Interpolation using Lagrange polynomials

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SUMMARY

The discussion focuses on two methods for polynomial interpolation using Lagrange polynomials: solving a NxN system of linear equations via Gaussian elimination and constructing the polynomial from Lagrange basis polynomials. While both methods require O(N^3) computational steps, Gaussian elimination is noted for its numerical instability, making it less reliable. The Lagrange basis method is preferred due to its explicit formulation, although the discussion raises questions about its accuracy compared to Gaussian elimination.

PREREQUISITES
  • Understanding of polynomial interpolation
  • Familiarity with Gaussian elimination
  • Knowledge of Lagrange basis polynomials
  • Basic concepts of numerical stability
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  • Research the numerical stability of Lagrange interpolation
  • Explore alternative interpolation methods such as Newton's divided differences
  • Learn about the computational complexity of polynomial interpolation techniques
  • Investigate practical applications of polynomial interpolation in data fitting
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Mathematicians, computer scientists, and engineers involved in numerical analysis, data fitting, and polynomial interpolation techniques.

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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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