Lagrangian, Piecewise Linear, and Cubic Spline Interpolation Methods

In numerical analysis, interpolation is a method of constructing a function through known data points to estimate values at unknown locations. Common interpolation methods include Lagrangian interpolation, piecewise linear interpolation, and cubic spline interpolation. This article presents MATLAB implementations of these three methods and analyzes how the number of nodes affects interpolation results.

### 1. Lagrangian Interpolation Lagrangian interpolation constructs a polynomial that exactly matches function values at all given nodes. Its advantage lies in simple formulation, but as the number of nodes increases, the polynomial degree rises, potentially causing severe oscillations (Runge phenomenon). In MATLAB implementation, the interpolation polynomial can be built by calculating Lagrangian basis functions, though high-order interpolation may introduce numerical instability. The algorithm typically involves nested loops to compute basis polynomials and their weighted sum.

### 2. Piecewise Linear Interpolation Piecewise linear interpolation connects adjacent nodes with straight lines, offering computational simplicity and stability, making it suitable for datasets with numerous points. Although the interpolation function remains continuous, its derivative becomes discontinuous at nodes, resulting in less smooth curves. MATLAB's `interp1` function easily implements piecewise linear interpolation by specifying the `'linear'` option, which performs linear interpolation between consecutive data points using simple slope calculations.

### 3. Cubic Spline Interpolation Cubic spline interpolation uses third-degree polynomials in each subinterval while ensuring continuity of function values, first derivatives, and second derivatives at nodes, producing smoother curves. This method avoids oscillation issues common in high-degree polynomial interpolation and suits high-precision fitting applications. In MATLAB, implementation can be achieved using either the dedicated `spline` function or the `'spline'` option in `interp1`, both solving tridiagonal systems of equations to determine polynomial coefficients that satisfy continuity conditions.

### Impact of Node Quantity on Interpolation Results Lagrangian interpolation: Increasing nodes raises polynomial degree, potentially introducing oscillations, particularly problematic in regions with rapid function changes. Piecewise linear interpolation: More nodes improve accuracy but maintain linear characteristics, resulting in generally non-smooth curves. Cubic spline interpolation: Additional nodes typically enhance fitting precision while preserving smoothness, making it suitable for most practical applications.

### Conclusion Different interpolation methods exhibit distinct advantages: Lagrangian interpolation works best with few nodes, piecewise linear interpolation offers simplicity but limited smoothness, while cubic spline interpolation balances accuracy and smoothness effectively. Practical applications should select appropriate methods based on data characteristics and optimize results by adjusting node quantities. MATLAB provides built-in functions for all three methods, with cubic spline generally offering the most robust performance for real-world datasets.