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

naginterfaces.library.matop.complex_gen_matrix_cond_exp(a)

complex_gen_matrix_cond_exp computes an estimate of the relative condition number ${\kappa }_{exp}\left(A\right)$ of the exponential of a complex $n\times n$ matrix $A$, in the $1$-norm. The matrix exponential $e^A$ is also returned.

For full information please refer to the NAG Library document for f01kg

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f01/f01kgf.html

Parameters

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

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

Returns

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

The $n\times n$ matrix exponential $e^A$.

condeafloat

An estimate of the relative condition number of the matrix exponential ${\kappa }_{exp}\left(A\right)$.

Raises

NagValueError

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $1$)

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

(errno $3$)

An unexpected internal error has occurred. Please contact NAG.

Warns

NagAlgorithmicWarning

(errno $2$)

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

Notes

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

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

The derivative describes the first-order effect of perturbations in $A$ on the exponential $e^A$.

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

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

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

To obtain the estimate of ${\kappa }_{exp}\left(A\right)$, complex_gen_matrix_cond_exp 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 $\gamma \le K$.

The algorithms used to compute ${\kappa }_{exp}\left(A\right)$ are detailed in the Al–Mohy and Higham (2009a) and Al–Mohy and Higham (2009b).

The matrix exponential $e^A$ is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is differentiated to obtain the Fréchet derivatives $L(A,E)$ 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