*> \brief \b SGGRQF * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SGGRQF + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SGGRQF( M, P, N, 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 *> *> SGGRQF 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 orthogonal matrix, Z is a P-by-P orthogonal *> 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**T *> *> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the *> 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 REAL 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 orthogonal *> 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 REAL array, dimension (min(M,N)) *> 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,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 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,P). *> \endverbatim *> *> \param[out] TAUB *> \verbatim *> TAUB is REAL array, dimension (min(P,N)) *> 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 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 SORMRQ. *> *> 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 INF0= -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. * *> \date December 2016 * *> \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(m,n). *> *> 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(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 SORGRQ. *> To use Q to update another matrix, use LAPACK subroutine SORMRQ. *> *> 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**T *> *> where taub is a real scalar, and v is a real 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 SORGQR. *> To use Z to update another matrix, use LAPACK subroutine SORMQR. *> \endverbatim *> * ===================================================================== SUBROUTINE SGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, \$ LWORK, INFO ) * * -- LAPACK computational routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. 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, SORMRQ, XERBLA * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters * INFO = 0 NB1 = ILAENV( 1, 'SGERQF', ' ', M, N, -1, -1 ) NB2 = ILAENV( 1, 'SGEQRF', ' ', P, N, -1, -1 ) NB3 = ILAENV( 1, 'SORMRQ', ' ', M, N, P, -1 ) NB = MAX( NB1, NB2, NB3 ) LWKOPT = MAX( N, M, P)*NB WORK( 1 ) = 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( 'SGGRQF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * RQ factorization of M-by-N matrix A: A = R*Q * CALL SGERQF( M, N, A, LDA, TAUA, WORK, LWORK, INFO ) LOPT = WORK( 1 ) * * Update B := B*Q**T * CALL SORMRQ( 'Right', '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 SGEQRF( P, N, B, LDB, TAUB, WORK, LWORK, INFO ) WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) ) * RETURN * * End of SGGRQF * END