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If we have $n + 1$ points $(x_i,y_i$), then we can use interpolation methods (Lagrange, ...etc) to find a polynomial of degree $n$: $$P_n(x) = a_0 + a_1x + a_2x^2 + \cdots a_nx^n$$

In curve fitting, we search for a polynomial that best fits the points (minimizes the squared error).

But what if we want to use polynomial curve fitting (a.k.a polynomial regression)to find the polynomial of degree $n$: $$Q_n(x) = b_0 + b_1x + b_2x^2 + \cdots b_nx^n$$

After testing this in Excel, I found that the interpolation and the polynomial regression of degree $n$ give the same polynomial.

So my conclusion is that: Using polynomial interpolation through $n+1$ points is equivalent to use polynomial curve fitting of degree $n$.

Am I right?

Thanks

(I am refering only to polynomial interpolation not piece-wise interpolation)

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1 Answer 1

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You are right. Note that in both cases, we have $n+1$ unknown coefficients. Finding the interpolation polynomial of degree $n$ means solving a system of $n+1$ linear equations; one corresponding to each point $(x_i,y_i)$. In general, when applying regression, there are many more data points than there are coefficients. The coefficients are then found by solving the least-squares system. However, in your case the polynomial can be fitted exactly, as there are just as much data points as there are coefficients.

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