version 1.3, 2010/08/06 15:28:49
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version 1.10, 2011/11/21 20:43:06
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*> \brief \b DTGEX2 |
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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 DTGEX2 + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgex2.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/dtgex2.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/dtgex2.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 DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, |
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* LDZ, J1, N1, N2, WORK, LWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* LOGICAL WANTQ, WANTZ |
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* INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2 |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), |
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* $ WORK( * ), Z( LDZ, * ) |
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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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*> DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) |
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*> of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair |
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*> (A, B) by an orthogonal equivalence transformation. |
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*> |
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*> (A, B) must be in generalized real Schur canonical form (as returned |
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*> by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 |
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*> diagonal blocks. B is upper triangular. |
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*> |
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*> Optionally, the matrices Q and Z of generalized Schur vectors are |
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*> updated. |
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*> |
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*> Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T |
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*> Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T |
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*> |
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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] WANTQ |
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*> \verbatim |
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*> WANTQ is LOGICAL |
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*> .TRUE. : update the left transformation matrix Q; |
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*> .FALSE.: do not update Q. |
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*> \endverbatim |
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*> |
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*> \param[in] WANTZ |
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*> \verbatim |
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*> WANTZ is LOGICAL |
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*> .TRUE. : update the right transformation matrix Z; |
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*> .FALSE.: do not update Z. |
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*> \endverbatim |
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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 matrices A and B. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in,out] A |
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*> \verbatim |
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*> A is DOUBLE PRECISION array, dimensions (LDA,N) |
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*> On entry, the matrix A in the pair (A, B). |
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*> On exit, the updated matrix A. |
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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[in,out] B |
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*> \verbatim |
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*> B is DOUBLE PRECISION array, dimensions (LDB,N) |
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*> On entry, the matrix B in the pair (A, B). |
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*> On exit, the updated matrix B. |
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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 >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in,out] Q |
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*> \verbatim |
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*> Q is DOUBLE PRECISION array, dimension (LDQ,N) |
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*> On entry, if WANTQ = .TRUE., the orthogonal matrix Q. |
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*> On exit, the updated matrix Q. |
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*> Not referenced if WANTQ = .FALSE.. |
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*> \endverbatim |
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*> |
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*> \param[in] LDQ |
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*> \verbatim |
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*> LDQ is INTEGER |
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*> The leading dimension of the array Q. LDQ >= 1. |
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*> If WANTQ = .TRUE., LDQ >= N. |
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*> \endverbatim |
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*> |
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*> \param[in,out] Z |
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*> \verbatim |
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*> Z is DOUBLE PRECISION array, dimension (LDZ,N) |
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*> On entry, if WANTZ =.TRUE., the orthogonal matrix Z. |
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*> On exit, the updated matrix Z. |
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*> Not referenced if WANTZ = .FALSE.. |
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*> \endverbatim |
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*> |
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*> \param[in] LDZ |
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*> \verbatim |
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*> LDZ is INTEGER |
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*> The leading dimension of the array Z. LDZ >= 1. |
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*> If WANTZ = .TRUE., LDZ >= N. |
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*> \endverbatim |
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*> |
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*> \param[in] J1 |
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*> \verbatim |
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*> J1 is INTEGER |
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*> The index to the first block (A11, B11). 1 <= J1 <= N. |
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*> \endverbatim |
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*> |
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*> \param[in] N1 |
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*> \verbatim |
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*> N1 is INTEGER |
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*> The order of the first block (A11, B11). N1 = 0, 1 or 2. |
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*> \endverbatim |
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*> |
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*> \param[in] N2 |
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*> \verbatim |
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*> N2 is INTEGER |
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*> The order of the second block (A22, B22). N2 = 0, 1 or 2. |
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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 (MAX(1,LWORK)). |
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*> \endverbatim |
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*> |
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*> \param[in] LWORK |
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*> \verbatim |
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*> LWORK is INTEGER |
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*> The dimension of the array WORK. |
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*> LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 ) |
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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 = 1, the transformed matrix (A, B) would be |
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*> too far from generalized Schur form; the blocks are |
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*> not swapped and (A, B) and (Q, Z) are unchanged. |
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*> The problem of swapping is too ill-conditioned. |
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*> <0: If INFO = -16: LWORK is too small. Appropriate value |
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*> for LWORK is returned in WORK(1). |
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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 November 2011 |
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* |
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*> \ingroup doubleGEauxiliary |
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* |
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*> \par Further Details: |
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* ===================== |
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*> |
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*> In the current code both weak and strong stability tests are |
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*> performed. The user can omit the strong stability test by changing |
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*> the internal logical parameter WANDS to .FALSE.. See ref. [2] for |
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*> details. |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> Bo Kagstrom and Peter Poromaa, Department of Computing Science, |
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*> Umea University, S-901 87 Umea, Sweden. |
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* |
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*> \par References: |
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* ================ |
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*> |
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*> \verbatim |
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*> |
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*> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the |
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*> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in |
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*> M.S. Moonen et al (eds), Linear Algebra for Large Scale and |
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*> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. |
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*> |
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*> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified |
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*> Eigenvalues of a Regular Matrix Pair (A, B) and Condition |
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*> Estimation: Theory, Algorithms and Software, |
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*> Report UMINF - 94.04, Department of Computing Science, Umea |
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*> University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working |
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*> Note 87. To appear in Numerical Algorithms, 1996. |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, |
SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, |
$ LDZ, J1, N1, N2, WORK, LWORK, INFO ) |
$ LDZ, J1, N1, N2, WORK, LWORK, INFO ) |
* |
* |
* -- LAPACK auxiliary routine (version 3.2) -- |
* -- LAPACK auxiliary routine (version 3.4.0) -- |
* -- 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..-- |
* November 2006 |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
LOGICAL WANTQ, WANTZ |
LOGICAL WANTQ, WANTZ |
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$ WORK( * ), Z( LDZ, * ) |
$ WORK( * ), Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) |
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* of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair |
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* (A, B) by an orthogonal equivalence transformation. |
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* |
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* (A, B) must be in generalized real Schur canonical form (as returned |
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* by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 |
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* diagonal blocks. B is upper triangular. |
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* |
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* Optionally, the matrices Q and Z of generalized Schur vectors are |
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* updated. |
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* |
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* Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)' |
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* Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)' |
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* |
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* |
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* Arguments |
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* ========= |
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* |
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* WANTQ (input) LOGICAL |
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* .TRUE. : update the left transformation matrix Q; |
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* .FALSE.: do not update Q. |
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* |
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* WANTZ (input) LOGICAL |
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* .TRUE. : update the right transformation matrix Z; |
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* .FALSE.: do not update Z. |
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* |
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* N (input) INTEGER |
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* The order of the matrices A and B. N >= 0. |
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* |
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* A (input/output) DOUBLE PRECISION arrays, dimensions (LDA,N) |
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* On entry, the matrix A in the pair (A, B). |
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* On exit, the updated matrix A. |
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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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* B (input/output) DOUBLE PRECISION arrays, dimensions (LDB,N) |
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* On entry, the matrix B in the pair (A, B). |
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* On exit, the updated matrix B. |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of the array B. LDB >= max(1,N). |
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* |
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* Q (input/output) DOUBLE PRECISION array, dimension (LDZ,N) |
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* On entry, if WANTQ = .TRUE., the orthogonal matrix Q. |
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* On exit, the updated matrix Q. |
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* Not referenced if WANTQ = .FALSE.. |
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* |
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* LDQ (input) INTEGER |
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* The leading dimension of the array Q. LDQ >= 1. |
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* If WANTQ = .TRUE., LDQ >= N. |
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* |
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* Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N) |
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* On entry, if WANTZ =.TRUE., the orthogonal matrix Z. |
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* On exit, the updated matrix Z. |
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* Not referenced if WANTZ = .FALSE.. |
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* |
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* LDZ (input) INTEGER |
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* The leading dimension of the array Z. LDZ >= 1. |
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* If WANTZ = .TRUE., LDZ >= N. |
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* |
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* J1 (input) INTEGER |
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* The index to the first block (A11, B11). 1 <= J1 <= N. |
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* |
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* N1 (input) INTEGER |
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* The order of the first block (A11, B11). N1 = 0, 1 or 2. |
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* |
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* N2 (input) INTEGER |
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* The order of the second block (A22, B22). N2 = 0, 1 or 2. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)). |
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* |
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* LWORK (input) INTEGER |
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* The dimension of the array WORK. |
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* LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 ) |
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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 = 1, the transformed matrix (A, B) would be |
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* too far from generalized Schur form; the blocks are |
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* not swapped and (A, B) and (Q, Z) are unchanged. |
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* The problem of swapping is too ill-conditioned. |
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* <0: If INFO = -16: LWORK is too small. Appropriate value |
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* for LWORK is returned in WORK(1). |
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* |
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* Further Details |
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* =============== |
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* |
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* Based on contributions by |
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* Bo Kagstrom and Peter Poromaa, Department of Computing Science, |
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* Umea University, S-901 87 Umea, Sweden. |
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* |
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* In the current code both weak and strong stability tests are |
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* performed. The user can omit the strong stability test by changing |
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* the internal logical parameter WANDS to .FALSE.. See ref. [2] for |
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* details. |
|
* |
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* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the |
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* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in |
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* M.S. Moonen et al (eds), Linear Algebra for Large Scale and |
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* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. |
|
* |
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* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified |
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* Eigenvalues of a Regular Matrix Pair (A, B) and Condition |
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* Estimation: Theory, Algorithms and Software, |
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* Report UMINF - 94.04, Department of Computing Science, Umea |
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* University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working |
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* Note 87. To appear in Numerical Algorithms, 1996. |
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* |
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* ===================================================================== |
* ===================================================================== |
* Replaced various illegal calls to DCOPY by calls to DLASET, or by DO |
* Replaced various illegal calls to DCOPY by calls to DLASET, or by DO |
* loops. Sven Hammarling, 1/5/02. |
* loops. Sven Hammarling, 1/5/02. |
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* .. Parameters .. |
* .. Parameters .. |
DOUBLE PRECISION ZERO, ONE |
DOUBLE PRECISION ZERO, ONE |
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) |
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) |
DOUBLE PRECISION TEN |
DOUBLE PRECISION TWENTY |
PARAMETER ( TEN = 1.0D+01 ) |
PARAMETER ( TWENTY = 2.0D+01 ) |
INTEGER LDST |
INTEGER LDST |
PARAMETER ( LDST = 4 ) |
PARAMETER ( LDST = 4 ) |
LOGICAL WANDS |
LOGICAL WANDS |
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CALL DLACPY( 'Full', M, M, T, LDST, WORK, M ) |
CALL DLACPY( 'Full', M, M, T, LDST, WORK, M ) |
CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM ) |
CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM ) |
DNORM = DSCALE*SQRT( DSUM ) |
DNORM = DSCALE*SQRT( DSUM ) |
THRESH = MAX( TEN*EPS*DNORM, SMLNUM ) |
* |
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* THRES has been changed from |
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* THRESH = MAX( TEN*EPS*SA, SMLNUM ) |
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* to |
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* THRESH = MAX( TWENTY*EPS*SA, SMLNUM ) |
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* on 04/01/10. |
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* "Bug" reported by Ondra Kamenik, confirmed by Julie Langou, fixed by |
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* Jim Demmel and Guillaume Revy. See forum post 1783. |
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* |
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THRESH = MAX( TWENTY*EPS*DNORM, SMLNUM ) |
* |
* |
IF( M.EQ.2 ) THEN |
IF( M.EQ.2 ) THEN |
* |
* |
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IF( WANDS ) THEN |
IF( WANDS ) THEN |
* |
* |
* Strong stability test: |
* Strong stability test: |
* F-norm((A-QL'*S*QR, B-QL'*T*QR)) <= O(EPS*F-norm((A,B))) |
* F-norm((A-QL**T*S*QR, B-QL**T*T*QR)) <= O(EPS*F-norm((A,B))) |
* |
* |
CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ), |
CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ), |
$ M ) |
$ M ) |
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* |
* |
* Compute orthogonal matrix QL: |
* Compute orthogonal matrix QL: |
* |
* |
* QL' * LI = [ TL ] |
* QL**T * LI = [ TL ] |
* [ 0 ] |
* [ 0 ] |
* where |
* where |
* LI = [ -L ] |
* LI = [ -L ] |
* [ SCALE * identity(N2) ] |
* [ SCALE * identity(N2) ] |
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* |
* |
* Compute orthogonal matrix RQ: |
* Compute orthogonal matrix RQ: |
* |
* |
* IR * RQ' = [ 0 TR], |
* IR * RQ**T = [ 0 TR], |
* |
* |
* where IR = [ SCALE * identity(N1), R ] |
* where IR = [ SCALE * identity(N1), R ] |
* |
* |
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IF( WANDS ) THEN |
IF( WANDS ) THEN |
* |
* |
* Strong stability test: |
* Strong stability test: |
* F-norm((A-QL*S*QR', B-QL*T*QR')) <= O(EPS*F-norm((A,B))) |
* F-norm((A-QL*S*QR**T, B-QL*T*QR**T)) <= O(EPS*F-norm((A,B))) |
* |
* |
CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ), |
CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ), |
$ M ) |
$ M ) |