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

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

complex_gen_matrix_cond_pow computes an estimate of the relative condition number ${\kappa }_{A^p}$ of the $p$th power (where $p$ is real) of a complex $n\times n$ matrix $A$, in the $1$-norm. The principal matrix power $A^p$ is also returned.

For full information please refer to the NAG Library document for f01ke

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f01/f01kef.html

Parameters

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

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

pfloat

The required power of $A$.

Returns

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

The $n\times n$ principal matrix $p$th power, $A^p$, unless $errno$ = 1, in which case a non-principal $p$th power is returned.

condpafloat

If the function exits successfully or $errno$ = 3, an estimate of the relative condition number of the matrix $p$th power, ${\kappa }_{A^p}$. Alternatively, if $errno$ = 4, the absolute condition number of the matrix $p$th power.

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 $5$)

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 in this case, so a non-principal power was 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.

(errno $4$)

The relative condition number is infinite. The absolute condition number was returned instead.

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 $\{z:-\pi <Im\left(z\right)<\pi \}$).

The Fréchet derivative of the matrix $p$th power of $A$ is the unique linear mapping $E⟼L(A,E)$ such that for any matrix $E$

$${(A+E)}^p-A^p-L(A,E)=o\left(\parallel E\parallel \right)\text{.}$$

The derivative describes the first-order effect of perturbations in $A$ on the matrix power $A^p$.

The relative condition number of the matrix $p$th power can be defined by

$${\kappa }_{A^p}=\frac{\parallel L\left(A\right)\parallel \parallel A\parallel }{\parallel A^p\parallel }\text{,}$$

where $\parallel L\left(A\right)\parallel$ is the norm of the Fréchet derivative of the matrix power at $A$.

complex_gen_matrix_cond_pow uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute ${\kappa }_{A^p}$ and $A^p$. 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, a Padé approximant and the scaling and squaring method.

To obtain the estimate of ${\kappa }_{A^p}$, complex_gen_matrix_cond_pow first estimates $\parallel L\left(A\right)\parallel$ by computing an estimate $\gamma$ of a quantity $K\in [n^{-1}{\parallel L\left(A\right)\parallel }_1,n{\parallel L\left(A\right)\parallel }_1]$, such that $\gamma \le K$. This requires multiple Fréchet derivatives to be computed. Fréchet derivatives of $A^q$ are obtained by differentiating the Padé approximant. Fréchet derivatives of $A^p$ are then computed using a combination of the chain rule and the product rule for Fréchet derivatives.

If $A$ is nonsingular but has negative real eigenvalues complex_gen_matrix_cond_pow will return a non-principal matrix $p$th power and its condition number.

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