ITERATIVE METHODS FOR LARGE INDEFINITE LINEAR SYSTEMS: A LANCZOS PROCESS APPROACH WITH ERROR ESTIMATION
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Date
2025
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Publisher
Saudi Digital Library
Abstract
This thesis describes a Krylov subspace iterative method designed for solving linear systems of
equations with a large, symmetric, nonsingular, and indefinite matrix. This method is tailored to
enable the evaluation of error estimates for the computed iterates. The availability of error estimates
makes it possible to terminate the iterative process when the estimated error is smaller than a use rspecified tolerance. The error estimates are calculated by leveraging the relationship between the
iterates and Gauss-type quadrature rules. Computed examples illustrate the performance of the
iterative method and the error estimates
Description
This study presents novel techniques for estimating the Euclidean norm of the error in approximate
solutions determined by an iterative Krylov subspace method, designed for solving linear systems of
equations with a symmetric, nonsingular indefinite matrix. The estimates are obtained by utilizing
the relationship between the Krylov subspace iterative method and Gauss-type quadrature rules.
The computed results demonstrate the performance of the iterative method and the estimated error
norms. Among the quadrature rules considered the rules (1.37) and (2.61) gave the most accurate
error norm estimates.
Keywords
Lanczos process, error norm estimate, Gauss-type quadrature rules, iterative method