# Multiple-rank Modification of Symmetric Eigenvalue Problem

## TL;DR

This paper introduces a novel rank-k modification algorithm for symmetric eigenvalue problems, improving computational efficiency and providing a specific rank-2 algorithm with O(n^1.5) complexity, along with a general rank-k update method.

## Contribution

It presents the first rank-k modification algorithm for symmetric eigenvalue problems, including a rank-2 algorithm and a general rank-k update based on the modified Sturm Theorem.

## Key findings

- Rank-2 modification algorithm with O(n^1.5) complexity.
- Comparison shows efficiency over direct eigenvalue decomposition.
- Proposed methods outperform perturbation approaches.

## Abstract

Rank-1 modifications in k-times (k > 1) often are performed to achieve rank-k modification. We propose a rank- k modification for enhancing computational efficiency. As the first step towards a rank- k modification, an algorithm to perform rank-2 modification is proposed and tested. The computation cost of our proposed algorithm is in O(n^1.5). We also propose a general rank- k update algorithm based upon the modified Sturm Theorem, and compare our results from those of the direct eigenvalue decomposition and of a perturbation method.

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Source: https://tomesphere.com/paper/1706.00773