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

naginterfaces.library.matop.complex_gen_matrix_cond_log(a)

complex_gen_matrix_cond_log computes an estimate of the relative condition number ${\kappa }_{log}\left(A\right)$ of the logarithm of a complex $n\times n$ matrix $A$, in the $1$-norm. The principal matrix logarithm $\log \left(A\right)$ is also returned.

For full information please refer to the NAG Library document for f01kj

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f01/f01kjf.html

Parameters

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

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

Returns

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

The $n\times n$ principal matrix logarithm, $\log \left(A\right)$. Alternatively, if $errno$ = 2, a non-principal logarithm is returned.

condlafloat

With the function exits successfully or $errno$ = 2 or 3, an estimate of the relative condition number of the matrix logarithm, ${\kappa }_{log}\left(A\right)$. Alternatively, if $errno$ = 4, contains the absolute condition number of the matrix logarithm.

Raises

NagValueError

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $1$)

$A$ is singular so the logarithm cannot be computed.

(errno $5$)

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

Warns

NagAlgorithmicWarning

(errno $2$)

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

(errno $3$)

$\log \left(A\right)$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

(errno $4$)

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, the principal matrix logarithm $\log \left(A\right)$ is the unique logarithm whose spectrum lies in the strip $\left\{z:-\pi <Im\left(\left(z\right)<\pi \right)\right\}$.

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

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

The derivative describes the first order effect of perturbations in $A$ on the logarithm $\log \left(A\right)$.

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

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

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

To obtain the estimate of ${\kappa }_{log}\left(A\right)$, complex_gen_matrix_cond_log 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 ${\kappa }_{log}\left(A\right)$ and $\log \left(A\right)$ are based on a Schur decomposition, the inverse scaling and squaring method and Padé approximants. Further details can be found in Al–Mohy and Higham (2011) and Al–Mohy et al. (2012).

If $A$ is nonsingular but has negative real eigenvalues, the principal logarithm is not defined, but complex_gen_matrix_cond_log will return a non-principal logarithm and its condition number.

References

Al–Mohy, A H and Higham, N J, 2011, Improved inverse scaling and squaring algorithms for the matrix logarithm, SIAM J. Sci. Comput. (34(4)), C152–C169

Al–Mohy, A H, Higham, N J and Relton, S D, 2012, Computing the Fréchet derivative of the matrix logarithm and estimating the condition number, SIAM J. Sci. Comput. (35(4)), C394–C410

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