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• W4214835925 abstract "In this paper, we derive upper bounds that characterize the rate of convergence of the SOR method for solving a linear system of the form <math xmlns=http://www.w3.org/1998/Math/MathML id=M1> <mi>G</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> </math> , where <math xmlns=http://www.w3.org/1998/Math/MathML id=M2> <mi>G</mi> </math> is a real symmetric positive semidefinite <math xmlns=http://www.w3.org/1998/Math/MathML id=M3> <mi>n</mi> <mo>×</mo> <mi>n</mi> </math> matrix. The bounds are given in terms of the condition number of <math xmlns=http://www.w3.org/1998/Math/MathML id=M4> <mi>G</mi> </math> , which is the ratio <math xmlns=http://www.w3.org/1998/Math/MathML id=M5> <mi>κ</mi> <mo>=</mo> <mi>α</mi> <mo>/</mo> <mi>β</mi> </math> , where <math xmlns=http://www.w3.org/1998/Math/MathML id=M6> <mi>α</mi> </math> is the largest eigenvalue of <math xmlns=http://www.w3.org/1998/Math/MathML id=M7> <mi>G</mi> </math> and <math xmlns=http://www.w3.org/1998/Math/MathML id=M8> <mi>β</mi> </math> is the smallest nonzero eigenvalue of <math xmlns=http://www.w3.org/1998/Math/MathML id=M9> <mi>G</mi> </math> . Let <math xmlns=http://www.w3.org/1998/Math/MathML id=M10> <mi>H</mi> </math> denote the related iteration matrix. Then, since <math xmlns=http://www.w3.org/1998/Math/MathML id=M11> <mi>G</mi> </math> has a zero eigenvalue, the spectral radius of <math xmlns=http://www.w3.org/1998/Math/MathML id=M12> <mi>H</mi> </math> equals 1, and the rate of convergence is determined by the size of <math xmlns=http://www.w3.org/1998/Math/MathML id=M13> <mi>η</mi> </math> , the largest eigenvalue of <math xmlns=http://www.w3.org/1998/Math/MathML id=M14> <mi>H</mi> </math> whose modulus differs from 1. The bound has the form <math xmlns=http://www.w3.org/1998/Math/MathML id=M15> <msup> <mrow> <mfenced open=| close=|> <mrow> <mi>η</mi> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mfenced open=( close=)> <mrow> <mi>κ</mi> <mi>c</mi> </mrow> </mfenced> </math> , where <math xmlns=http://www.w3.org/1998/Math/MathML id=M16> <mi>c</mi> <mo>=</mo> <mn>2</mn> <mo>+</mo> <msub> <mrow> <mi mathvariant=normal>log</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mi>n</mi> <mo>.</mo> </math> The main consequence from this bound is that small condition number forces fast convergence while large condition number allows slow convergence." @default.
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• W4214835925 date "2022-03-02" @default.
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• W4214835925 title "The Rate of Convergence of the SOR Method in the Positive Semidefinite Case" @default.
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