Universal Cubic Spline Interpolation Algorithm with MATLAB Implementation

In the following text, I will provide a complete MATLAB source code for a universal cubic spline interpolation algorithm, along with usage examples. Cubic spline interpolation is a widely-used numerical analysis method that constructs a smooth curve passing through known data points. This implementation includes the core algorithmic steps: calculating second derivatives using tridiagonal matrix solving, constructing piecewise cubic polynomials with continuity conditions (C² continuity), and handling natural/clamped boundary conditions. Key functions include data validation, coefficient calculation, and interpolation evaluation. The code demonstrates MATLAB's sparse matrix capabilities for efficient linear system solving and includes error handling for invalid inputs. Sample usage cases show how to apply the algorithm to both equally-spaced and irregular data points, with visualization examples plotting the interpolated curve against original data. This implementation provides practical insights into spline theory and numerical computation techniques.