Improved Complexity of a Homotopy Method for Locating an Approximate Zero

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1 Punjab University Journal of Mathematics (ISSN ) Vol. 5(2)(218) pp. 1-1 Improved Complexity of a Homotopy Method for Locating an Approximate Zero Ioannis K. Argyros Department of Mathematical Sciences, Cameron University, Lawton, OK 7355, USA. iargyros@cameron.edu Santhosh George Department of Mathematical Computational Sciences, NIT Karnataka, India. sgeorge@nitk.ac.in Received: 25 August, 217 / Accepted: 21 November, 217 / Published online: 18 January, 218 Abstract. The goal of this study is to extend the applicability of a homotopy method for locating an approximate zero using Newton s method. The improvements are obtained using more precise Lipschitz-type functions than in earlier works our new idea of restricted convergence regions. Moreover, these improvements are found under the same computational effort. AMS Subject Classifications: 65G99, 65H1, 47H17, 49M15, Key words: Approximate zero, Banach space, Homotopy method, Newton s method, local, semi-local convergence, Lipschitz-type conditions. 1 Introduction The convergence region error analysis of iterative methods are very pessimistic in general for both the semi-local local case [1 5, 11 16]. The aim of the paper is to extend the convergence region using the homotopy method. This goal is achieved using the same Lipschitz-type functions as before [4,6 1,13]. We achieve this goal, since we find a more precise location for the Newton iterates leading to at least as tight Lipschitz-type functions [4, 6, 7]. Let F : D B 1 B 2 be differentiable in the sense of Fréchet, D be a convex open subset of B 1 B 1, B 2 be Banach spaces. Let F is one-to-one onto, we introduce the Newton operator N F (x) := x F (x) 1 F (x) (1.1)

2 Cyclic Vector of the Weighted Mean Matrix Operator 2 the corresponding Newton iteration x n+1 = N F (x n ) for all n =, 1, 2,... (1.2) where x D is an initial point. We are concerned with the problem of approximating a regular (to be precised in Section 2) solution w of utilizing a homotopy method of the form F (x) = (1.3) H(x, t) := F (x) tf (x ) (1.4) where x D is a given initial point t [, 1]. Clearly this is a geometrical way of solving equation (1.3). Consider the line segment M = {tf (x ) : t [, 1]} the set F 1 (M). Suppose that F (x ) is one-to-one onto. Then, it follows by the implicit function theorem applied in a neighbourhood of x that there exists a curve x(t) solving the equation F (x(t)) = tf (x ) for t [1 ε, 1] ε >. This curve solves the initial value problem (IVP) ẋ(t) = DF (x(t)) 1 F (x ), x(1) = x. (1.5) It is well known that (1.5) has no solution on [, 1], in general. But if it has a solution, one must follow x(t) (numerically), which is given by H(x(t), t) = using the operator related to H(., t). That is consider the sequence {s n } given by s = 1 > s 1 >... > s n >... > such that x n+1 = N H(.,sn+1)(x k ) is an approximate zero of x(s n+1 ), with H(x(s n+1 ), s n+1 ) =. A convergence analysis of Newton sequence {x n } was given in the elegant work by Guttierrez et al. [1]. Here, we improve their results as already mentioned previously. The study is structured as: The convergence of Newton s method is presented in Section 2 whereas Section 3 contains the special cases. Finally, in Section 4, we present the numerical examples. 1 Convergence Analysis We need the Definition of an approximate zero. Definition 1.1 [14] A G regular ball is open so that G (x) is one-to-one onto. A point x is a regular approximate zero of G, provided there exists a ball G regular containing a zero w of G a sequence {x n } converging to w.

3 Cyclic Vector of the Weighted Mean Matrix Operator 3 Let L, L, L : [, + ) [, + ) be continuous non-decreasing functions. These functions are needed for the introduction of the Lipschitz conditions that follow (see (1.7), (1.9) (1.11)). We shall also suppose that there exists z D so G (z) is continuous, one-to-one, onto G (z) 1 exists. We need to introduce the following two Lipschitz conditions that follow. Definition 1.2 The function G (z) 1 G is L center Lipschitz at z if there exist positive quantities v satisfying for a D, γ ( a z ) v G (z) 1 (G (a) G (z)) γ := γ (G, z) (1.6) γ a z L (τ)dτ. (1.7) Definition 1.3 The function G (z) 1 G is L center Lipschitz restricted at z, if there exist positive quantities v satisfying for a, b D := D Ū(z, v γ ) γ := γ(g, z) (1.8) γ( a z + τ a b ) v G (z) 1 (G ((1 τ)a + τb) G (a)) for all τ [, 1]. γ( a z +τ b a ) γ a z L(τ)dτ (1.9) Definition 1.4 [1] The function G (y ) 1 G is L center Lipschitz at z if there exist positive quantities v satisfying for a, b D γ := γ(g, z) (1.1) γ( a z + τ a b ) v G (z) 1 (G ((1 τ)a + τb) G (a)) for each τ [, 1]. γ( a z +τ b a ) γ a z L(τ)dτ (1.11) REMARK 1.5 Notice that (1.11) implies (1.7) (1.9). We can certainly take v = v = v, L (τ) = L(τ) = L(τ) for each τ, so for all τ [, v] since D D. γ (τ) γ(τ) (1.12) γ(τ) γ(τ), (1.13)

4 Cyclic Vector of the Weighted Mean Matrix Operator 4 In what follows we shall assume that If instead of (1.14) γ (τ) γ(τ). (1.14) γ(τ) γ (τ), (1.15) holds then the following results are true with L replacing L in all of them. LEMMA 1.6 Suppose that v is the least positive number such that v Then F (x) is one-to-one, onto ( F (x) 1 F (z) 1 γ x z L (τ)dτ = 1. (1.16) L (τ)dτ ) 1 for all x U(z, v γ ). (1.17) The set U(z, v γ ) is called the γ ball of z. We define similarly, the γ γ balls. As in [1], we assume the existence of ϕ : [, v) [, + ) satisfying ϕ() = 1, where γ(f, x) = ϕ( γ(f, z) x z ) γ for each x in U(z, ). (1.18) v γ Moreover, for b = b(f, z) := F (z) 1 F (z) we set ᾱ := ᾱ(f, z) := γ b. (1.19) By simply using (1.17) instead of the less precise estimate (since γ (τ) γ(τ)) ( ) 1 γ x x F (x) 1 F (x ) 1 L(τ)dτ for all x U(x, v ). (1.2) γ as well as γ, v instead of γ, v, respectively, we can reproduce the proofs of the results of [1] in this setting. The following result improves Theorem 1 in [1] which in turn generalizes the corresponding result by Meyer [13]. THEOREM 1.7 Suppose: F (x ) 1 F is L L Lipschitz restricted at x D; ᾱ(f, x ) v L(τ)τ dτ (1.21) Ū(x, v) D, (1.22) where ᾱ is given by (1.19) v is the smallest positive number such that v L(τ)dτ = 1. (1.23)

5 Cyclic Vector of the Weighted Mean Matrix Operator 5 Then, the solution of the IVP (1.5) exists in U(x, v 1 γ ) for each t [, 1], where v 1 is the first positive root of gā(t) less than or equal to u L/ c where gā(t) = ā t + t L(τ)(t τ)dτ. Therefore, x() is a solution of equation (1.3). Condition (1.21) is the usual Newton-Kantorovich type criterion [2, 3, 15]. REMARK 1.8 If L (s) L(s) for all s [, v], then the results of Theorem 1.7 hold with L replacing L. The Theorem 1.7 does not apply, if ᾱ > v L(s)ds. That is why as in [1], we suppose that the solution of the IVP (1.5) is inside the γ ball of x. Then, we ask: How many k steps are needed to approximate the zero x k of F = h(., )? THEOREM 1.9 Let x be an element of the γ ball of z. Set v = γ x z for u < v, where v satisfies (1.23). Define function q on [, v] by Let u L be such that Let c 1 define function gā on [, v] by so that t q(t) = L(τ)dτ t(1 t L (τ)dτ). (1.24) q(u L) = 1. (1.25) t gā(t) = ā t + L(τ)(t τ)dτ, (1.26) u L/ c v min{u L/ c L(τ)(u L/ c τ)dτ, L(τ)τdτ} ā (1.27) with the smallest positive solution of equation gā(t) = is not exceeding u L/ c. Set p = ϕ(u)(ᾱ + v L(τ)(v τ)dτ + v ) (1 u L/ c L (τ)dτ)(1 u L (τ)dτ) q = u L/ c L(τ)(u L/ c τ)dτ + u L/ c 1 u, L/ c L (τ)dτ where ϕ is given in (1.18). Moreover, suppose x(t) is the solution of the IVP is inside the γ ball of z. Let us also define sequence {s n } by where s = 1, s n >, s n 1 s n > s n s n+1 >, n, lim s n =, (1.28) n + s 1 = 1 1 u L (τ)dτ ϕ(u)(ᾱ + v L(τ)(v τ)dτ + v ). Set w n such that F (w n ) = s n F (x ). Then, the following assertions hold:

6 Cyclic Vector of the Weighted Mean Matrix Operator 6 (i) Points w n w n+1, are such that γϕ(u) w n+1 w n ā. (ii) Newton sequence {x n } generated by (1.28) w n are such that γϕ(v ) x n w n u L/ c. L(τ)τdτ q v (iii) Set N = p. The steps n required for x n to be an approximate zero of w n exceeds or is equal to [ ] 1 N, if s n := max{, 1 n(1 s 1 )}, 1 s 1 [ ] log N, if s n := s n 1 logs 1 [ ( )] log N log 2 + 1, if s n := s 2k 1 1. logs 1 x n w q(ū) 2n 1 x w, where ū = γ(f, w) x w < u L q is given in (1.24). REMARK 1.1 If L = L = L, γ = γ = γ, then the preceding items coincide with the ones in [1]. But, if (1.12) or (1.8) hold as strict inequalities, then the new results constitute an improvement over the ones in [1]. These improvements are deduced using the same effort as in [1], because finding function L requires finding functions L L. If L > L, then, the preceding results hold with L replacing L. 2 Special Cases We consider specializations of the preceding results in the general (Kantorovich) case L(s) = 1 the analytic case L(s) 2 = (1 s), respectively. Examples, 3 where (1.14) (1.15) hold as strict inequalities in the Kantorovich case can be found in [2,3] whereas the examples in the analytic case can be found in [4]. To avoid repetitions, we refer the reader to [1], where α(f, x ), ϕ, v, N, L are replaced by ᾱ(f, x ), ϕ, u, N L, respectively. Next, we present the α γ Theorems improving the works in [1] which in turn improved the works by X. Wang [16] Traub Wozniakowski [15], respectively. THEOREM 2.1 Suppose: F (x ) 1 F is L L center-lipschitz restricted at x ; ᾱ(f, x ) v L(τ)τdτ,

7 Cyclic Vector of the Weighted Mean Matrix Operator 7 where v is given in (1.23). Specialize function ḡ α(f,x) by t ḡᾱ(f,x)(t) := ḡ(t) = ᾱ(f, x ) t + L(τ)(t τ)dτ. (2.1) Then, the following items hold (i) There exist ρ 1, ρ 2 R with ρ 1 ρ 2 such that ḡ(ρ 1 ) = ḡ(ρ 2 ) = with ḡ strictly convex where ψ(t) = ḡ(t) = (t ρ 1 )(t ρ 2 )ψ(t), 1 1 θ( L(1 θ) + θsρ 2 + θτt)dτdθ. for r =, lim n + r n = lim n + Nḡ(r n 1 ) = v 1. (ii) Equation F (x) = has a solution w which is unique in U(x, v γ(f,x ) ). (iii) Newton sequence {x n } defined by x n+1 = N F (x n ) exists, stays in Ū(x, converges to w, so that (iv) If ḡ(t) ᾱ(f,x ) ρ 1ρ 2, then THEOREM 2.2 Suppose: x n w x n w r n ρ 1 1 γ(f, x )z n ( ρ1 (i) w solves F (x) = is a regular solution: ρ 2 ) 2 n 1 ρ 1. ρ 1 γ(f,x ) ) v γ(f, w) ). (ii) F ( w) 1 F ( w) is L L center Lipschitz restricted for all x U( w, Then, Newton sequence {x n } generated by x = x, x n+1 = N F (x n ) converges to w for all x U( w, u L γ(f, w) ), where u L is given in (1.25). Moreover, we have the following: x n w q(ū) 2n 1 x w. REMARK 2.3 If L = L = L, γ = γ = γ, the two preceding results reduce to Theorem 3 Theorem 4 in [1], respectively, i.e., if (1.14) or (1.15) hold as strict inequalities, then the earlier results are improved (see also the numerical examples).

8 Cyclic Vector of the Weighted Mean Matrix Operator 8 3 Numerical examples We provide two examples for the Kantorovich case, where function has no positive roots. Hence the older results can not apply, but function ḡ has roots, so the new results apply to solve equations. EXAMPLE 3.1 Let B 1 = B 2 = R, x = 1, D = {x : x x λ}, λ [, 1/2) F defined by F (x) = x 3 λ. (3.2) Then, for L (τ) = L(τ) = L(τ) = 1, v = v = v = 1, we have 1 λ b = b =, γ (τ) = 3 λ, γ(τ) = 2(2 λ) γ(τ) = 2( λ ). Notice that γ < γ < γ. The functions g ḡ are then given, respectively by g(t) = t2 2 t + 2 (1 λ)(2 λ) 3 ḡ(t) = t2 2 t (1 λ)( λ ). The Newton-Kantorovich condition (i.e., the discriminant d g of g) is given by d g = 1 4 (1 λ)(2 λ) < for each λ [, 1/2) (3.3) 3 so function g has not positive roots. since the discriminant However, function ḡ has positive roots, dḡ = (1 λ)(1 + ) > for each λ I = [ , 1/2). (3.4) 3 3 λ Therefore, our Theorem 2.1 can be used to solve equation F (x) = for all λ I. EXAMPLE 3.2 Let B 1 = B 2 = C[, 1]. Let D = {x B 1 : x R} for R >. Define F on D by F (x)(s) = x(s) f(s) δ 1 where f B 1 is a fixed function λ is given by { (1 s)t, if t s, K(s, t) = s(1 t), if s t. K(s, t)x(t) 3 dt, x B 1, s [, 1], (3.5)

9 Cyclic Vector of the Weighted Mean Matrix Operator 9 Then, for each x D, F (x) is given by [F (x)(v)](s) = v(s) 3δ 1 K(s, t)x(t) 2 v(t)dt, v X, s [, 1]. Set x (s) = f(s) = 1. Then, we have I F (x ) 3 δ /8 if δ < 8/3, then F (x ) 1 exists F (x ) δ. Moreover, so F (x ) δ 8, b = F (x ) 1 F (x ) Furthermore, for x, y D, we have F (x) F (y) F (x) F (1) δ x + y δ ( x + 1) x 1 8 δ 8 3 δ. x y 1 + 6R δ x y δ (1 + R) x 1. 8 Let δ = R = 2, we have b = , γ(τ) = , γ (τ) = γ(τ) = , v = v = v = 1. Using these values, we get that the discriminant d g of g is but the discriminant dḡ of ḡ is d g = <, dḡ = >. Hence, lim n x n = x by Theorem 2.1, where x is a solution of equation F (x)(s) =, where F is given by (3.5). References [1] E. L. Allgower K. George, Numerical continuation methods, 33, Springer, Berlin 199. [2] I. K. Argyros, Computational theory of iterative methods. Series: Studies in Computational Mathematics, 15, Editors: C. K. Chui L. Wuytack, Elsevier Publ. Co. New York, U. S. A, 27. [3] I. K. Argyros S. Hilout, Weaker conditions for the convergence of Newton s method, J. Complexity, 28, (212)

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