naginterfaces.library.lapackeig.dtrsyl

naginterfaces.library.lapackeig.dtrsyl(trana, tranb, isgn, a, b, c)[source]

dtrsyl solves the real quasi-triangular Sylvester matrix equation.

For full information please refer to the NAG Library document for f08qh

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f08/f08qhf.html

Parameters
tranastr, length 1

Specifies the option .

.

or

.

tranbstr, length 1

Specifies the option .

.

or

.

isgnint

Indicates the form of the Sylvester equation.

The equation is of the form .

The equation is of the form .

afloat, array-like, shape

The upper quasi-triangular matrix in canonical Schur form, as returned by dhseqr().

bfloat, array-like, shape

The upper quasi-triangular matrix in canonical Schur form, as returned by dhseqr().

cfloat, array-like, shape

The right-hand side matrix .

Returns
cfloat, ndarray, shape

is overwritten by the solution matrix .

scalefloat

The value of the scale factor .

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: , or .

(errno )

On entry, error in parameter .

Constraint: , or .

(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Warns
NagAlgorithmicWarning
(errno )

and have common or close eigenvalues, perturbed values of which were used to solve the equation.

Notes

dtrsyl solves the real Sylvester matrix equation

where or , and the matrices and are upper quasi-triangular matrices in canonical Schur form (as returned by dhseqr()); is a scale factor () determined by the function to avoid overflow in ; is and is while the right-hand side matrix and the solution matrix are both . The matrix is obtained by a straightforward process of back-substitution (see Golub and Van Loan (1996)).

Note that the equation has a unique solution if and only if , where and are the eigenvalues of and respectively and the sign ( or ) is the same as that used in the equation to be solved.

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore

Higham, N J, 1992, Perturbation theory and backward error for , Numerical Analysis Report, University of Manchester