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.