Accurate and robust inverse Cholesky factorization

DOI Web Site 被引用文献2件 参考文献9件 オープンアクセス
  • Ogita Takeshi
    Division of Mathematical Sciences, School of Arts and Sciences, Tokyo Woman's Christian University JST, CREST
  • Oishi Shin'ichi
    Department of Applied Mathematics, Faculty of Science and Engineering, Waseda University JST, CREST

説明

In this paper, an algorithm for an accurate matrix factorization based on Cholesky factorization for extremely ill-conditioned matrices is proposed. The Cholesky factorization is widely used for solving a system of linear equations whose coefficient matrix is symmetric and positive definite. However, it sometimes breaks down by the presence of an imaginary root due to the accumulation of rounding errors, even if the matrix is actually positive definite. To overcome this, a completely stable algorithm named inverse Cholesky factorization is investigated, which never breaks down as long as the matrix is symmetric and positive definite. The proposed algorithm consists of standard numerical algorithms and an accurate algorithm for dot products. Moreover, it is shown that the algorithm can also verify the positive definiteness of a given real symmetric matrix. Numerical results are presented for illustrating the performance of the proposed algorithms.

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