Showing papers on "Tridiagonal matrix algorithm published in 1980"
TL;DR: The method is shown to be stable and for a large class of matrices it is, asymptotically, faster by an order of magnitude than theQR method.
Abstract: A method is given for calculating the eigenvalues of a symmetric tridiagonal matrix The method is shown to be stable and for a large class of matrices it is, asymptotically, faster by an order of magnitude than theQR method
460 citations
TL;DR: A general method is presented by which the initial value problem can be solved by iteration and the applicability of the method is demonstrated by calculating the eigenvalues of the laser Fokker-Planck operator.
Abstract: The applications of infinite systems of linear first order differential equations with 2L+1-term recursion formulas are discussed. It is shown that such systems can be written as a system of linear tridiagonal vector equations of dimensionL. A general method is presented by which the initial value problem can be solved by iteration. For special but physically important initial conditions the solution is given by a matrix continued fraction. The eigenvalues of the tridiagonal vector recurrence relations are obtained as the roots of aL×L determinant the elements of which are determined by a matrix continued fraction. The applicability of the method is demonstrated by calculating the eigenvalues of the laser Fokker-Planck operator.
43 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: It is shown that certain tridiagonal Toeplitz linear systems can be solved efficiently by an algorithm derived from the factorization of the coefficient matrix into two easily inverted matrices.
Abstract: A factorization method is described for the fast numerical solution of certain tridiagonal Toeplitz linear systems which occur repeatedly in the solution of linear partial differential equations under a variety of boundary conditions In this paper, we show that such special linear systems can be solved efficiently by an algorithm derived from the factorization of the coefficient matrix into two easily inverted matrices
21 citations
TL;DR: In this article, the usual Sturmanian sequence for finding the eigenvalues of a tridiagonal matrix arising from the radial Schroedinger equation is found to be unstable and a self-stabilising continued fraction approach is suggested.
10 citations
TL;DR: In this article, the inverse of this type of matrix is expressed in such a way that some of its theoretical properties may be obtained, such as symmetry, tridiagonal and Toeplitz.
Abstract: In some problems in numerical analysis one is faced with solving a linear system of equations in which the matrix of the linear system is symmetric, tridiagonal and Toeplitz, except for elements at or near the corners. The anomalous elements usually arise from boundary conditions in the original problem. This paper is concerned with expressing the inverse of this type of matrix in such a way that some of its theoretical properties may be obtained.
9 citations
1 Jan 1980
TL;DR: In this article, the forward error analysis of Gaussian and two-sided elimination of tridiagonal linear systems is described, and the rounding error analysis is based on a linearization method describing first-order approximations of the errors exactly.
Abstract: Publisher Summary This chapter describes the forward error analysis of Gaussian and two-sided elimination of tridiagonal linear systems.. The rounding error analysis is based on a linearization method describing first-order approximations of the errors exactly. Main new results of the analysis are optimal—that is, smallest intervals bounding the absolute errors of the solutions x under relatively uniform perturbations by data and rounding errors. The radii of the error intervals are given by data and rounding condition numbers. In addition, the error analysis yields fundamental results concerning the stability of the algorithms in the sense of Wilkinsons backward error analysis. Gaussian elimination, without pivoting, is well-conditioned or backward stable for tridiagonal M -matrices, for positive definite, and for elimination-regular diagonally dominant coefficient matrices. Two-sided elimination, without pivoting, is well-conditioned for all two-sided elimination-regular tridiagonal matrices.
4 citations
2 citations
TL;DR: The invariant imbedding technique is applied to the block tridiagonal systems in a general setting using, when necessary, the generalized Moore-Penrose inverse of a matrix.
1 citations
TL;DR: In this article, certain algebraic operations for directed graphs are developed for the Gaussian elimination process and some of their properties are derived for nonsingular and inverse graphs, respectively.
Abstract: Certain algebraic operations (in the Boolean sense) are developed for directed graphs. Nonsingular and inverse graphs are defined and some of their characteristics are derived. The results are applied for the Gaussian elimination process.
TL;DR: An ALGOL program is derived for a modified LR-algorithm, permitting the determination of two-sided approximations to the eigenvalues of a tridiagonal symmetric matrix.
Abstract: An ALGOL program is derived for a modified LR-algorithm, permitting the determination of two-sided approximations to the eigenvalues of a tridiagonal symmetric matrix. Test examples are considered.
TL;DR: In this article, separation, monotonicity, and growth rate theorems are obtained for the eigenvalues of a real symmetric tridiagonal matrix when some of its elements are varying.
Abstract: Separation, monotonicity, and growth rate theorems are obtained for the eigenvalues of a real symmetric tridiagonal matrix when some of its elements are varying.