# New Predictor-Corrector Methods with High Efficiency for Solving Nonlinear Systems

Authors: Alicia Cordero; Juan R. Torregrosa; María P. Vassileva
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

A new set of predictor-corrector iterative methods with increasing order of convergence is proposed in order to estimate the solution of nonlinear systems. Our aim is to achieve high order of convergence with few Jacobian and/or functional evaluations. Moreover, we pay special attention to the number of linear systems to be solved in the process, with different matrices of coefficients. On the other hand, by applying the pseudocomposition technique on each proposed scheme we get to increase their order of convergence, obtaining new efficient high-order methods. We use the classical efficiency index to compare the obtained procedures and make some numerical test, that allow us to confirm the theoretical results.


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## Body

## 1. Introduction

Many relationships in nature are inherently nonlinear, which according to these effects are not in direct proportion to their cause. Approximating a solutionξ of a nonlinear system, F(x)=0, is a classical problem that appears in different branches of science and engineering (see, e.g. [1]). In particular, the numerical solution of nonlinear equations and systems is needed in the study of dynamical models of chemical reactors [2] or in radioactive transfer [3]. Moreover, many of numerical applications use high precision in their computations; in [4], high-precision calculations are used to solve interpolation problems in astronomy; in [5] 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 [6], in particular for the calculation of zeros of nonlinear functions.Recently, many robust and efficient methods with high convergence order have been proposed to solve nonlinear equations, but in most of cases the schemes cannot be extended to multivariate problems. Few papers for the multidimensional case introduce methods with high order of convergence. The authors design in [7] a modified Newton-Jarrat scheme of sixth order; in [8] a third-order method is presented for computing real and complex roots of nonlinear systems; Shin et al. compare in [9] Newton-Krylov methods and Newton-like schemes for solving big-sized nonlinear systems; the authors in [10] and A. Iliev and I. Iliev in [11] show general procedures to design high-order methods by using frozen Jacobian and Taylor expansion, respectively. Special case of sparse Jacobian matrices is studied in [12].Dayton et al. in [13] formulate the multiplicity for the general nonlinear system at an isolated zero. They present an algorithm for computing the multiplicity structure, propose a depth-deflation method for accurate computation of multiple zeros, and introduce the basic algebraic theory of the multiplicity.In this paper, we present three new Newton-like schemes, of order of convergence four, six, and eight, respectively. After the analysis of convergence of the new methods, we apply the pseudocomposition technique in order to get higher-order procedures. This technique (see [14]) consists of the following: we consider a method of order of convergence p as a predictor, whose penultimate step is of order q, and then we use a corrector step based on the Gaussian quadrature. So, we obtain a family of iterative schemes whose order of convergence is min{q+p,3q}. This is a general procedure to improve the order of convergence of known methods.To analyze and compare the efficiency of the proposed methods we use the classic efficiency indexI=p1/d due to Ostrowski [15], where p is the order of convergence and d is the number of functional evaluations at each iteration.The convergence theorem in Section2 is demonstrated by means of the n-dimensional Taylor expansion of the functions involved. Let F:D⊆Rn→Rn be sufficiently Frechet differentiable in D. By using the notation introduced in [7], the qth derivative of F at u∈Rn, q≥1, is the q-linear function F(q)(u):Rn×⋯×Rn→Rn such that F(q)(u)(v1,…,vq)∈Rn. It is easy to observe that(1)
F(q)(u)(v1,…,vq-1,·)∈ℒ(Rn),(2)
F(q)(u)(vσ(1),…,vσ(q))=F(q)(u)(v1,…,vq), for all permutation σ of {1,2,…,q}.So, in the following we will denote:(a)
F(q)(u)(v1,…,vq)=F(q)(u)v1⋯vq,(b)
F(q)(u)vq-1F(p)vp=F(q)(u)F(p)(u)vq+p-1.It is well known that, forξ+h∈Rn lying in a neighborhood of a solution ξ of the nonlinear system F(x)=0, Taylor’s expansion can be applied (assuming that the jacobian matrix F′(ξ) is nonsingular), and
(1.1)F(ξ+h)=F′(ξ)[h+∑q=2p-1‍Cqhq]+O[hp],
where Cq=(1/q!)[F'(ξ)]-1F(q)(ξ), q≥2. We observe that Cqhq∈Rn since F(q)(ξ)∈ℒ(Rn×⋯×Rn,Rn) and [F'(ξ)]-1∈ℒ(Rn).In addition, we can express the Jacobian matrix ofF, F′ as
(1.2)F′(ξ+h)=F′(ξ)[I+∑q=2p-1‍qCqhq-1]+O[hp],
where I is the identity matrix. Therefore, qCqhq-1∈ℒ(Rn). From (1.2), we obtain
(1.3)[F′(ξ+h)]-1=[I+X2h+X3h2+X4h3+⋯][F′(ξ)]-1+O[hp],
where X2=-2C2,X3=4C22-3C3,….We denoteek=x(k)-ξ the error in the kth iteration. The equation e(k+1)=Lekp+O[ekp+1], where L is a p-linear function L∈ℒ(Rn×⋯×Rn,Rn), is called the error equation and p is the order of convergence.The rest of the paper is organized as follows: in the next section, we present the new methods of order four, six, and eight, respectively. Moreover, the convergence order is increased when the pseudocomposition technique is applied. Section3 is devoted to the comparison of the different methods by means of several numerical tests.


## 2. Design and Convergence Analysis of the New Methods

Let us introduce now a new Jarratt-type scheme of five steps which we will denote as M8. We will prove that its first three steps define a fourth-order scheme, denoted by M4, and its four first steps become a sixth-order method that will be denoted by M6. The coefficients involved have been obtained optimizing the order of the convergence, and the whole scheme requires three functional evaluations ofF and two of F′ to attain eighth order of convergence. Let us also note that the linear systems to be solved in first, second, and last step have the same matrix and also have the third and fourth steps, so the number of operations involved is not as high as it can seem.Theorem 2.1.
LetF:Ω⊆Rn→Rn be a sufficiently differentiable in a neighborhood of ξ∈Ω which is a solution of the nonlinear system F(x)=0, and let x(0) be an initial estimation close enough to the solution ξ. One also supposes that F′(x) is continuous and nonsingular at ξ. Then, the sequence {x(k)}k≥0 obtained by
(2.1)y(k)=x(k)-23[F′(x(k))]-1F(x(k)),z(k)=y(k)+16[F′(x(k))]-1F(x(k)),u(k)=z(k)+[F′(x(k))-3F′(y(k))]-1F(x(k)),v(k)=z(k)+[F′(x(k))-3F′(y(k))]-1[F(x(k))+2F(u(k))]x(k+1)=v(k)-12[F′(x(k))]-1[5F′(x(k))-3F'(y(k))][F'(x(k))]-1F(v(k))
converges to ξ with order of convergence eight. The error equation is
(2.2)ek+1=(C22-12C3)(2C23+2C3C2-2C2C3-209C4)ek8+O[ek9].Proof.
From (1.1) and (1.2) we obtain
(2.3)F(x(k))=F′(ξ)[ek+C2ek2+C3ek3+C4ek4+C5ek5+C6ek6+C7ek7+C8ek8]+O[ek9],F′(x(k))=F′(ξ)[I+2C2ek+3C3ek2+4C4ek3+5C5ek4+6C6ek5+7C7ek6+8C8ek7]+O[ek8].
As [F′(x(k))]-1F′(x(k))=I, we calculate
(2.4)[F′(x(k))]-1=[I+X2ek+X3ek2+X4ek3+X5ek4+X6ek5+X7ek6+X8ek7][F′(ξ)]-1+O[ek8],
where X1=I and Xs=-∑j=2s‍jXs-j+1Cj, for s=2,3,… So,
(2.5)[F′(x(k))]-1F(x(k))=ek+M2ek2+M3ek3+M4ek4+M5ek5+M6ek6+M7ek7+M8ek8+O[ek9],
where M2=C2+X4 and Ms=Cs+∑j=3s‍Xs-j+2Cj-1+Xs+2,s=3,4,….
Then,y(k)=ξ+(1/3)ek-(2/3)M and z(k)=ξ+(1/2)ek-(1/2)M, where M=M2ek2+M3ek3+M4ek4+M5ek5+M6ek6+M7ek7+M8ek8+O[ek9].
The Taylor expansion ofF′(y(k)) is
(2.6)F'(y(k))=F′(ξ)[I+Q1ek+Q2ek2+Q3ek3+Q4ek4+Q5ek5+Q6ek6+Q7ek7+Q8ek8]+O[ek9],
where
(2.7)Q1=23C2,Q2=13C3-43C2M2,Q3=427C4-43C3M2-43C2M3,Q4=581C5-89C4M2+43C3(M22-2M3)-43C2M4,Q5=281C6-4081C5M2+49C4(M22-M3)+43C3(M2M3+M3M2-M4)-43C2M5,Q6=7729C7-2081C6M2+4081C5(3M22-M3)+827C4(6M2M3+6M3M2-3M4-4M23)-C2M6,Q7=43C2M7+43C3(M2M5+M3M4+M4M3+M5M2+M6M2-M7)+827C4(6M2M4+6M32+6M4M2-3M5-4M22M3-4M2M3M2-4M3M22)+4081C5(3M2M3+3M3M2-M4-4M23)+2081C6(4M22-M3)-28243C7M2+82187C8,Q8=-43C2M8+43C3(M2M6+M3M5+M42+M5M3+M6M2-M7)+169C4(M2M5+M3M4+M4M3+M5M2-M6)+3227C4(M22M4+M2M32M4M2+M3M2M3+M32M2+M4M22)+4081C5(-M5-4M22M3-4M2M3M2-4M3M22+3M2M4+3M32+3M4M2+2M4)+2041C6(4M2M3+4M3M2-M4-8M23)+28243C7(5M22-M3)-1122187C8M2+1729C9.
We also obtain the Taylor expansion of F′(x(k))-3F′(y(k)):
(2.8)F′(x(k))-3F′(y(k))=F′(ξ)[-2I+A1ek+A2ek2+A3ek3+A4ek4+A5ek5+A6ek6+A7ek7+A8ek8]+O[ek9],
where As=(s+1)Cs+1-3Qs,s=1,2,…. As [F′(x(k))-3F′(y(k))]-1[F′(x(k))-3F′(y(k))]=I, we obtain
(2.9)[F′(x(k))-3F′(y(k))]-1=[-12I+Y2ek+Y3ek2+Y4ek3+Y5ek4+Y6ek5+Y7ek6+Y8ek7][F′(ξ)]-1+O[ek8],
where Y2=0 and Ys=(1/2)∑j=3s‍Ys-j+2Aj-2-(1/4)As-1,s=3,4,….
So,(2.10)[F′(x(k))-3F′(y(k))]-1F(x(k))=-12ek+R2ek2+R3ek3+R4ek4+R5ek5+R6ek6+R7ek7+R8ek8+O[ek9],
where R2=Y2-(1/2)C2 and Rs=Ys+∑j=3s‍Ys-j+2Cj-1-(1/2)Cs,s=3,4,….
We now calculateu(k)=z(k)+[F′(x(k))-3F′(y(k))]-1F(x(k)), and the error equation of the method at this step is
(2.11)eu(k)=12ek-12[M2ek2+M3ek3+M4ek4+M5ek5+M6ek6+M7ek7+M8ek8]-12ek+R2ek2+R3ek3+R4ek4+R5ek5+R6ek6+R7ek7+R8ek8+O[ek9]=P4ek4+P5ek5+P6ek6+P7ek7+P8ek8+O[ek9],
where Ps=-(1/2)Ms+Rs, s=4,5,… Then the first three steps define a fourth-order procedure, and
(2.12)F(u(k))=F′(ξ)[P4ek4+P5ek5+P6ek6+P7ek7+(P8+P42)ek8]+O[ek9],F(x(k))+2F(u(k))=F′(ξ)[ek+C2ek2+C3ek3+(C4+2P4)ek4+(C5+2P5)ek5+(C6+2P6)ek6+(C7+2P7)ek7+(C8+2P8)ek8]+O[ek9].
So,
(2.13)[F′(x(k))-3F′(y(k))]-1[F(x(k))+2F(u(k))]=-12ek+L2ek2+L3ek3+L4ek4+L5ek5+L6ek6+L7ek7+L8ek8+O[ek9],
where L2=Y2-(1/2)(C2+P2) and Ls=Ys+∑j=3s‍Ys-j+2Cj-1-(1/2)(Cs+Ps),s=3,4,….
At the fourth step, we calculate(2.14)ev(k)=zk-ξ+[F′(x(k))-3F′(y(k))]-1[F(x(k))+2F(u(k))]=N6ek6+N7ek7+N8ek8+O[ek9],
where Ns=Ls-(1/2)Ms and s=6,7,… This shows that the first four steps of the method define a sixth-order scheme. Indeed,
(2.15)F(v(k))=F′(ξ)[N6ek6+N7ek7+N8ek8]+O[ek9],(2.16)5F′(x(k))-3F′(y(k))=F′(ξ)[2I+B1ek+B2ek2+B3ek3+B4ek4+B5ek5+B6ek6+B7ek7+B8ek8]+O[ek9],
where Bs=5(s+1)Cs+1-3Qs and s=1,2… Then,
(2.17)[F′(x(k))]-1[5F′(x(k))-3F′(y(k))]=2I+S1ek+S2ek2+S3ek3+S4ek4+S5ek5+S6ek6+S7ek7+S8ek8+O[ek9],(2.18)[F′(x(k))]-1F(v(k))=2N6ek6+(N7+X2N6)ek7+(N8+X2N7+X3N6)ek8+O[ek9],
where S1=B1+2X2 and Ss=Bs+∑j=2s‍Xs-j+2Bj-1+2Xs+1, s=2,3,…. By multiplying (2.17) and (2.18),
(2.19)[F′(x(k))]-1[5F′(x(k))-3F′(y(k))][F′(x(k))]-1F(v(k))=2N6ek6+K7ek7+K8ek8+O[ek9],
where K7=2(N7+X2N6)+S1N6 and K8=2(N8+X2N7+X3N6)+S1(N7+X2N6)+S2N6.
Finally,(2.20)x(k+1)=ξ+[L8-12M8-12(2(N8+X2N7+X3N6)+S1(N7+X2N6)+S2N6)]ek8+O[ek9],
and the error equation is
(2.21)ek+1=(C22-12C3)(2C23+2C3C2-2C2C3-209C4)ek8+O[ek9],
proving that the order of convergence of the analyzed method is eight.In [14] the authors presented a new procedure to design higher-order schemes. This technique, called pseudocomposition, uses the two last steps of the predictor method to obtain a corrected scheme with higher order of convergence.Theorem 2.2 (see [14]).
LetF:Ω⊆Rn→Rn be differentiable enough Ω, let ξ∈Ω be a solution of the nonlinear system F(x)=0, and let x(0) be an initial estimation close enough to the solution ξ. We suppose that F′(x) is continuous and nonsingular at ξ. Let y(k) and z(k) be the penultimate and final steps of orders q and p, respectively, of a certain iterative method. Taking this scheme as a predictor we get a new approximation x(k+1) of ξ given by
(2.22)x(k+1)=y(k)-2[∑i=1m‍ωiF′(ηi(k))]-1F(y(k)),
where ηi(k)=(1/2)[(1+τi)z(k)+(1-τi)y(k)] and τi, ωi,i=1,2,…,m are the nodes and weights of the orthogonal polynomial corresponding to the Gaussian quadrature used. Then,(1)
the obtained set of families will have an order of convergence at leastq;(2)
ifσ=2 is satisfied, then the order of convergence will be at least 2q;(3)
if, also,σ1=0, the order of convergence will be min{p+q,3q},
where∑i=1n‍ωi=σ and ∑i=1n‍ωiτij/σ=σjwithj=1,2.Depending on the orthogonal polynomial corresponding to the Gaussian quadrature used in the corrector step, this procedure will determine a family of schemes. Furthermore, it is possible to obtain different methods in these families by using distinct number of nodes corresponding to the orthogonal polynomial used (see Table1). However, according to the proof of Theorem 2.2 the order of convergence of the obtained methods does not depend on the number of nodes used.Table 1
Values of the parameters for the different quadratures used.
Quadratures
Number of nodes
Chebyshev
Legendre
Lobatto
Radau
σ
σ
1
σ
σ
1
σ
σ
1
σ
σ
1
1
π
0
2
0
2
0
2
−1
2
π
0
2
0
2
0
2
0
3
π
0
2
0
2
0
2
0Let us note that these methods, obtained by means of Gaussian quadratures, seem to be known interpolation quadrature schemes such as midpoint, trapezoidal, or Simpson’s method (see [16]). It is only a similitude, as they are not applied on the last iteration x(k), and the last step of the predictor z(k), but on the two last steps of the predictor. In the following, we will use a midpoint-like as a corrector step, which corresponds to a Gauss-Legendre quadrature with one node; for this scheme the order of convergence will be at least min{q+p,3q}, by applying Theorem 2.2.The pseudocomposition can be applied to the proposed scheme M8 with iterative expression (2.1), but also to M6. By pseudocomposing on M6 and M8 there can be obtained two procedures of order of convergence 10 and 14 (denoted by PsM10 and PsM14), respectively. Let us note that it is also possible to pseudocompose on M4, but the resulting scheme would be of third order of convergence, which is worse than the original M4, so it will not be considered.Following the notation used in (2.1), the last step of PsM10 is
(2.23)x(k+1)=u(k)-2[F′(v(k)+u(k)2)]-1F(u(k)),
and the last three steps of psM14 can be expressed as
(2.24)v(k)=z(k)+[F′(x(k))-3F′(y(k))]-1[F(x(k))+2F(u(k))],w(k+1)=v(k)-12[F′(x(k))]-1[5F′(x(k))-3F′(y(k))][F′(x(k))]-1F(v(k)),x(k+1)=v(k)-2[F′(w(k)+v(k)2)]-1F(v(k)).In Figure1, we analyze the efficiency indices of the proposed methods, compared with Newton and Jarrat’s schemes and between themselves. There can be deduced the following conclusions: the new methods M4, M6, and M8 (and also the pseudocomposed PsM10 and PsM14) improve Newton and Jarratt’s schemes (in fact, the indices of M4 and Jarratt’s are equal). Indeed, for n≥3 the best index is that of M8. Nevertheless, none of the pseudocomposed methods improve the efficiency index of their original partners. Nevertheless, as we will see in the following section, the pseudocomposed schemes will show a very stable behavior that makes them worth.Figure 1
Efficiency index of the different methods for different sizes of the system.


## 3. Numerical Results

In order to illustrate the effectiveness of the proposed methods, we will compare them with other known schemes. Numerical computations have been performed in MATLAB R2011a by using variable-precision arithmetic, which uses floating-point representation of 2000 decimal digits of mantissa. The computer specifications are Intel(R) Core(TM) i5-2500 CPU @ 3.30 GHz with 16.00 GB of RAM. Each iteration is obtained from the former by means of an iterative expressionx(k+1)=x(k)-A-1b, where x(k)∈ℝn, A is a real matrix n×n and b∈ℝn. The matrix A and vector b are different according to the method used, but in any case, we calculate A-1b as the solution of the linear system Ay=b, with Gaussian elimination with partial pivoting. The stopping criterion used is ||x(k+1)-x(k)||<10-200 or ||F(x(k))||<10-200.Firstly, let us consider the following nonlinear systems of different sizes:(1)
F1=(f1(x),f2(x),…,fn(x)), where x=(x1,x2,…,xn)T and fi:ℝn→ℝ,i=1,2,…,n, such that
(3.1)fi(x)=xixi+1-1,i=1,2,…,n-1,fn(x)=xnx1-1.
When n is odd, the exact zeros of F1(x) are: ξ1=(1,1,…,1)T and ξ2=(-1,-1,…,-1)T.(2)
F2(x1,x2)=(x12-x1-x22-1,-sin(x1)+x2) and the solutions are ξ1≈(-0.845257,-0.748141)T and ξ2≈(1.952913,0.927877)T.(3)
F3(x1,x2)=(x12+x22-4,-exp(x1)+x2-1), being the solutions ξ1≈(1.004168,-1.729637)T and ξ2≈(-1.816264,0.837368)T.(4)
F4(x1,x2,x3)=(x12+x22+x32-9,x1x2x3-1,x1+x2-x32) with roots ξ1≈(2.14025,-2.09029,-0.223525)T, ξ2≈(2.491376,0.242746,1.653518)T and ξ1≈(0.242746,2.491376,1.653518)T.Table2 presents results showing the following information: the different iterative methods employed (Newton (NC), Jarratt (JT), the new methods M4, M6, and M8, and the pseudocomposed PsM10 and PsM14), the number of iterations Iter needed to converge to the solution Sol, the value of the stopping factors at the last step, and the computational order of convergence ρ (see [17]) approximated by the formula:
(3.2)ρ≈ln(∥x(k+1)-x(k)∥)/(∥x(k)-x(k-1)∥)ln(∥x(k)-x(k-1)∥)/(∥x(k-1)-x(k-2)∥).
The value of ρ which appears in Table 2 is the last coordinate of the vector ρ when the variation between their coordinates is small. Also the elapsed time, in seconds, appears in Table 2, being the mean execution time for 100 performances of the method (the command cputime of MATLAB has been used).Table 2
Numerical results for functionsF1 to F4.
Function
Method
Iter
Sol
∥
x
(
k
)
-
x
(
k
-
1
)
∥
∥
F
(
x
(
k
)
)
∥
ρ
e-time (sec)
NC
8
ξ
1
1.43
e
-
121
2.06
e
-
243
2.0000
8.6407
JT
4
ξ
1
1.69
e
-
60
2.06
e
-
243
4.0000
3.9347
F
1
M4
4
ξ
1
1.69
e
-
60
2.06
e
-
243
4.0000
3.7813
x
(
0
)
=
(
0.8
,
…
,
0.8
)
M6
4
ξ
1
6.94
e
-
193
4.33
e
-
1160
6.0000
5.3911
M8
3
ξ
1
9.40
e
-
50
3.51
e
-
4011
8.0913
5.0065
PsM10
3
ξ
1
1.28
e
-
91
9.54
e
-
921
10.0545
4.9061
PsM14
3
ξ
1
4.65
e
-
164
0
14.0702
6.1018
NC
17
ξ
1
3.37
e
-
340
1.14
e
-
340
—
9.2128
JT
9
ξ
1
8.18
e
-
085
1.14
e
-
340
4.0000
10.1416
F
1
M4
9
ξ
1
8.18
e
-
085
1.14
e
-
340
4.0000
10.9104
x
(
0
)
=
(
0.0015
,
…
,
0.0015
)
M6
7
ξ
1
1.40
e
-
035
9.46
e
-
216
—
12.3266
M8
19
ξ
1
9.50
e
-
030
1.29
e
-
240
—
59.4832
PsM10
6
ξ
1
3.02
e
-
102
5.23
e
-
1027
—
17.9957
PsM14
5
ξ
1
1.84
e
-
162
0
—
22.6130
NC
9
ξ
1
2.45
e
-
181
5.92
e
-
362
2.0148
0.2395
JT
5
ξ
1
9.48
e
-
189
8.13
e
-
754
4.0279
0.3250
F
2
M4
5
ξ
1
9.48
e
-
189
8.13
e
-
754
4.0279
0.1841
x
(
0
)
=
(
-
0.5
,
-
0.5
)
M6
4
ξ
1
1.34
e
-
146
2.14
e
-
878
5.9048
0.2744
M8
3
ξ
1
1.90
e
-
038
1.23
e
-
302
7.8530
0.3718
PsM10
3
ξ
1
6.72
e
-
72
2.68
e
-
714
9.9092
0.4674
PsM14
3
ξ
1
2.13
e
-
122
1.95
e
-
1706
13.9829
0.3187
NC
13
ξ
1
2.20
e
-
182
2.73
e
-
374
1.9917
0.3713
JT
7
ξ
1
2.10
e
-
179
4.51
e
-
716
3.9925
0.4001
F
2
M4
7
ξ
1
2.10
e
-
179
4.51
e
-
716
3.9925
0.7535
x
(
0
)
=
(
-
5
,
-
3
)
M6
8
ξ
1
2.55
e
-
036
5.81
e
-
216
—
0.9382
M8
>5000
PsM10
4
ξ
1
2.59
e
-
021
3.51
e
-
208
—
0.4363
PsM14
29
ξ
2
9.45
e
-
020
5.05
e
-
273
—
7.8090
NC
10
ξ
1
1.65
e
-
190
4.61
e
-
380
2.0000
1.4675
JT
5
ξ
1
8.03
e
-
113
7.59
e
-
450
3.9995
0.3151
F
3
M4
5
ξ
1
8.03
e
-
113
7.59
e
-
450
3.9995
0.3034
x
(
0
)
=
(
2
,
-
3
)
M6
4
ξ
1
1.25
e
-
082
2.83
e
-
493
6.0015
0.3696
M8
4
ξ
1
1.54
e
-
162
3.16
e
-
1296
7.9993
0.4463
PsM10
3
ξ
1
5.59
e
-
044
1.40
e
-
436
9.4708
0.4682
PsM14
3
ξ
1
3.46
e
-
068
3.45
e
-
948
13.1659
0.5925
NC
35
ξ
1
3.71
e
-
177
2.33
e
-
253
—
1.4828
JT
11
ξ
1
3.29
e
-
143
1.67
e
-
574
—
0.7781
F
3
M4
11
ξ
1
3.29
e
-
143
1.67
e
-
574
—
0.7535
x
(
0
)
=
(
0.2,0.1
)
M6
9
ξ
1
1.31
e
-
064
3.61
e
-
385
—
0.8001
M8
n.c.
ξ
1
PsM10
5
ξ
1
6.85
e
-
156
1.06
e
-
1555
—
0.6352
PsM14
8
ξ
2
7.87
e
-
155
0
—
1.1870
NC
10
ξ
1
1.03
e
-
135
1.55
e
-
270
1.9995
2.3263
JT
5
ξ
1
9.94
e
-
073
2.09
e
-
289
4.0066
0.5296
F
4
M4
5
ξ
1
9.94
e
-
073
2.09
e
-
289
4.0066
0.6340
x
(
0
)
=
(
1
,
-
1.5
,
-
0.5
)
M6
4
ξ
1
9.31
e
-
057
4.86
e
-
338
5.9750
0.7443
M8
4
ξ
1
4.43
e
-
046
1.08
e
-
364
—
0.8282
PsM10
3
ξ
1
1.43
e
-
031
1.04
e
-
311
9.6674
0.8100
PsM14
3
ξ
1
1.91
e
-
033
4.05
e
-
462
13.9954
1.0465
NC
12
ξ
1
1.08
e
-
192
1.55
e
-
384
1.9996
2.7271
JT
6
ξ
1
2.31
e
-
103
7.97
e
-
412
4.0090
0.7761
F
4
M4
6
ξ
1
2.31
e
-
103
7.97
e
-
412
4.0090
1.0301
x
(
0
)
=
(
7
,
-
5
,
-
5
)
M6
5
ξ
1
2.99
e
-
086
4.69
e
-
515
—
1.0090
M8
15
ξ
3
1.77
e
-
071
1.48
e
-
568
—
3.4007
PsM10
4
ξ
1
6.86
e
-
067
1.25
e
-
666
—
1.0245
PsM14
7
ξ
2
1.09
e
-
130
9.15
e
-
1825
—
1.8179We observe from Table2 that not only the order of convergence and the number of new functional evaluations and operations are important in order to obtain new efficient iterative methods to solve nonlinear systems of equations. A key factor is the range of applicability of the methods. Although they are slower than the original methods when the initial estimation is quite good, when we are far from the solution or inside a region of instability, the original schemes do not converge or do it more slowly, the corresponding pseudocomposed procedures usually still converge or do it faster.The advantage of pseudocomposition can be observed in Figures2(a) and 2(b) (methods M6 and PsM10) and Figures 3(a) and 3(b) (methods M8 and PsM14) where the dynamical plane on R2 is represented: we consider the system of two equations and two unknowns F2(x)=0, for any initial estimation in R2 represented by its position in the plane, a different color (blue or orange, as there exist only two solutions in this region) is used for the different solutions found (marked by a white point in the figure). Black color represents an initial point in which the method converges to infinity, and the green one means that no convergence is found (usually because any linear system cannot be solved). It is clear that when many initial estimations tend to infinity (see Figure 3(a)), the pseudocomposition “cleans” the dynamical plane, making the method more stable as it can find one of the solutions by using starting points that do not allow convergence with the original scheme (see Figure 3(b)).Real dynamical planes for systemF2(x)=0 and methods M6 and PsM10.
(a)
M6
(b)
PsM10Real dynamical planes for systemF2(x)=0 and methods M8 and PsM14.
(a)
M8
(b)
PsM14If an analogous study is made on systemF3(x)=0, similar conclusions can be obtained, as the effect of smoothness is clear when the real dynamical plane of a method and its pseudocomposed partner are compared. So, in Figure 4 the amount of points in the lower half of the plane that converge to one of the roots is higher after the pseudocomposition, and, in Figure 5, there is a big green region of no convergence for method M8 that shows to be convergent when pseudocomposition is applied in PsM14.Real dynamical planes for systemF3(x)=0 and methods M6 and PsM10.
(a)
M6
(b)
PsM10Real dynamical planes for systemF3(x)=0 and methods M8 and PsM14.
(a)
M8
(b)
PsM14We conclude that the presented schemes M4, M6, and M8 show to be excellent, in terms of order of convergence and efficiency, but also that the pseudocomposition technique achieves to transform them in competent and more robust new schemes.



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*Source: 709843-2012-09-29.xml*