naginterfaces.library.matop.complex_​herm_​matrix_​exp

naginterfaces.library.matop.complex_herm_matrix_exp(uplo, a)

complex_herm_matrix_exp computes the matrix exponential, $e^A$, of a complex Hermitian $n\times n$ matrix $A$.

For full information please refer to the NAG Library document for f01fd

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f01/f01fdf.html

Parameters

uplostr, length 1

If $uplo=\text{'U'}$, the upper triangle of the matrix $A$ is stored.

If $uplo=\text{'L'}$, the lower triangle of the matrix $A$ is stored.

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

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

Returns

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

If no exception or warning is raised, the upper or lower triangular part of the $n\times n$ matrix exponential, $e^A$.

Raises

NagValueError

(errno $-3$)

An internal error occurred when computing the spectral factorization. Please contact NAG.

(errno $-2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $-1$)

On entry, $uplo=⟨value⟩$.

Constraint: $uplo=\text{'L'}$ or $\text{'U'}$.

(errno $i>0$)

The computation of the spectral factorization failed to converge.

Notes

$e^A$ is computed using a spectral factorization of $A$

$$A=QD{Q}^H\text{,}$$

where $D$ is the diagonal matrix whose diagonal elements, $d_i$, are the eigenvalues of $A$, and $Q$ is a unitary matrix whose columns are the eigenvectors of $A$. $e^A$ is then given by

$$e^A=Q{e}^D{Q}^H\text{,}$$

where $e^D$ is the diagonal matrix whose $i$th diagonal element is $e^{d_i}$. See for example Section 4.5 of Higham (2008).

References

Higham, N J, 2005, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl. (26(4)), 1179–1193

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