naginterfaces.library.matop.real_​gen_​matrix_​cond_​exp

naginterfaces.library.matop.real_gen_matrix_cond_exp(a)[source]

real_gen_matrix_cond_exp computes an estimate of the relative condition number of the exponential of a real matrix , in the -norm. The matrix exponential is also returned.

For full information please refer to the NAG Library document for f01jg

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/f01/f01jgf.html

Parameters
afloat, array-like, shape

The matrix .

Returns
afloat, ndarray, shape

The matrix exponential .

condeafloat

An estimate of the relative condition number of the matrix exponential .

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 )

An unexpected internal error has occurred. Please contact NAG.

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 .

The relative condition number of the matrix exponential can be defined by

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

To obtain the estimate of , real_gen_matrix_cond_exp first estimates by computing an estimate of a quantity , such that .

The algorithms used to compute are detailed in the Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b).

The matrix exponential is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is differentiated to obtain the Fréchet derivatives which are used to estimate the condition number.

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