scholarly journals Relaxed Modulus-Based Matrix Splitting Methods for the Linear Complementarity Problem

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 503
Author(s):  
Shiliang Wu ◽  
Cuixia Li ◽  
Praveen Agarwal
Keyword(s):  
Fixed Point ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Sufficient Conditions ◽  
Linear Complementarity ◽  
Splitting Methods ◽  
Matrix Splitting ◽  
Numerical Examples ◽  
Iteration Methods ◽  
Point Form

In this paper, we obtain a new equivalent fixed-point form of the linear complementarity problem by introducing a relaxed matrix and establish a class of relaxed modulus-based matrix splitting iteration methods for solving the linear complementarity problem. Some sufficient conditions for guaranteeing the convergence of relaxed modulus-based matrix splitting iteration methods are presented. Numerical examples are offered to show the efficacy of the proposed methods.

scholarly journals General fixed-point method for solving the linear complementarity problem

2021 ◽  
Vol 6 (11) ◽  
pp. 11904-11920
Author(s):  
Xi-Ming Fang ◽  
Keyword(s):  
Fixed Point ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Sufficient Conditions ◽  
Positive Definite Matrix ◽  
Linear Complementarity ◽  
System Matrix ◽  
Fixed Point Equation ◽  
Point Equation ◽  
P Matrix

<abstract><p>In this paper, we consider numerical methods for the linear complementarity problem (LCP). By introducing a positive diagonal parameter matrix, the LCP is transformed into an equivalent fixed-point equation and the equivalence is proved. Based on such equation, the general fixed-point (GFP) method with two cases are proposed and analyzed when the system matrix is a $ P $-matrix. In addition, we provide several concrete sufficient conditions for the proposed method when the system matrix is a symmetric positive definite matrix or an $ H_{+} $-matrix. Meanwhile, we discuss the optimal case for the proposed method. The numerical experiments show that the GFP method is effective and practical.</p></abstract>

scholarly journals New error bound for linear complementarity problem of $ S $-$ SDDS $-$ B $ matrices

2022 ◽  
Vol 7 (2) ◽  
pp. 3239-3249
Author(s):  
Lanlan Liu ◽  
◽  
Pan Han ◽  
Feng Wang
Keyword(s):  
Error Bound ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Linear Complementarity ◽  
Numerical Examples

<abstract><p>$ S $-$ SDDS $-$ B $ matrices is a subclass of $ P $-matrices which contains $ B $-matrices. New error bound of the linear complementarity problem for $ S $-$ SDDS $-$ B $ matrices is presented, which improves the corresponding result in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. Numerical examples are given to verify the corresponding results.</p></abstract>

scholarly journals COMPONENTWISE SPLITTING METHODS FOR PRICING AMERICAN OPTIONS UNDER STOCHASTIC VOLATILITY

2007 ◽  
Vol 10 (02) ◽  
pp. 331-361 ◽  
Author(s):  
SAMULI IKONEN ◽  
JARI TOIVANEN
Keyword(s):  
Stochastic Volatility ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
American Options ◽  
Splitting Method ◽  
Linear Complementarity ◽  
Splitting Methods ◽  
Time Step ◽  
Operator Splitting Methods ◽  
Volatility Model

Efficient numerical methods for pricing American options using Heston's stochastic volatility model are proposed. Based on this model the price of a European option can be obtained by solving a two-dimensional parabolic partial differential equation. For an American option the early exercise possibility leads to a lower bound for the price of the option. This price can be computed by solving a linear complementarity problem. The idea of operator splitting methods is to divide each time step into fractional time steps with simpler operators. This paper proposes componentwise splitting methods for solving the linear complementarity problem. The basic componentwise splitting decomposes the discretized problem into three linear complementarity problems with tridiagonal matrices. These problems can be efficiently solved using the Brennan and Schwartz algorithm, which was originally introduced for American options under the Black and Scholes model. The accuracy of the componentwise splitting method is increased by applying the Strang symmetrization. The good accuracy and the computational efficiency of the proposed symmetrized splitting method are demonstrated by numerical experiments.

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