We present new high-order optimal iterative methods for solving a nonlinear equation, f(x)=0, by using Padé-like approximants. We compose optimal methods of order 4 with Newton’s step and substitute the derivative by using an appropriate rational approximant, getting optimal methods of order 8. In the same way, increasing the degree of the approximant, we obtain optimal methods of order 16. We also perform different numerical tests that confirm the theoretical results.
Ministerio de Ciencia e InnovaciónMTM2014-52016-C2-02-PGeneralitat ValencianaPROMETEO/2016/0891. Introduction
Many applied problems in different fields of science and technology require to find the solution of a nonlinear equation. Iterative methods are used to approximate its solutions. The performance of an iterative method can be measured by the efficiency index introduced by Ostrowski in [1]. In this sense, Kung and Traub conjectured in [2] that a multistep method without memory performing n+1 functional evaluations per iteration can have at most convergence order 2n, in which case it is said to be optimal.
Recently, different optimal eighth-order methods, with 4 functional evaluations per step, have been published. A very interesting survey can be found in [3]. Some of them are a generalization of the well-known Ostrowski’s optimal method of order four [4–7]. In [8] the authors start from a third-order method due to Potra-Pták, combine this scheme with Newton’s method using “frozen” derivative, and estimate the new functional evaluation. The procedure designed in [9] uses weight-functions and “frozen” derivative for the development of the schemes. As far as we know, beyond the family described by Kung and Traub in [2], only in [10] a general technique to obtain new optimal methods has been presented; the authors use inverse interpolation and methods of sixteenth order have also been obtained.
While computational engineering has achieved significant maturity, computational costs can be extremely large when high accuracy simulations are required. The development of a practical high-order solution method could diminish this problem by significantly decreasing the computational time required to achieve an acceptable error level (see, e.g., [11]).
The existence of an extensive literature on higher order methods (see, e.g., [3, 12] and the references therein) reveals that they are only limited by the nature of the problem to be solved: in particular, the numerical solutions of nonlinear equations and systems are needed in the study of dynamical models of chemical reactors [13], or in radioactive transfer [14]. Moreover, many of numerical applications use high precision in their computations; in [15], high-precision calculations are used to solve interpolation problems in Astronomy; in [16] the authors describe the use of arbitrary precision computations to improve the results obtained in climate simulations; the results of these numerical experiments show that the high-order methods associated with a multiprecision arithmetic floating point are very useful, because it yields a clear reduction in iterations. A motivation for an arbitrary precision in interval methods can be found in [17], in particular for the calculation of zeros of nonlinear functions.
The objective of this paper is to present a general procedure to obtain optimal order methods for n=3,4 starting from optimal order methods for n=2. The procedure consists in composing optimal methods of order 4 that use two evaluations of the function and one of the derivative, with Newton’s step and approximating the derivative in this last step by using an adequate rational function which allows duplicating the convergence order, introducing only one new functional evaluation per iteration.
In Section 2, we describe the process to generate the new eighth-order methods and establish their convergence order. In Section 3, the same procedure is used to obtain sixteenth-order methods by increasing the approximant degree. Finally, in Section 4, we collect several optimal methods of order 4 that are the starting point for our new methods and present numerical experiments that confirm the theoretical results.
2. Optimal Methods of Order 8
In this section, we describe a procedure that allows us to obtain new optimal methods of order 8, starting from optimal schemes of order 4. Let us denote by Ψ2m the set of iteration functions corresponding to optimal methods of order 2m.
Consider the three-step method given by (1)ϕ1xk=xk-fxkf′xk,ϕ2xk=ψfxk,ϕ3xk=ϕ2xk-fϕ2xkf′ϕ2xk,where ψf∈Ψ4.
In order to simplify the notation, we will omit the argument xk in the iterative process, so that we will write ϕi(xk) as ϕi,i=1,2,3 and ϕ0=xk.
Obviously, this three-step method has order 8, being a composition of schemes of orders 4 and 2, respectively (see [2], Th. 2.4), but the method is not optimal because it introduces two new functional evaluations in the last step.
Thus, to maintain the optimality, we substitute f′(ϕ2(xk)) with the derivative h2′(ϕ2(xk)) of the second-degree approximant (2)h2t=a02+a12t-ϕ0+a22t-ϕ021+b12t-ϕ0,verifying the conditions(3)h2ϕ0=fϕ0,(4)h2′ϕ0=f′ϕ0,(5)h2ϕ1=fϕ1,(6)h2ϕ2=fϕ2.
From the first condition one has a0(2)=f(ϕ0). Substituting in (4)–(6) we obtain the following linear system: (7)a12-b12fϕ0=f′ϕ0a12+a22ϕ1-ϕ0-b12fϕ1=fϕ0,ϕ1a12+a22ϕ2-ϕ0-b12fϕ2=fϕ0,ϕ2,where, as usual, f[x,y] denotes the divided difference of order 1, (fy-fx)/(y-x). Applying Gaussian elimination the following reduced system is obtained(8)a12-b12fϕ0=f′ϕ0a22-b12fϕ0,ϕ1=fϕ0,ϕ0,ϕ1-b12fϕ0,ϕ1,ϕ2=fϕ0,ϕ0,ϕ1,ϕ2.
In the divided differences with a repeated argument, one places the derivative instead of an undetermined quotient. The coefficients of the approximant are obtained by backward substitution. Then, the derivative of the approximant in ϕ2 is(9)h2′ϕ2=a12-a02b12+2a22ϕ2-ϕ0+a22b12ϕ2-ϕ021+b12ϕ2-ϕ02.
Substituting f′(ϕ2) by this value, we obtain an iterative method, M3, defined by (10)xk+1=ϕ3xk,where(11)ϕ1xk=xk-fxkf′xk,(12)ϕ2xk=ψfxk,(13)ϕ3xk=ϕ2xk-fϕ2xkh2′ϕ2xk.
This method only uses 4 functional evaluations per iteration. Showing that it is of order 8 we will prove that it is optimal in Kung-Traub’s sense.
Theorem 1.
Let α∈I be a simple root of a function f:I⊆R→R sufficiently differentiable in an open interval I. For an x0 close enough to α, the method defined by (11)–(13) has optimal convergence order 23.
Proof.
Let ϵm,k be the error of ϕm(xk); that is, ϵm,k=ϕm(xk)-α, m=0,1,2,3, for k=0,1,…. Then, by the definition of each step of the iterative method, we have(14)ϵ0,k=ϵk=xk-α,(15)ϵ1,k=ϕ1xk-α=Oϵk2,(16)ϵ2,k=ϕ2xk-α=Oϵk4,since ψf∈Ψ4.
Consider the expansion of f(ϕ0) around α(17)fϕ0=c1ϵk+c2ϵk2+c3ϵk3+c4ϵk4+c5ϵk5+c6ϵk6+c7ϵk7+c8ϵk8+Oϵk9,where cj=f(j)(α)/j!, for j=1,2,…; then,(18)f′ϕ0=c1+2c2ϵk+3c3ϵk2+4c4ϵk3+5c5ϵk4+6c6ϵk5+7c7ϵk6+8c8ϵk7+Oϵk8,so that using (11) and (14)(19)ϵ1,k=ϵk-fxkf′xk=c2c1ϵk2+2-c22+c1c3c12ϵk3+4c23-7c1c2c3+3c12c4c13ϵk4+⋯.
Substituting (19) in the expansion of f(ϕ1) around α we get(20)fϕ1=c2ϵk2+-2c22c1+2c3ϵk3+5c23c12-7c2c3c1+3c4ϵk4+⋯.
Using in (15) that ψf∈Ψ4, we write(21)ϵ2,k=i4ϵk4+i5ϵk5+i6ϵk6+i7ϵk7+i8ϵk8+Oϵk9,for some ij constants, j=4,5,…. Substituting (21) in Taylor’s expansion of f(ϕ2), we obtain(22)fϕ2=c1i4ϵk4+c1i5ϵk5+c1i6ϵk6+c1i7ϵk7+c2i42+c1i8ϵk8+Oϵk9.
Using (17), (18), (20), and (22) in the determination of the coefficients of the rational approximant and in the expression of its derivative (9) gives(23)h2′ϕ2=c1+-c32+c2c4+2c1i4c1ϵk4+⋯=f′α+Oϵk4.
Now, Taylor’s expansion of f′(ϕ2) in x=α gives (24)f′ϕ2=f′α+f′′αϕ2-α+Oϕ2-α2,and the fact that ϕ2(x) is of fourth order allows us to establish (25)f′ϕ2=f′α+Oϵk4.
Using this expression and (23) one can write(26)f′ϕ2=h2′ϕ21+Oϵk4.
The order of the method M3 is obtained by computing (27)ϵ3,k=ϕ3xk-α=ϕ2xk-α-fϕ2xkh2′ϕ2xk.
Using (26) we have(28)ϵ3,k=ϕ3xk-α=ϕ2xk-α-fϕ2xk1+Oϵk4f′ϕ2xk=ϕ2xk-α-fϕ2xkf′ϕ2xk-fϕ2xkf′ϕ2xkOϵk4.
From (19) it can be deduced that(29)fxkf′xk=ϵk-ϵ1,k=xk-α-c2c1xk-α2-⋯.So, it is clear that(30)fϕ2xkf′ϕ2xk=ϕ2xk-α-c2c1ϕ2xk-α2-⋯.
By substituting (30) in (28) and using that ψf∈Ψ4 one has (31)ϵ3,k=c2c1ϕ2xk-α2+⋯+ϕ2xk-αOϵk4+⋯=Oϵk8,which proves that method M3 has optimal order 23.
3. Optimal Methods of Order 16
The idea of this section is to extend the former process performing a new step to obtain optimal methods of order 2n starting from optimal methods of order 2n-1. For n=4 the method M4 can be defined as follows:(32)xk+1=ϕ4xk,k=0,1,…with (33)ϕ1x=x-fxf′x,ϕ2x=ψfx,ϕ3x=φfx,ϕ4x=ϕ3x-fϕ3xh3′ϕ3x,where ψf∈Ψ4 and φf∈Ψ8. (See [2, 5, 6, 8, 14, 16, 18] for some optimal eighth-order methods.)
Then, we start from a method that, in its first three steps, performs 4 functional evaluations and another additional evaluation in the last step f(ϕ3(x)) that allows us to construct the following rational approximant:(34)h3t=a03+a13t-ϕ0+a23t-ϕ02+a33t-ϕ031+b13t-ϕ0.
The coefficients are determined by imposing the following conditions: (35)h3ϕ0=fϕ0,(36)h3′ϕ0=f′ϕ0,(37)h3ϕ1=fϕ1,(38)h3ϕ2=fϕ2,(39)h3ϕ3=fϕ3.Similarly to the former case, a0(3)=f(ϕ0). Substituting in (36)–(39) we obtain the linear system(40)a13-b13fϕ0=f′ϕ0a13+a23ϕ1-ϕ0+a23ϕ1-ϕ02-b13fϕ1=fϕ0,ϕ1a13+a23ϕ2-ϕ0+a23ϕ2-ϕ02-b13fϕ2=fϕ0,ϕ2a13+a23ϕ3-ϕ0+a23ϕ3-ϕ02-b13fϕ3=fϕ0,ϕ3.
The remaining coefficients are obtained by reducing the system to triangular form and solving it by backward substitution(41)a13-b13fϕ0=f′ϕ0a23+a33ϕ1-ϕ0-b13fϕ0,ϕ1=fϕ0,ϕ0,ϕ1a33-b13fϕ0,ϕ0,ϕ2=fϕ0,ϕ0,ϕ1,ϕ2-b13fϕ0,ϕ0,ϕ2,ϕ3=fϕ0,ϕ0,ϕ1,ϕ2,ϕ3.
The derivative of the rational approximant in ϕ3 is(42)h3′ϕ3=a13-a03b13+2a23ϕ3-ϕ01+b13ϕ3-ϕ02+3a33+a23b13ϕ3-ϕ02+2a33b13ϕ3-ϕ031+b13ϕ3-ϕ02.
As in the previous case, this expression allows us to establish that(43)h3′ϕ3=f′α+Oϵk8,and taking into account the fact that (44)f′ϕ3=f′α+Oϵk8,we get (45)f′ϕ3=h3′ϕ31+Oϵk8.Similarly to the eighth-order case, from this expression it results that M4 has optimal convergence order 24.
4. Numerical Experiments
First of all, we consider some optimal four-order methods that we have used for developing high-order methods with the procedure described; all of them use Newton’s step as a predictor and another evaluation of function f.(46)yk=xk-fxkf′xk.
Ostrowski’s method (see [1])(47)xk+1=yk-fykxk-ykfxk-2fyk.
The family of King’s method (see [18])(48)xk+1=yk-fykf′xkfxk+bfykfxk+b-2fyk.
An optimal variant of Potra-Pták’s method (see [8]) (49)xk+1=xk-fxk+fykf′xk-fyk22fxk+fykfxk2f′xk.
Maheshwari method (see [19])(50)xk+1=xk-fxkf′xkfyk2fxk2-fxkfyk-fxk.
Now we check the performance of the methods M3 and M4 generated by (5) and (32), taking the different methods ψf∈Ψ4 described above.
We have chosen the following examples:
f(x)=(x-2)(x10+x+1)e-x-1,α=2.
f(x)=exsin(5x)-2,α≈1.36397318.
We have performed the computations in MATLAB in variable precision arithmetic with 1000 digits of mantissa.
Tables 1 and 2 show the distance |xk-α| for the first three iterations of the new order 8 and 16 methods, respectively. The last column, when we know the exact solution α, that is, for example, (a), depicts the computational convergence order p (see [20]) (51)p=lnxk+1-α/xk-αlnxk-α/xk-1-α,and for example (b), we compute the approximated computational convergence order ρ (see [21]) (52)ρ=lnxk+1-xk/xk-xk-1lnxk-xk-1/xk-1-xk-2.
Numerical results for f(x)=(x-2)(x10+x+1)e-x-1;α=2, with x0=2.1.
M3 with ψf
|x1-α|
|x2-α|
|x3-α|
p
Ostrowski
9.5688(-6)
3.1934(-37)
4.9152(-289)
8
Kingβ=-1
7.25(-5)
2.62(-29)
7.68(-225)
8
Kingβ=1
7.34(-5)
8.65(-29)
3.23(-220)
8
Opt. Potra
3.17(-5)
3.48(-33)
7.34(-257)
7.99
Maheshwari
1.03(-4)
2.56(-27)
3.72(-208)
8
Numerical results for f(x)=(x-2)(x10+x+1)e-x-1;α=2, with x0=2.1.
M4 with ψf
|x1-α|
|x2-α|
|x3-α|
p
Ostrowski
3.76(-10)
1.34(-143)
9.25(-2279)
15.8399
Kingβ=-1
2.08(-8)
5.55(-114)
3.83(-1803)
15.7977
Kingβ=1
2.17(-8)
1.02(-112)
5.72(-1782)
15.6564
Opt. Potra
3.94(-9)
1.56(-127)
5.93(-2022)
15.9907
Maheshwari
4.28(-8)
2.03(-107)
1.29(-1696)
15.5962
The results from Tables 3 and 4 correspond to an equation without exact solution, so that |xk+1-xk| is computed, instead of the actual error. In both cases, the numerical results support the optimality of the new methods, according to the proven theoretical results.
Numerical results for f(x)=exsin5x-2;α≈1.36397318⋯, with x0=1.2.
M3 with ψf
|x1-x0|
|x2-x1|
|x3-x2|
ρ
Ostrowski
0.00363
3.13(-17)
2.43(-128)
8
Kingβ=-1
0.00228
4.44(-18)
4.81(-135)
8
Kingβ=1
-0.00544
1.72(-16)
1.64(-122)
8
Opt. Potra
0.00669
4.15(-16)
1.74(-119)
7.99
Maheshwari
0.00305
1.5(-17)
7.5(-131)
7.99
Numerical results for f(x)=exsin5x-2;α≈1.36397318⋯, with x0=1.2.
M4 with ψf
|x1-x0|
|x2-x1|
|x3-x2|
ρ
Ostrowski
6.45(-5)
7.08(-75)
5.13(-1188)
16.0342
Kingβ=-1
4.02(-5)
-2.65(-78)
7.38(-1243)
15.9919
Kingβ=1
9.73(-5)
-5.81(-72)
2.3(-1141)
16.0731
Opt. Potra
5.41(-5)
3.43(-76)
4.58(-1209)
16.0182
Maheshwari
1.2(-4)
2.11(-70)
2.25(-1116)
16.0957
5. Conclusions
In this paper, we develop high-order iterative methods to solve nonlinear equations. The procedure to obtain the iteration functions is rigorously deduced and can be generalized. There are numerous applications where these schemes are needed because it is necessary to use high precision in their computations, as occurs in dynamical models of chemical reactors and in radioactive transfer and also high-precision calculations are used to solve interpolation problems in Astronomy and so forth. Moreover, the methods presented are optimal in terms of efficiency; this fact makes them very competitive.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work has been supported by Ministerio de Ciencia e Innovación de España MTM2014-52016-C2-02-P and Generalitat Valenciana PROMETEO/2016/089.
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