Interpolation

Interpolation is a numerical method used to estimate unknown values that fall within the range of a discrete set of known data points. Unlike regression (curve fitting), which finds a line of best fit that may not touch the data, an interpolating function must pass exactly through every given data point. Key concepts in engineering interpolation: 1. Linear Interpolation: The simplest and most common method (e.g., reading thermodynamic steam tables). It assumes a straight line between two adjacent points. 2. Polynomial Interpolation: Uses a single polynomial of degree N-1 to pass through N points (e.g., Lagrange or Newton polynomials). While exact at the nodes, high-degree polynomials often suffer from "Runge's phenomenon"—wild, uncontrolled oscillations between points, especially near the boundaries of the dataset. 3. Spline Interpolation: To avoid Runge's phenomenon, splines use piecewise low-degree polynomials between points. A cubic spline is standard; it guarantees that the function, its first derivative (slope), and its second derivative (curvature) are continuous across all internal points, creating a perfectly smooth curve. 4. Bilinear Interpolation: Used when a dependent variable relies on two independent variables arranged on a 2D grid (e.g., finding fluid properties given both pressure and temperature). 5. Extrapolation: Estimating values outside the known data range. This is mathematically distinct from interpolation and carries a much higher risk of extreme error.