*> \brief \b ZGGGLM * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZGGGLM + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, * INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDB, LWORK, M, N, P * .. * .. Array Arguments .. * COMPLEX*16 A( LDA, * ), B( LDB, * ), D( * ), WORK( * ), * $ X( * ), Y( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGGGLM solves a general Gauss-Markov linear model (GLM) problem: *> *> minimize || y ||_2 subject to d = A*x + B*y *> x *> *> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a *> given N-vector. It is assumed that M <= N <= M+P, and *> *> rank(A) = M and rank( A B ) = N. *> *> Under these assumptions, the constrained equation is always *> consistent, and there is a unique solution x and a minimal 2-norm *> solution y, which is obtained using a generalized QR factorization *> of the matrices (A, B) given by *> *> A = Q*(R), B = Q*T*Z. *> (0) *> *> In particular, if matrix B is square nonsingular, then the problem *> GLM is equivalent to the following weighted linear least squares *> problem *> *> minimize || inv(B)*(d-A*x) ||_2 *> x *> *> where inv(B) denotes the inverse of B. *> \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. 0 <= M <= N. *> \endverbatim *> *> \param[in] P *> \verbatim *> P is INTEGER *> The number of columns of the matrix B. P >= N-M. *> \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 upper triangular part of the array A contains *> the M-by-M upper triangular matrix R. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \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. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,N). *> \endverbatim *> *> \param[in,out] D *> \verbatim *> D is COMPLEX*16 array, dimension (N) *> On entry, D is the left hand side of the GLM equation. *> On exit, D is destroyed. *> \endverbatim *> *> \param[out] X *> \verbatim *> X is COMPLEX*16 array, dimension (M) *> \endverbatim *> *> \param[out] Y *> \verbatim *> Y is COMPLEX*16 array, dimension (P) *> *> On exit, X and Y are the solutions of the GLM problem. *> \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 >= M+min(N,P)+max(N,P)*NB, *> where NB is an upper bound for the optimal blocksizes for *> ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ. *> *> 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. *> = 1: the upper triangular factor R associated with A in the *> generalized QR factorization of the pair (A, B) is *> singular, so that rank(A) < M; the least squares *> solution could not be computed. *> = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal *> factor T associated with B in the generalized QR *> factorization of the pair (A, B) is singular, so that *> rank( A B ) < N; the least squares solution could not *> be computed. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup complex16OTHEReigen * * ===================================================================== SUBROUTINE ZGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, $ INFO ) * * -- LAPACK driver 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, * ), D( * ), WORK( * ), $ X( * ), Y( * ) * .. * * =================================================================== * * .. Parameters .. COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), $ CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3, $ NB4, NP * .. * .. External Subroutines .. EXTERNAL XERBLA, ZCOPY, ZGEMV, ZGGQRF, ZTRTRS, ZUNMQR, $ ZUNMRQ * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters * INFO = 0 NP = MIN( N, P ) LQUERY = ( LWORK.EQ.-1 ) IF( N.LT.0 ) THEN INFO = -1 ELSE IF( M.LT.0 .OR. M.GT.N ) THEN INFO = -2 ELSE IF( P.LT.0 .OR. P.LT.N-M ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF * * Calculate workspace * IF( INFO.EQ.0) THEN IF( N.EQ.0 ) THEN LWKMIN = 1 LWKOPT = 1 ELSE NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, M, -1, -1 ) NB2 = ILAENV( 1, 'ZGERQF', ' ', N, M, -1, -1 ) NB3 = ILAENV( 1, 'ZUNMQR', ' ', N, M, P, -1 ) NB4 = ILAENV( 1, 'ZUNMRQ', ' ', N, M, P, -1 ) NB = MAX( NB1, NB2, NB3, NB4 ) LWKMIN = M + N + P LWKOPT = M + NP + MAX( N, P )*NB END IF WORK( 1 ) = LWKOPT * IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN INFO = -12 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGGGLM', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * Compute the GQR factorization of matrices A and B: * * Q**H*A = ( R11 ) M, Q**H*B*Z**H = ( T11 T12 ) M * ( 0 ) N-M ( 0 T22 ) N-M * M M+P-N N-M * * where R11 and T22 are upper triangular, and Q and Z are * unitary. * CALL ZGGQRF( N, M, P, A, LDA, WORK, B, LDB, WORK( M+1 ), $ WORK( M+NP+1 ), LWORK-M-NP, INFO ) LOPT = WORK( M+NP+1 ) * * Update left-hand-side vector d = Q**H*d = ( d1 ) M * ( d2 ) N-M * CALL ZUNMQR( 'Left', 'Conjugate transpose', N, 1, M, A, LDA, WORK, $ D, MAX( 1, N ), WORK( M+NP+1 ), LWORK-M-NP, INFO ) LOPT = MAX( LOPT, INT( WORK( M+NP+1 ) ) ) * * Solve T22*y2 = d2 for y2 * IF( N.GT.M ) THEN CALL ZTRTRS( 'Upper', 'No transpose', 'Non unit', N-M, 1, $ B( M+1, M+P-N+1 ), LDB, D( M+1 ), N-M, INFO ) * IF( INFO.GT.0 ) THEN INFO = 1 RETURN END IF * CALL ZCOPY( N-M, D( M+1 ), 1, Y( M+P-N+1 ), 1 ) END IF * * Set y1 = 0 * DO 10 I = 1, M + P - N Y( I ) = CZERO 10 CONTINUE * * Update d1 = d1 - T12*y2 * CALL ZGEMV( 'No transpose', M, N-M, -CONE, B( 1, M+P-N+1 ), LDB, $ Y( M+P-N+1 ), 1, CONE, D, 1 ) * * Solve triangular system: R11*x = d1 * IF( M.GT.0 ) THEN CALL ZTRTRS( 'Upper', 'No Transpose', 'Non unit', M, 1, A, LDA, $ D, M, INFO ) * IF( INFO.GT.0 ) THEN INFO = 2 RETURN END IF * * Copy D to X * CALL ZCOPY( M, D, 1, X, 1 ) END IF * * Backward transformation y = Z**H *y * CALL ZUNMRQ( 'Left', 'Conjugate transpose', P, 1, NP, $ B( MAX( 1, N-P+1 ), 1 ), LDB, WORK( M+1 ), Y, $ MAX( 1, P ), WORK( M+NP+1 ), LWORK-M-NP, INFO ) WORK( 1 ) = M + NP + MAX( LOPT, INT( WORK( M+NP+1 ) ) ) * RETURN * * End of ZGGGLM * END