Matrix functions via linear systems built from continued fractions

by   Andreas Frommer, et al.

A widely used approach to compute the action f(A)v of a matrix function f(A) on a vector v is to use a rational approximation r for f and compute r(A)v instead. If r is not computed adaptively as in rational Krylov methods, this is usually done using the partial fraction expansion of r and solving linear systems with matrices A- τ I for the various poles τ of r. Here we investigate an alternative approach for the case that a continued fraction representation for the rational function is known rather than a partial fraction expansion. This is typically the case, for example, for Padé approximations. From the continued fraction, we first construct a matrix pencil from which we then obtain what we call the CF-matrix (continued fraction matrix), a block tridiagonal matrix whose blocks consist of polynomials of A with degree bounded by 1 for many continued fractions. We show that one can evaluate r(A)v by solving a single linear system with the CF-matrix and present a number of first theoretical results as a basis for an analysis of future, specific solution methods for the large linear system. While the CF-matrix approach is of principal interest on its own as a new way to compute f(A)v, it can in particular be beneficial when a partial fraction expansion is not known beforehand and computing its parameters is ill-conditioned. We report some numerical experiments which show that with standard preconditioners we can achieve fast convergence in the iterative solution of the large linear system.



There are no comments yet.


page 1

page 2

page 3

page 4


Computation of generalized matrix functions with rational Krylov methods

We present a class of algorithms based on rational Krylov methods to com...

Computing Markov functions of Toeplitz matrices

We investigate the problem of approximating the matrix function f(A) by ...

The global extended-rational Arnoldi method for matrix function approximation

The numerical computation of matrix functions such as f(A)V, where A is ...

Strongly minimal self-conjugate linearizations for polynomial and rational matrices

We prove that we can always construct strongly minimal linearizations of...

Infinite GMRES for parameterized linear systems

We consider linear parameter-dependent systems A(μ) x(μ) = b for many di...

Banded Matrix Fraction Representation of Triangular Input Normal Pairs

An input pair (A,B) is triangular input normal if and only if A is trian...

Well-Conditioned Methods for Ill-Conditioned Systems: Linear Regression with Semi-Random Noise

Classical iterative algorithms for linear system solving and regression ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.