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A system of nonlinear equations (power sum)

  1. Jan 16, 2012 #1
    greetings . i cam across this problem in a research paper i'm writing .

    [tex] x_1+x_2+.....+x_m=k_1 [/tex]
    [tex] x^2_1+x^2_2+.....+x^2_m=k_2 [/tex]
    [tex] .......................[/tex]
    [tex] .......................[/tex]
    [tex] x^m_1+x^m_2+.....+x^m_m=k_m [/tex]
    [itex]k_j[/itex] are constants

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


    is there a general solution(or algorithm to solve) for [itex] x_i [/itex] !?!?
     
  2. jcsd
  3. Jan 21, 2012 #2

    Stephen Tashi

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    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 [itex] [k_i] [/itex] when various transformations are applied to the vector [itex] [x_i] [/itex].

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

    The thing to look for would be some more complicated type of transformations. A interesting daydream would be to find some that leave [itex] \alpha_i [/itex] fixed for [itex] i = 1,2,..p [/itex] and change [itex] \alpha_p [/itex] to [itex] k_p [/itex] and do whatever they want to the remaining [itex] \alpha_i [/itex]. 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 [itex] [k_i] [/itex] that you have are the results of physical measurements or whether they might be integers or some other form of exact theoretical data.
     
  4. Jan 21, 2012 #3
    one way to solve it is the following .

    the elementary symmetric polynomials can be defined in terms of [itex] k_j [/itex] :

    [tex] 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}
    [/tex]

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

    [itex] ( x_1,x_2,....x_m )[/itex] are the solutions for the equation [itex] f(x)=0 [/itex]

    but i was hoping for some solution method that doesn't include polynomials and their roots !!
     
  5. Jan 23, 2012 #4

    Stephen Tashi

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    At least we can confine our attention to solutions on a m-sphere.


    Instead of the problem:

    [tex]\sum^{m}_{i=1}x^{n}_{i}=k_{n}, n=1,2...m[/tex]

    Let [itex] \lambda = \sqrt{k_2} [/itex] and consider the problem:

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

    For [itex] n = 2 [/itex] the equation requires that the [itex] \sum^{m}_{i=1} y^2_{i} = 1 [/itex]

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

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