greetings . is there a general method-algorithm to solve the following system !?
$\sum_{n=1}^{m} {x_{n}}^{j}= {k}_{j} $
$j=1,2,...,m$
$k_{j}$ are constants
thanks in advance
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1 Answer
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Solution of this system are ALL m roots of the polynomial equation P(x) = 0 in ONE variable. Where P(x) is defined as follows.
Power sum related to elementary symmetric functions by the so-called Newton formulas. $\sigma_i = Newton (p_i)$
http://en.wikipedia.org/wiki/Newton's_identities
So define $c_i= Newton(k_j)$.
And write polynom $P(x) = \sum_i c_i x^i$
Its zeros are yours numbers $x_i$.
To solve polynomial equation you may use Newton method or any other method.
answered Jan 23, 2012 at 17:08
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$\begingroup$ thank you , i was aware of this method . physicsforums.com/showthread.php?t=568099 . i was hoping for a different one . $\endgroup$ Commented Jan 23, 2012 at 20:37
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$\begingroup$ For practical solution I think this is the best method. Any way I think any method which will take into account specific structure of this system will be equivalent. Otherwise you may use general method like consider min( \sum_i |F_i(x)-k_i|^2 ) - this can be attacked by steepest descent. $\endgroup$ Commented Jan 23, 2012 at 20:47
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2$\begingroup$ For future reference, when you ask a question here, you should tell us what you already know, so people don't waste their time telling you things you already know. $\endgroup$ Commented Jan 23, 2012 at 22:29