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Many of the engineering problems are reduced to solve a nonlinear equation numerically, and as a result, an especial attention to suggest efficient and accurate root solvers is given in literature. Inspired and motivated by the research going on in this area, this paper establishes an efficient general class of root solvers, where per computing step, three evaluations of the function and one evaluation of the first-order derivative are used to achieve the optimal order of convergence eight. The without-memory methods from the developed class possess the optimal efficiency index 1.682. In order to show the applicability and validity of the class, some numerical examples are discussed.

Numerical solution of nonlinear scalar equations plays a crucial role in many optimization and engineering problems. For example, many engineering systems can be modeled as neutral delay differential equations (NDDEs) that involve a time delay in the derivative of the highest order, which are different from retarded delay differential equations (RDDEs) that do not involve a time delay in the derivative of the highest order. To illustrate more, a system, which consists of a mass mounted on a linear spring to which a pendulum is attached via a hinged massless rod, is used to predict the dynamic response of structures to external forces using a set of actuators, and it is modeled as an NDDE if the delay in actuators is taken into consideration [

There are numerical methods, which find one root at a time, such as Newton’s iteration or its variant, and the schemes, which find all the roots at a time, namely, simultaneous methods, such as Weierstrass method. Recently many journals such as Numerical Algorithms, Mathematical Problems in Engineering, Applied Mathematics and Computation, etc., have published new findings; see, for example, [

Noor et al. in [

In 2010, an eighth-order method is provided in [

Soleymani and Mousavi in [

For further reading, one may consult [

After providing a short background of this research in this section, we give the main contribution in Section

In order to contribute and give a general class of methods consistent with the optimality conjecture of Kung-Traub, an iteration eighth-order scheme without memory in this section should be constructed such that four evaluations per computing step are used. Such schemes are also known as predictor-corrector methods in which the first step is (Newton's step) predictor, while the other two steps correct the obtained solution. To achieve our goal, we consider the following three-step scheme on which the first two steps are the King's fourth-order family with one free parameter in real numbers,

Clearly in (

Solving the system of two linear equations with two unknowns, (

Considering (

Let

By defining

The class of three-step methods (

Future researches in this field of study can now be turned to finding optimal sixteenth-order four-step without-memory iterations based on the general class (

Typical forms of the weight functions satisfying (

Weight function | Form 1 | Form 2 | Form 3 |
---|---|---|---|

Interesting choices of

Method | Error equation | |||
---|---|---|---|---|

1 | −219 | |||

2 | 0 | −120 | ||

3 | −1 | −54 | ||

4 | 0 | 0 | ||

5 | 0 | −2 | −96 | |

The contribution given in Section

Test function, their roots, and the starting points.

Test functions | Roots | Guess |
---|---|---|

0.2588298273352688443917065956960 | 0.4 | |

2.55419595283704303782966617379187 | 2 | |

−1.25644837202179610636347166071080 | −1.3 | |

1.37318816571803393434603174397956 | 1.5 | |

−0.94316510784112785413792698729205 | −0.8 | |

−1.41582823891434741754690479611753 | −1.4 | |

−1.33663345183152516454641803558884 | −1.31 | |

0.931908190324955905662501231761206 | 1.5 | |

0.671280744928817559650708950129257 | 1 | |

−0.785398163397448309615660845819875 | −0.6 | |

0.714180223545189655580041854721814 | 1 | |

1.004983564135382106859677582107315 | 1 |

Comparison of different methods for finding the simple roots of test functions.

Test functions | Methods | NM | ( | ( | ( | ( | ( |
---|---|---|---|---|---|---|---|

IN | 8 | 4 | 3 | 3 | 3 | 3 | |

TNE | 16 | 32 | 12 | 12 | 12 | 12 | |

IN | 9 | 4 | 3 | 3 | 3 | 3 | |

TNE | 18 | 32 | 12 | 12 | 12 | 12 | |

IN | 9 | 4 | 3 | 3 | 3 | 3 | |

TNE | 18 | 32 | 12 | 12 | 12 | 12 | |

IN | 8 | 4 | 3 | 3 | 3 | 3 | |

TNI | 16 | 32 | 12 | 12 | 12 | 12 | |

IN | 8 | 5 | 3 | 3 | 3 | 3 | |

TNE | 16 | 40 | 12 | 12 | 12 | 12 | |

IN | 9 | 4 | 3 | 3 | 3 | 3 | |

TNE | 18 | 32 | 12 | 12 | 12 | 12 | |

IN | 9 | 4 | 3 | 3 | 3 | 3 | |

TNE | 18 | 32 | 12 | 12 | 12 | 12 | |

IN | 8 | 4 | 3 | 3 | 3 | 3 | |

TNE | 16 | 32 | 12 | 12 | 12 | 12 | |

IN | 9 | 4 | 3 | 3 | 3 | 3 | |

TNE | 18 | 32 | 12 | 12 | 12 | 12 | |

IN | 8 | 4 | 3 | 3 | 3 | 3 | |

TNE | 16 | 32 | 12 | 12 | 12 | 12 | |

IN | 8 | 4 | 3 | 3 | 3 | 3 | |

TNE | 16 | 32 | 12 | 12 | 12 | 12 | |

IN | 9 | 4 | 3 | 3 | 3 | 3 | |

TNE | 18 | 32 | 12 | 12 | 12 | 12 | |

All computations in this paper were performed in MATLAB 7.6 using variable precision arithmetic (VPA) to increase the number of significant digits. We have considered the following stopping criterion

If we need to solve a lot of equations from a large system of boundary-value problems, then the cost of function evaluations becomes important. Therefore, the proposed class (

In recent years, numerous works have been focusing on the development of more advanced and efficient methods for nonlinear scalar equations. Many methods have been developed, which improve the convergence rate of the Newton’s method. One practical drawback of so many methods is their slow rate of convergence. This paper has developed and established a rapid class of eighth-order iteration methods. Per iteration, the methods from our class require three evaluations of the function and one of its first derivatives; and therefore, the efficiency of the methods is equal to