Newton waveform relaxation method for solving algebraic nonlinear equations

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Abstract

We introduce a new notion for solving algebraic nonlinear equations, which is derived from the well known waveform relaxation iterative method, and is well suited to couple different numerical method for nonlinear equations, such as classical Newton’s method, quasi-Newton method, Conjugate-Gradient method, etc. We show in this paper that the arithmetic obtained by coupling the classical Newton’s method has essential capability for parallel computation and converges globally. Numerical results validate the theoretical analysis very well.

Introduction

Let us begin with the waveform relaxation method which is an important and effective numerical method for large scale differential systems [4], [5], [6], [7], [8], [9], [10], [11]. Consider the following systems:y(t)=g(t,y(t)),ta,y(a)=η,t=a,where yRn and g:R×RnRn. The key idea of waveform relaxation method for systems (1.1) is to chose some function G:R×Rn×RnRn and then solve the following equations successively:yk+1(t)=G(t,yk(t),yk+1(t)),ta,yk+1(a)=η,t=a,where y0(t) is a given initial approximation of y(t) which is the solver of (1.1). Then we obtain yk+1(t) from yk(t) by solving (1.2), and the corresponding process is called continuous-time waveform relaxation iteration.

The function G, which is called splitting function, is chosen to attempt to decouple systems (1.1) into easily solvable independent subsystems, which may then be solved separately. The function G is minimally assumed to satisfy a consistency condition, which ensures that the solution to (1.1) is a fixed one of (1.2), i.e.,G(t,v(t),v(t))=g(t,v(t)),for any function v(t)Rn.

Assume we have to solve nonlinear equationsf(x)=0,where f:DRnRn. Provided we have a starting guess x0 of the unknown solutions x at hand, like the continuous-time waveform relaxation iteration, we choose a function F:D×DRn which satisfiesF(x,x)=f(x),for any xRn.

And then we solve the following equations from xk to xk+1:F(xk,xk+1)=0.If xkxˆ as k+, by the consistency condition (1.4), we know that xˆ is a solution of nonlinear Eq. (1.3).

We see that it is well suited to couple different numerical methods to solve (1.5), like classical Newton’s method, quasi-Newton method, Conjugate-Gradient method, etc.

In this paper, we use classical Newton’s method to solve (1.5). This leads to the following arithmetic written compactly as:fork=0,1,with a given initial approximationx¯0ofxk+1,form=0,1,,M-1F2(xk,x¯m)x¯m=-F(xk,x¯m),x¯m+1=x¯m+x¯,endxk+1=x¯M,endhere and below:F2(x,y)=F(x,z)z|z=y.If we set x¯0=xk and M=1 in (1.6), by the consistency condition (1.4), arithmetic (1.6) is equivalent toF2(xk,xk)xk=-f(xk),xk+1=xk+xk,k=0,1,And we call method (1.8) Newton waveform relaxation method.

At a first glimpse the iterative formula (1.8) is very similar to the classic Newton’s method for (1.3) written asf(xk)xk=-f(xk),xk+1=xk+xk,k=0,1,provided f is at least once continuously differentiable. Newton’s method is an important and basic method [12], which converges quadratically and locally.

To know the key ideas of Newton’s method in the historical perspective as well as modern investigations and applications, we can refer the survey paper [13] and references therein.

But at a second glimpse, we notice that the splitting function F which satisfies (1.4) can be chosen broadly. Special choice of F can make F2(x,x) be a diagonal or block diagonal matrix and invertible in Rn. Therefore, the iterative method (1.8) can be processed simultaneously and stably by computer with multi-processor.

The organization of this paper is as follows: Section 2 provides the main result and its proof. Section 3 illustrates two numerical examples to validate our theoretic analysis.

Section snippets

Convergence analysis

At present moment, we give an affine covariant Lipschitz condition which is sufficient to guarantee the convergence of the method (1.8), and many others can be done in future.

Lemma 2.1

Let F:D×DRn be a continuously mapping with DRn open and convex and satisfy the consistent condition (1.4). Function F(x,y) is Fréchet differentiable with the second variable y and matrix F2(x,y) is invertible for any x,yD. Assume thatF2-1(x,x)(F(y,y)-F(x,y))αx-y,x,yD,F2-1(x,x)(F2(x,x)-F2(x,y))βx-y,x,yD,x0-x

Numerical results

In this section, we give two examples to illustrate the global convergence and the considerable computational efficiency of the method (1.8).

Example 1

Consider the following equations:f(x)=109arctan(x1109)+esin(x2)+3x1+x2-sin(3x1)=0.LetF(xk,xk+1)=109arctan(x1,k109)+[a(x1,k+1-x1,k)+1]esin(x2,k)+3x1,k+bx2,k+1+(1-b)x2,k-sin(3x1,k),and it is clear that F(x,x)=f(x), where a,bR and a,b0.

Routine calculation yieldsF2(xk,x)=aesin(x2,k)00b,and thus the Lipschitz constant β=0 in this example. It is clear from

Conclusions

We suggest a new notion for solving algebraic nonlinear equations, which is based on the waveform relaxation method. The key idea of this notion is to chose some splitting function F(x,x) which satisfies F(x,x)=f(x) for any xRn. The new notion coupled with the classical Newton’s method leads to the arithmetic (1.6), (1.8). By some special choice of F, the method (1.8) takes vantage of global convergence, less storage and simultaneous computation.

Acknowledgements

The author would like to thank Prof. Shi Baochang and Dr. Lu Zhihong from department of mathematics of Huazhong University of Science and Technology. This paper was much improved thanks to their many comments and suggestions.

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This work was supported by NSF of China (No. 10671078) and by Program for NCET, the State Education Ministry of China.

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