TY - JOUR
T1 - Convergence analysis of an algorithm for accurate inverse Cholesky factorization
AU - Yanagisawa, Yuka
AU - Ogita, Takeshi
AU - Oishi, Shin’ichi
N1 - Publisher Copyright:
© 2014, The JJIAM Publishing Committee and Springer Japan.
PY - 2014/11/1
Y1 - 2014/11/1
N2 - 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.
AB - 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.
KW - Accurate numerical algorithm
KW - Cholesky factorization
KW - Convergence analysis
KW - Ill-conditioned matrix
KW - Positive definiteness
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U2 - 10.1007/s13160-014-0154-4
DO - 10.1007/s13160-014-0154-4
M3 - Article
AN - SCOPUS:84940003635
VL - 31
SP - 461
EP - 482
JO - Japan Journal of Industrial and Applied Mathematics
JF - Japan Journal of Industrial and Applied Mathematics
SN - 0916-7005
IS - 3
ER -