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<p>This article presents a class of modulus-based relaxation methods to process the large and sparse implicit complementarity problem. By using two positive diagonal matrices, we formulate a fixed-point equation and prove that it is equivalent to implicit complementarity problem. We provide sufficient convergence conditions for the proposed methods when the system matrix is a <i>P</i>-matrix or an <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6743_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(H_+\)</EquationSource> </InlineEquation>-matrix. We discuss the convergence analysis when the system matrix is a nonsingular matrix. We show that the proposed method can process a subclass of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6743_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(P_0\)</EquationSource> </InlineEquation>-matrix with the help of numerical illustration. Several numerical examples are shown to illustrate the efficacy of the proposed methods which are superior than the earlier version of modulus-based matrix splitting iteration methods in terms of iteration steps and CPU time.</p>

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More on modulus based iterative method for solving large and sparse implicit complementarity problem

  • 期刊论文

  • 发表日期：

    2025年7月24日

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  • Bharat Kumar,
  • Deepmala,
  • A. K. Das

摘要

This article presents a class of modulus-based relaxation methods to process the large and sparse implicit complementarity problem. By using two positive diagonal matrices, we formulate a fixed-point equation and prove that it is equivalent to implicit complementarity problem. We provide sufficient convergence conditions for the proposed methods when the system matrix is a P-matrix or an \(H_+\) -matrix. We discuss the convergence analysis when the system matrix is a nonsingular matrix. We show that the proposed method can process a subclass of \(P_0\) -matrix with the help of numerical illustration. Several numerical examples are shown to illustrate the efficacy of the proposed methods which are superior than the earlier version of modulus-based matrix splitting iteration methods in terms of iteration steps and CPU time.