**Purpose**

To solve for X = op(U)'*op(U) either the stable non-negative definite continuous-time Lyapunov equation 2 op(A)'*X + X*op(A) = -scale *op(B)'*op(B) (1) or the convergent non-negative definite discrete-time Lyapunov equation 2 op(A)'*X*op(A) - X = -scale *op(B)'*op(B) (2) where op(K) = K or K' (i.e., the transpose of the matrix K), A is an N-by-N matrix in real Schur form, op(B) is an M-by-N matrix, U is an upper triangular matrix containing the Cholesky factor of the solution matrix X, X = op(U)'*op(U), and scale is an output scale factor, set less than or equal to 1 to avoid overflow in X. If matrix B has full rank then the solution matrix X will be positive-definite and hence the Cholesky factor U will be nonsingular, but if B is rank deficient then X may only be positive semi-definite and U will be singular. In the case of equation (1) the matrix A must be stable (that is, all the eigenvalues of A must have negative real parts), and for equation (2) the matrix A must be convergent (that is, all the eigenvalues of A must lie inside the unit circle).

SUBROUTINE SB03OU( DISCR, LTRANS, N, M, A, LDA, B, LDB, TAU, U, $ LDU, SCALE, DWORK, LDWORK, INFO ) C .. Scalar Arguments .. LOGICAL DISCR, LTRANS INTEGER INFO, LDA, LDB, LDU, LDWORK, M, N DOUBLE PRECISION SCALE C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), B(LDB,*), DWORK(*), TAU(*), U(LDU,*)

**Mode Parameters**

DISCR LOGICAL Specifies the type of Lyapunov equation to be solved as follows: = .TRUE. : Equation (2), discrete-time case; = .FALSE.: Equation (1), continuous-time case. LTRANS LOGICAL Specifies the form of op(K) to be used, as follows: = .FALSE.: op(K) = K (No transpose); = .TRUE. : op(K) = K**T (Transpose).

N (input) INTEGER The order of the matrix A and the number of columns in matrix op(B). N >= 0. M (input) INTEGER The number of rows in matrix op(B). M >= 0. A (input) DOUBLE PRECISION array, dimension (LDA,N) The leading N-by-N upper Hessenberg part of this array must contain a real Schur form matrix S. The elements below the upper Hessenberg part of the array A are not referenced. The 2-by-2 blocks must only correspond to complex conjugate pairs of eigenvalues (not to real eigenvalues). LDA INTEGER The leading dimension of array A. LDA >= MAX(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB,N) if LTRANS = .FALSE., and dimension (LDB,M), if LTRANS = .TRUE.. On entry, if LTRANS = .FALSE., the leading M-by-N part of this array must contain the coefficient matrix B of the equation. On entry, if LTRANS = .TRUE., the leading N-by-M part of this array must contain the coefficient matrix B of the equation. On exit, if LTRANS = .FALSE., the leading MIN(M,N)-by-MIN(M,N) upper triangular part of this array contains the upper triangular matrix R (as defined in METHOD), and the M-by-MIN(M,N) strictly lower triangular part together with the elements of the array TAU are overwritten by details of the matrix P (also defined in METHOD). When M < N, columns (M+1),...,N of the array B are overwritten by the matrix Z (see METHOD). On exit, if LTRANS = .TRUE., the leading MIN(M,N)-by-MIN(M,N) upper triangular part of B(1:N,M-N+1), if M >= N, or of B(N-M+1:N,1:M), if M < N, contains the upper triangular matrix R (as defined in METHOD), and the remaining elements (below the diagonal of R) together with the elements of the array TAU are overwritten by details of the matrix P (also defined in METHOD). When M < N, rows 1,...,(N-M) of the array B are overwritten by the matrix Z (see METHOD). LDB INTEGER The leading dimension of array B. LDB >= MAX(1,M), if LTRANS = .FALSE., LDB >= MAX(1,N), if LTRANS = .TRUE.. TAU (output) DOUBLE PRECISION array of dimension (MIN(N,M)) This array contains the scalar factors of the elementary reflectors defining the matrix P. U (output) DOUBLE PRECISION array of dimension (LDU,N) The leading N-by-N upper triangular part of this array contains the Cholesky factor of the solution matrix X of the problem, X = op(U)'*op(U). The array U may be identified with B in the calling statement, if B is properly dimensioned, and the intermediate results returned in B are not needed. LDU INTEGER The leading dimension of array U. LDU >= MAX(1,N). SCALE (output) DOUBLE PRECISION The scale factor, scale, set less than or equal to 1 to prevent the solution overflowing.

DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, or INFO = 1, DWORK(1) returns the optimal value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= MAX(1,4*N). 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; = 1: if the Lyapunov equation is (nearly) singular (warning indicator); if DISCR = .FALSE., this means that while the matrix A has computed eigenvalues with negative real parts, it is only just stable in the sense that small perturbations in A can make one or more of the eigenvalues have a non-negative real part; if DISCR = .TRUE., this means that while the matrix A has computed eigenvalues inside the unit circle, it is nevertheless only just convergent, in the sense that small perturbations in A can make one or more of the eigenvalues lie outside the unit circle; perturbed values were used to solve the equation (but the matrix A is unchanged); = 2: if matrix A is not stable (that is, one or more of the eigenvalues of A has a non-negative real part), if DISCR = .FALSE., or not convergent (that is, one or more of the eigenvalues of A lies outside the unit circle), if DISCR = .TRUE.; = 3: if matrix A has two or more consecutive non-zero elements on the first sub-diagonal, so that there is a block larger than 2-by-2 on the diagonal; = 4: if matrix A has a 2-by-2 diagonal block with real eigenvalues instead of a complex conjugate pair.

The method used by the routine is based on the Bartels and Stewart method [1], except that it finds the upper triangular matrix U directly without first finding X and without the need to form the normal matrix op(B)'*op(B) [2]. If LTRANS = .FALSE., the matrix B is factored as B = P ( R ), M >= N, B = P ( R Z ), M < N, ( 0 ) (QR factorization), where P is an M-by-M orthogonal matrix and R is a square upper triangular matrix. If LTRANS = .TRUE., the matrix B is factored as B = ( 0 R ) P, M >= N, B = ( Z ) P, M < N, ( R ) (RQ factorization), where P is an M-by-M orthogonal matrix and R is a square upper triangular matrix. These factorizations are used to solve the continuous-time Lyapunov equation in the canonical form 2 op(A)'*op(U)'*op(U) + op(U)'*op(U)*op(A) = -scale *op(F)'*op(F), or the discrete-time Lyapunov equation in the canonical form 2 op(A)'*op(U)'*op(U)*op(A) - op(U)'*op(U) = -scale *op(F)'*op(F), where U and F are N-by-N upper triangular matrices, and F = R, if M >= N, or F = ( R ), if LTRANS = .FALSE., or ( 0 ) F = ( 0 Z ), if LTRANS = .TRUE., if M < N. ( 0 R ) The canonical equation is solved for U.

[1] Bartels, R.H. and Stewart, G.W. Solution of the matrix equation A'X + XB = C. Comm. A.C.M., 15, pp. 820-826, 1972. [2] Hammarling, S.J. Numerical solution of the stable, non-negative definite Lyapunov equation. IMA J. Num. Anal., 2, pp. 303-325, 1982.

3 The algorithm requires 0(N ) operations and is backward stable.

The Lyapunov equation may be very ill-conditioned. In particular, if A is only just stable (or convergent) then the Lyapunov equation will be ill-conditioned. "Large" elements in U relative to those of A and B, or a "small" value for scale, are symptoms of ill-conditioning. A condition estimate can be computed using SLICOT Library routine SB03MD.

**Program Text**

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