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

naginterfaces.library.matop.complex_gen_matrix_cond_std(fun, a)

complex_gen_matrix_cond_std computes an estimate of the absolute condition number of a matrix function $f$ of a complex $n\times n$ matrix $A$ in the $1$-norm, where $f$ is either the exponential, logarithm, sine, cosine, hyperbolic sine (sinh) or hyperbolic cosine (cosh). The evaluation of the matrix function, $f\left(A\right)$, is also returned.

For full information please refer to the NAG Library document for f01ka

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f01/f01kaf.html

Parameters

funstr

Indicates which matrix function will be used.

$fun=\text{‘EXP'}$

The matrix exponential, $e^A$, will be used.

$fun=\text{‘SIN'}$

The matrix sine, $\sin \left(A\right)$, will be used.

$fun=\text{‘COS'}$

The matrix cosine, $\cos \left(A\right)$, will be used.

$fun=\text{‘SINH'}$

The hyperbolic matrix sine, $\sinh \left(A\right)$, will be used.

$fun=\text{‘COSH'}$

The hyperbolic matrix cosine, $\cosh \left(A\right)$, will be used.

$fun=\text{‘LOG'}$

The matrix logarithm, $\log \left(A\right)$, will be used.

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

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

Returns

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

The $n\times n$ matrix, $f\left(A\right)$.

condafloat

An estimate of the absolute condition number of $f$ at $A$.

normafloat

The $1$-norm of $A$.

normfafloat

The $1$-norm of $f\left(A\right)$.

Raises

NagValueError

(errno $-2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $-1$)

Input argument number $⟨value⟩$ is invalid.

(errno $1$)

An internal error occurred when estimating the norm of the Fréchet derivative of $f$ at $A$. Please contact NAG.

(errno $2$)

An internal error occurred when evaluating the matrix function $f\left(A\right)$. You can investigate further by calling complex_gen_matrix_exp(), complex_gen_matrix_log() or complex_gen_matrix_fun_std() with the matrix $A$.

Notes

The absolute condition number of $f$ at $A$, ${cond}_{abs}(f,A)$ is given by the norm of the Fréchet derivative of $f$, $L\left(A\right)$, which is defined by

$$\parallel L\left(X\right)\parallel :={max}_{E\ne 0}\frac{\parallel L(X,E)\parallel }{\parallel E\parallel }\text{,}$$

where $L(X,E)$ is the Fréchet derivative in the direction $E$. $L(X,E)$ is linear in $E$ and can, therefore, be written as

$$vec\left(L(X,E)\right)=K\left(X\right)vec\left(E\right)\text{,}$$

where the $vec$ operator stacks the columns of a matrix into one vector, so that $K\left(X\right)$ is $n^2\times n^2$. complex_gen_matrix_cond_std computes an estimate $\gamma$ such that $\gamma \le {\parallel K\left(X\right)\parallel }_1$, where ${\parallel K\left(X\right)\parallel }_1\in [n^{-1}{\parallel L\left(X\right)\parallel }_1,n{\parallel L\left(X\right)\parallel }_1]$. The relative condition number can then be computed via

$${cond}_{rel}(f,A)=\frac{{cond}_{abs}(f,A){\parallel A\parallel }_1}{{\parallel f\left(A\right)\parallel }_1}\text{.}$$

The algorithm used to find $\gamma$ is detailed in Section 3.4 of Higham (2008).

References

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