Purpose
To estimate the conditioning and compute an error bound on the
solution of the real discrete-time Lyapunov matrix equation
op(A)'*X*op(A) - X = scale*C
where op(A) = A or A' (A**T) and C is symmetric (C = C**T). The
matrix A is N-by-N, the right hand side C and the solution X are
N-by-N symmetric matrices, and scale is a given scale factor.
Specification
SUBROUTINE SB03SD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A,
$ LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEPD,
$ RCOND, FERR, 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, SEPD
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ),
$ T( LDT, * ), U( LDU, * ), X( LDX, * )
Arguments
Mode Parameters
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'C': Compute the reciprocal condition number only;
= 'E': Compute the error bound only;
= 'B': Compute both the 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 Lyapunov equations
should be solved in the iterative estimation process,
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.
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, X and C. N >= 0.
SCALE (input) DOUBLE PRECISION
The scale factor, scale, set by a Lyapunov solver.
0 <= SCALE <= 1.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If FACT = 'N' or LYAPUN = 'O', the leading N-by-N part of
this array must contain the original matrix A.
If FACT = 'F' and LYAPUN = 'R', A is not referenced.
LDA INTEGER
The leading dimension of the array A.
LDA >= MAX(1,N), if FACT = 'N' or LYAPUN = 'O';
LDA >= 1, if FACT = 'F' and LYAPUN = 'R'.
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.
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 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 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'.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,N).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
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'.
The array X is modified internally, but restored on exit.
LDX INTEGER
The leading dimension of the array X. LDX >= MAX(1,N).
SEPD (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'B', the estimated quantity
sepd(op(A),op(A)').
If N = 0, or X = 0, or JOB = 'E', SEPD is not referenced.
RCOND (output) DOUBLE PRECISION
If JOB = 'C' or JOB = 'B', an estimate of the reciprocal
condition number of the discrete-time Lyapunov equation.
If N = 0 or X = 0, RCOND is set to 1 or 0, respectively.
If JOB = 'E', RCOND is not referenced.
FERR (output) DOUBLE PRECISION
If JOB = 'E' or JOB = 'B', an 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.
If N = 0 or X = 0, FERR is set to 0.
If JOB = 'C', FERR is not referenced.
Workspace
IWORK INTEGER array, dimension (N*N)
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 dimension of the array DWORK.
LDWORK >= 1, if N = 0; else,
LDWORK >= MAX(3,2*N*N) + N*N, if JOB = 'C',
FACT = 'F';
LDWORK >= MAX(MAX(3,2*N*N) + N*N, 5*N), if JOB = 'C',
FACT = 'N';
LDWORK >= MAX(3,2*N*N) + N*N + 2*N, if JOB = 'E', or
JOB = 'B'.
For optimum performance LDWORK should sometimes be larger.
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;
> 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 DWORK(i+1:N) and DWORK(N+i+1:2*N)
contain the real and imaginary parts, respectively,
of the converged eigenvalues; this error is unlikely
to appear;
= N+1: if the matrix T has almost reciprocal eigenvalues;
perturbed values were used to solve Lyapunov
equations, but the matrix T, if given (for
FACT = 'F'), is unchanged.
Method
The condition number of the discrete-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*op(A) - W, Theta(W) = inv(Omega(op(W)'*X*op(A) + op(A)'*X*op(W))). The routine estimates the quantities sepd(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 [1].References
[1] 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 option LYAPUN = 'R' may occasionally produce slightly worse or better estimates, and it is much faster than the option 'O'. When SEPD is computed and it is zero, the routine returns immediately, with RCOND and FERR (if requested) set to 0 and 1, respectively. In this case, the equation is singular.Example
Program Text
* SB03SD 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( 3, 2*NMAX*NMAX ) +
$ NMAX*NMAX + 2*NMAX )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* .. Local Scalars ..
DOUBLE PRECISION FERR, RCOND, SCALE, SEPD
INTEGER I, INFO1, INFO2, J, N
CHARACTER*1 DICO, FACT, JOB, LYAPUN, TRANA, TRANAT, 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 DLACPY, MA02ED, MB01RU, SB03MD, SB03SD
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
DICO = 'D'
* 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
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
IF ( LSAME( FACT, 'F' ) ) READ ( NIN, FMT = * )
$ ( ( U(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,N )
CALL DLACPY( 'Full', N, N, A, LDA, T, LDT )
CALL DLACPY( 'Full', N, N, C, LDC, X, LDX )
* Solve the discrete-time Lyapunov matrix equation.
CALL SB03MD( DICO, 'X', FACT, TRANA, N, T, LDT, U, LDU, X, LDX,
$ SCALE, SEPD, FERR, DWORK(1), DWORK(N+1), IWORK,
$ DWORK(2*N+1), LDWORK-2*N, INFO1 )
*
IF ( INFO1.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99995 ) ( X(I,J), J = 1,N )
10 CONTINUE
IF ( LSAME( LYAPUN, 'R' ) ) THEN
IF( LSAME( TRANA, 'N' ) ) THEN
TRANAT = 'T'
ELSE
TRANAT = 'N'
END IF
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, X, LDX,
$ U, LDU, X, LDX, DWORK, N*N, INFO2 )
CALL MA02ED( UPLO, N, X, LDX )
CALL MB01RU( UPLO, TRANAT, N, N, ZERO, ONE, C, LDC,
$ U, LDU, C, LDC, DWORK, N*N, INFO2 )
END IF
* Estimate the condition and error bound on the solution.
CALL SB03SD( JOB, 'F', TRANA, UPLO, LYAPUN, N, SCALE, A,
$ LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEPD,
$ RCOND, FERR, IWORK, DWORK, LDWORK, INFO2 )
*
IF ( INFO2.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO2
ELSE
WRITE ( NOUT, FMT = 99993 ) SCALE
WRITE ( NOUT, FMT = 99992 ) SEPD
WRITE ( NOUT, FMT = 99991 ) RCOND
WRITE ( NOUT, FMT = 99990 ) FERR
END IF
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO1
END IF
END IF
STOP
*
99999 FORMAT (' SB03SD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB03MD =',I2)
99997 FORMAT (' INFO on exit from SB03SD =',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
SB03SD EXAMPLE PROGRAM DATA 3 B 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
SB03SD EXAMPLE PROGRAM RESULTS The solution matrix X is 2.0000 1.0000 1.0000 1.0000 3.0000 0.0000 1.0000 0.0000 4.0000 Scaling factor = 1.0000 Estimated separation = 5.2302 Estimated reciprocal condition number = 0.1832 Estimated error bound = 0.0000