Project Acronym: EFESTTitle: Efficient estimation of matrix functions with applications to statistics, networks and machine learningAffiliation: university of athensPi: Marilena MitrouliResearch Field: mathematics and computer sciences
Aitken's method for estimating bilinear forms arising in applications
by Fika, Paraskevi and Mitrouli, Marilena
Abstract:
In the present work we study how the Aitken's method can be applied to the sequence of moments $}{\$\backslash\c_k=(x, A^\k\x),\textasciitildek \backslashin \\backslashmathbb \Z\\\backslash\,$}{\$ \ c k = ( x , A k x ) , k ∈ Z \ , of a given symmetric positive definite matrix $}{\$A \backslashin \\backslashmathbb \R\\^\p \backslashtimes p\,$}{\$ A ∈ R p \texttimes p , for the prediction of possible unknown terms of this sequence. Direct estimation of $}{\$c_k$}{\$ c k leads to approximating bilinear quantities i.e. $}{\$(y, A^\k\x)$}{\$ ( y , A k x ) as well. By employing Taylor series expansion for appropriate f, prediction of $}{\$y^Tf(A)x$}{\$ y T f ( A ) x can be achieved. Estimates for many useful linear algebra quantities can be derived by appropriately selecting the vectors y and x. Numerical examples concerning such applications are presented. The estimates are illustrated through numerical examples executed on the high-performance computing system ARIS.
Reference:
Aitken's method for estimating bilinear forms arising in applications (Fika, Paraskevi and Mitrouli, Marilena), In Calcolo, volume 54, 2017.
Bibtex Entry:
@article{Fika2017,
author = {Fika, Paraskevi
and Mitrouli, Marilena},
title = {Aitken's method for estimating bilinear forms arising in applications},
journal = {Calcolo},
year = {2017},
bibyear = {2017},
month = {Mar},
day = {01},
volume = {54},
number = {1},
pages = {455--470},
abstract = {In the present work we study how the Aitken's method can be applied to the sequence of moments {\$}{\$}{\backslash}{\{}c{\_}k=(x, A^{\{}k{\}}x),{\textasciitilde}k {\backslash}in {\{}{\backslash}mathbb {\{}Z{\}}{\}}{\backslash}{\}},{\$}{\$} {\{} c k = ( x , A k x ) , k ∈ Z {\}} , of a given symmetric positive definite matrix {\$}{\$}A {\backslash}in {\{}{\backslash}mathbb {\{}R{\}}{\}}^{\{}p {\backslash}times p{\}},{\$}{\$} A ∈ R p {\texttimes} p , for the prediction of possible unknown terms of this sequence. Direct estimation of {\$}{\$}c{\_}k{\$}{\$} c k leads to approximating bilinear quantities i.e. {\$}{\$}(y, A^{\{}k{\}}x){\$}{\$} ( y , A k x ) as well. By employing Taylor series expansion for appropriate f, prediction of {\$}{\$}y^Tf(A)x{\$}{\$} y T f ( A ) x can be achieved. Estimates for many useful linear algebra quantities can be derived by appropriately selecting the vectors y and x. Numerical examples concerning such applications are presented. The estimates are illustrated through numerical examples executed on the high-performance computing system ARIS.},
issn = {1126-5434},
doi = {10.1007/s10092-016-0193-0},
url = {https://doi.org/10.1007/s10092-016-0193-0},
}