dgglse.f

          SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, 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 ..
          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