SEMI-LOCAL CONVERGENCE OF A DERIVATIVE-FREE METHOD FOR SOLVING EQUATIONS Текст научной статьи по специальности «Математика»

DOI: 10.15393/j3.art.2021.9230

UDC 517.988, 519.6, 531

G. Argyros, M. Argyros, I. K. Argyros, S. George

SEMI-LOCAL CONVERGENCE OF A DERIVATIVE-FREE METHOD FOR SOLVING EQUATIONS

Abstract. We present the semi-local convergence analysis of a two-step derivative-free method for solving Banach space valued equations. The convergence criteria are based only on the first derivative and our idea of recurrent functions.

Key words: Banach space, derivative-free method, semi-local convergence

2010 Mathematical Subject Classification: 65G99, 65J20, 49M15, 74G20, 41A25

1. Introduction. Let B, B2 stand for Banach spaces, U(x,p) denote a closed ball with center x E B1 and of radius p > 0. We denote the closure of U(x,p) by U(x,p).

We are dealing with the problem of approximating a solution x* of equation

F (x) = 0. (1)

Solving equation (1) is very important, because many problems are reduced to it by Mathematical modeling [1-8]. The solution methods are usually of iterative nature, since solutions in the closed form are rarely obtained. In this article, we develop a derivative-free method to generate a sequence approximating x* under certain conditions. The method is defined for all n ^ 0 as

yn xn Bn F (xn)

xn+1 = xn - A-1 F(xn), (2)

where A.n = [^n+nF] (n ^ 0), Bn = [Xn-1+yn-1 ,x.n; F] (n ^ 1), and x0,y0 E Q are initial points. Here, [x,y; F] : Q x Q —> L(X,Y) denotes a divided difference of order one for the operator F at the point

© Petrozavodsk State University, 2021

E D (see [2], [6], [8]). The method (2) is a useful alternative to third-order methods, such as the method of tangent hyperbolas (Halley) or the method of tangent parabolas (Euler-Chebysheff) [1-8]. However, these methods are very expensive, since they require the evaluation of the second Frechet derivative at each step. Discretized versions of these methods, such as Ulm's method, use divided differences of order one [1-8].

The rest of the paper is organized as follows. In Section 2, we present the semi-local convergence analysis of method (2), whereas in the concluding Section 3, we present the numerical examples.

2. Semi-local convergence. Semi-local convergence is based on the majorant sequence defined for n = 1, 2,... and some n ^ 0 and s ^ 0, as follows:

to = 0, ti = n ^ 0, so = s ^ 0,

= t I L(2(t™ — tn-i) + (sn—1 — tra-i))(tra — tra-i) (3)

Sn = tn + 2[1 - L0(tn-1 + Sn-1 + 2tn + s)] (3)

t = t 1 L(2(tn — tn— 1) + (sn—1 — tn — 1))(tn — tn— 1)

n+1 = n 2[1 - L0(3tn + s„ + s)] .

Define the scalar cubic polynomial p as

p(t) = Lot3 + 3Lot2 + 3Lt - 3L for some Lo > 0 and L > 0. (4)

By this definition, p(0) = — 3L and p(1) = 4Lo. It follows from the intermediate value theorem and the Descartes rule of signs, that polynomial p has a unique root 7 E (0,1). Moreover, define ao and a1 as

= L(2(t1 - to) + so - to) (_)

ao 2[1 - l20(3t1 + s1 + s)], ()

L(2(t1 - to) + so - to)

a = —;-^-TT , (6)

2[1 - f(to + so + 2t1 + s)]' V7

and set

Yo = max{ao, a1}. (7)

Next, we present a convergence result for the majorizing sequence {tn}. Lemma 1. Suppose that there exists 7, such that Los < 2 and

Lon

0 <70 ^ Y ^ 1 -

1 - f s'

Then, the sequence {tn} given by (33) is nondecreasing, bounded from

above by t** = 1——^, and converges to its unique least upper bounds t*, which satisfies n < t* < t**.

Proof. We shall show, using induction, that

0 < L(2(tfc+1 - tk) + (sk - 4) < Y (9)

2[1 - L0(3tfc+1 + sfc+1 + s)p ' W

and

0<»r/ w.1 . .<y. (10)

L(2(tfc+1 - tfc) + (sfc - tfc) 2[1 - L0(tk + s; + 2tfc+1 + s)]

Estimates (9) and (10) hold for k = 0 by (3), (5)-(8). Suppose that (9) and (10) hold for j = 1, 2,..., k - 1. Then, by (3), (9) and (10),

. 1 - Y k+1

0 < tk+1 - tk < Y(tfc - tk—1) < Ykn tfc+1 < —-n, (11)

k 1 - Yk k 1 - yk+1 0 < s; - tfc < y(tfc - tfc—1) < y n sfc < --n + Y n = —;-n.

1 - Y

1 ' Ykn = 1

1 - y 1 - Y

(12)

By (9) and (10), we must only complete the induction for (9). Evidently, this is true by (3), (11), and (12), provided that

L Y Ln 1 — Yk+1 1 — Y k+2

-(2y;n + Ykn) + V(3^—n + n + s) - Y < 0. (13)

2 2 1 - y 1 - Y

Estimate (13) suggests to introduce functions on [0,1) as

3L L

^ (t) = 32-tk—1n + y [3(1 + t + ... + tk) + (1 + t + ... + tk+1)]n. (14)

We seek for a relationship between two consecutive functions . We can write, in turn,

^k+1(t) =

ytk n + y (3(1 + t + ... + tk+1) + (1 + t + ... + tk+2))n + y

ytkn + y(3(1 + t + ... + tk+1) + (1 + t + ... + tk+2))n + ys - 1-()

- 3Ltk—1n- y (3(1+1 +... + tk) + (1 +1 +... +1^1)) n- y s + 1 + ^(t) =

tk—1n

^k (t)+ p(t)-^, (15)

where p(t) is given by (4). In particular, we get

№+i(Y) = № (Y) (16) by the definition of y. Therefore, (13) holds, provided that

<MY) ^ 0, (17)

where

<^(Y)=lim (y). (18)

k—yoo

However ;

(y ) = ,

1 - Y 2

2L0n L0

<£~(y) = ï--+ TT s - 1 (19)

by (13). Hence, (17) holds if

1 - y 2

or

2Lon + Ls - 1 ^ 0 (20)

Lon

Y ^ 1 - (21)

1 - "2"s

which is true, by (8). Then sequence {tn} is nondecreasing and, in view of (11), is bounded from above by t**. Hence, it converges to its unique least upper bound that satisfies n ^ t* ^ t**. □

Next, we present the semi-local convergence for the method (2). Theorem 1. Assume the following:

(i) F : Q C Bi —> B2 is a continuous operator with a standard divided difference of order one, such that [■ , ■] : Q x Q —y L(B1,B2) and x0, y0 E Q are such that A0 = [y" ,x0; F] is invertible. Let ||x1 — x0|| ^ n and ||y0 — x0|| ^ s.

(ii) Assumptions of Lemma 1 hold.

(iii) ||A-1([x,y; F] — A0)| ^ L0(||x — || + ||y — x0||) for all x,y E Q and some L0 > 0. Set p = 4(-2 — s) and Q0 = Q if U(x0,p).

(iv) || A-1([x, y; F] — [z, y; F]) || ^ L||x — z|| for all x, y, z E Q0 and some L >0 0.

(v) U(x0,t*) C Q.

Then there exists a limit point x* E U(x0,t*) of the sequence {xn}, such that F(x*) = 0.

Proof. We use mathematical induction to show the estimates

||xn+1 — xn! ^ tn+1 — tn (22)

and

||yn xn! ^ sn tn. (23)

These estimates are true due to the initial conditions and (3) for n = 0. Suppose that the initial conditions and (3) hold for n = 0. Also suppose that they are true for all k = 0,1, 2,... n — 1. Then we have, by (iii):

||A-1(Bk — A0)|| ^ L0(||xk-1 + yk-1 — ^^|| + ||xk — x0|) ^

^ L(|xk-1 — x01| + ||yk-1 — y0| + 2||xk — x01|) ^

^ L(2|xk-1 — x01| + ||yk-1 — xk-11| + ||y0 — x01| + 2|xk — x0|| ^

^ L(2(tk-1 — t0) + Sk-1 — tk-1 + 2(tk — t0) + s) =

= L (tk-1 + Sk-1 + 2tk + s) < 1, (24)

which, together with Banach's Lemma on invertible operators, show that Bk-1 is invertible and

||B--1 A0| ^ —-+1-+2t + ). (25)

1 — L" (tk-1 + sk-1 + 2tk + s)

Similarly,

||A-1(Ak — A0)|| ^ L0(|^^ — ^^|| + ||xk — x01|) ^ ^ y(|xk + yk — (x0 + y0)| + 2|xk — x0||) ^ ^ "20(||xk — x0| + ||yk — y0| + 2|xk — x0||) ^

^ 420(3|xk — x0^ + ^yk — y0|) ^

^ y(3||xk — x0| + ||yk — xk|| + ||xk — x0|| + ||x0 — y0|) ^

< L(4lxk - xo| + ||yk - xkII + Ilxo - yo|) <

2Lo

< -^(4tk + s; - tk + s)

= l2° (3tk + s; + s) < 1,

so

IIA—1 Aol < 1 L; (3t + + ). (26)

1 - L0 (3tk + s; + s) Using the method (2), we get the identity

F(x;) = F(x;) - F(xk—1) - [Xk—1 + yk—1, xk— 1; F](x; - xk—1) (27) so, by (iii) and (27),

II1F(xk)II < L(Ix; - xk—1 + yk—1 II + IIx; - xk—1I) < < LL II 2x; - (x;—1 + yk—1) IIIx; - xk—J < < L(IIxk - x;—1I + 11 x; - y;—1II)IIxk - x;—1I < < L ( 11 x; - x;—1II + 11 x; - x;—11 + IIyk— 1 - xk—J^x; - x;— J <

< L(2(tk - tk—1) + (s;—1 - tk—1))(tk - tk—1). (28) Then, by (3), (25), (26) and (28) we obtain IIy; - x;II = 1Ao][A0—1F(xk)]II < IB—1AoIIA0—1F(x;)II <

< L(2(tk - tk—1) + (sk—1 - tk—1))(tk - tk—1) = t (29)

< 2[1 - L0(tk—1 + sk—1 + 2tk + s)] = sk - k ( )

and

11 x;+1 - x; II = I[A°—1Ao][A—1F (x;)]II < II A—11 Ao III A—11F (x; )I <

L(2(tk - tk—1) + (sk—1 - tk—1 ))(tk - tk—1) _ , , /onx

< -^-L^-,-,-^- = tk+1 - tk (30)

2[1 - "21 (3tk + s; + s)] completing the induction for (22) and (23). We also have

llxfe+i - xoll ^ ||xfc+i - xfcII + ... + ||xi - ^

^ tfc+1 — tfc + ... + ti — to = tfc+1 < t*

and

||yk - Xo| ^ ||yk - xfcII + ||xfc - xoll ^ Sk - tk + tk - to = sfc < t*,

so yk, xk+1 G U(xo,t*). Moreover, the sequence {tk} is fundamental by Lemma 1. Hence, the sequence {xk} is fundamental too and, as such, it converges to some x* G U7(xo,t*). By sending k ^ ^ in (28) and using the continuity of F, we conclude F(x*) = 0. □

Concerning the uniqueness of the solution x*, we have:

Proposition 1. Under the assumptions of Theorem 1, assume further that

Lo(3t1 + t*) < 2 (31)

for some t1 ^ t*. Then, x* is the only solution of the equation F(x) = 0 in the set Q = Q n U(xo,t1).

Proof. Let x1 G Q1 with F(x1) = 0. Set T = [x*,x1; F]. Using (iii) and (31), we get

||A-1(T - Ao)| ^ Lo(|x* - || + ||x1 - xo||) ^

^ Lo(|x* - xo1 + l|x1 - xo" + ||x1 - xo||) ^ t +11

^ Lo(^ + t1) < 1, (32)

so x* = x1 is deduced, since T is invertible and

T(x* - x1) = F(x*) - F(x1) = 0 - 0 = 0. (33)

□

Remark. We can compute the computational order of convergence defined by

a = ]n( lxn+1 - x*,h l\n( |xn - x

\ ry< _ ry< f / \ rY' -1 _ rY'

1l ^ n 1l ll ^ n_1 ^ |

or the approximate computational order of convergence b = ]n / |xn+1 - xn| \ /]n ( |xn - xn-11

V|xn xn— 1|1' ' M|xn— 1 xn—2

This way, we obtain, in practice, the order of convergence in a way that avoids high Frechet derivatives for the operator F and Taylor series used in other studies.

3. Numerical Example.

Let B1 = B2 = R3, Q = U(0,1). Define F on Q by

e — 1

F(x) = F(u1,U2,U3) = (eui - 1,-^- V + «2,«3)T. (34)

For the points u = (u1 , u2,u3)T, the Frechet derivative is given by

eui 0 0 \

0 (e - 1)u2 + 10 I . 0 0 1 J

Using the norm of the maximum of the rows for xo = (10—3, 10—3,10—3)T, y = (10—4,10—4,10—4)T, we get Lo = 0.7(e- 1), L = ep, where p = 0.4118. Then we have s = 0.0156, n = 0.0015,

Yo = 0.0035 < y = 0.6245 < 1--^^ = 0.9985, t** = 0.0015.

1 - I0 s

We have verified all the conditions of Theorem 1. Hence, we conclude that lim xn = x* = (0, 0,0)T.

References

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[2] Argyros I. K. Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics 15 . Editors: Chui C.K. and Wuytack L. Elsevier Publ. Company, New York, 2007.

[3] Argyros I. K., George S. Mathematical modeling for the solution of equations and systems of equations with applications. Volume-IV, Nova Publishes, NY, 2020.

[4] Ezquerro J. A., Hernandez M. A. An improvement of the region of accessibility of Chebyshev's method from Newton's method. Math. Comput., 2009, vol. 78, pp. 1613-1627.

DOI: https://doi.org/10.1090/S0025-5718-09-02193-0

F» =

[5] Ezquerro J. A., Hernandez M. A., Magrenan A. A. Stating points for Newton's method under a center Lipschitz condition for the second derivative. J. Comput. Appl. Math., 2018, vol. 330, pp. 721-731.

DOI: https://doi .org/10.1016/j.cam.2016.12.013

[6] Kantorovich L. V., Akilov G. P. Functional Analysis. Oxford: Pergamon, 1982.

[7] Magrenan A. A., Argyros I. K. Two-step Newton methods. Journal of Complexity, 2014, vol. 30(4), pp. 533-553.

DOI: https://doi .org/10.1016/j.jco.2013.10.002

[8] Potra F. A, Ptak V. Nondiscrete induction and iterative processes, Research notes in Mathematics, 103, Pitman Boston, M.A, 1984.

Received October 8, 2020. In revised form, October 13, 2020. Accepted April 09, 2021. Published online April 19, 2021.

Gus Argyros"

E-mail: Gus.Argyros@cameron.edu Michael Argyros"

E-mail: Michael.Argyros@cameron.edu

Ioannis K. Argyrosv"

E-mail: iargyros@cameron.edu

Santhosh George6

E-mail: sgeorge@nitk.edu.in

a. Cameron University

2800 W. Gore Blvd. Lawton, OK 73505-6377, USA

b. National Institute of Technology Karnataka, India-575 025