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
1 Sep 1990
TL;DR: It is demonstrated how the efficiency of the general block tridiagonal multilevel algorithm can be improved by introducing the equivalent of two-way Gaussian elimination for the first and the last partitioning and by carefully balancing the load of the processors.
Abstract: This paper describes an efficient algorithm for the parallel solution of systems of linear equations with a block tridiagonal coefficient matrix. The algorithm comprises a multilevel LU-factorization based on block cyclic reduction and a corresponding solution algorithm. The paper includes a general presentation of the parallel multilevel LU-factorization and solution algorithms, but the main emphasis is on implementation principles for a message passing computer with hypercube topology. Problem partitioning, processor allocation and communication requirement are discussed for the general block tridiagonal algorithm. Band matrices can be cast into block tridiagonal form, and this special but important problem is dealt with in detail. It is demonstrated how the efficiency of the general block tridiagonal multilevel algorithm can be improved by introducing the equivalent of two-way Gaussian elimination for the first and the last partitioning and by carefully balancing the load of the processors. The presentation of the multilevel band solver is accompanied by detailed complexity analyses. The properties of the parallel band solver were evaluated by implementing the algorithm on an Intel iPSC hypercube parallel computer and solving a larger number of banded linear equations using 2 to 32 processors. The results of the evaluation include speed-up over a sequential processor, and the measure values are in good agreement with the theoretical values resulting from complexity analysis. It is found that the maximum asymptotic speed-up of the multilevel LU-factorization using p processors and load balancing is approximated well by the expression ( p +6)/4. Finally, the multilevel parallel solver is compared with solvers based on row and column interleaved organization.
28 citations
TL;DR: In this article, it was shown that the solution space of the matrix equation Hx = u is decomposed by a projection which leads to a recurrence for H -1.
28 citations
TL;DR: Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, the bound state spectrum as the eigenvalues of the associated analytical matrix representing the full Hamiltonian was computed in this article.
Abstract: Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate exactly the matrix elements of the Yukawa potential in this representation This enabled us to compute the bound state spectrum as the eigenvalues of the associated analytical matrix representing the full Hamiltonian We also used the complex scaling method to evaluate the resonance energies and compared our results with those obtained using the Gauss quadrature approach and the corresponding results from the literature
28 citations
1 Dec 1976
TL;DR: This paper gives a mathematical explanation of Stone's algorithms, and shows that recursive doubling can become a viable competitor to Gauss Elimination, and possible approaches to scaling are suggested.
Abstract: Recursive doubling is an algorithm proposed by Stone for the solution, on a parallel processor, of an n x n tridiagonal system of linear equations. Such systems arise frequently in physical applications, where n is typically large (e.g., n = 10,000) and many systems of this size need to be solved. An efficient method of solution is desirable. One method of solving Ax = y, where A is tridiagonal, is to decompose A into A = LU, where L is unit lower bidiagonal and U is upper bidiagonal. Then Lz = y is solved, followed by Ux = z. Stone proposed two algorithms; first, a parallel ''backsolver'' for the solution of the two bidiagonal systems, and second, a ''Q-finder'' which, together with three parallel divisions, produces L and U. This paper gives a mathematical explanation of Stone's algorithms which shows that they are two cases of a general algorithm. Then precise definition is given to the causes of numerical instability in the algorithm, and possible approaches to scaling are suggested. Particular applications are given to solving m-term linear recurrence relations, tridiagonal systems of equations, and the symmetric tridiagonal eigenvalue problem. For solving a tridiagonal system of equations, the special casemore » where the systems arise from operator splitting techniques for solving parabolic partial differential equations is also considered. It is shown that recursive doubling can become a viable competitor to Gauss Elimination. Pertinent timing information is given in most of the above applications. 3 tables. (RWR)« less
28 citations
TL;DR: This chapter discusses Gaussian elimination with partial pivoting, the universal method for solving a dense, unstructured n X n linear system of equations Ax = b on a serial computer.
Abstract: If numerical analysts understand anything, surely it must be Gaussian elimination. This is the oldest and truest of numerical algorithms. To be precise, I am speaking of Gaussian elimination with partial pivoting, the universal method for solving a dense, unstructured n X n linear system of equations Ax = b on a serial computer. This algorithm has been so successful that to many of us, Gaussian elimination and Ax = b are more or less synonymous. The chapter headings in the book by Golub and Van Loan [3] are typical -- along with "Orthogonalization and Least Squares Methods," "The Symetric Eigenvalue Problem," and the rest, one finds "Gaussian Elimination," not "Linear Systems of Equations."
28 citations
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| Year | Papers |
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| 2025 | 3 |
| 2024 | 7 |
| 2023 | 11 |
| 2022 | 22 |
| 2021 | 14 |
| 2020 | 11 |