In this paper, a one-step Steffensen-type method with super-cubic convergence for solving nonlinear equations is suggested. The convergence order 3.383 is proved theoretically and demonstrated numerically. This super-cubic convergence is obtained by self-accelerating second-order Steffensen’s method twice with memory, but without any new function evaluations. The proposed method is very efficient and convenient, since it is still a derivative-free two-point method. Its theoretical results and high computational efficiency is confirmed by Numerical examples.

Finding the root of a nonlinear equation

is a classical problem. It is well-known in scientific computation that Newton’s method (NM, see [

is widely used for root-finding, where

NM/SM converges quadratically and requires two function evaluations per iteration. The efficiency index of them is

Besides H.T. Kung and J.F. Traub conjectured that an iterative method based on

where

A lot of self-accelerating Steffensen-type methods were derived in the literature (see [

interpolatory polynomial

and M.S. Petkovića proposed a cubically convergent Steffensen-like method (see [

In this study, a one-step Steffensen-type method is proposed by doubly-self-accelerating in Section 2, its super-cubic convergence is proved in Section 3, and numerical examples are demonstrated in Section 4.

By the first-order Newtonian interpolatory polynomial

we have

where

So, with some

should be better than

Therefore, we suggest

where

vanish the asymptotic convergence constant, we establish a self-accelerating Steffensen’s method with super quadratic convergence as follows:

Furthermore, we propose a one-step Steffensen-type method with super cubic convergence by doubly-self- accelerating as follows:

Lemma 3.1

Proof. By Taylor formula, we have

So,

Then, the proof can be completed.

Theorem 3.2 Let

Proof. If

and if

Then

By Taylor formula and Lemma 3.1, we also have

So, comparing the exponents of

From its non-trivial solution

As the efficiency index is

and (9) are

Related one-step methods only using two function evaluations per iteration are showed in the following numeri- cal examples. The proposed method is a derivative-free two-point method with high computational efficiency.

Example 1. The numerical results of NM, SM, (4), (5) and (9) in

Example 2. The numerical results of NM, SM, (4), (5) and (9) are in

.

Methods | n | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

NM | . | 0.53279e−2 | 0.35561e−5 | 0.15808e−11 | 0.31235e−24 | 0.12195e−49 | 0.15890e−100 |

. | 2.25256 | 2.01691 | 0.15808e−11 | 2.00000 | 2.00000 | 2.00000 | |

SM | . | 0.28174e−1 | 0.51325e−3 | 0.16476e−6 | 0.16966e−13 | 0.17989e−27 | 0.20226e−55 |

. | 1.21776 | 2.04376 | 2.00830 | 2.00009 | 2.00000 | 2.00000 | |

(4) | . | 0.28174e−1 | 0.15996e−4 | 0.13132e−12 | 0.43283e−32 | 0.38442e−79 | 0.99936−193 |

. | 1.21776 | 3.81335 | 2.49109 | 2.40945 | 2.41512 | 2.41406 | |

(5) | . | 0.28174e−1 | 0.16560e−6 | 0.11521e−21 | 0.39821e−67 | 0.16444e−203 | 0.11580e−612 |

. | 1.21776 | 6.14536 | 2.89776 | 2.99925 | 3.00000 | 3.00000 | |

(9) | . | 0.28174e−1 | 0.43010e−7 | 0.21604e−27 | 0.23153e−94 | 0.20021e−321 | 0.69689e−1090 |

. | 1.21776 | 6.83322 | 3.49004 | 3.29917 | 3.39052 | 3.38434 |

. Numerical results for solving

Methods | NM | SM | (4) | (5) | (9) |
---|---|---|---|---|---|

. Numerical results for solving | 0.19785e−40 | 0.88156e−29 | 0.50439e−84 | 0.19314e−313 | 0.75162e−578 |

. Numerical results for solving | 2.0000 | 2.0000 | 2.4141 | 3.0000 | 3.3831 |

. Numerical results for solving | 0.32328e−44 | 0.42920e−26 | 0.19843e−85 | 0.57587e−282 | 0.13494e−706 |

. Numerical results for solving | 2.0000 | 2.0000 | 2.4141 | 3.0000 | 3.3825 |

. Numerical results for solving | 0.18813e−51 | 0.15758e−18 | 0.12013e−86 | 0.34524e−286 | 0.27679e−677 |

. Numerical results for solving | 2.0000 | 2.0000 | 2.4140 | 3.0000 | 3.3796 |

. Numerical results for solving | 0.35988e−79 | 0.96290e−84 | 0.16834e−248 | 0.21536e−597 | 0.25291e−1154 |

. Numerical results for solving | 2.0000 | 2.0000 | 2.4161 | 3.0000 | 3.3831 |

By theoretical analysis and numerical experiments, we confirm that the proposed method which is a derivative- free two-point method has high computational efficiency. Its convergence order is 3.383 and its efficiency index is 1.839. We can see that the suggested method is suitable to solve nonlinear equations and can also be used for solving boundary-value problems of nonlinear ordinary differential equations.