MB02UW

Solving a set of linear systems of order at most 2 with possible scaling and perturbation of system matrix

Purpose

  To solve a system of the form  A X = s B  or  A' X = s B  with
  possible scaling ("s") and perturbation of A.  (A' means
  A-transpose.)  A is an N-by-N real matrix, and X and B are
  N-by-M matrices.  N may be 1 or 2.  The scalar "s" is a scaling
  factor (.LE. 1), computed by this subroutine, which is so chosen
  that X can be computed without overflow.  X is further scaled if
  necessary to assure that norm(A)*norm(X) is less than overflow.

      SUBROUTINE MB02UW( LTRANS, N, M, PAR, A, LDA, B, LDB, SCALE,
     $                   IWARN )
C     .. Scalar Arguments ..
      LOGICAL            LTRANS
      INTEGER            IWARN, LDA, LDB, N, M
      DOUBLE PRECISION   SCALE, SMIN
C     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), PAR( * )

Mode Parameters

  LTRANS  LOGICAL
          Specifies if A or A-transpose is to be used, as follows:
          =.TRUE. :  A-transpose will be used;
          =.FALSE.:  A will be used (not transposed).

  N       (input) INTEGER
          The order of the matrix A.  It may (only) be 1 or 2.

  M       (input) INTEGER
          The number of right hand size vectors.

  PAR     (input) DOUBLE PRECISION array, dimension (3)
          Machine related parameters:
          PAR(1) =: PREC  (machine precision)*base, DLAMCH( 'P' );
          PAR(2) =: SFMIN safe minimum,             DLAMCH( 'S' );
          PAR(3) =: SMIN  The desired lower bound on the singular
                          values of A.  This should be a safe
                          distance away from underflow or overflow,
                          say, between (underflow/machine precision)
                          and (machine precision * overflow).

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The leading N-by-N part of this array must contain the
          matrix A.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= N.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
          On entry, the leading N-by-M part of this array must
          contain the matrix B (right-hand side).
          On exit, the leading N-by-M part of this array contains
          the N-by-M matrix X (unknowns).

  LDB     INTEGER
          The leading dimension of the array B.  LDB >= N.

  SCALE   (output) DOUBLE PRECISION
          The scale factor that B must be multiplied by to insure
          that overflow does not occur when computing X.  Thus,
          A X  will be SCALE*B, not B (ignoring perturbations of A).
          SCALE will be at most 1.

  IWARN   INTEGER
          = 0:  no warnings (A did not have to be perturbed);
          = 1:  A had to be perturbed to make its smallest (or only)
                singular value greater than SMIN (see below).

  Gaussian elimination with complete pivoting is used. The matrix A
  is slightly perturbed if it is (close to being) singular.

  If both singular values of A are less than SMIN, SMIN*identity
  will be used instead of A.  If only one singular value is less
  than SMIN, one element of A will be perturbed enough to make the
  smallest singular value roughly SMIN.  If both singular values
  are at least SMIN, A will not be perturbed.  In any case, the
  perturbation will be at most some small multiple of
  max( SMIN, EPS*norm(A) ), where EPS is the machine precision
  (see LAPACK Library routine DLAMCH).  The singular values are
  computed by infinity-norm approximations, and thus will only be
  correct to a factor of 2 or so.

  Note: all input quantities are assumed to be smaller than overflow
  by a reasonable factor.  (See BIGNUM.)  In the interests of speed,
  this routine does not check the inputs for errors.

Program Text

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