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 $k_j$ :

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

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 $1 \le \alpha_n \le m (\frac{1}{\sqrt{m}})^n$.

 

[blue_raver22]

I understand that this system of nonlinear equations is commonly known as a power sum. It involves a set of m variables (x_1, x_2, ..., x_m) and a set of m constants (k_1, k_2, ..., k_m). The equations are nonlinear because the variables are raised to different powers (1, 2, ..., m).

To find a solution for this system, we need to use a numerical method such as Newton's method or the bisection method. These methods involve using an initial guess for the values of the variables and then iteratively refining those values until they converge to a solution.

There is no general solution or algorithm that can solve all power sum equations, as the specific values of the constants and the number of variables can greatly affect the complexity of the problem. However, there are many numerical methods available that can be tailored to specific equations and can provide accurate solutions.

In summary, while there may not be a general solution or algorithm for solving all power sum equations, there are effective numerical methods that can be used to find solutions for specific cases.