Vector estimates for f(A)b via extrapolation

Marilena  Mitrouli; Paraskevi  Roupa

doi:doi:10.1553/etna_vol47s179

Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Vector estimates for f(A)b via extrapolation

Marilena Mitrouli,

Paraskevi Roupa

ETNA - Electronic Transactions on Numerical Analysis, pp. 179-196, 2017/11/15

doi: 10.1553/etna_vol47s179

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doi:10.1553/etna_vol47s179

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 approximatingthe action of $f(A)$ on a vector ${\bf b}\in\mathbb{R}^p$, i.e., $f(A){\bf b}$, without explicitlycomputing the matrix $f(A)$.In the present work, we derive families of one-term, two-term, and three-term inexpensive approximationsto 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 theform $W^Tf(A)U$, 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 effectivenessof our method for several functions $f$ for boththe quantity $f(A){\bf b}$ and the form $W^Tf(A)U$.

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

ETNA

Published Online: 2017/11/15 13:30:45

Object Identifier: 0xc1aa5572 0x00371070

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Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

Vector estimates for f(A)b via extrapolation

Abstract