# A modification of the Jacobi-Davidson method

## TL;DR

This paper introduces a modified Jacobi-Davidson method for large sparse eigenvalue problems, incorporating a new correction equation inspired by least squares, with analysis and numerical validation of its convergence and efficiency.

## Contribution

It proposes a novel modification to the Jacobi-Davidson method using a least squares motivated correction equation, enhancing convergence analysis for symmetric matrices.

## Key findings

- Convergence properties are established for symmetric matrices.
- Numerical experiments demonstrate the method's computational viability.
- The modified method offers potential improvements over existing approaches.

## Abstract

Each iteration in Jacobi-Davidson method for solving large sparse eigenvalue problems involves two phases, called subspace expansion and eigen pair extraction. The subspace expansion phase involves solving a correction equation. We propose a modification to this by introducing a related correction equation, motivated by the least squares. We call the proposed method as the Modified Jacobi-Davidson Method. When the subspace expansion is ignored as in the Simplified Jacobi- Davidson Method, the modified method is called as Modified Simplified Jacobi-Davidson Method. We analyze the convergence properties of the proposed method for Symmetric matrices. Numerical experiments have been carried out to check whether the method is computationally viable or not.