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- W1503638420 abstract "We investigate the nonlinear matrix equation<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:msup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mrow><mml:msubsup><mml:mo stretchy=false>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant=normal>1</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=normal>∗</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mi>I</mml:mi></mml:math>, where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:math>is a positive integer and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:mo stretchy=false>(</mml:mo><mml:mn>0,1</mml:mn><mml:mo stretchy=false>]</mml:mo><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mtext>for</mml:mtext><mml:mtext> </mml:mtext><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1,2</mml:mn><mml:mo>,</mml:mo><mml:mo>…,</mml:mo><mml:mi>m</mml:mi></mml:math>. We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. A sufficient condition for the equation to have a unique positive definite solution is established. An iterative algorithm is provided to compute the positive definite solutions for the equation and error estimate. Finally, some numerical examples are given to show the effectiveness and convergence of this algorithm." @default.
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- W1503638420 date "2015-01-01" @default.
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- W1503638420 title "Positive Definite Solutions of the Matrix EquationXr-∑i=1mAi∗X-δiAi=I" @default.
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- W1503638420 doi "https://doi.org/10.1155/2015/473965" @default.
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