Solvers for linear systems with tridiagonal coefficient matrices sometimes employ direct methods such as the Gauss elimination method or the cyclic reduction method. In each step of the cyclic reduction method, nonzero offdiagonal entries in the coefficient matrix move incrementally away from diagonal entries and eventually vanish. The steps of the cyclic reduction method are generalized as forms of the stride reduction method. For example, the 2-stride reduction method coincides with the 1st step of the cyclic reduction method which transforms tridiagonal linear systems into pentadiagonal systems. In this paper, we explain arbitrary-stride reduction for linear systems with coefficient matrices with three nonzero bands. We then show that arbitrary-stride reduction is equivalent to a combination of 2-stride reduction and row and column permutations. We thus clarify eigenvalue clustering of coefficient matrices in the step-by-step process of the stride reduction method. We also provide two examples verifying this property.

Eigenvalue clustering of coefficient matrices in the iterative stride reductions for linear systems

原著論文リスト[A1]: “The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-012-9607-5.”. [A2]: “The final publication is available at Springer via http://dx.doi.org/10.1007/s10092-013-0085-5.”, [A3]: DOI“10.1016/j.camwa.2015.11.022”

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/215686/1/yjohk00610.pdf

Studies on Accurate Singular Value Decomposition for Bidiagonal Matrices

It is to be hoped that other Viatores will embark, or have already embarked, on the task of producing similar books for other parts of Britain. In reading this one it becomes painfully clear how rapidly ancient remains, even tracks, are disappearing. The Viatores are explorers, but they are also fighting a rearguard action, for the visible evidence of much of the information they are salvaging is too often being swept away by motorways, housing developments and factories.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4FD8B2BD00EE029DC4AF9BC18AB73629/S007543580007057Xa.pdf/div-class-title-a-n-bryan-brown-and-others-span-class-italic-more-oxford-compositions-span-oxford-clarendon-press-1964-pp-xli-234-35-span-class-italic-s-span-div.pdf

A. N. Bryan-Brown and Others, More Oxford Compositions. Oxford: Clarendon Press, 1964. Pp. xli + 234. 35s.

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.

Roots of Characteristic Polynomial Sequences in Iterative Block Cyclic Reductions

The cyclic reduction method is a direct method for solving tridiagonal linear systems. At the first step of this method, a tridiagonal coefficient matrix is transformed into a pentadiagonal form. In this article, we prove that the condition number for eigenvalues of some classes of coefficient matrices always decreases after the first step of the cyclic reduction method.

On condition numbers in the cyclic reduction processes of a tridiagonal matrix

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On condition numbers in the cyclic reduction processes of a tridiagonal matrix