MB02YD

Solving the linear system A x = b, D x = 0, D diagonal

Purpose

  To determine a vector x which solves the system of linear
  equations

        A*x = b ,     D*x = 0 ,

  in the least squares sense, where A is an m-by-n matrix,
  D is an n-by-n diagonal matrix, and b is an m-vector.
  It is assumed that a QR factorization, with column pivoting, of A
  is available, that is, A*P = Q*R, where P is a permutation matrix,
  Q has orthogonal columns, and R is an upper triangular matrix
  with diagonal elements of nonincreasing magnitude.
  The routine needs the full upper triangle of R, the permutation
  matrix P, and the first n components of Q'*b (' denotes the
  transpose). The system A*x = b, D*x = 0, is then equivalent to

        R*z = Q'*b ,  P'*D*P*z = 0 ,                             (1)

  where x = P*z. If this system does not have full rank, then a
  least squares solution is obtained. On output, MB02YD also
  provides an upper triangular matrix S such that

        P'*(A'*A + D*D)*P = S'*S .

  The system (1) is equivalent to S*z = c , where c contains the
  first n components of the vector obtained by applying to
  [ (Q'*b)'  0 ]' the transformations which triangularized
  [ R'  P'*D*P ]', getting S.

      SUBROUTINE MB02YD( COND, N, R, LDR, IPVT, DIAG, QTB, RANK, X, TOL,
     $                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         COND
      INTEGER           INFO, LDR, LDWORK, N, RANK
      DOUBLE PRECISION  TOL
C     .. Array Arguments ..
      INTEGER           IPVT(*)
      DOUBLE PRECISION  DIAG(*), DWORK(*), QTB(*), R(LDR,*), X(*)

Mode Parameters

  COND    CHARACTER*1
          Specifies whether the condition of the matrix S should be
          estimated, as follows:
          = 'E' :  use incremental condition estimation and store
                   the numerical rank of S in RANK;
          = 'N' :  do not use condition estimation, but check the
                   diagonal entries of S for zero values;
          = 'U' :  use the rank already stored in RANK.

  N       (input) INTEGER
          The order of the matrix R.  N >= 0.

  R       (input/output) DOUBLE PRECISION array, dimension (LDR, N)
          On entry, the leading N-by-N upper triangular part of this
          array must contain the upper triangular matrix R.
          On exit, the full upper triangle is unaltered, and the
          strict lower triangle contains the strict upper triangle
          (transposed) of the upper triangular matrix S.

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

  IPVT    (input) INTEGER array, dimension (N)
          This array must define the permutation matrix P such that
          A*P = Q*R. Column j of P is column IPVT(j) of the identity
          matrix.

  DIAG    (input) DOUBLE PRECISION array, dimension (N)
          This array must contain the diagonal elements of the
          matrix D.

  QTB     (input) DOUBLE PRECISION array, dimension (N)
          This array must contain the first n elements of the
          vector Q'*b.

  RANK    (input or output) INTEGER
          On entry, if COND = 'U', this parameter must contain the
          (numerical) rank of the matrix S.
          On exit, if COND = 'E' or 'N', this parameter contains
          the numerical rank of the matrix S, estimated according
          to the value of COND.

  X       (output) DOUBLE PRECISION array, dimension (N)
          This array contains the least squares solution of the
          system A*x = b, D*x = 0.

  TOL     DOUBLE PRECISION
          If COND = 'E', the tolerance to be used for finding the
          rank of the matrix S. If the user sets TOL > 0, then the
          given value of TOL is used as a lower bound for the
          reciprocal condition number;  a (sub)matrix whose
          estimated condition number is less than 1/TOL is
          considered to be of full rank.  If the user sets TOL <= 0,
          then an implicitly computed, default tolerance, defined by
          TOLDEF = N*EPS,  is used instead, where EPS is the machine
          precision (see LAPACK Library routine DLAMCH).
          This parameter is not relevant if COND = 'U' or 'N'.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, the first N elements of this array contain the
          diagonal elements of the upper triangular matrix S, and
          the next N elements contain the solution z.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= 4*N, if COND =  'E';
          LDWORK >= 2*N, if COND <> 'E'.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value.

  Standard plane rotations are used to annihilate the elements of
  the diagonal matrix D, updating the upper triangular matrix R
  and the first n elements of the vector Q'*b. A basic least squares
  solution is computed.

  [1] More, J.J., Garbow, B.S, and Hillstrom, K.E.
      User's Guide for MINPACK-1.
      Applied Math. Division, Argonne National Laboratory, Argonne,
      Illinois, Report ANL-80-74, 1980.

                            2
  The algorithm requires 0(N ) operations and is backward stable.

  This routine is a LAPACK-based modification of QRSOLV from the
  MINPACK package [1], and with optional condition estimation.
  The option COND = 'U' is useful when dealing with several
  right-hand side vectors.

Program Text

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