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
19 Mar 2022
Abstract: We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the $n$-th power of a tridiagonal matrix and the enumeration of weighted paths of $n$ steps allows an easier combinatorial enumeration of the paths. It also seems promising for the theory of tridiagonal random matrices .
TL;DR: The model of bulk-synchronous parallel (BSP) computation is an emerging paradigm of general-purpose parallel computing and the BSP complexity of Gaussian elimination and related problems is studied.
Abstract: The model of bulk-synchronous parallel (BSP) computation is an emerging paradigm of general-purpose parallel computing. We study the BSP complexity of Gaussian elimination and related problems. First, we analyze the Gaussian elimination without pivoting, which can be applied to the LU decomposition of symmetric positive-definite or diagonally dominant real matrices. Then we analyze the Gaussian elimination with Schonhage's recursive local pivoting suitable for the LU decomposition of matrices over a finite field, and for the QR decomposition of real matrices by the Givens rotations. Both versions of Gaussian elimination can be performed with an optimal amount of local computation, but optimal communication and synchronization costs cannot be achieved simultaneously. The algorithms presented in the paper allow one to trade off communication and synchronization costs in a certain range, achieving optimal or near-optimal cost values at the extremes. Bibliography: 19 titles.
TL;DR: This work characterize certain properties of A, i.e., being an M-matrix or positive definite, in terms of the ui, vi,xi, and yi, and establishes a relation of zero row sums and zero column sums of A and pairwise constant ui.
Abstract: It is well known that the inverse C = [ci,j] of an irreducible nonsingular symmetric tridiagonal matrix is given by two sequences of real numbers, {ui} and {vi}, such that ci,j = u i vj for $i \leq j$. A similar result holds for nonsymmetric matrices A. There the inverse can be described by four sequences {ui},{vi}, {xi},$ and {vi} with u ivi = xiyi. Here we characterize certain properties of A, i.e., being an M-matrix or positive definite, in terms of the ui, vi,xi, and yi. We also establish a relation of zero row sums and zero column sums of A and pairwise constant ui,vi, xi, and yi. Moreover, we consider decay rates for the entries of the inverse of tridiagonal and block tridiagonal (banded) matrices. For diagonally dominant matrices we show that the entries of the inverse strictly decay along a row or column. We give a sharp decay result for tridiagonal irreducible M-matrices and tridiagonal positive definite matrices. We also give a decay rate for arbitrary banded M-matrices.
TL;DR: An efficient and fast computing method is given to obtain the elements of the inverse of a tridiagonal matrix by backward continued fractions by exploiting the relationships between the usual and backward continued fraction.
TL;DR: In this paper, a simple algorithm for inverting nonsymmetric tridiagonal matrices that leads immediately to closed forms when they exist is presented, and Ukita's theorem is extended to characterize the class of matrices with tridagonal inverses.
Abstract: This paper presents a simple algorithm for inverting nonsymmetric tridiagonal matrices that leads immediately to closed forms when they exist Ukita's theorem is extended to characterize the class of matrices that have tridiagonal inverses
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |