*> \brief CGGLSE solves overdetermined or underdetermined systems for OTHER matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CGGLSE + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, * INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDB, LWORK, M, N, P * .. * .. Array Arguments .. * COMPLEX A( LDA, * ), B( LDB, * ), C( * ), D( * ), * \$ WORK( * ), X( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CGGLSE solves the linear equality-constrained least squares (LSE) *> problem: *> *> minimize || c - A*x ||_2 subject to B*x = d *> *> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given *> M-vector, and d is a given P-vector. It is assumed that *> P <= N <= M+P, and *> *> rank(B) = P and rank( (A) ) = N. *> ( (B) ) *> *> These conditions ensure that the LSE problem has a unique solution, *> which is obtained using a generalized RQ factorization of the *> matrices (B, A) given by *> *> B = (0 R)*Q, A = Z*T*Q. *> \endverbatim * * Arguments: * ========== * *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrices A and B. N >= 0. *> \endverbatim *> *> \param[in] P *> \verbatim *> P is INTEGER *> The number of rows of the matrix B. 0 <= P <= N <= M+P. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX array, dimension (LDA,N) *> On entry, the M-by-N matrix A. *> On exit, the elements on and above the diagonal of the array *> contain the min(M,N)-by-N upper trapezoidal matrix T. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is COMPLEX array, dimension (LDB,N) *> On entry, the P-by-N matrix B. *> On exit, the upper triangle of the subarray B(1:P,N-P+1:N) *> contains the P-by-P upper triangular matrix R. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,P). *> \endverbatim *> *> \param[in,out] C *> \verbatim *> C is COMPLEX array, dimension (M) *> On entry, C contains the right hand side vector for the *> least squares part of the LSE problem. *> On exit, the residual sum of squares for the solution *> is given by the sum of squares of elements N-P+1 to M of *> vector C. *> \endverbatim *> *> \param[in,out] D *> \verbatim *> D is COMPLEX array, dimension (P) *> On entry, D contains the right hand side vector for the *> constrained equation. *> On exit, D is destroyed. *> \endverbatim *> *> \param[out] X *> \verbatim *> X is COMPLEX array, dimension (N) *> On exit, X is the solution of the LSE problem. *> \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,M+N+P). *> For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, *> where NB is an upper bound for the optimal blocksizes for *> CGEQRF, CGERQF, CUNMQR and 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. *> = 1: the upper triangular factor R associated with B in the *> generalized RQ factorization of the pair (B, A) is *> singular, so that rank(B) < P; the least squares *> solution could not be computed. *> = 2: the (N-P) by (N-P) part of the upper trapezoidal factor *> T associated with A in the generalized RQ factorization *> of the pair (B, A) is singular, so that *> rank( (A) ) < N; the least squares solution could not *> ( (B) ) *> be computed. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup complexOTHERsolve * * ===================================================================== SUBROUTINE CGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, \$ INFO ) * * -- LAPACK driver 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, * ), C( * ), D( * ), \$ WORK( * ), X( * ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX CONE PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER LOPT, LWKMIN, LWKOPT, MN, NB, NB1, NB2, NB3, \$ NB4, NR * .. * .. External Subroutines .. EXTERNAL CAXPY, CCOPY, CGEMV, CGGRQF, CTRMV, CTRTRS, \$ CUNMQR, CUNMRQ, XERBLA * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters * INFO = 0 MN = MIN( M, N ) LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( P.LT.0 .OR. P.GT.N .OR. P.LT.N-M ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, P ) ) 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, 'CGEQRF', ' ', M, N, -1, -1 ) NB2 = ILAENV( 1, 'CGERQF', ' ', M, N, -1, -1 ) NB3 = ILAENV( 1, 'CUNMQR', ' ', M, N, P, -1 ) NB4 = ILAENV( 1, 'CUNMRQ', ' ', M, N, P, -1 ) NB = MAX( NB1, NB2, NB3, NB4 ) LWKMIN = M + N + P LWKOPT = P + MN + MAX( M, N )*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( 'CGGLSE', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * * Compute the GRQ factorization of matrices B and A: * * B*Q**H = ( 0 T12 ) P Z**H*A*Q**H = ( R11 R12 ) N-P * N-P P ( 0 R22 ) M+P-N * N-P P * * where T12 and R11 are upper triangular, and Q and Z are * unitary. * CALL CGGRQF( P, M, N, B, LDB, WORK, A, LDA, WORK( P+1 ), \$ WORK( P+MN+1 ), LWORK-P-MN, INFO ) LOPT = INT( WORK( P+MN+1 ) ) * * Update c = Z**H *c = ( c1 ) N-P * ( c2 ) M+P-N * CALL CUNMQR( 'Left', 'Conjugate Transpose', M, 1, MN, A, LDA, \$ WORK( P+1 ), C, MAX( 1, M ), WORK( P+MN+1 ), \$ LWORK-P-MN, INFO ) LOPT = MAX( LOPT, INT( WORK( P+MN+1 ) ) ) * * Solve T12*x2 = d for x2 * IF( P.GT.0 ) THEN CALL CTRTRS( 'Upper', 'No transpose', 'Non-unit', P, 1, \$ B( 1, N-P+1 ), LDB, D, P, INFO ) * IF( INFO.GT.0 ) THEN INFO = 1 RETURN END IF * * Put the solution in X * CALL CCOPY( P, D, 1, X( N-P+1 ), 1 ) * * Update c1 * CALL CGEMV( 'No transpose', N-P, P, -CONE, A( 1, N-P+1 ), LDA, \$ D, 1, CONE, C, 1 ) END IF * * Solve R11*x1 = c1 for x1 * IF( N.GT.P ) THEN CALL CTRTRS( 'Upper', 'No transpose', 'Non-unit', N-P, 1, \$ A, LDA, C, N-P, INFO ) * IF( INFO.GT.0 ) THEN INFO = 2 RETURN END IF * * Put the solutions in X * CALL CCOPY( N-P, C, 1, X, 1 ) END IF * * Compute the residual vector: * IF( M.LT.N ) THEN NR = M + P - N IF( NR.GT.0 ) \$ CALL CGEMV( 'No transpose', NR, N-M, -CONE, A( N-P+1, M+1 ), \$ LDA, D( NR+1 ), 1, CONE, C( N-P+1 ), 1 ) ELSE NR = P END IF IF( NR.GT.0 ) THEN CALL CTRMV( 'Upper', 'No transpose', 'Non unit', NR, \$ A( N-P+1, N-P+1 ), LDA, D, 1 ) CALL CAXPY( NR, -CONE, D, 1, C( N-P+1 ), 1 ) END IF * * Backward transformation x = Q**H*x * CALL CUNMRQ( 'Left', 'Conjugate Transpose', N, 1, P, B, LDB, \$ WORK( 1 ), X, N, WORK( P+MN+1 ), LWORK-P-MN, INFO ) WORK( 1 ) = P + MN + MAX( LOPT, INT( WORK( P+MN+1 ) ) ) * RETURN * * End of CGGLSE * END