An ABS Algorithm for a Class of Systems of Stochastic Linear Equations

TL;DR

This work extends the ABS algorithm to stochastic linear systems of the form $A\xi=\eta$ with $\eta\sim N_m(v,I_m)$, defining the ABS-S method and proving that both step lengths $\alpha_i$ and iterates $\xi_i$ are Gaussian with explicit mean and covariance recurrences. The approach retains the finite-step solvability property of ABS while incorporating randomness in the right-hand side, yielding distributional guarantees for the solution $\xi_{i+1}$ as the algorithm progresses. The paper demonstrates the theory with a detailed example, showing that $\xi_{i+1}$ converges to a stochastic solution $\xi_{i+1}\sim N_n(U,\Sigma)$ that satisfies the first $i$ equations. This work provides a probabilistic interpretation and practical tool for solving stochastic linear equations where the right-hand side is normally distributed, with potential extensions to other distributions or random system matrices.

Abstract

This paper is to explore a model of the ABS Algorithms for dealing with a class of systems of linear stochastic equations A xi=eta satisfying eta sim N_m(v, I_{m}). It is shown that the iteration step alpha_{i} is N(V,π) and approximation solutions is xi_{i} \sim N_n(U,Σ) for this algorithm model. And some properties of (V,π)$ and $(U,Σ) are given.

Theorems & Definitions (10)

• Definition 2.1
• Definition 2.2
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 4.4
• Theorem 4.5
• Theorem 4.6
• Theorem 4.7
• Theorem 4.8