Interpolation
Interpolation is a numerical method for estimating values of a function at points between known data points.
Definition
Given a set of data points , interpolation finds a function such that::$$
f(x_i) = y_i \quad \text{for all } i = 0, 1, \ldots, n
The function $f(x)$ can then be used to estimate values at points $x$ where $x_i < x < x_{i+1}$. ## Purpose Interpolation is used when: - Data is available only at discrete points - The underlying function is unknown or too complex - We need to estimate intermediate values - Continuous representation of data is required ## Key Considerations When selecting an interpolation method, consider: 1. **Accuracy**: How close are the estimates to true values? 2. **Computational cost**: How expensive is the method? 3. **Smoothness**: Is the interpolant continuous? Differentiable? 4. **Data requirements**: How many points are needed? ## Types of Interpolation - **[[Linear Interpolation]]**: Connect points with straight lines - **[[Polynomial Interpolation]]**: Fit a polynomial through points - [[Newton Divided Difference]] - [[Lagrange Interpolation]] - **[[Spline Interpolation]]**: Piecewise polynomial fitting - **[[Bilinear Interpolation]]**: 2D interpolation ## Comparison to Extrapolation **Interpolation** estimates within the range of known data, while **extrapolation** estimates outside this range. Extrapolation is generally less reliable and should be used with caution.