An experimental study of the triangle algorithm with emphasis on solving a linear system

The triangle algorithm, Kalantari [4], is designed to solve the convex hull membership problem. It can also solve LP, and as shown in Kalantari[1] solve a square linear system. In this thesis we carry out some experimentation with the triangle algorithm both for solving convex hull problem and a linear system, however, with more emphasis on the latter problem. We first tested the triangle algorithm on the convex hull problem and made comparison with the Frank-Wolfe algorithm. The triangle algorithm outperformed the Frank-Wolfe for large scale problems, up to 10,000 points in dimensions up to 500. The triangle algorithm takes fewer iterations than the Frank-Wolfe algorithm. For linear systems, we implemented the incremental version of the triangle algorithm in [1] and made some comparison with SOR and Gauss-Seidel methods for systems of dimension up to 1000. The triangle algorithm is more efficient than these algorithms taking fewer iterations. We also tested the triangle algorithm for solving the PageRank matrix by converting it into a convex hull membership problem. We solved the problem in dimensions ranging from 200 to 2200. We made comparisons with the power method. The triangle algorithm took less iterations to reach the same accuracy. Additionally, we tested a large scale PageRank matrix problem of size of 281,903 due to S. Kamvar. Surprisingly, the triangle algorithm took only 1 iteration to obtain a solution with the accuracy of $10^{-10}$.M.S.Includes bibliographical referencesby Hao She