Tridiagonal matrix algorithm

Abstract: This work examines the relation between Gaussian elimination and the conjugate directions algorithm [Hestenes and Steifel, 1952]. Analysis is extended to the case where the sequence of the conjugated vectors is modified, which is shown to result in reordering of the solution vector. Based on these analyses an algorithm is described which combines Gaussian elimination with a look-ahead algorithm. The purpose of the algorithm is to employ Gaussian elimination on a system of smaller order and to use this solution to approximate the solution of the original system. The algorithm was tested on a range of linear systems and performed well when the components in the solution vector varied by large magnitude.

Abstract: Fast and efficient tridiagonal solvers are highly appreciated in scientific and engineering domain, but challenging optimization task for computer engineers. The state-of-the-art developments in multi-core computing paves the way to meet this challenge to an extent. The technical advances in multi-core computing provide opportunities to exploit lower levels of parallelism and concurrency for inherently sequential algorithms. In this article, the authors present an optimal performance pipelined parallel variant of the conventional Tridiagonal Matrix Algorithm (TDMA), aka the Thomas algorithm, on a multi-core CPU platform. The implementation, analysis and performance comparison of the proposed pipelined parallel TDMA and the conventional version are performed on an Intel SIMD multi-core architecture. The results are compared in terms of elapsed time, speedup, cache miss rate. For a system of ‘n' linear equations where n = 2^36 in presented pipelined parallel TDMA achieves speedup of 1.294X with a parallel efficiency of 43% initially and inclines towards linear speed up as the system grows.

Abstract: How much can be said about the location of the eigenvalues of a symmetric tridiagonal matrix just by looking at its diagonal entries? We use classical results on the eigenvalues of symmetric matrices to show that the diagonal entries are bounds for some of the eigenvalues regardless of the size of the off-diagonal entries. Numerical examples are given to illustrate that our arithmetic-free technique delivers useful information on the location of the eigenvalues. This research was financed by FEDER Funds through “Programa Operacional Factores de Competitividade COMPETE” and by Portuguese Funds through FCT “Fundação para a Ciência e a Tecnologia”, within the Project PEstOE/MAT/UI0013/2014. keywords: permutations, separation theorem.