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OpenBLAS/lapack-netlib/SRC/cggrqf.f

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*> \brief \b CGGRQF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CGGRQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cggrqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cggrqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cggrqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
* COMPLEX A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CGGRQF computes a generalized RQ factorization of an M-by-N matrix A
*> and a P-by-N matrix B:
*>
*> A = R*Q, B = Z*T*Q,
*>
*> where Q is an N-by-N unitary matrix, Z is a P-by-P unitary
*> matrix, and R and T assume one of the forms:
*>
*> if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
*> N-M M ( R21 ) N
*> N
*>
*> where R12 or R21 is upper triangular, and
*>
*> if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
*> ( 0 ) P-N P N-P
*> N
*>
*> where T11 is upper triangular.
*>
*> In particular, if B is square and nonsingular, the GRQ factorization
*> of A and B implicitly gives the RQ factorization of A*inv(B):
*>
*> A*inv(B) = (R*inv(T))*Z**H
*>
*> where inv(B) denotes the inverse of the matrix B, and Z**H denotes the
*> conjugate transpose of the matrix Z.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows of the matrix B. P >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, if M <= N, the upper triangle of the subarray
*> A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
*> if M > N, the elements on and above the (M-N)-th subdiagonal
*> contain the M-by-N upper trapezoidal matrix R; the remaining
*> elements, with the array TAUA, represent the unitary
*> matrix Q as a product of elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAUA
*> \verbatim
*> TAUA is COMPLEX array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors which
*> represent the unitary matrix Q (see Further Details).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX array, dimension (LDB,N)
*> On entry, the P-by-N matrix B.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(P,N)-by-N upper trapezoidal matrix T (T is
*> upper triangular if P >= N); the elements below the diagonal,
*> with the array TAUB, represent the unitary matrix Z as a
*> product of elementary reflectors (see Further Details).
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,P).
*> \endverbatim
*>
*> \param[out] TAUB
*> \verbatim
*> TAUB is COMPLEX array, dimension (min(P,N))
*> The scalar factors of the elementary reflectors which
*> represent the unitary matrix Z (see Further Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N,M,P).
*> For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
*> where NB1 is the optimal blocksize for the RQ factorization
*> of an M-by-N matrix, NB2 is the optimal blocksize for the
*> QR factorization of a P-by-N matrix, and NB3 is the optimal
*> blocksize for a call of CUNMRQ.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO=-i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggrqf
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix Q is represented as a product of elementary reflectors
*>
*> Q = H(1) H(2) . . . H(k), where k = min(m,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - taua * v * v**H
*>
*> where taua is a complex scalar, and v is a complex vector with
*> v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
*> A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
*> To form Q explicitly, use LAPACK subroutine CUNGRQ.
*> To use Q to update another matrix, use LAPACK subroutine CUNMRQ.
*>
*> The matrix Z is represented as a product of elementary reflectors
*>
*> Z = H(1) H(2) . . . H(k), where k = min(p,n).
*>
*> Each H(i) has the form
*>
*> H(i) = I - taub * v * v**H
*>
*> where taub is a complex scalar, and v is a complex vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
*> and taub in TAUB(i).
*> To form Z explicitly, use LAPACK subroutine CUNGQR.
*> To use Z to update another matrix, use LAPACK subroutine CUNMQR.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE CGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
$ LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
* ..
* .. External Subroutines ..
EXTERNAL CGEQRF, CGERQF, CUNMRQ, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
REAL SROUNDUP_LWORK
EXTERNAL ILAENV, SROUNDUP_LWORK
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
NB1 = ILAENV( 1, 'CGERQF', ' ', M, N, -1, -1 )
NB2 = ILAENV( 1, 'CGEQRF', ' ', P, N, -1, -1 )
NB3 = ILAENV( 1, 'CUNMRQ', ' ', M, N, P, -1 )
NB = MAX( NB1, NB2, NB3 )
LWKOPT = MAX( 1, MAX( N, M, P )*NB )
WORK( 1 ) = SROUNDUP_LWORK( LWKOPT )
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( P.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
INFO = -8
ELSE IF( LWORK.LT.MAX( 1, M, P, N ) .AND. .NOT.LQUERY ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGGRQF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* RQ factorization of M-by-N matrix A: A = R*Q
*
CALL CGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO )
LOPT = INT( WORK( 1 ) )
*
* Update B := B*Q**H
*
CALL CUNMRQ( 'Right', 'Conjugate Transpose', P, N, MIN( M, N ),
$ A( MAX( 1, M-N+1 ), 1 ), LDA, TAUA, B, LDB, WORK,
$ LWORK, INFO )
LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
*
* QR factorization of P-by-N matrix B: B = Z*T
*
CALL CGEQRF( P, N, B, LDB, TAUB, WORK, LWORK, INFO )
WORK( 1 ) = SROUNDUP_LWORK( MAX( LOPT, INT( WORK( 1 ) ) ) )
*
RETURN
*
* End of CGGRQF
*
END
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