Interpolation is one of the most practical tools in numerical analysis. It is the mathematical “art” of reading between the lines. If you have a set of known data points, interpolation allows you to estimate the value of a function for any point inside the range of those data points.
Core Concept: Connecting the Dots
Imagine you measured the temperature at 10:00 AM and 11:00 AM. Interpolation helps you estimate what the temperature was at 10:30 AM by fitting a curve (usually a polynomial) through your known points.
Key Formulas Mentioned in Your Image
The chapter objectives list several famous methods. Here is a quick guide to when to use which:
- Newton’s Forward/Backward Formula: Best used when your data points are equally spaced (e.g., measurements taken exactly every hour).
- Lagrange’s Interpolation: Use this when your data points are unequally spaced. It is a very flexible formula but can be computationally “expensive” if you have a lot of points.
- Spline Interpolation (Cubic Splines): Instead of one giant curve, this method connects points using smaller, smoother “mini-curves.” This is how modern computer graphics and fonts (like the ones on your screen) are rendered smoothly.
- Divided Differences: This is the foundation for Newton’s general interpolation and is used to build polynomials step-by-step.