Quarter-Sweep Projected Modified Gauss-Seidel Algorithm Applied to Linear Complementarity Problem

Abstract

Problem statement: Modified Gauss-Seidel (MGS) was developed in order to improve the convergence rate of classical iterative method in solving linear system. In solving linear system iteratively, it takes longer time when many computational points involved. It is known that by applying quarter-sweep iteration scheme, it can decrease the computational operations without altering the accuracy. In this study, we investigated the effectiveness of the new Quarter-Sweep Projected Modified Gauss-Seidel (QSPMGS) iterative method in solving a Linear Complementarity Problem (LCP). Approach: The LCP we looked into is the LCP arise in American option pricing problem. Actually, American option is a Partial Differential Complementarity Problem (PDCP). By using full-, half-and quarter-sweep Crank-Nicolson finite difference schemes, the problem was reduced to Linear Complementarity Problem (LCP). Results: Several numerical experiments were carried out to test the effectiveness of QSPMGS method in terms of number of iterations, computational time and Root Mean Square Error (RMSE). Comparisons were made with full-, half-and quarter-sweep algorithm based on Projected Gauss-Seidel (PGS) and Projected Modified Gauss-Seidel (PMGS) methods. Thus, the experimental results showed that the QSPMGS iterative method has the least number of iterations and shortest computational time. The RMSE of all tested methods are in good agreement. Conclusion: QSPMGS is the most effective among the tested iterative methods in solving LCP whereby it is fastest and the accuracy remains the same.

Numerical Solution for 2D European Option Pricing Using Quarter-Sweep Modified Gauss-Seidel Method

“…Previously, QSMGS method had been used to solve the one-dimensional Black-Scholes PDE in European (Koh and Sulaiman, 2009) and American option pricing (Koh et al, 2010a). The results obtained have pointed out that QSMGS iterative method has a better convergence rate.…”

“…Furthermore, will be the error tolerance. In computational finance, Root Mean Squared Relative Error (RMSE) is widely used to assess the accuracy of the iterative solutions such as in Zhao et al (2007), Jeong et al (2009) and Koh et al (2010a). The RMSE is defined by:…”

A Blind Adaptive SOR/JGS Iterative Kalman MUD Algorithm for Multiple Access Communication System

“…(12) is ( [20]. In summary, taking Jacobi iterative matrix J B and J f corresponding to B and f , respectively, and making dividing calculation for J B as J  B R L [21], setting the relaxation parameter is  when 0   , we have…”

A Blind Adaptive Kalman-PIC MUD Algorithm for the Multiple Access Communication Network

“…As an optimal estimation algorithm based on the linear minimum mean square error (MMSE) criterion [16], the Kalman algorithm can not only make on-line synchronous unbiased estimation of the unknown noise statistics characteristics in DS-CDMA system network while conducting state filtering, but also build a state space model for the MUD processing, so as to adaptive estimate the optimal decision vector by optimal filter [17], thus ensure algorithm converges to expected user. The combination of Kalman and PIC can restrict the detection error diffusion by the real-time estimation of channel, thus improve the dynamic environment tracking performance of detection algorithm [18].…”