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If we have $50$ points of known $(x,y)$ provided from a certain polynomial, ( note; this polynomial is unknown and we do not know it's degree). So is there any way to find the exact equation for this polynomial by using the value of $50$ points?( I am not looking here for the best fitting equation, I ask for the exact equation of the polynomial).

Thanks

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2 Answers 2

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The Lagrangian polynomial will interpolate your data points.

Then if your coordinates are exact and you compute the coefficients using exact arithmetic, the high degree terms will vanish by themselves.

If you coordinates are approximate, then you are in trouble as all coefficients will remain non-zero and you'll get a polynomial of the $49^{th}$ degree, as the method is highly unstable.

The Neville computation scheme (computing all coefficients, not just a value) should give you better insight as the intermediate polynomials should never exceed the actual degree and will tend to become equal.

A safer approach could be to compute fitting polynomials for increasing degrees and decide when the least-squares residue has dropped. It all depends on the quality of your data.

Make sure to know an upper bound on the degree !

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  • $\begingroup$ Ok, what i want to say that the 50 points provided from certanin equation (let's say y= 10 - x^6 - 4x^2, but we do not know this equation). So is there any way to find the equatios by using the 50 points. I am looking to find the exact equation not the best fit equation. $\endgroup$
    – Marwan
    May 7, 2015 at 13:02
  • $\begingroup$ Please spend some effort to read my answer. It directly addresses your request. $\endgroup$
    – user65203
    May 7, 2015 at 13:03
  • $\begingroup$ Thanks, but is this solution will give me the exact equation ( y= 10-x^6-4x^2)? $\endgroup$
    – Marwan
    May 7, 2015 at 13:16
  • $\begingroup$ This is answered in the second and third paragraphs. $\endgroup$
    – user65203
    May 7, 2015 at 13:34
  • $\begingroup$ Neville, newten or Lagrang or any other interpolation method will be accurate on the known points ( 50 points) but will not be accurate in the interpolation or extrapolation area, so i am looking for away to estimate the exact equation ( y=10-x^6-4x^4). $\endgroup$
    – Marwan
    May 7, 2015 at 13:47
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Given that each value of x only has one y value, f(x) can be a degree 50 polynomial.

$f(x) = a_1x^{49} + a_2x^{48} + ... + a_{49}x + a_{50}$

For all the values of (x,y), substitute the values of x and y and solve.

However, in the case x has more than one y value, f(x) is not a polynomial.

At the OP's request, here is an elaboration.

Let us say there is a point (3,5)

We substitute this value in and we get $a_1*3^{49} + a_2*3^{48} + ... + a_{49}*3 + a_{50} = 5$

After 50 values, we can get an answer.

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  • $\begingroup$ $49$ seems to be tha maximum degree (except if $a_{51}=0$), I guess. $\endgroup$
    – Claude Leibovici
    May 7, 2015 at 12:52
  • $\begingroup$ Ok, what i want to say that the 50 points provided from certanin equation (let's say y= 10 - x^6 - 4x^2, but we do not know this equation). So is there any way to find the equatios by using the 50 points? $\endgroup$
    – Marwan
    May 7, 2015 at 12:53
  • $\begingroup$ Hi, are you still with us and interested to answer this question? $\endgroup$
    – Marwan
    May 7, 2015 at 14:54
  • $\begingroup$ This method generally will solve your problem, but it may give more than one solution. $\endgroup$
    – the4seasons
    May 10, 2015 at 11:47

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