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<p>The class of SDD<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42967_2025_523_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(_1\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mn>1</mn> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> matrices is an important subclass of H-matrices. The strictly diagonally dominant (SDD) matrices, the doubly strictly diagonally dominant (DSDD) matrices, and the Dashnic Zusmanovich type (DZT) matrices are included in SDD<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42967_2025_523_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(_1\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mn>1</mn> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> matrices. This paper presents an infinity norm upper bound for the inverse of SDD<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42967_2025_523_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(_1\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mn>1</mn> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> matrices using the definition of the matrix norm. We prove that it is sharper than the well-known Varah’s bound for SDD matrices, and it generally performs better than the existing bounds for SDD<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42967_2025_523_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(_1\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mn>1</mn> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>, DSDD, and DZT matrices. As an application, an error bound for the linear complementarity problems (LCPs) of B<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42967_2025_523_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(_1\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mn>1</mn> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>-matrices is given.</p>

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An Infinity Norm Upper Bound for the Inverse of \(\hbox {SDD}_1\) Matrices and the Application in Linear Complementarity Problems

  • 期刊论文

  • 发表日期：

    2025年9月26日

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  • Yun Li,
  • Shiyun Wang

摘要

The class of SDD \(_1\) 1 matrices is an important subclass of H-matrices. The strictly diagonally dominant (SDD) matrices, the doubly strictly diagonally dominant (DSDD) matrices, and the Dashnic Zusmanovich type (DZT) matrices are included in SDD \(_1\) 1 matrices. This paper presents an infinity norm upper bound for the inverse of SDD \(_1\) 1 matrices using the definition of the matrix norm. We prove that it is sharper than the well-known Varah’s bound for SDD matrices, and it generally performs better than the existing bounds for SDD \(_1\) 1 , DSDD, and DZT matrices. As an application, an error bound for the linear complementarity problems (LCPs) of B \(_1\) 1 -matrices is given.