Applied Mathematical Sciences, Vol. 6, 2012, no. 77, Oana Bumbariu

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1 Applied Mathematical Sciences, Vol. 6, 2012, no. 77, A Convergence Result for the B-Algorithm Oana Bumbariu North University of Baia Mare Department of Mathematics and Computer Science Victoriei 76, , Baia Mare, Romania oanabumbariu@yahoo.com Abstract Our aim in this paper is to give a convergent result for the acceleration method that we introduced in [O. Bumbariu, An acceleration method for the Picard iteration, (to appear)], for solving a nonlinear equation, called the B-algorithm. Mathematics Subject Classification: 47H10, 65B99 Keywords: Picard iterations; Aitken s Δ 2 process; order of convergence 1 Introduction Aitken s Δ 2 process [1] is a powerful algorithm for accelerating the convergence speed of a sequence that converges linearly. Using the idea of Aitken we were able to develop a new acceleration method, called the B-algorithm (3) [2]. Because Aitken s Δ 2 process accelerates the convergence speed of sequences, it is natural to use these technique and the B-algorithm for solving a nonlinear equation of the form f(x) = 0. The iteration function for Aitken s Δ 2 process is the same as for the Steffensen s method, (4), [4], [5], because of that, we will use a similar approach in case of B-algorithm for proving the convergence. In the following section, we give the rule of the B-algorithm, then, under some assumptions, we prove that if the sequence of successive approximations, x n+1 = g(x n ), is of order p 2, then the B-algorithm is of order p 2 + p 1 and if p = 1, then the B-algorithm is of order 2. And if the equation f(x) = 0 has a root, α, with the multiplicity m 3, then the B-algorithm is of order one. 2 A convergence result for the B-algorithm Consider the nonlinear equation f(x) =0. (1)

2 3822 Oana Bumbariu We assume that α is a unique root of (1) in an interval [a, b], such that x [a, b], f(x) [a, b]. Staring from an initial approximation, x 0,ofα we can build a sequence of successive approximations x n+1 = g(x n ), n =0, 1,... (2) that converges to α, under appropriate assumptions. In order to improve the converge speed of the fixed point iterative method (2) we introduced in [2] a new acceleration technique, given by the following relation B n = S n+3 [ΔS n+1][δs n+2 ] ΔS n+1 ΔS n, n,k N (3) where {S n } is the sequence to be accelerated, Δ, the forward difference operator is defined by ΔS n = S n+1 S n and we denote by Δ, the two step forward difference operator as Δ=S n+2 S n. The acceleration method given by (3) is called the B-algorithm, it consists in composing the Picard iteration, (2), and the B sequence transformation as follow: Step 0: y 0 := x 0 Step n 1: (1) set S 0 = y 0, S 1 = f(s 0 ), S 2 = f(s 1 ), S 3 = f(s 2 ), (2) compute B n by the relation (3) from S 0, S 1,S 2,S 3, (3) set y n = B n. In what follows we remind the concept of convergence order that will be used in the paper. Let {x n } R be a sequence of real numbers convergent to α R. (which is obtained by iterating a fixed point equation) Definition 2.1 [6] Let {x n } converge to α. If there exist an integer constant p, and a real positive constant C such that lim x n+1 α n (x n α) p = C, then p is called the order and C the constant of convergence. The concept of rate of convergence given by Definition 2.1 is also known as the Q-order of convergence, see the monographs by Măruşter [6] and Ortega and Rheinboldt [8]. The iteration function of the Steffensen s method [5] is g(x) = xg(g(x)) (g(x))2 g(g(x)) 2g(x)+x, (4)

3 A convergence result for the B-algorithm 3823 and let G(x) be the iteration function of the B-algorithm defined by xg(g(g(x))) g(x)g(g(x)) G(x) = g(g(g(x))) g(g(x)) g(x)+x. (5) In the sequel we present a convergence result for the acceleration method (4). Theorem 2.2 [5] (1) If the functional iteration applied to x = g(x) is of order p 2 for some root α of (1) then Steffensen s method has order 2p 1. (2) If the functional iteration applied to x = g(x) is of first order (but not necessarily convergent) for a simple root α of (1) then Steffensen s method is of second order. (3) If as in (2), the root α of (1) has multiplicity m 2, then Steffensen s method is first order with the asymptotic convergence factor C =1 1 m. The proof can be found in [5], page 107. An similar result, but for the B-algorithm, is given by the following theorem. Theorem 2.3 (1) If the functional iteration applied to x = g(x) is of order p 2 for some root α of (1) then the B-algorithm has order p 2 + p 1. (2) If the functional iteration applied to x = g(x) is of first order (but not necessarily convergent) for a simple root α of (1) then the B-algorithm is of second order. (3) If as in (2), the root α of (1) has multiplicity m 3, then B-algorithm is first order and the asymptotic convergence factor C<1 1. m 1 Proof 2.4 We recall the fact that the iteration function for the B-algorithm is given by (5) and assume that x = α is a root of (1) and that: g =g =...= g (p 1) =0; g (p) =p!a 0; (6) g (p+1) in x α ρ. These conditions imply that g(x) determines a pth order method. From Taylor s theorem [7] and (6) for every ɛ such that ɛ ρ g(α + ɛ) =α + Aɛ p + g(p+1) (α + θɛ) ɛ p+1, 0 <θ<1 (p + 1)! = α + Aɛ p + Bɛ p+1 = α + δ, (7) where B = g(p+1) (α+θɛ) (p+1)! and δ =(A + Bɛ)ɛ p g(α + δ) =α + Aδ p + B δ p+1 = α + γ, (8)

4 3824 Oana Bumbariu where B = g(p+1) (α+θδ) (p+1)!, 0 < Θ < 1 and γ =(A + B δ)δ p. where B = g(p+1) (α+γγ) (p+1)!, 0 < Γ < 1. From (7)-(9) in (5) with x = α + ɛ and ɛ 0it results g(α + γ) =α + Aγ p + B γ p+1 (9) (α + ɛ)g(g(g(α + ɛ))) g(α + ɛ)g(g(α + ɛ)) G(α + ɛ) = g(g(g(g(α + ɛ)))) g(g(g(α + ɛ)g)) g(g(α + ɛ)) + (α + ɛ) = α δγ Aɛγp B ɛγ p+1. (10) ɛ δ γ + Aγ p + B γp+1 There are two cases to be considered p 2 and p =1. First, for p 2 the expression (10) can be written G(α + ɛ) =α [A + B (A + Bɛ)ɛ p ](A + Bɛ) p+1 ɛ p2 +p 1 ( 1 Aɛ[A + B (A + Bɛ)ɛ p ] p 1 (A + Bɛ) p2 p 1 ɛ p3 p 2 p 1 [A + B (A + Bɛ)ɛ p ](A + Bɛ) p ɛ p2 1 (A + Bɛ)ɛ p 1 + B ɛ[a + B (A + Bɛ)ɛ p ] p (A + Bɛ) p2 1 ɛ p3 p +A[A + B (A + Bɛ)ɛ p ] p (A + Bɛ) p2 ɛ p3 1 + B [A + B (A + Bɛ)ɛ p ] p+1 (A + Bɛ) p2 +p ɛ ). p3 +p 2 1 When ɛ approaches 0 the expression in the parenthesis approaches 1 and so the above relation, (11), it can be written (11) G(α + ɛ) =α A p+2 ɛ p2 +p 1 + O(ɛ p2 +p ), p 2. (12) For p =1the expression (10) can be written G(α + ɛ) =α [A + B (A + Bɛ)ɛ](A + Bɛ)ɛ 2 (B A 2 B ) (ABB +A 2 B B )ɛ 2ABB B ɛ 2 B 2 B B ɛ 3 ( (1 A 2 )(1 A)+(A 4 B +A 3 B + A 2 B B AB A 2 B )ɛ +(2A 3 BB +2A 4 B B +2A 2 BB 2ABB )ɛ 2 + +(6A 3 BB B +A 4 B 2 B +A 2 B 2 B +AB 2 B B 2 B )ɛ 3 +(6A 2 B 2 B B +4A 3 BB 2 B )ɛ 4 + +(5A 2 B 2 B 2 B +2AB 3 B B +AB 2 B 2 B )ɛ 5 +4AB 3 B 2 B ɛ 6 + B 4 B 2 ). (13) B ɛ7 In general, if A 1the expression in the parenthesis approaches B 1 A B and B approach B = g and so (13) can be written as 2 G(α + ɛ) =α A2 B when ɛ 0 because 1 A ɛ2 + O(ɛ 3 ), p =1,g =A 1. (14)

5 A convergence result for the B-algorithm 3825 If g =A =1and α has multiplicity m then G(α + ɛ) =α +[1+B (1 + Bɛ)ɛ](1 + Bɛ)ɛ B B +(BB +B B )ɛ +2BB B ɛ 2 + B 2 B B ɛ 3 ( B B +2(BB +B B )ɛ +(B 2 B +B 2 B +6BB B )ɛ 2 +(6B 2 B B +4BB 2 B )ɛ 3 + ). (15) +(6B 2 B 2 B +2B 3 B B )ɛ 4 +4B 3 B 2 B ɛ 5 + B 4 B 2 B ɛ6 Let g =α, g =1,g =... = g (m 1) =0and g (m) 0. In the hypothesis that g has derivatives of order 3m, g(α + ɛ) =α + ɛ + Bɛ 3, where m 3 and B(ɛ) = g(m) ɛ m 3 + g(m+1) (m + 1)! ɛm (16) and similarly, with δ = ɛ + Bɛ 3 we have g(α + δ) =α + δ + B δ 3 where B (δ) = g(m) δ m 3 + g(m+1) (m + 1)! δm (17) be observe that δ = ɛ + Bɛ 3 = ɛ + g(m) ɛ m +... and [ ] B (δ) = g(m) g ɛ m 3 (m) 2 +(m 3) ɛ 2m (18) where γ = δ + B δ 3 = ɛ +2 g(m) ɛ m + m B (γ) = g(m) ɛ m 3 +2(m 3) and therefore B (γ) = g(m) γ m 3 + g(m+1) (m + 1)! γm (19) [ ] 2 g (m) ɛ 2m then [ ] g (m) 2 [ ] g ɛ 2m 4 (m) 3 +(m 3)(2m 4) ɛ 3m 5 m)! m)! (20) [ ] g (m) 2 B B =2(m 3) ɛ 2m (21) [ ] g BB +B (m) 2 B =2 ɛ 2m (22) m)! Introducing (21) and (22) in (15) be obtain G(α + ɛ) =α + m 3+ɛ +... m 3+2ɛ +... ɛ<α+1 1 m 1. (23)

6 3826 Oana Bumbariu References [1] A. C. Aitken, On Bernoulli s numerical solution of algebraic equations, Proc. Roy. Soc. Edinburgh, 46 (1926), [2] O. Bumbariu, An acceleration method for the Picard iteration, (to appear) [3] C. Brezinski, Convergence acceleration during the 20th century, J. Comput. Appl. Math., 122 No. 1-2 (2000), [4] A. Fdil, A new method for solving nonlinear equation with error estimations, Appl. Numer. Math., 21 (1996), [5] E. Isaacson, H. B. Keller, Analysis of numerical methods, Wiley, New York, [6] Şt. Măruşter, Metode numerice în rezolvarea ecuaţiilor neliniare, Editura Tehnica, Bucuresti, [7] D.A. Murray, A first course in infinitesimal calculus, Classic reprint series, New York, [8] J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equation in Several Variables, Academic Press, New York, Received: March, 2012