Teacher’s Notes: Lagrange interpolation is a method for estimating values between known data points by constructing a single polynomial that passes through all given points. It uses a formula that builds the polynomial by combining special functions, called basis polynomials, which are zero at all but one data point. This technique is useful for small datasets where values need to be estimated smoothly, such as filling in missing temperature readings or predicting outcomes between observed data. Think of it as drawing a smooth curve that passes exactly through all the known points.

In numerical methods, one of the most fundamental problems is determining an approximating function that passes through a given set of data points. When the function values are known only at discrete points and we aim to estimate the function at other values within the domain, interpolation becomes an essential tool. Among the various interpolation techniques, Lagrange Interpolation stands out as one of the earliest and most straightforward methods. It constructs a polynomial that exactly fits a set of known points and does so without requiring solving equations or matrix operations.

Lagrange interpolation is used when we are given a table of values:

and we want to find a polynomial of degree that passes through all these points. The idea is to express as a sum of weighted polynomials. Each term is designed so that it is 1 at one specific value and 0 at all others. This means that each term in the polynomial only “activates” at its specific data point.

The formula is:

Where:

Each is called a Lagrange basis polynomial. It equals 1 when and 0 when for

. Because of this, the whole expression simplifies beautifully and ensures that the resulting polynomial passes through all the given points exactly. These basis polynomials act like filters that highlight one specific data point while ignoring the others during construction of the overall polynomial.

Teacher’s Notes: Each basis polynomial works like a “selector” that picks out only one y-value at a time. This keeps the overall equation neat and ensures accuracy at known points. Students may find it easier to understand if they imagine that the curve is being built one data point at a time.

Let us try a simple example. Suppose we are given three points:

We compute the Lagrange basis polynomials:

Then the final interpolating polynomial is:

This results in a second-degree polynomial that passes exactly through the three points. We can use this polynomial to estimate values between those points, like finding the value of the function at even though it’s not one of the original data points.

Applications:

• Estimating temperatures between measurements
• Creating smooth curves in graphics
• Filling in missing data in small datasets
• Reconstructing signals in digital communication
• Predicting intermediate results in experimental data

Teacher’s Notes: This method is accurate for small data sets but can become unstable for large ones. If you add or change a point, the entire polynomial changes. This is a key idea to understand why it’s not used for large data. Also, emphasize to students that while this technique gives exact matches at the known data points, it does not always produce good estimates outside of those points (extrapolation).

Despite its strengths, Lagrange interpolation has some weaknesses. It becomes inefficient when there are many data points, and if the data points are not spaced well, the polynomial might swing wildly (called Runge’s Phenomenon). For this reason, Lagrange is best used when the number of data points is small and well-distributed.

Even with these limitations, Lagrange interpolation remains a cornerstone of numerical methods. It introduces key ideas that are later used in more advanced topics like spline interpolation and Newton’s divided differences. By understanding Lagrange interpolation, students gain insight into how mathematics allows us to approximate unknown values from limited information. It teaches the principles of polynomial construction and provides intuition for function approximation.

Quick Check for Understanding:

1. What does each term do in the polynomial?
2. Why is Lagrange interpolation not good for many points?
3. What real-life situations could this be used for?
4. What happens to the entire polynomial if you change just one data point?