DOUBLE PRECISION FUNCTION DLAMCH( CMACH )
*
* -- LAPACK auxiliary routine (preliminary version) --
* Univ. of Tennessee, Oak Ridge National Lab, Argonne National Lab,
* Courant Institute, NAG Ltd., and Rice University
* March 26, 1990
*
* .. Scalar Arguments ..
CHARACTER CMACH
* ..
*
* Purpose
* =======
*
* DLAMCH determines double precision machine parameters.
*
*-----------------------------------------------------------------------
*
* The value returned by DLAMCH is determined by the parameter CMACH as
* follows:
*
* CMACH = 'E' or 'e', DLAMCH := eps
* CMACH = 'S' or 's , DLAMCH := sfmin
* CMACH = 'B' or 'b', DLAMCH := base
* CMACH = 'N' or 'n', DLAMCH := t
* CMACH = 'R' or 'r', DLAMCH := rnd
* CMACH = 'M' or 'm', DLAMCH := emin
* CMACH = 'U' or 'u', DLAMCH := rmin
* CMACH = 'L' or 'l', DLAMCH := emax
* CMACH = 'O' or 'o', DLAMCH := rmax
*
* where
*
* eps = relative machine precision
* sfmin = safe minimum, such that 1/sfmin does not overflow
* base = base of the machine
* t = number of (base) digits in the mantissa
* rnd = 1.0 when rounding occurs in addition, 0.0 otherwise
* emin = minimum exponent before (gradual) underflow
* rmin = underflow threshold - base**(emin-1)
* emax = largest exponent before overflow
* rmax = overflow threshold - (base**emax)*(1-eps)
*
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL FIRST, LRND
INTEGER BETA, IMAX, IMIN, IT
DOUBLE PRECISION BASE, EMAX, EMIN, EPS, RMACH, RMAX, RMIN, RND,
$ SFMIN, SMALL, T
* ..
* .. Save statement ..
SAVE FIRST, EPS, SFMIN, BASE, T, RND, EMIN, RMIN,
$ EMAX, RMAX
* ..
* .. External Subroutines ..
EXTERNAL DLAMC2
* ..
* .. Data statements ..
DATA FIRST / .TRUE. /
* ..
* .. Executable Statements ..
IF( FIRST ) THEN
FIRST = .FALSE.
CALL DLAMC2( BETA, IT, LRND, EPS, IMIN, RMIN, IMAX, RMAX )
BASE = BETA
T = IT
IF( LRND ) THEN
RND = ONE
EPS = ( BASE**( 1-IT ) ) / 2
ELSE
RND = ZERO
EPS = BASE**( 1-IT )
END IF
EMIN = IMIN
EMAX = IMAX
SFMIN = RMIN
SMALL = ONE / RMAX
IF( SMALL.GE.SFMIN ) THEN
*
* Use SMALL plus a bit, to avoid the possibility of rounding
* causing overflow when computing 1/sfmin.
*
SFMIN = SMALL*( ONE+EPS )
END IF
END IF
*
IF( ( CMACH.EQ.'E' ) .OR. ( CMACH.EQ.'e' ) ) THEN
RMACH = EPS
ELSE IF( ( CMACH.EQ.'S' ) .OR. ( CMACH.EQ.'s' ) ) THEN
RMACH = SFMIN
ELSE IF( ( CMACH.EQ.'B' ) .OR. ( CMACH.EQ.'b' ) ) THEN
RMACH = BASE
ELSE IF( ( CMACH.EQ.'N' ) .OR. ( CMACH.EQ.'n' ) ) THEN
RMACH = T
ELSE IF( ( CMACH.EQ.'R' ) .OR. ( CMACH.EQ.'r' ) ) THEN
RMACH = RND
ELSE IF( ( CMACH.EQ.'M' ) .OR. ( CMACH.EQ.'m' ) ) THEN
RMACH = EMIN
ELSE IF( ( CMACH.EQ.'U' ) .OR. ( CMACH.EQ.'u' ) ) THEN
RMACH = RMIN
ELSE IF( ( CMACH.EQ.'L' ) .OR. ( CMACH.EQ.'l' ) ) THEN
RMACH = EMAX
ELSE IF( ( CMACH.EQ.'O' ) .OR. ( CMACH.EQ.'o' ) ) THEN
RMACH = RMAX
END IF
*
DLAMCH = RMACH
RETURN
*
* End of DLAMCH
*
END
*
************************************************************************
*
SUBROUTINE DLAMC1( BETA, T, RND, IEEE1 )
*
* DLAMC1 returns the machine parameters given by:
*
* BETA - INTEGER.
* The base of the machine.
*
* T - INTEGER.
* The number of ( BETA ) digits in the mantissa.
*
* RND - LOGICAL.
* Whether proper rounding ( RND = .TRUE. ) or chopping
* ( RND = .FALSE. ) occurs in addition. This may not be a
* reliable guide to the way in which the machine perfoms
* its arithmetic.
*
* IEEE1 - LOGICAL.
* Whether rounding appears to be done in the IEEE 'round
* to nearest' style.
*
* The routine is based on the routine ENVRON by Malcolm and
* incorporates suggestions by Gentleman and Marovich. See
*
* Malcolm M. A. (1972) Algorithms to reveal properties of
* floating-point arithmetic. Comms. of the ACM, 15, 949-951.
*
* Gentleman W. M. and Marovich S. B. (1974) More on algorithms
* that reveal properties of floating point arithmetic units.
* Comms. of the ACM, 17, 276-277.
*
*
* .. Scalar Arguments ..
LOGICAL IEEE1, RND
INTEGER BETA, T
* ..
* .. Local Scalars ..
LOGICAL FIRST, LIEEE1, LRND
INTEGER LBETA, LT
DOUBLE PRECISION A, B, C, F, ONE, QTR, SAVEC, T1, T2
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3
EXTERNAL DLAMC3
* ..
* .. Save statement ..
SAVE FIRST, LIEEE1, LBETA, LRND, LT
* ..
* .. Data statements ..
*
DATA FIRST / .TRUE. /
* ..
* .. Executable Statements ..
*
IF( FIRST ) THEN
FIRST = .FALSE.
ONE = 1
*
* LBETA, LIEEE1, LT and LRND are the local values of BETA,
* IEEE1, T and RND.
*
* Throughout this routine we use the function DLAMC3 to ensure
* that relevant values are stored and not held in registers, or
* are not affected by optimizers.
*
* Compute a = 2.0**m with the smallest positive integer m such
* that
*
* fl( a + 1.0 ) = a.
*
A = 1
C = 1
*
*+ WHILE( C.EQ.ONE )LOOP
10 CONTINUE
IF( C.EQ.ONE ) THEN
A = 2*A
C = DLAMC3( A, ONE )
C = DLAMC3( C, -A )
GO TO 10
END IF
*+ END WHILE
*
* Now compute b = 2.0**m with the smallest positive integer m
* such that
*
* fl( a + b ) .gt. a.
*
B = 1
C = DLAMC3( A, B )
*
*+ WHILE( C.EQ.A )LOOP
20 CONTINUE
IF( C.EQ.A ) THEN
B = 2*B
C = DLAMC3( A, B )
GO TO 20
END IF
*+ END WHILE
*
* Now compute the base. a and c are neighbouring floating point
* numbers in the interval ( beta**t, beta**( t + 1 ) ) and so
* their difference is beta. Adding 0.25 to c is to ensure that it
* is truncated to beta and not ( beta - 1 ).
*
QTR = ONE / 4
SAVEC = C
C = DLAMC3( C, -A )
LBETA = C + QTR
*
* Now determine whether rounding or chopping occurs, by adding a
* bit less than beta/2 and a bit more than beta/2 to a.
*
B = LBETA
F = DLAMC3( B/2, -B/100 )
C = DLAMC3( F, A )
IF( C.EQ.A ) THEN
LRND = .TRUE.
ELSE
LRND = .FALSE.
END IF
F = DLAMC3( B/2, B/100 )
C = DLAMC3( F, A )
IF( ( LRND ) .AND. ( C.EQ.A ) )
$ LRND = .FALSE.
*
* Try and decide whether rounding is done in the IEEE 'round to
* nearest' style. B/2 is half a unit in the last place of the two
* numbers A and SAVEC. Furthermore, A is even, i.e. has last bit
* zero, and SAVEC is odd. Thus adding B/2 to A should not change
* A, but adding B/2 to SAVEC should change SAVEC.
*
T1 = DLAMC3( B/2, A )
T2 = DLAMC3( B/2, SAVEC )
LIEEE1 = ( T1.EQ.A ) .AND. ( T2.GT.SAVEC ) .AND. LRND
*
* Now find the mantissa, t. It should be the integer part of
* log to the base beta of a, however it is safer to determine t
* by powering. So we find t as the smallest positive integer for
* which
*
* fl( beta**t + 1.0 ) = 1.0.
*
LT = 0
A = 1
C = 1
*
*+ WHILE( C.EQ.ONE )LOOP
30 CONTINUE
IF( C.EQ.ONE ) THEN
LT = LT + 1
A = A*LBETA
C = DLAMC3( A, ONE )
C = DLAMC3( C, -A )
GO TO 30
END IF
*+ END WHILE
*
END IF
*
BETA = LBETA
T = LT
RND = LRND
IEEE1 = LIEEE1
RETURN
*
* End of DLAMC1
*
END
*
************************************************************************
*
SUBROUTINE DLAMC2( BETA, T, RND, EPS, EMIN, RMIN, EMAX, RMAX )
*
* DLAMC2 returns the machine parameters given by:
*
* BETA - INTEGER.
* The base of the machine.
*
* T - INTEGER.
* The number of ( BETA ) digits in the mantissa.
*
* RND - LOGICAL.
* Whether proper rounding ( RND = .TRUE. ) or chopping
* ( RND = .FALSE. ) occurs in addition. This may not be a
* reliable guide to the way in which the machine perfoms
* its arithmetic.
*
* EPS - DOUBLE PRECISION.
* The smallest positive number such that
*
* fl( 1.0 - EPS ) .LT. 1.0,
*
* where fl denotes the computed value.
*
* EMIN - INTEGER.
* The minimum exponent before (gradual) underflow occurs.
*
* RMIN - DOUBLE PRECISION.
* The smallest normalized number for the machine given by
* BASE**( EMIN - 1 ), where BASE is the floating point
* value of BETA.
*
* EMAX - INTEGER.
* The maximum exponent before overflow occurs.
*
* RMAX - DOUBLE PRECISION.
* The largest positive number for the machine given by
* BASE**EMAX * ( 1 - EPS ), where BASE is the floating
* point value of BETA.
*
*
* The computation of EPS is based on a routine, PARANOIA by
* W. Kahan of the University of California at Berkeley.
*
*
* .. Scalar Arguments ..
LOGICAL RND
INTEGER BETA, EMAX, EMIN, T
DOUBLE PRECISION EPS, RMAX, RMIN
* ..
* .. Local Scalars ..
LOGICAL FIRST, IEEE, IWARN, LIEEE1, LRND
INTEGER GNMIN, GPMIN, I, LBETA, LEMAX, LEMIN, LT,
$ NGNMIN, NGPMIN
DOUBLE PRECISION A, B, C, HALF, LEPS, LRMAX, LRMIN, ONE, RBASE,
$ SIXTH, SMALL, THIRD, TWO, ZERO
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3
EXTERNAL DLAMC3
* ..
* .. External Subroutines ..
EXTERNAL DLAMC1, DLAMC4, DLAMC5
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Save statement ..
SAVE FIRST, IWARN, LBETA, LEMAX, LEMIN, LEPS, LRMAX,
$ LRMIN, LT
* ..
* .. Data statements ..
DATA FIRST / .TRUE. / , IWARN / .FALSE. /
* ..
* .. Executable Statements ..
*
IF( FIRST ) THEN
FIRST = .FALSE.
ZERO = 0
ONE = 1
TWO = 2
*
* LBETA, LT, LRND, LEPS, LEMIN and LRMIN are the local values of
* BETA, T, RND, EPS, EMIN and RMIN.
*
* Throughout this routine we use the function DLAMC3 to ensure
* that relevant values are stored and not held in registers, or
* are not affected by optimizers.
*
* DLAMC1 returns the parameters LBETA, LT, LRND and LIEEE1.
*
CALL DLAMC1( LBETA, LT, LRND, LIEEE1 )
*
* Start to find EPS.
*
B = LBETA
A = B**( -LT )
LEPS = A
*
* Try some tricks to see whether or not this is the correct EPS.
*
B = TWO / 3
HALF = ONE / 2
SIXTH = DLAMC3( B, -HALF )
THIRD = DLAMC3( SIXTH, SIXTH )
B = DLAMC3( THIRD, -HALF )
B = DLAMC3( B, SIXTH )
B = ABS( B )
IF( B.LT.LEPS )
$ B = LEPS
*
LEPS = 1
*
*+ WHILE( ( LEPS.GT.B ).AND.( B.GT.ZERO ) )LOOP
10 CONTINUE
IF( ( LEPS.GT.B ) .AND. ( B.GT.ZERO ) ) THEN
LEPS = B
C = DLAMC3( HALF*LEPS, ( TWO**5 )*( LEPS**2 ) )
C = DLAMC3( HALF, -C )
B = DLAMC3( HALF, C )
C = DLAMC3( HALF, -B )
B = DLAMC3( HALF, C )
GO TO 10
END IF
*+ END WHILE
*
IF( A.LT.LEPS )
$ LEPS = A
*
* Computation of EPS complete.
*
* Now find EMIN. Let A = + or - 1, and + or - (1 + BASE**(-3)).
* Keep dividing A by BETA until (gradual) underflow occurs. This
* is detected when we cannot recover the previous A.
*
RBASE = ONE / LBETA
SMALL = ONE
DO 20 I = 1, 3
SMALL = DLAMC3( SMALL*RBASE, ZERO )
20 CONTINUE
A = DLAMC3( ONE, SMALL )
CALL DLAMC4( NGPMIN, ONE, LBETA )
CALL DLAMC4( NGNMIN, -ONE, LBETA )
CALL DLAMC4( GPMIN, A, LBETA )
CALL DLAMC4( GNMIN, -A, LBETA )
IEEE = .FALSE.
*
IF( ( NGPMIN.EQ.NGNMIN ) .AND. ( GPMIN.EQ.GNMIN ) ) THEN
IF( NGPMIN.EQ.GPMIN ) THEN
LEMIN = NGPMIN
* ( Non twos-complement machines, no gradual underflow;
* e.g., VAX )
ELSE IF( ( GPMIN-NGPMIN ).EQ.3 ) THEN
LEMIN = NGPMIN - 1 + LT
IEEE = .TRUE.
* ( Non twos-complement machines, with gradual underflow;
* e.g., IEEE standard followers )
ELSE
LEMIN = MIN( NGPMIN, GPMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
*
ELSE IF( ( NGPMIN.EQ.GPMIN ) .AND. ( NGNMIN.EQ.GNMIN ) ) THEN
IF( ABS( NGPMIN-NGNMIN ).EQ.1 ) THEN
LEMIN = MAX( NGPMIN, NGNMIN )
* ( Twos-complement machines, no gradual underflow;
* e.g., CYBER 205 )
ELSE
LEMIN = MIN( NGPMIN, NGNMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
*
ELSE IF( ( ABS( NGPMIN-NGNMIN ).EQ.1 ) .AND.
$ ( GPMIN.EQ.GNMIN ) ) THEN
IF( ( GPMIN-MIN( NGPMIN, NGNMIN ) ).EQ.3 ) THEN
LEMIN = MAX( NGPMIN, NGNMIN ) - 1 + LT
* ( Twos-complement machines with gradual underflow;
* no known machine )
ELSE
LEMIN = MIN( NGPMIN, NGNMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
*
ELSE
LEMIN = MIN( NGPMIN, NGNMIN, GPMIN, GNMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
***
* Comment out this if block if EMIN is ok
IF( IWARN ) THEN
FIRST = .TRUE.
WRITE( 6, FMT = 9999 )LEMIN
END IF
***
*
* Assume IEEE arithmetic if we found denormalised numbers above,
* or if arithmetic seems to round in the IEEE style, determined
* in routine DLAMC1. A true IEEE machine should have both things
* true; however, faulty machines may have one or the other.
*
IEEE = IEEE .OR. LIEEE1
*
* Compute RMIN by successive division by BETA. We could compute
* RMIN as BASE**( EMIN - 1 ), but some machines underflow during
* this computation.
*
LRMIN = 1
DO 30 I = 1, 1 - LEMIN
LRMIN = DLAMC3( LRMIN*RBASE, ZERO )
30 CONTINUE
*
* Finally, call DLAMC5 to compute EMAX and RMAX.
*
CALL DLAMC5( LBETA, LT, LEMIN, IEEE, LEMAX, LRMAX )
END IF
*
BETA = LBETA
T = LT
RND = LRND
EPS = LEPS
EMIN = LEMIN
RMIN = LRMIN
EMAX = LEMAX
RMAX = LRMAX
*
RETURN
*
9999 FORMAT( //' WARNING. The value EMIN may be incorrect:-',
$ ' EMIN = ', I8, /
$ ' If, after inspection, the value EMIN looks',
$ ' acceptable please comment out ',
$ /' the IF block as marked within the code of routine',
$ ' DLAMC2,', /' otherwise supply EMIN explicitly.', / )
*
* End of DLAMC2
*
END
*
************************************************************************
*
DOUBLE PRECISION FUNCTION DLAMC3( A, B )
* .. Scalar Arguments ..
DOUBLE PRECISION A, B
* ..
* .. Executable Statements ..
*
* DLAMC3 is intended to force A and B to be stored prior to doing
* the addition of A and B. For use in situations where optimizers
* might hold one of these in a register.
*
*
DLAMC3 = A + B
*
RETURN
*
* End of DLAMC3
*
END
*
************************************************************************
*
SUBROUTINE DLAMC4( EMIN, START, BASE )
*
* Service routine for DLAMC2.
*
*
* .. Scalar Arguments ..
INTEGER BASE, EMIN
DOUBLE PRECISION START
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION A, B1, B2, C1, C2, D1, D2, ONE, RBASE, ZERO
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3
EXTERNAL DLAMC3
* ..
* .. Executable Statements ..
*
A = START
ONE = 1
RBASE = ONE / BASE
ZERO = 0
EMIN = 1
B1 = DLAMC3( A*RBASE, ZERO )
C1 = A
C2 = A
D1 = A
D2 = A
*+ WHILE( ( C1.EQ.A ).AND.( C2.EQ.A ).AND.
* $ ( D1.EQ.A ).AND.( D2.EQ.A ) )LOOP
10 CONTINUE
IF( ( C1.EQ.A ) .AND. ( C2.EQ.A ) .AND. ( D1.EQ.A ) .AND.
$ ( D2.EQ.A ) ) THEN
EMIN = EMIN - 1
A = B1
B1 = DLAMC3( A/BASE, ZERO )
C1 = DLAMC3( B1*BASE, ZERO )
D1 = ZERO
DO 20 I = 1, BASE
D1 = D1 + B1
20 CONTINUE
B2 = DLAMC3( A*RBASE, ZERO )
C2 = DLAMC3( B2/RBASE, ZERO )
D2 = ZERO
DO 30 I = 1, BASE
D2 = D2 + B2
30 CONTINUE
GO TO 10
END IF
*+ END WHILE
*
RETURN
*
* End of DLAMC4
*
END
*
************************************************************************
*
SUBROUTINE DLAMC5( BETA, P, EMIN, IEEE, EMAX, RMAX )
*
* Given BETA, the base of floating-point arithmetic, P, the
* number of base BETA digits in the mantissa of a floating-point
* value, EMIN, the minimum exponent, and IEEE, a logical
* flag saying whether or not the arithmetic system is thought
* to comply with the IEEE standard, this routine attempts to
* compute RMAX, the largest machine floating-point number,
* without overflow. The routine assumes that EMAX + abs(EMIN)
* sum approximately to a power of 2. It will fail on machines
* where this assumption does not hold, for example the Cyber 205
* (EMIN = -28625, EMAX = 28718). It will also fail if the value
* supplied for EMIN is too large (i.e. too close to zero),
* probably with overflow.
*
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Scalar Arguments ..
LOGICAL IEEE
INTEGER BETA, EMAX, EMIN, P
DOUBLE PRECISION RMAX
* ..
* .. Local Scalars ..
INTEGER EXBITS, EXPSUM, I, LEXP, NBITS, TRY, UEXP
DOUBLE PRECISION OLDY, RECBAS, Y, Z
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3
EXTERNAL DLAMC3
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
* .. Executable Statements ..
*
* First compute LEXP and UEXP, two powers of 2 that bound
* abs(EMIN). We then assume that EMAX + abs(EMIN) will sum
* approximately to the bound that is closest to abs(EMIN).
* (EMAX is the exponent of the required number RMAX).
*
LEXP = 1
EXBITS = 1
10 CONTINUE
TRY = LEXP*2
IF( TRY.LE.( -EMIN ) ) THEN
LEXP = TRY
EXBITS = EXBITS + 1
GO TO 10
END IF
IF( LEXP.EQ.-EMIN ) THEN
UEXP = LEXP
ELSE
UEXP = TRY
EXBITS = EXBITS + 1
END IF
*
* Now -LEXP is less than or equal to EMIN, and -UEXP is greater
* than or equal to EMIN. EXBITS is the number of bits needed to
* store the exponent.
*
IF( ( UEXP+EMIN ).GT.( -LEXP-EMIN ) ) THEN
EXPSUM = 2*LEXP
ELSE
EXPSUM = 2*UEXP
END IF
*
* EXPSUM is the exponent range, approximately equal to
* EMAX - EMIN + 1 .
*
EMAX = EXPSUM + EMIN - 1
NBITS = 1 + EXBITS + P
*
* NBITS is the total number of bits needed to store a
* floating-point number.
*
IF( ( MOD( NBITS, 2 ).EQ.1 ) .AND. ( BETA.EQ.2 ) ) THEN
*
* Either there are an odd number of bits used to store a
* floating-point number, which is unlikely, or some bits are
* not used in the representation of numbers, which is possible,
* (e.g. Cray machines) or the mantissa has an implicit bit,
* (e.g. IEEE machines, Dec Vax machines), which is perhaps the
* most likely. We have to assume the last alternative.
* If this is true, then we need to reduce EMAX by one because
* there must be some way of representing zero in an implicit-bit
* system. On machines like Cray, we are reducing EMAX by one
* unnecessarily.
*
EMAX = EMAX - 1
END IF
*
IF( IEEE ) THEN
*
* Assume we are on an IEEE machine which reserves one exponent
* for infinity and NaN.
*
EMAX = EMAX - 1
END IF
*
* Now create RMAX, the largest machine number, which should
* be equal to (1.0 - BETA**(-P)) * BETA**EMAX .
*
* First compute 1.0 - BETA**(-P), being careful that the
* result is less than 1.0 .
*
RECBAS = ONE / BETA
Z = ONE
Y = ZERO
DO 20 I = 1, P
Z = Z*RECBAS
IF( Y.LT.ONE )
$ OLDY = Y
Y = DLAMC3( Y, Z )
20 CONTINUE
IF( Y.GE.ONE )
$ Y = OLDY
*
* Now multiply by BETA**EMAX to get RMAX.
*
DO 30 I = 1, EMAX
Y = DLAMC3( Y*BETA, ZERO )
30 CONTINUE
*
RMAX = Y
RETURN
*
* End of DLAMC5
*
END