The function may be called by the names: f08yhc, nag_lapackeig_dtgsyl or nag_dtgsyl.
3Description
f08yhc solves either the generalized real Sylvester equations
(1)
or the equations
(2)
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 function to avoid overflow in computing .
Equations (1) are equivalent to equations of the form
where
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 f08xcc, or f08xec with .
Optionally, the function 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 2.4.8.3 and 4.11.1.3 of Anderson et al. (1999) and Kågström and Poromaa (1996).
4References
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
5Arguments
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_TransTypeInput
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.
Normally, .
19: – double *Output
On exit: the estimate of . If , dif is not referenced.
20: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_EIGENVALUES
and have common or close eigenvalues and so no solution could be computed.
NE_ENUM_INT
On entry, and .
Constraint: if , .
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
7Accuracy
See Kågström (1994) for a perturbation analysis of the generalized Sylvester equation.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08yhc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
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.