SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, \$ INFO ) * * -- LAPACK driver routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, P * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( * ), D( * ), \$ WORK( * ), X( * ) * .. * * Purpose * ======= * * DGGLSE 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. * * Arguments * ========= * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * N (input) INTEGER * The number of columns of the matrices A and B. N >= 0. * * P (input) INTEGER * The number of rows of the matrix B. 0 <= P <= N <= M+P. * * A (input/output) DOUBLE PRECISION 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. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * B (input/output) DOUBLE PRECISION 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. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,P). * * C (input/output) DOUBLE PRECISION 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. * * D (input/output) DOUBLE PRECISION array, dimension (P) * On entry, D contains the right hand side vector for the * constrained equation. * On exit, D is destroyed. * * X (output) DOUBLE PRECISION array, dimension (N) * On exit, X is the solution of the LSE problem. * * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) 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 * DGEQRF, SGERQF, DORMQR and 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. * * INFO (output) 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. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER LOPT, LWKMIN, LWKOPT, MN, NB, NB1, NB2, NB3, \$ NB4, NR * .. * .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMV, DGGRQF, DORMQR, DORMRQ, \$ DTRMV, DTRTRS, 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, 'DGEQRF', ' ', M, N, -1, -1 ) NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 ) NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, P, -1 ) NB4 = ILAENV( 1, 'DORMRQ', ' ', 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( 'DGGLSE', -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' = ( 0 T12 ) P Z'*A*Q' = ( 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 * orthogonal. * CALL DGGRQF( P, M, N, B, LDB, WORK, A, LDA, WORK( P+1 ), \$ WORK( P+MN+1 ), LWORK-P-MN, INFO ) LOPT = WORK( P+MN+1 ) * * Update c = Z'*c = ( c1 ) N-P * ( c2 ) M+P-N * CALL DORMQR( 'Left', '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 DTRTRS( '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 DCOPY( P, D, 1, X( N-P+1 ), 1 ) * * Update c1 * CALL DGEMV( 'No transpose', N-P, P, -ONE, A( 1, N-P+1 ), LDA, \$ D, 1, ONE, C, 1 ) END IF * * Solve R11*x1 = c1 for x1 * IF( N.GT.P ) THEN CALL DTRTRS( '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 DCOPY( 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 DGEMV( 'No transpose', NR, N-M, -ONE, A( N-P+1, M+1 ), \$ LDA, D( NR+1 ), 1, ONE, C( N-P+1 ), 1 ) ELSE NR = P END IF IF( NR.GT.0 ) THEN CALL DTRMV( 'Upper', 'No transpose', 'Non unit', NR, \$ A( N-P+1, N-P+1 ), LDA, D, 1 ) CALL DAXPY( NR, -ONE, D, 1, C( N-P+1 ), 1 ) END IF * * Backward transformation x = Q'*x * CALL DORMRQ( 'Left', '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 DGGLSE * END