A SEMILOCAL CONVERGENCE ANALYSIS FOR THE METHOD OF TANGENT PARABOLAS

. We present a semilocal convergence analysis for the method of tangent parabolas (Euler–Chebyshev) using a combination of Lipschitz and center Lipschitz conditions on the Fr´echet derivatives involved. This way we produce a majorizing sequence which converges under weaker conditions than before. The error bounds obtained are more precise and the information of the location of the solution better than in earlier results.


INTRODUCTION
In this study we are concerned with the problem of approximating a locally unique solution x * of equation ( 1) where F is a twice-Fréchet differentiable operator on an open convex subset D of a Banach space X with values in a Banach space Y .The method of tangent parabolas (Euler-Chebyshev) (2) ) is one of the best known cubically convergent iterative procedures for solving nonlinear equations like (1).Here F (x n ) ∈ L(X, Y ), F (x n ) ∈ L(X, L(X, Y )) denote the first and second Fréchet derivatives of operator F evaluated at x = x n (n ≥ 0) [3], [7].
Here we provide a semilocal convergence analysis based on Lipschitz and center-Lipschitz conditions on the first and second Fréchet-derivatives of F .This way existing convergence conditions are finer and the information on the location of the solution more precise than before.

CONVERGENCE ANALYSIS
We need the following results on majorizing sequences.
Similarly we show the next two theorems: Theorem 2. Let η, 0 , 3 , 4 be non-negative parameters.Define scalar sequence {s n } (n ≥ 0) by , where where α 0 is the positive solution of quadratic equation Then, sequence {s n } (n ≥ 0) is non-decreasing, bounded above by s * * = 2η 0 , and converges to s * such that Moreover, the following error bounds hold for all n ≥ 0 where and parameter α by Assume: where α 0 is the positive solution of quadratic equation

Moreover, the following error bounds hold for all
We can show the main semilocal convergence theorem for method (2).
Estimates (28) and (29) imply that (21) and ( 22) hold for n = k + 1.By induction the proof of (21) and ( 22) is completed.Theorem 1 implies {t n } (n ≥ 0) is a Cauchy sequence.From (21) and (22) {x n } (n ≥ 0) becomes a Cauchy sequence too, and as such it converges to some x * ∈ U (x 0 , t * ) (since U (x 0 , t * ) is a closed set) such that (30) The combination of ( 21) and (30) yields F (x * ) = 0. Finally to show uniqueness let y * be a solution of equation F (x) = 0 in U (x 0 , R).It follows from (16) the estimate and the Banach Lemma on invertible operators that linear operator we deduce x * = y * .That completes the proof of the theorem.
Theorem 5. Let F : D ⊆ X → Y be a twice Fréchet-differentiable operator.Assume: there exist a point x 0 ∈ D and non-negative parameters η, 0 , 3 , 4 such that x − y for all x, y ∈ D.Moreover, hypotheses of Theorem 2 hold, and Then the method of tangent parabolas {x n } (n ≥ 0) generated by (2) is well defined, remains in U (x 0 , s * ) for all n ≥ 0 and converges to a solution x * ∈ U (x 0 , s * ) of equation F (x) = 0.
Moreover, the following error bounds hold for all n ≥ 0: and Proof.It follows along the lines of Theorem 4 but instead of (23) we use Moreover, instead of (26) we use which as in (27) leads to The rest follows as in Theorem 4 until the uniqueness part.Let y * be a solution of equation Hence, we get . By the above and the Banach Lemma on invertible operators L is invertible.
Using the identity we get x * = y * .That completes the proof of Theorem 5. Theorem 6.Let F : D ⊆ X → Y be a twice Fréchet-differentiable operator.Assume: there exist a point x 0 ∈ D and non-negative parameters η, 0 , 1 , 3 , x − y for all x, y ∈ D.Moreover, hypotheses of Theorem 1 hold, and Then the method of tangent parabolas {x n } (n ≥ 0) generated by (2) is well defined, remains in U (x 0 , v * ) for all n ≥ 0 and converges to a solution x * ∈ U (x 0 , v * ) of equation F (x) = 0. Moreover the following error bounds hold for all n ≥ 0: Furthermore, if there exists R 2 ≥ v * such that and Proof.Use (23) instead of (31).The rest follows as in Theorem 5 until the uniqueness part.Moreover the uniqueness part follows as in Theorem 4.
That completes the proof of Theorem 6.
Remark 1.In order for us to compare our results with earlier ones in [1], [6], [12] define sequences {δ n }, {M n }, {N n }, {β n } by and function ( 40) has one negative and two positive roots w * , w * * such that w * ≤ w * * and U (x 0 , w * ) ⊆ D or, equivalently, Then, the method of tangent parabolas {x n } (n ≥ 0) generated by ( 2) is well defined, remains in U (x 0 , w * ) for all n ≥ 0 and converges to a solution x * ∈ U (x 0 , w * ) of equation F (x) = 0. Moreover the following error bounds hold for all n ≥ 0: Furthermore, if: w * < w * * the solution is unique in U (x 0 , w * * ) otherwise the solution is unique in U (x 0 , w * ).
That is, our Theorem 5 provides more precise error bounds and a better information on the location of the solution x * .In the case of 3 = 4 , Theorem 5 reduces to earlier ones mentioned in this remark.
In the next example we show that 4  3 may be arbitrarily large.where c i , i = 0, 1, 2, 3 are given parameters.Using (44) and (45) we can easily see that for c 3 large and c 2 sufficiently small, 4 3 may be arbitrarily large.That is (43) holds as strict inequality and (41) may be violated whereas hypotheses of Theorem 5 may hold (see also Example 1).

Example 2 .
Let X = Y = R, x 0 = 0 and define function F on R by ) = c 0 t + c 1 + c 2 sin e c 3 t ,