Solving Linear System of Equations Via A Convex Hull Algorithm (1210.7858v1)

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.

• The paper introduces a novel iterative convex hull algorithm that transforms linear systems into convex optimization problems.
• It leverages the Triangle Algorithm and distance duality to approximate solutions with a theoretical complexity of O(n²ε⁻²).
• The approach bypasses traditional matrix constraints and promises scalable, parallelizable solvers for large or sparse systems.

An Iterative Convex Hull Approach to Solving Linear Systems of Equations

The paper by Bahman Kalantari introduces novel iterative algorithms for solving square linear systems $Ax = b$ through a transformation into convex hull problems. This is accomplished using the Triangle Algorithm, a fully polynomial-time approximation scheme (FPTAS) originally devised for determining if a point lies within the convex hull of a finite set of points in Euclidean space. The paper posits that, when converted into a convex hull issue, linear systems can be approximately solved with theoretical complexity bounds of $O(n^2\epsilon^{-2})$, where $\epsilon$ determines the approximation accuracy.

Methodology Overview

The Triangle Algorithm operates iteratively by leveraging a duality principle known as distance duality. The algorithm either confirms that a point lies within a convex hull or identifies a witness certifying the point's exclusion. Two silhouette approaches are proposed for solving linear systems:

1. Direct Transformation: The linear system is directly transformed into an equivalent convex hull problem. An approximate solution $x$ is produced, satisfying $|Ax - b| \leq \epsilon p$, where $p = \max(||a_1||, \ldots, ||a_n||, ||b||)$.
2. Incremental Application: The Triangle Algorithm is applied incrementally, adjusting through a sequence of closely related convex hull problems via a distance duality, allowing computations to adapt dynamically based on intermediate solution insights.

A noteworthy feature of the proposed methods is their lack of reliance on matrix $A$ structural constraints, diverging from traditional iterative techniques demanding specific decomposition or condition checks.

Implications and Further Developments

The implications of this research are multifaceted. Theoretically, it expands the intersection of convex geometry and numerical linear algebra by showing a novel way to reframe linear systems into geometry-oriented optimization problems. Practically, this conversion might spur new kinds of solvers for large or sparse linear systems, alternative to traditional methods like Gaussian elimination or Krylov subspace approaches. These convex hull-based solvers pose a potential for enhanced scalability and parallelizability, crucial for high-dimensional systems.

The decoupling from conventional preconditioning and matrix positivity constraints implies potential applicability across a broad range of problems. However, the paper appropriately notes that computational performance assessments are pending, earmarking this paper as a preparatory step for more exhaustive empirical evaluations.

Future developments in this area could aim at empirically validating these algorithmic approaches against established solvers, assessing the proposed methods' efficacy and efficiency across varied problem classes. Moreover, given its foundation in computational geometry, the move towards practical performance benchmarks could involve hybridizing the Triangle Algorithm with existing iterative frameworks, determining whether theoretical elegance translates into computational practicality.

Conclusion

Kalantari's paper opens a new frontier in the iterative solving of linear systems, shifting the computational geometry lens onto a classical mathematical problem. It provides a rigorous algorithmic foundation, alongside feasible computational complexity predictions, supporting future endeavors that might incubate these ideas into robust software tools for the mathematical and scientific computing community.