version 1.3, 2010/08/06 15:28:51
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version 1.12, 2012/12/14 12:30:28
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*> \brief \b ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. |
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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 ZGEHD2 + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgehd2.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/zgehd2.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/zgehd2.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 ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER IHI, ILO, INFO, LDA, N |
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* .. |
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* .. Array Arguments .. |
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* COMPLEX*16 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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*> ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H |
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*> by a unitary similarity transformation: Q**H * 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 ZGEBAL; otherwise they should be |
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*> set to 1 and N respectively. See Further Details. |
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*> 1 <= ILO <= IHI <= max(1,N). |
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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 COMPLEX*16 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 unitary 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 COMPLEX*16 array, dimension (N-1) |
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*> The scalar factors of the elementary reflectors (see Further |
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*> Details). |
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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 COMPLEX*16 array, dimension (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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*> \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 September 2012 |
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* |
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*> \ingroup complex16GEcomputational |
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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**H |
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*> |
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*> where tau is a complex scalar, and v is a complex 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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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO ) |
SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.4.2) -- |
* -- 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 |
* September 2012 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER IHI, ILO, INFO, LDA, N |
INTEGER IHI, ILO, INFO, LDA, N |
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COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) |
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H |
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* by a unitary 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 ZGEBAL; otherwise they should be |
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* set to 1 and N respectively. See Further Details. |
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* 1 <= ILO <= IHI <= max(1,N). |
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* |
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* A (input/output) COMPLEX*16 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 unitary 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) COMPLEX*16 array, dimension (N-1) |
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* The scalar factors of the elementary reflectors (see Further |
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* Details). |
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* |
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* WORK (workspace) COMPLEX*16 array, dimension (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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* |
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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 complex scalar, and v is a complex 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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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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CALL ZLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ), |
CALL ZLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ), |
$ A( 1, I+1 ), LDA, WORK ) |
$ A( 1, I+1 ), LDA, WORK ) |
* |
* |
* Apply H(i)' to A(i+1:ihi,i+1:n) from the left |
* Apply H(i)**H to A(i+1:ihi,i+1:n) from the left |
* |
* |
CALL ZLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, |
CALL ZLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, |
$ DCONJG( TAU( I ) ), A( I+1, I+1 ), LDA, WORK ) |
$ DCONJG( TAU( I ) ), A( I+1, I+1 ), LDA, WORK ) |