version 1.5, 2010/08/07 13:22:32
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version 1.14, 2016/08/27 15:34:48
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*> \brief \b ZGGQRF |
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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 ZGGQRF + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggqrf.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/zggqrf.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/zggqrf.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 ZGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, |
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* LWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER INFO, LDA, LDB, LWORK, M, N, P |
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* .. |
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* .. Array Arguments .. |
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* COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), |
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* $ 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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*> ZGGQRF computes a generalized QR factorization of an N-by-M matrix A |
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*> and an N-by-P matrix B: |
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*> |
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*> A = Q*R, B = Q*T*Z, |
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*> |
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*> where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, |
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*> and R and T assume one of the forms: |
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*> |
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*> if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, |
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*> ( 0 ) N-M N M-N |
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*> M |
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*> |
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*> where R11 is upper triangular, and |
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*> |
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*> if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, |
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*> P-N N ( T21 ) P |
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*> P |
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*> |
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*> where T12 or T21 is upper triangular. |
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*> |
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*> In particular, if B is square and nonsingular, the GQR factorization |
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*> of A and B implicitly gives the QR factorization of inv(B)*A: |
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*> |
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*> inv(B)*A = Z**H * (inv(T)*R) |
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*> |
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*> where inv(B) denotes the inverse of the matrix B, and Z**H denotes the |
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*> conjugate transpose of matrix Z. |
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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 number of rows of the matrices A and B. N >= 0. |
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*> \endverbatim |
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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 columns of the matrix A. M >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] P |
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*> \verbatim |
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*> P is INTEGER |
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*> The number of columns of the matrix B. P >= 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,M) |
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*> On entry, the N-by-M matrix A. |
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*> On exit, the elements on and above the diagonal of the array |
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*> contain the min(N,M)-by-M upper trapezoidal matrix R (R is |
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*> upper triangular if N >= M); the elements below the diagonal, |
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*> with the array TAUA, represent the unitary matrix Q as a |
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*> product of min(N,M) elementary reflectors (see Further |
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*> 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] TAUA |
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*> \verbatim |
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*> TAUA is COMPLEX*16 array, dimension (min(N,M)) |
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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[in,out] B |
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*> \verbatim |
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*> B is COMPLEX*16 array, dimension (LDB,P) |
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*> On entry, the N-by-P matrix B. |
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*> On exit, if N <= P, the upper triangle of the subarray |
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*> B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; |
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*> if N > P, the elements on and above the (N-P)-th subdiagonal |
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*> contain the N-by-P upper trapezoidal matrix T; the remaining |
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*> elements, with the array TAUB, represent the unitary |
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*> matrix Z as a product of elementary reflectors (see Further |
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*> Details). |
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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[out] TAUB |
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*> \verbatim |
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*> TAUB is COMPLEX*16 array, dimension (min(N,P)) |
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*> The scalar factors of the elementary reflectors which |
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*> represent the unitary matrix Z (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(1,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 dimension of the array WORK. LWORK >= max(1,N,M,P). |
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*> For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), |
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*> where NB1 is the optimal blocksize for the QR factorization |
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*> of an N-by-M matrix, NB2 is the optimal blocksize for the |
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*> RQ factorization of an N-by-P matrix, and NB3 is the optimal |
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*> blocksize for a call of ZUNMQR. |
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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 2011 |
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* |
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*> \ingroup complex16OTHERcomputational |
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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 elementary reflectors |
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*> |
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*> Q = H(1) H(2) . . . H(k), where k = min(n,m). |
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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 - taua * v * v**H |
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*> |
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*> where taua is a complex scalar, and v is a complex vector with |
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*> v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), |
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*> and taua in TAUA(i). |
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*> To form Q explicitly, use LAPACK subroutine ZUNGQR. |
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*> To use Q to update another matrix, use LAPACK subroutine ZUNMQR. |
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*> |
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*> The matrix Z is represented as a product of elementary reflectors |
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*> |
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*> Z = H(1) H(2) . . . H(k), where k = min(n,p). |
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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 - taub * v * v**H |
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*> |
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*> where taub is a complex scalar, and v is a complex vector with |
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*> v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in |
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*> B(n-k+i,1:p-k+i-1), and taub in TAUB(i). |
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*> To form Z explicitly, use LAPACK subroutine ZUNGRQ. |
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*> To use Z to update another matrix, use LAPACK subroutine ZUNMRQ. |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, |
SUBROUTINE ZGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, |
$ LWORK, INFO ) |
$ LWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational 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 .. |
INTEGER INFO, LDA, LDB, LWORK, M, N, P |
INTEGER INFO, LDA, LDB, LWORK, M, N, P |
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$ WORK( * ) |
$ WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZGGQRF computes a generalized QR factorization of an N-by-M matrix A |
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* and an N-by-P matrix B: |
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* |
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* A = Q*R, B = Q*T*Z, |
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* |
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* where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, |
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* and R and T assume one of the forms: |
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* |
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* if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, |
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* ( 0 ) N-M N M-N |
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* M |
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* |
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* where R11 is upper triangular, and |
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* |
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* if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, |
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* P-N N ( T21 ) P |
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* P |
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* |
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* where T12 or T21 is upper triangular. |
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* |
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* In particular, if B is square and nonsingular, the GQR factorization |
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* of A and B implicitly gives the QR factorization of inv(B)*A: |
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* |
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* inv(B)*A = Z'*(inv(T)*R) |
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* |
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* where inv(B) denotes the inverse of the matrix B, and Z' denotes the |
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* conjugate transpose of matrix Z. |
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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 number of rows of the matrices A and B. N >= 0. |
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* |
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* M (input) INTEGER |
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* The number of columns of the matrix A. M >= 0. |
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* |
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* P (input) INTEGER |
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* The number of columns of the matrix B. P >= 0. |
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* |
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* A (input/output) COMPLEX*16 array, dimension (LDA,M) |
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* On entry, the N-by-M matrix A. |
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* On exit, the elements on and above the diagonal of the array |
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* contain the min(N,M)-by-M upper trapezoidal matrix R (R is |
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* upper triangular if N >= M); the elements below the diagonal, |
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* with the array TAUA, represent the unitary matrix Q as a |
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* product of min(N,M) elementary reflectors (see Further |
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* 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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* TAUA (output) COMPLEX*16 array, dimension (min(N,M)) |
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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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* B (input/output) COMPLEX*16 array, dimension (LDB,P) |
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* On entry, the N-by-P matrix B. |
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* On exit, if N <= P, the upper triangle of the subarray |
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* B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; |
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* if N > P, the elements on and above the (N-P)-th subdiagonal |
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* contain the N-by-P upper trapezoidal matrix T; the remaining |
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* elements, with the array TAUB, represent the unitary |
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* matrix Z as a product of elementary reflectors (see Further |
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* Details). |
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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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* TAUB (output) COMPLEX*16 array, dimension (min(N,P)) |
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* The scalar factors of the elementary reflectors which |
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* represent the unitary matrix Z (see Further Details). |
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* |
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* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,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 dimension of the array WORK. LWORK >= max(1,N,M,P). |
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* For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), |
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* where NB1 is the optimal blocksize for the QR factorization |
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* of an N-by-M matrix, NB2 is the optimal blocksize for the |
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* RQ factorization of an N-by-P matrix, and NB3 is the optimal |
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* blocksize for a call of ZUNMQR. |
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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 elementary reflectors |
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* |
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* Q = H(1) H(2) . . . H(k), where k = min(n,m). |
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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 - taua * v * v' |
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* |
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* where taua is a complex scalar, and v is a complex vector with |
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* v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), |
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* and taua in TAUA(i). |
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* To form Q explicitly, use LAPACK subroutine ZUNGQR. |
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* To use Q to update another matrix, use LAPACK subroutine ZUNMQR. |
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* |
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* The matrix Z is represented as a product of elementary reflectors |
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* |
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* Z = H(1) H(2) . . . H(k), where k = min(n,p). |
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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 - taub * v * v' |
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* |
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* where taub is a complex scalar, and v is a complex vector with |
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* v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in |
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* B(n-k+i,1:p-k+i-1), and taub in TAUB(i). |
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* To form Z explicitly, use LAPACK subroutine ZUNGRQ. |
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* To use Z to update another matrix, use LAPACK subroutine ZUNMRQ. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Local Scalars .. |
* .. Local Scalars .. |
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CALL ZGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO ) |
CALL ZGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO ) |
LOPT = WORK( 1 ) |
LOPT = WORK( 1 ) |
* |
* |
* Update B := Q'*B. |
* Update B := Q**H*B. |
* |
* |
CALL ZUNMQR( 'Left', 'Conjugate Transpose', N, P, MIN( N, M ), A, |
CALL ZUNMQR( 'Left', 'Conjugate Transpose', N, P, MIN( N, M ), A, |
$ LDA, TAUA, B, LDB, WORK, LWORK, INFO ) |
$ LDA, TAUA, B, LDB, WORK, LWORK, INFO ) |