Showing papers on "Tridiagonal matrix algorithm published in 1988"
TL;DR: In this paper, a method for numerical solution of the one-dimensional Schrodinger equation based on a matrix formulation of Numerov's method is described, and an efficient algorithm for the slicing of the spectrum of the related eigenproblem for a pair of tridiagonal matrices is developed.
Abstract: A method for the numerical solution of the one‐dimensional Schrodinger equation based on a matrix formulation of Numerov’s method is described. Some matrix theory leading to an efficient algorithm for the slicing of the spectrum of the related eigenproblem for a pair of tridiagonal matrices is developed. Finally the application of defect correction to the discretization gives a method of order of accuracy eight. Results of example calculations are presented.
78 citations
TL;DR: In this paper, it was shown that if forcing is applied at an interior point, then the reconstruction is unique if that point is not a node of any eigenmode; if it is, there is a family of systems with the required properties.
Abstract: It is known that a simple spring-mass system may be reconstructed uniquely (apart from a single scaling factor) from the poles and zeros of the frequency response function corresponding to sinusoidal forcing at an end. The (squares of the) poles yield the eigenvalues of a tridiagonal matrix, A, while the zeros yield the eigenvalues of the matrix, A*, with the last row and column deleted. There are proven numerical methods for reconstructing A. The authors show that, if forcing is applied at an interior point, then the reconstruction is unique if that point is not a node of any eigenmode; if it is, there is a family of systems with the required properties. In either case the system may be constructed using modifications of proven techniques.
20 citations
7 Jun 1988
TL;DR: In this paper, the authors examined two approaches for reducing parallel sparse matrix solution time: the first based on pivot ordering algorithms for Gaussian elimination, and the second based on relaxation algorithms.
Abstract: The authors examine two approaches for reducing parallel sparse matrix solution time: the first based on pivot ordering algorithms for Gaussian elimination, and the second based on relaxation algorithms A pivot ordering algorithm is presented which increases the parallelism of Gaussian elimination compared to the commonly used Markowitz method The minimum number of parallel steps for the solution of a tridiagonal matrix is derived, and it is shown that this optimum is nearly achieved by the ordering heuristics which attempt to maximize parallelism Also presented is an optimality result about Gauss-Jacobi over Gauss-Seidel relaxation on parallel processors >
20 citations
TL;DR: In this paper, the authors considered the question: Is every n×n complex matrix unitarily similar to a tridiagonal matrix and showed that the answer is negative if n⩾6, and is affirmative if n=3.
17 citations
TL;DR: In this paper, a modified form of the alternating group explicit method for the iterative solution of tridiagonal linear systems is achieved by an alternative splitting of the coefficient matrix, which is shown to possess similar convergence and stability properties and reduced computational effort.
14 citations
TL;DR: It is shown that individual (state-to-state) S -matrix elements can be generated from the 1,1 element of the inverse of a small tridiagonal matrix.
12 citations
IBM1
TL;DR: A polyalgorithm can be derived which takes into account much better the given architecture of the IBM 3090 VF than the ‘pure’ cyclic reduction algorithm, as described for instance by Kershaw.
Abstract: Cyclic reduction, originally proposed by Hockney and Golub, is the most popular algorithm for solving tridiagonal linear systems on SIMD-type computers like CRAY-1 or CDC CYBER 205. That algorithm seems to be the adequate one for the IBM 3090 VF (uni-processor), too, although the overall expected speedup over Gaussian elimination, specialized for tridiagonal systems, is not as high as for the CRAY-1 or the CYBER 205. That is because the excellent scalar speed of the IBM 3090 makes its vector-to-scalar speed ratio relatively moderate. The idea of the cyclic reduction algorithm can be generalized and modified in various directions. A polyalgorithm can be derived which takes into account much better the given architecture of the IBM 3090 VF than the ‘pure’ cyclic reduction algorithm as described for instance by Kershaw. This is mainly achieved by introducing more locality into the formulae. For large systems of equations the well-known cache problems are prevented.
11 citations
TL;DR: In this paper, two different methods are developed for factoring Fn into products of tridiagonal and permutation matrices, one based on matrix identities associated with FFTs and the Rader prime algorithm, and the other based on a numerical technique, called minimal-variable oblique elimination.
8 citations
TL;DR: In this paper, a finite difference method has been used to solve the partial differential equation of u-momentum and a marching integration technique with the tridiagonal matrix algorithm is used.
7 citations
25 May 1988
TL;DR: The first step in the development of a chip set to support eigenvalue-eigenvector-based estimation algorithms is presented, based on the assumption that an averaging technique will produce a symmetric covariance matrix.
Abstract: The first step in the development of a chip set to support eigenvalue-eigenvector-based estimation algorithms is presented. It is based on the assumption that an averaging technique will produce a symmetric covariance matrix. Such a matrix can be reduced to a symmetric tridiagonal matrix, and hence the eigenvalues and eigenvectors can be found by successive iterations involving QR decomposition. The architecture is unique in that other architectures either solve only for the eigenvalues or use methods other than QR iteration. It has potential for use in a systolic computer for computer intensive digital signal processing based on modern spectral-analysis techniques. >
IBM1
TL;DR: Time measurements show impressive speed-ups over the domino ( ) function and the APL2 versions of the Gaussian elimination method, specialized for cyclic tridiagonal andtridiagonal systems.
Abstract: An APL2 function is presented for solving cyclic tridiagonal and tridiagonal systems of linear equations Those systems frequently occur in various areas, e g interpolation by spline functions, numerical solution of elliptic differential equations, etc The function is based on a modification of the cyclic reduction method [1], [2] Time measurements show impressive speed-ups over the domino ( ) function and the APL2 versions of the Gaussian elimination method, specialized for cyclic tridiagonal and tridiagonal systems
TL;DR: This parallel algorithm takes advantage of the multi-processor environment by carrying out N (number of processors) triangular factorizations of chosen shifted matrices in all processors concurrently and by minimizing communication between processors.
TL;DR: This paper develops optimal algorithms to multiply an n × n symmetric tridiagonal matrix by: (i) an arbitrary n × m matrix using 2nm − m multiplications; (ii) a symmetric TRD matrix using 6n − 7 multiplications'; and (iii) a tridi diagonal matrix using 7n −8 multiplications.
Abstract: This paper develops optimal algorithms to multiply an n × n symmetric tridiagonal matrix by: (i) an arbitrary n × m matrix using 2nm − m multiplications; (ii) a symmetric tridiagonal matrix using 6n − 7 multiplications; and (iii) a tridiagonal matrix using 7n −8 multiplications. Efficient algorithms are also developed to multiply a tridiagonal matrix by an arbitrary matrix, and to multiply two tridiagonal matrices.
TL;DR: In this paper, a computational procedure is presented for the rigorous calculation of multicomponent multistage separation for systems involving equilibria in ionization and chemical reactions in addition to the usual phase equilibrium.
TL;DR: In this article, it is shown that one-dimensional C 1 -collocation at the Gauss points may be viewed as a member of a family of 3-point finite difference schemes, which leads to a simple practical technique for the elimination of the derivative degrees of freedom.
1 Jan 1988
1 Oct 1988
TL;DR: The author describes an algorithm for computing the characteristic polynomial of a tridiagonal matrix which consists in the application of the divide-and-conquer technique to the evaluation of a three-term recurrence relation.
Abstract: The author describes an algorithm for computing the characteristic polynomial of a tridiagonal matrix. It is quite general and consists in the application of the divide-and-conquer technique to the evaluation of a three-term recurrence relation. The algorithm developed requires O(n log/sup 2/n) arithmetic operations as compared to the classical algorithm that requires O(n/sup 2/) arithmetic operations. >
TL;DR: An algorithm for solving a system with a tridiagonal matrix, which is obviously the fastest of the well-known algorithm for parallel computers, is described and the polynomial stability of the algorithm to the accumulation of rounding errors is established.
Abstract: An algorithm for solving a system with a tridiagonal matrix, which is obviously the fastest of the well-known algorithm for parallel computers, is described. The polynomial stability of the algorithm to the accumulation of rounding errors is established.
TL;DR: The method is an extension of the Thomas algorithm to a class of sparse matrices which is mainly tridiagonal but has some off-diagonal elements and finds a useful application in calculating the material balance of the interconnected cascade system with reflux and backmixing.
TL;DR: The explicit structure of the inverse of block tridiagonal matrices is presented in terms of blocks defined by linear recurrence relations, and parallel algorithms are shown which solve block second order linear recurrences without using commutativity.