2 & 1 Depts. (P.G.&U.G.) Of Mathematics, Hindu College,Guntur – 522 002, Andhra Pradesh, INDIA. Ph: 0863 – 350818 (Res) E-Mail: [email protected] “ABSTRACT” We present recursive algorithms for finding the inverse of any nonsingular square matrix and in particular a positive definite symmetric (PDS) matrix through single bordering and double bordering and obtain the operational counts fo...

A class of parallel incomplete factorization preconditionings for the solution of large linear systems is investigated. The approach may be regarded as a generalized domain decomposition method. Adjacent subdomains have to communicate during the setting up of the precon-ditioner, and during the application of the preconditioner. Overlap is not necessary to achieve high performance. Fill-in leve...

We discuss partitioning methods using hypergraphs to produce fill-reducing orderings of sparse matrices for Cholesky, LU and QR factorizations. For the Cholesky factorization, we investigate a recent result on pattern-wise decomposition of sparse matrices, generalize the result, and develop algorithmic tools to obtain more effective ordering methods. The generalized results help us to develop f...

This is a technical appendix to “Adaptive estimation of covariance matrices via Cholesky decomposition (arXiv:1010.1445). AMS 2000 subject classifications: Primary 62H12; secondary 62F35, 62J05.

The analysis of preconditioners based on incomplete Cholesky factorization in which the neglected (dropped) components are orthogonal to the approximations being kept is presented. General estimate for the condition number of the preconditioned system is given which only depends on the accuracy of individual approximations. The estimate is further improved if, for instance, only the newly compu...

Zeros in positive definite correlation matrices arise frequently in probability and statistics, and are intimately related to the notion of stochastic independence. The question of when zeros (i.e., sparsity) in a positive definite matrix A are preserved in its Cholesky decomposition, and vice versa, was addressed by Paulsen et al. [19] [see Journal of Functional Analysis, 85, 151-178]. In part...

We show that a novel class of preconditioners, designed by Pravin Vaidya in 1991 but never before implemented, is remarkably robust and can outperform incomplete-Cholesky preconditioners. Our test suite includes problems arising from finitedifferences discretizations of elliptic PDEs in two and three dimensions. On 2D problems, Vaidya’s preconditioners often outperform drop-tolerance incomplete...

This paper presents a su cient condition on sparsity patterns for the existence of the incomplete Cholesky factorization. Given the sparsity pattern P (A) of a matrix A, and a target sparsity pattern P satisfying the condition, incomplete Cholesky factorization successfully completes for all symmetric positive de nite matrices with the same pattern P (A). It is also shown that this condition is...

This paper proposes a hardware accelerator for Cholesky decomposition on FPGAs by designing a single triangular linear equation solver. Good performance is achieved by reordering the computation of Cholesky factorization algorithms and thus alleviating the data dependency. The dedicated hardware architecture for solving triangular linear equations is designed and implemented for different accur...