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: Efficient numerical methods for computing the Pfaffian of a skew-symmetric matrix in tridiagonal form based on Gaussian elimination are developed, and the equivalence of definitions based on the Hamiltonian and the scattering matrix is shown.
Abstract: Computing the Pfaffian of a skew-symmetric matrix is a problem that arises in various fields of physics. Both computing the Pfaffian and a related problem, computing the canonical form of a skew-symmetric matrix under unitary congruence, can be solved easily once the skew-symmetric matrix has been reduced to skew-symmetric tridiagonal form. We develop efficient numerical methods for computing this tridiagonal form based on Gauss transformations, using a skew-symmetric, blocked form of the Parlett-Reid algorithm, or based on unitary transformations, using block Householder transformations and Givens rotations, that are applicable to dense and banded matrices, respectively. We also give a complete and fully optimized implementation of these algorithms in Fortran, and also provide Python, Matlab and Mathematica implementations for convenience. Finally, we apply these methods to compute the topological charge of a class D nanowire, and show numerically the equivalence of definitions based on the Hamiltonian and the scattering matrix.
163 citations
TL;DR: In this paper, two canonical forms for Leonard pairs are introduced: the TD-D canonical form and the LB-UB canonical form, where the diagonal matrix of the matrix representing A is irreducible tridiagonal and the matrix of B is diagonal.
146 citations
TL;DR: The problem of solving tridmgonal linear systems on vector computers is considered and implementations of several direct and lterative methods are given for the Control Data Corporatlon STAR-100 computer.
Abstract: The problem of solving tridmgonal linear systems on vector computers is considered. In particular, implementations of several direct and lterative methods are given for the Control Data Corporatlon STAR-100 computer. The direct methods considered are Gaussian elimination, a parallel method due to Stone, and cyclic reduction; the iteratlve methods considered are Jacobi's method, successive overrelaxaUon, and a parallel method due to Traub. In addition, timing formulas for the methods based on current information are included to provide a basis for comparison In general, the direct methods are found to be superior to the iterative methods. The choice of direct methods depends on the size of the system, but for more than 125 equations, cyclic reduction is the fastest algorithm.
140 citations
TL;DR: In this paper, a complete analysis for general tridiagonal matrix inversion for both non-block and block cases is given, and some simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or Toeplitz tridimensional matrices.
Abstract: In this paper we give a complete analysis for general tridiagonal matrix inversion for both non-block and block cases, and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or Toeplitz tridiagonal matrices.
138 citations
TL;DR: In this paper, the eigenvalues of sequences of tridiagonal matrices that contain a Toeplitz matrix in the upper left block were determined, and they were shown to be linear.
138 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |