Purpose
To compute the Cholesky factor U of the matrix X, X = U**H * U or X = U * U**H, which is the solution of the generalized d-stable discrete-time Lyapunov equation H H 2 H A * X * A - E * X * E = - SCALE * B * B, (1) or the conjugate transposed equation H H 2 H A * X * A - E * X * E = - SCALE * B * B , (2) respectively, where A, E, B, and U are complex N-by-N matrices. The Cholesky factor U of the solution is computed without first finding X. The pencil A - lambda * E must be in complex generalized Schur form (A and E are upper triangular and the diagonal elements of E are non-negative real numbers). Moreover, it must be d-stable, i.e., the moduli of its eigenvalues must be less than one. B must be an upper triangular matrix with real non-negative entries on its main diagonal. The resulting matrix U is upper triangular. The entries on its main diagonal are non-negative. SCALE is an output scale factor set to avoid overflow in U.Specification
SUBROUTINE SG03BS( TRANS, N, A, LDA, E, LDE, B, LDB, SCALE, DWORK, $ ZWORK, INFO ) C .. Scalar Arguments .. CHARACTER TRANS DOUBLE PRECISION SCALE INTEGER INFO, LDA, LDB, LDE, N C .. Array Arguments .. DOUBLE PRECISION DWORK(*) COMPLEX*16 A(LDA,*), B(LDB,*), E(LDE,*), ZWORK(*)Arguments
Mode Parameters
TRANS CHARACTER*1 Specifies whether equation (1) or equation (2) is to be solved: = 'N': Solve equation (1); = 'C': Solve equation (2).Input/Output Parameters
N (input) INTEGER The order of the matrices. N >= 0. A (input/workspace) COMPLEX*16 array, dimension (LDA,N) The leading N-by-N upper triangular part of this array must contain the triangular matrix A. The lower triangular part is used as workspace, but the diagonal is restored. LDA INTEGER The leading dimension of the array A. LDA >= MAX(1,N). E (input/workspace) COMPLEX*16 array, dimension (LDE,N) The leading N-by-N upper triangular part of this array must contain the triangular matrix E. If TRANS = 'N', the strictly lower triangular part is used as workspace. LDE INTEGER The leading dimension of the array E. LDE >= MAX(1,N). B (input/output) COMPLEX*16 array, dimension (LDB,N) On entry, the leading N-by-N upper triangular part of this array must contain the matrix B. On exit, the leading N-by-N upper triangular part of this array contains the solution matrix U. LDB INTEGER The leading dimension of the array B. LDB >= MAX(1,N). SCALE (output) DOUBLE PRECISION The scale factor set to avoid overflow in U. 0 < SCALE <= 1.Workspace
DWORK DOUBLE PRECISION array, dimension LDWORK, where LDWORK = 0, if N <= 1; LDWORK = MAX(N-1,10), if N > 1. ZWORK COMPLEX*16, dimension MAX(3*N-3,0)Error Indicator
INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 3: the pencil A - lambda * E is not stable, i.e., there there are eigenvalues outside the open unit circle; = 4: the LAPACK routine ZSTEIN utilized to factorize M3 failed to converge. This error is unlikely to occur.Method
The method used by the routine is an extension of Hammarling's algorithm [1] to generalized Lyapunov equations. The real case is described in [2]. We present the method for solving equation (1). Equation (2) can be treated in a similar fashion. For simplicity, assume SCALE = 1. Since all matrices A, E, B, and U are upper triangular, we use the following partitioning ( A11 A12 ) ( E11 E12 ) A = ( ), E = ( ), ( 0 A22 ) ( 0 E22 ) ( B11 B12 ) ( U11 U12 ) B = ( ), U = ( ), (3) ( 0 B22 ) ( 0 U22 ) where the size of the (1,1)-blocks is 1-by-1. We compute U11, U12**H and U22 in three steps. Step I: From (1) and (3) we get the 1-by-1 equation H H H H H A11 * U11 * U11 * A11 - E11 * U11 * U11 * E11 = -B11 * B11. For brevity, details are omitted here. The technique for computing U11 is similar to those applied to standard Lyapunov equations in Hammarling's algorithm ([1], section 5). Furthermore, the auxiliary scalars M1 and M2 defined as follows M1 = A11 / E11 , M2 = B11 / E11 / U11 , are computed in a numerically reliable way. Step II: We solve for U12**H the linear system of equations, with scaling to prevent overflow, H H H ( M1 * A22 - E22 ) U12 = H H H = - M2 * B12 + U11 * ( E12 - M1 * A12 ) . Step III: One can show that H H H H A22 * U22 * U22 * A22 - E22 * U22 * U22 * E22 = H H - B22 * B22 - y * y (4) holds, where y is defined as follows H H H H y = ( B12 U11 * A12 + A22 * U12 ) * M3EV, where M3EV is a matrix which fulfils ( I - M2*M2 -M2*M1**H ) H M3 = ( ) = M3EV * M3EV . ( -M1*M2 I - M1*M1**H ) M3 is positive semidefinite and its rank is equal to 1. Therefore, a matrix M3EV can be found by solving the Hermitian eigenvalue problem for M3 such that y consists of one column. If B22_tilde is the square triangular matrix arising from the QR-factorization ( B22_tilde ) ( B22 ) Q * ( ) = ( ), ( 0 ) ( y**H ) then H H H - B22 * B22 - y * y = - B22_tilde * B22_tilde. Replacing the right hand side in (4) by the term - B22_tilde**H * B22_tilde leads to a generalized Lyapunov equation like (1), but of dimension N-1. The solution U of the equation (1) can be obtained by recursive application of the steps I to III.References
[1] Hammarling, S.J. Numerical solution of the stable, non-negative definite Lyapunov equation. IMA J. Num. Anal., 2, pp. 303-323, 1982. [2] Penzl, T. Numerical solution of generalized Lyapunov equations. Advances in Comp. Math., vol. 8, pp. 33-48, 1998.Numerical Aspects
The routine requires 2*N**3 flops. Note that we count a single floating point arithmetic operation as one flop.Further Comments
The Lyapunov equation may be very ill-conditioned. In particular, if the pencil A - lambda * E has a pair of almost reciprocal eigenvalues, then the Lyapunov equation will be ill-conditioned. Perturbed values were used to solve the equation. A condition estimate can be obtained from the routine SG03AD.Example
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