Lagrange Interpolation

Definition

Lagrange interpolation is a polynomial interpolation technique used to estimate values of a function at specific points by constructing a polynomial that passes through a given set of points. This method provides a straightforward way to find a polynomial that not only fits the data points but also allows for the evaluation of function values at other locations. It is particularly useful in numerical analysis for approximating functions when the actual function is unknown.

5 Must Know Facts For Your Next Test

1. Lagrange interpolation can exactly fit a polynomial of degree $n$ to $n+1$ distinct data points.
2. The Lagrange interpolating polynomial is expressed as a weighted sum of basis polynomials, with each weight depending on the x-coordinates of the data points.
3. It can be computationally expensive for large datasets because it requires recalculating all basis polynomials for every evaluation point.
4. The method is particularly sensitive to the choice of data points, which can lead to Runge's phenomenon when points are equidistant.
5. Lagrange interpolation is useful in applications like numerical integration and solving differential equations where function values are needed at non-sampled points.

Review Questions

• How does Lagrange interpolation ensure that the constructed polynomial passes through all given data points?
  • Lagrange interpolation constructs the interpolating polynomial using basis polynomials that are specifically designed to be equal to one at their corresponding data point and zero at all other points. This ensures that when combining these basis polynomials, the resulting polynomial will exactly match each of the provided data points. The formula combines these weighted basis polynomials to produce a single polynomial that seamlessly interpolates through all specified points.
• Discuss how Lagrange interpolation differs from Newton's divided differences in terms of application and computational efficiency.
  • Lagrange interpolation directly constructs a single polynomial from the given data points using explicit formulas for its basis polynomials, making it easy to understand conceptually. However, it can become inefficient with large datasets due to the need to recalculate all basis polynomials for each evaluation. In contrast, Newton's divided differences method builds the polynomial incrementally, allowing for easier updates if new data points are added and generally offering better computational efficiency in those scenarios.
• Evaluate the implications of Runge's phenomenon in Lagrange interpolation when using equidistant data points and suggest possible solutions.
  • Runge's phenomenon occurs when using Lagrange interpolation with equidistant data points, resulting in oscillations near the edges of the interval. This problem highlights how selecting inappropriate data points can lead to inaccurate approximations. To mitigate this effect, one could use Chebyshev nodes, which are strategically spaced to reduce oscillation or consider using piecewise interpolation methods like spline interpolation. These approaches provide more stable results while still accurately representing the underlying function.

Related terms

Newton's Divided Differences: A method for polynomial interpolation that uses divided differences to construct a polynomial based on given data points.

Barycentric Interpolation:

A form of Lagrange interpolation that enhances numerical stability and efficiency by reformulating the Lagrange basis polynomials.

Polynomial Degree:

The highest power of the variable in a polynomial, which determines the shape and complexity of the interpolating polynomial.