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

naginterfaces.library.matop.complex_gen_matrix_cond_pow(a, p)[source]

complex_gen_matrix_cond_pow computes an estimate of the relative condition number of the th power (where is real) of a complex matrix , in the -norm. The principal matrix power is also returned.

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

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/f01/f01kef.html

Parameters
acomplex, array-like, shape

The matrix .

pfloat

The required power of .

Returns
acomplex, ndarray, shape

The principal matrix th power, , unless = 1, in which case a non-principal th power is returned.

condpafloat

If the function exits successfully or = 3, an estimate of the relative condition number of the matrix th power, . Alternatively, if = 4, the absolute condition number of the matrix th power.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

is singular so the th power cannot be computed.

(errno )

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

Warns
NagAlgorithmicWarning
(errno )

has eigenvalues on the negative real line. The principal th power is not defined in this case, so a non-principal power was returned.

(errno )

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

(errno )

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

Notes

For a matrix with no eigenvalues on the closed negative real line, () can be defined as

where is the principal logarithm of (the unique logarithm whose spectrum lies in the strip ).

The Fréchet derivative of the matrix th power of is the unique linear mapping such that for any matrix

The derivative describes the first-order effect of perturbations in on the matrix power .

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

where is the norm of the Fréchet derivative of the matrix power at .

complex_gen_matrix_cond_pow uses the algorithms of Higham and Lin (2011) and Higham and Lin (2013) to compute and . The real number is expressed as where and . Then . The integer power is found using a combination of binary powering and, if necessary, matrix inversion. The fractional power is computed using a Schur decomposition, a Padé approximant and the scaling and squaring method.

To obtain the estimate of , complex_gen_matrix_cond_pow first estimates by computing an estimate of a quantity , such that . This requires multiple Fréchet derivatives to be computed. Fréchet derivatives of are obtained by differentiating the Padé approximant. Fréchet derivatives of are then computed using a combination of the chain rule and the product rule for Fréchet derivatives.

If is nonsingular but has negative real eigenvalues complex_gen_matrix_cond_pow will return a non-principal matrix 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