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.
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. 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 1.0Program 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