*> \brief \b ZGGQRF * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZGGQRF + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, * LWORK, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDB, LWORK, M, N, P * .. * .. Array Arguments .. * COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), * \$ WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGGQRF 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 unitary matrix, Z is a P-by-P unitary 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**H * (inv(T)*R) *> *> where inv(B) denotes the inverse of the matrix B, and Z**H denotes the *> conjugate transpose of 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 COMPLEX*16 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 unitary 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 COMPLEX*16 array, dimension (min(N,M)) *> 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*16 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 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,N). *> \endverbatim *> *> \param[out] TAUB *> \verbatim *> TAUB is COMPLEX*16 array, dimension (min(N,P)) *> The scalar factors of the elementary reflectors which *> represent the unitary matrix Z (see Further Details). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX*16 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 ZUNMQR. *> *> 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. * *> \date December 2016 * *> \ingroup complex16OTHERcomputational * *> \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**H *> *> where taua is a complex scalar, and v is a complex 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 ZUNGQR. *> To use Q to update another matrix, use LAPACK subroutine ZUNMQR. *> *> 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**H *> *> where taub is a complex scalar, and v is a complex 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 ZUNGRQ. *> To use Z to update another matrix, use LAPACK subroutine ZUNMRQ. *> \endverbatim *> * ===================================================================== SUBROUTINE ZGGQRF( N, M, P, 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 .. COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), \$ WORK( * ) * .. * * ===================================================================== * * .. Local Scalars .. LOGICAL LQUERY INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3 * .. * .. External Subroutines .. EXTERNAL XERBLA, ZGEQRF, ZGERQF, ZUNMQR * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters * INFO = 0 NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, M, -1, -1 ) NB2 = ILAENV( 1, 'ZGERQF', ' ', N, P, -1, -1 ) NB3 = ILAENV( 1, 'ZUNMQR', ' ', 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( 'ZGGQRF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * QR factorization of N-by-M matrix A: A = Q*R * CALL ZGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO ) LOPT = WORK( 1 ) * * Update B := Q**H*B. * CALL ZUNMQR( 'Left', 'Conjugate 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 ZGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO ) WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) ) * RETURN * * End of ZGGQRF * END