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: In this article, the Gaussian fluctuations and deviations of the traces of tridiagonal random matrices were studied and it was shown that the traces are approximately normally distributed. And the large deviation principle and the moderate deviation principle for the traces were obtained under independent and identically distributed conditions.
Abstract: This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrices. Under quite general assumptions, we prove that the traces are approximately normally distributed. A Multi-dimensional central limit theorem is also obtained here. These results have several applications to various physical models and random matrix models, such as the Anderson model, the random birth–death Markov kernel, the random birth–death Q matrix and the $$\beta $$
-Hermite ensemble. Furthermore, under an independent-and-identically-distributed condition, we also prove the large deviation principle as well as the moderate deviation principle for the traces.
5 citations
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TL;DR: A fundamental result is obtained that connects the condition number of a common block tridiagonal system to properties of the individual blocks and shows how unstable Kalman smoothing problems can arise, and how to easily stabilize them.
Abstract: Block tridiagonal systems play a key role in all Kalman smoothing applications, including the classic Rauch-Tung-Striebel smoother, as well as more modern variants that incorporate nonlinear models, inequality constraints on the state, and robust penalties on state and measurement components. The paper begins with a fundamental result that connects the condition number of a common block tridiagonal system to properties of the individual blocks. As a consequence, we obtain sufficient conditions for the stability of Kalman smoothing formulations. The result also illustrates how unstable Kalman smoothing problems can arise, and how to easily stabilize them. Then, turning our attention to algorithms, it is shown that the classic Rauch-Tung-Striebel smoother is an implementation of the forward Thomas algorithm for block tridiagonal systems. A flaw in the existing theory for characterizing the stability of the algorithm is revealed. A remedy is provided by proving a new stability result which shows that the condition number of every recursively modified block is bounded by the condition number of the whole system as the algorithm proceeds. Finally, a new backward block tridiagonal Thomas algorithm is presented. It is demonstrate that this algorithm behaves stably independent of the condition number of the full block tridiagonal system.
5 citations
TL;DR: In this paper, a recurrence solution is used to solve the matrix equation A U = W, where A is a block tridiagonal matrix where the boundary conditions are periodic, and the resulting matrix A is block trdiagonal with addiOperating system: NOS tional blocks in the upper right and left corners, referred to here as block perdiagonal.
5 citations
TL;DR: In this note a comparison of the odd/even and stride of 3 reduction algorithms is presented for a periodic tridiagonal linear system.
Abstract: In this note a comparison of the odd/even and stride of 3 reduction algorithms is presented for a periodic tridiagonal linear system.
5 citations
TL;DR: Algorithms BANDSQ and BDSQMX are presented which require N2b in-core words but minimize the number of retrieving and restoring the direct-access records during the Gaussian elimination of the structural stiffness matrix.
Abstract: Based on direct-access programming, algorithms have been developed for the generation, and solution by Gaussian elimination of the structural stiffness matrix equation resulted from application of the finite element method in engineering analyses. A large disk storage is used to store the rows of the stiffness matrix as directly accessible records. The developed algorithm BAND2R requires only 2Nb in-core words in implementing the Gaussian elimination where Nb is the semi-band width of the stiffness matrix. Algorithms BANDSQ and BDSQMX are presented which require N2b in-core words but minimize the number of retrieving and restoring the direct-access records during the Gaussian elimination. BANDSQ has the direct-access feature in both the elimination and backward-substitution steps whereas BDSQMX has the direct-access feature only in the backward-substitution step of the Gaussian elimination. Illustrative applications of the developed algorithms are given and the computer core and time requirements for BAND2R, BANDSQ and BDSQMX are compared to those for the conventional Gaussian elimination of using sequential, in-core storages. Methods for reducing the semi-band width Nb of the structural stiffness matrix are also discussed.
5 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |