Purpose
To solve the real continuous-time Lyapunov matrix equation
op(A)'*X + X*op(A) = scale*C,
estimate the conditioning, and compute an error bound on the
solution X, where op(A) = A or A' (A**T), the matrix A is N-by-N,
the right hand side C and the solution X are N-by-N symmetric
matrices (C = C', X = X'), and scale is an output scale factor,
set less than or equal to 1 to avoid overflow in X.
Specification
SUBROUTINE SB03TD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A,
$ LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEP,
$ RCOND, FERR, WR, WI, IWORK, DWORK, LDWORK,
$ INFO )
C .. Scalar Arguments ..
CHARACTER FACT, JOB, LYAPUN, TRANA, UPLO
INTEGER INFO, LDA, LDC, LDT, LDU, LDWORK, LDX, N
DOUBLE PRECISION FERR, RCOND, SCALE, SEP
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ T( LDT, * ), U( LDU, * ), WI( * ), WR( * ),
$ X( LDX, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'X': Compute the solution only;
= 'S': Compute the separation only;
= 'C': Compute the reciprocal condition number only;
= 'E': Compute the error bound only;
= 'A': Compute all: the solution, separation, reciprocal
condition number, and the error bound.
FACT CHARACTER*1
Specifies whether or not the real Schur factorization
of the matrix A is supplied on entry, as follows:
= 'F': On entry, T and U (if LYAPUN = 'O') 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 T and U (if
LYAPUN = 'O').
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).
UPLO CHARACTER*1
Specifies which part of the symmetric matrix C is to be
used, as follows:
= 'U': Upper triangular part;
= 'L': Lower triangular part.
LYAPUN CHARACTER*1
Specifies whether or not the original or "reduced"
Lyapunov equations should be solved, as follows:
= 'O': Solve the original Lyapunov equations, updating
the right-hand sides and solutions with the
matrix U, e.g., X <-- U'*X*U;
= 'R': Solve reduced Lyapunov equations only, without
updating the right-hand sides and solutions.
This means that a real Schur form T of A appears
in the equation, instead of A.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X, and C. N >= 0.
SCALE (input or output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'E', SCALE is an input argument:
the scale factor, set by a Lyapunov solver.
0 <= SCALE <= 1.
If JOB = 'X' or JOB = 'A', SCALE is an output argument:
the scale factor, scale, set less than or equal to 1 to
prevent the solution overflowing.
If JOB = 'S', this argument is not used.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If FACT = 'N' or (LYAPUN = 'O' and JOB <> 'X'), the
leading N-by-N part of this array must contain the
original matrix A.
If FACT = 'F' and (LYAPUN = 'R' or JOB = 'X'), A is
not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if FACT = 'N' or LYAPUN = 'O' and
JOB <> 'X';
LDA >= 1, otherwise.
T (input/output) DOUBLE PRECISION array, dimension
(LDT,N)
If FACT = 'F', then on entry the leading N-by-N upper
Hessenberg part of this array must contain the upper
quasi-triangular matrix T in Schur canonical form from a
Schur factorization of A.
If FACT = 'N', then this array need not be set on input.
On exit, (if INFO = 0 or INFO = N+1, for FACT = 'N') the
leading N-by-N upper Hessenberg part of this array
contains the upper quasi-triangular matrix T in Schur
canonical form from a Schur factorization of A.
The contents of array T is not modified if FACT = 'F'.
LDT INTEGER
The leading dimension of the array T. LDT >= MAX(1,N).
U (input or output) DOUBLE PRECISION array, dimension
(LDU,N)
If LYAPUN = 'O' and FACT = 'F', then U is an input
argument and on entry, the leading N-by-N part of this
array must contain the orthogonal matrix U from a real
Schur factorization of A.
If LYAPUN = 'O' and 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 a real Schur
factorization of A.
If LYAPUN = 'R', the array U is not referenced.
LDU INTEGER
The leading dimension of the array U.
LDU >= 1, if LYAPUN = 'R';
LDU >= MAX(1,N), if LYAPUN = 'O'.
C (input) DOUBLE PRECISION array, dimension (LDC,N)
If JOB <> 'S' and UPLO = 'U', the leading N-by-N upper
triangular part of this array must contain the upper
triangular part of the matrix C of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
If JOB <> 'S' and UPLO = 'L', the leading N-by-N lower
triangular part of this array must contain the lower
triangular part of the matrix C of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
The remaining strictly triangular part of this array is
used as workspace.
If JOB = 'X', then this array may be identified with X
in the call of this routine.
If JOB = 'S', the array C is not referenced.
LDC INTEGER
The leading dimension of the array C.
LDC >= 1, if JOB = 'S';
LDC >= MAX(1,N), otherwise.
X (input or output) DOUBLE PRECISION array, dimension
(LDX,N)
If JOB = 'C' or 'E', then X is an input argument and on
entry, the leading N-by-N part of this array must contain
the symmetric solution matrix X of the original Lyapunov
equation (with matrix A), if LYAPUN = 'O', or of the
reduced Lyapunov equation (with matrix T), if
LYAPUN = 'R'.
If JOB = 'X' or 'A', then X is an output argument and on
exit, if INFO = 0 or INFO = N+1, the leading N-by-N part
of this array contains the symmetric solution matrix X of
of the original Lyapunov equation (with matrix A), if
LYAPUN = 'O', or of the reduced Lyapunov equation (with
matrix T), if LYAPUN = 'R'.
If JOB = 'S', the array X is not referenced.
LDX INTEGER
The leading dimension of the array X.
LDX >= 1, if JOB = 'S';
LDX >= MAX(1,N), otherwise.
SEP (output) DOUBLE PRECISION
If JOB = 'S' or JOB = 'C' or JOB = 'A', and INFO = 0 or
INFO = N+1, SEP contains the estimated separation of the
matrices op(A) and -op(A)', sep(op(A),-op(A)').
If N = 0, or X = 0, or JOB = 'X' or JOB = 'E', SEP is not
referenced.
RCOND (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'A', an estimate of the reciprocal
condition number of the continuous-time Lyapunov equation.
If N = 0 or X = 0, RCOND is set to 1 or 0, respectively.
If JOB = 'X' or JOB = 'S' or JOB = 'E', RCOND is not
referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'E' or JOB = 'A', 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 N = 0 or X = 0, FERR is set to 0.
If JOB = 'X' or JOB = 'S' or JOB = 'C', 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.
If JOB = 'X', then
LDWORK >= MAX(1,N*N), if FACT = 'F';
LDWORK >= MAX(1,MAX(N*N,3*N)), if FACT = 'N'.
If JOB = 'S' or JOB = 'C', then
LDWORK >= MAX(1,2*N*N), if FACT = 'F';
LDWORK >= MAX(1,2*N*N,3*N), if FACT = 'N'.
If JOB = 'E', or JOB = 'A', and LYAPUN = 'O', then
LDWORK >= MAX(1,3*N*N);
If JOB = 'E', or JOB = 'A', and LYAPUN = 'R', then
LDWORK >= MAX(1,3*N*N+N-1).
For optimum performance LDWORK should sometimes 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, i <= N, the QR algorithm failed to
complete the reduction to Schur canonical form (see
LAPACK Library routine DGEES); on exit, the matrix
T(i+1:N,i+1:N) contains the partially converged
Schur form, and the elements i+1:n of WR and WI
contain the real and imaginary parts, respectively,
of the converged eigenvalues; this error is unlikely
to appear;
= N+1: if the matrices T and -T' have common or very
close eigenvalues; perturbed values were used to
solve Lyapunov equations, but the matrix T, if given
(for FACT = 'F'), is unchanged.
Method
After reducing matrix A to real Schur canonical form (if needed), 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. The condition number of the continuous-time Lyapunov equation is estimated as cond = (norm(Theta)*norm(A) + norm(inv(Omega))*norm(C))/norm(X), where Omega and Theta are linear operators defined by Omega(W) = op(A)'*W + W*op(A), Theta(W) = inv(Omega(op(W)'*X + X*op(W))). The routine estimates the quantities sep(op(A),-op(A)') = 1 / norm(inv(Omega)) and norm(Theta) using 1-norm condition estimators. The forward error bound is estimated using a practical error bound similar to the one proposed in [2].References
[1] 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.
[2] Higham, N.J.
Perturbation theory and backward error for AX-XB=C.
BIT, vol. 33, pp. 124-136, 1993.
Numerical Aspects
3 The algorithm requires 0(N ) operations. The accuracy of the estimates obtained depends on the solution accuracy and on the properties of the 1-norm estimator.Further Comments
The separation of op(A) and -op(A)' can also be defined as
sep( 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( I(N), op(A)' ) + kprod( op(A)', I(N) ).
I(N) is an N-by-N identity matrix, and kprod denotes the Kronecker
product. The routine 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.
Example
Program Text
* SB03TD 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, LDC, LDT, LDU, LDX
PARAMETER ( LDA = NMAX, LDC = NMAX, LDT = NMAX,
$ LDU = NMAX, LDX = NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX*NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( 1, 3*NMAX*NMAX + NMAX - 1 ) )
* .. Local Scalars ..
DOUBLE PRECISION FERR, RCOND, SCALE, SEP
INTEGER I, INFO, J, N
CHARACTER*1 DICO, FACT, JOB, LYAPUN, TRANA, UPLO
* .. Local Arrays ..
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), C(LDC,NMAX), DWORK(LDWORK),
$ T(LDT,NMAX), U(LDU,NMAX), X(LDX,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL SB03TD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
DICO = 'C'
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, JOB, FACT, TRANA, UPLO, LYAPUN
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'E' ) )
$ READ ( NIN, FMT = * ) SCALE
IF ( LSAME( FACT, 'N' ) .OR. ( LSAME( LYAPUN, 'O' ) .AND.
$ .NOT.LSAME( JOB, 'X') ) )
$ READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACT, 'F' ) ) THEN
READ ( NIN, FMT = * ) ( ( T(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( LYAPUN, 'O' ) )
$ READ ( NIN, FMT = * ) ( ( U(I,J), J = 1,N ), I = 1,N )
END IF
IF ( .NOT.LSAME( JOB, 'S' ) )
$ READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'E' ) )
$ READ ( NIN, FMT = * ) ( ( X(I,J), J = 1,N ), I = 1,N )
* Solve the continuous-time Lyapunov matrix equation and/or
* estimate the condition and error bound on the solution.
CALL SB03TD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A, LDA,
$ T, LDT, U, LDU, C, LDC, X, LDX, SEP, RCOND, FERR,
$ DWORK(1), DWORK(N+1), IWORK, DWORK(2*N+1),
$ LDWORK-2*N, INFO )
*
IF ( INFO.EQ.0 ) THEN
IF ( LSAME( JOB, 'X' ) .OR. LSAME( JOB, 'A' ) ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( X(I,J), J = 1,N )
10 CONTINUE
WRITE ( NOUT, FMT = 99993 ) SCALE
END IF
IF ( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'C' )
$ .OR. LSAME( JOB, 'A' ) )
$ WRITE ( NOUT, FMT = 99992 ) SEP
IF ( LSAME( JOB, 'C' ) .OR. LSAME( JOB, 'A' ) )
$ WRITE ( NOUT, FMT = 99991 ) RCOND
IF ( LSAME( JOB, 'E' ) .OR. LSAME( JOB, 'A' ) )
$ WRITE ( NOUT, FMT = 99990 ) FERR
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB03TD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB03TD =',I2)
99996 FORMAT (' The solution matrix X is')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' Scaling factor = ',F8.4)
99992 FORMAT (/' Estimated separation = ',F8.4)
99991 FORMAT (/' Estimated reciprocal condition number = ',F8.4)
99990 FORMAT (/' Estimated error bound = ',F8.4)
END
Program Data
SB03TD EXAMPLE PROGRAM DATA 3 A N N U O 3.0 1.0 1.0 1.0 3.0 0.0 0.0 0.0 3.0 25.0 24.0 15.0 24.0 32.0 8.0 15.0 8.0 40.0Program Results
SB03TD EXAMPLE PROGRAM RESULTS The solution matrix X is 3.2604 2.7187 1.8616 2.7187 4.4271 0.5699 1.8616 0.5699 6.0461 Scaling factor = 1.0000 Estimated separation = 4.9068 Estimated reciprocal condition number = 0.3611 Estimated error bound = 0.0000