## Enhanced matrix function approximation

Nasim Eshghi, Lothar Reichel, and Miodrag M. Spalević

### Abstract

Matrix functions of the form $f(A)v$, where $A$ is a large symmetric matrix, $f$ is a function, and $v\ne 0$ is a vector, are commonly approximated by first applying a few, say $n$, steps of the symmetric Lanczos process to $A$ with the initial vector $v$ in order to determine an orthogonal section of $A$. The latter is represented by a (small) $n\times n$ tridiagonal matrix to which $f$ is applied. This approach uses the $n$ first Lanczos vectors provided by the Lanczos process. However, $n$ steps of the Lanczos process yield $n+1$ Lanczos vectors. This paper discusses how the $(n+1)$st Lanczos vector can be used to improve the quality of the computed approximation of $f(A)v$. Also the approximation of expressions of the form $v^Tf(A)v$ is considered.

Full Text (PDF) [216 KB], BibTeX

### Key words

matrix function, symmetric Lanczos process, Gauss quadrature

### AMS subject classifications

65D32, 65F10, 65F60

### Links to the cited ETNA articles

 [11] Vol. 45 (2016), pp. 371-404 Sotirios E. Notaris: Gauss-Kronrod quadrature formulae - A survey of fifty years of research

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