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Influence of the Center Condition on the Two-Step Secant Method.

1. Introduction

Consider the problem to approximate a locally unique solution x* of

G(x) = 0, (1)

where G : D [subset or equal to] X [right arrow] Y is a nonlinear operator. X, Y are Banach spaces and D is an open nonempty convex subset of X. This is one of the very important problems in applied mathematics and engineering science. Many real life problems in diverse areas such as equilibrium theory and elasticity often reduce to solving these equations depending on one or more parameters. Mathematical modeling of many problems uses integral equations, boundary value problems, differential equations, and so forth, whose solutions are obtained by solving scalar equations or a system of equations. Many nonlinear differential equations can be solved by transforming them to matrix equations which give a system of nonlinear equations in R". Many researchers [1-4] have extensively studied these problems and many methods, both direct and iterative, are developed for their solutions. Good convergence properties, efficiency, and numerical stability are the requirements of all these methods. It is a common problem to choose the good starting points for the iterative methods which ensure the convergence of the iterative method. The semilocal convergence [2, 5, 6] uses information given at the initial point whereas local convergence [7, 8] uses information around the solution. The quadratically convergent Newton's iteration [9, 10] is used to solve (1). It is defined for n [greater than or equal to] 0 by

[x.sub.n+1] = [x.sub.n] - [[GAMMA].sub.n]G ([x.sub.n]), (2)

where [x.sub.0] [member of] D is the starting point and [[GAMMA].sub.n] = G'[([x.sub.n]).sup.-1] [member of] L(Y, X) (the set of bounded linear operators from Y into X). Sufficient conditions for the semilocal convergence with existence ball and error estimates of (2) are given in [11]. The secant iteration [1, 12,13] is the simplification of (2) used to solve (1) and is given for n [greater than or equal to] 0 by

[x.sub.n+1] = [x.sub.n] - [[[x.sub.n-1], [x.sub.n];G].sup.-1] G{[x.sub.n]), (3)

where [x.sub.-1], [x.sub.0] [member of] D are two starting points and [x, y; G] is the divided difference of order one for G on the points x, y [member of] D and satisfies the equality [x, y; G](x - y) = G(x) - G(y). In case of operators, this equality does not hold uniquely unless X is one-dimensional. In [R.sup.n], it is defined by a matrix

[[x,y,G].sub.j,k] = [G.sub.j]([x.sub.1], ..., [x.sub.k], [y.sub.k+1], [y.sub.n])- [G.sub.j] ([x.sub.1], ..., [x.sub.k-1], [y.sub.k], ..., [y.sub.n)]/[x.sub.k] - [y.sub.k] j,k = 1, ..., n, (4)

for x, y [member of] [R.sup.n] and G [member of] L([R.sup.n], [R.sup.n]). So, many real life problems that require the solution of matrix equations can also be solved by the abovementioned methods.

Recently an iteration known as the King-Werner iteration originally proposed by King [14] is discussed in [15,16] along with its local and semilocal convergence using majorizing sequences under the Lipschitz continuous Frechet derivative of G. It is given for n [greater than or equal to] 1 by

[x.sub.n] = [x.sub.n+1] - G' [([x.sub.n-1] + [y.sub.n-1]/2).sup.-1] G ([x.sub.n-1]), [y.sub.n] = [x.sub.n] - G' [([x.sub.n-1] + [y.sub.n-1]/2).sup.-1] G([x.sub.n]), (5)

where, [x.sub.0], [y.sub.0] [member of] D are the starting iterates. Its order is equal to 1 + [square root of 2]. A two-step secant iteration with order of convergence same as (5) with its semilocal and local convergence under combination of Lipschitz and center-Lipschitz continuous divided differences of order one using majorizing sequences for solving (1) is described in Banach space setting in [17]. It is defined for n [greater than or equal to] 0 by

[x.sub.n+1] = [x.sub.n] - [[[x.sub.n], [y.sub.n]; G].sup.-1] G([x.sub.n]), [y.sub.n+1] = [x.sub.n+1] - [[[x.sub.n], [y.sub.n]; G].sup.-1] G ([x.sub.n+1]), (6)

where [x.sub.0], [y.sub.0] [member of] D are starting iterates.

In this paper, iteration (6) is considered for solving (1) along with its semilocal and local convergence analysis under weaker Lipschitz continuity condition on divided differences of order one on the involved operator G in Banach space setting. The influence on the domain by our approach is shown by some numerical examples. It provides the improved error estimations along with the better information on the location of solutions. Semilocal convergence of (6) is studied, which improves the applicability of the method corresponding to some earlier study [17, 18]. It is shown by our work that earlier studies for (6) do not hold while the new convergence criteria hold. For local convergence analysis, weaker center-Lipschitz continuity condition is used in place of a combination of Lipschitz and center-Lipschitz continuity conditions. Larger convergence ball is obtained through this study in comparison to the older one.

The paper is arranged as follows. Introduction forms Section 1. In Section 2, the semilocal convergence analysis of (6) under weaker convergence conditions on divided differences of operator G is established. In Section 3, local convergence analysis of (6) is established using only center-Lipschitz continuity condition on divided differences. In Section 4, numerical examples are given to validate the theoretical results obtained by us. Finally, conclusions and references are included in Section 5.

2. Semilocal Convergence

In this section, firstly, we provide a lemma that will be used to provide the semilocal convergence theorem of (6).

Lemma 1. Let [k.sub.0], k, [k.sub.1], [k.sub.2], [eta], and s be nonnegative parameters and a be the unique root of the polynomial defined by

g (t) = k[t.sup.3] + [k.sub.0][t.sup.2] + ([k.sub.1] + [k.sub.2]) t - ([k.sub.1] + [k.sub.2]) (7)

and sequences {[t.sub.n]} and {[s.sub.n]} defined for n [greater than or equal to] 0 by [t.sub.0] 0, [t.sub.1] = [eta],

[mathematical expression not reproducible], (8)

and for all n [greater than or equal to] 1 by

[mathematical expression not reproducible] (9)

Supposing

0 < [k.sub.0][t.sub.1] + [ks.sub.0]/1 - ([k.sub.0][t.sub.1] + [ks.sub.1]) [less than or equal to] [alpha] [less than or equal to] 1 - ([k.sub.0] + k)[t.sub.1], (10)

then sequences {[s.sub.n]}, {[t.sub.n]} are well defined, increasing, and bounded above by t** = [t.sub.1]/(1 - [alpha]) and converge to their least upper bound t* which satisfies [t.sub.1] [less than or equal to] t* [less than or equal to] t**. Moreover, the following estimates hold for all n [less than or equal to] 1:

0 < [s.sub.n+1] - [t.sub.n+1] [less than or equal to] [alpha] ([t.sub.n+1] - [t.sub.n]), 0 [less than or equal to] [t.sub.n+2] - [t.sub.n+1] [less than or equal to] ([t.sub.n+1] - [t.sub.n]). (11)

Proof. From (7), g(0) = -([k.sub.1] + [k.sub.2]) and g(1) = (k + [k.sub.0]). Using intermediate value theorem, g has at least one root in (0, 1); also, it is increasing in this interval. So, it has a unique root in this interval which is denoted by a. Suppose [t.sub.1] = 0; then, all terms of sequences {[t.sub.n]} and {[s.sub.n]} will be equal to 0 and Lemma 1 holds in this case. Taking [t.sub.1] = [eta] > 0, then (11) is true if

[mathematical expression not reproducible] (12)

for each n = 1, 2, ...

This implies that 0 [less than or equal to] [s.sub.n] - [t.sub.n] [less than or equal to] [[alpha].sup.n] ([t.sub.1] - [t.sub.0]) and 0 [less than or equal to] [t.sub.n+1] - [t.sub.n] [less than or equal to] [[alpha].sup.n] ([t.sub.1] - [t.sub.0]). Now, instead of showing (12), it will be sufficient to show that

([k.sub.1] + [k.sub.2])[t.sub.1] [[alpha].sup.n-1] + [k.sub.0][t.sub.1]/1 - [alpha] (1 - [[alpha.sup.]n+1]) + [kt.sub.1]/1 - [alpha](1 - [[alpha].sup.n+2]) [less than or equal to] 1. (13)

From (13), we are motivated to construct a recurrent polynomial

[f.sub.n](t) = ([k.sub.1] + [k.sub.2]) [t.sub.1][t.sup.n-1] + [k.sub.0][t.sub.1]/1 - t (1 - [t.sup.n+1]) + [kt.sub.1]/1 - t (1 - [t.sup.n+2]) - 1. (14)

Replacing n by n+1 in (14), this gives

[f.sub.n+1](t) = ([k.sub.1] + [k.sub.2]) [t.sub.1][t.sup.n] + [k.sub.0][t.sub.1]/1 - t (1 - [t.sup.n+2]) + [kt.sub.1]/1 - t (1 - [t.sup.n+3]) - 1. (15)

Now, from (7), (14), and (15) and the help of some algebraic manipulations, we have

[f.sub.n+1] (t) = [f.sub.n] (t) + g (t) [t.sup.n-1] [t.sub.1]. (16)

Using (16), we get [f.sub.n+1] ([alpha]) = [f.sub.n]([alpha]); also, [f.sub.n](t) is an increasing function in (0, 1). Let us define a function R(t) on (0, 1) by

[mathematical expression not reproducible]. (17)

Now, we need to show only R([alpha]) [less than or equal to] 0. Using (10), this assertion can be proved easily and, thus, Lemma 1 is established.

Next, we provide a semilocal convergence theorem followed by Lemma 1 for (6).

Theorem 2. Let G : D [subset or equal to] X [right arrow] Y be a nonlinear operator; [k.sub.0], k, [k.sub.1], [k.sub.2], [eta], and s are given parameters. Denote [A.sub.n] = [[x.sub.n], [y.sub.n]; G] for n [greater than or equal to] 0. Under the hypothesis of Lemma 1, the following assumptions hold in [bar.B([x.sub.0], t*)] [subset or equal to] D:

[mathematical expression not reproducible]. (18)

Staring with suitable [x.sub.0], [y.sub.0] [member of] [bar.B([x.sub.0],t*)], sequences {[x.sub.n]} and {[y.sub.n]} defined in (6) are well defined, remain in [bar.B([x.sub.0], t*)], and converge to a solution x* in [bar.B([x.sub.0], t*))] of (1). Moreover, the following estimates hold for each n [greater than or equal to] 0:

[parallel][y.sub.n] - [x.sub.n][parallel] [less than or equal to] [s.sub.n] - [t.sub.n], (19)

[parallel][x.sub.n+1] - [x.sub.n][parallel] [less than or equal to] [t.sub.n+1] - [t.sub.n], [parallel]x* - [x.sub.n][parallel][less than or equal to] t* - [t.sub.n]. (20)

Further, if there exists R > t* such that B([x.sub.0], R) [subset or equal to] D and [k.sub.0] t* + kR < 1, then x* is the only solution of (1) in B([x.sub.0], R) [intersection] D.

Proof. Using mathematical induction on n, we shall show that (19) hold true. For n =1, this follows directly from (18) which shows that [x.sub.1] [member of] [bar.B([x.sub.0], t*))]. Using Banach lemma and (18), we get [parallel][A.sub.0.sup.-1] G'([x.sub.0])[parallel] [less than or equal to] 1 (1 - ks). Next,

[mathematical expression not reproducible]. (21)

This shows that [y.sub.1] [member of] [bar.B([x.sub.0], t*)]

[mathematical expression not reproducible]. (22)

Using Banach lemma [4] on invertible operators, we get

[parallel][A.sub.1.sup.-1] G' ([x.sub.0])[parallel] [less than or equal to] 1/1 - ([k.sub.0][t.sub.1] + [ks.sub.1]). (23)

Now,

[mathematical expression not reproducible], (24)

This implies [x.sub.2] [member of] [bar.B([x.sub.0], t*)] and thus (19) is true for n = 1. Now, from (6),

[mathematical expression not reproducible] (25)

This shows that [x.sub.2], [y.sub.2] [member of] [bar.B([x.sub.0], t*)]. Thus, replacing [x.sub.1], [y.sub.1], [x.sub.2], [y.sub.2] by [x.sub.n+1], [y.sub.n+1], [x.sub.n+2], [y.sub.n+2] and proceeding in a similar manner, this gives the notion that {[x.sub.n]} is a complete sequence in Banach space X such that it converges to some x* [member of] [bar.B([x.sub.0], t*))]. Now, to show that x* is a solution of (1),

[mathematical expression not reproducible]. (26)

So, G(x*) = 0. Suppose y* is another solution of (1) such that G(y*) = 0. Let T = [x*, y* ; G] be an operator and

[mathematical expression not reproducible]. (27)

It follows that x* = y* and this establishes Theorem 2.

To make the paper self-content, we present the lemma and semilocal convergence theorem of (6) that can be found in [18].

Lemma 3 (see [18]). Let [mathematical expression not reproducible], and s be nonnegative parameters and [alpha] be the unique root of the polynomial defined by

[mathematical expression not reproducible] (28)

and sequences {[l.sub.n]} and {[r.sub.n]} defined for n [greater than or equal to] 0, by [l.sub.0] = 0, [l.sub.1] = [eta] [r.sub.0] = s

[mathematical expression not reproducible], (29)

and for all n [greater than or equal to] 1 by

[mathematical expression not reproducible]. (30)

Supposing

[mathematical expression not reproducible]. (31)

then sequences {[r.sub.n]}, {[l.sub.n]} are well defined, increasing, and bounded above by l** = [l.sub.1]/(1 - [alpha]) and converge to their least upper bound l* which satisfies [l.sub.1] [less than or equal to] l* < l**. Moreover, the following estimates hold for all n [greater than or equal to] 1:

[mathematical expression not reproducible]. (32)

Theorem 4 (see [18]). Let G : D [subset or equal to] X [right arrow] Y be a nonlinear operator; [k.sub.0], k, [k.sub.1], [k.sub.2], [eta], and s are given parameters. Denote [A.sub.n] = [[x.sub.n], [y.sub.n]; G] for n [greater than or equal to] 0. Under the hypothesis of Lemma 3, the following assumptions hold in [bar.B([x.sub.0], l*)] [subset or equal to] D:

[mathematical expression not reproducible]. (33)

Starting with suitable [x.sub.0], [y.sub.0] [member of] [bar.B([x.sub.0], l*)], sequences {[x.sub.n]} and {[y.sub.n]} defined in (6) are well defined, remain in [bar.B([x.sub.0], l*)], and converge to a unique solution x* in [bar.B([x.sub.0], l*)] of (1). Moreover, the following estimates hold for each n [greater than or equal to] 0:

[mathematical expression not reproducible]. (34)

Further, if there exists [??] > l* such that [mathematical expression not reproducible], then x* is the only solution of(1) in B([x.sub.0], [??]) [intersection] D.

Previous assertions [17] are made for iteration (6) as follows:

[mathematical expression not reproducible]. (35)

One can easily see that our conditions are more general than (35), and with conditions (35), the following majorizing sequences are obtained:

[mathematical expression not reproducible]. (36)

3. Local Convergence

In this section, the local convergence of (6) to solve (1) is established under center-Lipschitz condition on divided differences of order one of the involved operator G.

Theorem5. Let [l.sub.0], l [greater than or equal to] 0 be given parameters and G : D x D [right arrow] Y be a nonlinear divided difference operator such that for all x, y [member of] D

[mathematical expression not reproducible] (37)

and [bar.B(x*, r*)] [subset or equal to] D, where r* = 1/(2l + 3[l.sub.0]). Then, sequences {[x.sub.n]}, {[y.sub.n]} of (6) starting from [x.sub.0], [y.sub.0] [member of] B(x*, r*) are well defined, remain in [bar.B(x*, r*))] for each n = 0, 1, 2, ..., and converge to x*. Moreover, the following error estimates hold:

[mathematical expression not reproducible]. (38)

Additionally, if there exists R* > r* such that [bar.B(x*, R*)] [subset or equal to] D and R* < 1/l, then x* is the only solution of (1) [bar.B(x*, R*))] [intersection] D.

Proof. For [x.sub.0], [y.sub.0] [member of] B(x*, r*) and using (37), we get

[mathematical expression not reproducible]. (39)

So, by Banach lemma, [A.sup.-1.sub.0] exists and

[parallel][A.sup.-1.sub.0]G' (x*([parallel] [less than or equal to] 1/1 - ([l.sub.0] + l)/ (2l + 3l0) = 2l + [3l.sub.0]/l + [2l.sub.0]. (40)

Using (37) and (40) and hypothesis of Theorem 5, we have

[mathematical expression not reproducible] (41)

Again, using (37)-(41) and hypothesis of Theorem 5, we get

[mathematical expression not reproducible]. (42)

This shows that [x.sub.1], [y.sub.1] [member of] B(x*, r*). Clearly, using induction on n,

[mathematical expression not reproducible] (43)

This shows (38). Now, let y* be another solution of (1) in B (x*,R*) such that G(y*) = 0. Using (37), this gives

[parallel]G' [(x*).sup.-1] ([x*, y*; G]-G' (x*))[parallel] [less than or equal to] l[parallel] [less than or equal to] l [parallel]y* - x*[parallel] (44)

This shows that x* is the unique solution of (1) in [bar.B(x*, R*))] [intersection] D.

4. Numerical Examples

In this section, some numerical examples are given to show the effectiveness of the present study. We have used approach-1, approach-2, and approach-3 by (18), (33), and (35), respectively.

Example 1. Let X = Y = R, D = B([x.sub.0], 1) and define function G on D by

G(x) = [x.sup.3] - [theta]. (45)

We take [x.sub.0] = 1 and [y.sub.0] free in order to find a relation between [theta] and [y.sub.0] for which all the criteria for ensuring the convergence are satisfied. In Figure 1, we have taken horizontal axis for [y.sub.0] and vertical axis for [theta]. With the help of (6), we obtain s = [absolute value of (1 - [y.sub.0])], [eta] = [absolute value (1 - [theta])]/(1 + [y.sub.0] + [y.sup.2.sub.0]), [k.sub.0] = 2, k = 1, [k.sub.1] = 8/3, [k.sub.2] = 4/3, [k.sub.0] = 6, [??] = 3, [[??].sub.1] = 4, [K.sub.1] = [[??].sub.2] = 8, and K = 4. The efficacy of our approach can be seen in Figure 1.

For comparing the error estimation where all approaches including the older approach are satisfied, we take the domain D = B ([x.sub.0], 0.3) and fix [theta] = 0.75, [y.sub.0] = 0.95. In this case, we get

G(x) = [x.sup.3] - 0.75 (46)

and = 0.05, [eta] = 0.0876424, [k.sub.0] = 1.53333, k = 0.766666, [k.sub.1] = 1.73333, [k.sub.2] = 0.86666, [??] = 0.8063, K = [[??].sub.0] = 1.6126, [[??].sub.1] = 0.9115, and [K.sub.1] = [[??].sub.2] = 1.8230. Comparison of error estimations with different approaches is given in Table 1.

From Figure 1, it can be seen that there exists some combination of [y.sub.0] and [theta] where the condition used earlier fails. When all conditions hold, then it gives the precise error bounds. Thus, the claim made by us in the abstract and the Introduction is justified here.

Example 2. Let X = Y = C [0, 1], the space of all continuous functions defined in [0, 1] equipped with the max-norm. Let D = {x [member of] C [0,1]; [parallel]x[parallel] [less than or equal to] R} such that R =1 and define G on D by

G(x) = x (s) - f (s) - [1/8] [[integral].sup.1.sub.0] [G.sub.1], (s, t) x [(t).sup.3] dt, x [member of] C [0,1], s [member of] [0, 1], (47)

where f [member of] C[0, 1] is a given function and the kernel [G.sub.1] is Green's function:

[mathematical expression not reproducible]. (48)

Now, one can represent a linear operator G'(x) by

(G' (x)) v(s) = v (s)- [3/8] [[integral].sup.1.sub.0] [G.sub.1] (s, t) x[(t).sup.2] v (t) dt, v [member of] C [0, 1], s [member of] [0, 1] (49)

Choose [x.sub.0](s) = f(s) = s and [y.sub.0](s) = 2s; we obtain

[parallel]G([x.sub.0])[parallel] [less than or equal to] 1/64. (50)

It can be easily seen that

[mathematical expression not reproducible]. (51)

[eta] = 0.1404, [k.sub.0] = 0.0984, k = 0.0492, [k.sub.1] = 0.1311, [k.sub.2] = 0.0565, [??] = 0.0526, K = [[??].sub.0] = 0.1228, [[??].sub.1] = 0.1404, [K.sub.1] = [[??].sub.2] = 0.0702, and s =1. Comparison of the error estimation with the older one is given in Table 2.

Example 3. Consider the partial differential equation

[DELTA]u = [u.sup.3], (52)

where

[DELTA] [[partial derivative].sup.2]/[partial derivative][[xi].sup.2.sub.1] + [[partial derivative].sup.2]/ [partial derivative][[xi].sup.2.sub.2] (53)

is the two-dimensional Laplace operator. These types of equations arise in the theory of gas dynamics [19]. Assume that (52) is satisfied in the rectangular domain D = {([[xi].sub.1], [[xi].sub.2]) [member of] [R.sup.2]; 0 [less than or equal to] 1,0 [less than or equal to] [[xi].sub.2] [less than or equal to] 1} with the Dirichlet boundary condition given by

u([[xi].sub.1], 0) = 2[[xi].sup.2.sub.1] - [[xi].sub.1] + 1, u ([[xi].sub.1], 1) = 2, u(0,[[xi].sub.2]) = 2[[xi].sup.2.sub.2] - [[xi].sub.2] + 1, u(1, [[xi].sub.2]) = 2. (54)

In order to transform (52) into a system of nonlinear equations, central divided differences scheme has been used. This leads to a nonlinear system of equations:

[u.sub.i+1,j-] [4u.sub.i,j] + [u.sub.i,j+1] + [u.sub.i,j-1] - [h.sup.2] [u.sub.i,j] = 0 i = 1, ..., n, j = 1, ..., m. (55)

u(i, j) denotes the estimation u([[xi].sub.1,i], [[xi].sub.2,j]). Now, taking n = m = 5, this generates a 6 x 6 mesh. Boundary values can be obtained from (54), and to find interior points, we transform interior values as [x.sub.1] = [u.sub.1,1], [x.sub.2] = [u.sub.2,1], [x.sub.3] = [u.sub.3,1], [x.sub.4] = [u.sub.4,1], [x.sub.5] = [u.sub.1,2], [x.sub.6] = [u.sub.2,2], = [u.sub.3,2], = [x.sub.8] = [u.sub.4,2], = [x.sub.9] [u.sub.1,3], [x.sub.10] = [u.sub.2,3], [x.sub.11] = [u.sub.3,3], [x.sub.12] = [u.sub.4,3], [x.sub.13] = [u.sub.1,4], [x.sub.14] = [u.sub.2,4], [x.sub.15] = [u.sub.3,4], [x.sub.16] = [u.sub.4,4]. So, for X = Y = [R.sup.16], the system can be expressed as

G (x) = Ax + [h.sup.2] [phi] (x)-b = 0, (56)

where A, h, and b are given by

[mathematical expression not reproducible]. (57)

Now,

[mathematical expression not reproducible]. (58)

So,

[mathematical expression not reproducible]. (59)

We choose [x.sub.0] = [(4/5, 4/5, ..., 4/5).sup.T] and [y.sub.0](i) = [x.sub.0](i) + 0.1, and we get [k.sub.0] = 0.2169, k = 0.1084, [k.sub.1] = 0.2168, [k.sub.2] = 0.1205, s = 0.1, [eta] = 1.0425, [??] = 0.1070813, K = [[??].sub.0] = 0.2201, [[??].sub.1] = 0.2379, and [K.sub.1] = [[??].sub.2] = 0.1189. In this example, approach-3 does not hold but approach-1 and approach-2 hold well. Now, we compare approach-1 and approach-2 for this example and a comparison is given in Table 3. Next, we use (6) to solve (56) and the approximate solution is given with stopping criterion [parallel][x.sub.n] - [x.sub.n-1][parallel] [less than or equal to] [10.sup.- 15]. The approximate solution is then given in Table 4.

Interpolating the value of Table 4, we get the numerical approximation of the solution which can be seen in Figure 2.

Example 4. Let X = Y = [R.sup.3], D = [R.sup.3], and x* = [(0, 0, 0).sup.T] and define a function G on D by

G(x, y, z) = [([e.sup.x] -1, [y.sup.2] + y, z).sup.T]. (60)

For t = [(x, y, z).sup.T],

[mathematical expression not reproducible]. (61)

Using max-norm of rows, one can easily find that G'(x*) = diag {1,1,1}, [l.sub.0] = e/2, and l = 1, r* = 0.164543. So, from Theorem 5, (6) starting from [x.sub.0], [y.sub.0] [member of] B(x* ,r*) converges to x*.

For comparison of the radius of ball convergence with a previous study, we take another example which satisfies the previous condition given in [17].

Example 5. Let X = Y = R, D = R, x* = 0 and define a function G on D by

G(x) = [e.sup.x] -1. (62)

Clearly, G'(x*) = 1. In this case, we get [l.sub.0] = l = (e - 1)/2, and using Theorem 5, we get r* = 0.23279 which is bigger than the corresponding radius 0.20002 in [17].

Example 6. Let X = Y = C[0,1], the space of continuous functions defined on [0,1], equipped with the max-norm and D = [bar.B(0, 1)]. Define G on D, given by

G (x) (s) = x (s) -5 [[integral].sup.1.sub.0] st[x.sup.3] (t) dt, (63)

s [member of] [0, 1]. We obtain [l.sub.0] = l = 3.75, and using this value, we find that r* = 0.05333 which is bigger than the corresponding radius 0.039052 in [17].

5. Conclusions

In this work, the semilocal and local convergence analysis for two-step secant method is established. A comparison is established on different types of center conditions used earlier for the convergence analysis. It is shown that the approach used in this paper gives precise error bounds along with the better information to the solution. It is also shown that sometimes earlier condition fails to converge but the sufficient conditions used in this paper hold. Moreover, it gives precise error bounds. Finally, some numerical examples including gas dynamics and integral equations validate the theoretical results obtained in this study.

https://doi.org/10.1155/2017/7364236

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

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[3] P. K. Parida and D. K. Gupta, "Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces," Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 350-361, 2008.

[4] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970.

[5] I. K. Argyros, A. Cordero, A. Margenen, and J. R. Torregrosa, "On the convergence of a damped Newton-like method with modified right hand side vector," Applied Mathematics and Computation, vol. 266, Article ID 21280, pp. 927-936, 2015.

[6] P. Maroju, R. Behl, and S. S. Motsa, "Convergence of a parameter based iterative method for solving nonlinear equations in Banach spaces," in S.S. Convergence of a parameter based iterative method for solving nonlinear equations in Banach spaces, pp. 10-1007, II. Ser, Rend. Circ. Mat. Palermo, 2016.

[7] I. K. Argyros and S. George, "Local convergence of deformed Halley method in Banach space under HOLder continuity conditions," Journal of Nonlinear Science and its Applications. JNSA, vol. 8, no. 3, pp. 246-254, 2015.

[8] S. Amat, S. Busquier, and J. M. Gutierrez, "On the local convergence of secant-type methods," International Journal of Computer Mathematics, vol. 81, no. 9, pp. 1153-1161, 2004.

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Abhimanyu Kumar, (1) D. K. Gupta, (1) and Shwetabh Srivastava (2)

(1) Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India

(2) Department of Mathematics, School of Arts and Sciences, Amrita Vishwa Vidyapeetham (Amrita University), Amritapuri, India

Correspondence should be addressed to Shwetabh Srivastava; shwetabhiit@gmail.com

Received 23 June 2017; Accepted 8 August 2017; Published 24 September 2017

Academic Editor: Shamsul Qamar

Caption: Figure 1: Domain of (1).

Caption: Figure 2: Approximated solution of (52).
Table 1: Comparison of error estimation for (46).

[t.sub.n+1] -     [s.sub.n] -     [l.sub.n+1] -    [r.sub.n] -
[t.sub.n]           [t.sub.n]       [l.sub.n]       [l.sub.n]

1.9250e - 02      1.5700e - 02     2.1668e - 02    1.5922e - 02
1.2013e - 03      1.1508e - 03     1.6866e - 03    1.5928e - 03
4.9236e - 06      4.9113e - 06     1.1047e - 05    1.1002e - 05
8.3814e - 11      8.3813e - 11     4.8224e - 10    4.8222e - 10
0                       0               0               0

[t.sub.n+1] -     [P.sub.n+1] -    [q.sub.n] -
[t.sub.n]           [P.sub.n]       [P.sub.n]

1.9250e - 02       3.2362e - 02    2.1993e - 02
1.2013e - 03       6.1234e - 03    5.3343e - 03
4.9236e - 06       2.4979e - 04    2.4428e - 04
8.3814e - 11       4.3972e - 07    4.3937e - 07
0                  1.3772e - 12    1.3772e - 12

Table 2: Comparison of error estimation for Example 2.

[t.sub.n+1] -     [s.sub.n]     [l.sub.n+1] -      [r.sub.n]
[t.sub.n]         [t.sub.n]       [l.sub.n]       [l.sub.n]

9.3052e - 03      9.0388e - 03   1.0231e - 02     9.4607e - 03
1.6579e - 05      1.6594e - 05   2.3267e - 05     2.3239e - 05
5.5371e - 11      5.5371e - 11   1.2339e - 10     1.2339e - 10

[t.sub.n+1] -     [P.sub.n+1] -     [q.sub.n] -
[t.sub.n]            [P.sub.n]       [P.sub.n]

9.3052e - 03       1.9542e - 02     2.2479e - 02
1.6579e - 05       1.3390e - 04     1.3363e - 04
5.5371e - 11       5.8417e - 09     5.8416e - 09

Table 3: Comparison of error estimation for Example 3.

[t.sub.n+1] -     [s.sub.n] -     [l.sub.n+1] -      [r.sub.n]
[t.sub.n]          [t.sub.n]      [l.sub.n]          [l.sub.n]

0.3897               0.2497       0.4029               0.2504
9.1893e - 02      7.6239e - 02    9.9774e - 02      8.1506e - 02
5.7163e - 03      5.4747e - 03    6.9079e - 03      6.5728e - 03
2.3179e - 05      2.3121e - 05    3.4807e - 05      3.4695e - 05
3.8625e - 10      3.8624e - 10    8.9732e - 10      8.9730e - 10

Table 4: Approximate solution of (52).

i         [x*.sub.i]

1      0.967514648571165
2      1.073142808305482
3      1.255308661675940
4      1.547504427760980
5      1.073142808305482
6      1.199182696602124
7      1.359712017969179
8      1.602945733655613
9      1.255308661675940
10     1.359712017969179
11     1.481965315289151
12     1.669313085344323
13     1.547504427760980
14     1.602945733655613
15     1.669313085344323
16     1.778410018624668
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Title Annotation:Research Article
Author:Kumar, Abhimanyu; Gupta, D.K.; Srivastava, Shwetabh
Publication:International Journal of Analysis
Article Type:Report
Date:Jan 1, 2017
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