## SB02PD

### Solution of continuous-time algebraic Riccati equations (matrix sign function method) with error bounds and condition estimates

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

Purpose

      To solve the real continuous-time matrix algebraic Riccati
      equation

         op(A)'*X + X*op(A) + Q - X*G*X = 0,

      where op(A) = A or A' = A**T and G, Q are symmetric (G = G**T,
      Q = Q**T). The matrices A, G and Q are N-by-N and the solution X
      is an N-by-N symmetric matrix.

      An error bound on the solution and a condition estimate are also
      optionally provided.

      It is assumed that the matrices A, G and Q are such that the
      corresponding Hamiltonian matrix has N eigenvalues with negative
      real parts.

Specification

          SUBROUTINE SB02PD( JOB, TRANA, UPLO, N, A, LDA, G, LDG, Q, LDQ, X,
         $                   LDX, RCOND, FERR, WR, WI, IWORK, DWORK, LDWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER          JOB, TRANA, UPLO
          INTEGER            INFO, LDA, LDG, LDQ, LDWORK, LDX, N
          DOUBLE PRECISION   FERR, RCOND
    C     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), DWORK( * ), G( LDG, * ),
         $                   Q( LDQ, * ), WI( * ), WR( * ), X( LDX, * )

Arguments

Mode Parameters

      JOB     CHARACTER*1
              Specifies the computation to be performed, as follows:
              = 'X':  Compute the solution only;
              = 'A':  Compute all: the solution, reciprocal condition
                      number, and the error bound.

      TRANA   CHARACTER*1
              Specifies the option op(A):
              = 'N':  op(A) = A    (No transpose);
              = 'T':  op(A) = A**T (Transpose);
              = 'C':  op(A) = A**T (Conjugate transpose = Transpose).

      UPLO    CHARACTER*1
              Specifies which triangle of the matrices G and Q is
              stored, as follows:
              = 'U':  Upper triangles of G and Q are stored;
              = 'L':  Lower triangles of G and Q are stored.

Input/Output Parameters

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

      A       (input) DOUBLE PRECISION array, dimension (LDA,N)
              The leading N-by-N part of this array must contain the
              coefficient matrix A of the equation.

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

      G       (input) DOUBLE PRECISION array, dimension (LDG,N)
              If UPLO = 'U', the leading N-by-N upper triangular part of
              this array must contain the upper triangular part of the
              matrix G.
              If UPLO = 'L', the leading N-by-N lower triangular part of
              this array must contain the lower triangular part of the
              matrix G.

      LDG     INTEGER
              The leading dimension of the array G.  LDG >= max(1,N).

      Q       (input) DOUBLE PRECISION array, dimension (LDQ,N)
              If UPLO = 'U', the leading N-by-N upper triangular part of
              this array must contain the upper triangular part of the
              matrix Q.
              If UPLO = 'L', the leading N-by-N lower triangular part of
              this array must contain the lower triangular part of the
              matrix Q.

      LDQ     INTEGER
              The leading dimension of the array Q.  LDQ >= max(1,N).

      X       (output) DOUBLE PRECISION array, dimension (LDX,N)
              If INFO = 0, INFO = 2, or INFO = 4, the leading N-by-N
              part of this array contains the symmetric solution matrix
              X of the algebraic Riccati equation.

      LDX     INTEGER
              The leading dimension of the array X.  LDX >= max(1,N).

      RCOND   (output) DOUBLE PRECISION
              If JOB = 'A', the estimate of the reciprocal condition
              number of the Riccati equation.

      FERR    (output) DOUBLE PRECISION
              If JOB = 'A', the estimated forward error bound for the
              solution X. If XTRUE is the true solution, FERR bounds the
              magnitude of the largest entry in (X - XTRUE) divided by
              the magnitude of the largest entry in X.

      WR      (output) DOUBLE PRECISION array, dimension (N)
      WI      (output) DOUBLE PRECISION array, dimension (N)
              If JOB = 'A' and TRANA = 'N', WR and WI contain the real
              and imaginary parts, respectively, of the eigenvalues of
              the matrix A - G*X, i.e., the closed-loop system poles.
              If JOB = 'A' and TRANA = 'T' or 'C', WR and WI contain the
              real and imaginary parts, respectively, of the eigenvalues
              of the matrix A - X*G, i.e., the closed-loop system poles.
              If JOB = 'X', these arrays are not referenced.

Workspace

      IWORK   INTEGER array, dimension (LIWORK), where
              LIWORK >= 2*N,          if JOB = 'X';
              LIWORK >= max(2*N,N*N), if JOB = 'A'.

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0 or INFO = 2, DWORK(1) contains the
              optimal value of LDWORK. If JOB = 'A', then DWORK(2:N*N+1)
              and DWORK(N*N+2:2*N*N+1) contain a real Schur form of the
              closed-loop system matrix, Ac = A - G*X (if TRANA = 'N')
              or Ac = A - X*G (if TRANA = 'T' or 'C'), and the
              orthogonal matrix which reduced Ac to real Schur form,
              respectively.

      LDWORK  INTEGER
              The dimension of the array DWORK.
              LDWORK >= 4*N*N + 8*N + 1,               if JOB = 'X';
              LDWORK >= max( 4*N*N + 8*N + 1, 6*N*N ), if JOB = 'A'.
              For good performance, LDWORK should be larger, e.g.,
              LDWORK >= 4*N*N + 6*N +( 2*N+1 )*NB,     if JOB = 'X',
              where NB is the optimal blocksize.

              If LDWORK = -1, then a workspace query is assumed;
              the routine only calculates the optimal size of the
              DWORK array, returns this value as the first entry of
              the DWORK array, and no error message related to LDWORK
              is issued by XERBLA.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the Hamiltonian matrix has eigenvalues on the
                    imaginary axis, so the solution and error bounds
                    could not be computed;
              = 2:  the iteration for the matrix sign function failed to
                    converge after 50 iterations, but an approximate
                    solution and error bounds (if JOB = 'A') have been
                    computed;
              = 3:  the system of linear equations for the solution is
                    singular to working precision, so the solution and
                    error bounds could not be computed;
              = 4:  the matrix A-G*X (or A-X*G) cannot be reduced to
                    Schur canonical form and condition number estimate
                    and forward error estimate have not been computed.

Method

      The Riccati equation is solved by the matrix sign function
      approach [1], [2], implementing a scaling which enhances the
      numerical stability [4].

References

      [1] Bai, Z., Demmel, J., Dongarra, J., Petitet, A., Robinson, H.,
          and Stanley, K.
          The spectral decomposition of nonsymmetric matrices on
          distributed memory parallel computers.
          SIAM J. Sci. Comput., vol. 18, pp. 1446-1461, 1997.

      [2] Byers, R., He, C., and Mehrmann, V.
          The matrix sign function method and the computation of
          invariant subspaces.
          SIAM J. Matrix Anal. Appl., vol. 18, pp. 615-632, 1997.

      [3] Higham, N.J.
          Perturbation theory and backward error for AX-XB=C.
          BIT, vol. 33, pp. 124-136, 1993.

      [4] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.,
          DGRSVX and DMSRIC: Fortran 77 subroutines for solving
          continuous-time matrix algebraic Riccati equations with
          condition and accuracy estimates.
          Preprint SFB393/98-16, Fak. f. Mathematik, Technical
          University Chemnitz, May 1998.

Numerical Aspects

      The solution accuracy can be controlled by the output parameter
      FERR.

Further Comments

      The condition number of the Riccati equation is estimated as

      cond = ( norm(Theta)*norm(A) + norm(inv(Omega))*norm(Q) +
                  norm(Pi)*norm(G) ) / norm(X),

      where Omega, Theta and Pi are linear operators defined by

      Omega(W) = op(Ac)'*W + W*op(Ac),
      Theta(W) = inv(Omega(op(W)'*X + X*op(W))),
         Pi(W) = inv(Omega(X*W*X)),

      and the matrix Ac (the closed-loop system matrix) is given by
         Ac = A - G*X, if TRANA = 'N', or
         Ac = A - X*G, if TRANA = 'T' or 'C'.

      The program estimates the quantities

      sep(op(Ac),-op(Ac)') = 1 / norm(inv(Omega)),

      norm(Theta) and norm(Pi) using 1-norm condition estimator.

      The forward error bound is estimated using a practical error bound
      similar to the one proposed in [3].

Example

Program Text

    *     SB02PD EXAMPLE PROGRAM TEXT.
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 20 )
          INTEGER          LDA, LDG, LDQ, LDX
          PARAMETER        ( LDA = NMAX, LDG = NMAX, LDQ = NMAX,
         $                   LDX = NMAX )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = MAX( 2*NMAX, NMAX*NMAX ) )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX( 4*NMAX*NMAX + 8*NMAX,
         $                                 6*NMAX*NMAX ) + 1 )
    *     .. Local Scalars ..
          DOUBLE PRECISION FERR, RCOND
          INTEGER          I, INFO, J, N
          CHARACTER        JOB, TRANA, UPLO
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), G(LDG,NMAX),
         $                 Q(LDQ,NMAX), WI(NMAX), WR(NMAX),
         $                 X(LDX,NMAX)
          INTEGER          IWORK(LIWORK)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         SB02PD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, JOB, TRANA, UPLO
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99995 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
             READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,N ), I = 1,N )
    *        Find the solution matrix X.
             CALL SB02PD( JOB, TRANA, UPLO, N, A, LDA, G, LDG, Q, LDQ, X,
         $                LDX, RCOND, FERR, WR, WI, IWORK, DWORK, LDWORK,
         $                INFO )
    *
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             END IF
             IF ( INFO.EQ.0 .OR. INFO.EQ.2 .OR. INFO.EQ.4 ) THEN
                 WRITE ( NOUT, FMT = 99997 )
                 DO 20 I = 1, N
                    WRITE ( NOUT, FMT = 99996 ) ( X(I,J), J = 1,N )
       20        CONTINUE
                 IF ( LSAME( JOB, 'A' ) .AND. INFO.NE.4 ) THEN
                    WRITE ( NOUT, FMT = 99994 ) RCOND
                    WRITE ( NOUT, FMT = 99993 ) FERR
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' SB02PD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from SB02PD = ',I2)
    99997 FORMAT (' The solution matrix X is ')
    99996 FORMAT (20(1X,F8.4))
    99995 FORMAT (/' N is out of range.',/' N = ',I5)
    99994 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
    99993 FORMAT (/' Estimated error bound = ',F20.16)
          END

Program Data

     SB02PD EXAMPLE PROGRAM DATA
       2     A     N     U
       0.0   1.0
       0.0   0.0
       1.0   0.0
       0.0   2.0
       0.0   0.0
       0.0   1.0

Program Results

     SB02PD EXAMPLE PROGRAM RESULTS

     The solution matrix X is
       2.0000   1.0000
       1.0000   2.0000

     Estimated reciprocal condition number =   0.1333

     Estimated error bound =   0.0000000000000063

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