Volume 35, pp. 118-128, 2009.

Monotone convergence of the Lanczos approximations to matrix functions of Hermitian matrices

Andreas Frommer

Abstract

When $A$ is a Hermitian matrix, the action $f(A)b$ of a matrix function $f(A)$ on a vector $b$ can efficiently be approximated via the Lanczos method. In this note we use $M$-matrix theory to establish that the $2$-norm of the error of the sequence of approximations is monotonically decreasing if $f$ is a Stieltjes transform and $A$ is positive definite. We discuss the relation of our approach to a recent, more general monotonicity result of Druskin for Laplace transforms. We also extend the class of functions to certain product type functions. This yields, for example, monotonicity when approximating $\text{sign}(A)b$ with $A$ indefinite if the Lanczos method is performed for $A^2$ rather than $A$.

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Key words

matrix functions, Lanczos method, Galerkin approximation, monotone convergence, error estimates

AMS subject classifications

6530, 65F10, 65F50

Links to the cited ETNA articles

[22]	Vol. 13 (2002), pp. 56-80 Zdeněk Strakoš and Petr Tichý: On error estimation in the conjugate gradient method and why it works in finite precision computations