MB02UD

Minimum norm least squares solution of op(R) X = alpha B, or X op(R) = alpha B, with R upper triangular, using singular value decomposition

Purpose

  To compute the minimum norm least squares solution of one of the
  following linear systems

     op(R)*X = alpha*B,                                          (1)
     X*op(R) = alpha*B,                                          (2)

  where alpha is a real scalar, op(R) is either R or its transpose,
  R', R is an L-by-L real upper triangular matrix, B is an M-by-N
  real matrix, and L = M for (1), or L = N for (2). Singular value
  decomposition, R = Q*S*P', is used, assuming that R is rank
  deficient.

      SUBROUTINE MB02UD( FACT, SIDE, TRANS, JOBP, M, N, ALPHA, RCOND,
     $                   RANK, R, LDR, Q, LDQ, SV, B, LDB, RP, LDRP,
     $                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         FACT, JOBP, SIDE, TRANS
      INTEGER           INFO, LDB, LDQ, LDR, LDRP, LDWORK, M, N, RANK
      DOUBLE PRECISION  ALPHA, RCOND
C     .. Array Arguments ..
      DOUBLE PRECISION  B(LDB,*), DWORK(*), Q(LDQ,*), R(LDR,*),
     $                  RP(LDRP,*), SV(*)

Mode Parameters

  FACT    CHARACTER*1
          Specifies whether R has been previously factored or not,
          as follows:
          = 'F':  R has been factored and its rank and singular
                  value decomposition, R = Q*S*P', are available;
          = 'N':  R has not been factored and its singular value
                  decomposition, R = Q*S*P', should be computed.

  SIDE    CHARACTER*1
          Specifies whether op(R) appears on the left or right
          of X as follows:
          = 'L':  Solve op(R)*X = alpha*B  (op(R) is on the left);
          = 'R':  Solve X*op(R) = alpha*B  (op(R) is on the right).

  TRANS   CHARACTER*1
          Specifies the form of op(R) to be used as follows:
          = 'N':  op(R) = R;
          = 'T':  op(R) = R';
          = 'C':  op(R) = R'.

  JOBP    CHARACTER*1
          Specifies whether or not the pseudoinverse of R is to be
          computed or it is available as follows:
          = 'P':  Compute pinv(R), if FACT = 'N', or
                  use pinv(R),     if FACT = 'F';
          = 'N':  Do not compute or use pinv(R).

  M       (input) INTEGER
          The number of rows of the matrix B.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix B.  N >= 0.

  ALPHA   (input) DOUBLE PRECISION
          The scalar alpha. When alpha is zero then B need not be
          set before entry.

  RCOND   (input) DOUBLE PRECISION
          RCOND is used to determine the effective rank of R.
          Singular values of R satisfying Sv(i) <= RCOND*Sv(1) are
          treated as zero. If RCOND <= 0, then EPS is used instead,
          where EPS is the relative machine precision (see LAPACK
          Library routine DLAMCH).  RCOND <= 1.
          RCOND is not used if FACT = 'F'.

  RANK    (input or output) INTEGER
          The rank of matrix R.
          RANK is an input parameter when FACT = 'F', and an output
          parameter when FACT = 'N'.  L >= RANK >= 0.

  R       (input/output) DOUBLE PRECISION array, dimension (LDR,L)
          On entry, if FACT = 'F', the leading L-by-L part of this
          array must contain the L-by-L orthogonal matrix P' from
          singular value decomposition, R = Q*S*P', of the matrix R;
          if JOBP = 'P', the first RANK rows of P' are assumed to be
          scaled by inv(S(1:RANK,1:RANK)).
          On entry, if FACT = 'N', the leading L-by-L upper
          triangular part of this array must contain the upper
          triangular matrix R.
          On exit, if INFO = 0, the leading L-by-L part of this
          array contains the L-by-L orthogonal matrix P', with its
          first RANK rows scaled by inv(S(1:RANK,1:RANK)), when
          JOBP = 'P'.

  LDR     INTEGER
          The leading dimension of array R.  LDR >= MAX(1,L).

  Q       (input or output) DOUBLE PRECISION array, dimension
          (LDQ,L)
          On entry, if FACT = 'F', the leading L-by-L part of this
          array must contain the L-by-L orthogonal matrix Q from
          singular value decomposition, R = Q*S*P', of the matrix R.
          If FACT = 'N', this array need not be set on entry, and
          on exit, if INFO = 0, the leading L-by-L part of this
          array contains the orthogonal matrix Q.

  LDQ     INTEGER
          The leading dimension of array Q.  LDQ >= MAX(1,L).

  SV      (input or output) DOUBLE PRECISION array, dimension (L)
          On entry, if FACT = 'F', the first RANK entries of this
          array must contain the reciprocal of the largest RANK
          singular values of the matrix R, and the last L-RANK
          entries of this array must contain the remaining singular
          values of R sorted in descending order.
          If FACT = 'N', this array need not be set on input, and
          on exit, if INFO = 0, the first RANK entries of this array
          contain the reciprocal of the largest RANK singular values
          of the matrix R, and the last L-RANK entries of this array
          contain the remaining singular values of R sorted in
          descending order.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
          On entry, if ALPHA <> 0, the leading M-by-N part of this
          array must contain the matrix B.
          On exit, if INFO = 0 and RANK > 0, the leading M-by-N part
          of this array contains the M-by-N solution matrix X.

  LDB     INTEGER
          The leading dimension of array B.  LDB >= MAX(1,M).

  RP      (input or output) DOUBLE PRECISION array, dimension
          (LDRP,L)
          On entry, if FACT = 'F', JOBP = 'P', and RANK > 0, the
          leading L-by-L part of this array must contain the L-by-L
          matrix pinv(R), the Moore-Penrose pseudoinverse of R.
          On exit, if FACT = 'N', JOBP = 'P', and RANK > 0, the
          leading L-by-L part of this array contains the L-by-L
          matrix pinv(R), the Moore-Penrose pseudoinverse of R.
          If JOBP = 'N', this array is not referenced.

  LDRP    INTEGER
          The leading dimension of array RP.
          LDRP >= MAX(1,L), if JOBP = 'P'.
          LDRP >= 1,        if JOBP = 'N'.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK;
          if INFO = i, 1 <= i <= L, then DWORK(2:L) contain the
          unconverged superdiagonal elements of an upper bidiagonal
          matrix D whose diagonal is in SV (not necessarily sorted).
          D satisfies R = Q*D*P', so it has the same singular
          values as R, and singular vectors related by Q and P'.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= MAX(1,L),   if FACT = 'F';
          LDWORK >= MAX(1,5*L), if FACT = 'N'.
          For optimum performance LDWORK should be larger than
          MAX(1,L,M*N),   if FACT = 'F';
          MAX(1,5*L,M*N), if FACT = 'N'.

          If LDWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          DWORK array, returns this value as the first entry of
          the DWORK array, and no error message related to LDWORK
          is issued by XERBLA.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          > 0:  if INFO = i, i = 1:L, the SVD algorithm has failed
                to converge. In this case INFO specifies how many
                superdiagonals did not converge (see the description
                of DWORK); this failure is not likely to occur.

  The L-by-L upper triangular matrix R is factored as  R = Q*S*P',
  if FACT = 'N', using SLICOT Library routine MB03UD, where Q and P
  are L-by-L orthogonal matrices and S is an L-by-L diagonal matrix
  with non-negative diagonal elements, SV(1), SV(2), ..., SV(L),
  ordered decreasingly. Then, the effective rank of R is estimated,
  and matrix (or matrix-vector) products and scalings are used to
  compute X. If FACT = 'F', only matrix (or matrix-vector) products
  and scalings are performed.

  Option JOBP = 'P' should be used only if the pseudoinverse is
  really needed. Usually, it is possible to avoid the use of
  pseudoinverse, by computing least squares solutions.
  The routine uses BLAS 3 calculations if LDWORK >= M*N, and BLAS 2
  calculations, otherwise. No advantage of any additional workspace
  larger than L is taken for matrix products, but the routine can
  be called repeatedly for chunks of columns of B, if LDWORK < M*N.

Program Text

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