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• W2023256767 abstract "The paper is concerned with variants of the successive overrelaxation method (SOR method) for solving the linear system <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A u equals b> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Au = b</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Necessary and sufficient conditions are given for the convergence of the symmetric and unsymmetric SOR methods when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is symmetric. The modified SOR, symmetric SOR, and unsymmetric SOR methods are also considered for systems of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D 1 u 1 minus upper C Subscript upper U Baseline u 2 equals b 1 comma minus upper C Subscript upper L Baseline u 1 plus upper D 2 u 2 equals b 2> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>U</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{D_1}{u_1} - {C_U}{u_2} = {b_1}, - {C_L}{u_1} + {D_2}{u_2} = {b_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D 1> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{D_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D 2> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{D_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are square diagonal matrices. Different values of the relaxation factor are used on each set of equations. It is shown that if the matrix corresponding to the Jacobi method of iteration has real eigenvalues and has spectral radius <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu overbar greater-than 1> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>bar mu > 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the spectral radius of the matrix <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with any of the methods is not less than that of the ordinary SOR method with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=omega equals 2 left-parenthesis 1 plus left-parenthesis 1 minus mu overbar squared right-parenthesis Superscript 1 slash 2 Baseline right-parenthesis Superscript negative 1> <mml:semantics> <mml:mrow> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>omega = 2{(1 + {(1 - {bar mu ^2})^{1/2}})^{ - 1}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, if the eigenvalues of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are real then no improvement is possible by the use of semi-iterative methods." @default.
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• W2023256767 title "Convergence properties of the symmetric and unsymmetric successive overrelaxation methods and related methods" @default.
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