cggglm.f

          SUBROUTINE CGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK,
         $                   INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LDB, LWORK, M, N, P
    *     ..
    *     .. Array Arguments ..
          COMPLEX            A( LDA, * ), B( LDB, * ), D( * ), WORK( * ),
         $                   X( * ), Y( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  CGGGLM 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.
    *
    *  Arguments
    *  =========
    *
    *  N       (input) INTEGER
    *          The number of rows of the matrices A and B.  N >= 0.
    *
    *  M       (input) INTEGER
    *          The number of columns of the matrix A.  0 <= M <= N.
    *
    *  P       (input) INTEGER
    *          The number of columns of the matrix B.  P >= N-M.
    *
    *  A       (input/output) COMPLEX 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.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A. LDA >= max(1,N).
    *
    *  B       (input/output) COMPLEX 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.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B. LDB >= max(1,N).
    *
    *  D       (input/output) COMPLEX array, dimension (N)
    *          On entry, D is the left hand side of the GLM equation.
    *          On exit, D is destroyed.
    *
    *  X       (output) COMPLEX array, dimension (M)
    *  Y       (output) COMPLEX array, dimension (P)
    *          On exit, X and Y are the solutions of the GLM problem.
    *
    *  WORK    (workspace/output) COMPLEX 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,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
    *          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.
    *
    *  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 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.
    *
    *  ===================================================================
    *
    *     .. Parameters ..
          COMPLEX            CZERO, CONE
          PARAMETER          ( CZERO = ( 0.0E+0, 0.0E+0 ),
         $                   CONE = ( 1.0E+0, 0.0E+0 ) )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            LQUERY
          INTEGER            I, LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3,
         $                   NB4, NP
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           CCOPY, CGEMV, CGGQRF, CTRTRS, CUNMQR, CUNMRQ,
         $                   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, 'CGEQRF', ' ', N, M, -1, -1 )
                NB2 = ILAENV( 1, 'CGERQF', ' ', N, M, -1, -1 )
                NB3 = ILAENV( 1, 'CUNMQR', ' ', N, M, P, -1 )
                NB4 = ILAENV( 1, 'CUNMRQ', ' ', 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( 'CGGGLM', -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'*A = ( R11 ) M,    Q'*B*Z' = ( 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 CGGQRF( 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'*d = ( d1 ) M
    *                                             ( d2 ) N-M
    *
          CALL CUNMQR( '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 CTRTRS( '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 CCOPY( 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 CGEMV( '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 CTRTRS( '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 CCOPY( M, D, 1, X, 1 )
          END IF
    *
    *     Backward transformation y = Z'*y
    *
          CALL CUNMRQ( '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 CGGGLM
    *
          END