Interpolation using Lagrange polynomials

In summary, there are two solutions to calculating a polynomial of degree N-1 that crosses N known points in the plane: Gauss elimination and construction from Lagrange basis polynomials. While Gauss elimination may be considered inferior by some, both methods require O(N^3) steps and it is uncertain if the Lagrange basis technique is more accurate or stable.
  • #1
Lojzek
249
1
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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  • #2
Gaussian elimination is numerically unstable, but I don't know if the Lagrange basis technique is any better.
 

Related to Interpolation using Lagrange polynomials

1. What is interpolation using Lagrange polynomials?

Interpolation using Lagrange polynomials is a mathematical method used to approximate a curve or function based on a set of known data points. It involves constructing a polynomial function that passes through all the given data points, allowing for the estimation of values at points between the known data points.

2. How does Lagrange interpolation differ from other interpolation methods?

Lagrange interpolation differs from other methods in that it uses a single polynomial function to interpolate between data points, rather than using a series of polynomial segments. This results in a smoother and more accurate curve.

3. What are the advantages of using Lagrange polynomials for interpolation?

One advantage of using Lagrange polynomials is that they are easy to compute and do not require complex calculations. They also provide a simple and intuitive way to visualize and understand the interpolated curve. Additionally, Lagrange interpolation guarantees an exact fit to the given data points.

4. Are there any limitations to using Lagrange polynomials for interpolation?

One limitation of Lagrange interpolation is that it can be computationally expensive for a large number of data points. Additionally, if the data points are not evenly spaced, the accuracy of the interpolated curve may be affected. Lagrange interpolation also does not provide an estimate of the error in the interpolated values.

5. Can Lagrange interpolation be used for both one-dimensional and multi-dimensional data?

Yes, Lagrange interpolation can be used for both one-dimensional and multi-dimensional data. In one-dimensional interpolation, a single polynomial function is used to approximate the curve, while in multi-dimensional interpolation, a separate polynomial function is used for each dimension. The resulting curve is a surface instead of a curve.

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