View of A convergence analysis of an iterative algorithm of order \(1.839\ldots\) under weak assumptions

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A CONVERGENCE ANALYSIS OF AN ITERATIVE ALGORITHM OF ORDER 1.839. . . UNDER WEAK ASSUMPTIONS

IOANNIS K. ARGYROS^∗

Abstract. We provide new and weaker sufficient local and semilocal conditions for the convergence of a certain iterative method of order 1.839. . .to a solution of an equation in a Banach space. The new idea is to use center-Lipschitz/Lipschitz conditions instead of just Lipschitz conditions on the divided differences of the operator involved. This way we obtain finer error bounds and a better informa- tion on the location of the solution under weaker assumptions than before.

MSC 2000. 65B05, 65G99, 65J15, 65N30, 65N35, 47H17, 49M15.

Keywords. Banach space, majorizing sequence, Halley method, Euler–Cheby- shev method, divided differences of order one and two, Fr´echet-derivative, R- order of convergence, convergence radius.

1. INTRODUCTION

In this study we are concerned with the problem of approximating a solution x^∗ of equation

(1) F(x) = 0,

whereF is a Fr´echet differentiable operator on an open convex subset Dof a Banach spaceX with values in a Banach spaceY.

The iterations

xn+1 =xn−L⁻¹_n F(xn), (2)

Ln= [xn, xn−1] + [xn−2, xn]−[xn−2, xn−1], n≥0,

have been already used to generate a sequence converging to x^∗ withR-order 1.839. . . [4], [12] and [13]. Here, [x, y] ∈ L(X, Y), [x, y, z] ∈ L(X, L(X, Y)) denote divided differences of order one and two respectively of operator F satisfying

(3) [x, y](y−x) =F(y)−F(x)

and

(4) [x, y, z](y−z) = [x, y]−[x, z]

for all x, y, z∈D[4].

∗Department of Mathematics, Cameron University, Lawton, OK 73505, USA, e-mail:

[email protected].

Method (2) is considered to be a discretized version of the famous cubi- cally convergent methods of Euler–Chebyshev (tangent hyperbola) or Halley (parabola) [1]–[3], [6], [10] and [11]. Discretized versions of the above meth- ods using divided differences of order one and two or just one have also been considered in [5], [7] and [9].

Here we provide a new local and semilocal convergence analysis for method (2). Using Lipschitz and center Lipschitz conditions on the divided differences of operatorF instead of just Lipschitz conditions we introduce weaker sufficient convergence conditions than before. Moreover we obtain finer error bounds on the distances involved as well as a better information on the location of the solutionx^∗. Furthermore in the case of local analysis a larger convergence radius is obtained.

2. SEMILOCAL ANALYSIS

Let d_i, i = 0,1, . . . ,5, η₀, η₁ be non-negative parameters and δ ∈ [0,1).

Define the parameters α0,α1,α2,β0,β1 by α₂ = (1−δ²)d₅,

(5)

α₁ = δδ(d₁+δd₀)−(1−δ²)(d₃+δd₂) , (6)

α0 = −δ(1−δ)(η0+η1)d5η1, (7)

β1 = d3+δ(d0+d2), (8)

β₀ = (η₀+η₁)η₁(d₅+δd₄)−δ, (9)

and functionsf,g by

f(t) = α₂t²+α₁t+α₀, (10)

g(t) = β1t+β0. (11)

We can show the following result on majorizing sequences.

Theorem 1. Let η be a non-negative parameter such that

(12) η≤







min{α₃, β2}, β₂=−^β_β⁰

1, ifβ16= 0,

0, ifα₀ = 0,

α3, ifβ1= 0,

provided

(13) β₀ ≤0,

where α₃, β₂ are the non-negative zeros of functions f and g respectively.

Then

(a) Iteration{t_n}, n≥ −2,given by t−2 =0,

t−1 =η0, t0 =η0+η1, t₁ =η₀+η₁+η,

t_n+2 =t_n+1+_1−d ^{d3(tn+1−tn)+d5(tn−tn−2)(tn−tn−1)}

4η1(η0+η1)−d₀(tn+1−t₀)−d₁(tn−t₀)−d₂(tn+1−tn)(t_n+1−t_n), (14)

n≥0,

is non-decreasing, bounded above by

(15) t^∗∗= η

1−δ +η0+η1

and converges tot^∗ such that

(16) 0≤t^∗ ≤t^∗∗.

Moreover, the following error bounds hold for alln≥0 (17) 0≤t_n+2−t_n+1≤δ(t_n+1−t_n)≤δⁿ⁺¹η.

(b) Iteration{s_n}, n≥ −2,given by s−2−s−1=η₁,

s−1−s₀=η₀, s0−s1=η,

sn+1−sn+2=_1−d ^{d3(sn−sn+1)+d5(sn−2−sn)(sn−1−sn)}

4η1(η0+η1)−d0(s0−sn+1)−d1(s0−sn)−d2(sn−sn+1)(sn−sn+1), (18)

n≥0,

for s−1, s0, s1≥0 is non-increasing, bounded below by

(19) s^∗∗=s₀− η

1−δ and converges to some s^∗ such that

(20) 0≤s^∗∗≤s^∗.

Moreover, the following error bounds hold for alln≥0:

(21) 0≤sn+1−sn+2≤δ(sn−sn+1)≤δⁿ⁺¹η.

Proof. (a) We must show:

(d₃+δd₂)(t_k+1−t_k) +d₅(t_k−tk−2)(t_k−tk−1) +δd₀(t_k+1−t₀)+

+δd1(tk−t0) +δd4η1(η0+η1) ≤ δ (22)

1−d₄η₁(η₀+η₁)−d₀t_k+1−d₁t_k−d₂(t_k+1−t_k) > 0 (23)

for all k≥0.

Inequalities (22) and (23) hold fork= 0 by the initial conditions. But then (14) gives

(24) 0≤t₂−t₁ ≤δ(t₁−t₀).

Let us assume (22), (23) and (17) hold for all k ≤ n+ 1. By the induction hypotheses we get

(d₃+δd₂)(t_k+2−t_k+1) +d₅(t_k+1−tk−1)(t_k+1−t_k) +δd₀(t_k+2−t₀) +δd₁(t_k+1−t₀) +δd₄η₁(η₀+η₁)≤ (25)

≤(d₃+δd₂)δ^k+1η+d₅(δ^k+δ^k−1)η²δ^k+δd_01−δ^η +δd_11−δ^η +δd₄η₁(η₀+η₁).

It is clear that (25) will be bounded above byδ if

(d₃+δd₂)δ^k+1η+d₅(δ^k+δ^k−1)η²δ^k+δd_01−δ^η +δd_11−δ^η +δd₄η₁(η₀+η₁)≤

≤(d3+δd2)η+d5(η0+η1)η1+δd0η+δd4η1(η0+η1) or

(d₃+δd₂)(1−δ)δ^k+1η+d₅δ^k−1(1−δ²)η²δ^k+δd₀η+δd₁η ≤

≤(1−δ)(d₃+δd₂)η+ (1−δ)d₅(η₀+η₁)η₁+ (1−δ)δd₀η or, for k≥0,

(d3+δd2)δ(1−δ)η+^1−δ_δ²d5η²+δd1η+δd0η ≤

≤(d3+δd2)(1−δ)η+d5(1−δ)(η0+η1)η1+ (1−δ)δd0η or, for δ6= 0,

(26) α₂η²+α₁η+α₀≤0,

which is true by the choice ofη. By the same proof as above we show (23) for k=n+ 1.

We must also show:

(27) tk ≤t^∗∗, k≥1.

For k = 1,2 we have t₁ ≤ t^∗ and t₂ ≤ t₁ +δη = η₀+η₁+ (1 +δ)η ≤ t^∗∗. Assume (27) holds for all k≤n+ 1. It follows from (17) that

tk+2 ≤ tk+1+δ(tk+1−tk)

≤ t_k+δ(t_k−tk−1) +δ(t_k+1−t_k)

· · ·

≤ t₁+δ(t₁−t₀) +· · ·+δ(t_k+1−t_k)

≤ η0+η1+η+δη+· · ·+δ^k+1η

= η₀+η₁+ ^1−δ_1−δ^k+2η

< t^∗∗.

That is, {t_n}, n≥0,is bounded abovet^∗∗. By (22) and (23) we get

(28) t_k+2−t_k+1 ≥0.

Hence sequence {t_n}, n ≥0,is also non-decreasing and as such it converges to somet^∗ satisfying (16).

(b) The proof follows along the lines of part (a).

That completes the proof of Theorem 1.

We show the following semilocal convergence theorem for method (2).

Theorem 2. Let F:D ⊆X → Y be a differentiable operator with divided differences of the first and second order denoted by [·,·], [·,·,·] respectively.

Assume:

– there exist pointsx−2, x−1, x0 ∈DsoL0 is invertible and non-negative numbers η, d_i, i= 0,1, . . . ,5 such that

L⁻¹₀ [x₀, x₀]−[y, x₀] ≤ d₀kx₀−yk, (29)

L⁻¹₀ [x, x0]−[x, y] ≤ d1kx₀−yk, (30)

L⁻¹₀ [x, y]−[x, z] ≤ d₂ky−zk, (31)

L⁻¹₀ [x, y]−[x, x] ≤ d₃kx−yk, (32)

L⁻¹₀ [x, y, x0]−[z, y, x0] ≤ d4kx−zk, (33)

L⁻¹₀ [x, x, y]−[z, x, y] ≤ d₅kx−zk, for all x, y, z∈D, (34)

kx₋₁−x0k ≤η1, kx₋₁−x−2k ≤ η0, kx₁−x0k ≤η;

(35)

– hypotheses of Theorem1 hold, and

(36) U(x₀, t^∗) =x∈X :kx−x₀k ≤t^∗ ⊆D.

Then method {x_n}, n ≥ 0, generated by (2) is well defined, remains in U(x₀, t^∗) for all n≥0 and converges to a solution x^∗ ∈U(x₀, t^∗) of equation F(x) = 0. Moreover, the following error bounds hold for all n≥0:

(37) kx_n+2−xn+1k ≤tn+2−tn+1

and

(38) kx_n−x^∗k ≤t^∗−tn. Furthermore, if there exists R≥t^∗ such that

(39) U(x₀, R)⊆D,

(40) L⁻¹₀ [x, y]−[z, w]≤d6 kx−zk+ky−wk, for all x, y, z, w∈D, and

(41) d₆(R+t^∗+ 2η₀+ 2η₁)≤1 or [·,·] is symmetric and

(42) d₆(R+t^∗+η₀+η₁) +d₁(η₀+η₁)≤1, the solution x^∗ is unique in U(x₀, R).

Proof. Let us prove

(43) kx_k+1−x_kk ≤t_k+1−t_k k≥ −2.

For k=−2,−1,0 (43) holds by the initial conditions. Assume (43) holds for all n≤k. Using (29), (30), (31), (33) and (43) we obtain

L⁻¹₀ L0−Lk+1 =

= L⁻¹₀ L₀−[x₀, x₀] + [x₀, x₀]−[x_k+1, x₀] + [x_k+1, x₀]

−[x_k+1, x_k] + [x_k+1, x_k]−L_k+1

≤ L⁻¹₀ [x₀, x−1, x₀]−[x−2, x−1, x₀](x−1−x₀)k

+L⁻¹₀ [x0, x0]−[xk+1, x0]+L⁻¹₀ [xk+1, x0]−[xk+1, xk] +L⁻¹₀ [xk−1, x_k]−[xk−1, x_k+1]

≤ d4kx₋₁−x0k kx₀−x−2k+d0kx_k+1−x0k +d₁kx_k−x₀k+d₂kx_k−x_k+1k

≤ d₄η₁(η₀+η₁) +d₀(t_k+1−t₀) +d(t_k−t₀) +d₂(t_k+1−t_k)

≤ d4η1(η0+η1) +d0(t^∗−t0) +d1(t^∗−t0) +d2δ^kη

< 1.

(44)

by the choice ofδ and (12).

It follows by (44) and the Banach Lemma on invertible operators [8] that L⁻¹_k+1 is invertible and

kL⁻¹_k+1L0k ≤

≤1−(d4kx−1−x0k kx₀−x−2k+d0kx_k+1−x0k+d1kx_k−x0k + d₂kx_k+1−x_kk)⁻¹

≤[1−d4η1(η0+η1)−d0(tk+1−t0)−d1(tk−t0)−d2(tk+1−tk)]⁻¹. (45)

Moreover, we have

L⁻¹₀ [x_k, x_k+1]−L_k

=L⁻¹₀ [x_k, x_k+1]−[x_k, x_k] + [x_k, x_k]−L_k

≤L⁻¹₀ [x_k, x_k+1]−[x_k, x_k]

+L⁻¹₀ [x_k, x_k, xk−1]−[xk−2, x_k, xk−1](x_k−xk−1)

≤d3kx_k+1−xkk+d5kx_k−xk−2k kx_k−xk−1k

≤d₃(t_k+1−t_k) +d₅(t_k−tk−2)(t_k−tk−1).

(46)

Furthermore using (2), (43), and (46) we get (47)

kx_k+2−x_k+1k ≤ kL⁻¹_k+1L₀kL⁻¹₀ [x_k, x_k+1]−L_kkx_k+1−x_kk ≤t_k+2−t_k+1, which shows (43).

Theorem 1 and (47) imply {x_n}, n≥0,is a Cauchy sequence in a Banach space X and as such it converges to some x^∗ ∈U(x0, t^∗), sinceU(x0, t^∗) is a closed set.

By (43) we get

(48) kx_n−xmk ≤tm−tn, −2≤n≤m, while by lettingm→ ∞ in (48) we obtain (38).

Finally, by lettingk→ ∞ in (47), we get F(x^∗) = 0. To show uniqueness, let y^∗ ∈U(x₀, R) be a solution of equationF(x) = 0. We can have in turn

L⁻¹₀ [y^∗, x^∗]−L₀≤

≤L⁻¹₀ [y^∗, x^∗]−[x₀, x−2] +L⁻¹₀ [x−2, x−1]−[x−1, x₀]

≤d₆(ky^∗−x₀k+kx^∗−x−2k+kx₋₂−x−1k+kx₋₁−x₀k)

< d₆(R+t^∗+ 2η₀+ 2η₁)

≤1.

(49)

It follows by (49) and the Banach Lemma on invertible operators that [y^∗, x^∗] is invertible. Hence, from

(50) F(x^∗)−F(y^∗) = [y^∗, x^∗](x^∗−y^∗), we deduce thatx^∗=y^∗.

If [·,·] is symmetric as in (49) we get

(51) L⁻¹₀ [y^∗, x^∗]−L₀< d₆(R+t^∗+η₀+η₁) +d₁(η₀+η₁)≤1.

We conclude again that x^∗ =y^∗.

That completes the proof of Theorem 2.

The proof of the following result follows exactly as in Theorem 2 but using part (b) of Theorem 1.

Theorem 3. Assume hypotheses of Theorems1 and 2 hold.

Then method {x_n}, n ≥ 0, generated by (2) is well defined, remains in U(x₀, s^∗) for all n≥0 and converges to a solution x^∗∈U(x₀, s^∗) of equation F(x) = 0. Moreover, the following error bounds hold for all n≥0:

(52) kx_n+2−xn+1k ≤sn+1−sn+2

and

(53) kx_n−x^∗k ≤s_n−s^∗.

Furthermore if there exists R1≥s^∗ such that, together with (40),

(54) U(x0, R1)⊆D holds,

and

(55) d₆[R₁+s^∗+ 2(η₀+R₁)]≤1

or

[·,·] is symmetric and

(56) d₆(R₁+s^∗+η₀+η₁) +d₁(η₀+η₁)≤1, the solution x^∗ is unique in U(x₀, R).

Remark 1. In [12, Th. 5.1], condition (40) was used together with (57) L⁻¹₀ [x, y, z]−[u, y, z]≤d7kx−uk

for all x, y, z, v∈D to show convergence of method (2).

The following error bounds were found

(58) kx_n+1−xnk ≤vn−vn+1

and

(59) kx_n−x^∗k ≤vn−v^∗, where,

(60) v^∗ = lim

n→∞v_n,

and {v_n}is similar to{s_n}but usingd₆,d₇,η₀,η₁,ηinstead of d₀,d₁,d₂,d₃, d4,d5,η0,η1,η. Note also that in general

(61) d0 ≤d1≤d3 ≤d2≤d6

and

(62) d4 ≤d5 ≤d7.

Hence we can easily obtain by induction

(63) s_n−s_n+1 ≤v_n−v_n+1

and

(64) s_n−s^∗≤v_n−v^∗.

That is, under weaker convergence conditions we obtain finer error bounds.

3. LOCAL ANALYSIS

We can show the following local results for method (2).

Theorem 4. Let F:D ⊆X → Y be a differentiable operator. Assume F has divided differences of the first and second order such that:

F⁰(x^∗) = [x^∗, x^∗], (65)

F⁰(x^∗)⁻¹ [x^∗, x^∗]−[x, x^∗] ≤ a₀kx−x^∗k, (66)

F⁰(x^∗)⁻¹ [x, x^∗]−[x, y] ≤ b₀kx−x^∗k, (67)

F⁰(x^∗)⁻¹ [x, x^∗, y]−[z, x^∗, y] ≤ c0kx−zk, (68)

F⁰(x₀)⁻¹ [u, x, y]−[v, x, y] ≤ cku−vk, (69)

U(x^∗, r^∗) ⊆ D, (70)

for all x, y, z, u, v∈D, where x^∗ is a simple zero ofF, and

(71) r^∗ = 2

a0+ 2b0+^p(a0+ 2b0)²+ 8(c+c0).

Then method (2) is well defined, remains in U(x^∗, r^∗) for all n ≥ 0, and converges to x^∗ provided that x−1, x−2,x0∈U(x^∗, r^∗).

Moreover, the following error bounds hold for all n≥0:

kx_n+1−x^∗k ≤

≤ ^b0kxn−x

∗k+c kxn−x^∗k+kxn−2−x^∗k

kxn−1−x^∗k+kxn−x^∗k

1−(a0+b0)kxn−x^∗k−c0 kxn−x^∗k+kxn−2−x^∗k

kxn−1−x^∗k kx_n−x^∗k=αn. (72)

Proof. We first show linear operator

L≡L(x, y, z) = [x, y] + [z, x]−[z, y], x, y, z ∈U(x^∗, r^∗) is invertible. By (65), (67), (69), we get in turn

F⁰(x^∗)⁻¹ F⁰(x^∗)−L=

=F⁰(x^∗)⁻¹[x^∗, x^∗]−[x, x^∗] + [z, x^∗]−[z, x]

+ [x, x^∗]−[x, y]−[z, x^∗] + [z, y]

≤F⁰(x^∗)⁻¹ [x^∗, x^∗]−[x, x^∗]+F⁰(x^∗)⁻¹ [z, x^∗]−[z, x]

+F⁰(x^∗)⁻¹ [x, x^∗, y]−[z, x^∗, y](x^∗−y)k

≤(a₀+b₀)kx−x^∗k+ckx−zk · kx^∗−yk

≤(a₀+b₀)kx−x^∗k+c(kx−x^∗k+kz−x^∗k)ky−x^∗k

<(a0+b0)r^∗+ 2c(r^∗)²

<1, (73)

by the choice ofr^∗. It follows from (73) and the Banach Lemma on invertible operators that L is invertible.

Assume xk−2, xk−1, x_k ∈ U(x^∗, r^∗) and set x_k = x, xk−1 = y, xk−2 = z, k= 0,1,2, . . . , n,L_k =L(x_k, xk−1, xk−2). We have

kL⁻¹_k F⁰(x^∗)k ≤1−(a₀+b₀)kx_k−x^∗k (74)

−c₀ kx_k−x^∗k+kx_k−2−x^∗kkx_k−1−x^∗k⁻¹. Moreover, by (67) and (69) we get

F⁰(x^∗)⁻¹ [xk, x^∗]−Lk =

=F⁰(x^∗)⁻¹ [x_k, x^∗]−[x_k, x_k] + [x_k, x_k]−[x_k, xk−1]

−[xk−2, x_k] + [xk−2, xk−1]

≤F⁰(x^∗)⁻¹ [x_k, x^∗]−[x_k, x_k]

+F⁰(x^∗)⁻¹ [xk, xk, xk−1]−[xk−2, xk, xk−1](xk−xk−1)

≤b₀kx_k−x^∗k+ckx_k−xk−2k kx_k−xk−1k

≤b₀kx_k−x^∗k+c kx_k−x^∗k+kx^∗−xk−2k kx_k−x^∗k+kx^∗−xk−1k. (75)

By (2) we can write

kx_k+1−x^∗k = x_k−x^∗−L⁻¹_k F(x_k)−F(x^∗)

= −L⁻¹_k [xk, x^∗]−Lk

(xk−x^∗)

≤ kL⁻¹_k F⁰(x^∗)kF⁰(x^∗)⁻¹ [x_k, x^∗]−L_kkx_k−x^∗k.

(76)

Estimate (72) now follows from (74), (75) and (76). By the choice of r^∗ and (76) we get

kx_k+1−x^∗k<kx_k−x^∗k< r^∗, from which it follows x_k+1∈U(x^∗, r^∗) and lim

k→∞x_k=x^∗.

That completes the proof of Theorem 3.

Remark 2. In the elegant paper [12] the following conditions were used:

(77) F⁰(x^∗)⁻¹ [x, y]−[u, v]≤b kx−uk+ky−vk and

(78) F⁰(x^∗)⁻¹ [u, x, y]−[v, x, y]≤cku−vk

for all x, y, u, v∈D. Note that (77) impliesF⁰(x^∗) = [x^∗, x^∗], [4], [8].

The convergence radius is given by

(79) r_p = 2

3b+√

9b²+ 16c.

Moreover the corresponding error bounds are kx_n+1−x^∗k ≤

(80)

≤β_n

= ^bkxn−x

∗k+c kx_n−x^∗k+kxn−2−x^∗k

kxn−1−x^∗k+kx_n−x^∗k

1−2bkxn−x^∗k−c kxn−x^∗k+kxn−2−x^∗k

kxn−1−x^∗k kx_n−x^∗k, n≥0.

It can easily be seen that conditions (65)–(69) are weaker than (77) and (78).

Moreover in general

(81) a₀≤b₀≤b

and

(82) c0≤c.

Hence,

(83) r_p ≤r^∗

and

(84) αn≤βn, n≥0.

In case strict inequality holds in one of (81) or (82) then (83), (84) hold as strict inequalities also. That is under weaker conditions we provide finer error bounds and a wider choice of initial guesses. This observation is important in computational mathematics.

Finally note that it was also shown in [12] that theR-order of convergence of method (2) is the unique positive root of equation

t³−t²−t−1 = 0, being approximately 1.839. . . .

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Received by the editors: October 24, 2002.