C------------------------------------------------------------------
C FORTRAN 77 program to test GAMMA or DGAMMA
C
C Method:
C
C Accuracy tests compare function values against values
C generated with the duplication formula.
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 significant of a floating-point number
C EPS - The smallest positive floating-point
C number such that 1.0+EPS .NE. 1.0
C XMIN - The smallest non-vanishing floating-point
C integral power of the radix
C XMAX - The largest finite floating-point number
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, INT, LOG, MAX, REAL, SQRT
C
C Reference: "Performance evaluation of programs related
C to the real gamma function", W. J. Cody,
C submitted for publication.
C
C Latest modification: April 13, 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--------------------------------------------------------------------
INTEGER I,IBETA,IEXP,IOUT,IRND,IT,J,JT,K1,K2,K3,
1 MACHEP,MAXEXP,MINEXP,N,NEGEP,NGRD,NX
CS REAL GAMMA,
CD DOUBLE PRECISION DGAMMA,
1 A,AIT,ALBETA,ALNX,B,BETA,C,CL,CONV,C1,C2,DEL,EPS,
2 EPSNEG,FUNC,HALF,ONE,REN,R6,R7,TEN,TWO,W,X,XC,XL,
3 XMAX,XMIN,XMINV,XN,XNUM,XXN,XP,XPH,X99,XP99,Y,Z,ZERO,ZZ
CS DATA C1,C2/2.8209479177387814347E-1,9.1893853320467274178E-1/,
CS 1 ZERO,HALF,ONE,TWO,TEN/0.0E0,0.5E0,1.0E0,2.0E0,10.0E0/,
CS 2 X99,XP99/-999.0E0,0.99E0/
CD DATA C1,C2/2.8209479177387814347D-1,9.1893853320467274178D-1/,
CD 1 ZERO,HALF,ONE,TWO,TEN/0.0D0,0.5D0,1.0D0,2.0D0,10.0D0/,
CD 2 X99,XP99/-999.0D0,0.99D0/
DATA IOUT/6/
C------------------------------------------------------------------
C Statement functions for conversion
C------------------------------------------------------------------
CS CONV(I) = REAL(I)
CD CONV(I) = DBLE(I)
CS FUNC(X) = GAMMA(X)
CD FUNC(X) = DGAMMA(X)
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)
A = ZERO
B = TWO
N = 2000
XN = CONV(N)
JT = 0
C-----------------------------------------------------------------
C Determine smallest argument for GAMMA
C-----------------------------------------------------------------
IF (XMIN*XMAX .LT. ONE) THEN
XMINV = ONE/XMAX
ELSE
XMINV = XMIN
END IF
C-----------------------------------------------------------------
C Determine largest argument for GAMMA by Newton iteration
C-----------------------------------------------------------------
CL = LOG(XMAX)
XP = HALF * CL
CL = C2 - CL
50 X = XP
ALNX = LOG(X)
XNUM = (X-HALF)*ALNX - X + CL
XP = X - XNUM/(ALNX-HALF/X)
IF (ABS(XP-X)/X .GE. TEN*EPS) GO TO 50
CL = XP
C-----------------------------------------------------------------
C Random argument accuracy tests
C-----------------------------------------------------------------
DO 300 J = 1, 4
K1 = 0
K3 = 0
XC = ZERO
R6 = ZERO
R7 = ZERO
DEL = (B - A) / XN
XL = A
DO 200 I = 1, N
X = DEL * REN(JT) + XL
C-----------------------------------------------------------------
C Use duplication formula for X not close to the zero
C-----------------------------------------------------------------
XPH = X * HALF + HALF
XP = XPH - HALF
X = XP + XP
NX = INT(X)
XXN = CONV(NX)
C = (TWO ** NX) * (TWO ** (X-XXN))
Z = FUNC(X)
ZZ = ((C*C1)*FUNC(XP)) * FUNC(XPH)
C--------------------------------------------------------------------
C Accumulate results
C--------------------------------------------------------------------
W = (Z - ZZ) / Z
IF (W .GT. ZERO) THEN
K1 = K1 + 1
ELSE IF (W .LT. ZERO) THEN
K3 = K3 + 1
END IF
W = ABS(W)
IF (W .GT. R6) THEN
R6 = W
XC = X
END IF
R7 = R7 + W * W
XL = XL + DEL
200 CONTINUE
C------------------------------------------------------------------
C Gather and print statistics for test
C------------------------------------------------------------------
K2 = N - K3 - K1
R7 = SQRT(R7/XN)
WRITE (IOUT,1000)
WRITE (IOUT,1010) N,A,B
WRITE (IOUT,1011) K1,K2,K3
WRITE (IOUT,1020) IT,IBETA
IF (R6 .NE. ZERO) THEN
W = LOG(ABS(R6))/ALBETA
ELSE
W = X99
END IF
WRITE (IOUT,1021) R6,IBETA,W,XC
W = MAX(AIT+W,ZERO)
WRITE (IOUT,1022) IBETA,W
IF (R7 .NE. ZERO) THEN
W = LOG(ABS(R7))/ALBETA
ELSE
W = X99
END IF
WRITE (IOUT,1023) R7,IBETA,W
W = MAX(AIT+W,ZERO)
WRITE (IOUT,1022) IBETA,W
C------------------------------------------------------------------
C Initialize for next test
C------------------------------------------------------------------
A = B
IF (J .EQ. 1) THEN
B = TEN
ELSE IF (J .EQ. 2) THEN
B = CL - HALF
ELSE
A = -(TEN-HALF) * HALF
B = A + HALF
END IF
300 CONTINUE
C-----------------------------------------------------------------
C Special tests
C First test with special arguments
C-----------------------------------------------------------------
WRITE (IOUT,1025)
WRITE (IOUT,1040)
X = -HALF
Y = FUNC(X)
WRITE (IOUT,1041) X, Y
X = XMINV / XP99
Y = FUNC(X)
WRITE (IOUT,1041) X, Y
X = ONE
Y = FUNC(X)
WRITE (IOUT,1041) X, Y
X = TWO
Y = FUNC(X)
WRITE (IOUT,1041) X, Y
X = CL * XP99
Y = FUNC(X)
WRITE (IOUT,1041) X, Y
C-----------------------------------------------------------------
C Test of error returns
C-----------------------------------------------------------------
WRITE (IOUT,1050)
X = -ONE
WRITE (IOUT,1052) X
Y = FUNC(X)
WRITE (IOUT,1061) Y
X = ZERO
WRITE (IOUT,1052) X
Y = FUNC(X)
WRITE (IOUT,1061) Y
X = XMINV * (ONE - EPS)
WRITE (IOUT,1052) X
Y = FUNC(X)
WRITE (IOUT,1061) Y
X = CL * (ONE + EPS)
WRITE (IOUT,1052) X
Y = FUNC(X)
WRITE (IOUT,1061) Y
WRITE (IOUT,1100)
STOP
C-----------------------------------------------------------------
1000 FORMAT('1Test of GAMMA(X) vs Duplication Formula'//)
1010 FORMAT(I7,' Random arguments were tested from the interval (',
1 F7.3,',',F7.3,')'//)
1011 FORMAT(' GAMMA(X) was larger',I6,' times,'/
1 13X,' agreed',I6,' times, and'/
2 9X,'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)
1025 FORMAT('1Special Tests'//)
1040 FORMAT(//' Test of special arguments'//)
1041 FORMAT(' GAMMA (',E13.6,') = ',E13.6//)
1050 FORMAT('1Test of Error Returns'///)
1052 FORMAT(' GAMMA will be called with the argument',E13.6,/
1 ' This should trigger an error message'//)
1061 FORMAT(' GAMMA returned the value',E13.6///)
1100 FORMAT(' This concludes the tests')
C---------- Last line of GAMMA 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