# naginterfaces.library.matop.complex_​gen_​matrix_​frcht_​exp¶

naginterfaces.library.matop.complex_gen_matrix_frcht_exp(a, e)[source]

complex_gen_matrix_frcht_exp computes the Fréchet derivative of the matrix exponential of a complex matrix applied to the complex matrix . The matrix exponential is also returned.

For full information please refer to the NAG Library document for f01kh

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/f01/f01khf.html

Parameters
acomplex, array-like, shape

The matrix .

ecomplex, array-like, shape

The matrix

Returns
acomplex, ndarray, shape

The matrix exponential .

ecomplex, ndarray, shape

The Fréchet derivative

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

The linear equations to be solved for the Padé approximant are singular; it is likely that this function has been called incorrectly.

(errno )

Warns
NagAlgorithmicWarning
(errno )

The arithmetic precision is higher than that used for the Padé approximant computed matrix exponential.

Notes

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

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

complex_gen_matrix_frcht_exp uses the algorithms of Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b) to compute and . The matrix exponential is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is then differentiated in order to obtain the Fréchet derivative .

References

Al–Mohy, A H and Higham, N J, 2009, A new scaling and squaring algorithm for the matrix exponential, SIAM J. Matrix Anal. (31(3)), 970–989

Al–Mohy, A H and Higham, N J, 2009, Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation, SIAM J. Matrix Anal. Appl. (30(4)), 1639–1657

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

Moler, C B and Van Loan, C F, 2003, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev. (45), 3–49