How do you compare and contrast Lagrange and Newton interpolation methods?

1 Lagrange interpolation

Lagrange interpolation is a method that uses a linear combination of basis functions, called Lagrange polynomials, to construct an interpolating polynomial. Each Lagrange polynomial is zero at all data points except one, where it is one. The interpolating polynomial is then the weighted sum of the Lagrange polynomials, where the weights are the function values at the data points. Lagrange interpolation has the advantage of being easy to implement and understand, and it does not require solving a system of equations or finding derivatives. However, Lagrange interpolation can suffer from high degree, oscillations, and numerical instability, especially when the data points are not evenly spaced or close together.

2 Newton interpolation

Newton interpolation is a method that uses a nested form of a polynomial function, called Newton polynomial, to construct an interpolating polynomial. The Newton polynomial is built by adding terms that are products of differences of function values and powers of differences of data points. The difference of function values, called divided differences, can be computed recursively using a table or a formula. Newton interpolation has the advantage of being efficient and stable, and it allows adding more data points without changing the previous terms. However, Newton interpolation can be difficult to implement and understand, and it requires finding divided differences and storing them in memory.

3 Comparison criteria

When comparing the Lagrange and Newton interpolation methods, one can use certain criteria to measure their performance and suitability for different situations. These criteria include accuracy, complexity, flexibility, and stability. Accuracy is determined by how well the interpolating polynomial approximates the true function within and outside the data interval. Complexity examines the number of operations and memory needed to construct and evaluate the interpolating polynomial. Flexibility is evaluated by how easy it is to modify or extend the interpolating polynomial when new data points are added or removed. Lastly, stability assesses how sensitive the interpolating polynomial is to errors or changes in the data points or function values.

5 Application examples

To demonstrate how Lagrange and Newton interpolation methods can be utilized to solve different numerical analysis problems, let's look at some examples. For instance, if you need to find the value of a function at a point that is not given by a table or a formula, you can use either Lagrange or Newton interpolation depending on the available data points and their spacing. For example, if you have a table of values of sin(x) at equally spaced points from 0 to pi/2, you can use Lagrange interpolation to find sin(0.3) with an appropriately low degree polynomial. On the other hand, if you have a table of values of exp(x) at unevenly spaced points from 0 to 1, then Newton interpolation is more suitable for finding exp(0.5) with a more stable polynomial. Similarly, when aiming to smooth or fit a curve to noisy or scattered data, you can choose between Lagrange or Newton interpolation based on the degree of smoothness and the number of data points. Lastly, when trying to solve a differential equation by converting it into a system of algebraic equations, the type of boundary conditions and the order of accuracy will determine which method is most suitable.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?