SOLVING (n×n) LINEAR EQUATIONS WITH PARAMETERS COVARIANCES

Abstract

The aim of this paper is to propose numerical models and methods for solving (n×n) linear equations, AX=b where parameters in the matrix A and the vector b are random and correlated. In case of a probabilistic b and a deterministic A, the amount of second order co-variances computation is equivalent to solving the (n×n) problem sequentially n times with additional efforts of performing a Cholesky factorization plus a symmetric matrix multiplication of size n×n while the complexity for kth order correlation can be proven to be O(nk+1). For randomness in both A and b with an assumption of multidimensional Gaussian distribution, two systems of linear equations with size n2(n+1) and n(n+1)/2 are derived and can be solved sequentially to obtain the desired statistical parameters up to the second order of X. Both approaches are coded as Matlab M-files, complied and run to test their efficiency. For higher order correlation with non-Gaussian distribution, an approximation scheme is proposed. Two applications in optimization are discussed.