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297 lines
9.1 KiB
297 lines
9.1 KiB
2 years ago
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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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*> \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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* =====================================================================
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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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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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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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*
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* .. Local Scalars ..
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LOGICAL LQUERY
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INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
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* ..
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* .. External Subroutines ..
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EXTERNAL XERBLA, ZGEQRF, ZGERQF, ZUNMQR
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* ..
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* .. External Functions ..
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INTEGER ILAENV
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EXTERNAL ILAENV
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC INT, MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters
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*
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INFO = 0
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NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, M, -1, -1 )
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NB2 = ILAENV( 1, 'ZGERQF', ' ', N, P, -1, -1 )
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NB3 = ILAENV( 1, 'ZUNMQR', ' ', N, M, P, -1 )
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NB = MAX( NB1, NB2, NB3 )
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LWKOPT = MAX( N, M, P )*NB
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WORK( 1 ) = LWKOPT
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LQUERY = ( LWORK.EQ.-1 )
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IF( N.LT.0 ) THEN
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INFO = -1
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ELSE IF( M.LT.0 ) THEN
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INFO = -2
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ELSE IF( P.LT.0 ) THEN
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INFO = -3
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -5
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -8
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ELSE IF( LWORK.LT.MAX( 1, N, M, P ) .AND. .NOT.LQUERY ) THEN
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INFO = -11
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZGGQRF', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* QR factorization of N-by-M matrix A: A = Q*R
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*
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CALL ZGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO )
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LOPT = INT( WORK( 1 ) )
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*
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* Update B := Q**H*B.
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*
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CALL ZUNMQR( 'Left', 'Conjugate Transpose', N, P, MIN( N, M ), A,
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$ LDA, TAUA, B, LDB, WORK, LWORK, INFO )
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LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
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*
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* RQ factorization of N-by-P matrix B: B = T*Z.
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*
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CALL ZGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO )
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WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
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*
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RETURN
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*
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* End of ZGGQRF
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*
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END
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