I sometimes encounter a problem where I need to find coefficients of a polynomial given planar points, which lie on it. I usually plug the points in the polynomial and solve the equations.

I noticed that it is possible to construct the polynomial directly as a sum of polynomials, which have the property that if one plugs in a x-value then the first term gives the correct y-value, while all the other terms are zeros. For the next x-value the second term gives the correct y-value, while all the other terms give noughts again. And so on. The coefficients can be obtained by simply calculating all the multiplications and additions arduously.

Am I using a not so easy to prove -theorem implicitly?

  • 1
    $\begingroup$ that's Interpolation by Lagrange's Polynomials $\endgroup$
    – G Cab
    Jun 10 '17 at 13:35
  • $\begingroup$ To be clear, are you looking for an exact polynomial solution, or will an approximate solution suffice? $\endgroup$
    – Cataline
    Jun 10 '17 at 13:38
  • $\begingroup$ Lagrange method is what I had in mind. $\endgroup$
    – Bezdomnyi
    Jun 10 '17 at 13:49

This kind of problem comes under the category of curve-fitting, where we need to adjust given planar points to a required degree of polynomial.

The methods to find the polynomial comes under this category:

  1. Graphical method
  2. Method of least squares
  3. Method of group averages
  4. Methods of moments And so on..

I myself prefer Method of least squares since it makes the calculation easier. ( method of least squares. )

The method you are enquiring about may be the superposition principle which is one of the common techniques used in calculation of electrical circuits parameters and have similar use in mathematics too.

  • $\begingroup$ If you need further help., don't hesitate to ask. $\endgroup$
    – MIB
    Jun 10 '17 at 12:59

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