Solving (n × n) linear equations with parameters correlation by a first order approximation approach

Abstract

The aim of this paper is to propose approximated 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. The resulting of X is expressed in statistical forms of a mean vector E[X] and a variance-covariance matrix K[XX^T ]. Additionally, co-variances between A and X can also be estimated. The modeling approach is simply based on approximations by the first order Taylor’s series expansion. Then, a linear statistical matrix equations for finding E [X] and K[XX^T ] can be obtained by taking the expected value on the approximated results. The complexity of the approach can be proven to be within O(n ⁴). In the special case of deterministic A and probabilistic b, the complexity is reduced to O(n ³). A numerical example is illustrated and solved by a sample Matlab code. The code is extended to solve a set of randomly generated problems with various problem sizes. Computational results are reported and discussed. Then, the approach is preliminary extended for handling higher order correlations in case of certain A and random b.

Keywords:

• Statistical linear algebra
• computational methods
• distributed computing
• higher order correlation