SB04PX

Solving a discrete-time Sylvester equation for an m-by-n matrix X, 1 <= m,n <= 2

Purpose

  To solve for the N1-by-N2 matrix X, 1 <= N1,N2 <= 2, in

         op(TL)*X*op(TR) + ISGN*X = SCALE*B,

  where TL is N1-by-N1, TR is N2-by-N2, B is N1-by-N2, and ISGN = 1
  or -1.  op(T) = T or T', where T' denotes the transpose of T.

      SUBROUTINE SB04PX( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
     $                   LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
C     .. Scalar Arguments ..
      LOGICAL            LTRANL, LTRANR
      INTEGER            INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
      DOUBLE PRECISION   SCALE, XNORM
C     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
     $                   X( LDX, * )

Mode Parameters

  LTRANL  LOGICAL
          Specifies the form of op(TL) to be used, as follows:
          = .FALSE.:  op(TL) = TL,
          = .TRUE. :  op(TL) = TL'.

  LTRANR  LOGICAL
          Specifies the form of op(TR) to be used, as follows:
          = .FALSE.:  op(TR) = TR,
          = .TRUE. :  op(TR) = TR'.

  ISGN    INTEGER
          Specifies the sign of the equation as described before.
          ISGN may only be 1 or -1.

  N1      (input) INTEGER
          The order of matrix TL.  N1 may only be 0, 1 or 2.

  N2      (input) INTEGER
          The order of matrix TR.  N2 may only be 0, 1 or 2.

  TL      (input) DOUBLE PRECISION array, dimension (LDTL,N1)
          The leading N1-by-N1 part of this array must contain the
          matrix TL.

  LDTL    INTEGER
          The leading dimension of array TL.  LDTL >= MAX(1,N1).

  TR      (input) DOUBLE PRECISION array, dimension (LDTR,N2)
          The leading N2-by-N2 part of this array must contain the
          matrix TR.

  LDTR    INTEGER
          The leading dimension of array TR.  LDTR >= MAX(1,N2).

  B       (input) DOUBLE PRECISION array, dimension (LDB,N2)
          The leading N1-by-N2 part of this array must contain the
          right-hand side of the equation.

  LDB     INTEGER
          The leading dimension of array B.  LDB >= MAX(1,N1).

  SCALE   (output) DOUBLE PRECISION
          The scale factor. SCALE is chosen less than or equal to 1
          to prevent the solution overflowing.

  X       (output) DOUBLE PRECISION array, dimension (LDX,N2)
          The leading N1-by-N2 part of this array contains the
          solution of the equation.
          Note that X may be identified with B in the calling
          statement.

  LDX     INTEGER
          The leading dimension of array X.  LDX >= MAX(1,N1).

  XNORM   (output) DOUBLE PRECISION
          The infinity-norm of the solution.

  INFO    INTEGER
          = 0:  successful exit;
          = 1:  if TL and -ISGN*TR have almost reciprocal
                eigenvalues, so TL or TR is perturbed to get a
                nonsingular equation.

          NOTE: In the interests of speed, this routine does not
                check the inputs for errors.

  The equivalent linear algebraic system of equations is formed and
  solved using Gaussian elimination with complete pivoting.

  [1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
      Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
      Ostrouchov, S., and Sorensen, D.
      LAPACK Users' Guide: Second Edition.
      SIAM, Philadelphia, 1995.

  The algorithm is stable and reliable, since Gaussian elimination
  with complete pivoting is used.

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Program Text

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