naginterfaces.library.matop.complex_​gen_​matrix_​pow

naginterfaces.library.matop.complex_gen_matrix_pow(a, p)

complex_gen_matrix_pow computes an abitrary real power $A^p$ of a complex $n\times n$ matrix $A$.

For full information please refer to the NAG Library document for f01fq

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f01/f01fqf.html

Parameters

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

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

pfloat

The required power of $A$.

Returns

acomplex, ndarray, shape $(n,n)$

If no exception or warning is raised, the $n\times n$ matrix $p$th power, $A^p$. Alternatively, if $errno$ = 1, contains an $n\times n$ non-principal power of $A$.

Raises

NagValueError

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $2$)

$A$ is singular so the $p$th power cannot be computed.

(errno $4$)

An unexpected internal error occurred. This failure should not occur and suggests that the function has been called incorrectly.

Warns

NagAlgorithmicWarning

(errno $1$)

$A$ has eigenvalues on the negative real line. The principal $p$th power is not defined so a non-principal power is returned.

(errno $3$)

$A^p$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

Notes

For a matrix $A$ with no eigenvalues on the closed negative real line, $A^p$ ($p\in R$) can be defined as

$$A^p=exp\left(p\log \left(A\right)\right)$$

where $\log \left(A\right)$ is the principal logarithm of $A$ (the unique logarithm whose spectrum lies in the strip $\{z:-\pi <Im\left(z\right)<\pi \}$).

$A^p$ is computed using the Schur–Padé algorithm described in Higham and Lin (2011) and Higham and Lin (2013).

The real number $p$ is expressed as $p=q+r$ where $q\in (-1,1)$ and $r\in Z$. Then $A^p=A^q{A}^r$. The integer power $A^r$ is found using a combination of binary powering and, if necessary, matrix inversion. The fractional power $A^q$ is computed using a Schur decomposition and a Padé approximant.

References

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

Higham, N J and Lin, L, 2011, A Schur–Padé algorithm for fractional powers of a matrix, SIAM J. Matrix Anal. Appl. (32(3)), 1056–1078

Higham, N J and Lin, L, 2013, An improved Schur–Padé algorithm for fractional powers of a matrix and their Fréchet derivatives, SIAM J. Matrix Anal. Appl. (34(3)), 1341–1360