SB03MV

Solving a discrete-time Lyapunov equation for a 2-by-2 matrix

Purpose

  To solve for the 2-by-2 symmetric matrix X in

         op(T)'*X*op(T) - X = SCALE*B,

  where T is 2-by-2, B is symmetric 2-by-2, and op(T) = T or T',
  where T' denotes the transpose of T.

      SUBROUTINE SB03MV( LTRAN, LUPPER, T, LDT, B, LDB, SCALE, X, LDX,
     $                   XNORM, INFO )
C     .. Scalar Arguments ..
      LOGICAL            LTRAN, LUPPER
      INTEGER            INFO, LDB, LDT, LDX
      DOUBLE PRECISION   SCALE, XNORM
C     .. Array Arguments ..
      DOUBLE PRECISION   B( LDB, * ), T( LDT, * ), X( LDX, * )

Mode Parameters

  LTRAN   LOGICAL
          Specifies the form of op(T) to be used, as follows:
          = .FALSE.:  op(T) = T,
          = .TRUE. :  op(T) = T'.

  LUPPER  LOGICAL
          Specifies which triangle of the matrix B is used, and
          which triangle of the matrix X is computed, as follows:
          = .TRUE. :  The upper triangular part;
          = .FALSE.:  The lower triangular part.

  T       (input) DOUBLE PRECISION array, dimension (LDT,2)
          The leading 2-by-2 part of this array must contain the
          matrix T.

  LDT     INTEGER
          The leading dimension of array T.  LDT >= 2.

  B       (input) DOUBLE PRECISION array, dimension (LDB,2)
          On entry with LUPPER = .TRUE., the leading 2-by-2 upper
          triangular part of this array must contain the upper
          triangular part of the symmetric matrix B and the strictly
          lower triangular part of B is not referenced.
          On entry with LUPPER = .FALSE., the leading 2-by-2 lower
          triangular part of this array must contain the lower
          triangular part of the symmetric matrix B and the strictly
          upper triangular part of B is not referenced.

  LDB     INTEGER
          The leading dimension of array B.  LDB >= 2.

  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,2)
          On exit with LUPPER = .TRUE., the leading 2-by-2 upper
          triangular part of this array contains the upper
          triangular part of the symmetric solution matrix X and the
          strictly lower triangular part of X is not referenced.
          On exit with LUPPER = .FALSE., the leading 2-by-2 lower
          triangular part of this array contains the lower
          triangular part of the symmetric solution matrix X and the
          strictly upper triangular part of X is not referenced.
          Note that X may be identified with B in the calling
          statement.

  LDX     INTEGER
          The leading dimension of array X.  LDX >= 2.

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

  INFO    INTEGER
          = 0:  successful exit;
          = 1:  if T has almost reciprocal eigenvalues, so T
                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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