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Is it a polynomial or rational polynomial or else?

$y = \dfrac{a}{x^4} + \dfrac {b}{x^2} + c$

I need to fit a curve to a discrete data of that form, so I need to know what fitting to use.

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  • $\begingroup$ And what do you think? $\endgroup$ – TZakrevskiy Apr 17 '15 at 14:28
  • $\begingroup$ That I need to take class of elementary mathematics... $\endgroup$ – Trenera Apr 17 '15 at 14:29
  • $\begingroup$ Try rewriting as a single fraction, that may give better insight. $\endgroup$ – kmeis Apr 17 '15 at 14:32
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We have $$\frac{a}{x^4}+\frac{b}{x^2}+c=\frac{a}{x^4}+\frac{bx^2}{x^4}+\frac{cx^4}{x^4}=\frac{a+bx^2+cx^4}{x^4}$$ This is a rational function.

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The equation $y-a/x^4-b/x^2-c=0$ yields $$ f(x,y)= - a - bx^2 - cx^4 + x^4y=0, $$ which is an affine algebraic curve, see here.

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If we consider that the problem is to fit data using $(x_i,y_i)$ data points for a model $$y = \dfrac{a}{x^4} + \dfrac {b}{x^2} + c$$ define two variables $u_i=\dfrac{1}{x_i^4}$ and $v_i=\dfrac{1}{x_i^2}$ which make $$y=a u+b v +c$$ and the problem is just to perform a multilinear regression for which the normal equations are very simple to establish and use (otherwise, matrix formulation).

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