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
1 / 2
Papers
1 Dec 2011
TL;DR: In this article, the tridiagonal-to-banded back transformation was proposed to improve the parallel efficiency for large numbers of processors as well as the per-processor utilization.
Abstract: The computation of selected eigenvalues and eigenvectors of a symmetric (Hermitian) matrix is an important subtask in many contexts, for example in electronic structure calculations. If a significant portion of the eigensystem is required then typically direct eigensolvers are used. The central three steps are: reduce the matrix to tridiagonal form, compute the eigenpairs of the tridiagonal matrix, and transform the eigenvectors back. To better utilize memory hierarchies, the reduction may be effected in two stages: full to banded, and banded to tridiagonal. Then the back transformation of the eigenvectors also involves two stages. For large problems, the eigensystem calculations can be the computational bottleneck, in particular with large numbers of processors. In this paper we discuss variants of the tridiagonal-to-banded back transformation, improving the parallel efficiency for large numbers of processors as well as the per-processor utilization. We also modify the divide-and-conquer algorithm for symmetric tridiagonal matrices such that it can compute a subset of the eigenpairs at reduced cost. The effectiveness of our modifications is demonstrated with numerical experiments.
221 citations
TL;DR: Roundoff error m the solution of near algebraic systems is studied using a more reahstsc notion of what st means to perturb a problem, namely, that each datum is subject to a relatwely small change.
Abstract: Roundoff error m the solution of hnear algebraic systems is stud,ed using a more reahstsc notion of what st means to perturb a problem, namely, that each datum :s subject to a relatwely small change Th:s ,s particularly appropriate for sparse linear systems The condition number :s determined for th:s approach The effect of scahng on the stabdlty of Gaussmn ellmmat,on is stud:ed, and st is d:scovered that the proper way to scale a system depends on the right-hand s:de However, ff only the norm of the error is of concern, then there ~s a good way to scale that does not depend on the right-hand stde
208 citations
TL;DR: In this paper, a formula for the inverse of a general tridiagonal matrix is given in terms of the principal minors of the matrix, where the principal minor is defined by the principal matrix.
202 citations
TL;DR: The solution of a general block tridiagonal linear system by a cyclic odd-even reduction algorithm is considered, under conditions of diagonal dominance, norms describing the off-diagonal blocks relative to the diagonal blocks decrease quadratically with each reduction.
Abstract: The solution of a general block tridiagonal linear system by a cyclic odd-even reduction algorithm is considered. Under conditions of diagonal dominance, norms describing the off-diagonal blocks relative to the diagonal blocks decrease quadratically with each reduction. This may allow early termination of the reduction when an approximate solution is desired. The algorithm is well-suited for parallel computation.
174 citations
TL;DR: The bisect procedure was designed to replace the tridibi 1 and 2 given in [5] and is based on the following theorem as discussed by the authors, where bisect is defined as follows:
Abstract: The procedure bisect is designed to replace the procedures tridibi 1 and 2 given in [5]. All three procedures are based essentially on the following theorem.
167 citations
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| Year | Papers |
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| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |