- Email: [email protected]
Local convergence of iterative methods for solving equations and system of equations using weight function techniques
Applied Mathematics and Computation 347 (2019) 891–902
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Local convergence of iterative methods for solving equations and system of equations using weight function techniques Ioannis K. Argyros a, Ramandeep Behl b,∗, J.A. Tenreiro Machado c, Ali Saleh Alshomrani b a
Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia c Dept. of Electrical Engineering, ISEP-Institute of Engineering,Polytechnic of Porto, Porto 431, 4294-015, Portugal b
a r t i c l e
i n f o
2010 MSC: 65G99 65H10 47J25 47J05 Keywords: Newton-like method Local convergence Banach space Lipschitz constant Radius of convergence Nonlinear system
a b s t r a c t This paper analyzes the local convergence of several iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space. It is shown that the local convergence of these methods depends of hypotheses requiring the first-order derivative and the Lipschitz condition. The new approach expands the applicability of previous methods and formulates their theoretical radius of convergence. Several numerical examples originated from real world problems illustrate the applicability of the technique in a wide range of nonlinear cases where previous methods can not be used. © 2018 Published by Elsevier Inc.
1. Introduction One of the most fundamental and important problems of numerical analysis involves approximating a locally unique solution x∗ of an equation of the form
F ( x ) = 0,
(1.1)
where F is a Fréchet-differentiable operator defined on a convex subset D of a Banach space X with value in a Banach space Y. Analytical methods for such type of problems are difficult to find in the literature. Therefore, it is only possible to approximate solutions by relying upon numerical iterative methods. The convergence of iterative methods is usually divided into two categories: semi-local and local convergence analysis. The semi-local convergence is based on the information around an initial point to obtain criteria ensuring the convergence of iteration procedures. On the other hand, the local convergence is based on the information around a solution to find estimates of the radii of the convergence balls. An important problem in the study of iterative procedures is their convergence domains and radius of convergence.
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (R. Behl).
https://doi.org/10.1016/j.amc.2018.09.060 0 096-30 03/© 2018 Published by Elsevier Inc.
892
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
Herein, we study the local convergence of the uni-dimensional two step method defined for n = 0, 1, 2, . . . , by the expressions:
yn = xn − α F (xn )−1 F (xn ),
α∈R
(1.2)
xn+1 = yn − h(μ(xn ))F (xn )−1 F (yn ), where h(μ(xn )) is a weight function of real variable μ(xn ) =
a 1 F ( x n )+ a 2 F ( y n ) b 1 F ( x n )+ b 2 F ( y n )
and a1 , a2 , b1 , and b2 are real numbers with
b1 = 0. The fourth order of convergence was shown in [3], using the Taylor series expansion, several hypotheses namely h, but also others, in particular about the regularity of derivatives up to the fourth of the considered function F (although only the first derivative appears in Eq. (1.2)). However, the hypothesis on the derivatives of F restricts the applicability of method (1.2). As a motivational example, we define function F on X = Y = R, D = [− 52 , 12 ] such as:
F (x ) =
x3 ln x2 + x5 − x4 , 0,
x = 0 . x=0
(1.3)
We have that
F (x ) = 3x2 ln x2 + 5x4 − 4x3 + 2x2 , F (x ) = 12x ln x2 + 20x3 − 12x2 + 10x, F (x ) = 12 ln x2 + 60x2 − 12x + 22. We verify that F (x) is unbounded on D and that the results in [3], cannot be applied for studying the convergence of Eq. (1.2). In the literature, we have a large number of iteration functions [1–3,5,7,8,10–16,18,19,21] for finding the approximate solutions of nonlinear equations and we know that for guaranteeing convergence the initial guess must be sufficient close to the required solution. However, nothing is said about the how close the initial guess must be required to guarantee convergence. In other words, it is not provided any information on the radius of the ball convergence for the corresponding method. Moreover, the counter example (1.3) reveals the same type of problem in the earlier studies [1–3,5,7,8,10–16,18,19,21]. Therefore, the local convergence is important, since it shows the degree of difficulty in choosing the initial guesses. In the present study, we expand the applicability of method (1.2) using only hypothesis on the first order derivative of F. Furthermore, we propose a scheme for deriving the radii of convergence and the error bounds based on the Lipschitz constants. We address the range of initial guesses x∗ that gives information about how close it must be to guarantee the convergence of Eq. (1.2), extending, therefore, previous results [3]. We study the local convergence of the extension of the method (1.2) on a Banach space setting, defined for n = 0, 1, 2, . . . , as follows
Yn = Xn − F (Xn )−1 F (Xn ),
Xn+1 = Yn − H (μ ¯ (Xn ))F (Xn )−1 F (Yn ),
(1.4)
where H : L(X, X ) → L(X, X ), μ ¯ : D → L(X, X ), are continuous operators and
−1 μ¯ (xn ) = b1 + b2 − b2 F (xn )−1 [xn , yn ; F ] a1 + a2 − a2 F (xn )−1 [xn , yn ; F ] ,
where [·, · ; F ] : D × D → L(X, Y ) is a divided difference of order one for operator F [1,2]. The method (1.4) was studied in [3] for the special case of X = Rm (m ∈ N). The paper is organized as follows. In Sections 2 and 3, the local convergence analysis of methods (1.2) and (1.4) are presented, respectively. In Section 4, several illustrative numerical examples are discussed. Finally, in Section 5, the main conclusions are outlined. 2. Local convergence: one dimensional case Here, we present the local convergence analysis of method (1.2) for X = Y = R. Let α , a1 , a2 , b1 , b2 ∈ R, L0 > 0, L > 0 and M ≥ 1 with b1 = 0 be given parameters. Let us also assume h : R → R and μ : D → R to be continuous functions such that for x ∈ D, we have
|h(μ(x ))| ≤ h(|μ(x )| )
(2.1)
and h: [0, ∞) → [0, ∞) is nondecreasing. Furthermore, let us suppose that
b2 M |1 − α| < 1 b1 2
(2.2)
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
and
M (|a1 | + |a2 |M|1 − α| )
|b1 | 1 − M2 |1 − α| bb21
1 + Mh
893
M|1 − α| < 1.
(2.3)
It is convenient for the local convergence
analysis in the follow-up to introduce several functions and parameters. Define 1 functions g1 , p and p¯ in the interval 0, L such that we have 0
g1 (t ) =
1 (Lt + 2M|1 − α| ), 2 ( 1 − L0 t )
1 p(t ) = |b1 |
L0 |b1 |t + M|b2 |g1 (t ) , 2
p¯ (t ) = p(t ) − 1 and parameter r1 by the expression
r1 =
2(1 − M|1 − α| ) . 2L0 + L
It is straightforward to verify that the functions g1 and p are (strictly) monotonic increasing. Then, we have from b 2 Eq. (2.3) that g1 (r1 ) = 1 and 0 ≤ g1 (t) < 1, for each t ∈ [0, r1 ). We also obtain that p¯ (0 ) = M b2 |1 − α| − 1 < 0 using Eq. (2.2) and that p¯ (t ) → ∞ as t →
0,
in the interval
1 L0
1− L0 .
1
Then, by the intermediate value theorem, we can say that the function p¯ has zeros
. Let us say that rp is the smallest such zero of p¯ . Moreover, let us define functions g2 and g¯ 2 in the
interval [0, rp ) by the expressions
M M (|a1 | + |a2 |g1 (t )) g2 (t ) = 1 + h 1 − L0 t |b1 |(1 − p(t ))
g1 (t ),
g¯ 2 (t ) = g2 (t ) − 1. Then, it follows from Eq. (2.3) that g¯ 2 (0 ) < 0 and g¯ 2 (t ) → ∞ as t → r − p . Now, we assume that r2 be the smallest zero of function g¯ 2 on the interval (0, rp ). Define
r = min{r1 , r2 }.
(2.4)
Then, we have that for each t ∈ [0, r)
0 ≤ g1 (t ) < 1,
(2.5)
0 ≤ p(t ) < 1,
(2.6)
0 ≤ g2 (t ) < 1.
(2.7)
Denote by U (γ , s ) and U¯ (γ , s ) the open and closed balls in X with center γ ∈ X and of radius s > 0 and assume that X = R. Let us assume that L0 > 0 and define D0 := D ∩ U x∗ , L1 . Notice that D0 ⊆ D. In Lipschitz condition (2.9) x ∈ D, but in 0
Eq. (2.10) and (2.11) x ∈ D0 . That is L0 = L0 (D ), but L = L(D0 , L0 ) and M = M (D0 , L0 ). Condition (2.9) help us to define L and M. If the expressions (2.10) and (2.11) are defined for x ∈ D, then the corresponding L1 and M1 are such that L ≤ L1 and M ≤ M1 , leading to a less precise convergence analysis (see also Remark 3.2(b) and (c)). Next, we present the local convergence analysis of method (1.2) using the preceding notations. Theorem 2.1. Let F : D ⊆ R → R be a differentiable function. Suppose that functions h : R → R and μ : D → R satisfy estimate (2.1); there exist α , a1 , a2 , b1 , b2 ∈ R with b1 = 0 such that together with function h follow Eqs. (2.2) and (2.3); there exist x∗ ∈ D, L0 > 0, L > 0 and M ≥ 1 such that
F ( x∗ ) = 0,
F ( x∗ ) = 0,
|F (x∗ )−1 (F (x ) − F (x∗ )| ≤ L0 |x − x∗ |, for each x ∈ D |F (x∗ )−1 F (x ) − F (y ) | ≤ L|x − y|, for each x, y ∈ D0
(2.8) (2.9) (2.10)
894
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
|F (x∗ )−1 F (x )| ≤ M, for each x ∈ D0 ,
(2.11)
U¯ (x∗ , r ) ⊆ D,
(2.12)
where the radius of convergence r is defined by Eq. (2.4). Then, sequence {xn } generated for x0 ∈ U (x∗ , r ) − {x∗ } by method (1.2) is well defined, remains in U(x∗ , r) for each n = 0, 1, 2, . . . and converges to x∗ . Moreover, the following estimates hold
|yn − x∗ | ≤ g1 (|xn − x∗ | )|xn − x∗ | ≤ |xn − x∗ | < r,
(2.13)
and
|xn+1 − x∗ | ≤ g2 (|xn − x∗ | )|xn − x∗ | < |xn − x∗ |, where the functions g1 and g2 are defined previously. Also, for T ∈ [r, F (x ) = 0 in D1 , where D1 := U¯ (x∗ , T ) ∩ D.
(2.14) 2 L0
), the limit point
x∗
is the only solution of equation
Proof. We show that the estimates (2.13) and (2.14) hold with the help of mathematical induction. Using the hypothesis x0 ∈ U (x∗ , r ) − {x∗ } and expressions (2.4) and (2.9), we obtain
|F (x∗ )−1 (F (x0 ) − F (x∗ ))| ≤ L0 |x0 − x∗ | < L0 r < 1.
(2.15)
It follows from Eq. (2.15) and the Banach Lemma on invertible operators [1,18] that F (x0 ) = 0 and
|F (x0 )−1 F (x∗ )| ≤
1 . 1 − L0 |x0 − x∗ |
(2.16)
By the mean value theorem (see [13, page 64]) and Eq. (2.8), we can write
F ( x0 ) = F ( x0 ) − F ( x∗ ) =
0
1
F (x∗ + θ (x0 − x∗ ))(x0 − x∗ )dθ .
(2.17)
Notice that |x∗ + θ (x0 − x∗ ) − x∗ | = θ |x0 − x∗ | < r. Hence, x∗ + θ (x0 − x∗ ) ∈ U (x∗ , r ). Using Eqs. (2.11) and (2.17), we have
1 |F (x∗ )−1 F (x0 )| = F (x∗ )−1 F (x∗ + θ (x0 − x∗ ))(x0 − x∗ )dθ ≤ M|x0 − x∗ |.
(2.18)
0
Inserting the expressions (2.4), (2.5), (2.10), (2.16) and (2.18) in the first substep of method (1.2) for n = 0, we obtain in turn that
|y0 − x∗ | = | x0 − x∗ − F (x0 )−1 F (x0 ) + (1 − α )F (x0 )−1 F (x0 )| 1 ∗ −1 ∗ ∗ −1 ∗ ∗ ≤ |F ( x0 ) F ( x )| F (x ) F (x + θ (x0 − x )) − F (x0 ) (x0 − x )dθ 0 +|1 − α||F (x0 )−1 F (x∗ )||F (x∗ )−1 F (x0 )| L|x0 − x∗ |2 M|1 − α||x0 − x∗ | + ∗ 2 ( 1 − L0 |x0 − x | ) 1 − L0 |x0 − x∗ | ∗ ∗ = g1 (|x0 − x | )|x0 − x | < g1 (r )|x0 − x∗ | < |x0 − x∗ | < r, ≤
(2.19)
∈ U(x ∗
which leads to Eq. (2.13) for n = 0 and y0 r). We also prove that b1 F (x0 ) + b2 F (y0 ) = 0 for b1 = 0 and x0 = x∗ . Using Eqs. (2.4), (2.6), (2.8), (2.9), (2.18) and (2.19), we have
−1 b1 F (x∗ )(x0 − x∗ ) b1 F (x0 ) − F (x∗ ) − F (x∗ )(x0 − x∗ ) + b2 F (y0 )
1 L |b | ≤ |x0 − x∗ |−1 0 1 |x0 − x∗ |2 + M|b2 ||y0 − x∗ | |b1 | 2
1 L |b | ≤ |x0 − x∗ |−1 0 1 |x0 − x∗ |2 + M|b2 |g1 (|x0 − x∗ | )|x0 − x∗ | |b1 | 2 ∗ = p( | x 0 − x | ) < p( r ) < 1 .
Hence, it results
(2.20)
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
(b1 F (x0 ) + b2 F (y0 ) )−1 F (x∗ ) ≤
1
|b1 ||x0 − x∗ |(1 − p(|x0 − x∗ | ) )
.
895
(2.21)
It is also needed an estimation of |h(μ(x0 ))|. We can obtain an estimate of |h(μ(x0 ))| by means of expressions (2.18), (2.19), and (2.21), resulting
M ( |a1 | + |a2 |g1 |x0 − x∗ | ) |x0 − x∗ | |h(μ(x0 ))| ≤ h . |b1 ||x0 − x∗ |(1 − p(|x0 − x∗ | ) )
(2.22)
Now, using the second sub step of method (1.2) for n = 0 and with the help of Eqs. (2.7), (2.16), (2.18) and (2.22), it yields
|x1 − x∗ | ≤ |y0 − x∗ | + |h(μ(x0 ))||F (x0 )−1 F (x∗ )||F (x∗ )−1 F (y0 )|
M M (|a1 | + |a2 |g1 (|x0 − x∗ | )) |y0 − x∗ | ≤ 1+ h 1 − L|x0 − x∗ | | b 1 | ( 1 − p( | x 0 − x ∗ | ) ) ≤ g2 (|x0 − x∗ | )|x0 − x∗ | ≤ |x0 − x∗ | < r,
(2.23)
∈ U(x∗ , r).
which shows Eq. (2.14) for n = 0 and x1 Replacing x0 , y0 , z0 and x1 , by xk , yk , zk and xk+1 , respectively, in the preceding estimates we arrive at Eqs. (2.13)–(2.14). Using the estimate |xk+1 − x∗ | ≤ c|xk − x∗ | < r, c = g2 (|x0 − x∗ | ) ∈ [0, 1 ), we deduce that lim xk = x∗ and xk+1 ∈ U (x∗ , r ). k→∞ 1 Finally, to show the uniqueness part, let Q = 0 F (y∗ + θ (x∗ − y∗ ))dθ for some y∗ ∈ U¯ (x∗ , T ) with F (y∗ ) = 0. Using Eq. (2.9), we get that
|F (x∗ )−1 (Q − F (x∗ ))| ≤ | ≤
1
∗ ∗ 0 ( 1 − θ )|y − x |d θ ≤
1 0
L0 |y∗ + θ ( x∗ − y∗ ) − x∗ |d θ (2.24)
L0 T < 1. 2
It follows from Eq. (2.24) that Q is invertible. Then, in view of the identity 0 = F (x∗ ) − F (y∗ ) = Q (x∗ − y∗ ), we conclude that x∗ = y∗ . 3. Local convergence: multi-dimensional case In this section, we present the local convergence of method (1.4). Let a1 , a2 , b1 , b2 ∈ R, L0 > 0, L > 0, M ≥ 1 and N ≥ 0, with b1 + b2 = 0 be given parameters. In addition, we also assume that the operators H and μ ¯ given in the introduction satisfy the following estimate
H (μ¯ (X )) ≤ H0 ( (μ¯ (X ) ), for each X ∈ D,
(3.1)
for some function H0 : [0, ∞) → [0, ∞) which is nondecreasing and continuous. Suppose that
b2 b1 + b2 N < 1.
(3.2)
Now, we define the functions g1 , q, and q¯ in the interval 0,
g1 (t ) =
1 L0
by
Lt , 2 ( 1 − L0 t )
b2 N q(t ) = , b1 + b2 1 − L0 t q¯ (t ) = q(t ) − 1 and parameter rA as
rA =
2 . 2L0 + L
1 ¯ L0 , g1 (rA ) = 1 and 0 ≤ g1 (t) < 1 for each t ∈ [0, rA ). Moreover, from Eq. (3.2) we get that q (0 ) = − b2 b1 +b2 N − 1 < 0 and q¯ (t ) → ∞ as t → 1L0 . Denote by rq the smallest zero of function q¯ in the interval 0, L10 . Further-
We have that 0 < rA <
more, define functions g2 and g¯ 2 on the interval (0, rq ) as follows:
g2 (t ) =
M 1+ H0 1 − L0 t
|a1 + a2 | + 1|a−L2 |0Nt |b1 + b2 |(1 − q(t ))
g1 (t )
896
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
and
g¯ 2 (t ) = g2 (t ) − 1. We have that g¯ 2 (0 ) = −1 < 0. Notice that if rA < rq ⇒ g¯ 2 (rA ) > 0, since g1 (rA ) = 1. (C) if rq ≤ rA suppose that g¯ 2 (t ) → a positive number or infinity as t → r¯q− . In either case denote by r2 the smallest zero of function g¯ 2 in (0, rA ) (if rA < rq ) or (0, rq ) (if condition (C) holds and rq ≤ rA ). Finally, if we define the radius of convergence r by
r = min{rA , r2 }.
(3.3)
Then, we have that
0 < r ≤ rA
(3.4)
and for each t ∈ [0, r)
0 ≤ g1 (t ) < 1
(3.5)
0 ≤ q(t ) < 1,
(3.6)
0 ≤ g2 (t ) < 1.
Set D0 := D ∩ U X ∗ ,
1 L0
(3.7)
and L0 > 0.
Theorem 3.1. Let F : D ⊆ X → Y be a Fréchet-differentiable operator. In addition [·, · ; F ] : D × D → L(X, Y ) is a divided difference of order one, H, H0 and μ ¯ are as previously and satisfying Eq. (3.1). Furthermore, suppose that there exist X ∗ ∈ D, a1 , a2 , b1 , b2 ∈ R, L0 > 0, L > 0, M ≥ 1 and N ≥ 0 with b1 + b2 = 0 such that
F ( X ∗ ) = 0,
F (X ∗ )−1 ∈ L(Y, X ),
(3.8)
F (X ∗ )−1 (F (X ) − F (X0 ) ≤ L0 X − X0 , for each X ∈ D,
(3.9)
F (X ∗ )−1 F (X ) − F (Y ) ≤ L X − Y , for each X, Y ∈ D0 ,
(3.10)
F (X ∗ )−1 F (X ) ≤ M, for each X ∈ D0 ,
(3.11)
F (X ∗ )−1 [X, Y ; F ] ≤ N for each X, Y ∈ D0 ,
(3.12)
U¯ (X ∗ , r ) ⊆ D,
(3.13)
expression (3.2) and condition (C) hold, if rq ≤ rA , where the radius of convergence r is defined by Eq. (3.3). The sequence {Xn } generated for X0 ∈ U (X ∗ , r ) − {X ∗ } by method (1.4) is well defined, remains in U(X∗ , r) for each n = 0, 1, 2, . . . and converges to X∗ . Moreover, the following estimates hold
Yn − X ∗ ≤ g1 ( Xn − X ∗ ) Xn − X ∗ ≤ Xn − X ∗ < r,
(3.14)
and
Xn+1 − X ∗ ≤ g2 ( Xn − X ∗ ) Xn − X ∗ < Xn − X ∗ , where the functions g1 and g2 were defined previously. Furthermore, for T ∈ [r, equation F (X ) = 0 in D1 , where D1 = U (X ∗ , T ) ∩ D.
(3.15) 2 L0
), the limit point
X∗
is the only solution of
Proof. As in the proof of Theorem 2.1, we arrive at the estimates
F (X0 )−1 F (X ∗ ) ≤
1 , 1 − L0 X0 − X ∗
F (X0 )−1 F (X0 ) ≤ M X0 − X ∗ ,
(3.16) (3.17)
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
and also estimate Eq. (3.14) (for α = 1). Moreover, we need to show that linear operator I − invertible. We have that
b2 F (X0 )−1 [X0 , b1 +b2
b2 −1 b1 + b2 F (X0 ) [X0 , Y0 ; F ] b2 F (X0 )−1 F (X ∗ ) F (X ∗ )−1 [X0 , Y0 ; F ] ≤ b1 + b2 b2 N ≤ = q( X0 − X ∗ ) < 1. b1 + b2 1 − L0 X0 − X ∗
897
Y0 ; F ] is
(3.18)
Hence, we deduce that
−1 b2 1 −1 F (X0 ) [X0 , Y0 ; F ] . I− ≤ ∗ b + b 1 − q ( X 1 2 0 − X ))
(3.19)
Notice that
−1 b2 1 −1 μ(X0 ) ≤ I− F (X0 ) [X0 , Y0 ; F ] a1 + a2 − a2 F (X0 )−1 [X0 , Y0 ; F ] b1 + b2 b1 + b2 ≤
|a1 + a2 | + 1−L0| a2X|0N−X ∗ . |b1 + b2 |(1 − q( X0 − X ∗ ))
(3.20)
Then, by using the last sub step of the method (1.4) for n = 0, expressions (3.1), (3.3), (3.7), (3.14), (3.16), (3.17) (for Y0 = X0 ) and (3.20), we have
X1 − X ∗ ≤ Y0 − X ∗ + H (μ¯ (X0 )) F (X0 )−1 F (X ∗ ) F (X ∗ )−1 F (Y0 ) N |a1 + a2 | + 1−L| aX20|−X ∗ M ≤ 1+ H0 Y0 − X ∗ 1 − L X0 − X ∗ |b1 + b2 |(1 − q( X0 − X ∗ )) ≤ g2 ( X0 − X ∗ ) X0 − X ∗ ≤ c X0 − X ∗ < r,
(3.21) − X ∗|
which shows Eq. (3.15) for n = 0 and X1 Using the estimate Xk+1 ≤ c Xk < r, c = g2 ( X0 − X ∗ ) ∈ ∗ ∗ [0, 1 ), we deduce that lim Xk = X and Xk+1 ∈ U (X , r ). k→∞ 1 Finally, to show the uniqueness part, let Q = 0 F (Y ∗ + θ (X ∗ − Y ∗ ))dθ for some Y ∗ ∈ U¯ (X ∗ , T ) with F (Y ∗ ) = 0. Using Eq. (3.9), we get that ∈ U(X∗ , r).
− X ∗
1 ∗ −1 ∗ ∗ ∗ ∗ L Y + θ ( X − Y ) − X d θ ( ) F ( X ) Q − F ( X ∗ ) ≤ 0 0
≤
1 0
(1 − θ ) Y ∗ − X ∗ dθ ≤
L0 T < 1. 2
(3.22)
It follows from Eq. (3.22) that Q is invertible. Then, in view of the identity 0 = F (X ∗ ) − F (Y ∗ ) = Q (X ∗ − Y ∗ ), we conclude that X ∗ = Y ∗ . Remark 3.2. (a) In view of Eq. (2.9) (or (3.10)) and the estimate
F (X ∗ )−1 F (X ) = F (X ∗ )−1 (F (X ) − F (X ∗ )) + I ≤ 1 + F (X ∗ )−1 (F (X ) − F (X ∗ )) ≤ 1 + L0 X0 − X ∗ condition (2.11) (or (3.11)) can be dropped and M can be replaced by
M (t ) = 1 + L0 t or by M (t ) = M = 2, since t ∈ [0, L1 ). 0 (b) The results obtained here can be used for operators F satisfying the autonomous differential equation [1,2] of the form
F (x ) = P (F (x )), where P is a known continuous operator. Since F (x∗ ) = P (F (x∗ )) = P (0 ), we can apply the results without actually knowing the solution x∗ . For example, with F (x ) = ex − 1, we can choose P (x ) = x + 1.
898
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902 Table 1 Different radii of convergence for multi-factor effect problem. Case
Values of parameters that satisfy Theorem 2.1 a1
a2
b1
b2
1 2
0 1
2 0
1 2
0 −1
1
h(μ) 6
6−3μ−μ
2
2μ − 1
rA
r2
r
x0
n
ρ
0.405594 0.405594
0.10784 0.091777
0.10784 0.091777
−0.399 −0.219
5 5
4.0 0 0 0 4.0 0 0 0
Table 2 Different radii of convergence for Example (4.2). Case
Values of parameters that satisfy Theorem 2.1 a1
1 2
0 1
a2
b1
2 0
1 2
1
b2 0 −1
h(μ) 6
6−3μ−μ2
2μ − 1
rA
r2
r
x0
n
ρ
0.66667 0.66667
0.27145 0.31615
0.27145 0.31615
0.26 0.30
4 4
5.0 0 0 0 5.0 0 0 0
(c) The value rA = 2L 2+L was shown in [1,2] to be the convergence radius for the Newton’s method under conditions (2.9) and 0 (2.10) (or (3.9) and (3.10)). It follows from Eq. (2.4) (or (3.3)) and the definition of rA that the convergence radius r of the method (1.2) (or method (1.4)) cannot be larger than the convergence radius rA of the second order Newton’s method. The radius rA is at least large as the convergence ball given by Rheinboldt [18]
rT R =
2 , 3L1
where L1 is given above Theorem 2.1. In particular, for L0 < L1 we have that
rT R < rA and
rT R 1 → rA 3
L0 → 0. L1
as
That is, our convergence ball rA is at most three times larger than the one derived by Rheinboldt. The same value for rt R 1
given by Traub [19]. In the case of the example in (b), we have L0 = e − 1, L = e e−1 and L1 = e, so that L0 < L < L1 and rT R = 2.24 < rA = 2.38. (d) It is worth noticing that method (1.4) does not change if we use the conditions of Theorems 2.1 and 3.1 instead of the stronger conditions given in [3]. We adopt the following formulas for calculating the computational order of convergence (COC) [8] −X ln XXn+1 ∗ n −X ∗
ρ=
ln XXn −X −X ∗ ∗
,
for each n = 1, 2, . . .
(3.23)
n−1
or the approximate computational order of convergence (ACOC)
ρ∗ =
−Xn ln XXnn+1 −X n−1
n−1 ln XXn −X −X n−1
,
for each n = 2, 3, . . .
(3.24)
n−2
With this method, we obtain the order of convergence that avoids the bounds involving estimates higher than the first Fréchet derivative. (e) Methods (1.2) and (1.4) generate the single and two step methods introduced in [3–5,7,8,10–13,16–19,21], because they can be reduced to those methods by specializing α , h, μ, H and μ ¯ . As an example, choose α = 1, μ(x ) = 1 and h(x ) = 1 in method (1.2) to obtain two step Newton-methods considered in [17] (similar choices for the Banach space case). Other choices are also possible [1,2]. Notice that in the earlier studies (as already mentioned in the introduction) the convergence criteria are stronger than those presented here, limiting the applicability of such methods.
4. Numerical examples In this section, we illustrate numerically the theoretical results developed previously. We consider four examples for nonlinear scalar equations: the first one is a real world problem; the second and third are standard academic problems; the fourth one is a motivational problem, described in Eqs. (4.1)–(4.4). We also choose some of the weight functions mentioned in Tables 1, 2, 3, 4 to solve these equations. Furthermore, in the tables we present the initial guesses of the considered problem, the radius of convergence and the minimum number of iterations required to get the desired in zeros of F(x) and
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
899
Table 3 Different radii of convergence for Example (4.3). Case
Values of parameters which satisfy Theorem 2.1 a1
a2
b1
b2
1 2
0 1
2 0
1 2
0 −1
1
h(μ) 6
6−3μ−μ
2
2μ − 1
rA
r2
r
x0
n
ρ
0.382692 0.382692
0.10809 0.098237
0.10809 0.098237
0.09 −0.08
5 5
4.0 0 0 0 4.0 0 0 0
Table 4 Different radii of convergence for Example (4.4). Case
Values of parameters which satisfy Theorem 2.1 a1
1 2
0 1
a2
b1
2 0
1 2
h(μ)
b2
6
0 −1
1
6−3μ−μ2
2μ − 1
rA
r2
r
x0
n
ρ
0.0068968 0.0068968
0.0018337 0.0015606
0.0018337 0.0015606
1.001 0.9986
4 4
4.0 0 0 0 4.0 0 0 0
Table 5 Different radii of convergence for Van der Pol equation. Case
Values of parameters that satisfy Theorem 3.1 a1
a2
b1
b2
rA
r2
r
x0
n
ρ
1 2
1
0 0
2 2
1
0.026482 0.026482
0.10872 0.11917
0.026482 0.026482
(0.8,1.48,2.15,3.09,7.79) (0.85, 1.54, 2.44, 3.14, 7.84 )
4 5
4.0077 4.0026
1 2
1 2
Table 6 The values of ψ i and φ i (in radians) for Example (4.6). i
ψi
φi
0 1 2 3
1.3954170041747090114 1.7444828545735749268 2.0656234369405315689 2.460 067847891250 0533
1.7461756494150842271 2.0364691127919609051 2.2390977868265978920 2.4600678409809344550
Table 7 Different radii of convergence for Kinematic synthesis problem. Case
Values of parameters that satisfy Theorem 3.1 a1
a2
b1
b2
rA
r2
r
x0
n
ρ
1 2
1
0 0
2 2
1
0.034861 0.034861
0.14311 0.15687
0.034861 0.034861
(0.872,0.665,0.618) (0.938,0.731,0.684)
5 5
4.0070 4.0047
1 2
1 2
Table 8 Different radii of convergence. Case
1 2
Different values of parameters that satisfy Theorem 3.1 a1
a2
b1
b2
rA
r2
r
x0
n
ρ
1
0 0
2 2
1
0.38798 0.38798
1.5054 1.5054
0.38798 1.6429
(0.37, 0.36, 0.36) (0.27, 0.26, 0.26)
5 5
4.0 0 01 4.075
1 2
1 2
we verify the theoretical order of convergence. Therefore, we calculate the COC given by Eq. (3.23) or the ACOC given by formula (3.24). The computations are performed by means of Mathematica 9 with multiple precision arithmetic. We adopt = 10−500 as a tolerance error and the following stopping criteria are used: (i ) |xn+1 − xn | < and (ii ) |F (xn+1 )| < . In what regards nonlinear systems, we also assume three problems. Examples 4.5 and 4.6 are applied science exercises and Example 4.7 is a standard academic problem. We choose some weight functions in these examples. In Tables 5, 7, 8, we list the initial guesses, radius of convergence and minimum number of iterations required to get the desire accuracy to the corresponding zeros of the functions. Furthermore, we calculate the COC or ACOC by means of the formulas (3.23) or (3.24) to verify the theoretical order of convergence of the nonlinear system. We adopt = 10−100 as a tolerance error for the nonlinear system, and the following stopping criterion is used: (i ) X (n+1) − X (n) < and (ii ) F (X (n+1) ) < . 1 Moreover, we consider [X, Y ; F ] = F Y + θ (X − Y ) dθ in all examples. 0
900
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
Example 4.1. In the study addressing the multi-factor effect [9], the trajectory of an electron in the air gap between two parallel plates is given by
x(t ) = x0 +
v0 + e
E0 E0 sin(ωt0 + α ) (t − t0 ) + e cos(ωt + α ) + sin(ω + α ) , mω mω2
(4.1)
where t is time, e and m denote the charge and the mass of the electron at rest, x0 and v0 represent the position and velocity of the electron at time t0 and E0 sin(ωt + α ) is the RF electric field between the plates. We choose the particular parameters in expression (4.1) in order to obtain a simpler expression, as follows:
f (x ) = x −
1 π cos(x ) + . 2 4
(4.2)
The required zero of the above function (4.2) is x∗ = −0.309093271541794952741986808924. We refer the reader to Hueso et al. [9] for the expression (4.2), where the computational details can be found. We get L0 = L = 1.643681148, M = 2 and we choose α = 1. We obtain different radii of convergence by using the parameters values shown in Table 1. Example 4.2. Let S = R, D = [−1, 1], x∗ = 0 and define function F on D by
F (x ) = sin(x ).
(4.3)
Then, we get that = 0, L0 = L = M = 1 and α = 1. We obtain distinct radius of convergence by using different values of parameters listed in the Table 2. x∗
Example 4.3. Let S = R, D = [−1, 1], x∗ = 0 and define function F on D by
F ( x ) = ex − 1.
(4.4)
Then, we get L0 = e − 1, L = M = e values exhibited in Table 3.
1 e−1
and consider α = 1. We obtain different radii of convergence by using the parameters
Example 4.4. Returning back to the motivation example at the introduction of this paper, we have x∗ = 1, L = L0 = 96.662907, M = 2 and we select α = 1. We obtain distinct radii of convergence by using the parameters depicted in Table 4. Example 4.5. Let us consider the Van der Pol equation [6], that is defined as follows:
y − μ(y2 − 1 )y + y = 0,
μ > 0,
(4.5)
and models the flow of current in a vacuum tube, with the boundary conditions y(0 ) = 0, y(2 ) = 1. We consider the partition of the given interval [0, 2], that is given by
x0 = 0 < x1 < x2 < x3 < · · · < xn , where xi = x0 + ih, h =
2 , i = 1, 2, . . . , n. n
Moreover, we assume that
y0 = y(x0 ) = 0, y1 = y(x1 ), . . . , yn−1 = y(xn−1 ), yn = y(xn ) = 1. If we discretize the above problem (4.5) by using the second order divided difference for the first and second derivatives, that are given by
yk =
yk+1 − yk−1 y − 2yk + yk+1 , yk = k−1 , k = 1, 2, . . . , n − 1, 2h h2
then, we obtain a system of nonlinear equations of order (n − 1 ) × (n − 1 )
2h2 xk − hμ x2k − 1 (xk+1 − xk−1 ) + 2(xk−1 + xk+1 − 2xk ) = 0, k = 1, 2, . . . , n − 1. In this problem, let us consider the value n = 6 that leads to a 5 × 5 system of nonlinear equations. The solution of this problem is
X ∗ = (0.8243131 . . . , 1.516531 . . . , 2.187958 . . . , 3.123402 . . . , 7.824242 . . . )T . Then, we get L0 = L = 25.1739325, M = N = 2, H (μ ¯ (xn )) = I +
2b21 a2 b1 −a1 b2
μ¯ (xn ) −
a1 I b1
, where μ ¯ (xn ) was defined earlier, and
we consider α = 1. By these using values, we obtain different radii of convergence in Table 5. Example 4.6. Consider the kinematic synthesis for the steering problem described in [4,20], written as system of nonlinear equations, given by
[Ei (x2 sin (ψi ) − x3 ) − Fi (x2 sin (φi ) − x3 )] + [Fi (x2 cos (φi ) + 1 ) − Fi (x2 cos (ψi ) − 1 )] 2
2
− [x1 (x2 sin (ψi ) − x3 )(x2 cos (φi ) + 1 ) − x1 (x2 cos (ψi ) − x3 )(x2 sin (φi ) − x3 )] = 0, for i = 1, 2, 3, 2
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
901
where
Ei = −x3 x2 (sin (φi ) − sin (φ0 ) ) − x1 (x2 sin (φi ) − x3 ) + x2 (cos (φi ) − cos (φ0 ) ), i = 1, 2, 3 and
Fi = −x3 x2 sin (ψi ) + (−x2 ) cos (ψi ) + (x3 − x1 )x2 sin (ψ0 ) + x2 cos (ψ0 ) + x1 x3 , i = 1, 2, 3. We display the values of ψ i and φ i (in radians) in Table 6. We consider the initial guess x(0 ) = (0.7, 0.7, 0.7 ). We obtain the following approximated solution
X ∗ = (0.9051567 . . . , 0.6977417 . . . , 0.6508335 . . . )T . Then, we get L0 = L = 19.1237412, M = N = 2, H (μ ¯ (xn )) = I +
2b21 a2 b1 −a1 b2
μ¯ (xn ) −
a1 I b1
, where μ ¯ (xn ) was defined earlier
and we take α = 1. By using these values, we obtain different radii of convergence as shown in Table 7. Example 4.7. Let X = Y = R3 , D = U¯ (0, 1 ). Define F on D for v = (x, y, z )T by means of
F ( v ) = ex − 1,
e−1 2 y + y, z 2
T
.
(4.6)
Then, the Fréchet-derivative is given by
F (v ) =
ex 0 0
0 ( e − 1 )y + 1 0
0 0 . 1 1
Notice that X ∗ = (0, 0, 0 ), F (x∗ ) = F (x∗ )−1 = diag{1, 1, 1}, L0 = e − 1, L = M = N = e e−1 ,
H (μ ¯ (xn )) = I +
2b21 a2 b1 − a1 b2
μ¯ (xn ) −
a1 I , b1
where μ ¯ (xn ) was defined earlier and we take α = 1. Hence, we obtain different radii of convergence by using the parameters in Table 8. 5. Conclusion The aim of this paper was to generate high convergence order methods to solve systems of equations by extending unidimensional methods. In particular, we extended methods introduced in [3] where Taylor expansions and derivatives up to fifth order were used to determine the fourth order of convergence. The requirement for these derivatives limit the applicability of the method. Moreover, no computable radii convergence or error bounds were given. To address these problems we introduced the multidimensional analog of the method proposed in [3] and we demonstrated its convergence under hypothesis on the first derivative only. Moreover, we found computable radii of convergence as well as error bounds using Lipschitz-type conditions. The convergence order is determined using the COC or ACOC indices that do not require derivatives with order higher than one. This way we expand the applicability of these methods in the multidimensional case and avoid high order derivatives. We presented seven numerical examples for solving systems of equations appearing in the electricity, kinematics and in the solution of the Van der Pol boundary value problems that governs the flow of current in a vacuum tube. In all cases the proposed method revealed good performance. References [1] I.K. Argyros, Convergence and Application of Newton-type Iterations, Springer, 2008. [2] I.K. Argyros, S. Hilout, Numerical methods in nonlinear analysis, World Scientific Publishing Co., New Jersey, 2013. [3] S. Artidiello, A. Cordero, J.R. Torregrosa, M.P. Vassileva, Multidimensional generalization of iterative methods for solving nonlinear problems by means of weigh-function procedure, Appl. Math. Comput. 268 (2015) 1064–1071. [4] F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numer. Algorithm 54 (2010) 395–409. [5] R. Behl, S.S. Motsa, Geometric construction of eighth-order optimal families of Ostrowski’s method, Wor. Sci. J. 2015 (2015) 11. Article ID 614612. [6] R.L. Burden, J.D. Faires, Numerical Analysis, PWS Publishing Company, Boston, 2001. [7] A. Cordero, J. R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas, Appl. Math. Comput. 190 (1) (2007) 686–698. [8] J.A. Ezquerro, M.A. Hernández, New iterations of R-order four with reduced computational cost, BIT Numer. Math. 49 (2009) 325–342. [9] J.L. Hueso, E. Martínez, C. Teruel, Determination of multiple roots of nonlinear equations and applications, J. Math. Chem. 53 (2015) 880–892. [10] V. Kanwar, R. Behl, K.K. Sharma, Simply constructed family of a Ostrowski’s method with optimal order of convergence, Comput. Math. Appl. 62 (11) (2011) 4021–4027. [11] Á. A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput. 233 (2014a) 29–38. [12] A.A. Magreñán, A new tool to study real dynamics: the convergence plane, Appl. Math. Comput. 248 (2014b) 215–224. [13] J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New-York, 1970. [14] A.M. Ostrowski, Solution of equations and systems of equations, in: Pure and Applied Mathematics, Vol. I, X. Academic Press, New York-London, 1960. [15] A.M. Ostrowski, Solution of equations in euclidean and Banach spaces, in: Pure and Applied Mathematics, Vol. 9, Academic Press, New York-London, 1973. [16] M.S. Petkovic, B. Neta, L. Petkovic, J. Džunicˇ , Multipoint methods for solving nonlinear equations, Elsevier, 2013. [17] F.A. Potra, V. Pták, On a class of modified Newton processes, Numer. Funct. Anal. Optim. 2 (1980) 107–120.
902
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
[18] W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, 3, Polish Academy of Science, Banach Center Publication, 1978, pp. 129–142. [19] J.F. Traub, Iterative methods for the solution of equations, in: Prentice Hall Series in Automatic Computation, 1964. Englewood Cliffs, N.J. [20] I.G. Tsoulos, A. Stavrakoudis, On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods, Nonlinear Anal. Real World Appl. 11 (2010) 2465–2471. [21] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third order convergence, Appl. Math. Lett. 13 (20 0 0) 87–93.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Local convergence of iterative methods for solving equations and system of equations using weight function techniques Ioannis K. Argyros a, Ramandeep Behl b,∗, J.A. Tenreiro Machado c, Ali Saleh Alshomrani b a
Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia c Dept. of Electrical Engineering, ISEP-Institute of Engineering,Polytechnic of Porto, Porto 431, 4294-015, Portugal b
a r t i c l e
i n f o
2010 MSC: 65G99 65H10 47J25 47J05 Keywords: Newton-like method Local convergence Banach space Lipschitz constant Radius of convergence Nonlinear system
a b s t r a c t This paper analyzes the local convergence of several iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space. It is shown that the local convergence of these methods depends of hypotheses requiring the first-order derivative and the Lipschitz condition. The new approach expands the applicability of previous methods and formulates their theoretical radius of convergence. Several numerical examples originated from real world problems illustrate the applicability of the technique in a wide range of nonlinear cases where previous methods can not be used. © 2018 Published by Elsevier Inc.
1. Introduction One of the most fundamental and important problems of numerical analysis involves approximating a locally unique solution x∗ of an equation of the form
F ( x ) = 0,
(1.1)
where F is a Fréchet-differentiable operator defined on a convex subset D of a Banach space X with value in a Banach space Y. Analytical methods for such type of problems are difficult to find in the literature. Therefore, it is only possible to approximate solutions by relying upon numerical iterative methods. The convergence of iterative methods is usually divided into two categories: semi-local and local convergence analysis. The semi-local convergence is based on the information around an initial point to obtain criteria ensuring the convergence of iteration procedures. On the other hand, the local convergence is based on the information around a solution to find estimates of the radii of the convergence balls. An important problem in the study of iterative procedures is their convergence domains and radius of convergence.
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (R. Behl).
https://doi.org/10.1016/j.amc.2018.09.060 0 096-30 03/© 2018 Published by Elsevier Inc.
892
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
Herein, we study the local convergence of the uni-dimensional two step method defined for n = 0, 1, 2, . . . , by the expressions:
yn = xn − α F (xn )−1 F (xn ),
α∈R
(1.2)
xn+1 = yn − h(μ(xn ))F (xn )−1 F (yn ), where h(μ(xn )) is a weight function of real variable μ(xn ) =
a 1 F ( x n )+ a 2 F ( y n ) b 1 F ( x n )+ b 2 F ( y n )
and a1 , a2 , b1 , and b2 are real numbers with
b1 = 0. The fourth order of convergence was shown in [3], using the Taylor series expansion, several hypotheses namely h, but also others, in particular about the regularity of derivatives up to the fourth of the considered function F (although only the first derivative appears in Eq. (1.2)). However, the hypothesis on the derivatives of F restricts the applicability of method (1.2). As a motivational example, we define function F on X = Y = R, D = [− 52 , 12 ] such as:
F (x ) =
x3 ln x2 + x5 − x4 , 0,
x = 0 . x=0
(1.3)
We have that
F (x ) = 3x2 ln x2 + 5x4 − 4x3 + 2x2 , F (x ) = 12x ln x2 + 20x3 − 12x2 + 10x, F (x ) = 12 ln x2 + 60x2 − 12x + 22. We verify that F (x) is unbounded on D and that the results in [3], cannot be applied for studying the convergence of Eq. (1.2). In the literature, we have a large number of iteration functions [1–3,5,7,8,10–16,18,19,21] for finding the approximate solutions of nonlinear equations and we know that for guaranteeing convergence the initial guess must be sufficient close to the required solution. However, nothing is said about the how close the initial guess must be required to guarantee convergence. In other words, it is not provided any information on the radius of the ball convergence for the corresponding method. Moreover, the counter example (1.3) reveals the same type of problem in the earlier studies [1–3,5,7,8,10–16,18,19,21]. Therefore, the local convergence is important, since it shows the degree of difficulty in choosing the initial guesses. In the present study, we expand the applicability of method (1.2) using only hypothesis on the first order derivative of F. Furthermore, we propose a scheme for deriving the radii of convergence and the error bounds based on the Lipschitz constants. We address the range of initial guesses x∗ that gives information about how close it must be to guarantee the convergence of Eq. (1.2), extending, therefore, previous results [3]. We study the local convergence of the extension of the method (1.2) on a Banach space setting, defined for n = 0, 1, 2, . . . , as follows
Yn = Xn − F (Xn )−1 F (Xn ),
Xn+1 = Yn − H (μ ¯ (Xn ))F (Xn )−1 F (Yn ),
(1.4)
where H : L(X, X ) → L(X, X ), μ ¯ : D → L(X, X ), are continuous operators and
−1 μ¯ (xn ) = b1 + b2 − b2 F (xn )−1 [xn , yn ; F ] a1 + a2 − a2 F (xn )−1 [xn , yn ; F ] ,
where [·, · ; F ] : D × D → L(X, Y ) is a divided difference of order one for operator F [1,2]. The method (1.4) was studied in [3] for the special case of X = Rm (m ∈ N). The paper is organized as follows. In Sections 2 and 3, the local convergence analysis of methods (1.2) and (1.4) are presented, respectively. In Section 4, several illustrative numerical examples are discussed. Finally, in Section 5, the main conclusions are outlined. 2. Local convergence: one dimensional case Here, we present the local convergence analysis of method (1.2) for X = Y = R. Let α , a1 , a2 , b1 , b2 ∈ R, L0 > 0, L > 0 and M ≥ 1 with b1 = 0 be given parameters. Let us also assume h : R → R and μ : D → R to be continuous functions such that for x ∈ D, we have
|h(μ(x ))| ≤ h(|μ(x )| )
(2.1)
and h: [0, ∞) → [0, ∞) is nondecreasing. Furthermore, let us suppose that
b2 M |1 − α| < 1 b1 2
(2.2)
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
and
M (|a1 | + |a2 |M|1 − α| )
|b1 | 1 − M2 |1 − α| bb21
1 + Mh
893
M|1 − α| < 1.
(2.3)
It is convenient for the local convergence
analysis in the follow-up to introduce several functions and parameters. Define 1 functions g1 , p and p¯ in the interval 0, L such that we have 0
g1 (t ) =
1 (Lt + 2M|1 − α| ), 2 ( 1 − L0 t )
1 p(t ) = |b1 |
L0 |b1 |t + M|b2 |g1 (t ) , 2
p¯ (t ) = p(t ) − 1 and parameter r1 by the expression
r1 =
2(1 − M|1 − α| ) . 2L0 + L
It is straightforward to verify that the functions g1 and p are (strictly) monotonic increasing. Then, we have from b 2 Eq. (2.3) that g1 (r1 ) = 1 and 0 ≤ g1 (t) < 1, for each t ∈ [0, r1 ). We also obtain that p¯ (0 ) = M b2 |1 − α| − 1 < 0 using Eq. (2.2) and that p¯ (t ) → ∞ as t →
0,
in the interval
1 L0
1− L0 .
1
Then, by the intermediate value theorem, we can say that the function p¯ has zeros
. Let us say that rp is the smallest such zero of p¯ . Moreover, let us define functions g2 and g¯ 2 in the
interval [0, rp ) by the expressions
M M (|a1 | + |a2 |g1 (t )) g2 (t ) = 1 + h 1 − L0 t |b1 |(1 − p(t ))
g1 (t ),
g¯ 2 (t ) = g2 (t ) − 1. Then, it follows from Eq. (2.3) that g¯ 2 (0 ) < 0 and g¯ 2 (t ) → ∞ as t → r − p . Now, we assume that r2 be the smallest zero of function g¯ 2 on the interval (0, rp ). Define
r = min{r1 , r2 }.
(2.4)
Then, we have that for each t ∈ [0, r)
0 ≤ g1 (t ) < 1,
(2.5)
0 ≤ p(t ) < 1,
(2.6)
0 ≤ g2 (t ) < 1.
(2.7)
Denote by U (γ , s ) and U¯ (γ , s ) the open and closed balls in X with center γ ∈ X and of radius s > 0 and assume that X = R. Let us assume that L0 > 0 and define D0 := D ∩ U x∗ , L1 . Notice that D0 ⊆ D. In Lipschitz condition (2.9) x ∈ D, but in 0
Eq. (2.10) and (2.11) x ∈ D0 . That is L0 = L0 (D ), but L = L(D0 , L0 ) and M = M (D0 , L0 ). Condition (2.9) help us to define L and M. If the expressions (2.10) and (2.11) are defined for x ∈ D, then the corresponding L1 and M1 are such that L ≤ L1 and M ≤ M1 , leading to a less precise convergence analysis (see also Remark 3.2(b) and (c)). Next, we present the local convergence analysis of method (1.2) using the preceding notations. Theorem 2.1. Let F : D ⊆ R → R be a differentiable function. Suppose that functions h : R → R and μ : D → R satisfy estimate (2.1); there exist α , a1 , a2 , b1 , b2 ∈ R with b1 = 0 such that together with function h follow Eqs. (2.2) and (2.3); there exist x∗ ∈ D, L0 > 0, L > 0 and M ≥ 1 such that
F ( x∗ ) = 0,
F ( x∗ ) = 0,
|F (x∗ )−1 (F (x ) − F (x∗ )| ≤ L0 |x − x∗ |, for each x ∈ D |F (x∗ )−1 F (x ) − F (y ) | ≤ L|x − y|, for each x, y ∈ D0
(2.8) (2.9) (2.10)
894
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
|F (x∗ )−1 F (x )| ≤ M, for each x ∈ D0 ,
(2.11)
U¯ (x∗ , r ) ⊆ D,
(2.12)
where the radius of convergence r is defined by Eq. (2.4). Then, sequence {xn } generated for x0 ∈ U (x∗ , r ) − {x∗ } by method (1.2) is well defined, remains in U(x∗ , r) for each n = 0, 1, 2, . . . and converges to x∗ . Moreover, the following estimates hold
|yn − x∗ | ≤ g1 (|xn − x∗ | )|xn − x∗ | ≤ |xn − x∗ | < r,
(2.13)
and
|xn+1 − x∗ | ≤ g2 (|xn − x∗ | )|xn − x∗ | < |xn − x∗ |, where the functions g1 and g2 are defined previously. Also, for T ∈ [r, F (x ) = 0 in D1 , where D1 := U¯ (x∗ , T ) ∩ D.
(2.14) 2 L0
), the limit point
x∗
is the only solution of equation
Proof. We show that the estimates (2.13) and (2.14) hold with the help of mathematical induction. Using the hypothesis x0 ∈ U (x∗ , r ) − {x∗ } and expressions (2.4) and (2.9), we obtain
|F (x∗ )−1 (F (x0 ) − F (x∗ ))| ≤ L0 |x0 − x∗ | < L0 r < 1.
(2.15)
It follows from Eq. (2.15) and the Banach Lemma on invertible operators [1,18] that F (x0 ) = 0 and
|F (x0 )−1 F (x∗ )| ≤
1 . 1 − L0 |x0 − x∗ |
(2.16)
By the mean value theorem (see [13, page 64]) and Eq. (2.8), we can write
F ( x0 ) = F ( x0 ) − F ( x∗ ) =
0
1
F (x∗ + θ (x0 − x∗ ))(x0 − x∗ )dθ .
(2.17)
Notice that |x∗ + θ (x0 − x∗ ) − x∗ | = θ |x0 − x∗ | < r. Hence, x∗ + θ (x0 − x∗ ) ∈ U (x∗ , r ). Using Eqs. (2.11) and (2.17), we have
1 |F (x∗ )−1 F (x0 )| = F (x∗ )−1 F (x∗ + θ (x0 − x∗ ))(x0 − x∗ )dθ ≤ M|x0 − x∗ |.
(2.18)
0
Inserting the expressions (2.4), (2.5), (2.10), (2.16) and (2.18) in the first substep of method (1.2) for n = 0, we obtain in turn that
|y0 − x∗ | = | x0 − x∗ − F (x0 )−1 F (x0 ) + (1 − α )F (x0 )−1 F (x0 )| 1 ∗ −1 ∗ ∗ −1 ∗ ∗ ≤ |F ( x0 ) F ( x )| F (x ) F (x + θ (x0 − x )) − F (x0 ) (x0 − x )dθ 0 +|1 − α||F (x0 )−1 F (x∗ )||F (x∗ )−1 F (x0 )| L|x0 − x∗ |2 M|1 − α||x0 − x∗ | + ∗ 2 ( 1 − L0 |x0 − x | ) 1 − L0 |x0 − x∗ | ∗ ∗ = g1 (|x0 − x | )|x0 − x | < g1 (r )|x0 − x∗ | < |x0 − x∗ | < r, ≤
(2.19)
∈ U(x ∗
which leads to Eq. (2.13) for n = 0 and y0 r). We also prove that b1 F (x0 ) + b2 F (y0 ) = 0 for b1 = 0 and x0 = x∗ . Using Eqs. (2.4), (2.6), (2.8), (2.9), (2.18) and (2.19), we have
−1 b1 F (x∗ )(x0 − x∗ ) b1 F (x0 ) − F (x∗ ) − F (x∗ )(x0 − x∗ ) + b2 F (y0 )
1 L |b | ≤ |x0 − x∗ |−1 0 1 |x0 − x∗ |2 + M|b2 ||y0 − x∗ | |b1 | 2
1 L |b | ≤ |x0 − x∗ |−1 0 1 |x0 − x∗ |2 + M|b2 |g1 (|x0 − x∗ | )|x0 − x∗ | |b1 | 2 ∗ = p( | x 0 − x | ) < p( r ) < 1 .
Hence, it results
(2.20)
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
(b1 F (x0 ) + b2 F (y0 ) )−1 F (x∗ ) ≤
1
|b1 ||x0 − x∗ |(1 − p(|x0 − x∗ | ) )
.
895
(2.21)
It is also needed an estimation of |h(μ(x0 ))|. We can obtain an estimate of |h(μ(x0 ))| by means of expressions (2.18), (2.19), and (2.21), resulting
M ( |a1 | + |a2 |g1 |x0 − x∗ | ) |x0 − x∗ | |h(μ(x0 ))| ≤ h . |b1 ||x0 − x∗ |(1 − p(|x0 − x∗ | ) )
(2.22)
Now, using the second sub step of method (1.2) for n = 0 and with the help of Eqs. (2.7), (2.16), (2.18) and (2.22), it yields
|x1 − x∗ | ≤ |y0 − x∗ | + |h(μ(x0 ))||F (x0 )−1 F (x∗ )||F (x∗ )−1 F (y0 )|
M M (|a1 | + |a2 |g1 (|x0 − x∗ | )) |y0 − x∗ | ≤ 1+ h 1 − L|x0 − x∗ | | b 1 | ( 1 − p( | x 0 − x ∗ | ) ) ≤ g2 (|x0 − x∗ | )|x0 − x∗ | ≤ |x0 − x∗ | < r,
(2.23)
∈ U(x∗ , r).
which shows Eq. (2.14) for n = 0 and x1 Replacing x0 , y0 , z0 and x1 , by xk , yk , zk and xk+1 , respectively, in the preceding estimates we arrive at Eqs. (2.13)–(2.14). Using the estimate |xk+1 − x∗ | ≤ c|xk − x∗ | < r, c = g2 (|x0 − x∗ | ) ∈ [0, 1 ), we deduce that lim xk = x∗ and xk+1 ∈ U (x∗ , r ). k→∞ 1 Finally, to show the uniqueness part, let Q = 0 F (y∗ + θ (x∗ − y∗ ))dθ for some y∗ ∈ U¯ (x∗ , T ) with F (y∗ ) = 0. Using Eq. (2.9), we get that
|F (x∗ )−1 (Q − F (x∗ ))| ≤ | ≤
1
∗ ∗ 0 ( 1 − θ )|y − x |d θ ≤
1 0
L0 |y∗ + θ ( x∗ − y∗ ) − x∗ |d θ (2.24)
L0 T < 1. 2
It follows from Eq. (2.24) that Q is invertible. Then, in view of the identity 0 = F (x∗ ) − F (y∗ ) = Q (x∗ − y∗ ), we conclude that x∗ = y∗ . 3. Local convergence: multi-dimensional case In this section, we present the local convergence of method (1.4). Let a1 , a2 , b1 , b2 ∈ R, L0 > 0, L > 0, M ≥ 1 and N ≥ 0, with b1 + b2 = 0 be given parameters. In addition, we also assume that the operators H and μ ¯ given in the introduction satisfy the following estimate
H (μ¯ (X )) ≤ H0 ( (μ¯ (X ) ), for each X ∈ D,
(3.1)
for some function H0 : [0, ∞) → [0, ∞) which is nondecreasing and continuous. Suppose that
b2 b1 + b2 N < 1.
(3.2)
Now, we define the functions g1 , q, and q¯ in the interval 0,
g1 (t ) =
1 L0
by
Lt , 2 ( 1 − L0 t )
b2 N q(t ) = , b1 + b2 1 − L0 t q¯ (t ) = q(t ) − 1 and parameter rA as
rA =
2 . 2L0 + L
1 ¯ L0 , g1 (rA ) = 1 and 0 ≤ g1 (t) < 1 for each t ∈ [0, rA ). Moreover, from Eq. (3.2) we get that q (0 ) = − b2 b1 +b2 N − 1 < 0 and q¯ (t ) → ∞ as t → 1L0 . Denote by rq the smallest zero of function q¯ in the interval 0, L10 . Further-
We have that 0 < rA <
more, define functions g2 and g¯ 2 on the interval (0, rq ) as follows:
g2 (t ) =
M 1+ H0 1 − L0 t
|a1 + a2 | + 1|a−L2 |0Nt |b1 + b2 |(1 − q(t ))
g1 (t )
896
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
and
g¯ 2 (t ) = g2 (t ) − 1. We have that g¯ 2 (0 ) = −1 < 0. Notice that if rA < rq ⇒ g¯ 2 (rA ) > 0, since g1 (rA ) = 1. (C) if rq ≤ rA suppose that g¯ 2 (t ) → a positive number or infinity as t → r¯q− . In either case denote by r2 the smallest zero of function g¯ 2 in (0, rA ) (if rA < rq ) or (0, rq ) (if condition (C) holds and rq ≤ rA ). Finally, if we define the radius of convergence r by
r = min{rA , r2 }.
(3.3)
Then, we have that
0 < r ≤ rA
(3.4)
and for each t ∈ [0, r)
0 ≤ g1 (t ) < 1
(3.5)
0 ≤ q(t ) < 1,
(3.6)
0 ≤ g2 (t ) < 1.
Set D0 := D ∩ U X ∗ ,
1 L0
(3.7)
and L0 > 0.
Theorem 3.1. Let F : D ⊆ X → Y be a Fréchet-differentiable operator. In addition [·, · ; F ] : D × D → L(X, Y ) is a divided difference of order one, H, H0 and μ ¯ are as previously and satisfying Eq. (3.1). Furthermore, suppose that there exist X ∗ ∈ D, a1 , a2 , b1 , b2 ∈ R, L0 > 0, L > 0, M ≥ 1 and N ≥ 0 with b1 + b2 = 0 such that
F ( X ∗ ) = 0,
F (X ∗ )−1 ∈ L(Y, X ),
(3.8)
F (X ∗ )−1 (F (X ) − F (X0 ) ≤ L0 X − X0 , for each X ∈ D,
(3.9)
F (X ∗ )−1 F (X ) − F (Y ) ≤ L X − Y , for each X, Y ∈ D0 ,
(3.10)
F (X ∗ )−1 F (X ) ≤ M, for each X ∈ D0 ,
(3.11)
F (X ∗ )−1 [X, Y ; F ] ≤ N for each X, Y ∈ D0 ,
(3.12)
U¯ (X ∗ , r ) ⊆ D,
(3.13)
expression (3.2) and condition (C) hold, if rq ≤ rA , where the radius of convergence r is defined by Eq. (3.3). The sequence {Xn } generated for X0 ∈ U (X ∗ , r ) − {X ∗ } by method (1.4) is well defined, remains in U(X∗ , r) for each n = 0, 1, 2, . . . and converges to X∗ . Moreover, the following estimates hold
Yn − X ∗ ≤ g1 ( Xn − X ∗ ) Xn − X ∗ ≤ Xn − X ∗ < r,
(3.14)
and
Xn+1 − X ∗ ≤ g2 ( Xn − X ∗ ) Xn − X ∗ < Xn − X ∗ , where the functions g1 and g2 were defined previously. Furthermore, for T ∈ [r, equation F (X ) = 0 in D1 , where D1 = U (X ∗ , T ) ∩ D.
(3.15) 2 L0
), the limit point
X∗
is the only solution of
Proof. As in the proof of Theorem 2.1, we arrive at the estimates
F (X0 )−1 F (X ∗ ) ≤
1 , 1 − L0 X0 − X ∗
F (X0 )−1 F (X0 ) ≤ M X0 − X ∗ ,
(3.16) (3.17)
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
and also estimate Eq. (3.14) (for α = 1). Moreover, we need to show that linear operator I − invertible. We have that
b2 F (X0 )−1 [X0 , b1 +b2
b2 −1 b1 + b2 F (X0 ) [X0 , Y0 ; F ] b2 F (X0 )−1 F (X ∗ ) F (X ∗ )−1 [X0 , Y0 ; F ] ≤ b1 + b2 b2 N ≤ = q( X0 − X ∗ ) < 1. b1 + b2 1 − L0 X0 − X ∗
897
Y0 ; F ] is
(3.18)
Hence, we deduce that
−1 b2 1 −1 F (X0 ) [X0 , Y0 ; F ] . I− ≤ ∗ b + b 1 − q ( X 1 2 0 − X ))
(3.19)
Notice that
−1 b2 1 −1 μ(X0 ) ≤ I− F (X0 ) [X0 , Y0 ; F ] a1 + a2 − a2 F (X0 )−1 [X0 , Y0 ; F ] b1 + b2 b1 + b2 ≤
|a1 + a2 | + 1−L0| a2X|0N−X ∗ . |b1 + b2 |(1 − q( X0 − X ∗ ))
(3.20)
Then, by using the last sub step of the method (1.4) for n = 0, expressions (3.1), (3.3), (3.7), (3.14), (3.16), (3.17) (for Y0 = X0 ) and (3.20), we have
X1 − X ∗ ≤ Y0 − X ∗ + H (μ¯ (X0 )) F (X0 )−1 F (X ∗ ) F (X ∗ )−1 F (Y0 ) N |a1 + a2 | + 1−L| aX20|−X ∗ M ≤ 1+ H0 Y0 − X ∗ 1 − L X0 − X ∗ |b1 + b2 |(1 − q( X0 − X ∗ )) ≤ g2 ( X0 − X ∗ ) X0 − X ∗ ≤ c X0 − X ∗ < r,
(3.21) − X ∗|
which shows Eq. (3.15) for n = 0 and X1 Using the estimate Xk+1 ≤ c Xk < r, c = g2 ( X0 − X ∗ ) ∈ ∗ ∗ [0, 1 ), we deduce that lim Xk = X and Xk+1 ∈ U (X , r ). k→∞ 1 Finally, to show the uniqueness part, let Q = 0 F (Y ∗ + θ (X ∗ − Y ∗ ))dθ for some Y ∗ ∈ U¯ (X ∗ , T ) with F (Y ∗ ) = 0. Using Eq. (3.9), we get that ∈ U(X∗ , r).
− X ∗
1 ∗ −1 ∗ ∗ ∗ ∗ L Y + θ ( X − Y ) − X d θ ( ) F ( X ) Q − F ( X ∗ ) ≤ 0 0
≤
1 0
(1 − θ ) Y ∗ − X ∗ dθ ≤
L0 T < 1. 2
(3.22)
It follows from Eq. (3.22) that Q is invertible. Then, in view of the identity 0 = F (X ∗ ) − F (Y ∗ ) = Q (X ∗ − Y ∗ ), we conclude that X ∗ = Y ∗ . Remark 3.2. (a) In view of Eq. (2.9) (or (3.10)) and the estimate
F (X ∗ )−1 F (X ) = F (X ∗ )−1 (F (X ) − F (X ∗ )) + I ≤ 1 + F (X ∗ )−1 (F (X ) − F (X ∗ )) ≤ 1 + L0 X0 − X ∗ condition (2.11) (or (3.11)) can be dropped and M can be replaced by
M (t ) = 1 + L0 t or by M (t ) = M = 2, since t ∈ [0, L1 ). 0 (b) The results obtained here can be used for operators F satisfying the autonomous differential equation [1,2] of the form
F (x ) = P (F (x )), where P is a known continuous operator. Since F (x∗ ) = P (F (x∗ )) = P (0 ), we can apply the results without actually knowing the solution x∗ . For example, with F (x ) = ex − 1, we can choose P (x ) = x + 1.
898
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902 Table 1 Different radii of convergence for multi-factor effect problem. Case
Values of parameters that satisfy Theorem 2.1 a1
a2
b1
b2
1 2
0 1
2 0
1 2
0 −1
1
h(μ) 6
6−3μ−μ
2
2μ − 1
rA
r2
r
x0
n
ρ
0.405594 0.405594
0.10784 0.091777
0.10784 0.091777
−0.399 −0.219
5 5
4.0 0 0 0 4.0 0 0 0
Table 2 Different radii of convergence for Example (4.2). Case
Values of parameters that satisfy Theorem 2.1 a1
1 2
0 1
a2
b1
2 0
1 2
1
b2 0 −1
h(μ) 6
6−3μ−μ2
2μ − 1
rA
r2
r
x0
n
ρ
0.66667 0.66667
0.27145 0.31615
0.27145 0.31615
0.26 0.30
4 4
5.0 0 0 0 5.0 0 0 0
(c) The value rA = 2L 2+L was shown in [1,2] to be the convergence radius for the Newton’s method under conditions (2.9) and 0 (2.10) (or (3.9) and (3.10)). It follows from Eq. (2.4) (or (3.3)) and the definition of rA that the convergence radius r of the method (1.2) (or method (1.4)) cannot be larger than the convergence radius rA of the second order Newton’s method. The radius rA is at least large as the convergence ball given by Rheinboldt [18]
rT R =
2 , 3L1
where L1 is given above Theorem 2.1. In particular, for L0 < L1 we have that
rT R < rA and
rT R 1 → rA 3
L0 → 0. L1
as
That is, our convergence ball rA is at most three times larger than the one derived by Rheinboldt. The same value for rt R 1
given by Traub [19]. In the case of the example in (b), we have L0 = e − 1, L = e e−1 and L1 = e, so that L0 < L < L1 and rT R = 2.24 < rA = 2.38. (d) It is worth noticing that method (1.4) does not change if we use the conditions of Theorems 2.1 and 3.1 instead of the stronger conditions given in [3]. We adopt the following formulas for calculating the computational order of convergence (COC) [8] −X ln XXn+1 ∗ n −X ∗
ρ=
ln XXn −X −X ∗ ∗
,
for each n = 1, 2, . . .
(3.23)
n−1
or the approximate computational order of convergence (ACOC)
ρ∗ =
−Xn ln XXnn+1 −X n−1
n−1 ln XXn −X −X n−1
,
for each n = 2, 3, . . .
(3.24)
n−2
With this method, we obtain the order of convergence that avoids the bounds involving estimates higher than the first Fréchet derivative. (e) Methods (1.2) and (1.4) generate the single and two step methods introduced in [3–5,7,8,10–13,16–19,21], because they can be reduced to those methods by specializing α , h, μ, H and μ ¯ . As an example, choose α = 1, μ(x ) = 1 and h(x ) = 1 in method (1.2) to obtain two step Newton-methods considered in [17] (similar choices for the Banach space case). Other choices are also possible [1,2]. Notice that in the earlier studies (as already mentioned in the introduction) the convergence criteria are stronger than those presented here, limiting the applicability of such methods.
4. Numerical examples In this section, we illustrate numerically the theoretical results developed previously. We consider four examples for nonlinear scalar equations: the first one is a real world problem; the second and third are standard academic problems; the fourth one is a motivational problem, described in Eqs. (4.1)–(4.4). We also choose some of the weight functions mentioned in Tables 1, 2, 3, 4 to solve these equations. Furthermore, in the tables we present the initial guesses of the considered problem, the radius of convergence and the minimum number of iterations required to get the desired in zeros of F(x) and
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
899
Table 3 Different radii of convergence for Example (4.3). Case
Values of parameters which satisfy Theorem 2.1 a1
a2
b1
b2
1 2
0 1
2 0
1 2
0 −1
1
h(μ) 6
6−3μ−μ
2
2μ − 1
rA
r2
r
x0
n
ρ
0.382692 0.382692
0.10809 0.098237
0.10809 0.098237
0.09 −0.08
5 5
4.0 0 0 0 4.0 0 0 0
Table 4 Different radii of convergence for Example (4.4). Case
Values of parameters which satisfy Theorem 2.1 a1
1 2
0 1
a2
b1
2 0
1 2
h(μ)
b2
6
0 −1
1
6−3μ−μ2
2μ − 1
rA
r2
r
x0
n
ρ
0.0068968 0.0068968
0.0018337 0.0015606
0.0018337 0.0015606
1.001 0.9986
4 4
4.0 0 0 0 4.0 0 0 0
Table 5 Different radii of convergence for Van der Pol equation. Case
Values of parameters that satisfy Theorem 3.1 a1
a2
b1
b2
rA
r2
r
x0
n
ρ
1 2
1
0 0
2 2
1
0.026482 0.026482
0.10872 0.11917
0.026482 0.026482
(0.8,1.48,2.15,3.09,7.79) (0.85, 1.54, 2.44, 3.14, 7.84 )
4 5
4.0077 4.0026
1 2
1 2
Table 6 The values of ψ i and φ i (in radians) for Example (4.6). i
ψi
φi
0 1 2 3
1.3954170041747090114 1.7444828545735749268 2.0656234369405315689 2.460 067847891250 0533
1.7461756494150842271 2.0364691127919609051 2.2390977868265978920 2.4600678409809344550
Table 7 Different radii of convergence for Kinematic synthesis problem. Case
Values of parameters that satisfy Theorem 3.1 a1
a2
b1
b2
rA
r2
r
x0
n
ρ
1 2
1
0 0
2 2
1
0.034861 0.034861
0.14311 0.15687
0.034861 0.034861
(0.872,0.665,0.618) (0.938,0.731,0.684)
5 5
4.0070 4.0047
1 2
1 2
Table 8 Different radii of convergence. Case
1 2
Different values of parameters that satisfy Theorem 3.1 a1
a2
b1
b2
rA
r2
r
x0
n
ρ
1
0 0
2 2
1
0.38798 0.38798
1.5054 1.5054
0.38798 1.6429
(0.37, 0.36, 0.36) (0.27, 0.26, 0.26)
5 5
4.0 0 01 4.075
1 2
1 2
we verify the theoretical order of convergence. Therefore, we calculate the COC given by Eq. (3.23) or the ACOC given by formula (3.24). The computations are performed by means of Mathematica 9 with multiple precision arithmetic. We adopt = 10−500 as a tolerance error and the following stopping criteria are used: (i ) |xn+1 − xn | < and (ii ) |F (xn+1 )| < . In what regards nonlinear systems, we also assume three problems. Examples 4.5 and 4.6 are applied science exercises and Example 4.7 is a standard academic problem. We choose some weight functions in these examples. In Tables 5, 7, 8, we list the initial guesses, radius of convergence and minimum number of iterations required to get the desire accuracy to the corresponding zeros of the functions. Furthermore, we calculate the COC or ACOC by means of the formulas (3.23) or (3.24) to verify the theoretical order of convergence of the nonlinear system. We adopt = 10−100 as a tolerance error for the nonlinear system, and the following stopping criterion is used: (i ) X (n+1) − X (n) < and (ii ) F (X (n+1) ) < . 1 Moreover, we consider [X, Y ; F ] = F Y + θ (X − Y ) dθ in all examples. 0
900
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
Example 4.1. In the study addressing the multi-factor effect [9], the trajectory of an electron in the air gap between two parallel plates is given by
x(t ) = x0 +
v0 + e
E0 E0 sin(ωt0 + α ) (t − t0 ) + e cos(ωt + α ) + sin(ω + α ) , mω mω2
(4.1)
where t is time, e and m denote the charge and the mass of the electron at rest, x0 and v0 represent the position and velocity of the electron at time t0 and E0 sin(ωt + α ) is the RF electric field between the plates. We choose the particular parameters in expression (4.1) in order to obtain a simpler expression, as follows:
f (x ) = x −
1 π cos(x ) + . 2 4
(4.2)
The required zero of the above function (4.2) is x∗ = −0.309093271541794952741986808924. We refer the reader to Hueso et al. [9] for the expression (4.2), where the computational details can be found. We get L0 = L = 1.643681148, M = 2 and we choose α = 1. We obtain different radii of convergence by using the parameters values shown in Table 1. Example 4.2. Let S = R, D = [−1, 1], x∗ = 0 and define function F on D by
F (x ) = sin(x ).
(4.3)
Then, we get that = 0, L0 = L = M = 1 and α = 1. We obtain distinct radius of convergence by using different values of parameters listed in the Table 2. x∗
Example 4.3. Let S = R, D = [−1, 1], x∗ = 0 and define function F on D by
F ( x ) = ex − 1.
(4.4)
Then, we get L0 = e − 1, L = M = e values exhibited in Table 3.
1 e−1
and consider α = 1. We obtain different radii of convergence by using the parameters
Example 4.4. Returning back to the motivation example at the introduction of this paper, we have x∗ = 1, L = L0 = 96.662907, M = 2 and we select α = 1. We obtain distinct radii of convergence by using the parameters depicted in Table 4. Example 4.5. Let us consider the Van der Pol equation [6], that is defined as follows:
y − μ(y2 − 1 )y + y = 0,
μ > 0,
(4.5)
and models the flow of current in a vacuum tube, with the boundary conditions y(0 ) = 0, y(2 ) = 1. We consider the partition of the given interval [0, 2], that is given by
x0 = 0 < x1 < x2 < x3 < · · · < xn , where xi = x0 + ih, h =
2 , i = 1, 2, . . . , n. n
Moreover, we assume that
y0 = y(x0 ) = 0, y1 = y(x1 ), . . . , yn−1 = y(xn−1 ), yn = y(xn ) = 1. If we discretize the above problem (4.5) by using the second order divided difference for the first and second derivatives, that are given by
yk =
yk+1 − yk−1 y − 2yk + yk+1 , yk = k−1 , k = 1, 2, . . . , n − 1, 2h h2
then, we obtain a system of nonlinear equations of order (n − 1 ) × (n − 1 )
2h2 xk − hμ x2k − 1 (xk+1 − xk−1 ) + 2(xk−1 + xk+1 − 2xk ) = 0, k = 1, 2, . . . , n − 1. In this problem, let us consider the value n = 6 that leads to a 5 × 5 system of nonlinear equations. The solution of this problem is
X ∗ = (0.8243131 . . . , 1.516531 . . . , 2.187958 . . . , 3.123402 . . . , 7.824242 . . . )T . Then, we get L0 = L = 25.1739325, M = N = 2, H (μ ¯ (xn )) = I +
2b21 a2 b1 −a1 b2
μ¯ (xn ) −
a1 I b1
, where μ ¯ (xn ) was defined earlier, and
we consider α = 1. By these using values, we obtain different radii of convergence in Table 5. Example 4.6. Consider the kinematic synthesis for the steering problem described in [4,20], written as system of nonlinear equations, given by
[Ei (x2 sin (ψi ) − x3 ) − Fi (x2 sin (φi ) − x3 )] + [Fi (x2 cos (φi ) + 1 ) − Fi (x2 cos (ψi ) − 1 )] 2
2
− [x1 (x2 sin (ψi ) − x3 )(x2 cos (φi ) + 1 ) − x1 (x2 cos (ψi ) − x3 )(x2 sin (φi ) − x3 )] = 0, for i = 1, 2, 3, 2
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
901
where
Ei = −x3 x2 (sin (φi ) − sin (φ0 ) ) − x1 (x2 sin (φi ) − x3 ) + x2 (cos (φi ) − cos (φ0 ) ), i = 1, 2, 3 and
Fi = −x3 x2 sin (ψi ) + (−x2 ) cos (ψi ) + (x3 − x1 )x2 sin (ψ0 ) + x2 cos (ψ0 ) + x1 x3 , i = 1, 2, 3. We display the values of ψ i and φ i (in radians) in Table 6. We consider the initial guess x(0 ) = (0.7, 0.7, 0.7 ). We obtain the following approximated solution
X ∗ = (0.9051567 . . . , 0.6977417 . . . , 0.6508335 . . . )T . Then, we get L0 = L = 19.1237412, M = N = 2, H (μ ¯ (xn )) = I +
2b21 a2 b1 −a1 b2
μ¯ (xn ) −
a1 I b1
, where μ ¯ (xn ) was defined earlier
and we take α = 1. By using these values, we obtain different radii of convergence as shown in Table 7. Example 4.7. Let X = Y = R3 , D = U¯ (0, 1 ). Define F on D for v = (x, y, z )T by means of
F ( v ) = ex − 1,
e−1 2 y + y, z 2
T
.
(4.6)
Then, the Fréchet-derivative is given by
F (v ) =
ex 0 0
0 ( e − 1 )y + 1 0
0 0 . 1 1
Notice that X ∗ = (0, 0, 0 ), F (x∗ ) = F (x∗ )−1 = diag{1, 1, 1}, L0 = e − 1, L = M = N = e e−1 ,
H (μ ¯ (xn )) = I +
2b21 a2 b1 − a1 b2
μ¯ (xn ) −
a1 I , b1
where μ ¯ (xn ) was defined earlier and we take α = 1. Hence, we obtain different radii of convergence by using the parameters in Table 8. 5. Conclusion The aim of this paper was to generate high convergence order methods to solve systems of equations by extending unidimensional methods. In particular, we extended methods introduced in [3] where Taylor expansions and derivatives up to fifth order were used to determine the fourth order of convergence. The requirement for these derivatives limit the applicability of the method. Moreover, no computable radii convergence or error bounds were given. To address these problems we introduced the multidimensional analog of the method proposed in [3] and we demonstrated its convergence under hypothesis on the first derivative only. Moreover, we found computable radii of convergence as well as error bounds using Lipschitz-type conditions. The convergence order is determined using the COC or ACOC indices that do not require derivatives with order higher than one. This way we expand the applicability of these methods in the multidimensional case and avoid high order derivatives. We presented seven numerical examples for solving systems of equations appearing in the electricity, kinematics and in the solution of the Van der Pol boundary value problems that governs the flow of current in a vacuum tube. In all cases the proposed method revealed good performance. References [1] I.K. Argyros, Convergence and Application of Newton-type Iterations, Springer, 2008. [2] I.K. Argyros, S. Hilout, Numerical methods in nonlinear analysis, World Scientific Publishing Co., New Jersey, 2013. [3] S. Artidiello, A. Cordero, J.R. Torregrosa, M.P. Vassileva, Multidimensional generalization of iterative methods for solving nonlinear problems by means of weigh-function procedure, Appl. Math. Comput. 268 (2015) 1064–1071. [4] F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numer. Algorithm 54 (2010) 395–409. [5] R. Behl, S.S. Motsa, Geometric construction of eighth-order optimal families of Ostrowski’s method, Wor. Sci. J. 2015 (2015) 11. Article ID 614612. [6] R.L. Burden, J.D. Faires, Numerical Analysis, PWS Publishing Company, Boston, 2001. [7] A. Cordero, J. R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas, Appl. Math. Comput. 190 (1) (2007) 686–698. [8] J.A. Ezquerro, M.A. Hernández, New iterations of R-order four with reduced computational cost, BIT Numer. Math. 49 (2009) 325–342. [9] J.L. Hueso, E. Martínez, C. Teruel, Determination of multiple roots of nonlinear equations and applications, J. Math. Chem. 53 (2015) 880–892. [10] V. Kanwar, R. Behl, K.K. Sharma, Simply constructed family of a Ostrowski’s method with optimal order of convergence, Comput. Math. Appl. 62 (11) (2011) 4021–4027. [11] Á. A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput. 233 (2014a) 29–38. [12] A.A. Magreñán, A new tool to study real dynamics: the convergence plane, Appl. Math. Comput. 248 (2014b) 215–224. [13] J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New-York, 1970. [14] A.M. Ostrowski, Solution of equations and systems of equations, in: Pure and Applied Mathematics, Vol. I, X. Academic Press, New York-London, 1960. [15] A.M. Ostrowski, Solution of equations in euclidean and Banach spaces, in: Pure and Applied Mathematics, Vol. 9, Academic Press, New York-London, 1973. [16] M.S. Petkovic, B. Neta, L. Petkovic, J. Džunicˇ , Multipoint methods for solving nonlinear equations, Elsevier, 2013. [17] F.A. Potra, V. Pták, On a class of modified Newton processes, Numer. Funct. Anal. Optim. 2 (1980) 107–120.
902
I.K. Argyros, R. Behl and J.A. Tenreiro Machado et al. / Applied Mathematics and Computation 347 (2019) 891–902
[18] W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, 3, Polish Academy of Science, Banach Center Publication, 1978, pp. 129–142. [19] J.F. Traub, Iterative methods for the solution of equations, in: Prentice Hall Series in Automatic Computation, 1964. Englewood Cliffs, N.J. [20] I.G. Tsoulos, A. Stavrakoudis, On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods, Nonlinear Anal. Real World Appl. 11 (2010) 2465–2471. [21] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third order convergence, Appl. Math. Lett. 13 (20 0 0) 87–93.