Interpolation using Lagrange polynomials

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The discussion centers on two methods for interpolating a polynomial of degree N-1 through N known points: solving a system of linear equations via Gaussian elimination and constructing the polynomial using Lagrange basis polynomials. While both methods require O(N^3) computational steps, concerns are raised about the numerical stability of Gaussian elimination, which is considered inferior. The Lagrange method is more explicit, but its accuracy compared to Gaussian elimination remains uncertain. Participants are exploring whether the Lagrange approach offers any advantages in terms of stability or accuracy. The conversation highlights the need for a deeper understanding of the numerical properties of both methods.
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Problem: We want to calculate a polynomial of degree N-1 that crosses N known points in the plane.

Solution A: solving a NxN system of linear equation (Gauss elimination)

Solution B: construction from Lagrange basis polynomials.

One of my professors said that the first solution is inferior and I am trying to find out why.

Of course method B is more explicit, but the required time for calculation of all coefficients is probably similar, since both methods require O(N^3) steps. Is there any difference in accuracy or any other reason that would make method B better?
 
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Gaussian elimination is numerically unstable, but I don't know if the Lagrange basis technique is any better.
 

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