dggglm.f

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*> \brief \b DGGGLM
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGGLM( 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 ..
*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
*      $                   X( * ), Y( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGGLM 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.
*>
*> Callers of this subroutine should note that the singularity/rank-deficiency checks
*> implemented in this subroutine are rudimentary. The DTRTRS subroutine called by this
*> subroutine only signals a failure due to singularity if the problem is exactly singular.
*>
*> It is conceivable for one (or more) of the factors involved in the generalized QR
*> factorization of the pair (A, B) to be subnormally close to singularity without this
*> subroutine signalling an error. The solutions computed for such almost-rank-deficient
*> problems may be less accurate due to a loss of numerical precision.
*> 
*> \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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (M)
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*>          Y is DOUBLE PRECISION array, dimension (P)
*>
*>          On exit, X and Y are the solutions of the GLM problem.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION 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
*>          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.
*> \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 exactly
*>                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 exactly 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.
*
*> \ingroup ggglm
*
*  =====================================================================
      SUBROUTINE dggglm( N, M, P, A, LDA, B, LDB, D, X, Y, 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 ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
     $                   X( * ), Y( * )
*     ..
*
*  ===================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3,
     $                   nb4, np
*     ..
*     .. External Subroutines ..
      EXTERNAL           dcopy, dgemv, dggqrf, dormqr, dormrq,
     $                   dtrtrs,
     $                   xerbla
*     ..
*     .. 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, 'DGEQRF', ' ', n, m, -1, -1 )
            nb2 = ilaenv( 1, 'DGERQF', ' ', n, m, -1, -1 )
            nb3 = ilaenv( 1, 'DORMQR', ' ', n, m, p, -1 )
            nb4 = ilaenv( 1, 'DORMRQ', ' ', 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( 'DGGGLM', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 ) THEN
         DO i = 1, m
            x(i) = zero
         END DO
         DO i = 1, p
            y(i) = zero
         END DO
         RETURN
      END IF
*
*     Compute the GQR factorization of matrices A and B:
*
*          Q**T*A = ( R11 ) M,    Q**T*B*Z**T = ( 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
*     orthogonal.
*
      CALL dggqrf( n, m, p, a, lda, work, b, ldb, work( m+1 ),
     $             work( m+np+1 ), lwork-m-np, info )
      lopt = int( work( m+np+1 ) )
*
*     Update left-hand-side vector d = Q**T*d = ( d1 ) M
*                                               ( d2 ) N-M
*
      CALL dormqr( 'Left', '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 dtrtrs( '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 dcopy( 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 ) = zero
   10 CONTINUE
*
*     Update d1 = d1 - T12*y2
*
      CALL dgemv( 'No transpose', m, n-m, -one, b( 1, m+p-n+1 ), ldb,
     $            y( m+p-n+1 ), 1, one, d, 1 )
*
*     Solve triangular system: R11*x = d1
*
      IF( m.GT.0 ) THEN
         CALL dtrtrs( '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 dcopy( m, d, 1, x, 1 )
      END IF
*
*     Backward transformation y = Z**T *y
*
      CALL dormrq( 'Left', '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 DGGGLM
*
      END