Theory

Interpolation is a fundamental numerical technique used to estimate unknown values of a function within the range of known data points. Unlike extrapolation, which predicts values outside the given range, interpolation focuses on filling gaps in existing data with a high degree of reliability. This makes it indispensable in scientific computing, engineering, data science, image processing, climate modeling, and numerical simulations, where continuous information must be inferred from discrete observations.

The central idea of interpolation is to construct a mathematical function that approximates the underlying relationship between the given data points. Several methods are commonly employed, each with distinct characteristics, advantages, and limitations:

1. Linear Interpolation

• The simplest method, where straight lines are drawn between consecutive data points.
• Provides quick estimates but lacks smoothness and may be inaccurate for rapidly varying functions.

2. Polynomial Interpolation

• Constructs a single polynomial of degree (n – 1) passing through n given points.
• Includes techniques like Lagrange Interpolation and Newton’s Divided Difference Method.
• Useful for small datasets but may suffer from Runge’s phenomenon (oscillations) when many points are used.

3. Newton’s Forward and Backward Interpolation

• Specifically designed for evenly spaced data points.
• Utilizes forward and backward differences to generate efficient polynomial approximations.
• Suitable for tabulated functions in physics, statistics, and engineering experiments.

4. Spline Interpolation

• Uses piecewise polynomials (commonly cubic splines) to ensure smoothness and continuity.
• Provides greater accuracy and stability compared to high-degree single polynomials.
• Widely applied in computer graphics, CAD systems, and curve fitting.

5. Nearest-Neighbor Interpolation

• Assigns the value of the closest known data point to the unknown point.
• Extremely fast but may lead to discontinuities, making it less suitable for precise scientific tasks.

Evenly vs. Unevenly Spaced Data

• For evenly spaced points, Newton’s Forward and Backward methods are efficient and computationally straightforward.
• For unevenly spaced points, generalized approaches such as Lagrange and Newton’s Divided Difference methods are more appropriate.

Learning Outcomes from This Experiment

By performing this experiment, learners will be able to:

• Analyze the accuracy of different interpolation methods.
• Compare their computational efficiency.
• Select the most suitable method depending on the dataset type.
• Understand real-world applications in signal processing, image scaling, weather forecasting, biomedical data, and machine learning preprocessing.

Interpolation is not just a mathematical tool but a bridge between discrete data and continuous analysis, making it a cornerstone in modern computational sciences.