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<p>Based on the Hermitian and skew Hermitian splitting of the coefficient matrices, we demonstrate a shift-splitting hierarchical identification (SSHI) iterative algorithm to solve the matrix equation <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2024_3849_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>A</mi> <mi>X</mi> <mo>−</mo> <mover accent="true"> <mi>X</mi> <mo>‾</mo> </mover> <mi>B</mi> <mo>=</mo> <mi>C</mi> </math></EquationSource> <EquationSource Format="TEX">$AX - \overline{X} B=C$</EquationSource> </InlineEquation>. For any initial value, the suggested method converges to the exact solution under certain conditions. Three numerical examples are presented to demonstrate the effectiveness of the shift-splitting hierarchical identification (SSHI) iterative method and to compare it to the Jacobi-gradient iterative algorithm (JGI) (Bayoumi in <i>Appl. Math. Inf. Sci.</i> (<CitationRef CitationID="CR17">2021</CitationRef>)) and the gradient iterative algorithm (GI).</p>

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A shift-splitting hierarchical identification iterative algorithm for solving the matrix equation \(AX - \overline{X} B=C\)

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  • 发表日期：

    2025年5月14日

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  • Ahmed M. E. Bayoumi

摘要

Based on the Hermitian and skew Hermitian splitting of the coefficient matrices, we demonstrate a shift-splitting hierarchical identification (SSHI) iterative algorithm to solve the matrix equation A X − X ‾ B = C $AX - \overline{X} B=C$ . For any initial value, the suggested method converges to the exact solution under certain conditions. Three numerical examples are presented to demonstrate the effectiveness of the shift-splitting hierarchical identification (SSHI) iterative method and to compare it to the Jacobi-gradient iterative algorithm (JGI) (Bayoumi in Appl. Math. Inf. Sci. (2021)) and the gradient iterative algorithm (GI).