Bounds for the error of linear systems of equations using the theory of moments
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TL;DR
It is shown that by examining δi = xi + 1 − xi, it is possible to construct upper and lower bounds for ∥ xi − x ∥, which indicates the euclidean norm.
Abstract
Consider the system, of linear equations Ax = b where A is an n × n real symmetric, positive definite matrix and b is a known vector. Suppose we are given an approximation to x, ξ, and we wish to determine upper and lower bounds for ∥ x − ξ ∥ where ∥ ··· ∥ indicates the euclidean norm. Given the sequence of vectors {ri}ik = 0, where ri = Ari − 1 and r0 = b − Aξ, it is shown how to construct a sequence of upper and lower bounds for ∥ x − ξ ∥ using the theory of moments. In addition, consider the Jacobi algorithm for solving the system x = Mx + b, viz., xi + 1 = Mxi + b. It is shown that by examining δi = xi + 1 − xi, it is possible to construct upper and lower bounds for ∥ xi − x ∥.