Abstract: Matrix-partition algorithms for solving tridiagonal systems of equations are highly adaptable to parallel processors. In this paper, a matrix-partition algorithm for generating a block-Cholesky factorization of a permuted form of a block tridiagonal system is presented. A preconditioning system based on an incomplete application of this algorithm is then described. Under certain dominance conditions, it is shown that the computations within a partition can be performed independently thus yielding a highly parallel incomplete-Cholesky factorization particularly suitable for multi-processing architectures.

Abstract: As shown in the next chapter, the Fokker-Planck equation describing the Brownian motion in arbitrary potentials, i.e., the Kramers equation, can be cast into a tridiagonal vector recurrence relation by suitable expansion of the distribution function. In this chapter we shall investigate the solutions of tridiagonal vector recurrence relations. As it turns out, the Laplace transform of these solutions as well as the eigenvalues and eigenfunctions can be obtained in terms of matrix continued fractions. Therefore, the corresponding solutions of the Kramers equation can also be given in terms of matrix continued fractions. This method has the advantage that no detailed balance condition is needed for its application. This matrix continued-fraction method is especially suitable for numerical calculations and for some problems it seems to be the most accurate and fastest method, as will be discussed in other chapters. Besides its advantage for numerical purposes, the matrix continued-fraction solutions are also very useful for analytical evaluations. By a proper Taylor series expansion of the matrix continued fractions we obtain, for instance, in Sect. 10.4 the high-friction limit solutions of the Kramers equation.

Abstract: We consider the calculation of r(ξ), where ξ is a given number, and where {r(x)=p(x)/q(x); xe IR} is a generalized rational function whose coefficients should satisfy some interpolation conditions. We study a procedure that obtains r(ξ)=p(ξ)/q(ξ) by applying Gaussian elimination to remove the unknown coefficients from a system of linear equations. It is shown that the procedure breaks down only if r(ξ) or the coefficients of the rational function are not properly defined. It is proved that the intermediate equations of Gaussian elimination are related to rational interpolating functions that depend on subsets of the coefficients and data. A numerical example demonstrates the procedure.

Abstract: In this paper we given an algorithm of low computational complexity which determines the eigenvalues of a symmetric tridiagonal matrix.

Abstract: The problem of computing the eigenvectors and eigenvalues of the operator Tf( X) • fa -a sin( b( x-y) ) (x.;..y) f(y)dy -a :l!i: x :l!i: a, appears in communications theory, electrical engineering, and other contexts. we call T a time-band limited operator, because it is the composition of projection operators in both real space and Fourier space. The spectral analysis ofT is facilitated by the existence of a commuting differential operator L • d~(a x )d~b x . An operator analogous to T, based on the eigenfunctions of Gegenbauer•s equation 2 ( -0 + 2 ) ~ -= >-IZL . 2 Sl.n X ~(O) = ~(rr) • o, has been investigated by P. A. Grunbaum, who has shown that this analogous operator (which we also call a time-band limited operator) again enjoys the existence of a commuting differential operator. A dual, discrete, version of this phenomenon also exists; instead of a commuting differential operator, one has a commuting tridiagonal matrix commuting with a time-band limited matrix.

Abstract: In this paper, we present a divide-and-conquer approach to the evaluation of the characteristic polynomial of a symmetric tridiagonal matrix for a real argument. Here, the problem is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem. We give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation. We show that the three-term recurrence is a specific case of our equations. Furthermore, this set of recursions can be implemented on a regular tree structure. If the concurrency exhibited by order is increased by one at every step.