KDY * KDY IFC = IFC + INFC+2*NCOMP IF ( I .LT. N ) GO TO 320 326 IF ( IZET .GT. MSTAR ) GO TO 320 DELZ(IZ-1+IZET) = RHS(NDMZ+IZET) IZET = IZET + 1 GO TO 326 320 CONTINUE C C... perform forward and backward substitution for mode=0,2, or 3. C CALL SBBLOK (G, INTEGS, N, IPVTG, DELZ) C C... finally find deldmz C CALL DMZSOL (KDY, MSTAR, N, V, DELZ, DELDMZ) C IF ( MODE .NE. 1 ) RETURN C C... project current iterate into current pp-space C DO 321 L = 1, NDMZ DMZ(L) = DMZO(L) 321 CONTINUE IZ = 1 IDMZ = 1 IW = 1 IFC = 1 IZET = 1 DO 350 I=1, N NROW = INTEGS(1,I) IZETA = NROW + 1 - MSTAR IF ( I .EQ. N ) IZETA = IZSAVE 330 IF ( IZET .EQ. IZETA ) GO TO 340 Z(IZ-1+IZET) = DGZ(IZET) IZET = IZET + 1 GO TO 330 340 H = XI(I+1) - XI(I) CALL GBLOCK (H, G(1), NROW, IZETA, W(IW), DF, KDY, 1 Z(IZ), DMZ(IDMZ), IPVTW(IDMZ), 2, MODE, + XI1, ZVAL, YVAL, FC(IFC+INFC+NCOMP), + DF, CB, IPVTCB, FC(IFC), DFSUB, ISING, + NCOMP, NYCB, NCY ) IZ = IZ + MSTAR IDMZ = IDMZ + KDY IW = IW + KDY * KDY IFC = IFC + INFC+2*NCOMP IF ( I .LT. N ) GO TO 350 342 IF ( IZET .GT. MSTAR ) GO TO 350 Z(IZ-1+IZET) = DGZ(IZET) IZET = IZET + 1 GO TO 342 350 CONTINUE CALL SBBLOK (G, INTEGS, N, IPVTG, Z) C C... finally find dmz C CALL DMZSOL (KDY, MSTAR, N, V, Z, DMZ) C RETURN END SUBROUTINE GDERIV ( GI, NROW, IROW, ZVAL, DGZ, MODE, DGSUB) C C********************************************************************** C C purpose: C C construct a collocation matrix row according to mode: C mode = 1 - a row corresponding to a initial condition C (i.e. at the left end of the subinterval). C mode = 2 - a row corresponding to a condition at aright. C C variables: C C gi - the sub-block of the global bvp matrix in C which the equations are to be formed. C nrow - no. of rows in gi. C irow - the row in gi to be used for equations. C zval - z(xi) C dg - the derivatives of the side condition. C C********************************************************************** IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION GI(NROW,1), ZVAL(1), DGZ(1), DG(40) C COMMON /COLORD/ KDUM, NDUM, NY, NCY, MSTAR, KD, KDY, MMAX, M(20) COMMON /COLSID/ ZETA(40), ALEFT, ARIGHT, IZETA, IDUM COMMON /COLNLN/ NONLIN, ITER, LIMIT, ICARE, IGUESS, INDEX C C... zero jacobian dg C DO 10 J=1,MSTAR 10 DG(J) = 0.D0 C C... evaluate jacobian dg C CALL DGSUB (IZETA, ZVAL, DG) C C... evaluate dgz = dg * zval once for a new mesh C IF (NONLIN .EQ. 0 .OR. ITER .GT. 0) GO TO 30 DOT = 0.D0 DO 20 J = 1, MSTAR 20 DOT = DOT + DG(J) * ZVAL(J) DGZ(IZETA) = DOT C C... branch according to m o d e C 30 IF ( MODE .EQ. 2 ) GO TO 50 C C... provide coefficients of the j-th linearized side condition. C... specifically, at x=zeta(j) the j-th side condition reads C... dg(1)*z(1) + ... +dg(mstar)*z(mstar) + g = 0 C C C... handle an initial condition C DO 40 J = 1, MSTAR GI(IROW,J) = DG(J) 40 GI(IROW,MSTAR+J) = 0.D0 RETURN C C... handle a final condition C 50 DO 60 J= 1, MSTAR GI(IROW,J) = 0.D0 60 GI(IROW,MSTAR+J) = DG(J) RETURN END SUBROUTINE VWBLOK (XCOL, HRHO, JJ, WI, VI, IPVTW, KDY, ZVAL, 1 YVAL, DF, ACOL, DMZO, NCY, DFSUB, MSING) C C********************************************************************** C C purpose: C C construct a group of ncomp rows of the matrices wi and vi. C corresponding to an interior collocation point. C C C variables: C C xcol - the location of the collocation point. C jj - xcol is the jj-th of k collocation points C in the i-th subinterval. C wi,vi - the i-th block of the collocation matrix C before parameter condensation. C kdy - no. of rows in vi and wi . C zval - z(xcol) C yval - y(xcol) C df - the jacobian at xcol . C jcomp - counter for the component being dealt with. C C********************************************************************** IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION WI(KDY,1), VI(KDY,1), ZVAL(1), DMZO(1), DF(NCY,1) DIMENSION IPVTW(1), HA(7,4), ACOL(7,4), BASM(5), YVAL(1) C COMMON /COLORD/ K, NCOMP, NY, NDM, MSTAR, KD, KDYM, MMAX, M(20) COMMON /COLNLN/ NONLIN, ITER, LIMIT, ICARE, IGUESS, INDEX C C... initialize wi C I1 = (JJ-1) * NCY DO 10 ID = 1+I1, NCOMP+I1 WI(ID,ID) = 1.D0 10 CONTINUE C C... calculate local basis C 30 FACT = 1.D0 DO 35 L=1,MMAX FACT = FACT * HRHO / FLOAT(L) BASM(L) = FACT DO 35 J=1,K HA(J,L) = FACT * ACOL(J,L) 35 CONTINUE C C... zero jacobian C DO 40 JCOL = 1, MSTAR+NY DO 40 IR = 1, NCY 40 DF(IR,JCOL) = 0.D0 C C... build ncy rows for interior collocation point x. C... the linear expressions to be constructed are: C... (m(id)) C... u - df(id,1)*z(1) - ... - df(id,mstar)*z(mstar) - C... id C... - df(id,mstar+1)*u(1) - ... - df(id,mstar+ny)*y(ny) C... for id = 1 to ncy (m(id)=0 for id > ncomp). C CALL DFSUB (XCOL, ZVAL, YVAL, DF) I0 = (JJ-1) * NCY I1 = I0 + 1 I2 = I0 + NCY C C... evaluate dmzo = dmzo - df * (zval,yval) once for a new mesh C IF (NONLIN .EQ. 0 .OR. ITER .GT. 0) GO TO 60 DO 50 J = 1, MSTAR+NY IF (J .LE. MSTAR) THEN FACT = - ZVAL(J) ELSE FACT = - YVAL(J-MSTAR) END IF DO 50 ID = 1, NCY DMZO(I0+ID) = DMZO(I0+ID) + FACT * DF(ID,J) 50 CONTINUE C C... loop over the ncomp expressions to be set up for the C... current collocation point. C 60 DO 70 J = 1, MSTAR DO 70 ID = 1, NCY VI(I0+ID,J) = DF(ID,J) 70 CONTINUE JN = 1 DO 140 JCOMP = 1, NCOMP MJ = M(JCOMP) JN = JN + MJ DO 130 L = 1, MJ JV = JN - L JW = JCOMP DO 90 J = 1, K AJL = - HA(J,L) DO 80 IW = I1, I2 WI(IW,JW) = WI(IW,JW) + AJL * VI(IW,JV) 80 CONTINUE 90 JW = JW + NCY LP1 = L + 1 IF ( L .EQ. MJ ) GO TO 130 DO 110 LL = LP1, MJ JDF = JN - LL BL = BASM(LL-L) DO 100 IW = I1, I2 VI(IW,JV) = VI(IW,JV) + BL * VI(IW,JDF) 100 CONTINUE 110 CONTINUE 130 CONTINUE 140 CONTINUE C C... loop for the algebraic solution components C DO 150 JCOMP = 1,NY JD = NCOMP+JCOMP DO 150 ID = 1,NCY WI(I0+ID,I0+JD) = -DF(ID,MSTAR+JCOMP) 150 CONTINUE IF ( JJ .LT. K ) RETURN C C ...decompose the wi block and solve for the mstar columns of vi C C C... do parameter condensation C MSING = 0 CALL DGEFA (WI, KDY, KDY, IPVTW, MSING) C C... check for singularity C IF ( MSING .NE. 0 ) RETURN DO 250 J= 1,MSTAR CALL DGESL (WI, KDY, KDY, IPVTW, VI(1,J), 0) 250 CONTINUE RETURN END SUBROUTINE GBLOCK (H, GI, NROW, IROW, WI, VI, KDY, 1 RHSZ, RHSDMZ, IPVTW, MODE, MODL, + XI1, ZVAL, YVAL, F, DF, CB, IPVTCB, + FC, DFSUB, ISING, NCOMP, NYCB, NCY ) C C********************************************************************** C C purpose: C C construct collocation matrix rows according to mode: C mode = 1 - a group of mstar rows corresponding C to a mesh interval. C = 2 - compute condensed form of rhs C modl = mode of lsyslv C C variables: C C h - the local stepsize. C gi - the sub-block of the collocation matrix in C which the equations are to be formed. C wi - the sub-block of noncondensed collocation equations, C left-hand side part. C vi - the sub-block of noncondensed collocation equations, C right-hand side part. C rhsdmz - the inhomogenous term of the uncondensed collocation C equations. C rhsz - the inhomogenous term of the condensed collocation C equations. C nrow - no. of rows in gi. C irow - the first row in gi to be used for equations. C xi1 - next mesh point (right end) C zval,yval - current solution at xi1 C fc - projection matrices C C********************************************************************** IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION HB(7,4), BASM(5) DIMENSION GI(NROW,1), WI(1), VI(KDY,1) DIMENSION RHSZ(1), RHSDMZ(1), IPVTW(1) DIMENSION ZVAL(1), YVAL(1), F(1), DF(NCY,1), CB(NYCB,NYCB), + IPVTCB(1), FC(NCOMP,1), BCOL(40), U(400), V(400) C COMMON /COLORD/ K, NCD, NY, NCYD, MSTAR, KD, KDUM, MMAX, M(20) COMMON /COLBAS/ B(7,4), ACOL(28,7), ASAVE(28,4) COMMON /COLNLN/ NONLIN, ITER, LIMIT, ICARE, IGUESS, INDEX C C... compute local basis C FACT = 1.D0 BASM(1) = 1.D0 DO 30 L=1,MMAX FACT = FACT * H / FLOAT(L) BASM(L+1) = FACT DO 20 J=1,K 20 HB(J,L) = FACT * B(J,L) 30 CONTINUE C C... branch according to m o d e C GO TO (40, 120 ), MODE C C... set right gi-block to identity C 40 CONTINUE IF (MODL .EQ. 2) GO TO 110 DO 60 J = 1, MSTAR DO 50 IR = 1, MSTAR GI(IROW-1+IR,J) = 0.D0 50 GI(IROW-1+IR,MSTAR+J) = 0.D0 60 GI(IROW-1+J,MSTAR+J) = 1.D0 C C... compute the block gi C IR = IROW DO 100 ICOMP = 1, NCOMP MJ = M(ICOMP) IR = IR + MJ DO 90 L = 1, MJ ID = IR - L DO 80 JCOL = 1, MSTAR IND = ICOMP RSUM = 0.D0 DO 70 J = 1, K RSUM = RSUM - HB(J,L) * VI(IND,JCOL) 70 IND = IND + NCY GI(ID,JCOL) = RSUM 80 CONTINUE JD = ID - IROW DO 85 LL = 1, L GI(ID,JD+LL) = GI(ID,JD+LL) - BASM(LL) 85 CONTINUE 90 CONTINUE 100 CONTINUE IF (INDEX .EQ. 1 .OR. NY .EQ. 0) RETURN C C... projected collocation C... set up projection matrix and update gi-block C CALL DFSUB (XI1, ZVAL, YVAL, DF) C C... if index=2 then form projection matrices directly C... otherwise use svd to define appropriate projection C IF (INDEX .EQ. 0) THEN CALL PRJSVD (FC,DF,CB,U,V,NCOMP,NCY,NY,IPVTCB,ISING,1) IF (ISING .NE. 0) RETURN ELSE C C... form cb C DO 102 I=1,NY DO 102 J=1,NY FACT = 0 ML = 0 DO 101 L=1,NCOMP ML = ML + M(L) FACT = FACT + DF(I+NCOMP,ML) * DF(L,MSTAR+J) 101 CONTINUE CB(I,J) = FACT 102 CONTINUE C C... decompose cb C CALL DGEFA (CB, NY, NY, IPVTCB, ISING) IF (ISING .NE. 0) RETURN C C... form columns of fc C DO 105 J=1,MSTAR+NY IF (J .LE. MSTAR) THEN DO 103 I=1,NY 103 BCOL(I) = DF(I+NCOMP,J) ELSE DO 203 I=1,NY 203 BCOL(I) = 0.0D0 BCOL(J-MSTAR) = 1.D0 END IF CALL DGESL (CB, NY, NY, IPVTCB, BCOL, 0) DO 105 I=1,NCOMP FACT = 0.0D0 DO 104 L=1,NY FACT = FACT + DF(I,L+MSTAR) * BCOL(L) 104 CONTINUE FC(I,J) = FACT 105 CONTINUE C END IF C C... update gi C DO 108 J = 1,MSTAR DO 107 I=1,NCOMP FACT = 0 DO 106 L=1,MSTAR FACT = FACT + FC(I,L) * GI(IROW-1+L,J) 106 CONTINUE BCOL(I) = FACT 107 CONTINUE ML = 0 DO 108 I = 1,NCOMP ML = ML + M(I) GI(IROW-1+ML,J) = GI(IROW-1+ML,J) - BCOL(I) 108 CONTINUE C C... prepare extra rhs piece; two if new mesh C 110 IF (INDEX .EQ. 1 .OR. NY .EQ. 0 ) RETURN DO 115 JCOL=1,2 DO 112 I=1,NCOMP FACT = 0 DO 111 L=1,NY FACT = FACT + FC(I,L+MSTAR) * F(L+NCOMP) 111 CONTINUE FC(I,JCOL+MSTAR+NY) = FACT 112 CONTINUE C IF (MODL .NE. 1 .OR. JCOL .EQ. 2) RETURN DO 113 I = 1+NCOMP,NY+NCOMP 113 F(I) = 0 DO 114 J=1,MSTAR FACT = -ZVAL(J) DO 114 I = 1+NCOMP,NY+NCOMP F(I) = F(I) + DF(I,J) * FACT 114 CONTINUE 115 CONTINUE RETURN C C... compute the appropriate piece of rhsz C 120 CONTINUE CALL DGESL (WI, KDY, KDY, IPVTW, RHSDMZ, 0) IR = IROW DO 140 JCOMP = 1, NCOMP MJ = M(JCOMP) IR = IR + MJ DO 130 L = 1, MJ IND = JCOMP RSUM = 0.D0 DO 125 J = 1, K RSUM = RSUM + HB(J,L) * RHSDMZ(IND) 125 IND = IND + NCY RHSZ(IR-L) = RSUM 130 CONTINUE 140 CONTINUE IF (INDEX .EQ. 1 .OR. NY .EQ. 0) RETURN C C... projected collocation C... calculate projected rhsz C DO 160 I=1,NCOMP FACT = 0 DO 150 L=1,MSTAR FACT = FACT + FC(I,L) * RHSZ(L+IROW-1) 150 CONTINUE BCOL(I) = FACT 160 CONTINUE ML = 0 DO 170 I = 1,NCOMP ML = ML + M(I) RHSZ(IROW-1+ML) = RHSZ(IROW-1+ML) - BCOL(I) - F(I) 170 CONTINUE C RETURN END SUBROUTINE PRJSVD (FC,DF,D,U,V,NCOMP,NCY,NY,IPVTCB,ISING,MODE) C C********************************************************************** C C purpose: C C construct projection matrices in fc in the general case C where the problem may have a higher index but is not in C a Hessenberg index-2 form C C calls the linpack routine dsvdc for a singular value C decomposition. C C mode = 1 - called from gblock C = 2 - called from newmsh C (then fc consists of only ncomp columns) C C********************************************************************** IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION FC(NCOMP,1), DF(NCY,1), D(NY,NY), U(NY,NY), V(NY,NY) DIMENSION WORK(20), S(21), E(20), IPVTCB(1) COMMON /COLORD/ K, NCD, NYD, NCYD, MSTAR, KD, KDUM, MMAX, M(20) COMMON /COLOUT/ PRECIS, IOUT, IPRINT COMMON /COLEST/ TOL(40), WGTMSH(40), WGTERR(40), TOLIN(40), 1 ROOT(40), JTOL(40), LTOL(40), NTOL c C... compute the maximum tolerance C CHECK = 0.D0 DO 5 I = 1, NTOL 5 CHECK = DMAX1 ( TOLIN(I), CHECK ) C C... construct d and find its svd C DO 10 I=1,NY DO 10 J=1,NY D(I,J) = DF(I+NCOMP,J+MSTAR) 10 CONTINUE JOB = 11 CALL DSVDC (D,NY,NY,NY,S,E,U,NY,V,NY,WORK,JOB,INFO) C C... determine rank of d C S(NY+1) = 0 IRANK = 0 20 IF (S(IRANK+1) .LT. CHECK) GO TO 30 IRANK = IRANK + 1 GO TO 20 C C... if d has full rank then no projection is needed C 30 IF (IRANK .EQ. NY) THEN DO 35 I=1,NCOMP DO 35 J=1,MSTAR+NY 35 FC(I,J) = 0.D0 RETURN ELSE C C... form projected cb C IR = NY-IRANK DO 50 I=1,NY DO 50 J=1,NY FACT = 0 ML = 0 DO 40 L=1,NCOMP ML = ML + M(L) FACT = FACT + DF(I+NCOMP,ML) * DF(L,MSTAR+J) 40 CONTINUE D(I,J) = FACT 50 CONTINUE DO 70 I=1,NY DO 60 J=1,IR WORK(J) = 0 DO 60 L=1,NY WORK(J) = WORK(J) + D(I,L)*V(L,J+IRANK) 60 CONTINUE DO 70 J=1,IR D(I,J) = WORK(J) 70 CONTINUE DO 90 I=1,IR DO 80 J=1,IR WORK(J) = 0 DO 80 L=1,NY WORK(J) = WORK(J) + U(L,I+IRANK)*D(L,J) 80 CONTINUE DO 90 J=1,IR D(I,J) = WORK(J) 90 CONTINUE C C... decompose projected cb C CALL DGEFA (D, NY, IR, IPVTCB, ISING) IF (ISING .NE. 0) RETURN C C... form columns of fc C DO 130 J=MSTAR+1,MSTAR+NY DO 100 I=1,IR 100 WORK(I) = U(J-MSTAR,I+IRANK) CALL DGESL (D, NY, IR, IPVTCB, WORK, 0) DO 110 I=1,NY U(J-MSTAR,I) = 0 DO 110 L=1,IR U(J-MSTAR,I) = U(J-MSTAR,I) + V(I,L+IRANK)*WORK(L) 110 CONTINUE DO 130 I=1,NCOMP FACT = 0 DO 120 L=1,NY FACT = FACT + DF(I,MSTAR+L)*U(J-MSTAR,L) 120 CONTINUE FC(I,J) = FACT 130 CONTINUE C IF (MODE .EQ. 1) THEN C DO 150 I=1,NCOMP DO 150 J=1,MSTAR FACT = 0 DO 140 L=1,NY FACT = FACT + FC(I,L+MSTAR) * DF(L+NCOMP,J) 140 CONTINUE FC(I,J) = FACT 150 CONTINUE C ELSE C DO 160 I=1,NCOMP MJ = 0 DO 160 J=1,NCOMP MJ = MJ + M(J) FACT = 0 DO 155 L=1,NY FACT = FACT + FC(I,L+MSTAR) * DF(L+NCOMP,MJ) 155 CONTINUE FC(I,J) = FACT 160 CONTINUE END IF C END IF RETURN END C---------------------------------------------------------------------- C p a r t 4 C polynomial and service routines C---------------------------------------------------------------------- C SUBROUTINE APPSLN (X, Z, Y, FSPACE, ISPACE) C C***************************************************************** C C purpose C C set up a standard call to approx to evaluate the C approximate solution z = z( u(x) ), y = y(x) at a C point x (it has been computed by a call to coldae ). C the parameters needed for approx are retrieved C from the work arrays ispace and fspace . C C***************************************************************** C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION Z(1), Y(1), FSPACE(1), ISPACE(1), A(28), DUMMY(1) IS6 = ISPACE(7) IS5 = ISPACE(1) + 2 IS4 = IS5 + ISPACE(5) * (ISPACE(1) + 1) I = 1 CALL APPROX (I, X, Z, Y, A, FSPACE(IS6), FSPACE(1), ISPACE(1), 1 FSPACE(IS5), FSPACE(IS4), ISPACE(2), ISPACE(3), 2 ISPACE(4), ISPACE(6), ISPACE(9), ISPACE(5), 2, 3 DUMMY, 1) RETURN END SUBROUTINE APPROX (I, X, ZVAL, YVAL, A, COEF, XI, N, Z, DMZ, K, 1 NCOMP, NY, MMAX, M, MSTAR, MODE, DMVAL, MODM) C C********************************************************************** C C purpose C (1) (m1-1) (mncomp-1) C evaluate z(u(x))=(u (x),u (x),...,u (x),...,u (x) ) C 1 1 1 mncomp C as well as optionally y(x) and dmval(x) at one point x. C C variables C a - array of mesh independent rk-basis coefficients C xi - the current mesh (having n subintervals) C z - the current solution vector (differential components). C it is convenient to imagine z as a two-dimensional C array with dimensions mstar x (n+1). then C z(j,i) = the jth component of z at the ith mesh point C dmz - the array of mj-th derivatives of the current solution C plus algebraic solution components at collocation points C it is convenient to imagine dmz as a 3-dimensional C array with dimensions ncy x k x n. then C dmz(l,j,i) = a solution value at the jth collocation C point in the ith mesh subinterval: if l <= ncomp then C dmz(l,j,i) is the ml-th derivative of ul, while if C l > ncomp then dmz(l,j,i) is the value of the current C (l-ncomp)th component of y at this collocation point C mode - determines the amount of initialization needed C = 4 forms z(u(x)) using z, dmz and ha C = 3 as in =4, but computes local rk-basis C = 2 as in =3, but determines i such that C xi(i) .le. x .lt. xi(i+1) (unless x=xi(n+1)) C = 1 retrieve z=z(u(x(i))) directly C modm = 0 evaluate only zval C = 1 evaluate also yval C = 2 evaluate in addition dmval C output C zval - the solution vector z(u(x)) (differential components) C yval - the solution vector y(x) (algebraic components) C dmval - the mth derivatives of u(x) C C********************************************************************** C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION ZVAL(1), DMVAL(1), XI(1), M(1), A(7,1), DM(7) DIMENSION Z(1), DMZ(1), BM(4), COEF(1), YVAL(1) C COMMON /COLOUT/ PRECIS, IOUT, IPRINT C GO TO (10, 30, 80, 90), MODE C C... mode = 1 , retrieve z( u(x) ) directly for x = xi(i). C 10 X = XI(I) IZ = (I-1) * MSTAR DO 20 J = 1, MSTAR IZ = IZ + 1 ZVAL(J) = Z(IZ) 20 CONTINUE RETURN C C... mode = 2 , locate i so xi(i) .le. x .lt. xi(i+1) C 30 CONTINUE IF ( X .GE. XI(1)-PRECIS .AND. X .LE. XI(N+1)+PRECIS ) 1 GO TO 40 IF (IPRINT .LT. 1) WRITE(IOUT,900) X, XI(1), XI(N+1) IF ( X .LT. XI(1) ) X = XI(1) IF ( X .GT. XI(N+1) ) X = XI(N+1) 40 IF ( I .GT. N .OR. I .LT. 1 ) I = (N+1) / 2 ILEFT = I IF ( X .LT. XI(ILEFT) ) GO TO 60 DO 50 L = ILEFT, N I = L IF ( X .LT. XI(L+1) ) GO TO 80 50 CONTINUE GO TO 80 60 IRIGHT = ILEFT - 1 DO 70 L = 1, IRIGHT I = IRIGHT + 1 - L IF ( X .GE. XI(I) ) GO TO 80 70 CONTINUE C C... mode = 2 or 3 , compute mesh independent rk-basis. C 80 CONTINUE S = (X - XI(I)) / (XI(I+1) - XI(I)) CALL RKBAS ( S, COEF, K, MMAX, A, DM, MODM ) C C... mode = 2, 3, or 4 , compute mesh dependent rk-basis. C 90 CONTINUE BM(1) = X - XI(I) DO 95 L = 2, MMAX BM(L) = BM(1) / FLOAT(L) 95 CONTINUE C C... evaluate z( u(x) ). C 100 IR = 1 NCY = NCOMP + NY IZ = (I-1) * MSTAR + 1 IDMZ = (I-1) * K * NCY DO 140 JCOMP = 1, NCOMP MJ = M(JCOMP) IR = IR + MJ IZ = IZ + MJ DO 130 L = 1, MJ IND = IDMZ + JCOMP ZSUM = 0.D0 DO 110 J = 1, K ZSUM = ZSUM + A(J,L) * DMZ(IND) 110 IND = IND + NCY DO 120 LL = 1, L LB = L + 1 - LL 120 ZSUM = ZSUM * BM(LB) + Z(IZ-LL) 130 ZVAL(IR-L) = ZSUM 140 CONTINUE IF ( MODM .EQ. 0 ) RETURN C C... for modm = 1 evaluate y(j) = j-th component of y. C DO 150 JCOMP = 1, NY 150 YVAL(JCOMP) = 0.D0 DO 170 J = 1, K IND = IDMZ + (J-1)*NCY + NCOMP + 1 FACT = DM(J) DO 160 JCOMP = 1, NY YVAL(JCOMP) = YVAL(JCOMP) + FACT * DMZ(IND) IND = IND + 1 160 CONTINUE 170 CONTINUE IF ( MODM .EQ. 1 ) RETURN C C... for modm = 2 evaluate dmval(j) = mj-th derivative of uj. C DO 180 JCOMP = 1, NCOMP 180 DMVAL(JCOMP) = 0.D0 DO 200 J = 1, K IND = IDMZ + (J-1)*NCY + 1 FACT = DM(J) DO 190 JCOMP = 1, NCOMP DMVAL(JCOMP) = DMVAL(JCOMP) + FACT * DMZ(IND) IND = IND + 1 190 CONTINUE 200 CONTINUE RETURN C-------------------------------------------------------------------- 900 FORMAT(37H ****** DOMAIN ERROR IN APPROX ****** 1 /4H X =,D20.10, 10H ALEFT =,D20.10, 2 11H ARIGHT =,D20.10) END SUBROUTINE RKBAS (S, COEF, K, M, RKB, DM, MODE) C C********************************************************************** C C purpose C evaluate mesh independent runge-kutta basis for given s C C variables C s - argument, i.e. the relative position for which C the basis is to be evaluated ( 0. .le. s .le. 1. ). C coef - precomputed derivatives of the basis C k - number of collocatin points per subinterval C m - maximal order of the differential equation C rkb - the runge-kutta basis (0-th to (m-1)-th derivatives ) C dm - basis elements for m-th derivative C these are evaluated if mode > 0. C C********************************************************************** C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION COEF(K,1), RKB(7,1), DM(1), T(10) C IF ( K .EQ. 1 ) GO TO 70 KPM1 = K + M - 1 DO 10 I = 1, KPM1 10 T(I) = S / FLOAT(I) DO 40 L = 1, M LB = K + L + 1 DO 30 I = 1, K P = COEF(1,I) DO 20 J = 2, K P = P * T(LB-J) + COEF(J,I) 20 CONTINUE RKB(I,L) = P 30 CONTINUE 40 CONTINUE IF ( MODE .EQ. 0 ) RETURN DO 60 I = 1, K P = COEF(1,I) DO 50 J = 2, K 50 P = P * T(K+1-J) + COEF(J,I) DM(I) = P 60 CONTINUE RETURN 70 RKB(1,1) = 1.0D0 DM(1) = 1.0D0 RETURN END SUBROUTINE VMONDE ( RHO, COEF, K ) C C********************************************************************** C C purpose C solve vandermonde system v * x = e C with v(i,j) = rho(j)**(i-1)/(i-1)! . C C********************************************************************** C INTEGER K, I,IFAC,J,KM1,KMI DOUBLE PRECISION RHO(K), COEF(K) C IF ( K .EQ. 1 ) RETURN KM1 = K - 1 DO 10 I = 1, KM1 KMI = K - I DO 10 J = 1, KMI COEF(J) = (COEF(J+1) - COEF(J)) / (RHO(J+I) - RHO(J)) 10 CONTINUE C IFAC = 1 DO 40 I = 1, KM1 KMI = K + 1 - I DO 30 J = 2, KMI 30 COEF(J) = COEF(J) - RHO(J+I-1) * COEF(J-1) COEF(KMI) = FLOAT(IFAC) * COEF(KMI) IFAC = IFAC * I 40 CONTINUE COEF(1) = FLOAT(IFAC) * COEF(1) RETURN END SUBROUTINE HORDER (I, UHIGH, HI, DMZ, NCOMP, NCY, K) C C********************************************************************** C C purpose C determine highest order (piecewise constant) derivatives C of the current collocation solution C C variables C hi - the stepsize, hi = xi(i+1) - xi(i) C dmz - vector of mj-th derivative of the solution C uhigh - the array of highest order (piecewise constant) C derivatives of the approximate solution on C (xi(i),xi(i+1)), viz, C (k+mj-1) C uhigh(j) = u (x) on (xi(i),xi(i+1)) C j C C********************************************************************** C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION UHIGH(1), DMZ(1) C COMMON /COLLOC/ RHO(7), COEF(49) C DN = 1.D0 / HI**(K-1) C C... loop over the ncomp solution components C DO 10 ID = 1, NCOMP UHIGH(ID) = 0.D0 10 CONTINUE KIN = 1 IDMZ = (I-1) * K * NCY + 1 DO 30 J = 1, K FACT = DN * COEF(KIN) DO 20 ID = 1, NCOMP UHIGH(ID) = UHIGH(ID) + FACT * DMZ(IDMZ) IDMZ = IDMZ + 1 20 CONTINUE KIN = KIN + K 30 CONTINUE RETURN END SUBROUTINE DMZSOL (KDY, MSTAR, N, V, Z, DMZ) C C********************************************************************** C C purpose C compute dmz in a blockwise manner C dmz(i) = dmz(i) + v(i) * z(i), i = 1,...,n C C********************************************************************** C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION V(KDY,1), DMZ(KDY,1), Z(1) C JZ = 1 DO 30 I = 1, N DO 20 J = 1, MSTAR FACT = Z(JZ) DO 10 L = 1, KDY DMZ(L,I) = DMZ(L,I) + FACT * V(L,JZ) 10 CONTINUE JZ = JZ + 1 20 CONTINUE 30 CONTINUE RETURN END C---------------------------------------------------------------------- C p a r t 5 C we list here a modified (column oriented, faster) C version of the package solveblok of de boor - weiss [7]. C we also give a listing of the linpack C routines dgefa und dgesl used by coldae. C---------------------------------------------------------------------- C SUBROUTINE FCBLOK (BLOKS, INTEGS, NBLOKS, IPIVOT, SCRTCH, INFO) C C C calls subroutines factrb and shiftb . C C fcblok supervises the plu factorization with pivoting of C scaled rows of the almost block diagonal matrix stored in the C arrays bloks and integs . C C factrb = subprogram which carries out steps 1,...,last of gauss C elimination (with pivoting) for an individual block. C shiftb = subprogram which shifts the remaining rows to the top of C the next block C C parameters C bloks an array that initially contains the almost block diago- C nal matrix a to be factored, and on return contains the C computed factorization of a . C integs an integer array describing the block structure of a . C nbloks the number of blocks in a . C ipivot an integer array of dimension sum (integs(3,n) ; n=1, C ...,nbloks) which, on return, contains the pivoting stra- C tegy used. C scrtch work area required, of length max (integs(1,n) ; n=1, C ...,nbloks). C info output parameter; C = 0 in case matrix was found to be nonsingular. C otherwise, C = n if the pivot element in the nth gauss step is zero. C C********************************************************************** C INTEGER INTEGS(3,NBLOKS),IPIVOT(1),INFO, I,INDEX,INDEXN,LAST, 1 NCOL,NROW DOUBLE PRECISION BLOKS(1),SCRTCH(1) INFO = 0 INDEXX = 1 INDEXN = 1 I = 1 C C... loop over the blocks. i is loop index C 10 INDEX = INDEXN NROW = INTEGS(1,I) NCOL = INTEGS(2,I) LAST = INTEGS(3,I) C C... carry out elimination on the i-th block until next block C... enters, i.e., for columns 1,...,last of i-th block. C CALL FACTRB ( BLOKS(INDEX), IPIVOT(INDEXX), SCRTCH, NROW, 1 NCOL, LAST, INFO) C C... check for having reached a singular block or the last block C IF ( INFO .NE. 0 ) GO TO 20 IF ( I .EQ. NBLOKS ) RETURN I = I+1 INDEXN = NROW * NCOL + INDEX INDEXX = INDEXX + LAST C C... put the rest of the i-th block onto the next block C CALL SHIFTB ( BLOKS(INDEX), NROW, NCOL, LAST, 1 BLOKS(INDEXN), INTEGS(1,I), INTEGS(2,I) ) GO TO 10 20 INFO = INFO + INDEXX - 1 RETURN END SUBROUTINE FACTRB ( W, IPIVOT, D, NROW, NCOL, LAST, INFO) C C******************************************************************** C C adapted from p.132 of element.numer.analysis by conte-de boor C C constructs a partial plu factorization, corresponding to steps C 1,..., last in gauss elimination, for the matrix w of C order ( nrow , ncol ), using pivoting of scaled rows. C C parameters C w contains the (nrow,ncol) matrix to be partially factored C on input, and the partial factorization on output. C ipivot an integer array of length last containing a record of C the pivoting strategy used; explicit interchanges C are used for pivoting. C d a work array of length nrow used to store row sizes C temporarily. C nrow number of rows of w. C ncol number of columns of w. C last number of elimination steps to be carried out. C info on output, zero if the matrix is found to be non- C singular, in case a zero pivot was encountered in row C n, info = n on output. C C********************************************************************** C INTEGER IPIVOT(NROW),NCOL,LAST,INFO, I,J,K,L,KP1 DOUBLE PRECISION W(NROW,NCOL),D(NROW), COLMAX,T,S DOUBLE PRECISION DABS,DMAX1 C C... initialize d C DO 10 I = 1, NROW D(I) = 0.D0 10 CONTINUE DO 20 J = 1, NCOL DO 20 I = 1, NROW D(I) = DMAX1( D(I) , DABS(W(I,J))) 20 CONTINUE C C... gauss elimination with pivoting of scaled rows, loop over C... k=1,.,last C K = 1 C C... as pivot row for k-th step, pick among the rows not yet used, C... i.e., from rows k ,..., nrow , the one whose k-th entry C... (compared to the row size) is largest. then, if this row C... does not turn out to be row k, interchange row k with this C... particular row and redefine ipivot(k). C 30 CONTINUE IF ( D(K) .EQ. 0.D0 ) GO TO 90 IF (K .EQ. NROW) GO TO 80 L = K KP1 = K+1 COLMAX = DABS(W(K,K)) / D(K) C C... find the (relatively) largest pivot C DO 40 I = KP1, NROW IF ( DABS(W(I,K)) .LE. COLMAX * D(I) ) GO TO 40 COLMAX = DABS(W(I,K)) / D(I) L = I 40 CONTINUE IPIVOT(K) = L T = W(L,K) S = D(L) IF ( L .EQ. K ) GO TO 50 W(L,K) = W(K,K) W(K,K) = T D(L) = D(K) D(K) = S 50 CONTINUE C C... if pivot element is too small in absolute value, declare C... matrix to be noninvertible and quit. C IF ( DABS(T)+D(K) .LE. D(K) ) GO TO 90 C C... otherwise, subtract the appropriate multiple of the pivot C... row from remaining rows, i.e., the rows (k+1),..., (nrow) C... to make k-th entry zero. save the multiplier in its place. C... for high performance do this operations column oriented. C T = -1.0D0/T DO 60 I = KP1, NROW 60 W(I,K) = W(I,K) * T DO 70 J=KP1,NCOL T = W(L,J) IF ( L .EQ. K ) GO TO 62 W(L,J) = W(K,J) W(K,J) = T 62 IF ( T .EQ. 0.D0 ) GO TO 70 DO 64 I = KP1, NROW 64 W(I,J) = W(I,J) + W(I,K) * T 70 CONTINUE K = KP1 C C... check for having reached the next block. C IF ( K .LE. LAST ) GO TO 30 RETURN C C... if last .eq. nrow , check now that pivot element in last row C... is nonzero. C 80 IF( DABS(W(NROW,NROW))+D(NROW) .GT. D(NROW) ) RETURN C C... singularity flag set C 90 INFO = K RETURN END SUBROUTINE SHIFTB (AI, NROWI, NCOLI, LAST, AI1, NROWI1, NCOLI1) C C********************************************************************* C C shifts the rows in current block, ai, not used as pivot rows, if C any, i.e., rows (last+1),..., (nrowi), onto the first mmax = C = nrow-last rows of the next block, ai1, with column last+j of C ai going to column j , j=1,...,jmax=ncoli-last. the remaining C columns of these rows of ai1 are zeroed out. C C picture C C original situation after results in a new block i+1 C last = 2 columns have been created and ready to be C done in factrb (assuming no factored by next factrb C interchanges of rows) call. C 1 C x x 1x x x x x x x x C 1 C 0 x 1x x x 0 x x x x C block i 1 --------------- C nrowi = 4 0 0 1x x x 0 0 1x x x 0 01 C ncoli = 5 1 1 1 C last = 2 0 0 1x x x 0 0 1x x x 0 01 C ------------------------------- 1 1 new C 1x x x x x 1x x x x x1 block C 1 1 1 i+1 C block i+1 1x x x x x 1x x x x x1 C nrowi1= 5 1 1 1 C ncoli1= 5 1x x x x x 1x x x x x1 C ------------------------------- 1-------------1 C 1 C C********************************************************************* C INTEGER LAST, J,JMAX,JMAXP1,M,MMAX DOUBLE PRECISION AI(NROWI,NCOLI),AI1(NROWI1,NCOLI1) MMAX = NROWI - LAST JMAX = NCOLI - LAST IF (MMAX .LT. 1 .OR. JMAX .LT. 1) RETURN C C... put the remainder of block i into ai1 C DO 10 J=1,JMAX DO 10 M=1,MMAX 10 AI1(M,J) = AI(LAST+M,LAST+J) IF (JMAX .EQ. NCOLI1) RETURN C C... zero out the upper right corner of ai1 C JMAXP1 = JMAX + 1 DO 20 J=JMAXP1,NCOLI1 DO 20 M=1,MMAX 20 AI1(M,J) = 0.D0 RETURN END SUBROUTINE SBBLOK ( BLOKS, INTEGS, NBLOKS, IPIVOT, X ) C C********************************************************************** C C calls subroutines subfor and subbak . C C supervises the solution (by forward and backward substitution) of C the linear system a*x = b for x, with the plu factorization of C a already generated in fcblok . individual blocks of C equations are solved via subfor and subbak . C C parameters C bloks, integs, nbloks, ipivot are as on return from fcblok. C x on input: the right hand side, in dense storage C on output: the solution vector C C********************************************************************* C INTEGER INTEGS(3,NBLOKS),IPIVOT(1), I,INDEX,INDEXX,J,LAST, 1 NBP1,NCOL,NROW DOUBLE PRECISION BLOKS(1), X(1) C C... forward substitution pass C INDEX = 1 INDEXX = 1 DO 10 I = 1, NBLOKS NROW = INTEGS(1,I) LAST = INTEGS(3,I) CALL SUBFOR ( BLOKS(INDEX), IPIVOT(INDEXX), NROW, LAST, 1 X(INDEXX) ) INDEX = NROW * INTEGS(2,I) + INDEX 10 INDEXX = INDEXX + LAST C C... back substitution pass C NBP1 = NBLOKS + 1 DO 20 J = 1, NBLOKS I = NBP1 - J NROW = INTEGS(1,I) NCOL = INTEGS(2,I) LAST = INTEGS(3,I) INDEX = INDEX - NROW * NCOL INDEXX = INDEXX - LAST 20 CALL SUBBAK ( BLOKS(INDEX), NROW, NCOL, LAST, X(INDEXX) ) RETURN END SUBROUTINE SUBFOR ( W, IPIVOT, NROW, LAST, X ) C C********************************************************************** C C carries out the forward pass of substitution for the current C block, i.e., the action on the right side corresponding to the C elimination carried out in factrb for this block. C C parameters C w, ipivot, nrow, last are as on return from factrb. C x(j) is expected to contain, on input, the right side of j-th C equation for this block, j=1,...,nrow. C x(j) contains, on output, the appropriately modified right C side of equation (j) in this block, j=1,...,last and C for j=last+1,...,nrow. C C********************************************************************* C INTEGER IPIVOT(LAST), IP,K,KP1,LSTEP DOUBLE PRECISION W(NROW,LAST), X(NROW), T C IF ( NROW .EQ. 1 ) RETURN LSTEP = MIN0( NROW-1 , LAST ) DO 20 K = 1, LSTEP KP1 = K + 1 IP = IPIVOT(K) T = X(IP) X(IP) = X(K) X(K) = T IF ( T .EQ. 0.D0 ) GO TO 20 DO 10 I = KP1, NROW 10 X(I) = X(I) + W(I,K) * T 20 CONTINUE 30 RETURN END SUBROUTINE SUBBAK ( W, NROW, NCOL, LAST, X ) C C********************************************************************* C C carries out backsubstitution for current block. C C parameters C w, ipivot, nrow, ncol, last are as on return from factrb. C x(1),...,x(ncol) contains, on input, the right side for the C equations in this block after backsubstitution has been C carried up to but not including equation (last). C means that x(j) contains the right side of equation (j) C as modified during elimination, j=1,...,last, while C for j .gt. last, x(j) is already a component of the C solution vector. C x(1),...,x(ncol) contains, on output, the components of the C solution corresponding to the present block. C C********************************************************************* C INTEGER LAST, I,J,K,KM1,LM1,LP1 DOUBLE PRECISION W(NROW,NCOL),X(NCOL), T C LP1 = LAST + 1 IF ( LP1 .GT. NCOL ) GO TO 30 DO 20 J = LP1, NCOL T = - X(J) IF ( T .EQ. 0.D0 ) GO TO 20 DO 10 I = 1, LAST 10 X(I) = X(I) + W(I,J) * T 20 CONTINUE 30 IF ( LAST .EQ. 1 ) GO TO 60 LM1 = LAST - 1 DO 50 KB = 1, LM1 KM1 = LAST - KB K = KM1 + 1 X(K) = X(K)/W(K,K) T = - X(K) IF ( T .EQ. 0.D0 ) GO TO 50 DO 40 I = 1, KM1 40 X(I) = X(I) + W(I,K) * T 50 CONTINUE 60 X(1) = X(1)/W(1,1) RETURN END