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: An algorithm for solving systems of block bidiagonal triangular linear equations was given, which is not necessary to treat with the zero elements out of banded systems and can quicken the speed of ray-tracing.
Abstract: Beginning with the method of whole path iterative ray-tracing and according to the positive definiteness of the coefficient matrix of the systems of linear equations, a symmetry block tridiagonal matrix was decomposed into the product of block bidiagonal triangular matrix and its transpose by means of Cholesky decomposition. Then an algorithm for solving systems of block bidiagonal triangular linear equations was given, which is not necessary to treat with the zero elements out of banded systems. A fast algorithm for solving the systems of symmetry block tridiagonal linear equations was deduced, which can quicken the speed of ray-tracing. Finally, the simulation based on this algorithm for ray-tracing in three dimensional media was carried out. Meanwhile, the segmentally-iterative ray-tracing method and banded method for solving the systems of block tridiagonal linear equations were compared in the same model mentioned above. The convergence condition was assumed that the L-2 norm summation for mk, 1 and mk, 2 in the whole ray path was limited in 10−6. And the calculating speeds of these methods were compared. The results show that the calculating speed of this algorithm is faster than that of conventional method and the calculated results are accurate enough. In addition, its precision can be controlled according to the requirement of ray-tracing
2 citations
TL;DR: In this paper, the spectra of certain tridiagonal matrices arising from differential equations commonly used for modeling flocking behavior were described and applied to demonstrate how asymptotic stability for consensus and flocking systems depends on the imposed boundary condition.
Abstract: We describe the spectra of certain tridiagonal matrices arising from differential equations commonly used for modeling flocking behavior. In particular we consider systems resulting from allowing an arbitrary boundary condition for the end of a one-dimensional flock. We apply our results to demonstrate how asymptotic stability for consensus and flocking systems depends on the imposed boundary condition.
2 citations
TL;DR: In this paper , it was shown that the cp-rank of any completely positive irreducible tridiagonal doubly stochastic matrix is equal to its rank, and that the same is true for symmetric pentadiagonal matrices.
Abstract: Abstract We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix A A is completely positive. Our decomposition can be applied to a wide range of matrices. We give alternate proofs for a number of related results found in the literature in a simple, straightforward manner. We show that the cp-rank of any completely positive irreducible tridiagonal doubly stochastic matrix is equal to its rank. We then consider symmetric pentadiagonal matrices, proving some analogous results and providing two different decompositions sufficient for complete positivity. We illustrate our constructions with a number of examples.
2 citations
TL;DR: In this article, the authors considered the problem of partitioning tridiagonal matrices and proposed a serial algorithm based on LU-factorization of the coefficient matrix A. In this paper, we extend the LU factorization of A to partitioned diagonal blocks.
Abstract: For the direct solution of tridiagonal linear systems Ax = d, the best known serial algorithm is based on LU-factorization of the coefficient matrix A. In the present paper we consider extending the idea to partitioned tridiagonal matrices. Let A be partitioned: A = (A (i, j)) so that the diagonal blocks A( i,i )are tridiagonal. We seek a factorization of A into L = (L (i j) and U = (U (i j) ), partitioned conformally. For the diagonal blocks of A we require the classical factorization: A( ii ) = L (i,i) U (i,i) , L (i,i) unit lower bidiagonal and U (i,i) upper bidiagonal. But, because of the presence of a non-zero element in each of the off-diagonal blocks of A, it is necessary to have Lupper block bidiagonal and U lower block bidiagonal, with only last row of L(i,i+1) and last column of U(i,i-1) filled. To avoid any interlocking/updating during/after the factorization stage, each of these last row and column in each block are required to have their last elements as zeros. On completion of the determinat...
2 citations
TL;DR: A fully scalable parallel algorithm is presented for solving symmetric tridiagonal eigenvalue problems using the quasi-Laguerre's method and seems to be the best for distributed memory parallel architecture.
Abstract: In this article, a fully scalable parallel algorithm is presented for solving symmetric tridiagonal eigenvalue problems using the quasi-Laguerre's method. The algorithm is implemented using Parallel Virtual Machine and is tested on a variety of matrices with a load balancing scheme. Test results show that the algorithm has high parallel efficiency. Compared with other existing algorithms, our algorithm seems to be the best for distributed memory parallel architecture.
2 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |