naginterfaces.library.matop.real_​symm_​matrix_​fun

naginterfaces.library.matop.real_symm_matrix_fun(uplo, a, f, data=None)

real_symm_matrix_fun computes the matrix function, $f\left(A\right)$, of a real symmetric $n\times n$ matrix $A$. $f\left(A\right)$ must also be a real symmetric matrix.

For full information please refer to the NAG Library document for f01ef

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f01/f01eff.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.

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

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

fcallable fx = f(x, data=None)

The function $f$ evaluates $f\left(z_i\right)$ at a number of points $z_i$.

Parameters

xfloat, ndarray, shape $\left(n\right)$

The $n$ points $x_1,x_2,\dots ,x_n$ at which the function $f$ is to be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

fxfloat, array-like, shape $\left(n\right)$

The $n$ function values. $fx[\text{i}-1]$ should return the value $f\left(x_{\text{i}}\right)$, for $\text{i}=1,2,\dots ,n$.

dataarbitrary, optional

User-communication data for callback functions.

Returns

afloat, ndarray, shape $(n,n)$

If no exception or warning is raised, the upper or lower triangular part of the $n\times n$ matrix function, $f\left(A\right)$.

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.

Warns

NagCallbackTerminateWarning

(errno $-6$)

Termination requested in $f$.

Notes

$f\left(A\right)$ is computed using a spectral factorization of $A$

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

where $D$ is the diagonal matrix whose diagonal elements, $d_i$, are the eigenvalues of $A$, and $Q$ is an orthogonal matrix whose columns are the eigenvectors of $A$. $f\left(A\right)$ is then given by

$$f\left(A\right)=Qf\left(D\right)Q^T\text{,}$$

where $f\left(D\right)$ is the diagonal matrix whose $i$th diagonal element is $f\left(d_i\right)$. See for example Section 4.5 of Higham (2008). $f\left(d_i\right)$ is assumed to be real.

References

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