version 1.4, 2010/12/21 13:53:53
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version 1.14, 2018/05/29 07:18:32
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SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO ) |
*> \brief \b ZPOEQUB |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* =========== DOCUMENTATION =========== |
* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- |
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* -- Jason Riedy of Univ. of California Berkeley. -- |
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* -- November 2008 -- |
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* |
* |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* Online html documentation available at |
* -- Univ. of California Berkeley and NAG Ltd. -- |
* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download ZPOEQUB + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpoequb.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpoequb.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpoequb.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER INFO, LDA, N |
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* DOUBLE PRECISION AMAX, SCOND |
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* .. |
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* .. Array Arguments .. |
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* COMPLEX*16 A( LDA, * ) |
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* DOUBLE PRECISION S( * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> ZPOEQUB computes row and column scalings intended to equilibrate a |
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*> Hermitian positive definite matrix A and reduce its condition number |
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*> (with respect to the two-norm). S contains the scale factors, |
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*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with |
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*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This |
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*> choice of S puts the condition number of B within a factor N of the |
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*> smallest possible condition number over all possible diagonal |
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*> scalings. |
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*> |
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*> This routine differs from ZPOEQU by restricting the scaling factors |
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*> to a power of the radix. Barring over- and underflow, scaling by |
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*> these factors introduces no additional rounding errors. However, the |
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*> scaled diagonal entries are no longer approximately 1 but lie |
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*> between sqrt(radix) and 1/sqrt(radix). |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The order of the matrix A. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] A |
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*> \verbatim |
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*> A is COMPLEX*16 array, dimension (LDA,N) |
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*> The N-by-N Hermitian positive definite matrix whose scaling |
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*> factors are to be computed. Only the diagonal elements of A |
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*> are referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] S |
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*> \verbatim |
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*> S is DOUBLE PRECISION array, dimension (N) |
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*> If INFO = 0, S contains the scale factors for A. |
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*> \endverbatim |
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*> |
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*> \param[out] SCOND |
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*> \verbatim |
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*> SCOND is DOUBLE PRECISION |
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*> If INFO = 0, S contains the ratio of the smallest S(i) to |
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*> the largest S(i). If SCOND >= 0.1 and AMAX is neither too |
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*> large nor too small, it is not worth scaling by S. |
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*> \endverbatim |
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*> |
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*> \param[out] AMAX |
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*> \verbatim |
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*> AMAX is DOUBLE PRECISION |
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*> Absolute value of largest matrix element. If AMAX is very |
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*> close to overflow or very close to underflow, the matrix |
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*> should be scaled. |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: successful exit |
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*> < 0: if INFO = -i, the i-th argument had an illegal value |
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*> > 0: if INFO = i, the i-th diagonal element is nonpositive. |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \date December 2016 |
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* |
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*> \ingroup complex16POcomputational |
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* |
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* ===================================================================== |
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SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO ) |
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* |
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* -- LAPACK computational routine (version 3.7.0) -- |
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* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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* December 2016 |
* |
* |
IMPLICIT NONE |
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* .. |
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* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER INFO, LDA, N |
INTEGER INFO, LDA, N |
DOUBLE PRECISION AMAX, SCOND |
DOUBLE PRECISION AMAX, SCOND |
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DOUBLE PRECISION S( * ) |
DOUBLE PRECISION S( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZPOEQUB computes row and column scalings intended to equilibrate a |
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* symmetric positive definite matrix A and reduce its condition number |
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* (with respect to the two-norm). S contains the scale factors, |
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* S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with |
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* elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This |
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* choice of S puts the condition number of B within a factor N of the |
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* smallest possible condition number over all possible diagonal |
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* scalings. |
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* |
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* Arguments |
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* ========= |
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* |
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* N (input) INTEGER |
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* The order of the matrix A. N >= 0. |
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* |
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* A (input) COMPLEX*16 array, dimension (LDA,N) |
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* The N-by-N symmetric positive definite matrix whose scaling |
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* factors are to be computed. Only the diagonal elements of A |
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* are referenced. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max(1,N). |
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* |
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* S (output) DOUBLE PRECISION array, dimension (N) |
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* If INFO = 0, S contains the scale factors for A. |
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* |
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* SCOND (output) DOUBLE PRECISION |
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* If INFO = 0, S contains the ratio of the smallest S(i) to |
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* the largest S(i). If SCOND >= 0.1 and AMAX is neither too |
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* large nor too small, it is not worth scaling by S. |
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* |
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* AMAX (output) DOUBLE PRECISION |
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* Absolute value of largest matrix element. If AMAX is very |
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* close to overflow or very close to underflow, the matrix |
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* should be scaled. |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if INFO = -i, the i-th argument had an illegal value |
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* > 0: if INFO = i, the i-th diagonal element is nonpositive. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* .. Local Scalars .. |
* .. Local Scalars .. |
INTEGER I |
INTEGER I |
DOUBLE PRECISION SMIN, BASE, TMP |
DOUBLE PRECISION SMIN, BASE, TMP |
COMPLEX*16 ZDUM |
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* .. |
* .. |
* .. External Functions .. |
* .. External Functions .. |
DOUBLE PRECISION DLAMCH |
DOUBLE PRECISION DLAMCH |
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* .. Intrinsic Functions .. |
* .. Intrinsic Functions .. |
INTRINSIC MAX, MIN, SQRT, LOG, INT, REAL, DIMAG |
INTRINSIC MAX, MIN, SQRT, LOG, INT, REAL, DIMAG |
* .. |
* .. |
* .. Statement Functions .. |
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DOUBLE PRECISION CABS1 |
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* .. |
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* .. Statement Function Definitions .. |
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CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) |
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* .. |
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* .. Executable Statements .. |
* .. Executable Statements .. |
* |
* |
* Test the input parameters. |
* Test the input parameters. |