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澳洲论文代写-澳洲墨尔本大学代写数学essay:COMPLEX NUMBERS AND NEWTON'S METHOD

澳洲论文代写-澳洲墨尔本大学代写数学essay:COMPLEX NUMBERS AND NEWTON
  • 国家 : 澳洲
  • 级别 : 硕士
  • 专业 :
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COMPLEX NUMBERS AND NEWTON'S METHOD

Abstract

Newton's Method for finding roots of an equation, or more generally solutions to systems of equations, is useful in essentially any area that requires solving equationsfrom engineering to economics. It has also fueled major areas of ongoing research.This project explores the extension of Newton's Method, roots of complex functions. First, introduce the method of written mapping in C in terms of the coordinates x and y, where z=x + iy. Then finding the roots of complex functions, in fact, equals to finding the roots of real functions in. At last, how Newton’s method converges to the solution(s) of the equation will be discussed.

 

1.COMPLEX NUMBERS

(1)Since a complex number written as x + iy has two real components, it can be thought of as a point in the real plane (which can also be labeled as C in this context): the number z = x + iy corresponds to the point (x , y). Any function CC is therefore also a mapping.

So, the mapping can be written in terms of the coordinates x and y,where z=x + iy:

Since,, so 

In a similar way, the mappingcan be written in terms of x and y as below:

, so 

 

(2)The Jacobian matrix in the coordinates x and y for  is: 

The Jacobian matrix in the coordinates x and y for  is:

 

 

 

3. What I learned

The Newton’s method can be applied on the complex numbers. All we have to do is to think a complex number as a real point (x , y), where z = x + iy. But the situations on the complex numbers are much complicate. Because the simple mapping on C plane will turn to be very complicate when it is written in terms of the coordinates x and y. Thus, cause the complexity of the iteration of the Newton’s method.

From the results above, we found that if the equation only has real roots, like. Then the process is stable, and any given point on the plane will converge to one of the solutions eventually. And it even goes the same when we only consider the real root of, which is 1.

However, when complex roots are to be considered, the Newton’s method might not be applicable. And sometimes, there will be a phenomenon called chaos, which is the unstable part of the iteration. But, many interesting pictures are generated by the chaos, and they are used in many areas like decorating.

 

以下附带百度翻译(机器翻译)帮助理解

复数和牛顿方法
摘要
求方程的根牛顿法,或更一般的解方程组,可基本上任何地区,需要解决equationsfrom工程经济学。这也助长了正在进行的研究的主要领域,该项目探讨了牛顿方法的扩展,复杂的功能的根源。首先,介绍了坐标x和y表示C写的映射方法,其中z = x + iy。然后,寻找复变函数的根源,实际上,就等于找到了实函数的根。最后,牛顿的方法收敛到该方程的解的收敛性。
1.complex数
(1)从一个复数写成x + iy有两个真正的组件,它可以被认为是真正的平面上的一个点(也可称为C,在这种情况下):数z = x + iy对应的点(x,y)。因此,任何函数都是一个映射。
因此,该映射可以被写在x和y坐标,其中z = x + iy:
因为,所以
以类似的方式,这是写在mappingcan x和y表示如下:
,所以
(2)在坐标系中的矩阵的矩阵:
在坐标系中的矩阵的矩阵:
3。我学到了什么
牛顿的方法可以应用于复数。我们要做的是想一个复数作为一个真正的点(x,y),其中z = x + iy。但是,复杂的数字的情况复杂得多。因为在平面上的简单映射将变得非常复杂,当它被写在坐标和Y,从而导致了牛顿的方法迭代的复杂性。
从上面的结果,我们发现,如果方程只有真正的根,如。然后,该过程是稳定的,并且在平面上的任何给定的点,最终收敛到一个解决方案。它甚至在我们只考虑真正的根,这是1。
然而,当复杂的根是要考虑的,牛顿的方法可能不适用。有时,会有一种叫做“混沌”的现象,这是迭代过程中不稳定的部分。但是,许多有趣的图片是由混乱产生的,他们在许多领域都喜欢装饰。
以下附带百度翻译(机器翻译)帮助理解
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