MATLAB Implementation of Cubic Spline Interpolation

Cubic spline interpolation is a numerical method that constructs smooth curves through piecewise cubic polynomials. Widely used in engineering and scientific computing, it ensures exact matching at interpolation points while maintaining continuous derivatives. MATLAB provides comprehensive mathematical tools to implement this algorithm efficiently. Core Computational Steps Data Preparation: Input discrete node data (x,y) with strictly increasing x-values Boundary Conditions: Select natural boundary conditions (second derivative equals 0) or fixed boundary conditions (specified endpoint derivatives) based on practical requirements Equation System Construction: Use the three-moment method to establish a linear equation system for second derivatives, ensuring continuity of function values, first derivatives, and second derivatives at nodes through continuity conditions Matrix Solution: Employ efficient algorithms like the Thomas algorithm to solve tridiagonal systems, obtaining second derivative values at all nodes MATLAB Implementation Key Points The built-in `spline` function can directly perform cubic spline interpolation Manual implementation requires constructing tridiagonal coefficient matrices and establishing equations using continuity conditions When calculating function values at interpolation points, determine the containing interval and substitute into the corresponding polynomial expression First and second derivatives are obtained through polynomial differentiation formulas Application Extensions This method not only provides interpolated function values but also enables derivative calculations for: Velocity and acceleration analysis in physical motion studies Gradient information extraction in engineering optimization Contour smoothing in image processing The efficient implementation of cubic splines in MATLAB makes them essential tools for connecting discrete data with continuous models, particularly in scenarios requiring guaranteed curve smoothness. The algorithm's tridiagonal matrix structure ensures computational efficiency while maintaining mathematical robustness.