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
TL;DR: The theoretical background relevant to any method for producing a tridiagonal matrix similar to an arbitrary square matrix is presented and some suggestions for restarting are evaluated.
Abstract: This paper presents the theoretical background relevant to any method for producing a tridiagonal matrix similar to an arbitrary square matrix. Gragg’s work on factoring Hankel matrices and the Kalman–Gilbert structure theorem from systems theory both find a place in the development.Tridiagonalization is equivalent to the application of the generalized Gram–Schmidt process to a pair of Krylov sequences. In Euclidean space proper normalization allows one to monitor a tight lower bound on the condition number of the transformation. The various possibilities for breakdown find a natural classification by the ranks of certain matrices.The theory is illustrated by some small examples and some suggestions for restarting are evaluated
138 citations
TL;DR: In this paper, a numerical method is proposed for the numerical solution of a coupled system of viscous Burgers' equation with appropriate initial and boundary conditions, by using the cubic B-spline collocation scheme on the uniform mesh points.
120 citations
TL;DR: A new algorithm is described that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations.
Abstract: Recently Laurie presented a new algorithm for the computation of (2n+1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n + 1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation phase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible. Numerical examples illustrate the performance of the algorithm.
119 citations
116 citations
TL;DR: A symbolic algorithm for finding the inverse of any general nonsingular tridiagonal matrix, suited for implementation using Computer Algebra Systems (CAS) such as MACSYMA, MAPLE and MATHEMATICA.
112 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |