This chapter is mostly a review, discussing unique factorization of positive integers, unique factorization of polynomials whose coefficients are rational or real or complex, signs of permutations, and matrix algebra. Sections 1–2 concern unique factorization of positive integers. Section 1 proves the division and Euclidean algorithms, used to compute greatest common divisors. Section 2 establi...

Real-world emission factors for different vehicle types and their contributions to roadside air pollution are needed air-quality management. Tunnel measurements have been used estimate several using linear regression or receptor-based source apportionment. However, the accuracy uncertainties of these methods not sufficiently discussed. This study applies four derive from tunnel in Hong Kong, Ch...

In this paper, we present several high performance variants of the classical Schur algorithm to factor various Toeplitz matrices. For positive definite block Toeplitz matrices, we show how hyperbolic Householder transformations may be blocked to yield a block Schur algorithm. This algorithm uses BLAS3 primitives and makes efficient use of a memory hierarchy. We present three algorithms for inde...

A method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix A is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations. It is proved that in exact arithmetic the preconditioner is well defined if A is an H-matrix. The results of numerical experiments are pre...

Different approaches to the decomposition of a nonsingular totally positive matrix as a product of bidiagonal matrices are studied. Special attention is paid to the interpretation of the factorization in terms of the Neville elimination process of the matrix and in terms of corner cutting algorithms of Computer Aided Geometric Design. Conditions of uniqueness for the decomposition are also given.

In this paper, we describe a scalable parallel algorithm for sparse matrix factorization, analyze their performance and scalability, and present experimental results for up to 1024 processors on a Cray T3D parallel computer. Through our analysis and experimental results, we demonstrate that our algorithm substantially improves the state of the art in parallel direct solution of sparse linear sy...

The LDL software package is a set of short, concise routines for factorizing symmetric positive-definite sparse matrices, with some applicability to symmetric indefinite matrices. Its primary purpose is to illustrate much of the basic theory of sparse matrix algorithms in as concise a code as possible, including an elegant method of sparse symmetric factorization that computes the factorization...