naginterfaces.library.matop.real_​gen_​matrix_​fun_​std

naginterfaces.library.matop.real_gen_matrix_fun_std(fun, a)

real_gen_matrix_fun_std computes the matrix exponential, sine, cosine, sinh or cosh, of a real $n\times n$ matrix $A$ using the Schur–Parlett algorithm.

For full information please refer to the NAG Library document for f01ek

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f01/f01ekf.html

Parameters

funstr

Indicates which matrix function will be computed.

$fun=\text{‘EXP'}$

The matrix exponential, $e^A$, will be computed.

$fun=\text{‘SIN'}$

The matrix sine, $\sin \left(A\right)$, will be computed.

$fun=\text{‘COS'}$

The matrix cosine, $\cos \left(A\right)$, will be computed.

$fun=\text{‘SINH'}$

The hyperbolic matrix sine, $\sinh \left(A\right)$, will be computed.

$fun=\text{‘COSH'}$

The hyperbolic matrix cosine, $\cosh \left(A\right)$, will be computed.

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

The $n\times n$ matrix $A$.

Returns

afloat, ndarray, shape $(n,n)$

The $n\times n$ matrix, $f\left(A\right)$.

imnormfloat

If $A$ has complex eigenvalues, real_gen_matrix_fun_std will use complex arithmetic to compute the matrix function. The imaginary part is discarded at the end of the computation, because it will theoretically vanish. $imnorm$ contains the $1$-norm of the imaginary part, which should be used to check that the routine has given a reliable answer.

If $A$ has real eigenvalues, real_gen_matrix_fun_std uses real arithmetic and $imnorm=0$.

Raises

NagValueError

(errno $-2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $-1$)

Input argument $fun$ is invalid.

(errno $1$)

A Taylor series failed to converge.

(errno $2$)

An unexpected internal error occurred when evaluating the function at a point. Please contact NAG.

(errno $3$)

There was an error whilst reordering the Schur form of $A$.

Note: this failure should not occur and suggests that the function has been called incorrectly.

(errno $4$)

The function was unable to compute the Schur decomposition of $A$.

Note: this failure should not occur and suggests that the function has been called incorrectly.

(errno $5$)

An unexpected internal error occurred. Please contact NAG.

(errno $6$)

The linear equations to be solved are nearly singular and the Padé approximant used to compute the exponential may have no correct figures.

Note: this failure should not occur and suggests that the function has been called incorrectly.

Notes

$f\left(A\right)$, where $f$ is either the exponential, sine, cosine, sinh or cosh, is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003).

References

Davies, P I and Higham, N J, 2003, A Schur–Parlett algorithm for computing matrix functions, SIAM J. Matrix Anal. Appl. (25(2)), 464–485

Higham, N J, 2008, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, PA, USA