*> \brief \b SGGQRF
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE SGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, P
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGGQRF computes a generalized QR factorization of an N-by-M matrix A
*> and an N-by-P matrix B:
*>
*> A = Q*R, B = Q*T*Z,
*>
*> where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
*> matrix, and R and T assume one of the forms:
*>
*> if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N,
*> ( 0 ) N-M N M-N
*> M
*>
*> where R11 is upper triangular, and
*>
*> if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P,
*> P-N N ( T21 ) P
*> P
*>
*> where T12 or T21 is upper triangular.
*>
*> In particular, if B is square and nonsingular, the GQR factorization
*> of A and B implicitly gives the QR factorization of inv(B)*A:
*>
*> inv(B)*A = Z**T*(inv(T)*R)
*>
*> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
*> transpose of the matrix Z.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of columns of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of columns of the matrix B. P >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA,M)
*> On entry, the N-by-M matrix A.
*> On exit, the elements on and above the diagonal of the array
*> contain the min(N,M)-by-M upper trapezoidal matrix R (R is
*> upper triangular if N >= M); the elements below the diagonal,
*> with the array TAUA, represent the orthogonal matrix Q as a
*> product of min(N,M) elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAUA
*> \verbatim
*> TAUA is REAL array, dimension (min(N,M))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q (see Further Details).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is REAL array, dimension (LDB,P)
*> On entry, the N-by-P matrix B.
*> On exit, if N <= P, the upper triangle of the subarray
*> B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
*> if N > P, the elements on and above the (N-P)-th subdiagonal
*> contain the N-by-P upper trapezoidal matrix T; the remaining
*> elements, with the array TAUB, represent the orthogonal
*> 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,N).
*> \endverbatim
*>
*> \param[out] TAUB
*> \verbatim
*> TAUB is REAL array, dimension (min(N,P))
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Z (see Further Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL 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 QR factorization
*> of an N-by-M matrix, NB2 is the optimal blocksize for the
*> RQ factorization of an N-by-P matrix, and NB3 is the optimal
*> blocksize for a call of SORMQR.
*>
*> 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 realOTHERcomputational
*
*> \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(n,m).
*>
*> Each H(i) has the form
*>
*> H(i) = I - taua * v * v**T
*>
*> where taua is a real scalar, and v is a real vector with
*> v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
*> and taua in TAUA(i).
*> To form Q explicitly, use LAPACK subroutine SORGQR.
*> To use Q to update another matrix, use LAPACK subroutine SORMQR.
*>
*> The matrix Z is represented as a product of elementary reflectors
*>
*> Z = H(1) H(2) . . . H(k), where k = min(n,p).
*>
*> Each H(i) has the form
*>
*> H(i) = I - taub * v * v**T
*>
*> where taub is a real scalar, and v is a real vector with
*> v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
*> B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
*> To form Z explicitly, use LAPACK subroutine SORGRQ.
*> To use Z to update another matrix, use LAPACK subroutine SORMRQ.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SGGQRF( N, M, P, 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 ..
REAL A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3
* ..
* .. External Subroutines ..
EXTERNAL SGEQRF, SGERQF, SORMQR, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
NB1 = ILAENV( 1, 'SGEQRF', ' ', N, M, -1, -1 )
NB2 = ILAENV( 1, 'SGERQF', ' ', N, P, -1, -1 )
NB3 = ILAENV( 1, 'SORMQR', ' ', N, M, P, -1 )
NB = MAX( NB1, NB2, NB3 )
LWKOPT = MAX( N, M, P )*NB
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( P.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LWORK.LT.MAX( 1, N, M, P ) .AND. .NOT.LQUERY ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGGQRF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* QR factorization of N-by-M matrix A: A = Q*R
*
CALL SGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO )
LOPT = WORK( 1 )
*
* Update B := Q**T*B.
*
CALL SORMQR( 'Left', 'Transpose', N, P, MIN( N, M ), A, LDA, TAUA,
$ B, LDB, WORK, LWORK, INFO )
LOPT = MAX( LOPT, INT( WORK( 1 ) ) )
*
* RQ factorization of N-by-P matrix B: B = T*Z.
*
CALL SGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO )
WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) )
*
RETURN
*
* End of SGGQRF
*
END