Tridiagonal matrix algorithm

Abstract: For real symmetric or Hermitian matrices with tridiagonal form, the secular equation may be written as a continued fraction equation f(λ)=0. f(λ) is a member of a recursively defined sequence R(n)(λ) of n continued fractions if the secular equation is of the nth order. The basis for a new method of computing the eigenvalues of such tridiagonal matrices is given. The method requires the determination of an integervalues function Pn(γ) for a succession of values of γ, where Pn(γ) is a function only of n and the signs of the n terms in R(n)(γ).

Abstract: In this paper we present new formulas for characterizing the sensitivity of tridiagonal systems that are independent of the condition number of the underlying matrix. We also introduce efficient algorithms for solving tridiagonal systems of linear equations which are stable and reliable (namely, stable in the backward sense and little sensitive to perturbations in the coefficients).

Abstract: Block tridiagonal systems of linear equations arise in a wide variety of scientific and engineering applications. Recursive doubling algorithm is a well-known prefix computation-based numerical algorithm that requires O(M3(N/P + logP)) work to compute the solution of a block tridiagonal system with N block rows and block size M on P processors. In real-world applications, solutions of tridiagonal systems are most often sought with multiple, often hundreds and thousands, of different right hand sides but with the same tridiagonal matrix. Here, we show that a recursive doubling algorithm is sub-optimal when computing solutions of block tridiagonal systems with multiple right hand sides and present a novel algorithm, called the accelerated recursive doubling algorithm, that delivers O(R) improvement when solving block tridiagonal systems with R distinct right hand sides. Since R is typically ~ 102--104, this improvement translates to very significant speedups in practice. Detailed complexity analyses of the new algorithm with empirical confirmation of runtime improvements are presented. To the best of our knowledge, this algorithm has not been reported before in the literature.

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.