Solving Linear System of Equations Via A Convex Hull Algorithm

TL;DR

This paper introduces new iterative algorithms that convert linear systems into convex hull problems and solve them using the Triangle Algorithm, providing an efficient approximation method with strong theoretical guarantees.

Contribution

The paper develops novel convex hull-based iterative algorithms for solving linear systems, leveraging the Triangle Algorithm and sensitivity analysis, with no restrictions on matrix structure.

Findings

Achieves approximate solutions in $O(n^2\,\epsilon^{-2})$ operations.

Provides theoretical complexity bounds for the algorithms.

Offers potential practical alternatives for large-scale linear systems.

Abstract

We present new iterative algorithms for solving a square linear system $Ax=b$ in dimension $n$ by employing the {\it Triangle Algorithm} \cite{kal12}, a fully polynomial-time approximation scheme for testing if the convex hull of a finite set of points in a Euclidean space contains a given point. By converting $Ax=b$ into a convex hull problem and solving via the Triangle Algorithm, together with a {\it sensitivity theorem}, we compute in $O(n^2\epsilon^{-2})$ arithmetic operations an approximate solution satisfying $\Vert Ax_\epsilon - b \Vert \leq \epsilon \rho$, where $\rho= \max \{\Vert a_1 \Vert,..., \Vert a_n \Vert, \Vert b \Vert \}$, and $a_i$ is the $i$-th column of $A$. In another approach we apply the Triangle Algorithm incrementally, solving a sequence of convex hull problems while repeatedly employing a {\it distance duality}. The simplicity and theoretical complexity bounds of the proposed algorithms, requiring no structural restrictions on the matrix $A$, suggest their potential practicality, offering alternatives to the existing exact and iterative methods, especially for large scale linear systems. The assessment of computational performance however is the subject of future experimentations.