## SB03PD

### Solution of discrete-time Lyapunov equations and separation estimation

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

Purpose

      To solve the real discrete Lyapunov matrix equation

             op(A)'*X*op(A) - X = scale*C

      and/or estimate the quantity, called separation,

          sepd(op(A),op(A)') = min norm(op(A)'*X*op(A) - X)/norm(X)

      where op(A) = A or A' (A**T) and C is symmetric (C = C').
      (A' denotes the transpose of the matrix A.) A is N-by-N, the right
      hand side C and the solution X are N-by-N, and scale is an output
      scale factor, set less than or equal to 1 to avoid overflow in X.

Specification

          SUBROUTINE SB03PD( JOB, FACT, TRANA, N, A, LDA, U, LDU, C, LDC,
         $                   SCALE, SEPD, FERR, WR, WI, IWORK, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          FACT, JOB, TRANA
          INTEGER            INFO, LDA, LDC, LDU, LDWORK, N
          DOUBLE PRECISION   FERR, SCALE, SEPD
    C     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), DWORK( * ),
         $                   U( LDU, * ), WI( * ), WR( * )

Arguments

Mode Parameters

      JOB     CHARACTER*1
              Specifies the computation to be performed, as follows:
              = 'X':  Compute the solution only;
              = 'S':  Compute the separation only;
              = 'B':  Compute both the solution and the separation.

      FACT    CHARACTER*1
              Specifies whether or not the real Schur factorization
              of the matrix A is supplied on entry, as follows:
              = 'F':  On entry, A and U contain the factors from the
                      real Schur factorization of the matrix A;
              = 'N':  The Schur factorization of A will be computed
                      and the factors will be stored in A and U.

      TRANA   CHARACTER*1
              Specifies the form of op(A) to be used, as follows:
              = 'N':  op(A) = A    (No transpose);
              = 'T':  op(A) = A**T (Transpose);
              = 'C':  op(A) = A**T (Conjugate transpose = Transpose).

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrices A, X, and C.  N >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the matrix A. If FACT = 'F', then A contains
              an upper quasi-triangular matrix in Schur canonical form.
              On exit, if INFO = 0 or INFO = N+1, the leading N-by-N
              part of this array contains the upper quasi-triangular
              matrix in Schur canonical form from the Shur factorization
              of A. The contents of array A is not modified if
              FACT = 'F'.

      LDA     INTEGER
              The leading dimension of array A.  LDA >= MAX(1,N).

      U       (input or output) DOUBLE PRECISION array, dimension
              (LDU,N)
              If FACT = 'F', then U is an input argument and on entry
              it must contain the orthogonal matrix U from the real
              Schur factorization of A.
              If FACT = 'N', then U is an output argument and on exit,
              if INFO = 0 or INFO = N+1, it contains the orthogonal
              N-by-N matrix from the real Schur factorization of A.

      LDU     INTEGER
              The leading dimension of array U.  LDU >= MAX(1,N).

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry with JOB = 'X' or 'B', the leading N-by-N part of
              this array must contain the symmetric matrix C.
              On exit with JOB = 'X' or 'B', if INFO = 0 or INFO = N+1,
              the leading N-by-N part of C has been overwritten by the
              symmetric solution matrix X.
              If JOB = 'S', C is not referenced.

      LDC     INTEGER
              The leading dimension of array C.
              LDC >= 1,        if JOB = 'S';
              LDC >= MAX(1,N), otherwise.

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

      SEPD    (output) DOUBLE PRECISION
              If JOB = 'S' or JOB = 'B', and INFO = 0 or INFO = N+1,
              SEPD contains the estimate in the 1-norm of
              sepd(op(A),op(A)').
              If JOB = 'X' or N = 0, SEPD is not referenced.

      FERR    (output) DOUBLE PRECISION
              If JOB = 'B', and INFO = 0 or INFO = N+1, FERR contains
              an estimated forward error bound for the solution X.
              If XTRUE is the true solution, FERR bounds the relative
              error in the computed solution, measured in the Frobenius
              norm:  norm(X - XTRUE)/norm(XTRUE).
              If JOB = 'X' or JOB = 'S', FERR is not referenced.

      WR      (output) DOUBLE PRECISION array, dimension (N)
      WI      (output) DOUBLE PRECISION array, dimension (N)
              If FACT = 'N', and INFO = 0 or INFO = N+1, WR and WI
              contain the real and imaginary parts, respectively, of the
              eigenvalues of A.
              If FACT = 'F', WR and WI are not referenced.

Workspace

      IWORK   INTEGER array, dimension (N*N)
              This array is not referenced if JOB = 'X'.

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0 or INFO = N+1, DWORK(1) returns the
              optimal value of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.  LDWORK >= 1 and
              If JOB = 'X' then
                 If FACT = 'F', LDWORK >= MAX(N*N,2*N);
                 If FACT = 'N', LDWORK >= MAX(N*N,3*N).
              If JOB = 'S' or JOB = 'B' then
                 LDWORK >= 2*N*N + 2*N.
              For optimum performance LDWORK should be larger.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              > 0:  if INFO = i, the QR algorithm failed to compute all
                    the eigenvalues (see LAPACK Library routine DGEES);
                    elements i+1:n of WR and WI contain eigenvalues
                    which have converged, and A contains the partially
                    converged Schur form;
              = N+1:  if matrix A has almost reciprocal eigenvalues;
                    perturbed values were used to solve the equation
                    (but the matrix A is unchanged).

Method

      After reducing matrix A to real Schur canonical form (if needed),
      a discrete-time version of the Bartels-Stewart algorithm is used.
      A set of equivalent linear algebraic systems of equations of order
      at most four are formed and solved using Gaussian elimination with
      complete pivoting.

References

      [1] Barraud, A.Y.                   T
          A numerical algorithm to solve A XA - X = Q.
          IEEE Trans. Auto. Contr., AC-22, pp. 883-885, 1977.

      [2] Bartels, R.H. and Stewart, G.W.  T
          Solution of the matrix equation A X + XB = C.
          Comm. A.C.M., 15, pp. 820-826, 1972.

Numerical Aspects

                                3
      The algorithm requires 0(N ) operations.

Further Comments

      SEPD is defined as

             sepd( op(A), op(A)' ) = sigma_min( T )

      where sigma_min(T) is the smallest singular value of the
      N*N-by-N*N matrix

         T = kprod( op(A)', op(A)' ) - I(N**2).

      I(N**2) is an N*N-by-N*N identity matrix, and kprod denotes the
      Kronecker product. The program estimates sigma_min(T) by the
      reciprocal of an estimate of the 1-norm of inverse(T). The true
      reciprocal 1-norm of inverse(T) cannot differ from sigma_min(T) by
      more than a factor of N.

      When SEPD is small, small changes in A, C can cause large changes
      in the solution of the equation. An approximate bound on the
      maximum relative error in the computed solution is

                             EPS * norm(A)**2 / SEPD

      where EPS is the machine precision.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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