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version 1.11, 2012/08/22 09:48:31
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*> \brief \b ZGGRQF |
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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 ZGGRQF + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggrqf.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/zggrqf.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/zggrqf.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 ZGGRQF( M, P, N, 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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*> ZGGRQF computes a generalized RQ factorization of an M-by-N matrix A |
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*> and a P-by-N matrix B: |
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*> |
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*> A = R*Q, B = Z*T*Q, |
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*> |
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*> where Q is an N-by-N unitary matrix, Z is a P-by-P unitary |
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*> matrix, and R and T assume one of the forms: |
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*> |
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*> if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N, |
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*> N-M M ( R21 ) N |
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*> N |
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*> |
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*> where R12 or R21 is upper triangular, and |
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*> |
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*> if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P, |
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*> ( 0 ) P-N P N-P |
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*> N |
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*> |
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*> where T11 is upper triangular. |
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*> |
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*> In particular, if B is square and nonsingular, the GRQ factorization |
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*> of A and B implicitly gives the RQ factorization of A*inv(B): |
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*> |
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*> A*inv(B) = (R*inv(T))*Z**H |
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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 the 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] M |
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*> \verbatim |
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*> M is INTEGER |
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*> The number of rows 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 rows of the matrix B. P >= 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 of the matrices A and B. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in,out] A |
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*> \verbatim |
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*> A is COMPLEX*16 array, dimension (LDA,N) |
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*> On entry, the M-by-N matrix A. |
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*> On exit, if M <= N, the upper triangle of the subarray |
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*> A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; |
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*> if M > N, the elements on and above the (M-N)-th subdiagonal |
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*> contain the M-by-N upper trapezoidal matrix R; the remaining |
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*> elements, with the array TAUA, represent the unitary |
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*> matrix Q 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] 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] TAUA |
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*> \verbatim |
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*> TAUA 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[in,out] B |
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*> \verbatim |
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*> B is COMPLEX*16 array, dimension (LDB,N) |
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*> On entry, the P-by-N matrix B. |
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*> On exit, the elements on and above the diagonal of the array |
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*> contain the min(P,N)-by-N upper trapezoidal matrix T (T is |
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*> upper triangular if P >= N); the elements below the diagonal, |
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*> with the array TAUB, represent the unitary matrix Z as a |
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*> product of elementary reflectors (see Further 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,P). |
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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(P,N)) |
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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 RQ factorization |
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*> of an M-by-N matrix, NB2 is the optimal blocksize for the |
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*> QR factorization of a P-by-N matrix, and NB3 is the optimal |
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*> blocksize for a call of ZUNMRQ. |
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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(m,n). |
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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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in |
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*> A(m-k+i,1:n-k+i-1), and taua in TAUA(i). |
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*> To form Q explicitly, use LAPACK subroutine ZUNGRQ. |
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*> To use Q to update another matrix, use LAPACK subroutine ZUNMRQ. |
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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(p,n). |
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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(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i), |
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*> and taub in TAUB(i). |
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*> To form Z explicitly, use LAPACK subroutine ZUNGQR. |
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*> To use Z to update another matrix, use LAPACK subroutine ZUNMQR. |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, |
SUBROUTINE ZGGRQF( M, P, N, 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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* ZGGRQF computes a generalized RQ factorization of an M-by-N matrix A |
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* and a P-by-N matrix B: |
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* |
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* A = R*Q, B = Z*T*Q, |
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* |
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* where Q is an N-by-N unitary matrix, Z is a P-by-P unitary |
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* matrix, and R and T assume one of the forms: |
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* |
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* if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N, |
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* N-M M ( R21 ) N |
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* N |
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* |
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* where R12 or R21 is upper triangular, and |
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* |
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* if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P, |
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* ( 0 ) P-N P N-P |
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* N |
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* |
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* where T11 is upper triangular. |
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* |
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* In particular, if B is square and nonsingular, the GRQ factorization |
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* of A and B implicitly gives the RQ factorization of A*inv(B): |
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* |
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* A*inv(B) = (R*inv(T))*Z' |
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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 the matrix Z. |
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* |
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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 of the matrix A. M >= 0. |
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* |
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* P (input) INTEGER |
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* The number of rows of the matrix B. P >= 0. |
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* |
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* N (input) INTEGER |
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* The number of columns of the matrices A and B. 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 matrix A. |
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* On exit, if M <= N, the upper triangle of the subarray |
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* A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; |
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* if M > N, the elements on and above the (M-N)-th subdiagonal |
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* contain the M-by-N upper trapezoidal matrix R; the remaining |
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* elements, with the array TAUA, represent the unitary |
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* matrix Q as a product of 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,M). |
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* |
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* TAUA (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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* B (input/output) COMPLEX*16 array, dimension (LDB,N) |
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* On entry, the P-by-N matrix B. |
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* On exit, the elements on and above the diagonal of the array |
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* contain the min(P,N)-by-N upper trapezoidal matrix T (T is |
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* upper triangular if P >= N); the elements below the diagonal, |
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* with the array TAUB, represent the unitary matrix Z as a |
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* product of elementary reflectors (see Further 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,P). |
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* |
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* TAUB (output) COMPLEX*16 array, dimension (min(P,N)) |
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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 RQ factorization |
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* of an M-by-N matrix, NB2 is the optimal blocksize for the |
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* QR factorization of a P-by-N matrix, and NB3 is the optimal |
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* blocksize for a call of ZUNMRQ. |
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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(m,n). |
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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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in |
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* A(m-k+i,1:n-k+i-1), and taua in TAUA(i). |
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* To form Q explicitly, use LAPACK subroutine ZUNGRQ. |
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* To use Q to update another matrix, use LAPACK subroutine ZUNMRQ. |
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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(p,n). |
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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(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i), |
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* and taub in TAUB(i). |
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* To form Z explicitly, use LAPACK subroutine ZUNGQR. |
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* To use Z to update another matrix, use LAPACK subroutine ZUNMQR. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Local Scalars .. |
* .. Local Scalars .. |
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CALL ZGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO ) |
CALL ZGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO ) |
LOPT = WORK( 1 ) |
LOPT = WORK( 1 ) |
* |
* |
* Update B := B*Q' |
* Update B := B*Q**H |
* |
* |
CALL ZUNMRQ( 'Right', 'Conjugate Transpose', P, N, MIN( M, N ), |
CALL ZUNMRQ( 'Right', 'Conjugate Transpose', P, N, MIN( M, N ), |
$ A( MAX( 1, M-N+1 ), 1 ), LDA, TAUA, B, LDB, WORK, |
$ A( MAX( 1, M-N+1 ), 1 ), LDA, TAUA, B, LDB, WORK, |