Recent Developments in Algorithms for Solving Dense Eigenproblems (I) : Algorithm of Multiple Relatively Robust Representations(<Special Issue>Algorithms for Matrix・Eigenvalue Problems and their Applications)

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  • Yamamoto Yusaku
    Department of Computational Science & Engineering, Nagoya University

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  • 密行列固有値解法の最近の発展(I) : Multiple Relatively Robust Representationsアルゴリズム(<特集>行列・固有値問題における線形計算アルゴリズムとその応用)
  • 特集:行列・固有値問題における線形計算アルゴリズムとその応用
  • トクシュウ ギョウレツ コユウチ モンダイ ニ オケル センケイ ケイサン アルゴリズム ト ソノ オウヨウ

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Abstract

The Algorithm of Multiple Relatively Robust Representations (MR^3) is a new algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem proposed by I. Dhillon in 1997. It has attracted much attention because it can compute all the eigenvectors of an n×n matrix in only O(n^2) work and is easy to parallelize. In this article, we survey the papers related to the MR^3 algorithm and try to present a simple and easily understandable picture of the algorithm by explaining, one by one, its key ingredients such as the relatively robust representations of a symmetric tridiagonal matrix, the dqds algorithm for computing accurate eigenvalues and the twisted factorization for computing accurate eigenvectors. Limitations of the algorithm and directions for future research are also discussed.

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