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version 1.19, 2023/08/07 08:38:54
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*> \brief \b DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow. |
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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 DLAG2 + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlag2.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/dlag2.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/dlag2.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 DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, |
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* WR2, WI ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER LDA, LDB |
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* DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2 |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), B( LDB, * ) |
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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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*> DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue |
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*> problem A - w B, with scaling as necessary to avoid over-/underflow. |
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*> |
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*> The scaling factor "s" results in a modified eigenvalue equation |
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*> |
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*> s A - w B |
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*> |
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*> where s is a non-negative scaling factor chosen so that w, w B, |
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*> and s A do not overflow and, if possible, do not underflow, either. |
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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] A |
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*> \verbatim |
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*> A is DOUBLE PRECISION array, dimension (LDA, 2) |
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*> On entry, the 2 x 2 matrix A. It is assumed that its 1-norm |
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*> is less than 1/SAFMIN. Entries less than |
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*> sqrt(SAFMIN)*norm(A) are subject to being treated as zero. |
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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 >= 2. |
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*> \endverbatim |
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*> |
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*> \param[in] B |
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*> \verbatim |
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*> B is DOUBLE PRECISION array, dimension (LDB, 2) |
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*> On entry, the 2 x 2 upper triangular matrix B. It is |
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*> assumed that the one-norm of B is less than 1/SAFMIN. The |
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*> diagonals should be at least sqrt(SAFMIN) times the largest |
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*> element of B (in absolute value); if a diagonal is smaller |
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*> than that, then +/- sqrt(SAFMIN) will be used instead of |
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*> that diagonal. |
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*> \endverbatim |
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*> |
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*> \param[in] LDB |
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*> \verbatim |
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*> LDB is INTEGER |
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*> The leading dimension of the array B. LDB >= 2. |
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*> \endverbatim |
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*> |
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*> \param[in] SAFMIN |
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*> \verbatim |
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*> SAFMIN is DOUBLE PRECISION |
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*> The smallest positive number s.t. 1/SAFMIN does not |
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*> overflow. (This should always be DLAMCH('S') -- it is an |
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*> argument in order to avoid having to call DLAMCH frequently.) |
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*> \endverbatim |
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*> |
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*> \param[out] SCALE1 |
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*> \verbatim |
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*> SCALE1 is DOUBLE PRECISION |
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*> A scaling factor used to avoid over-/underflow in the |
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*> eigenvalue equation which defines the first eigenvalue. If |
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*> the eigenvalues are complex, then the eigenvalues are |
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*> ( WR1 +/- WI i ) / SCALE1 (which may lie outside the |
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*> exponent range of the machine), SCALE1=SCALE2, and SCALE1 |
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*> will always be positive. If the eigenvalues are real, then |
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*> the first (real) eigenvalue is WR1 / SCALE1 , but this may |
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*> overflow or underflow, and in fact, SCALE1 may be zero or |
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*> less than the underflow threshold if the exact eigenvalue |
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*> is sufficiently large. |
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*> \endverbatim |
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*> |
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*> \param[out] SCALE2 |
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*> \verbatim |
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*> SCALE2 is DOUBLE PRECISION |
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*> A scaling factor used to avoid over-/underflow in the |
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*> eigenvalue equation which defines the second eigenvalue. If |
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*> the eigenvalues are complex, then SCALE2=SCALE1. If the |
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*> eigenvalues are real, then the second (real) eigenvalue is |
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*> WR2 / SCALE2 , but this may overflow or underflow, and in |
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*> fact, SCALE2 may be zero or less than the underflow |
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*> threshold if the exact eigenvalue is sufficiently large. |
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*> \endverbatim |
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*> |
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*> \param[out] WR1 |
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*> \verbatim |
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*> WR1 is DOUBLE PRECISION |
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*> If the eigenvalue is real, then WR1 is SCALE1 times the |
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*> eigenvalue closest to the (2,2) element of A B**(-1). If the |
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*> eigenvalue is complex, then WR1=WR2 is SCALE1 times the real |
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*> part of the eigenvalues. |
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*> \endverbatim |
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*> |
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*> \param[out] WR2 |
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*> \verbatim |
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*> WR2 is DOUBLE PRECISION |
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*> If the eigenvalue is real, then WR2 is SCALE2 times the |
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*> other eigenvalue. If the eigenvalue is complex, then |
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*> WR1=WR2 is SCALE1 times the real part of the eigenvalues. |
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*> \endverbatim |
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*> |
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*> \param[out] WI |
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*> \verbatim |
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*> WI is DOUBLE PRECISION |
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*> If the eigenvalue is real, then WI is zero. If the |
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*> eigenvalue is complex, then WI is SCALE1 times the imaginary |
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*> part of the eigenvalues. WI will always be non-negative. |
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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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*> \ingroup doubleOTHERauxiliary |
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* |
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* ===================================================================== |
SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, |
SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, |
$ WR2, WI ) |
$ WR2, WI ) |
* |
* |
* -- LAPACK auxiliary routine (version 3.2) -- |
* -- LAPACK auxiliary routine -- |
* -- 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..-- |
* November 2006 |
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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER LDA, LDB |
INTEGER LDA, LDB |
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DOUBLE PRECISION A( LDA, * ), B( LDB, * ) |
DOUBLE PRECISION A( LDA, * ), B( LDB, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue |
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* problem A - w B, with scaling as necessary to avoid over-/underflow. |
|
* |
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* The scaling factor "s" results in a modified eigenvalue equation |
|
* |
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* s A - w B |
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* |
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* where s is a non-negative scaling factor chosen so that w, w B, |
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* and s A do not overflow and, if possible, do not underflow, either. |
|
* |
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* Arguments |
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* ========= |
|
* |
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* A (input) DOUBLE PRECISION array, dimension (LDA, 2) |
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* On entry, the 2 x 2 matrix A. It is assumed that its 1-norm |
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* is less than 1/SAFMIN. Entries less than |
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* sqrt(SAFMIN)*norm(A) are subject to being treated as zero. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= 2. |
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* |
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* B (input) DOUBLE PRECISION array, dimension (LDB, 2) |
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* On entry, the 2 x 2 upper triangular matrix B. It is |
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* assumed that the one-norm of B is less than 1/SAFMIN. The |
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* diagonals should be at least sqrt(SAFMIN) times the largest |
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* element of B (in absolute value); if a diagonal is smaller |
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* than that, then +/- sqrt(SAFMIN) will be used instead of |
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* that diagonal. |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of the array B. LDB >= 2. |
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* |
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* SAFMIN (input) DOUBLE PRECISION |
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* The smallest positive number s.t. 1/SAFMIN does not |
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* overflow. (This should always be DLAMCH('S') -- it is an |
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* argument in order to avoid having to call DLAMCH frequently.) |
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* |
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* SCALE1 (output) DOUBLE PRECISION |
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* A scaling factor used to avoid over-/underflow in the |
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* eigenvalue equation which defines the first eigenvalue. If |
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* the eigenvalues are complex, then the eigenvalues are |
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* ( WR1 +/- WI i ) / SCALE1 (which may lie outside the |
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* exponent range of the machine), SCALE1=SCALE2, and SCALE1 |
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* will always be positive. If the eigenvalues are real, then |
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* the first (real) eigenvalue is WR1 / SCALE1 , but this may |
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* overflow or underflow, and in fact, SCALE1 may be zero or |
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* less than the underflow threshhold if the exact eigenvalue |
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* is sufficiently large. |
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* |
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* SCALE2 (output) DOUBLE PRECISION |
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* A scaling factor used to avoid over-/underflow in the |
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* eigenvalue equation which defines the second eigenvalue. If |
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* the eigenvalues are complex, then SCALE2=SCALE1. If the |
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* eigenvalues are real, then the second (real) eigenvalue is |
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* WR2 / SCALE2 , but this may overflow or underflow, and in |
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* fact, SCALE2 may be zero or less than the underflow |
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* threshhold if the exact eigenvalue is sufficiently large. |
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* |
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* WR1 (output) DOUBLE PRECISION |
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* If the eigenvalue is real, then WR1 is SCALE1 times the |
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* eigenvalue closest to the (2,2) element of A B**(-1). If the |
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* eigenvalue is complex, then WR1=WR2 is SCALE1 times the real |
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* part of the eigenvalues. |
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* |
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* WR2 (output) DOUBLE PRECISION |
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* If the eigenvalue is real, then WR2 is SCALE2 times the |
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* other eigenvalue. If the eigenvalue is complex, then |
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* WR1=WR2 is SCALE1 times the real part of the eigenvalues. |
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* |
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* WI (output) DOUBLE PRECISION |
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* If the eigenvalue is real, then WI is zero. If the |
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* eigenvalue is complex, then WI is SCALE1 times the imaginary |
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* part of the eigenvalues. WI will always be non-negative. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* Note: the test of R in the following IF is to cover the case when |
* Note: the test of R in the following IF is to cover the case when |
* DISCR is small and negative and is flushed to zero during |
* DISCR is small and negative and is flushed to zero during |
* the calculation of R. On machines which have a consistent |
* the calculation of R. On machines which have a consistent |
* flush-to-zero threshhold and handle numbers above that |
* flush-to-zero threshold and handle numbers above that |
* threshhold correctly, it would not be necessary. |
* threshold correctly, it would not be necessary. |
* |
* |
IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN |
IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN |
SUM = PP + SIGN( R, PP ) |
SUM = PP + SIGN( R, PP ) |