naginterfaces.library.lapackeig.dtrsyl

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

dtrsyl solves the real quasi-triangular Sylvester matrix equation.

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

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08qhf.html

Parameters

tranastr, length 1

Specifies the option $op\left(A\right)$.

$trana=\text{'N'}$

$op\left(A\right)=A$.

$trana=\text{'T'}$ or $\text{'C'}$

$op\left(A\right)=A^T$.

tranbstr, length 1

Specifies the option $op\left(B\right)$.

$tranb=\text{'N'}$

$op\left(B\right)=B$.

$tranb=\text{'T'}$ or $\text{'C'}$

$op\left(B\right)=B^T$.

isgnint

Indicates the form of the Sylvester equation.

$isgn=+1$

The equation is of the form $op\left(A\right)X+Xop\left(B\right)=\alpha C$.

$isgn=-1$

The equation is of the form $op\left(A\right)X-Xop\left(B\right)=\alpha C$.

afloat, array-like, shape $(m,m)$

The $m\times m$ upper quasi-triangular matrix $A$ in canonical Schur form, as returned by dhseqr().

bfloat, array-like, shape $(n,n)$

The $n\times n$ upper quasi-triangular matrix $B$ in canonical Schur form, as returned by dhseqr().

cfloat, array-like, shape $(m,n)$

The $m\times n$ right-hand side matrix $C$.

Returns

cfloat, ndarray, shape $(m,n)$

$c$ is overwritten by the solution matrix $X$.

scalefloat

The value of the scale factor $\alpha$.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $trana$.

Constraint: $trana=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.

(errno $-2$)

On entry, error in parameter $tranb$.

Constraint: $tranb=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.

(errno $-3$)

On entry, error in parameter $isgn$.

Constraint: $isgn=+1$ or $-1$.

(errno $-4$)

On entry, error in parameter $\text{m}$.

Constraint: $m\ge 0$.

(errno $-5$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $1$)

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

Notes

dtrsyl solves the real Sylvester matrix equation

$$op\left(A\right)X\pm Xop\left(B\right)=\alpha C\text{,}$$

where $op\left(A\right)=A$ or $A^T$, and the matrices $A$ and $B$ are upper quasi-triangular matrices in canonical Schur form (as returned by dhseqr()); $\alpha$ is a scale factor ($\le 1$) determined by the function to avoid overflow in $X$; $A$ is $m\times m$ and $B$ is $n\times n$ while the right-hand side matrix $C$ and the solution matrix $X$ are both $m\times n$. The matrix $X$ 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 ${\alpha }_i\pm {\beta }_j\ne 0$, where $\left\{{\alpha }_i\right\}$ and $\left\{{\beta }_j\right\}$ are the eigenvalues of $A$ and $B$ 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 $AX-XB=C$, Numerical Analysis Report, University of Manchester