Topic
Tridiagonal matrix algorithm
About: Tridiagonal matrix algorithm is a research topic. Over the lifetime, 1070 publications have been published within this topic receiving 21084 citations.
Papers published on a yearly basis
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Papers
TL;DR: A short comment on a paper involving the minimal eigenvalues of a specialized version of 2-Toeplitz tridiagonal matrices.
2 citations
1 Jul 1977
TL;DR: Three relatively brief papers are presented on the following subjects: element growth in Gaussian elimination, estimating cond(A) with subroutine DGECO, and consideration of an error bound inGaussian elimination.
Abstract: Three relatively brief papers are presented on the following subjects: element growth in Gaussian elimination, estimating cond(A) with subroutine DGECO, and consideration of an error bound in Gaussian elimination. (RWR)
2 citations
Posted Content•
TL;DR: In this article, it was shown that small coordinates of eigenvectors of symmetric tridiagonal matrices can be evaluated with high relative accuracy under certain conditions, e.g. small coordinates whose magnitude is below the machine precision.
Abstract: Evaluation of the eigenvectors of symmetric tridiagonal matrices is one of the most basic tasks in numerical linear algebra. It is a widely known fact that, in the case of well separated eigenvalues, the eigenvectors can be evaluated with high relative accuracy. Nevertheless, in general, each coordinate of the eigenvector is evaluated with only high $absolute$ accuracy. In particular, those coordinates whose magnitude is below the machine precision are not expected to be evaluated with any accuracy whatsoever.
It turns out that, under certain conditions, frequently ecountered in applications, small (e.g. $10^{-50}$) coordinates of eigenvectors of symmetric tridiagonal matrices can be evaluated with high $relative$ accuracy. In this paper, we investigate such conditions, carry out the analysis, and describe the resulting numerical schemes. While our schemes can be viewed as a modification of already existing (and well known) numerical algorithms, the related error analysis appears to be new.
Our results are illustrated via several numerical examples.
2 citations
1 Jan 1994
TL;DR: An error analysis is given showing that Huard's elimination method is as stable as Gaussian elimination with appropriate pivoting strategy, and is proven in a similar way as the proof of stability for Gauss-Jordan given in [4].
Abstract: This paper considers elimination methods to solve dense linear systems, in particular a variant due to Huard of Gaussian elimination [13]. This variant reduces the system to an equivalent diagonal system just as GaussJordan elimination, but does not require more floating-point operations than Gaussian elimination. Huard's method may be advantageous for use in computers with hierarchical memory, such as cache, and in distributedmemory systems. An error analysis is given showing that Huard's elimination method is as stable as Gauss-Jordan elimination with appropriate pivoting strategy. This result was announced in [5] and is proven in a similar way as the proof of stability for Gauss-Jordan given in [4].
2 citations
TL;DR: In this article, an iterative reduction based on Krylov sequences as a direct tridiagonalization procedure via Householder transformations is reconsidered, which is then applied to the normal case and equality of the absolute value between the super and subdiagonals is proved.
2 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |