We introduce a novel subclass of H-matrices called <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S^k$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation>-strictly diagonally dominant(<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S^k$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation>-SDD) matrices, where k is any positive integer. These matrices generalize SDD matrices, S-SDD matrices, and generalized SDD<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$_1$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mn>1</mn> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> matrices. We provide a method for constructing scaling matrices for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S^k$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation>-SDD matrices, ensuring that their multiplication with the scaling matrix results in an SDD matrix. By decomposing the scaling matrix into a product of two matrices, we establish an upper bound on the infinity norm of the inverse matrix for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$S^k$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation>-SDD matrices. Moreover, based on the decomposition of the scaling matrix, we derive an error bound for the linear complementarity problem associated with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$S^k$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation>-SDD matrices. Significantly, our error bound represents a theoretical enhancement over the findings reported by P.F. Dai, Y.T. Li, and C.J. Lu in their paper titled “Error bounds for linear complementarity problems for SB-matrices” (Numerical Algorithms, 61(1): 121-139, 2012). Furthermore, we substantiate the effectiveness and superiority of our findings through numerical experiments conducted with randomly generated matrices.

Construction and decomposition of scaling matrices for $S^k$-SDD matrices and their application to linear complementarity problems

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发表日期：

2025年3月13日

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Wenlong Zeng,
Qing-Wen Wang,
Jianzhou Liu

摘要

We introduce a novel subclass of H-matrices called $S^k$ $S^{k}$ -strictly diagonally dominant( $S^k$ $S^{k}$ -SDD) matrices, where k is any positive integer. These matrices generalize SDD matrices, S-SDD matrices, and generalized SDD $_1$ $_{1}$ matrices. We provide a method for constructing scaling matrices for $S^k$ $S^{k}$ -SDD matrices, ensuring that their multiplication with the scaling matrix results in an SDD matrix. By decomposing the scaling matrix into a product of two matrices, we establish an upper bound on the infinity norm of the inverse matrix for $S^k$ $S^{k}$ -SDD matrices. Moreover, based on the decomposition of the scaling matrix, we derive an error bound for the linear complementarity problem associated with $S^k$ $S^{k}$ -SDD matrices. Significantly, our error bound represents a theoretical enhancement over the findings reported by P.F. Dai, Y.T. Li, and C.J. Lu in their paper titled “Error bounds for linear complementarity problems for SB-matrices” (Numerical Algorithms, 61(1): 121-139, 2012). Furthermore, we substantiate the effectiveness and superiority of our findings through numerical experiments conducted with randomly generated matrices.

Construction and decomposition of scaling matrices for \(S^k\)-SDD matrices and their application to linear complementarity problems

摘要