C--------------------------------------------------------------------
C FORTRAN 77 program to test RKBESL
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 five
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 MINEXP - the largest in magnitude negative
C integer such that FLOAT(IBETA)**MINEXP
C is a positive floating-point number
C EPS - the smallest positive floating-point
C number such that 1.0+EPS .NE. 1.0
C EPSNEG - the smallest positive floating-point
C number such that 1.0-EPSNEG .NE. 1.0
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, EXP, LOG, MAX, REAL, SQRT
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 Authors: W. J. Cody and L. Stoltz
C Mathematics and Computer Science Division
C Argonne National Laboratory
C Argonne, IL 60439
C
C--------------------------------------------------------------------
LOGICAL SFLAG,TFLAG
INTEGER I,IBETA,IEXP,II,IND,IOUT,IRND,IT,IZE,IZ1,J,JT,K1,K2,
1 K3,MACHEP,MAXEXP,MB,MINEXP,N,NCALC,NDUM,NEGEP,NGRD
CS REAL
CD DOUBLE PRECISION
1 A,AIT,ALBETA,ALPHA,AMAXEXP,A1,B,BETA,C,CONV,D,DEL,EIGHT,EPS,
2 EPSNEG,FIVE,HALF,HUND,ONE,OVRCHK,REN,R6,R7,SIXTEN,SUM,T,TEN,
3 TINYX,T1,U,U2,W,X,XL,XLAM,XMAX,XMB,XMIN,XN,XNINE,X1,X99,Y,Z,
4 ZERO,ZZ
DIMENSION U(560),U2(560)
CS DATA ZERO,HALF,ONE,FIVE,EIGHT/0.0E0,0.5E0,1.0E0,5.0E0,8.0E0/
CS DATA XNINE,TEN,HUND/9.0E0,10.0E0,1.0E2/
CS DATA X99,SIXTEN,XLAM/-999.0E0,1.6E1,0.9375E0/
CS DATA C/2.2579E-1/,TINYX/1.0E-10/
CD DATA ZERO,HALF,ONE,FIVE,EIGHT/0.0D0,0.5D0,1.0D0,5.0D0,8.0D0/
CD DATA XNINE,TEN,HUND/9.0D0,10.0D0,1.0D2/
CD DATA X99,SIXTEN,XLAM/-999.0D0,1.6D1,0.9375D0/
CD DATA C/2.2579D-1/,TINYX/1.0D-10/
DATA IOUT/6/
C--------------------------------------------------------------------
C Define statement function for integer to float conversion
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)
AIT = CONV(IT)
ALBETA = LOG(BETA)
AMAXEXP = CONV(MAXEXP)
A = EPS
B = ONE
JT = 0
C-------------------------------------------------------------------
C Random argument accuracy tests
C-------------------------------------------------------------------
DO 300 J = 1, 3
SFLAG = ((J .EQ. 1) .AND. (AMAXEXP/AIT .LE. FIVE))
N = 2000
XN = CONV(N)
K1 = 0
K2 = 0
K3 = 0
X1 = ZERO
A1 = ZERO
R6 = ZERO
R7 = ZERO
DEL = (B - A) / XN
XL = A
DO 200 I = 1, N
X = DEL * REN(JT) + XL
ALPHA = REN(JT)
C--------------------------------------------------------------------
C Accuracy test is based on the multiplication theorem
C--------------------------------------------------------------------
IZE = 1
MB = 3
C--------------------------------------------------------------------
C Carefully purify arguments
C--------------------------------------------------------------------
Y = X/XLAM
W = SIXTEN * Y
T1 = W + Y
Y = T1 - W
X = Y * XLAM
CALL RKBESL(Y,ALPHA,MB,IZE,U2,NCALC)
TFLAG = SFLAG .AND. (Y .LT. HALF)
IF (TFLAG) THEN
U2(1) = U2(1) * EPS
U2(2) = U2(2) * EPS
END IF
MB = 1
XMB = ZERO
W = (ONE-XLAM) * (ONE+XLAM) * HALF
D = W*Y
T = U2(1) + D * U2(2)
T1 = EPS / HUND
C--------------------------------------------------------------------
C Generate terms using recurrence
C--------------------------------------------------------------------
DO 110 II = 3, 35
XMB = XMB + ONE
T1 = XMB * T1 / D
Z = U2(II-1)
OVRCHK = (XMB+ALPHA)/(Y*HALF)
IF (Z/T1 .LT. T) THEN
GO TO 120
ELSE IF (U2(II-1) .GT. ONE) THEN
IF (OVRCHK .GT. (XMAX/U2(II-1))) THEN
XL = XL + DEL
A = XL
GO TO 200
END IF
END IF
U2(II) = OVRCHK * U2(II-1) + U2(II-2)
IF (T1 .GT. ONE/EPS) THEN
T = T * T1
T1 = ONE
END IF
MB = MB + 1
110 CONTINUE
C--------------------------------------------------------------------
C Accurate Summation
C--------------------------------------------------------------------
XMB = XMB + ONE
120 SUM = U2(MB+1)
IND = MB
DO 155 II = 1, MB
SUM = SUM * D / XMB + U2(IND)
IND = IND - 1
XMB = XMB - ONE
155 CONTINUE
ZZ = SUM
IF (TFLAG) ZZ = ZZ / EPS
ZZ = ZZ * XLAM ** ALPHA
MB = 2
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
Z = U(1)
Y = Z
IF (U2(1) .GT. Y) Y = U2(1)
C--------------------------------------------------------------------
C Accumulate Results
C--------------------------------------------------------------------
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
A1 = ALPHA
IZ1 = IZE
END IF
R7 = R7 + W * W
XL = XL + DEL
200 CONTINUE
C--------------------------------------------------------------------
C Gather and print statistics for test
C--------------------------------------------------------------------
N = K1 + K2 + K3
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 = X99
IF (R6 .NE. ZERO) W = LOG(R6)/ALBETA
IF (J .EQ. 3) THEN
WRITE (IOUT,1024) R6,IBETA,W,X1,A1,IZ1
ELSE
WRITE (IOUT,1021) R6,IBETA,W,X1,A1,IZ1
END IF
W = MAX(AIT+W,ZERO)
WRITE (IOUT,1022) IBETA,W
W = X99
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
C--------------------------------------------------------------------
C Initialize for next test
C--------------------------------------------------------------------
A = B
B = B + B
IF (J .EQ. 1) B = TEN
300 CONTINUE
C-------------------------------------------------------------------
C Test of error returns
C First, test with bad parameters
C-------------------------------------------------------------------
WRITE (IOUT, 2006)
X = -ONE
ALPHA = HALF
MB = 5
IZE = 2
U(1) = ZERO
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
X = -X
ALPHA = ONE + HALF
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
ALPHA = HALF
MB = -MB
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
MB = -MB
IZE = 5
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
C-------------------------------------------------------------------
C Last tests are with extreme parameters
C-------------------------------------------------------------------
X = XMIN
ALPHA = ZERO
MB = 2
IZE = 2
U(1) = ZERO
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
X = TINYX*(ONE-SQRT(EPS))
MB = 20
U(1) = ZERO
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
X = TINYX*(ONE+SQRT(EPS))
MB = 20
U(1) = ZERO
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
C-------------------------------------------------------------------
C Determine largest safe argument for unscaled functions
C-------------------------------------------------------------------
Z = LOG(XMIN)
W = Z-C
ZZ = -Z-TEN
350 Z = ZZ
ZZ = ONE/(EIGHT*Z)
A = Z+LOG(Z)*HALF+ZZ*(ONE-XNINE*HALF*ZZ)+W
B = ONE+(HALF-ZZ*(ONE-XNINE*ZZ))/Z
ZZ = Z-A/B
IF (ABS(Z-ZZ) .GT. HUND*EPS*Z) GO TO 350
X = ZZ*XLAM
IZE = 1
MB = 2
U(1) = ZERO
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
X = ZZ
MB = 2
U(1) = ZERO
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
X = TEN / EPS
IZE = 2
U(1) = ZERO
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
X = XMAX
U(1) = ZERO
CALL RKBESL(X,ALPHA,MB,IZE,U,NCALC)
WRITE (IOUT, 2011) X,ALPHA,MB,IZE,U(1),NCALC
STOP
C-------------------------------------------------------------------
1000 FORMAT('1Test of K(X,ALPHA) vs Multiplication Theorem'//)
1010 FORMAT(I7,' Random arguments were tested from the interval ',
1 '(',F5.2,',',F5.2,')'//)
1011 FORMAT(' K(X,ALPHA) was larger',I6,' times,'/
1 15X,' agreed',I6,' times, and'/
1 11X,'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,', NU =',E13.6,
2 ' and IZE =',I2)
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,', NU =',E13.6,
2 ' and IZE =',I2)
1025 FORMAT(' The root mean square absolute error was',E15.4,
1 ' = ',I4,' **',F7.2)
2006 FORMAT('1Check of Error Returns'///
1 ' The following summarizes calls with indicated parameters'//
2 ' NCALC different from MB indicates some form of error'//
3 ' See documentation for RKBESL for details'//
4 7X,'ARG',12X,'ALPHA',6X,'MB',3X,'IZ',7X,'RES',6X,'NCALC'//)
2011 FORMAT(2E15.7,2I5,E15.7,I5//)
C---------- Last line of RKBESL test program ----------
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