version 1.5, 2010/08/07 13:22:21
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version 1.16, 2017/06/17 10:53:57
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SUBROUTINE DLASQ1( N, D, E, WORK, INFO ) |
*> \brief \b DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr. |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download DLASQ1 + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq1.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/dlasq1.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/dlasq1.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 DLASQ1( N, D, E, WORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER INFO, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION D( * ), E( * ), WORK( * ) |
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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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*> DLASQ1 computes the singular values of a real N-by-N bidiagonal |
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*> matrix with diagonal D and off-diagonal E. The singular values |
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*> are computed to high relative accuracy, in the absence of |
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*> denormalization, underflow and overflow. The algorithm was first |
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*> presented in |
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*> |
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*> "Accurate singular values and differential qd algorithms" by K. V. |
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*> Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, |
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*> 1994, |
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*> |
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*> and the present implementation is described in "An implementation of |
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*> the dqds Algorithm (Positive Case)", LAPACK Working Note. |
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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 number of rows and columns in the matrix. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in,out] D |
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*> \verbatim |
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*> D is DOUBLE PRECISION array, dimension (N) |
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*> On entry, D contains the diagonal elements of the |
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*> bidiagonal matrix whose SVD is desired. On normal exit, |
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*> D contains the singular values in decreasing order. |
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*> \endverbatim |
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*> |
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*> \param[in,out] E |
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*> \verbatim |
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*> E is DOUBLE PRECISION array, dimension (N) |
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*> On entry, elements E(1:N-1) contain the off-diagonal elements |
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*> of the bidiagonal matrix whose SVD is desired. |
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*> On exit, E is overwritten. |
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*> \endverbatim |
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*> |
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*> \param[out] WORK |
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*> \verbatim |
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*> WORK is DOUBLE PRECISION array, dimension (4*N) |
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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: the algorithm failed |
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*> = 1, a split was marked by a positive value in E |
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*> = 2, current block of Z not diagonalized after 100*N |
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*> iterations (in inner while loop) On exit D and E |
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*> represent a matrix with the same singular values |
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*> which the calling subroutine could use to finish the |
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*> computation, or even feed back into DLASQ1 |
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*> = 3, termination criterion of outer while loop not met |
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*> (program created more than N unreduced blocks) |
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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. |
* |
* |
* -- LAPACK routine (version 3.2) -- |
*> \date December 2016 |
* |
* |
* -- Contributed by Osni Marques of the Lawrence Berkeley National -- |
*> \ingroup auxOTHERcomputational |
* -- Laboratory and Beresford Parlett of the Univ. of California at -- |
* |
* -- Berkeley -- |
* ===================================================================== |
* -- November 2008 -- |
SUBROUTINE DLASQ1( N, D, E, WORK, INFO ) |
* |
* |
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* -- LAPACK computational routine (version 3.7.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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* December 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER INFO, N |
INTEGER INFO, N |
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DOUBLE PRECISION D( * ), E( * ), WORK( * ) |
DOUBLE PRECISION D( * ), E( * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DLASQ1 computes the singular values of a real N-by-N bidiagonal |
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* matrix with diagonal D and off-diagonal E. The singular values |
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* are computed to high relative accuracy, in the absence of |
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* denormalization, underflow and overflow. The algorithm was first |
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* presented in |
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* |
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* "Accurate singular values and differential qd algorithms" by K. V. |
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* Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, |
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* 1994, |
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* |
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* and the present implementation is described in "An implementation of |
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* the dqds Algorithm (Positive Case)", LAPACK Working Note. |
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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 number of rows and columns in the matrix. N >= 0. |
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* |
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* D (input/output) DOUBLE PRECISION array, dimension (N) |
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* On entry, D contains the diagonal elements of the |
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* bidiagonal matrix whose SVD is desired. On normal exit, |
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* D contains the singular values in decreasing order. |
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* |
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* E (input/output) DOUBLE PRECISION array, dimension (N) |
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* On entry, elements E(1:N-1) contain the off-diagonal elements |
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* of the bidiagonal matrix whose SVD is desired. |
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* On exit, E is overwritten. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension (4*N) |
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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: the algorithm failed |
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* = 1, a split was marked by a positive value in E |
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* = 2, current block of Z not diagonalized after 30*N |
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* iterations (in inner while loop) |
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* = 3, termination criterion of outer while loop not met |
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* (program created more than N unreduced blocks) |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* |
* |
INFO = 0 |
INFO = 0 |
IF( N.LT.0 ) THEN |
IF( N.LT.0 ) THEN |
INFO = -2 |
INFO = -1 |
CALL XERBLA( 'DLASQ1', -INFO ) |
CALL XERBLA( 'DLASQ1', -INFO ) |
RETURN |
RETURN |
ELSE IF( N.EQ.0 ) THEN |
ELSE IF( N.EQ.0 ) THEN |
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CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 ) |
CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 ) |
CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1, |
CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1, |
$ IINFO ) |
$ IINFO ) |
* |
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* Compute the q's and e's. |
* Compute the q's and e's. |
* |
* |
DO 30 I = 1, 2*N - 1 |
DO 30 I = 1, 2*N - 1 |
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D( I ) = SQRT( WORK( I ) ) |
D( I ) = SQRT( WORK( I ) ) |
40 CONTINUE |
40 CONTINUE |
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO ) |
CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO ) |
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ELSE IF( INFO.EQ.2 ) THEN |
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* |
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* Maximum number of iterations exceeded. Move data from WORK |
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* into D and E so the calling subroutine can try to finish |
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* |
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DO I = 1, N |
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D( I ) = SQRT( WORK( 2*I-1 ) ) |
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E( I ) = SQRT( WORK( 2*I ) ) |
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END DO |
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CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO ) |
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CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, E, N, IINFO ) |
END IF |
END IF |
* |
* |
RETURN |
RETURN |