## SB03MV

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

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

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.

Specification

          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, * )

Arguments

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.

Input/Output Parameters

      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.

Error Indicator

      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.

Method

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

References

      [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.

Numerical Aspects

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

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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