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: Examples are presented that arise naturally from integral and differential equations and that lead to disastrous error growth in Gaussian elimination with partial pivoting.
Abstract: Even though Gaussian elimination with partial pivoting is very widely used, $n \times n$ matrices can be constructed where the error growth in the algorithm is proportional to $2^{n-1}$. Thus for moderate or large $n$, in theory, there is a potential for disastrous error growth. However, prior to 1993 no reports of such an example in a practical application had appeared in the literature. Examples are presented that arise naturally from integral and differential equations and that lead to disastrous error growth in Gaussian elimination with partial pivoting.
56 citations
1 Jul 1991
TL;DR: A divide and conquer algorithm which solves linear tridiagonal systems with one right-hand side, especially suited for parallel computers, and can be combined with recursive doubling, cyclic reduction or Wang's partition method in order to increase the degree of parallelism and vectorizability.
Abstract: We describe a divide and conquer algorithm which solves linear tridiagonal systems with one right-hand side, especially suited for parallel computers. The algorithm is flexible, permits multiprocessing or a combination of vector and multiprocessor implementations, and is adaptable to a wide range of parallelism granularities. This algorithm can also be combined with recursive doubling, cyclic reduction or Wang's partition method, in order to increase the degree of parallelism and vectorizability. The divide and conquer method will be explained. Some results of time measurements on a CRAY X-MP/28, on an Alliant FX/8, and on a Sequent Symmetry S81b, as well as comparisons with the cyclic reduction algorithm and Gaussian elimination will be presented. Finally, numerical results are given.
55 citations
1 Jul 1997
TL;DR: It is shown that there is an exponential length lower bound on the operands for a well-deflned variant of Gaussian elimination when applied to Smith and Hermite normal form calculation, and the analysis provides guidance as to how integer matrix algorithms based onGaussian elimination may be further developed for better performance.
Abstract: Gaussian elimination is the basis for classical algorithms for computing canonical forms of integer matrices. Experimental results have shown that integer Gaussian elimination may lead to rapid growth of intermediate entries. On the other hand various polynomial time algorithms do exist for such computations, but these algorithms are relatively complicated to describe and understand. Gaussian elimination provides the simplest descriptions of algorithms for this purpose. These algorithms have a nice polynomial number of steps, but the steps deal with long operands. Here we show that there is an exponential length lower bound on the operands for a well-deflned variant of Gaussian elimination when applied to Smith and Hermite normal form calculation. We present explicit matrices for which this variant produces exponential length entries. Thus, Gaussian elimination has worst-case exponential space and time complexity for such applications. The analysis provides guidance as to how integer matrix algorithms based on Gaussian elimination may be further developed for better performance, which is important since many practical algorithms for computing canonical forms are so based.
55 citations
1 Aug 1995
TL;DR: A detailed study of the PDD algorithm is given, which is extended to solve periodic tridiagonal systems and its scalability is studied, and the reduced PDD algorithms are proposed, which has a smaller operation count than that of the conventional sequential algorithm for many applications.
Abstract: The Parallel Diagonal Dominant (PDD) algorithm is an efficient tridiagonal solver. In this paper, a detailed study of the PDD algorithm is given. First the PDD algorithm is extended to solve periodic tridiagonal systems and its scalability is studied. Then the reduced PDD algorithm, which has a smaller operation count than that of the conventional sequential algorithm for many applications, is proposed. Accuracy analysis is provided for a class of tridiagonal systems, the symmetric and skew-symmetric Toeplitz tridiagonal systems. Implementation results show that the analysis gives a good bound on the relative error, and the PDD and reduced PDD algorithms are good candidates for emerging massively parallel machines.
54 citations
TL;DR: In this article, the alternative of simply altering the pivot element is examined and the alteration, which amounts to a rank one modification of the matrix, is undone at a later stage by means of the well-known formula for the inverse of a modified matrix.
Abstract: The rounding-error analysis of Gaussian elimination shows that the method is stable only when the elements of the matrix do not grow excessively in the course of the reduction. Usually such growth is prevented by interchanging rows and columns of the matrix so that the pivot element is acceptably large. In this paper the alternative of simply altering the pivot element is examined. The alteration, which amounts to a rank one modification of the matrix, is undone at a later stage by means of the well-known formula for the inverse of a modified matrix. The technique should prove useful in applications in which the pivoting strategy has been fixed, say to preserve sparseness in the reduction.
51 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |