MB02UW - SLICOT Library Routine Documentation

## MB02UW

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

[Specification] [Arguments] [Method] [References] [Comments] [Example]

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.

Specification

          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( * )

Arguments

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).

Input/Output Parameters

      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.

Warning Indicator

      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).

Method

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

Further Comments

      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.

Example

Program Text

      None

Program Data

      None

Program Results

      None

Return to Supporting Routines index