operator.matrix_function

operator.matrix_function(A, fun='identity', deg=20, rtol=None, orth=0, **kwargs)

Constructs a LinearOperator approximating the action of the matrix function fun(A).

This function uses the Lanczos method to approximate the action of matrix function: v \mapsto f(A)v, \quad f(A) = U f(\lambda) U^T The resulting operator may be used in conjunction with other methods, such as hutch or xtrace.

The resulting operator also supports fast evaluation of f(A)’s quadratic form: v \mapsto v^T f(A) v The weights of the quadrature may be computed using either the Golub-Welsch (GW) or Forward Three Term Recurrence algorithms (FTTR) (see the method parameter).

For a description of all other parameters, see the Lanczos function.

Note

To compute the weights of the quadrature, the GW computation uses implicit symmetric QR steps with Wilkinson shifts, while the FTTR algorithm uses the explicit expression for orthogonal polynomials. While both require O(\mathrm{deg}^2) time to execute, the former requires O(\mathrm{deg}^2) space but is highly accurate, while the latter uses only O(1) space at the cost of stability. If deg is large, fttr is preferred.

Parameters

Name	Type	Description	Default
A	scipy.sparse.linalg.LinearOperator or ndarray or sparray	real symmetric operator.	required
fun	str or typing.Callable	real-valued function defined on the spectrum of A.	"identity"
deg	int	Degree of the Krylov expansion.	20
rtol	float	Relative tolerance to consider two Lanczos vectors are numerically orthogonal.	1e-8
orth	int	Number of additional Lanczos vectors to orthogonalize against when building the Krylov basis.	0
kwargs	dict	additional key-values to parameterize the chosen function ‘fun’.	{}

Returns

Type	Description
scipy.sparse.linalg.LinearOperator	a LinearOperator approximating the action of fun on the spectrum of A