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For a polynomial of order n with unknown coefficients, what are the ways to find the coefficients from n+1 points on its plot?

I remember one way is to construct a fractional for each point, and the polynomial is the sum of the fractionals, s.t. for each fractional, its value is the desired value for its corresponding point and zero for other points. Is there a name for the method?


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up vote 2 down vote accepted

The method you are talking about is called Lagrange Interpolation. If you want to find out the individual coefficients in the original "basis", then you would need to solve a $(n+1) \times (n+1)$ Linear system.

What we are essentially doing in Lagrange Interpolation is to shift to a new "basis" which depends on the points at which we observe the value, so that the system we need to solve is just the Identity matrix and hence we do not need to invert it as such to get the coefficients.

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