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

54 Béla FintaConsequently:‖y‖ 1 ≤ ω ∗ λ‖x‖ 1 + ω ∗ λ‖y‖ 1 + ω ∗∗ ‖x‖ 1 ,which is equivalent to‖y‖ 1≤ ω∗ λ + ω ∗∗‖x‖ 1 1 − ω ∗ λ .This means that‖B ω x‖ 1 ‖y‖ 1‖B ω ‖ 1 = sup = sup ≤ ω∗ λ + ω ∗∗x≠θ l‖x‖1 1 x≠θ l‖x‖1 1 1 − ω ∗ λ < 1.Now we can apply the Banach fixed point theorem for the iteration map Φ:l 1 → l 1 , Φ(x) =B ω x + c. Indeed, Φ is a contraction, because‖Φ(x) − Φ(y)‖ 1 = ‖(B ω x + c) − (B ω y + c)‖ 1 = ‖B ω (x − y)‖ 1 ≤‖B ω ‖ 1 ‖x − y‖ 1 .This means that the sequence (x k ) k∈N is convergent in l 1 for every x 0 ∈ l 1 andits limit point x ∗ ∈ l 1 is the unique fixed point of Φ in l 1 , i.e. Φ(x ∗ )=x ∗ . SoB ω x ∗ + c = x ∗ , which is equivalent to Ax ∗ = b.□Corollary 5.4. If for the matrix A =(a ij ) i,j∈N we have a ij =0when i>n,j>nand b i =0for i>n,n∈ N, then we reobtain the linear system with finite number ofequations and finite number of unknowns. In this way from Theorem 5.3 we obtain theclassical SOR iterative numerical method to solve finite systems of linear equations [7].In the following we consider the particular case when ω i = ω, for every i ∈ N.So, from (5.1) we can deduce: let us choose x 0 ∈ s and we generate the sequence(x k ) k∈N ⊂ s by the following iterative formula:(5.2)⎧⎪⎨⎪⎩.∞ x k+10 =(1− ω)x k 0 − ω ∑ a 0jx k j + ω b 0a 00j=1x k+11 = −ω a 10a 11x k+10 +(1− ω)x k 1 − ωx k+12 = −ω a 20x k+10 − ω a 21x k+1a 22 a 22..x k+1i∑i−1= −ωj=0a 00∞∑a 1jx k j + ω b 1aj=2 11 a 11∞1 +(1− ω)x k 2 − ω ∑a ija iix k+1j +(1− ω)x k i − ω ∞ ∑In this case from Theorem 5.3 we obtain:j=i+1j=3a ija iix k j + ω b ia iia 2ja 22x k j + ω b 2a 22|ω|λ + |1 − ω|Corollary 5.5. If < 1, (|ω|λ < 1) then the corresponding iterative1 −|ω|λsequence (x k ) k∈N given by (5.2) is convergent in l 1 for every x 0 ∈ l 1 . The limit pointx ∗ ∈ l 1 is the unique solution of the linear system Ax = b.