The SOR method for infinite systems of linear equations (III)

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52 Béla Finta5. THE SOR METHOD FOR INFINITE SYSTEMS OF LINEAR EQUATIONSLet us consider the infinite system of linear equations Ax = b, where A ∈ Mand x, b ∈ s.Definition 5.2. For a given A ∈ M and b ∈ s we will say that x ∗ ∈ s is a solutionof the infinite system of linear equations Ax = b if we have Ax ∗ = b.∞∑∞∑This means that the series a ij x ∗ j is convergent and we have a ij x ∗ j = b i ,j=0for every i ∈ N.Let us suppose that a ii ≠0for every i ∈ N and let us consider the constantsω i ∈ R \{0} for every i ∈ N. The initial system of linear equations Ax = b isequivalent to the following iterative system of linear equations:⎧∑∞ a 0j b 0x 0 =(1− ω 0 )x 0 − ω 0 x j + ω 0a 00⎪⎨⎪⎩.j=1x 1 = −ω 1a 10a 11x 0 +(1− ω 1 )x 1 − ω 1a 00∑∞j=2a 1ja 11x j + ω 1b 1x 2 = −ω 2a 20a 22x 0 − ω 2a 21a 22x 1 +(1− ω 2 )x 2 − ω 2.∑i−1a ijx i = −ω i x j +(1− ω i )x i − ω ia iij=0∞ ∑j=i+1a 11∑∞j=3a ija iix j + ω ib ia iij=0a 2ja 22x j + ω 2b 2a 22Using this system of linear equations, let us choose x 0 ∈ s and we generate thesequence (x k ) k∈N ⊂ s by the following iterative formula:⎧∞∑x k+10 =(1− ω 0 )x k 0 − ω a 0j0 x k ja + ω b 0000(5.1)⎪⎨⎪⎩.j=1a 00x k+1 a 10∑∞1 = −ω 1 x k+10 +(1− ω 1 )x k a 1j1 − ω 1 x k b 1j + ω 1a 11 a 11 a 11x k+12 = −ω 2a 20a 22x k+10 −ω 2a 21a 22tx k+1.x k+1i∑i−1a ij= −ω i x k+1ja iij=0j=21 +(1−ω 2 )x k 2 −ω 2 ∞ ∑+(1− ω i )x k i − ω i∞∑j=i+1a ijj=3a iix k j + ω ia 2ja 22x k j +ω 2b 2a 22Consequently, starting from the vector x k , we generate the vector x k+1 by therecursion formula x k+1 = B ω · x k + c.b ia ii
The SOR method for infinite systems of linear equations (III) 53Definition 5.3. The matrix A =(a ij ) i,j∈N is l 1 diagonal dominated if there existsthe positive real number λ > 0 such that for every j ∈ N we have∞∑λ|a jj | > |a ij |.i=0i≠jIt is immediately that A is l 1 diagonal dominated if and only if supj∈Na finite real number.Let us denote by λ := supj∈N∞∑∣ a ij ∣∣∣∣ isa jji=0i≠j∣ ∞∑a ij ∣∣∣∣ ,ω ∗ =sup{|ω i |/i ∈ N} ∈R and ω ∗∗ =a jji=0i≠jsup{|1 − ω i |/i∈ N} ∈R. Let us suppose ω ∗ · λ
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The SOR method for infinite systems of linear equations (III)

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