**Purpose**

To solve the real discrete-time Lyapunov matrix equation op(A)'*X*op(A) - X = 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.

SUBROUTINE SB03UD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A, $ LDA, T, LDT, U, LDU, C, LDC, X, LDX, SEPD, $ 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, SEPD C .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), C( LDC, * ), DWORK( * ), $ T( LDT, * ), U( LDU, * ), WI( * ), WR( * ), $ X( LDX, * )

**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.

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. SEPD (output) DOUBLE PRECISION If JOB = 'S' or JOB = 'C' or JOB = 'A', and INFO = 0 or INFO = N+1, SEPD contains the estimated separation of the matrices op(A) and op(A)', sepd(op(A),op(A)'). If N = 0, or X = 0, or JOB = 'X' or JOB = 'E', SEPD 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.

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,2*N), if FACT = 'F'; LDWORK >= MAX(1,N*N,3*N), if FACT = 'N'. If JOB = 'S', then LDWORK >= MAX(3,2*N*N). If JOB = 'C', then LDWORK >= MAX(3,2*N*N) + N*N. If JOB = 'E', or JOB = 'A', then LDWORK >= MAX(3,2*N*N) + N*N + 2*N. For optimum performance LDWORK should sometimes be larger.

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 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.

After reducing matrix A to real Schur canonical form (if needed), a discrete-time version of 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 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 [3].

[1] Barraud, A.Y. T A numerical algorithm to solve A XA - X = Q. IEEE Trans. Auto. Contr., AC-22, pp. 883-885, 1977. [2] 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. [3] 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 "separation" sepd of op(A) and op(A)' can also be defined as sepd( 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( op(A)', op(A)' ) - I(N**2). I(N**2) is an N*N-by-N*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.

**Program Text**

* SB03UD 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 ) * .. Local Scalars .. DOUBLE PRECISION FERR, RCOND, SCALE, SEPD 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 SB03UD * .. 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 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 discrete-time Lyapunov matrix equation and/or * estimate the condition and error bound on the solution. CALL SB03UD( JOB, FACT, TRANA, UPLO, LYAPUN, N, SCALE, A, LDA, $ T, LDT, U, LDU, C, LDC, X, LDX, SEPD, 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 ) SEPD 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 (' SB03UD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from SB03UD =',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

SB03UD 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 40.0

SB03UD 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