C-------------------------------------------------------------------- C FORTRAN 77 program to test BESK1 C C Data required C C None C C Subprograms required from this package C C MACHAR - an environmental inquiry program providing C information on the floating-point arithmetic C system. Note that the call to MACHAR can C be deleted provided the following three C parameters are assigned the values indicated C C IBETA - the radix of the floating-point system C IT - the number of base-IBETA digits in the C significand of a floating-point number C MAXEXP - the smallest positive power of BETA C that overflows C EPS - the smallest positive floating-point C number such that 1.0+EPS .NE. 1.0 C XMIN - the smallest non-vanishing normalized C floating-point power of the radix, i.e., C XMIN = FLOAT(IBETA) ** MINEXP C XMAX - the largest finite floating-point number. C In particular XMAX = (1.0-EPSNEG) * C FLOAT(IBETA) ** MAXEXP C C REN(K) - a function subprogram returning random real C numbers uniformly distributed over (0,1) C C C Intrinsic functions required are: C C ABS, DBLE, LOG, MAX, REAL, SQRT C C User defined functions C C BOT, TOP C C Reference: "Performance evaluation of programs for certain C Bessel functions", W. J. Cody and L. Stoltz, C ACM Trans. on Math. Software, Vol. 15, 1989, C pp 41-48. C C Latest modification: May 30, 1989 C C Author - Laura Stoltz C Mathematics and Computer Science Division C Argonne National Laboratory C Argonne, IL 60439 C-------------------------------------------------------------------- LOGICAL SFLAG,TFLAG INTEGER I,IBETA,IEXP,II,IND,IOUT,IRND,IT,J,JT,K1,K2,K3, 1 MACHEP,MAXEXP,MB,MINEXP,N,NDUM,NEGEP,NGRD CS REAL CD DOUBLE PRECISION 1 A,AIT,ALBETA,ALL9,AMAXEXP,ATETEN,B,BESEK1,BESK0,BESK1,BETA, 2 BOT,C,CONST,CONV,DEL,EIGHT,EPS,EPSNEG,FIFTEN,FIVE,FOUR8, 3 HALF,HUND,ONE,ONENEG,ONE28,PI,REN,R6,R7,SUM,T,T1,THIRTY,THREE, 4 TOP,TWENTY,TWO,U,W,X,XA,XB,XDEN,XL,XLAM,XLARGE,XLEAST,XMAX, 5 XMB,XMIN,XN,XNINE,X1,Y,Z,ZERO,ZZ DIMENSION U(0:559) C-------------------------------------------------------------------- C Mathematical constants C-------------------------------------------------------------------- CS DATA ZERO,HALF,ONE,TWO,EIGHT/0.0E0,0.5E0,1.0E0,2.0E0,8.0E0/, CS 1 XNINE,ATETEN,TWENTY,ONE28/9.0E0,18.0E0,20.0E0,128.0E0/, CS 2 FIVE,ONENEG,XDEN,ALL9/5.0E0,-1.0E0,16.0E0,-999.0E0/, CS 3 THREE,FIFTEN,THIRTY,FOUR8/3.0E0,15.0E0,30.0E0,48.0E0/, CS 4 PI/3.141592653589793E0/,HUND/100.0E0/ CD DATA ZERO,HALF,ONE,TWO,EIGHT/0.0D0,0.5D0,1.0D0,2.0D0,8.0D0/, CD 1 XNINE,ATETEN,TWENTY,ONE28/9.0D0,18.0D0,20.0D0,128.0D0/, CD 2 FIVE,ONENEG,XDEN,ALL9/5.0D0,-1.0D0,16.0D0,-999.0D0/, CD 3 THREE,FIFTEN,THIRTY,FOUR8/3.0D0,15.0D0,30.0D0,48.0D0/, CD 4 PI/3.141592653589793D0/,HUND/100.0D0/ C--------------------------------------------------------------------- C Machine-dependent constant C--------------------------------------------------------------------- CS DATA XLEAST/1.18E-38/ CD DATA XLEAST/2.23D-308/ DATA IOUT/6/ TOP(X) = -X - HALF*LOG(TWO*X) + LOG(ONE+(THREE/EIGHT-FIFTEN/ 1 ONE28/X)/X) BOT(X) = - ONE - HALF/X + ((- FOUR8*X + THIRTY) / 1 (((ONE28*X+FOUR8)*X-FIFTEN)*X)) C--------------------------------------------------------------------- C Statement functions for conversion between integer and float C--------------------------------------------------------------------- CS CONV(NDUM) = REAL(NDUM) CD CONV(NDUM) = DBLE(NDUM) C-------------------------------------------------------------------- C Determine machine parameters and set constants C-------------------------------------------------------------------- CALL MACHAR(IBETA,IT,IRND,NGRD,MACHEP,NEGEP,IEXP,MINEXP, 1 MAXEXP,EPS,EPSNEG,XMIN,XMAX) BETA = CONV(IBETA) ALBETA = LOG(BETA) AIT = CONV(IT) AMAXEXP = CONV(MAXEXP) JT = 0 B = EPS XLAM = (XDEN - ONE) / XDEN CONST = HALF * LOG(PI) - LOG(XMIN) C-------------------------------------------------------------------- C Random argument accuracy tests C-------------------------------------------------------------------- DO 300 J = 1, 3 SFLAG = ((J .EQ. 1) .AND. (AMAXEXP/AIT .LE. FIVE)) K1 = 0 K2 = 0 K3 = 0 X1 = ZERO R6 = ZERO R7 = ZERO N = 2000 A = B IF (J .EQ. 1) THEN B = ONE ELSE IF (J .EQ. 2) THEN B = EIGHT ELSE B = TWENTY END IF XN = CONV(N) DEL = (B - A) / XN XL = A C--------------------------------------------------------------------- C Accuracy test is based on the multiplication theorem C--------------------------------------------------------------------- DO 200 I = 1, N X = DEL * REN(JT) + XL Y = X / XLAM W = XDEN * Y Y = (W + Y) - W X = Y * XLAM U(0) = BESK0(Y) U(1) = BESK1(Y) TFLAG = SFLAG .AND. (Y .LT. HALF) IF (TFLAG) THEN U(0) = U(0) * EPS U(1) = U(1) * EPS END IF MB = 1 XMB = ONE Y = Y * HALF W = (ONE-XLAM) * (ONE+XLAM) C = W *Y T = U(0) + C * U(1) T1 = EPS / HUND DO 110 II = 2, 60 Z = U(II-1) IF (Z/T1 .LT. T) THEN GO TO 120 ELSE IF (U(II-1) .GT. ONE) THEN IF ((XMB/Y) .GT. (XMAX/U(II-1))) THEN XL = XL + DEL A = XL GO TO 200 END IF END IF U(II) = XMB/Y * U(II-1) + U(II-2) IF (T1 .GT. ONE/EPS) THEN T = T * T1 T1 = ONE END IF T1 = XMB * T1 / C XMB = XMB + ONE MB = MB + 1 110 CONTINUE 120 SUM = U(MB) IND = MB MB = MB - 1 DO 155 II = 1, MB XMB = XMB - ONE IND = IND - 1 SUM = SUM * W * Y / XMB + U(IND) 155 CONTINUE ZZ = SUM IF (TFLAG) ZZ = ZZ / EPS ZZ = ZZ * XLAM Z = BESK1(X) Y = Z IF (U(0) .GT. Y) Y= U(0) W = (Z - ZZ) / Y IF (W .GT. ZERO) THEN K1 = K1 + 1 ELSE IF (W .LT. ZERO) THEN K3 = K3 + 1 ELSE K2 = K2 + 1 END IF W = ABS(W) IF (W .GT. R6) THEN R6 = W X1 = X END IF R7 = R7 + W * W XL = XL + DEL 200 CONTINUE N = K1 + K2 + K3 XN = CONV(N) R7 = SQRT(R7/XN) WRITE (IOUT,1000) WRITE (IOUT,1010) N,A,B WRITE (IOUT,1011) K1,K2,K3 WRITE (IOUT,1020) IT,IBETA W = ALL9 IF (R6 .NE. ZERO) W = LOG(R6)/ALBETA IF (J .EQ. 3) THEN WRITE (IOUT,1024) R6,IBETA,W,X1 ELSE WRITE (IOUT,1021) R6,IBETA,W,X1 END IF W = MAX(AIT+W,ZERO) WRITE (IOUT,1022) IBETA,W W = ALL9 IF (R7 .NE. ZERO) W = LOG(R7)/ALBETA IF (J .EQ. 3) THEN WRITE (IOUT,1025) R7,IBETA,W ELSE WRITE (IOUT,1023) R7,IBETA,W END IF W = MAX(AIT+W,ZERO) WRITE (IOUT,1022) IBETA,W 300 CONTINUE C-------------------------------------------------------------------- C Special tests C-------------------------------------------------------------------- WRITE (IOUT,1030) WRITE (IOUT,1031) Y = BESK1(XLEAST) WRITE (IOUT,1032) Y Y = BESK1(XMIN) WRITE (IOUT,1036) Y Y = BESK1(ZERO) WRITE (IOUT,1033) 0,Y X = REN(JT) * ONENEG Y = BESK1(X) WRITE (IOUT,1034) X,Y Y = BESEK1(XMAX) WRITE (IOUT,1035) Y XA = LOG(XMAX) 330 XB = XA - (TOP(XA)+CONST) / BOT(XA) IF (ABS(XB-XA)/XB .LE. EPS) THEN GO TO 350 ELSE XA = XB GO TO 330 END IF 350 XLARGE = XB * XLAM Y = BESK1(XLARGE) WRITE (IOUT,1034) XLARGE,Y XLARGE = XB * (XNINE / EIGHT) Y = BESK1(XLARGE) WRITE (IOUT,1034) XLARGE,Y C-------------------------------------------------------------------- C Test of error returns C-------------------------------------------------------------------- STOP 1000 FORMAT('1Test of K1(X) vs Multiplication Theorem'//) 1010 FORMAT(I7,' random arguments were tested from the interval (', 1 F5.1,',',F5.1,')'//) 1011 FORMAT(' ABS(K1(X)) was larger',I6,' times,'/ 1 20X,' agreed',I6,' times, and'/ 1 16X,'was smaller',I6,' times.'//) 1020 FORMAT(' There are',I4,' base',I4, 1 ' significant digits in a floating-point number.'//) 1021 FORMAT(' The maximum relative error of',E15.4,' = ',I4,' **', 1 F7.2/4X,'occurred for X =',E13.6) 1022 FORMAT(' The estimated loss of base',I4, 1 ' significant digits is',F7.2//) 1023 FORMAT(' The root mean square relative error was',E15.4, 1 ' = ',I4,' **',F7.2) 1024 FORMAT(' The maximum absolute error of',E15.4,' = ',I4,' **', 1 F7.2/4X,'occurred for X =',E13.6) 1025 FORMAT(' The root mean square absolute error was',E15.4, 1 ' = ',I4,' **',F7.2) 1030 FORMAT('1Special Tests'//) 1031 FORMAT(//' Test with extreme arguments'/) 1032 FORMAT(' K1(XLEAST) = ',E24.17/) 1033 FORMAT(' K1(',I1,') = ',E24.17/) 1034 FORMAT(' K1(',E24.17,' ) = ',E24.17/) 1035 FORMAT(' E**X * K1(XMAX) = ',E24.17/) 1036 FORMAT(' K1(XMIN) = ',E24.17/) C---------- Last line of BESK1 test program ---------- END SUBROUTINE CALCK0(ARG,RESULT,JINT) C-------------------------------------------------------------------- C C This packet computes modified Bessel functions of the second kind C and order zero, K0(X) and EXP(X)*K0(X), for real C arguments X. It contains two function type subprograms, BESK0 C and BESEK0, and one subroutine type subprogram, CALCK0. C the calling statements for the primary entries are C C Y=BESK0(X) C and C Y=BESEK0(X) C C where the entry points correspond to the functions K0(X) and C EXP(X)*K0(X), respectively. The routine CALCK0 is C intended for internal packet use only, all computations within C the packet being concentrated in this routine. The function C subprograms invoke CALCK0 with the statement C CALL CALCK0(ARG,RESULT,JINT) C where the parameter usage is as follows C C Function Parameters for CALCK0 C Call ARG RESULT JINT C C BESK0(ARG) 0 .LT. ARG .LE. XMAX K0(ARG) 1 C BESEK0(ARG) 0 .LT. ARG EXP(ARG)*K0(ARG) 2 C C The main computation evaluates slightly modified forms of near C minimax rational approximations generated by Russon and Blair, C Chalk River (Atomic Energy of Canada Limited) Report AECL-3461, C 1969. This transportable program is patterned after the C machine-dependent FUNPACK packet NATSK0, but cannot match that C version for efficiency or accuracy. This version uses rational C functions that theoretically approximate K-SUB-0(X) to at C least 18 significant decimal digits. The accuracy achieved C depends on the arithmetic system, the compiler, the intrinsic C functions, and proper selection of the machine-dependent C constants. C C******************************************************************* C******************************************************************* C C Explanation of machine-dependent constants C C beta = Radix for the floating-point system C minexp = Smallest representable power of beta C maxexp = Smallest power of beta that overflows C XSMALL = Argument below which BESK0 and BESEK0 may C each be represented by a constant and a log. C largest X such that 1.0 + X = 1.0 to machine C precision. C XINF = Largest positive machine number; approximately C beta**maxexp C XMAX = Largest argument acceptable to BESK0; Solution to C equation: C W(X) * (1-1/8X+9/128X**2) = beta**minexp C where W(X) = EXP(-X)*SQRT(PI/2X) C C C Approximate values for some important machines are: C C C beta minexp maxexp C C CRAY-1 (S.P.) 2 -8193 8191 C Cyber 180/185 C under NOS (S.P.) 2 -975 1070 C IEEE (IBM/XT, C SUN, etc.) (S.P.) 2 -126 128 C IEEE (IBM/XT, C SUN, etc.) (D.P.) 2 -1022 1024 C IBM 3033 (D.P.) 16 -65 63 C VAX D-Format (D.P.) 2 -128 127 C VAX G-Format (D.P.) 2 -1024 1023 C C C XSMALL XINF XMAX C C CRAY-1 (S.P.) 3.55E-15 5.45E+2465 5674.858 C Cyber 180/855 C under NOS (S.P.) 1.77E-15 1.26E+322 672.788 C IEEE (IBM/XT, C SUN, etc.) (S.P.) 5.95E-8 3.40E+38 85.337 C IEEE (IBM/XT, C SUN, etc.) (D.P.) 1.11D-16 1.79D+308 705.342 C IBM 3033 (D.P.) 1.11D-16 7.23D+75 177.852 C VAX D-Format (D.P.) 6.95D-18 1.70D+38 86.715 C VAX G-Format (D.P.) 5.55D-17 8.98D+307 706.728 C C******************************************************************* C******************************************************************* C C Error returns C C The program returns the value XINF for ARG .LE. 0.0, and the C BESK0 entry returns the value 0.0 for ARG .GT. XMAX. C C C Intrinsic functions required are: C C EXP, LOG, SQRT C C Latest modification: March 19, 1990 C C Authors: W. J. Cody and Laura Stoltz C Mathematics and Computer Science Division C Argonne National Laboratory C Argonne, IL 60439 C C-------------------------------------------------------------------- INTEGER I,JINT CS REAL CD DOUBLE PRECISION 1 ARG,F,G,ONE,P,PP,Q,QQ,RESULT,SUMF,SUMG,SUMP,SUMQ,TEMP, 2 X,XINF,XMAX,XSMALL,XX,ZERO DIMENSION P(6),Q(2),PP(10),QQ(10),F(4),G(3) C-------------------------------------------------------------------- C Mathematical constants C-------------------------------------------------------------------- CS DATA ONE/1.0E0/,ZERO/0.0E0/ CD DATA ONE/1.0D0/,ZERO/0.0D0/ C-------------------------------------------------------------------- C Machine-dependent constants C-------------------------------------------------------------------- CS DATA XSMALL/5.95E-8/,XINF/3.40E+38/,XMAX/ 85.337E0/ CD DATA XSMALL/1.11D-16/,XINF/1.79D+308/,XMAX/705.342D0/ C-------------------------------------------------------------------- C C Coefficients for XSMALL .LE. ARG .LE. 1.0 C C-------------------------------------------------------------------- CS DATA P/ 5.8599221412826100000E-04, 1.3166052564989571850E-01, CS 1 1.1999463724910714109E+01, 4.6850901201934832188E+02, CS 2 5.9169059852270512312E+03, 2.4708152720399552679E+03/ CS DATA Q/-2.4994418972832303646E+02, 2.1312714303849120380E+04/ CS DATA F/-1.6414452837299064100E+00,-2.9601657892958843866E+02, CS 1 -1.7733784684952985886E+04,-4.0320340761145482298E+05/ CS DATA G/-2.5064972445877992730E+02, 2.9865713163054025489E+04, CS 1 -1.6128136304458193998E+06/ CD DATA P/ 5.8599221412826100000D-04, 1.3166052564989571850D-01, CD 1 1.1999463724910714109D+01, 4.6850901201934832188D+02, CD 2 5.9169059852270512312D+03, 2.4708152720399552679D+03/ CD DATA Q/-2.4994418972832303646D+02, 2.1312714303849120380D+04/ CD DATA F/-1.6414452837299064100D+00,-2.9601657892958843866D+02, CD 1 -1.7733784684952985886D+04,-4.0320340761145482298D+05/ CD DATA G/-2.5064972445877992730D+02, 2.9865713163054025489D+04, CD 1 -1.6128136304458193998D+06/ C-------------------------------------------------------------------- C C Coefficients for 1.0 .LT. ARG C C-------------------------------------------------------------------- CS DATA PP/ 1.1394980557384778174E+02, 3.6832589957340267940E+03, CS 1 3.1075408980684392399E+04, 1.0577068948034021957E+05, CS 2 1.7398867902565686251E+05, 1.5097646353289914539E+05, CS 3 7.1557062783764037541E+04, 1.8321525870183537725E+04, CS 4 2.3444738764199315021E+03, 1.1600249425076035558E+02/ CS DATA QQ/ 2.0013443064949242491E+02, 4.4329628889746408858E+03, CS 1 3.1474655750295278825E+04, 9.7418829762268075784E+04, CS 2 1.5144644673520157801E+05, 1.2689839587977598727E+05, CS 3 5.8824616785857027752E+04, 1.4847228371802360957E+04, CS 4 1.8821890840982713696E+03, 9.2556599177304839811E+01/ CD DATA PP/ 1.1394980557384778174D+02, 3.6832589957340267940D+03, CD 1 3.1075408980684392399D+04, 1.0577068948034021957D+05, CD 2 1.7398867902565686251D+05, 1.5097646353289914539D+05, CD 3 7.1557062783764037541D+04, 1.8321525870183537725D+04, CD 4 2.3444738764199315021D+03, 1.1600249425076035558D+02/ CD DATA QQ/ 2.0013443064949242491D+02, 4.4329628889746408858D+03, CD 1 3.1474655750295278825D+04, 9.7418829762268075784D+04, CD 2 1.5144644673520157801D+05, 1.2689839587977598727D+05, CD 3 5.8824616785857027752D+04, 1.4847228371802360957D+04, CD 4 1.8821890840982713696D+03, 9.2556599177304839811D+01/ C-------------------------------------------------------------------- X = ARG IF (X .GT. ZERO) THEN IF (X .LE. ONE) THEN C-------------------------------------------------------------------- C 0.0 .LT. ARG .LE. 1.0 C-------------------------------------------------------------------- TEMP = LOG(X) IF (X .LT. XSMALL) THEN C-------------------------------------------------------------------- C Return for small ARG C-------------------------------------------------------------------- RESULT = P(6)/Q(2) - TEMP ELSE XX = X * X SUMP = ((((P(1)*XX + P(2))*XX + P(3))*XX + 1 P(4))*XX + P(5))*XX + P(6) SUMQ = (XX + Q(1))*XX + Q(2) SUMF = ((F(1)*XX + F(2))*XX + F(3))*XX + F(4) SUMG = ((XX + G(1))*XX + G(2))*XX + G(3) RESULT = SUMP/SUMQ - XX*SUMF*TEMP/SUMG - TEMP IF (JINT .EQ. 2) RESULT = RESULT * EXP(X) END IF ELSE IF ((JINT .EQ. 1) .AND. (X .GT. XMAX)) THEN C-------------------------------------------------------------------- C Error return for ARG .GT. XMAX C-------------------------------------------------------------------- RESULT = ZERO ELSE C-------------------------------------------------------------------- C 1.0 .LT. ARG C-------------------------------------------------------------------- XX = ONE / X SUMP = PP(1) DO 120 I = 2, 10 SUMP = SUMP*XX + PP(I) 120 CONTINUE SUMQ = XX DO 140 I = 1, 9 SUMQ = (SUMQ + QQ(I))*XX 140 CONTINUE SUMQ = SUMQ + QQ(10) RESULT = SUMP / SUMQ / SQRT(X) IF (JINT .EQ. 1) RESULT = RESULT * EXP(-X) END IF ELSE C-------------------------------------------------------------------- C Error return for ARG .LE. 0.0 C-------------------------------------------------------------------- RESULT = XINF END IF C-------------------------------------------------------------------- C Update error counts, etc. C-------------------------------------------------------------------- RETURN C---------- Last line of CALCK0 ---------- END CS REAL CD DOUBLE PRECISION 1 FUNCTION BESK0(X) C-------------------------------------------------------------------- C C This function program computes approximate values for the C modified Bessel function of the second kind of order zero C for arguments 0.0 .LT. ARG .LE. XMAX (see comments heading C CALCK0). C C Authors: W. J. Cody and Laura Stoltz C C Latest Modification: January 19, 1988 C C-------------------------------------------------------------------- INTEGER JINT CS REAL CD DOUBLE PRECISION 1 X, RESULT C-------------------------------------------------------------------- JINT = 1 CALL CALCK0(X,RESULT,JINT) BESK0 = RESULT RETURN C---------- Last line of BESK0 ---------- END CS REAL CD DOUBLE PRECISION 1 FUNCTION BESEK0(X) C-------------------------------------------------------------------- C C This function program computes approximate values for the C modified Bessel function of the second kind of order zero C multiplied by the Exponential function, for arguments C 0.0 .LT. ARG. C C Authors: W. J. Cody and Laura Stoltz C C Latest Modification: January 19, 1988 C C-------------------------------------------------------------------- INTEGER JINT CS REAL CD DOUBLE PRECISION 1 X, RESULT C-------------------------------------------------------------------- JINT = 2 CALL CALCK0(X,RESULT,JINT) BESEK0 = RESULT RETURN C---------- Last line of BESEK0 ---------- END SUBROUTINE MACHAR(IBETA,IT,IRND,NGRD,MACHEP,NEGEP,IEXP,MINEXP, 1 MAXEXP,EPS,EPSNEG,XMIN,XMAX) C---------------------------------------------------------------------- C This Fortran 77 subroutine is intended to determine the parameters C of the floating-point arithmetic system specified below. The C determination of the first three uses an extension of an algorithm C due to M. Malcolm, CACM 15 (1972), pp. 949-951, incorporating some, C but not all, of the improvements suggested by M. Gentleman and S. C Marovich, CACM 17 (1974), pp. 276-277. An earlier version of this C program was published in the book Software Manual for the C Elementary Functions by W. J. Cody and W. Waite, Prentice-Hall, C Englewood Cliffs, NJ, 1980. C C The program as given here must be modified before compiling. If C a single (double) precision version is desired, change all C occurrences of CS (CD) in columns 1 and 2 to blanks. C C Parameter values reported are as follows: C C IBETA - the radix for the floating-point representation C IT - the number of base IBETA digits in the floating-point C significand C IRND - 0 if floating-point addition chops C 1 if floating-point addition rounds, but not in the C IEEE style C 2 if floating-point addition rounds in the IEEE style C 3 if floating-point addition chops, and there is C partial underflow C 4 if floating-point addition rounds, but not in the C IEEE style, and there is partial underflow C 5 if floating-point addition rounds in the IEEE style, C and there is partial underflow C NGRD - the number of guard digits for multiplication with C truncating arithmetic. It is C 0 if floating-point arithmetic rounds, or if it C truncates and only IT base IBETA digits C participate in the post-normalization shift of the C floating-point significand in multiplication; C 1 if floating-point arithmetic truncates and more C than IT base IBETA digits participate in the C post-normalization shift of the floating-point C significand in multiplication. C MACHEP - the largest negative integer such that C 1.0+FLOAT(IBETA)**MACHEP .NE. 1.0, except that C MACHEP is bounded below by -(IT+3) C NEGEPS - the largest negative integer such that C 1.0-FLOAT(IBETA)**NEGEPS .NE. 1.0, except that C NEGEPS is bounded below by -(IT+3) C IEXP - the number of bits (decimal places if IBETA = 10) C reserved for the representation of the exponent C (including the bias or sign) of a floating-point C number C MINEXP - the largest in magnitude negative integer such that C FLOAT(IBETA)**MINEXP is positive and normalized C MAXEXP - the smallest positive power of BETA that overflows C EPS - FLOAT(IBETA)**MACHEP. C EPSNEG - FLOAT(IBETA)**NEGEPS. C XMIN - the smallest non-vanishing normalized floating-point C power of the radix, i.e., XMIN = FLOAT(IBETA)**MINEXP C XMAX - the largest finite floating-point number. In C particular XMAX = (1.0-EPSNEG)*FLOAT(IBETA)**MAXEXP C Note - on some machines XMAX will be only the C second, or perhaps third, largest number, being C too small by 1 or 2 units in the last digit of C the significand. C C Latest modification: May 30, 1989 C C Author: W. J. Cody C Mathematics and Computer Science Division C Argonne National Laboratory C Argonne, IL 60439 C C---------------------------------------------------------------------- INTEGER I,IBETA,IEXP,IRND,IT,ITEMP,IZ,J,K,MACHEP,MAXEXP, 1 MINEXP,MX,NEGEP,NGRD,NXRES CS REAL CD DOUBLE PRECISION 1 A,B,BETA,BETAIN,BETAH,CONV,EPS,EPSNEG,ONE,T,TEMP,TEMPA, 2 TEMP1,TWO,XMAX,XMIN,Y,Z,ZERO C---------------------------------------------------------------------- CS CONV(I) = REAL(I) CD CONV(I) = DBLE(I) ONE = CONV(1) TWO = ONE + ONE ZERO = ONE - ONE C---------------------------------------------------------------------- C Determine IBETA, BETA ala Malcolm. C---------------------------------------------------------------------- A = ONE 10 A = A + A TEMP = A+ONE TEMP1 = TEMP-A IF (TEMP1-ONE .EQ. ZERO) GO TO 10 B = ONE 20 B = B + B TEMP = A+B ITEMP = INT(TEMP-A) IF (ITEMP .EQ. 0) GO TO 20 IBETA = ITEMP BETA = CONV(IBETA) C---------------------------------------------------------------------- C Determine IT, IRND. C---------------------------------------------------------------------- IT = 0 B = ONE 100 IT = IT + 1 B = B * BETA TEMP = B+ONE TEMP1 = TEMP-B IF (TEMP1-ONE .EQ. ZERO) GO TO 100 IRND = 0 BETAH = BETA / TWO TEMP = A+BETAH IF (TEMP-A .NE. ZERO) IRND = 1 TEMPA = A + BETA TEMP = TEMPA+BETAH IF ((IRND .EQ. 0) .AND. (TEMP-TEMPA .NE. ZERO)) IRND = 2 C---------------------------------------------------------------------- C Determine NEGEP, EPSNEG. C---------------------------------------------------------------------- NEGEP = IT + 3 BETAIN = ONE / BETA A = ONE DO 200 I = 1, NEGEP A = A * BETAIN 200 CONTINUE B = A 210 TEMP = ONE-A IF (TEMP-ONE .NE. ZERO) GO TO 220 A = A * BETA NEGEP = NEGEP - 1 GO TO 210 220 NEGEP = -NEGEP EPSNEG = A C---------------------------------------------------------------------- C Determine MACHEP, EPS. C---------------------------------------------------------------------- MACHEP = -IT - 3 A = B 300 TEMP = ONE+A IF (TEMP-ONE .NE. ZERO) GO TO 320 A = A * BETA MACHEP = MACHEP + 1 GO TO 300 320 EPS = A C---------------------------------------------------------------------- C Determine NGRD. C---------------------------------------------------------------------- NGRD = 0 TEMP = ONE+EPS IF ((IRND .EQ. 0) .AND. (TEMP*ONE-ONE .NE. ZERO)) NGRD = 1 C---------------------------------------------------------------------- C Determine IEXP, MINEXP, XMIN. C C Loop to determine largest I and K = 2**I such that C (1/BETA) ** (2**(I)) C does not underflow. C Exit from loop is signaled by an underflow. C---------------------------------------------------------------------- I = 0 K = 1 Z = BETAIN T = ONE + EPS NXRES = 0 400 Y = Z Z = Y * Y C---------------------------------------------------------------------- C Check for underflow here. C---------------------------------------------------------------------- A = Z * ONE TEMP = Z * T IF ((A+A .EQ. ZERO) .OR. (ABS(Z) .GE. Y)) GO TO 410 TEMP1 = TEMP * BETAIN IF (TEMP1*BETA .EQ. Z) GO TO 410 I = I + 1 K = K + K GO TO 400 410 IF (IBETA .EQ. 10) GO TO 420 IEXP = I + 1 MX = K + K GO TO 450 C---------------------------------------------------------------------- C This segment is for decimal machines only. C---------------------------------------------------------------------- 420 IEXP = 2 IZ = IBETA 430 IF (K .LT. IZ) GO TO 440 IZ = IZ * IBETA IEXP = IEXP + 1 GO TO 430 440 MX = IZ + IZ - 1 C---------------------------------------------------------------------- C Loop to determine MINEXP, XMIN. C Exit from loop is signaled by an underflow. C---------------------------------------------------------------------- 450 XMIN = Y Y = Y * BETAIN C---------------------------------------------------------------------- C Check for underflow here. C---------------------------------------------------------------------- A = Y * ONE TEMP = Y * T IF (((A+A) .EQ. ZERO) .OR. (ABS(Y) .GE. XMIN)) GO TO 460 K = K + 1 TEMP1 = TEMP * BETAIN IF ((TEMP1*BETA .NE. Y) .OR. (TEMP .EQ. Y)) THEN GO TO 450 ELSE NXRES = 3 XMIN = Y END IF 460 MINEXP = -K C---------------------------------------------------------------------- C Determine MAXEXP, XMAX. C---------------------------------------------------------------------- IF ((MX .GT. K+K-3) .OR. (IBETA .EQ. 10)) GO TO 500 MX = MX + MX IEXP = IEXP + 1 500 MAXEXP = MX + MINEXP C---------------------------------------------------------------------- C Adjust IRND to reflect partial underflow. C---------------------------------------------------------------------- IRND = IRND + NXRES C---------------------------------------------------------------------- C Adjust for IEEE-style machines. C---------------------------------------------------------------------- IF (IRND .GE. 2) MAXEXP = MAXEXP - 2 C---------------------------------------------------------------------- C Adjust for machines with implicit leading bit in binary C significand, and machines with radix point at extreme C right of significand. C---------------------------------------------------------------------- I = MAXEXP + MINEXP IF ((IBETA .EQ. 2) .AND. (I .EQ. 0)) MAXEXP = MAXEXP - 1 IF (I .GT. 20) MAXEXP = MAXEXP - 1 IF (A .NE. Y) MAXEXP = MAXEXP - 2 XMAX = ONE - EPSNEG IF (XMAX*ONE .NE. XMAX) XMAX = ONE - BETA * EPSNEG XMAX = XMAX / (BETA * BETA * BETA * XMIN) I = MAXEXP + MINEXP + 3 IF (I .LE. 0) GO TO 520 DO 510 J = 1, I IF (IBETA .EQ. 2) XMAX = XMAX + XMAX IF (IBETA .NE. 2) XMAX = XMAX * BETA 510 CONTINUE 520 RETURN C---------- Last line of MACHAR ---------- END FUNCTION REN(K) C--------------------------------------------------------------------- C Random number generator - based on Algorithm 266 by Pike and C Hill (modified by Hansson), Communications of the ACM, C Vol. 8, No. 10, October 1965. C C This subprogram is intended for use on computers with C fixed point wordlength of at least 29 bits. It is C best if the floating-point significand has at most C 29 bits. C C Latest modification: May 30, 1989 C C Author: W. J. Cody C Mathematics and Computer Science Division C Argonne National Laboratory C Argonne, IL 60439 C C--------------------------------------------------------------------- INTEGER IY,J,K CS REAL CONV,C1,C2,C3,ONE,REN CD DOUBLE PRECISION CONV,C1,C2,C3,ONE,REN DATA IY/100001/ CS DATA ONE,C1,C2,C3/1.0E0,2796203.0E0,1.0E-6,1.0E-12/ CD DATA ONE,C1,C2,C3/1.0D0,2796203.0D0,1.0D-6,1.0D-12/ C--------------------------------------------------------------------- C Statement functions for conversion between integer and float C--------------------------------------------------------------------- CS CONV(J) = REAL(J) CD CONV(J) = DBLE(J) C--------------------------------------------------------------------- J = K IY = IY * 125 IY = IY - (IY/2796203) * 2796203 REN = CONV(IY) / C1 * (ONE + C2 + C3) RETURN C---------- Last card of REN ---------- END