^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

We present the Steffensen method in

Let

In recent years, some authors [

which is given for solving algebraic equation (only in one dimension) [

Newton's method can be extended to high dimension directly, nevertheless the Steffensen method cannot. So the Steffensen method in high dimension is seldom researched. In [

In this paper, we discuss another form of the Steffensen method in

Compared with Newton's iteration, the advantage of this form avoids evaluation of the derivative but has the same convergence order. In the paper, we also establish the convergence theorem under Kantorovich condition and give the error estimate.

This paper is organized as follows. In Section

Let

Applying the iteration (

When

Under the assumptions of previous Lemma, the real sequence

Suppose that the sequence

Let

Let

Let

Let

For an initial value

If

Let

If the sequence

By (

If

The conclusions (a), (b), and (c) are obviously true for

By Lemma

Since

By (b) in Lemma

We establish a theorem of convergence for the iteration (

We consider the following equation in 2D:

The initial vector is

The numerical comparison between Newton's method and Steffensen method.

Step | Steffensen method | Newton's method |
---|---|---|

1 | ||

5 | ||

10 |

We consider the following nonlinear boundary value problem of second order [

To solve this problem by finite differences, we start by drawing the usual grid line with

The numerical comparison between Newton's method and Steffensen method of

Steffensen method | Newton's method | |
---|---|---|

1 | ||

2 |

We consider the following nonlinear boundary value problem of Burgers-Huxley equation:

This equation's exact solution is

We solve system (

The error of the approximation solution _{n} and the exact solution

Method | Step | Error |
---|---|---|

Steffensen | 7 | |

Newton | 10 |

The derivatives may prevent the application of the methods especially when they are not easy to find. In this paper, we show that the Steffensen method only uses evaluations of the function but maintains quadratic convergence and can converge. The new iterative method seems to work well in our numerical results, since we have obtained optimal order of convergence without any stability problem. With the different choice of