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C **********
C
C THIS PROGRAM CHECKS THE CONSTANTS OF MACHINE PRECISION AND
C SMALLEST AND LARGEST MACHINE REPRESENTABLE NUMBERS SPECIFIED IN
C FUNCTION SPMPAR, AGAINST THE CORRESPONDING HARDWARE-DETERMINED
C MACHINE CONSTANTS OBTAINED BY SMCHAR, A SUBROUTINE DUE TO
C W. J. CODY.
C
C DATA STATEMENTS IN SPMPAR CORRESPONDING TO THE MACHINE USED MUST
C BE ACTIVATED BY REMOVING C IN COLUMN 1.
C
C THE PRINTED OUTPUT CONSISTS OF THE MACHINE CONSTANTS OBTAINED BY
C SMCHAR AND COMPARISONS OF THE SPMPAR CONSTANTS WITH THEIR
C SMCHAR COUNTERPARTS. DESCRIPTIONS OF THE MACHINE CONSTANTS ARE
C GIVEN IN THE PROLOGUE COMMENTS OF SMCHAR.
C
C SUBPROGRAMS CALLED
C
C MINPACK-SUPPLIED ... SMCHAR,SPMPAR
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. MARCH 1980.
C BURTON S. GARBOW, KENNETH E. HILLSTROM, JORGE J. MORE
C
C **********
INTEGER IBETA,IEXP,IRND,IT,MACHEP,MAXEXP,MINEXP,NEGEP,NGRD,
* NWRITE
REAL DWARF,EPS,EPSMCH,EPSNEG,GIANT,XMAX,XMIN
REAL RERR(3)
REAL SPMPAR
C
C LOGICAL OUTPUT UNIT IS ASSUMED TO BE NUMBER 6.
C
DATA NWRITE /6/
C
C DETERMINE THE MACHINE CONSTANTS DYNAMICALLY FROM SMCHAR.
C
CALL SMCHAR(IBETA,IT,IRND,NGRD,MACHEP,NEGEP,IEXP,MINEXP,MAXEXP,
* EPS,EPSNEG,XMIN,XMAX)
C
C COMPARE THE SPMPAR CONSTANTS WITH THEIR SMCHAR COUNTERPARTS AND
C STORE THE RELATIVE DIFFERENCES IN RERR.
C
EPSMCH = SPMPAR(1)
DWARF = SPMPAR(2)
GIANT = SPMPAR(3)
RERR(1) = (EPSMCH - EPS)/EPSMCH
RERR(2) = (DWARF - XMIN)/DWARF
RERR(3) = (XMAX - GIANT)/GIANT
C
C WRITE THE SMCHAR CONSTANTS.
C
WRITE (NWRITE,10)
* IBETA,IT,IRND,NGRD,MACHEP,NEGEP,IEXP,MINEXP,MAXEXP,EPS,
* EPSNEG,XMIN,XMAX
C
C WRITE THE SPMPAR CONSTANTS AND THE RELATIVE DIFFERENCES.
C
WRITE (NWRITE,20) EPSMCH,RERR(1),DWARF,RERR(2),GIANT,RERR(3)
STOP
10 FORMAT (17H1SMCHAR CONSTANTS /// 8H IBETA =, I6 // 8H IT =,
* I6 // 8H IRND =, I6 // 8H NGRD =, I6 // 9H MACHEP =,
* I6 // 8H NEGEP =, I6 // 7H IEXP =, I6 // 9H MINEXP =,
* I6 // 9H MAXEXP =, I6 // 6H EPS =, E15.7 // 9H EPSNEG =,
* E15.7 // 7H XMIN =, E15.7 // 7H XMAX =, E15.7)
20 FORMAT ( /// 42H SPMPAR CONSTANTS AND RELATIVE DIFFERENCES ///
* 9H EPSMCH =, E15.7 / 10H RERR(1) =, E15.7 //
* 8H DWARF =, E15.7 / 10H RERR(2) =, E15.7 // 8H GIANT =,
* E15.7 / 10H RERR(3) =, E15.7)
C
C LAST CARD OF DRIVER.
C
END
SUBROUTINE SMCHAR(IBETA,IT,IRND,NGRD,MACHEP,NEGEP,IEXP,MINEXP,
1 MAXEXP,EPS,EPSNEG,XMIN,XMAX)
C
INTEGER I,IBETA,IEXP,IRND,IT,IZ,J,K,MACHEP,MAXEXP,MINEXP,
1 MX,NEGEP,NGRD
REAL A,B,BETA,BETAIN,BETAM1,EPS,EPSNEG,ONE,XMAX,XMIN,Y,Z,ZERO
C
C THIS SUBROUTINE IS INTENDED TO DETERMINE THE CHARACTERISTICS
C OF THE FLOATING-POINT ARITHMETIC SYSTEM THAT ARE SPECIFIED
C BELOW. THE FIRST THREE ARE DETERMINED ACCORDING TO AN
C ALGORITHM DUE TO M. MALCOLM, CACM 15 (1972), PP. 949-951,
C INCORPORATING SOME, BUT NOT ALL, OF THE IMPROVEMENTS
C SUGGESTED BY M. GENTLEMAN AND S. MAROVICH, CACM 17 (1974),
C PP. 276-277.
C
C
C IBETA - THE RADIX OF 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
C NGRD - THE NUMBER OF GUARD DIGITS FOR MULTIPLICATION. IT IS
C 0 IF IRND=1, OR IF IRND=0 AND ONLY IT BASE IBET
C DIGITS PARTICIPATE IN THE POST NORMALIZATION SHIFT
C OF THE FLOATING-POINT SIGNIFICAND IN MULTIPLICATION
C 1 IF IRND=0 AND MORE THAN IT BASE IBETA DIGITS
C PARTICIPATE IN THE POST NORMALIZATION SHIFT OF THE
C FLOATING-POINT 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 A POSITIVE FLOATING-POINT
C NUMBER
C MAXEXP - THE LARGEST POSITIVE INTEGER EXPONENT FOR A FINITE
C FLOATING-POINT NUMBER
C EPS - THE SMALLEST POSITIVE FLOATING-POINT NUMBER SUCH
C THAT 1.0+EPS .NE. 1.0. IN PARTICULAR, IF EITHER
C IBETA = 2 OR IRND = 0, EPS = FLOAT(IBETA)**MACHEP.
C OTHERWISE, EPS = (FLOAT(IBETA)**MACHEP)/2
C EPSNEG - A SMALL POSITIVE FLOATING-POINT NUMBER SUCH THAT
C 1.0-EPSNEG .NE. 1.0. IN PARTICULAR, IF IBETA = 2
C OR IRND = 0, EPSNEG = FLOAT(IBETA)**NEGEPS.
C OTHERWISE, EPSNEG = (IBETA**NEGEPS)/2. BECAUSE
C NEGEPS IS BOUNDED BELOW BY -(IT+3), EPSNEG MAY NOT
C BE THE SMALLEST NUMBER WHICH CAN ALTER 1.0 BY
C SUBTRACTION.
C XMIN - THE SMALLEST NON-VANISHING FLOATING-POINT POWER OF TH
C RADIX. IN PARTICULAR, 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 REVISION - OCTOBER 22, 1979
C
C AUTHOR - W. J. CODY
C ARGONNE NATIONAL LABORATORY
C
C-----------------------------------------------------------------
ONE = FLOAT(1)
ZERO = 0.0E0
C-----------------------------------------------------------------
C DETERMINE IBETA,BETA ALA MALCOLM
C-----------------------------------------------------------------
A = ONE
10 A = A + A
IF (((A+ONE)-A)-ONE .EQ. ZERO) GO TO 10
B = ONE
20 B = B + B
IF ((A+B)-A .EQ. ZERO) GO TO 20
IBETA = INT((A+B)-A)
BETA = FLOAT(IBETA)
C-----------------------------------------------------------------
C DETERMINE IT, IRND
C-----------------------------------------------------------------
IT = 0
B = ONE
100 IT = IT + 1
B = B * BETA
IF (((B+ONE)-B)-ONE .EQ. ZERO) GO TO 100
IRND = 0
BETAM1 = BETA - ONE
IF ((A+BETAM1)-A .NE. ZERO) IRND = 1
C-----------------------------------------------------------------
C DETERMINE NEGEP, EPSNEG
C-----------------------------------------------------------------
NEGEP = IT + 3
BETAIN = ONE / BETA
A = ONE
C
DO 200 I = 1, NEGEP
A = A * BETAIN
200 CONTINUE
C
B = A
210 IF ((ONE-A)-ONE .NE. ZERO) GO TO 220
A = A * BETA
NEGEP = NEGEP - 1
GO TO 210
220 NEGEP = -NEGEP
EPSNEG = A
IF ((IBETA .EQ. 2) .OR. (IRND .EQ. 0)) GO TO 300
A = (A*(ONE+A)) / (ONE+ONE)
IF ((ONE-A)-ONE .NE. ZERO) EPSNEG = A
C-----------------------------------------------------------------
C DETERMINE MACHEP, EPS
C-----------------------------------------------------------------
300 MACHEP = -IT - 3
A = B
310 IF((ONE+A)-ONE .NE. ZERO) GO TO 320
A = A * BETA
MACHEP = MACHEP + 1
GO TO 310
320 EPS = A
IF ((IBETA .EQ. 2) .OR. (IRND .EQ. 0)) GO TO 350
A = (A*(ONE+A)) / (ONE+ONE)
IF ((ONE+A)-ONE .NE. ZERO) EPS = A
C-----------------------------------------------------------------
C DETERMINE NGRD
C-----------------------------------------------------------------
350 NGRD = 0
IF ((IRND .EQ. 0) .AND. ((ONE+EPS)*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
400 Y = Z
Z = Y * Y
C-----------------------------------------------------------------
C CHECK FOR UNDERFLOW HERE
C-----------------------------------------------------------------
A = Z * ONE
IF ((A+A .EQ. ZERO) .OR. (ABS(Z) .GE. Y)) 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 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
IF (((A+A) .EQ. ZERO) .OR. (ABS(Y) .GE. XMIN)) GO TO 460
K = K + 1
GO TO 450
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 FOR MACHINES WITH IMPLICIT LEADING
C BIT IN BINARY SIGNIFICAND AND MACHINES WITH
C RADIX POINT AT EXTREME 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
C
DO 510 J = 1, I
IF (IBETA .EQ. 2) XMAX = XMAX + XMAX
IF (IBETA .NE. 2) XMAX = XMAX * BETA
510 CONTINUE
C
520 RETURN
C ---------- LAST CARD OF SMCHAR ----------
END
|
