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.
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TL;DR: In this paper, the concept of cyclic tridiagonal pairs is introduced and explicit examples are given, and the algebra generated by the pair of divided polynomials is identified as a higher-order generalization of the Onsager algebra.
Abstract: The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N, we associate a pair of `divided polynomials'. The properties of this pair generalize the ones of tridiagonal pairs of Racah type. The algebra generated by the pair of divided polynomials is identified as a higher-order generalization of the Onsager algebra. It can be viewed as a subalgebra of the q-Onsager algebra for a proper specialization at q the primitive 2Nth root of unity. Orthogonal polynomials beyond the Leonard duality are revisited in light of this framework. In particular, certain second-order Dunkl shift operators provide a realization of the divided polynomials at N=2 or q=i.
8 citations
TL;DR: In this paper, two different methods are developed for factoring Fn into products of tridiagonal and permutation matrices, one based on matrix identities associated with FFTs and the Rader prime algorithm, and the other based on a numerical technique, called minimal-variable oblique elimination.
8 citations
TL;DR: In this article, it was shown that the restriction that all pivots are to be chosen along the main diagonal can be removed without loss of generality, and the restriction on the number of pivots that can be chosen in a symmetric matrices can also be removed.
8 citations
TL;DR: In this paper, a class of two-level high-order compact finite difference implicit schemes are proposed for solving the Burgers' equations, and the stability of the scheme is analyzed by using the Fourier analysis method.
8 citations
TL;DR: In this article, the authors presented another method which is very efficient and convenient for ordinary physical problems, such as the backward finite difference method for solving the heat equation, which requires the solution of equations (2) for each step where A is fixed and d is changed from step to step.
Abstract: A~1 so obtained, however, is no longer a tridiagonal matrix. If 1 is a n X ft matrix, we will have in general n2 elements of A-1. When n is large, this method becomes unwieldy, even with an electronic computer. Linear equations similar to Eqs. (1) are frequently encountered in problems of mathematical physics. For instance, the backward finite difference method for solving the heat equation requires the solution of equations (2) for each step where A is fixed and d is changed from step to step. Heat equations also appear in the study of longitudinal impact on visco-plastic rods. The solution of other problems, such as discretely loaded strings and the application of the three moment theorem to continuous beams, result in equations of the form of (2). Methods are known which enable one to determine x of (2) more efficiently than by using Eq. (3) (see [1], [2] for example). In the following, we will present another method which is very efficient and convenient for ordinary physical problems. The comparison with the traditional triangular decomposition method is presented in the Appendix.
7 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |