The routine may be called by the names f08yhf, nagf_lapackeig_dtgsyl or its LAPACK name dtgsyl.
f08yhf solves either the generalized real Sylvester equations
or the equations
where the pair are given matrices in real generalized Schur form, are given matrices in real generalized Schur form and are given matrices. The pair are the solution matrices, and is an output scaling factor determined by the routine to avoid overflow in computing .
Equations (1) are equivalent to equations of the form
and is the Kronecker product. Equations (2) are then equivalent to
The pair are in real generalized Schur form if is block upper triangular with and diagonal blocks on the diagonal and is upper triangular as returned, for example, by f08xcf, or f08xef with .
Optionally, the routine estimates , the separation between the matrix pairs and , which is the smallest singular value of . The estimate can be based on either the Frobenius norm, or the -norm. The -norm estimate can be three to ten times more expensive than the Frobenius norm estimate, but makes the condition estimation uniform with the nonsymmetric eigenproblem. The Frobenius norm estimate provides a low cost, but equally reliable estimate. For more information see Sections 18.104.22.168 and 22.214.171.124 of Anderson et al. (1999) and Kågström and Poromaa (1996).
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Kågström B (1994) A perturbation analysis of the generalized Sylvester equation SIAM J. Matrix Anal. Appl. 15 1045–1060
Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs ACM Trans. Math. Software22 78–103
1: – Character(1)Input
On entry: if , solve the generalized Sylvester equation (1).
If , c and f hold the solutions and , respectively, to a slightly perturbed system but the input arrays a, b, d and e have not been changed.
If , c and f hold the solutions and , respectively, to the homogeneous system with . In this case dif is not referenced.
18: – Real (Kind=nag_wp)Output
On exit: the estimate of . If , dif is not referenced.
19: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
20: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08yhf is called.
If , a workspace query is assumed; the routine only calculates the minimum size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
if and or , ;
21: – Integer arrayWorkspace
22: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
and have common or close eigenvalues and so no solution could be computed.
See Kågström (1994) for a perturbation analysis of the generalized Sylvester equation.
8Parallelism and Performance
f08yhf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations needed to solve the generalized Sylvester equations is approximately . The Frobenius norm estimate of does not require additional significant computation, but the -norm estimate is typically five times more expensive.