MATLAB Implementation of Common Interpolation Algorithms

In scientific computing and engineering applications, interpolation algorithms serve as fundamental and essential numerical analysis methods for estimating values at unknown points based on known data points. MATLAB, as a powerful numerical computation tool, offers various built-in interpolation functions while also enabling users to implement custom interpolation algorithms. Common interpolation methods primarily include: Linear Interpolation: The simplest approach using straight lines between adjacent data points Polynomial Interpolation: Constructing polynomial functions to fit data points Spline Interpolation: Utilizing piecewise polynomials to ensure smooth interpolation curves Nearest Neighbor Interpolation: Directly adopting values from the closest data points Cubic Hermite Interpolation: Maintaining continuity of both function values and derivatives MATLAB's built-in interp1 function supports these interpolation methods through method parameter specification. For instance, linear interpolation corresponds to the 'linear' option, while cubic spline interpolation uses the 'spline' option. For two-dimensional and higher-dimensional data, corresponding interp2 and interpn functions are available. Custom interpolation implementations typically address specific application requirements, such as handling non-uniformly sampled data, improving computational efficiency, or meeting special boundary conditions. These custom programs often provide greater flexibility than built-in functions, allowing adjustments and optimizations based on practical problem constraints. When selecting appropriate interpolation algorithms in practical applications, considerations must include data characteristics, accuracy requirements, and computational efficiency. MATLAB's comprehensive toolset and open environment facilitate convenient implementation and testing of interpolation algorithms through straightforward function calls and parameter configurations. Key implementation aspects include: - Using interp1 with method parameters like 'linear', 'spline', 'nearest' for 1D data - Implementing cubic Hermite interpolation with pchip function for shape preservation - Handling 2D/3D data with interp2/interp3 using meshgrid-generated coordinate matrices - Custom algorithm development for specialized boundary conditions or irregular grids - Performance optimization through vectorization and pre-allocation techniques