Solving Linear System of Equations Via A Convex Hull Algorithm
This work offers a new theoretical approach to solving linear systems that may be useful for large-scale problems, but it is currently untested in practice.
The authors propose iterative algorithms for solving square linear systems by converting them into convex hull problems and applying the Triangle Algorithm, achieving an approximate solution in O(n²ε⁻²) operations with no structural restrictions on the matrix.
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 $O(n^2ε^{-2})$ arithmetic operations an approximate solution satisfying $\Vert Ax_ε- b \Vert \leq ερ$, where $ρ= \max \{\Vert a_1 \Vert,..., \Vert a_n \Vert, \Vert b \Vert \}$, 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.