Convergence analysis of an algorithm for accurate inverse Cholesky factorization

Yuka Yanagisawa, Takeshi Ogita, Shinichi Oishi

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

This paper is concerned with factorization of symmetric and positive definite matrices which are extremely ill-conditioned. Following the results by Rump (1990), Oishi et al. (2007, 2009) and Ogita (2010), Ogita and Oishi (2012) derived an iterative algorithm for an accurate inverse matrix factorization based on Cholesky factorization for such ill-conditioned matrices. We analyze the behavior of the algorithm in detail and give reasons for convergency by the use of numerical error analysis. Main analysis is that each iteration reduces the condition number of a preconditioned matrix by a factor around the relative rounding error unit until convergence. This behavior is consistent with the numerical results.

Original languageEnglish
Pages (from-to)461-482
Number of pages22
JournalJapan Journal of Industrial and Applied Mathematics
Volume31
Issue number3
DOIs
Publication statusPublished - 2014 Nov 1

Fingerprint

Cholesky factorisation
Factorization
Convergence Analysis
Matrix Factorization
Inverse matrix
Positive definite matrix
Rounding error
Relative Error
Condition number
Error Analysis
Iterative Algorithm
Numerical Analysis
Iteration
Numerical Results
Unit
Error analysis

Keywords

  • Accurate numerical algorithm
  • Cholesky factorization
  • Convergence analysis
  • Ill-conditioned matrix
  • Positive definiteness

ASJC Scopus subject areas

  • Applied Mathematics
  • Engineering(all)

Cite this

Convergence analysis of an algorithm for accurate inverse Cholesky factorization. / Yanagisawa, Yuka; Ogita, Takeshi; Oishi, Shinichi.

In: Japan Journal of Industrial and Applied Mathematics, Vol. 31, No. 3, 01.11.2014, p. 461-482.

Research output: Contribution to journalArticle

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