Roots of Characteristic Polynomial Sequences in Iterative Block Cyclic Reductions

Masato Shinjo, Tan Wang, Masashi Iwasaki and Yoshimasa Nakamura
Additional contact information
Masato Shinjo: Faculty of Science and Engineering, Doshisha University, Kyotanabe 610-0394, Japan
Tan Wang: Digital Technology & Innovation, Siemens Healthineers Digital Technology (Shanghai) Co., Ltd., Shanghai 201318, China
Masashi Iwasaki: Faculty of Life and Environmental Science, Kyoto Prefectural University, Kyoto 606-8522, Japan
Yoshimasa Nakamura: Department of Informatics and Mathematical Science, Osaka Seikei University, Osaka 533-0007, Japan

Mathematics, 2021, vol. 9, issue 24, 1-17

Abstract: The block cyclic reduction method is a finite-step direct method used for solving linear systems with block tridiagonal coefficient matrices. It iteratively uses transformations to reduce the number of non-zero blocks in coefficient matrices. With repeated block cyclic reductions, non-zero off-diagonal blocks in coefficient matrices incrementally leave the diagonal blocks and eventually vanish after a finite number of block cyclic reductions. In this paper, we focus on the roots of characteristic polynomials of coefficient matrices that are repeatedly transformed by block cyclic reductions. We regard each block cyclic reduction as a composition of two types of matrix transformations, and then attempt to examine changes in the existence range of roots. This is a block extension of the idea presented in our previous papers on simple cyclic reductions. The property that the roots are not very scattered is a key to accurately solve linear systems in floating-point arithmetic. We clarify that block cyclic reductions do not disperse roots, but rather narrow their distribution, if the original coefficient matrix is symmetric positive or negative definite.

Keywords: block cyclic reduction; block tridiagonal matrix; characteristic polynomial; linear system (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
https://www.mdpi.com/2227-7390/9/24/3213/pdf (application/pdf)
https://www.mdpi.com/2227-7390/9/24/3213/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:9:y:2021:i:24:p:3213-:d:700644

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().