Solving the nonlinear equations by the Newton-homotopy continuation method with adjustable auxiliary homotopy function

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Abstract

In this paper, the concept of adjustable auxiliary homotopy function for the Newton-homotopy continuation method is presented. By means of adjusting the auxiliary function, we can solve the nonlinear equations and guarantee the solutions exactly without divergence rather than the traditional numerical methods such as the Newton–Raphson method and so on.

Introduction

Some uncontrollable situations, such as overflow, divergence, etc., usually happen in solving the nonlinear equations. Bad structure of equations or initial guesses will impel these situations. Homotopy continuation method was known as early as in the 1930s. This method was used by kinematician in the 1960s at US for solving mechanism synthesis problems. The latest development was done by Morgan [1], [2] at GM. We also have two important literatures: Garcia [3] and Allgower [4]. Continuation method gives a set of certain answers and some simple iteration process to obtain our solutions more exactly. Wu [6] focused on the numerical system and developed some useful rules by the homotopy continuation method to avoid the uncontrollable situations. However, the study only dealt with the nonlinear equation not the simultaneous nonlinear equations. Wu [5] also applied the homotopy continuation method to search all the roots of inverse kinematics problem of robot and obtained more but not all convergence answers than the traditional numerical methods. This paper extends these studies to the general nonlinear equations system by a new concept “adjustable auxiliary homotopy function” to solve all the nonlinear equations.

Section snippets

The Newton-homotopy continuation method and its adjustable auxiliary homotopy function

As we known, we have two kinds of nonlinear equations: they are nonlinear equation and simultaneous nonlinear equations, shown as Eqs. (1), (2), respectivelyf(x)=0andF(X)=0i.e.f(x,y,,z)=0,g(x,y,,z)=0,h(x,y,,z)=0.

To solve these equations, we have many different numerical methods such as the Newton–Raphson method and so on. The numerical iteration formula of Newton’s method for solving these equations are given asxn+1=xn-f(xn)f(xn)andf(xn,yn,)xf(xn,yn,)yg(xn,yn,)xg(xn,yn,)yxn

Conclusion

Typically, it is usually a big trouble and disadvantage for us to do the algebraic operation, for example, solving the nonlinear equations. Fortunately, by the aid of computer science, the nonlinear equations will be solved no more difficulty. With the computer improved and the numerical technique developed, the solutions of nonlinear equations become easier. We can use current high-speed processor to determine the solutions quickly.

Although, we already have many different numerical methods can

References (6)

  • A.P. Morgan, A Method for Computing all Solutions to Systems of Polynomial Equations, GM Research Publication, GMR...
  • A.P. Morgan

    Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems

    (1987)
  • C.B. Garcia et al.

    Pathways to Solutions, Fixed Points, and Equilibria

    (1981)
There are more references available in the full text version of this article.

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    However, it should be noted that even with the use of damping and Jacobian updates at each or some iterations, the Newton method can diverge from some starting points. Homotopy and continuation methods offer a way to address the divergence problems of Newton methods [30,31]. Several researchers have investigated the basin of attraction of the Newton method for nonlinear equations and systems of nonlinear equations [32–35].

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