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Published in:
30-05-2016
Aitken’s method for estimating bilinear forms arising in applications
Published in: Calcolo | Issue 1/2017
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Abstract
In the present work we study how the Aitken’s method can be applied to the sequence of moments \(\{c_k=(x, A^{k}x),~k \in {\mathbb {Z}}\},\) of a given symmetric positive definite matrix \(A \in {\mathbb {R}}^{p \times p},\) for the prediction of possible unknown terms of this sequence. Direct estimation of \(c_k\) leads to approximating bilinear quantities i.e. \((y, A^{k}x)\) as well. By employing Taylor series expansion for appropriate f, prediction of \(y^Tf(A)x\) 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.