A system of nonlinear equations (power sum)

[mmzaj]

greetings . i cam across this problem in a research paper I'm writing .

$$x_1+x_2+...+x_m=k_1$$
$$x^2_1+x^2_2+...+x^2_m=k_2$$
$$......$$
$$......$$
$$x^m_1+x^m_2+...+x^m_m=k_m$$
$k_j$ are constants

in compact form :
$$\sum^{m}_{i=1}x^{n}_{i}=k_{n}, n=1,2...m$$


is there a general solution(or algorithm to solve) for $x_i$ ??

 

[Stephen Tashi]

I don't know a general solution, but it's an interesing problem. The thread at least deserves a bump. It's the kind of problem that I suspect has been the subject of some mathematical publications, but I don't know how to formulate good search terms for it.

If this arises in solving some applied math problem, can't you find papers related to that problem that deal with it?

A more general problem than yours is the problem of solving simultaneous polynomial equations in several variables. There is a method, which I have not studied in detail, called "Wu's elimination method" that supposedly does that.


If we tackle this as a math research problem, one approach would be to examine the transformation properties of the vector $[k_i]$ when various transformations are applied to the vector $[x_i]$.

For example, let $[y_i]$ be some initial guess for a solution and let $[\alpha_i]$ be the vector $[\sum_j y_j^i ]$. If we multiply $[y_i]$ by the constant $\lambda$ then we change $[\alpha_i]$ to $[ \lambda^i \alpha_i]$.

The thing to look for would be some more complicated type of transformations. A interesting daydream would be to find some that leave $\alpha_i$ fixed for $i = 1,2,..p$ and change $\alpha_p$ to $k_p$ and do whatever they want to the remaining $\alpha_i$. You could use a series of such transformations to change the results of an initial guess to the desired result.

If we tackle this problem as a problem in numerical methods, there are probably many ways to do it, but I'm not sure whether that's the sort of approach you are looking for. For example, I don't know whether the $[k_i]$ that you have are the results of physical measurements or whether they might be integers or some other form of exact theoretical data.

 

[mmzaj]

one way to solve it is the following .

the elementary symmetric polynomials can be defined in terms of $k_j$ :

$$e_{j}=\frac{1}{j!} \begin{vmatrix}<br /> k_{1} & 1 & 0 & ... & & & & \\ <br /> k_{2} & k_{1} & 2 & 0 & ... & & & \\ <br /> . & ... & k_{1} & ... & & & & \\ <br /> . & & & & . & & & \\ <br /> . & & & & & . & & \\ <br /> . & & & & & & & \\ <br /> k_{j-1}&k_{j-2} & ... & & & & k_{1} & j-1 \\ <br /> k_{j} &k_{j-1} & ... & & & & k_{2} & k_{1}<br /> \end{vmatrix}$$

define the polynomial :
$$f(x) = \sum_{j=0}^{m}(-1)^je_{j}x^{m-j}$$

$( x_1,x_2,...x_m )$ are the solutions for the equation $f(x)=0$

but i was hoping for some solution method that doesn't include polynomials and their roots !

 

[Stephen Tashi]

At least we can confine our attention to solutions on a m-sphere.Instead of the problem:

$$\sum^{m}_{i=1}x^{n}_{i}=k_{n}, n=1,2...m$$

Let $\lambda = \sqrt{k_2}$ and consider the problem:

$$\sum^{m}_{i=1}y^{n}_{i}=k_{n}/ \lambda^n = \alpha_{n}, n=1,2...m$$

For $n = 2$ the equation requires that the $\sum^{m}_{i=1} y^2_{i} = 1$

If we can solve the second problem then the solution to the first problem is given by
$x_i = \lambda y_i$.

We should be able to put bounds on the possible values of the $\alpha_n$ that allow a solution. For even n, I'd guess $1 \le \alpha_n \le m (\frac{1}{\sqrt{m}})^n$.