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296 lines
9.1 KiB
296 lines
9.1 KiB
*> \brief \b SGGRQF
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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 SGGRQF + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sggrqf.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/sggrqf.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/sggrqf.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 SGGRQF( 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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* REAL 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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*> SGGRQF 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 orthogonal matrix, Z is a P-by-P orthogonal
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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**T
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*>
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*> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
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*> 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 REAL 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 orthogonal
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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 REAL array, dimension (min(M,N))
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*> The scalar factors of the elementary reflectors which
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*> represent the orthogonal 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 REAL 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 orthogonal 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 REAL array, dimension (min(P,N))
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*> The scalar factors of the elementary reflectors which
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*> represent the orthogonal 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 REAL 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 SORMRQ.
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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 INF0= -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 realOTHERcomputational
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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**T
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*>
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*> where taua is a real scalar, and v is a real 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 SORGRQ.
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*> To use Q to update another matrix, use LAPACK subroutine SORMRQ.
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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**T
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*>
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*> where taub is a real scalar, and v is a real 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 SORGQR.
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*> To use Z to update another matrix, use LAPACK subroutine SORMQR.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE SGGRQF( M, P, N, 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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REAL 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 SGEQRF, SGERQF, SORMRQ, XERBLA
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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, 'SGERQF', ' ', M, N, -1, -1 )
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NB2 = ILAENV( 1, 'SGEQRF', ' ', P, N, -1, -1 )
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NB3 = ILAENV( 1, 'SORMRQ', ' ', M, N, 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( M.LT.0 ) THEN
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INFO = -1
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ELSE IF( P.LT.0 ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -5
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ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
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INFO = -8
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ELSE IF( LWORK.LT.MAX( 1, M, P, N ) .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( 'SGGRQF', -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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* RQ factorization of M-by-N matrix A: A = R*Q
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*
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CALL SGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO )
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LOPT = INT( WORK( 1 ) )
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*
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* Update B := B*Q**T
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*
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CALL SORMRQ( 'Right', 'Transpose', P, N, MIN( M, N ),
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$ A( MAX( 1, M-N+1 ), 1 ), LDA, TAUA, B, LDB, WORK,
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$ LWORK, INFO )
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LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
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*
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* QR factorization of P-by-N matrix B: B = Z*T
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*
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CALL SGEQRF( P, N, 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 SGGRQF
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*
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END
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