Newton waveform relaxation method for solving algebraic nonlinear equations☆
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:where and . The key idea of waveform relaxation method for systems (1.1) is to chose some function and then solve the following equations successively:where is a given initial approximation of which is the solver of (1.1). Then we obtain from 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.,for any function .
Assume we have to solve nonlinear equationswhere . Provided we have a starting guess of the unknown solutions at hand, like the continuous-time waveform relaxation iteration, we choose a function which satisfiesfor any .
And then we solve the following equations from to :If as , by the consistency condition (1.4), we know that 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:here and below:If we set and in (1.6), by the consistency condition (1.4), arithmetic (1.6) is equivalent toAnd 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 asprovided 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 be a diagonal or block diagonal matrix and invertible in . 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 be a continuously mapping with open and convex and satisfy the consistent condition (1.4). Function is Fréchet differentiable with the second variable y and matrix is invertible for any . Assume that
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:Letand it is clear that , where and . Routine calculation yieldsand thus the Lipschitz constant 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 which satisfies for any . 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.