**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.

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, * )

**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.

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.

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.

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.

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].

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

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.

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.

**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

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 40.0

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