American Journal of Applied Sciences 4 (5): 300-306, 2007 ISSN 1546-9239 © 2007 Science Publications Corresponding Author: Peerayuth Charnsethikul, Operations Research and Management Science Units, Industrial Engineering Department, Kasetsart University, Bangkok 10903, Thailand 300 Parallel Approaches for Intervals Analysis of Variable Statistics in Large and Sparse Linear Equations with RHS Ranges Peerayuth Charnsethikul Operations Research and Management Science Units, Industrial Engineering Department Kasetsart University, Bangkok 10903, Thailand Abstract: This study proposes an algorithm capable of working in parallel for solving large and sparse linear equations under given right hand side (RHS) ranges. A comparative study to the direct linear programming method is reported theoretically, computationally and discussed. Moreover, the approach can be adapted for the system under domain decompositions structure leading to a better efficiency experimentally. Key words: RHS ranges, large and sparse linear equations, parallel approaches, domain decompositions INTRODUCTION This paper considers a system of linear equations, AX = b with det(A) ≠ 0 and l ≤ b ≤ u where A = [a ij ] is an n×n matrix, b = [b i ] is an n×1 uncertain vector lying between vectors l = [l j ] and u = [u j ]. The problem is to solve for the possible range of X = [x j ]. Mathematically, it can be formulated as the following linear programming (LP) problems. Min (Max) x k , k = 1,2,….,n (P1) Subject to n ∑ a ij x j – b i = 0 , i = 1,2,…...,n j=1 x j unrestricted, l j ≤ b j ≤ u j , j = 1,2,…..,n The above model becomes more complex when n is large. Usually, this situation arises in approximately solving linear boundary partial differential equations. Applying finite approximation schemes normally leads to the result of very large and sparse linear equations. By representing uncertainties of equation right hand sides and boundary conditions as a set of statistical confidence intervals, the solution as a set of related variable ranges can be solved using the above formulation. Directly, the problem can be solved as a set of independent 2n LP problems. However, this paper will present another approach theoretically equivalent but computationally much faster as n grows and the proposed method is suitable for use in parallel architecture. Additionally, the approach is illustrated for further applications on obtaining confidence intervals of variable variance/covariance matrix for a given ranges of right hand side variance/covariance. Linear programming (LP) has been one of the major tools in solving real world resource allocation problems. Its origin and early development historically were described and presented extensively by Dantzig [1] . The problem of determining the confidence interval of statistics also has long been studied by statisticians [2] . A large and sparse system of linear equations normally arises in finite approximations scheme for solving boundary value ordinary and partial differential equation basically found in engineering analysis and computational physics. The resulting system is sometimes too large to be handled by the direct approach and the iterative indirect methods are frequently more appropriate [3,4] . The integration of these three aspects has not yet been studied systematically but its application is motivated by industrial nature of dynamic diffusion or transport phenomena where repetitive operations cause a chance effect leading to a stationary uncertainty of system input or initial/ boundary conditions. For a more complex system where the resulting equations size can be very large, parallel computations can be more efficient. Some early works on parallel and vector solutions for large linear systems were presented in Heller [5] , Ortega [6] and Stone [7] . Distributed-memory based parallel algorithms on basic matrix-vector multiplication especially in case of large block- distributed matrices were initially studied in Dekel et al. [8] . This research area has been received much attention and several new approaches on different