version 1.1, 2010/01/26 15:22:45
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version 1.16, 2017/06/17 10:54:08
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*> \brief \b ZGEBD2 reduces a general matrix to bidiagonal 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 ZGEBD2 + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgebd2.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/zgebd2.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/zgebd2.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 ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER INFO, LDA, M, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION D( * ), E( * ) |
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* COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), 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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*> ZGEBD2 reduces a complex general m by n matrix A to upper or lower |
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*> real bidiagonal form B by a unitary transformation: Q**H * A * P = B. |
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*> |
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*> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. |
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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] M |
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*> \verbatim |
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*> M is INTEGER |
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*> The number of rows in the matrix A. M >= 0. |
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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 number of columns in the matrix A. 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 COMPLEX*16 array, dimension (LDA,N) |
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*> On entry, the m by n general matrix to be reduced. |
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*> On exit, |
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*> if m >= n, the diagonal and the first superdiagonal are |
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*> overwritten with the upper bidiagonal matrix B; the |
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*> elements below the diagonal, with the array TAUQ, represent |
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*> the unitary matrix Q as a product of elementary |
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*> reflectors, and the elements above the first superdiagonal, |
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*> with the array TAUP, represent the unitary matrix P as |
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*> a product of elementary reflectors; |
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*> if m < n, the diagonal and the first subdiagonal are |
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*> overwritten with the lower bidiagonal matrix B; the |
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*> elements below the first subdiagonal, with the array TAUQ, |
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*> represent the unitary matrix Q as a product of |
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*> elementary reflectors, and the elements above the diagonal, |
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*> with the array TAUP, represent the unitary matrix P as |
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*> a product of elementary reflectors. |
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*> 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,M). |
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*> \endverbatim |
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*> |
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*> \param[out] D |
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*> \verbatim |
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*> D is DOUBLE PRECISION array, dimension (min(M,N)) |
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*> The diagonal elements of the bidiagonal matrix B: |
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*> D(i) = A(i,i). |
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*> \endverbatim |
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*> |
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*> \param[out] E |
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*> \verbatim |
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*> E is DOUBLE PRECISION array, dimension (min(M,N)-1) |
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*> The off-diagonal elements of the bidiagonal matrix B: |
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*> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; |
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*> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. |
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*> \endverbatim |
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*> |
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*> \param[out] TAUQ |
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*> \verbatim |
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*> TAUQ is COMPLEX*16 array dimension (min(M,N)) |
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*> The scalar factors of the elementary reflectors which |
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*> represent the unitary matrix Q. See Further Details. |
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*> \endverbatim |
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*> |
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*> \param[out] TAUP |
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*> \verbatim |
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*> TAUP is COMPLEX*16 array, dimension (min(M,N)) |
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*> The scalar factors of the elementary reflectors which |
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*> represent the unitary matrix P. See Further 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 (max(M,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 December 2016 |
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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 matrices Q and P are represented as products of elementary |
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*> reflectors: |
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*> |
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*> If m >= n, |
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*> |
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*> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) |
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*> |
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*> Each H(i) and G(i) has the form: |
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*> |
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*> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H |
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*> |
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*> where tauq and taup are complex scalars, and v and u are complex |
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*> vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in |
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*> A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in |
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*> A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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*> |
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*> If m < n, |
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*> |
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*> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) |
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*> |
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*> Each H(i) and G(i) has the form: |
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*> |
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*> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H |
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*> |
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*> where tauq and taup are complex scalars, v and u are complex vectors; |
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*> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); |
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*> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); |
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*> tauq is stored in TAUQ(i) and taup in TAUP(i). |
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*> |
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*> The contents of A on exit are illustrated by the following examples: |
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*> |
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*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): |
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*> |
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*> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) |
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*> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) |
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*> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) |
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*> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) |
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*> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) |
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*> ( v1 v2 v3 v4 v5 ) |
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*> |
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*> where d and e denote diagonal and off-diagonal elements of B, vi |
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*> denotes an element of the vector defining H(i), and ui an element of |
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*> the vector defining G(i). |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) |
SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.7.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
* December 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER INFO, LDA, M, N |
INTEGER INFO, LDA, M, N |
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COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) |
COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZGEBD2 reduces a complex general m by n matrix A to upper or lower |
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* real bidiagonal form B by a unitary transformation: Q' * A * P = B. |
|
* |
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* If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. |
|
* |
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* Arguments |
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* ========= |
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* |
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* M (input) INTEGER |
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* The number of rows in the matrix A. M >= 0. |
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* |
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* N (input) INTEGER |
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* The number of columns in the matrix A. N >= 0. |
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* |
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* A (input/output) COMPLEX*16 array, dimension (LDA,N) |
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* On entry, the m by n general matrix to be reduced. |
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* On exit, |
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* if m >= n, the diagonal and the first superdiagonal are |
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* overwritten with the upper bidiagonal matrix B; the |
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* elements below the diagonal, with the array TAUQ, represent |
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* the unitary matrix Q as a product of elementary |
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* reflectors, and the elements above the first superdiagonal, |
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* with the array TAUP, represent the unitary matrix P as |
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* a product of elementary reflectors; |
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* if m < n, the diagonal and the first subdiagonal are |
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* overwritten with the lower bidiagonal matrix B; the |
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* elements below the first subdiagonal, with the array TAUQ, |
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* represent the unitary matrix Q as a product of |
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* elementary reflectors, and the elements above the diagonal, |
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* with the array TAUP, represent the unitary matrix P as |
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* a product of elementary reflectors. |
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* 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,M). |
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* |
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* D (output) DOUBLE PRECISION array, dimension (min(M,N)) |
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* The diagonal elements of the bidiagonal matrix B: |
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* D(i) = A(i,i). |
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* |
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* E (output) DOUBLE PRECISION array, dimension (min(M,N)-1) |
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* The off-diagonal elements of the bidiagonal matrix B: |
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* if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; |
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* if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. |
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* |
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* TAUQ (output) COMPLEX*16 array dimension (min(M,N)) |
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* The scalar factors of the elementary reflectors which |
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* represent the unitary matrix Q. See Further Details. |
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* |
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* TAUP (output) COMPLEX*16 array, dimension (min(M,N)) |
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* The scalar factors of the elementary reflectors which |
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* represent the unitary matrix P. See Further Details. |
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* |
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* WORK (workspace) COMPLEX*16 array, dimension (max(M,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 matrices Q and P are represented as products of elementary |
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* reflectors: |
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* |
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* If m >= n, |
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* |
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* Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) |
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* |
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* Each H(i) and G(i) has the form: |
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* |
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* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
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* |
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* where tauq and taup are complex scalars, and v and u are complex |
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* vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in |
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* A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in |
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* A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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* |
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* If m < n, |
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* |
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* Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) |
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* |
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* Each H(i) and G(i) has the form: |
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* |
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* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
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* |
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* where tauq and taup are complex scalars, v and u are complex vectors; |
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* v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); |
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* u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); |
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* tauq is stored in TAUQ(i) and taup in TAUP(i). |
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* |
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* The contents of A on exit are illustrated by the following examples: |
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* |
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* m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): |
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* |
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* ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) |
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* ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) |
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* ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) |
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* ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) |
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* ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) |
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* ( v1 v2 v3 v4 v5 ) |
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* |
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* where d and e denote diagonal and off-diagonal elements of B, vi |
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* denotes an element of the vector defining H(i), and ui an element of |
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* the vector defining G(i). |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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D( I ) = ALPHA |
D( I ) = ALPHA |
A( I, I ) = ONE |
A( I, I ) = ONE |
* |
* |
* Apply H(i)' to A(i:m,i+1:n) from the left |
* Apply H(i)**H to A(i:m,i+1:n) from the left |
* |
* |
IF( I.LT.N ) |
IF( I.LT.N ) |
$ CALL ZLARF( 'Left', M-I+1, N-I, A( I, I ), 1, |
$ CALL ZLARF( 'Left', M-I+1, N-I, A( I, I ), 1, |
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E( I ) = ALPHA |
E( I ) = ALPHA |
A( I+1, I ) = ONE |
A( I+1, I ) = ONE |
* |
* |
* Apply H(i)' to A(i+1:m,i+1:n) from the left |
* Apply H(i)**H to A(i+1:m,i+1:n) from the left |
* |
* |
CALL ZLARF( 'Left', M-I, N-I, A( I+1, I ), 1, |
CALL ZLARF( 'Left', M-I, N-I, A( I+1, I ), 1, |
$ DCONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA, |
$ DCONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA, |