version 1.3, 2010/08/06 15:28:36
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version 1.14, 2015/11/26 11:44:15
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*> \brief \b DGEHRD |
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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 DGEHRD + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgehrd.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/dgehrd.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/dgehrd.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 DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER IHI, ILO, INFO, LDA, LWORK, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), TAU( * ), 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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*> DGEHRD reduces a real general matrix A to upper Hessenberg form H by |
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*> an orthogonal similarity transformation: Q**T * A * Q = H . |
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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] ILO |
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*> \verbatim |
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*> ILO is INTEGER |
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*> \endverbatim |
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*> |
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*> \param[in] IHI |
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*> \verbatim |
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*> IHI is INTEGER |
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*> |
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*> It is assumed that A is already upper triangular in rows |
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*> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally |
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*> set by a previous call to DGEBAL; otherwise they should be |
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*> set to 1 and N respectively. See Further Details. |
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*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if 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, dimension (LDA,N) |
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*> On entry, the N-by-N general matrix to be reduced. |
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*> On exit, the upper triangle and the first subdiagonal of A |
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*> are overwritten with the upper Hessenberg matrix H, and the |
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*> elements below the first subdiagonal, with the array TAU, |
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*> represent the orthogonal matrix Q as a product of elementary |
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*> reflectors. See Further Details. |
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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] TAU |
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*> \verbatim |
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*> TAU is DOUBLE PRECISION array, dimension (N-1) |
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*> The scalar factors of the elementary reflectors (see Further |
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*> Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to |
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*> zero. |
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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 (LWORK) |
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*> On exit, if INFO = 0, WORK(1) returns the optimal 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 length of the array WORK. LWORK >= max(1,N). |
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*> For good performance, LWORK should generally be larger. |
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*> |
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*> If LWORK = -1, then a workspace query is assumed; the routine |
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*> only calculates the optimal size of the WORK array, returns |
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*> this value as the first entry of the WORK array, and no error |
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*> message related to LWORK is issued by XERBLA. |
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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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*> \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 2015 |
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* |
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*> \ingroup doubleGEcomputational |
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* |
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*> \par Further Details: |
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* ===================== |
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*> |
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*> \verbatim |
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*> |
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*> The matrix Q is represented as a product of (ihi-ilo) elementary |
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*> reflectors |
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*> |
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*> Q = H(ilo) H(ilo+1) . . . H(ihi-1). |
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*> |
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*> Each H(i) has the form |
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*> |
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*> H(i) = I - tau * v * v**T |
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*> |
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*> where tau is a real scalar, and v is a real vector with |
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*> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on |
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*> exit in A(i+2:ihi,i), and tau in TAU(i). |
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*> |
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*> The contents of A are illustrated by the following example, with |
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*> n = 7, ilo = 2 and ihi = 6: |
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*> |
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*> on entry, on exit, |
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*> |
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*> ( a a a a a a a ) ( a a h h h h a ) |
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*> ( a a a a a a ) ( a h h h h a ) |
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*> ( a a a a a a ) ( h h h h h h ) |
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*> ( a a a a a a ) ( v2 h h h h h ) |
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*> ( a a a a a a ) ( v2 v3 h h h h ) |
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*> ( a a a a a a ) ( v2 v3 v4 h h h ) |
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*> ( a ) ( a ) |
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*> |
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*> where a denotes an element of the original matrix A, h denotes a |
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*> modified element of the upper Hessenberg matrix H, and vi denotes an |
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*> element of the vector defining H(i). |
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*> |
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*> This file is a slight modification of LAPACK-3.0's DGEHRD |
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*> subroutine incorporating improvements proposed by Quintana-Orti and |
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*> Van de Geijn (2006). (See DLAHR2.) |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) |
SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2.1) -- |
* -- LAPACK computational routine (version 3.6.0) -- |
* -- 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..-- |
* -- April 2009 -- |
* November 2015 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER IHI, ILO, INFO, LDA, LWORK, N |
INTEGER IHI, ILO, INFO, LDA, LWORK, N |
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DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) |
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DGEHRD reduces a real general matrix A to upper Hessenberg form H by |
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* an orthogonal similarity transformation: Q' * A * Q = H . |
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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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* ILO (input) INTEGER |
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* IHI (input) INTEGER |
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* It is assumed that A is already upper triangular in rows |
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* and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally |
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* set by a previous call to DGEBAL; otherwise they should be |
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* set to 1 and N respectively. See Further Details. |
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* 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. |
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* |
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* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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* On entry, the N-by-N general matrix to be reduced. |
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* On exit, the upper triangle and the first subdiagonal of A |
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* are overwritten with the upper Hessenberg matrix H, and the |
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* elements below the first subdiagonal, with the array TAU, |
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* represent the orthogonal matrix Q as a product of elementary |
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* reflectors. See Further Details. |
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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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* TAU (output) DOUBLE PRECISION array, dimension (N-1) |
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* The scalar factors of the elementary reflectors (see Further |
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* Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to |
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* zero. |
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* |
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* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) |
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* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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* |
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* LWORK (input) INTEGER |
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* The length of the array WORK. LWORK >= max(1,N). |
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* For optimum performance LWORK >= N*NB, where NB is the |
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* optimal blocksize. |
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* |
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* If LWORK = -1, then a workspace query is assumed; the routine |
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* only calculates the optimal size of the WORK array, returns |
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* this value as the first entry of the WORK array, and no error |
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* message related to LWORK is issued by XERBLA. |
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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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* |
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* Further Details |
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* =============== |
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* |
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* The matrix Q is represented as a product of (ihi-ilo) elementary |
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* reflectors |
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* |
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* Q = H(ilo) H(ilo+1) . . . H(ihi-1). |
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* |
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* Each H(i) has the form |
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* |
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* H(i) = I - tau * v * v' |
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* |
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* where tau is a real scalar, and v is a real vector with |
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* v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on |
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* exit in A(i+2:ihi,i), and tau in TAU(i). |
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* |
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* The contents of A are illustrated by the following example, with |
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* n = 7, ilo = 2 and ihi = 6: |
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* |
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* on entry, on exit, |
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* |
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* ( a a a a a a a ) ( a a h h h h a ) |
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* ( a a a a a a ) ( a h h h h a ) |
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* ( a a a a a a ) ( h h h h h h ) |
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* ( a a a a a a ) ( v2 h h h h h ) |
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* ( a a a a a a ) ( v2 v3 h h h h ) |
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* ( a a a a a a ) ( v2 v3 v4 h h h ) |
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* ( a ) ( a ) |
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* |
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* where a denotes an element of the original matrix A, h denotes a |
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* modified element of the upper Hessenberg matrix H, and vi denotes an |
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* element of the vector defining H(i). |
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* |
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* This file is a slight modification of LAPACK-3.0's DGEHRD |
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* subroutine incorporating improvements proposed by Quintana-Orti and |
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* Van de Geijn (2006). (See DLAHR2.) |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
INTEGER NBMAX, LDT |
INTEGER NBMAX, LDT, TSIZE |
PARAMETER ( NBMAX = 64, LDT = NBMAX+1 ) |
PARAMETER ( NBMAX = 64, LDT = NBMAX+1, |
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$ TSIZE = LDT*NBMAX ) |
DOUBLE PRECISION ZERO, ONE |
DOUBLE PRECISION ZERO, ONE |
PARAMETER ( ZERO = 0.0D+0, |
PARAMETER ( ZERO = 0.0D+0, |
$ ONE = 1.0D+0 ) |
$ ONE = 1.0D+0 ) |
* .. |
* .. |
* .. Local Scalars .. |
* .. Local Scalars .. |
LOGICAL LQUERY |
LOGICAL LQUERY |
INTEGER I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB, |
INTEGER I, IB, IINFO, IWT, J, LDWORK, LWKOPT, NB, |
$ NBMIN, NH, NX |
$ NBMIN, NH, NX |
DOUBLE PRECISION EI |
DOUBLE PRECISION EI |
* .. |
* .. |
* .. Local Arrays .. |
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DOUBLE PRECISION T( LDT, NBMAX ) |
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* .. |
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* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL DAXPY, DGEHD2, DGEMM, DLAHR2, DLARFB, DTRMM, |
EXTERNAL DAXPY, DGEHD2, DGEMM, DLAHR2, DLARFB, DTRMM, |
$ XERBLA |
$ XERBLA |
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* Test the input parameters |
* Test the input parameters |
* |
* |
INFO = 0 |
INFO = 0 |
NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) ) |
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LWKOPT = N*NB |
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WORK( 1 ) = LWKOPT |
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LQUERY = ( LWORK.EQ.-1 ) |
LQUERY = ( LWORK.EQ.-1 ) |
IF( N.LT.0 ) THEN |
IF( N.LT.0 ) THEN |
INFO = -1 |
INFO = -1 |
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ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN |
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN |
INFO = -8 |
INFO = -8 |
END IF |
END IF |
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* |
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IF( INFO.EQ.0 ) THEN |
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* |
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* Compute the workspace requirements |
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* |
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NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) ) |
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LWKOPT = N*NB + TSIZE |
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WORK( 1 ) = LWKOPT |
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END IF |
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* |
IF( INFO.NE.0 ) THEN |
IF( INFO.NE.0 ) THEN |
CALL XERBLA( 'DGEHRD', -INFO ) |
CALL XERBLA( 'DGEHRD', -INFO ) |
RETURN |
RETURN |
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* |
* |
NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) ) |
NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) ) |
NBMIN = 2 |
NBMIN = 2 |
IWS = 1 |
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IF( NB.GT.1 .AND. NB.LT.NH ) THEN |
IF( NB.GT.1 .AND. NB.LT.NH ) THEN |
* |
* |
* Determine when to cross over from blocked to unblocked code |
* Determine when to cross over from blocked to unblocked code |
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* |
* |
* Determine if workspace is large enough for blocked code |
* Determine if workspace is large enough for blocked code |
* |
* |
IWS = N*NB |
IF( LWORK.LT.N*NB+TSIZE ) THEN |
IF( LWORK.LT.IWS ) THEN |
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* |
* |
* Not enough workspace to use optimal NB: determine the |
* Not enough workspace to use optimal NB: determine the |
* minimum value of NB, and reduce NB or force use of |
* minimum value of NB, and reduce NB or force use of |
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* |
* |
NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI, |
NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI, |
$ -1 ) ) |
$ -1 ) ) |
IF( LWORK.GE.N*NBMIN ) THEN |
IF( LWORK.GE.(N*NBMIN + TSIZE) ) THEN |
NB = LWORK / N |
NB = (LWORK-TSIZE) / N |
ELSE |
ELSE |
NB = 1 |
NB = 1 |
END IF |
END IF |
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* |
* |
* Use blocked code |
* Use blocked code |
* |
* |
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IWT = 1 + N*NB |
DO 40 I = ILO, IHI - 1 - NX, NB |
DO 40 I = ILO, IHI - 1 - NX, NB |
IB = MIN( NB, IHI-I ) |
IB = MIN( NB, IHI-I ) |
* |
* |
* Reduce columns i:i+ib-1 to Hessenberg form, returning the |
* Reduce columns i:i+ib-1 to Hessenberg form, returning the |
* matrices V and T of the block reflector H = I - V*T*V' |
* matrices V and T of the block reflector H = I - V*T*V**T |
* which performs the reduction, and also the matrix Y = A*V*T |
* which performs the reduction, and also the matrix Y = A*V*T |
* |
* |
CALL DLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT, |
CALL DLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), |
$ WORK, LDWORK ) |
$ WORK( IWT ), LDT, WORK, LDWORK ) |
* |
* |
* Apply the block reflector H to A(1:ihi,i+ib:ihi) from the |
* Apply the block reflector H to A(1:ihi,i+ib:ihi) from the |
* right, computing A := A - Y * V'. V(i+ib,ib-1) must be set |
* right, computing A := A - Y * V**T. V(i+ib,ib-1) must be set |
* to 1 |
* to 1 |
* |
* |
EI = A( I+IB, I+IB-1 ) |
EI = A( I+IB, I+IB-1 ) |
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* |
* |
CALL DLARFB( 'Left', 'Transpose', 'Forward', |
CALL DLARFB( 'Left', 'Transpose', 'Forward', |
$ 'Columnwise', |
$ 'Columnwise', |
$ IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT, |
$ IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, |
$ A( I+1, I+IB ), LDA, WORK, LDWORK ) |
$ WORK( IWT ), LDT, A( I+1, I+IB ), LDA, |
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$ WORK, LDWORK ) |
40 CONTINUE |
40 CONTINUE |
END IF |
END IF |
* |
* |
* Use unblocked code to reduce the rest of the matrix |
* Use unblocked code to reduce the rest of the matrix |
* |
* |
CALL DGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO ) |
CALL DGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO ) |
WORK( 1 ) = IWS |
WORK( 1 ) = LWKOPT |
* |
* |
RETURN |
RETURN |
* |
* |