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In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of nonlinear ODEs are shown as well in the numerical examples.

We consider a system of nonlinear equations as follows:

where,

1) There exists an

2)

3)

How to efficiently solve the system of nonlinear Equation (1) is a typical issue in scientific computation and engineering field. The most famous method is probably Newton’s method as follows (see [

where x_{0} is an initial guess of the root. This is one-step method, and it uses two evaluations of the function and derivative to achieve second-order convergence by iteration.

Besides, third-order iterative equations, such as the Halley iterative method and Chebyshev iterative method, are often used. In recent years, a few similar methods [

Especially, M. T. Darvish & A. Barati [

Frontini & Sormani [

These are classic two-step Newton-type methods to achieve cubic convergence to approximate the root of a system of nonlinear equations.

In this paper, we propose a new variant of two-step Newton’s method with the third-order convergence by quadrature formulae in Section 2, some numerical examples using this new method for solving systems of nonlinear equations and boundary-value problems of nonlinear ODEs in Section 3, and finally make conclusions in Section 4.

Based on the multivariable mean-value theorem

we use the left rectangular integral rule

and use

substituting

Now, we applied the quadrature formula

to construct the following new scheme:

We state and prove the convergence theorem as follows:

Theorem Let _{0} be close to

Proof. As

By Taylor’s expansion, we have

From

where

We have

By (6), therefore

And

where

By Toylor’s expansion

Furthermore, by (8) and (10), we have

Finally, using (7), (11) and (6), we obtain the error equation as

That is

Using (9), so,

This shows that the method is the third-order convergence.

The new scheme of two-step Newton-type method (6) uses three evaluations of the function and derivative per iteration to achieve third-order convergence for solving a simple root of systems of nonlinear equations. As the efficiency index is

The iterative method (6) is demonstrated by solving some systems of nonlinear equations and boundary-value problems of nonlinear ODE.

Example 1. Consider the system of nonlinear equation:

where

According to results of the numerical experiments, the new iterative method (6) can achieve super third-order convergence for some systems of nonlinear equations.

Example 2. We have numerical results in

1 | 3.444444444444444039576788 | −1.025641025641025617667891 | 1.446e−0 | 1.971e−0 |

2 | 1.999998516331195414738740 | −0.9999996492101374900753236 | 1.524e−6 | 1.854e−6 |

3 | 2.000000000000000000004496 | −1.0000000000000000000008992 | 4.585e−21 | 5.395e−21 |

4 | 1.999999999999999999999999 | −0.999999999999999999999999 | 7.725e−65 | 9.091e−65 |

1 | 0.499565371550889536 | 0.006147092477281110 | −0.521628814405284191 | 1.07e−1 |

2 | 0.500000035199424988 | 0.000005327105559032 | −0.523598502730812839 | 8.61e−5 |

3 | 0.500000000000000033 | 0.000000000000003291 | −0.523598775598308161 | 1.97e−13 |

4 | 0.500000000000000000 | 0.000000000000000000 | −0.523598775598298873 | 1.09e−31 |

1 | 2 | 3 | 4 | |
---|---|---|---|---|

2.439e−2 | 1.238e−10 | 1.612e−14 | 1.612e−14 | |

1.996e−3 | 6.128e−12 | 7.926e−37 | 1.729e−112 |

^{*}Corresponding author.

where

The above numerical results agree with the theoretical analysis on the convergence, and the iterative efficiency is comparative high.

Example 3. Consider solving the following boundary-value problem of nonlinear ODE:

Discretize the nonlinear ODE (15) with the finite difference method. Taking nodes

where

The results for the system of nonlinear equations of ODE (15) are shown in

In this paper, we presented a new two-step iterative method of cubic convergence to solve systems of nonlinear equations. Through theoretical analysis and numerical experiments, we believe that the new variant of Newton- type method is so efficient and fast convergent as to be able to find solutions with required accuracy. To sum up, this method is suitable for solving systems of nonlinear equations, and can be used to resolve boundary-value problems of nonlinear ordinary differential equations as well.

The work is funded by the Science & Technology Program of Beijing Municipal Commission of Education (No. KM201511417012).

ZhongliLiu,QuanyouFang, (2015) A New Newton-Type Method with Third-Order for Solving Systems of Nonlinear Equations. Journal of Applied Mathematics and Physics,03,1256-1261. doi: 10.4236/jamp.2015.310154