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Suppose i have a function f defined on the positives reals that i can only evaluate values from : for a given x, i have an algo that compute f(x) but that's it (and it's costly..)

I want to approximate by another function $g$ with parameters a_i and b_i and the following form :

g(x) = sum_{i=1}^n [ a_i/(b_i - x) ]

For a given number n of couples (a_i,b_i).

What algorithm could i use to do such thing ?

lrnv
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  • I don't think that's possible. If you want to get an approximation over a relatively small interval, you might approximate your function using taylor-young devlopment though. – Maxime B. Dec 05 '19 at 09:26
  • Approximate must be defined correctly here... But **g** has 2N parameters, thus you may determine them (if it is possible as we know nothing about **f**) by evaluating **f** at least 2N times. This will give you a system of 2N equations with 2N variables and a function **g** (if system solvable) that will pass over 2N very same points of **f**. – Jean-Baptiste Yunès Dec 05 '19 at 09:39
  • Approximate means that $g$ and $f$ must be close, say in a $L_2$ sense. Using 2N evaluations will give me a possible solution. But using more evaluation and a square distance between the two function will be better right ? – lrnv Dec 05 '19 at 09:42
  • You may have a look at Padé approximant. There are many variants.I understood it is an active area of research. https://en.wikipedia.org/wiki/Pad%C3%A9_approximant – Damien Dec 05 '19 at 10:08
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    You are just asking for nonlinear curve fitting it seems (unfortunately you cannot linearize). Have a look at this answer of mine: https://math.stackexchange.com/questions/3463981/is-there-a-general-way-to-determine-the-best-combination-of-parameters-to-fit-po/3464013#3464013 –  Dec 05 '19 at 10:53

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