## Vector estimates for f(A)b via extrapolation

Marilena Mitrouli and Paraskevi Roupa

### Abstract

Let $A\in\mathbb{R}^{p\times p}$ be a diagonalizable matrix and $f$ a smooth function. We are interested in the problem of approximating the action of $f(A)$ on a vector ${\bf b}\in\mathbb{R}^p$, i.e., $f(A){\bf b}$, without explicitly computing the matrix $f(A)$. In the present work, we derive families of one-term, two-term, and three-term inexpensive approximations to the quantity $f(A){\bf b}$ via an extrapolation procedure. For a given diagonalizable matrix $A$, the proposed families of vector estimates allow us to approximate the form $W^Tf(A)U$, for any matrices $W,U\in\mathbb{R}^{p\times m}$, $1 \leq m \ll p$, not necessarily biorthogonal. We present several numerical examples to illustrate the effectiveness of our method for several functions $f$ for both the quantity $f(A){\bf b}$ and the form $W^Tf(A)U$.

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### Key words

f(A)b, vector estimates, vector moments, extrapolation, diagonalizable matrices

### AMS subject classifications

65F15, 65F30, 65F60, 65B05, 15A18

### Links to the cited ETNA articles

 [6] Vol. 39 (2012), pp. 144-155 Claude Brezinski, Paraskevi Fika, and Marilena Mitrouli : Estimations of the trace of powers of positive self-adjoint operators by extrapolation of the moments [14] Vol. 43 (2014-2015), pp. 70-89 Paraskevi Fika, Marilena Mitrouli, and Paraskevi Roupa: Estimates for the bilinear form $x^T A^{-1} y$ with applications to linear algebra problems [19] Vol. 37 (2010), pp. 147-165 Bernard N. Sheehan, Yousef Saad, and Roger B. Sidje: Computing $\exp(-\tau A)b$ with Laguerre polynomials

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