1. Introduction
In the paper, we develop several novel iterative schemes to solve the nonlinear equation:
which is frequently occurred in engineering and scientific applications. For solving Equation (
1),
is a well-known Newton iterative method. A lot of methods were modified from the Newton iterative method [
1,
2,
3,
4,
5,
6,
7,
8], and they are effective to solve nonlinear equations. We are going to replace
in the denominator of Equation (
2) by a linear function of
, like as
for some constants
a and
b. In doing so, a major drawback of the sensitivity to the initial guess of the Newton iterative method can be avoided, and at the same time, a major advantage is that
is no longer needed.
The iterative schemes in [
2,
9,
10,
11,
12,
13,
14] were based on the quadratures, which are of two-step-type iterative schemes with third-order convergence, needing the first Newton step to generate a trial solution, and then a correction at the second step by some quadrature rules. Our fractional iterative scheme is of the one-step type and also of the third-order convergence, which saves much computation of function per iteration.
Weerakoon and Fernando [
2] resorted to a trapezoidal quadrature rule to derive an arithmetic mean Newton method with third-order convergence of the iterative scheme. After that, third-order iterative schemes based on different quadrature methods were developed in [
9,
10,
11,
12,
15,
16], of which the evaluations of
with
are required per iteration. They have the same order
and have the same efficiency index (E.I.)
. However, the optimal order and efficiency index of the iterative scheme based on
are
and E.I. = 1.5874, according to the conjecture of Kung and Traub [
17].
For the fourth-order optimal iterative scheme (FOIS) with
performing at each iteration, there were many methods [
1,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29]. Recently, Liu and Liu [
30] derived a double-weight function technique to derive the FOIS. The iterative schemes using difference modifications of Potra-Ptak’s method with optimal fourth and eighth orders of convergence were performed by Cordero et al. [
31]. However, the two-step FOIS based on
is not yet developed. We are going to propose a simple method based on generalized quadratures to derive the FOIS, which is the optimal combination of two third-order iterative schemes to be developed in the paper.
Besides the quadratures and the finite difference approximation as used in the Ostrowski method and its modifications in [
32,
33,
34,
35,
36], the function interpolation technique is often used to generate the high-order iterative scheme. Data interpolation is a mathematical process to construct the interpolant from the given data, of which the differential at each point is inserted to set up the Hermite interpolant, which was used in higher-order iterative schemes [
30,
37,
38,
39,
40,
41,
42,
43]. Also the three-point generalized Hermite interpolation techniques, and a new class of three-step eighth-order optimal iterative schemes (EOIS) is constructed in the paper, which involves four evaluations of functions that are optimal in the sense of Kung and Traub.
Previously, Zhanlav et al. [
44] used the generating function method for constructing new EOIS iterations, by which our technique is quite different.
The scalar equation obtained in the engineering application is usually an implicit function of the unknown variable. For instance, the target equation used in the shooting method to solve the nonlinear boundary value problem is an implicit equation with being an implicit function of x, and under this situation, it is hard to obtain the derivative term . When the Newton iterative method cannot be applied to solve this sort problem, the proposed fractional iterative schemes have an advantage to solve this type of problem without using .
The paper is organized as follows. A two-dimensional variant of Newton’s iterative method is developed in
Section 2. We derive a fractional iterative scheme, whose convergence criterion is proven. The convergence behavior analysis of the fractional iterative scheme is carried out in
Section 3. We verify the performance of the proposed iterative schemes in
Section 4 by computing several numerical examples. In
Section 5, we combine the fractional iterative scheme to the quadrature methods to generate the FOIS. We reduce the fractional iterative schemes to some derivative-free iterative schemes in
Section 6, and the convergence is identified. The Hermite interpolation is introduced in
Section 7, and a three-point interpolation formula is derived. The results are used in
Section 8 to derive the three-point generalized Hermite interpolations. In
Section 9, we construct the EOIS by using the weight functions obtained from the three-point generalized Hermite interpolations, and examples are given. Finally, we draw the conclusions in
Section 10. The abbreviations are listed in the Abbreviations.
2. Two-Dimensional Generalization of Newton’s Method
To motivate the present study, we begin with
where
r is a simple solution of
with
and
. Inserting Equation (
3) for
into Equation (
2) and using
derived from Equation (4) with higher-order terms being neglected yields
This iterative scheme is a variant near to the Newton iterative scheme (
2). We will prove that the iterative scheme (
5), like Equation (
2), is quadratically convergent as to be stated by Theorem 3 in
Section 3. Below we will derive an iterative scheme with a similar form to Equation (
5), but it is cubically convergent, rather than the quadratic convergence of Equation (
5).
When a new variable is defined by
where we suppose that
can be decomposed as
, from Equation (
6) the following identity holds:
By Equations (
1) and (
7), we have
where
a is an accelerating parameter and
is a splitting function to be discussed below. While
is added on both sides of the equation
in Equation (
8), we add
on both sides of Equation (
7) to render Equation (9). Herein, the problem of finding the solution of Equation (
1) is mathematically transformed to a coupled system of quasi-linear Equations (
8) and (9) in the two-dimensional space
.
The splitting technique of
in Equation (9) is used to solve Equation (
1) in [
45]. Then, Liu et al. [
46] proposed a derivative-free iterative scheme using
. We further carry out a theoretical analysis in the two-dimensional space directly for
.
When
is obtained at the
nth step, the linearizations around
for Equations (
8) and (9) are
which are two-dimensional linear system for
. We take
and
for shorting the notations.
From Equations (
10) and (11), we can obtain
at the next step by
which renders
Both the nominator and denominator on the right-hand side of Equation (
12) are multiplied by
, and using Equation (
6), it can be refined to
Remark 1. Mathematically, after adding on both sides, Equation (1) is equivalent towhich is then multiplied by x, If is already known, we can seek the next by Upon taking the same notation with , Equation (15) goes back to Equation (14). However, this one-dimensional approach cannot generate Equation (13); without setting the problem in the two-dimensional space as carried out in Equations (8) and (9), it is hard to determine a and , and this proves Theorem 1 given below. We are going to show that without resorting on the derivative term, the third-order convergence of the iterative scheme can also be realized. Our aim is reducing the number of function evaluations to one per iteration, without using the derivative term to maintain the order of convergence to be three.
Letting
where
b is a constant, we can cancel
in the fractional term of Equation (
14), and with the aid of Equation (
16), we can achieve
which, including two constant parameters
a and
b, is a novel iterative scheme to solve Equation (
1). For use in the later, the iterative scheme (
17) is called a fractional iterative scheme.
Theorem 1. For solving , the iterative scheme (17) is convergent, if Proof. In Equation (13),
is replaced by
and
is replaced by
. In view of Equations (
6) and (
16), we have
Let
be a contraction factor. By Equation (
18), we have
which implies a strictly monotonically decreasing sequence of
Hence, the absolute convergence of the iterative scheme (
17) was proved. Equation (
20) can be written as
If the criterion in Equation (
19) is satisfied, from Equation (
23), we can derive the inequality in Equation (
22). The proof is completed. □
Remark 2. Although we have made a decomposition of in Equation (6), the final results in Equations (17) and (20) are independent to . However, the decomposition technique in Equation (6) can help us to derive the two-dimensional approach to the Newton method and the fractional iterative scheme. 3. Convergence Analysis of Fractional Iterative Scheme
The convergence analysis of Equation (
17) is given below.
Theorem 2. The iterative scheme (17) for solving has third-order convergence, with the parameters given by Proof. For the proof of convergence, let
r be a simple solution of
, i.e.,
and
. Thus, by giving
it follows that
where
Inserting Equation (26) into Equation (
17) yields
where we have used the first one in Equation (
24), and
,
and
are given by
Inserting Equation (
27) into Equation (
17) and using Equation (
25) yields
If
holds as that given in second one in Equation (
24), we have
and at the same time, Equation (
29) reduces to
Equation (
30) indicates the third-order convergence. □
Theorem 3. The iterative scheme (5) for solving has second-order convergence. Proof. Upon comparing to Equation (
17), the parameters in Equation (
5) are given by
Inserting
b into
in Equation (
28) yields
By Equation (
29),
proves this theorem. □
In practice, the iterative scheme (
5) is a variant of the Newton iterative scheme (
2). Both orders of convergence are two.
Notice that the Newton method (
2) is a single-point second-order optimal iterative scheme, with two function operations of
and
. Halley [
47] derived the following extension of the Newton method to a third-order iterative scheme:
However, because it needs three function operations on
,
and
, it is not an optimal iterative scheme. Besides the Halley method, there are many two-point iterative schemes which are of third-order convergence. Liu and Lee [
48] generalized many quadrature-type third-order iterative schemes to
Based on three function operations of
,
and
, Equation (
32) is not an optimal iterative scheme. It is interesting that upon comparing Equation (
31) to Equation (
17), these two iterative schemes are the same if we take
and
. But merely with
a and
b given by Equation (
24), the iterative scheme (
17) is of third-order convergence. It must be emphasized that we do not need
,
and two-point operation to achieve the third-order convergence. Therefore, the key issue of Equations (
17) and (
24) is that we need to give a precise estimation of
a and
b, without using the information from
and
.
4. Numerical Verifications
Some examples are used to evaluate the iterative scheme (
17), which is subjected to the convergence criteria:
We fix
for all numerical tests. The numerically computed order of convergence (COC) is defined in [
2]:
where
.
The presently computed results are compared to those obtained by the Newton method (NM), the Halley method (HM) [
47] in Equation (
31) and the method of Li (LM) [
18]:
The orders of convergence for the NM, HM and LM are two, three and four, respectively.
We first use the following example:
to present the monotonically decreasing sequence of
, which is generated by the method in Equation (
17). The parameters
a and
b are specified, not that given by Equation (
24).
There are three solutions
,
and
of Equation (
35). We consider four cases:
We take
,
and
for (a);
,
and
for (b);
,
and
for (c);
,
and
for (d). Cases (a) and (b) tend to the solution
; case (c) tends to
; and case (d) tends to
.
Due to the monotonically decreasing sequence of
, all the COCs are near to the third-order and they converge very fast. In the last column of
Table 1, we list the NIs by using the LM of [
18]. Starting from the same initial values, for the first two cases, the scheme (
17) is convergent slightly faster than the LM. As mentioned by Li [
18], the iterative scheme (
34) requires two evaluations of the function and one first-order derivative per iteration. Therefore, the scheme (
17) with one evaluation of the function saves much of the computational cost.
Other test examples are given by
Table 2 lists
,
a and
b and the NIs for different methods. We can observe that the present iterative scheme converges faster than the NM and HM. NM and HM are not good for the solution of
with a worse initial guess
, and cannot be applied to solve
in Equation (39).
5. Fourth-Order Optimal Iterative Schemes
Now, we propose some new FOIS by a constantly weighting combination of the third-order iterative schemes from Equations (
17) and (
24), as well as the following one. Before that, we cite the following result [
48].
Lemma 1. The following two-step iterative scheme has third-order convergence:where W satisfies The error equation reads as Proof. The proof can refer to [
48]. □
Theorem 4. The following iterative scheme as an optimal combination of Equations (17) and (40):is of fourth-order convergence, where , , , and Proof. The combination of Equations (
17) and (
40) is given by Equation (
42), whose weighting factors
and
are subjected to
Then, we consider the weighting combination of the error equations in Equations (
30) and (
41), such that the combined coefficient preceding
is zero:
Solving Equations (
44) and (
46), we can derive Equation (
43). Thus, the error equation of the optimally combined iterative scheme in Equation (
42) is
This completes the proof of Theorem 4. □
As an application of Theorem 4, we consider [
49]:
which in the form of Equation (
40) leads to
When we take
, the conditions
,
,
in Theorem 4 are satisfied. In
Table 3, we solve Equation (
35) by using the FOIS (
42), and list the results for three different solutions
, which show large values of the COC.
Let
be a parameter. With
in Equation (
40), we have
The best value of
is chosen such that
is an FOIS.
In
Table 4, we solve Equation (
36) and list the results for the solution
with four different initial guesses:
, which show large values of the COC.
6. Derivative-Free Iterative Schemes
In this section, we approximate
a and
b in Equation (
24). With two initial guesses
and
satisfying
to render
,
is taken. By a finite difference approximation of
a and
b, we take
Inserting
a and
b in Equation (
48) into Equation (
17), we solve Equation (
35) and the related data are tabulated in
Table 5.
We solve Equations (
36)–(39), and the related data are tabulated in
Table 6.
Remark 3. For the solution of , it does not exist and such that , due to . However, we place and on the right-side of the solution and the present iterative scheme is applicable to find the solution with 14 iterations, as shown in Table 6. If the curve of
vs.
x is available, we can observe a rough position of the solution
r, and then the slope
and the curvature
can be estimated roughly. Intuitively, we can estimate
a and
b by the slope and curvature. In order to maintain the fast convergence, we must choose
and
quite close to the solution
r, such that
are very close to
a and
b in Equation (
24), where
. For Equation (
35) with the solution
, if we take
and
, NI is greatly reduced from 11 to 5 and COC = 2.904; for
, NI reduces to 6 and COC = 2.862; and for
, NI reduces to 5 and COC = 2.936. Here, NI and COC are improved by comparing to that in
Table 5.
A greed search such as the 2D golden section search algorithm in the given range can help us to obtain the optimal values of
a and
b for fast convergence. However, it would spend much more computations. Instead of a greed search in the plane
, we discuss the influence of
and
in Equation (
49) by giving
and
. For different
c,
and
are different. In
Table 7, we list the results. Obviously, COC defined in Equation (
33) is sensitive to the values of
and
, when they approach to the optimal values, but NI does not have a large variation. When
and
tend to optimal values
and
, COC = 2.9041 tends to the theoretical one with COC = 3, as listed in
Table 1.
Like that performed in Equations (
48) and (
17), we introduce a derivative-free modification of the Newton variant in Equation (
5) to
where
For Equations (
36)–(39) solved by the first derivative-free Newton method (FDFNM), the related data are tabulated in
Table 8.
By neglecting the higher-order terms in Equations (
2) and (
3), we have
By using
and neglecting the higher-order terms, it follows from Equation (
53) a quadratic equation for
:
Inserting Equation (
54) into Equation (
52), we can derive the second modified Newton method:
which is different from the first modified Newton method (
5). Let
and we can obtain the second derivative-free Newton method (SDFNM):
We employ the SDFNM to solve Equations (
36)–(39), and the related data are tabulated in
Table 9.
Upon comparing
Table 6,
Table 8 and
Table 9, the performance of the presented method in Equations (
48) and (
17), as well as the FDFNM in Equations (
50) and (
51) and the SDFNM in Equations (
56) and (
57), are almost the same.
As a practical application of the proposed iterative schemes, let us consider a nonlinear boundary value problem:
The conventional shooting method is assumed to be an unknown initial slope
, and we integrate Equation (
58) with the initial conditions
and
, which results an implicit equation
to be solved. The exact solution is
.
We apply the fourth-order Runge–Kutta method to integrate Equation (
58) with
steps, and fix
and
. By using Equations (
48) and (
17), NI is 16, the error of
x is
and the maximum error of
u is
. While we use Equations (
50) and (
51), NI is increased to 18, and the errors are the same.
7. Hermite Interpolation
To be the extensions of the one-step Newton method (
2), there are, respectively, two-step and three-step methods of double Newton and triple Newton:
However, due to the low efficiency index (E.I.) = 1.414 of these iterative schemes, they are rarely used in the solution of nonlinear equations. Below, we will employ the generalized Hermite interpolation techniques to raise the values of E.I.
The Hermite function
for the interpolation of the data of a function
at two points
and
is such that
If
is a polynomial to match these four conditions in Equation (
63), it is at least a second-order function of
x, denoted as
. When
is computed from Equation (
61), it is not independent to
; hence, there exists an Hermite interpolation formula to predict
from the data
. The two-point Hermite interpolation formula was generalized in [
30], involving a weight function.
The second-order Hermite polynomial is constructed according to the Hermite interpolation conditions [
30]:
Wang and Liu [
36] derived
Then, it follows from
and Equations (
61) and (62) that
which expressed a certain two-point generalized Hermite interpolation.
Definition 1. A two-point generalized Hermite interpolation of in terms of and is depicted bywhere the weight function satisfies Equations (65) and (66) include Equation (64) as a special case. If one replaces
in the first equation in Equation (
60) by that in Equation (
65), it generates an FOIS [
17,
37,
38]:
The E.I. of Equation (
67) is now raised to E.I. = 1.587, which is better than E.I. = 1.414 of the double Newton method.
This fact encourages us also to replace
in Equation (
60) by the following three-point interpolation formula:
such that the combination of Equations (
60), (
65) and (
68) leads to
With certain conditions on
and
, the E.I. of the iterative scheme (
69) can be further raised to E.I. = 1.682. They are three-point EOISs.
8. Three-Point Generalized Hermite Interpolations
Using the third-order Hermite polynomial for the three-point
interpolation, Wang and Liu [
36] and Petković [
37] derived
where
and
and
are defined by the same fashion.
From the first two equations in Equation (
69) and Equation (62), we can derive the following divided differences:
Using Equations (
70), (
71) and (72) and through some manipulations, we can derive
where
Definition 2. A three-point generalized Hermite interpolation of in terms of and is defined by Equations (73) and (74) aswhere the weight functions , and satisfy For the two-point Hermite interpolation function in Equation (64), by Equation (74), we can derive Equation (77) is a special case of Equation (76). Unlike that in Equation (74), for the generalized interpolation in Equation (75), can be independent to . The function is subjected to four conditions, in which it is somewhat not easy to obtain . We attempt to construct it by a function with merely two conditions.
Lemma 2. A function , withcan be obtained fromwhere Proof. Inserting
into Equation (
79),
follows directly. It follows from Equation (
79) that
of which after inserting
and using Equation (
80), we can derive the last three conditions in Equation (
78). □
Taking advantage of Lemma 2, we can replace Equation (
76) by
There appear different interpolation techniques. Using a rational function, Sharma and Sharma [
50] derived
which in terms of Equation (
71) can be written as
Based on the Taylor series expansion,
was derived by Bi et al. [
34], which with the aid of Equations (
71) and (72) can be recast to
Both interpolations (
81) and (
82) are special cases of the generalized interpolation in Equation (
75).
To observe the accuracy of Equation (
75) and the other two interpolations (
81) and (
82), we consider two definite functions:
of which
and
. We define
to be the relative error of the interpolation of
, where
and
are, respectively, the value calculated from Equation (
75) and the exact value.
The following cases are with simple weight functions:
Table 10 lists the REs.
For , the accuracy of Cases (a)–(d) is much better than others because the interpolant is itself a third-order polynomial; however, for , the accuracy of all cases are of the levels and .
9. Three-Point Eighth-Order Optimal Iterative Schemes
In this section, we combine the two-point and three-point generalized Hermite interpolations to generate some eighth-order optimal iterative schemes (EOIS).
Theorem 5. Equation (69) has eighth-order convergence, if satisfiesand , and insatisfy the conditions in Equation (76). Proof. Before giving the proof, we emphasize that for the special case with
and
,
and
given in Equation (
77), Wang and Liu [
36] have proven the eighth-order convergence of the iterative scheme (
69) and derived the corresponding error equation. For saving space, the details are not written here, and the error equation is not written out explicitly.
Let
,
,
and
. As shown in [
36],
where
In view of Equations (
88)–(92),
in Equation (62) have the following asymptotic estimations:
Due to
and Equation (90), the term
in Equation (
69) has the following asymptotic estimation:
Therefore, we merely need to expand
g to the third order by
where Equation (
93) was taken into account.
Inserting
and Equations (
94) and (
95) into the last one in Equation (
69) yields
since
satisfies Equation (
86);
,
and
have the same values at
as listed in Equation (
76) with those in Equation (
77); and according to [
36], the coefficients preceding
to
are zeros for the iterative scheme (
69). Equation (
96) indicates that Equation (
69) has eighth-order convergence. Because we do not derive the error equation in an explicit form, many processes were omitted. □
Although, the details of the derivation of the corresponding error equation is not given here, we give two examples to verify the performance of the iterative scheme (
69), of which
and
are independent, not like that in [
36] as shown by Case (a) in Equation (
85), wherein
g is related to
h by Equations (
75) and (
74). We abbreviate the method in [
36] as WLM. For the purpose of comparison, the WLM is written as follows:
This iterative scheme was proved in [
36] to be eighth-order convergence, which is optimal, because there are four function operations on
. Compared to Equation (
69), which is also eighth-order convergence, and also with the same function operations on
, the two methods are different in their construction techniques.
The iterative scheme (
69), which involves
and
, is definitely more general than that in Equations (
97) and (98). The theoretical basis of Equation (
69) is the generalizations of the two-point and three-point Hermite interpolation methods.
Substituting the functions in Cases (a)–(f) into the iterative scheme (
69), we have different algorithms; in particular, using the functions in Equations (
81) and (
82), we recover the iterative schemes developed in [
34,
50], which are shortened as SSM and BRWM in
Table 11. The convergence criteria are
and
. It can be seen that these iterative schemes have the same performance.