Implementation of Classic Mathematical Algorithms in MATLAB Software

In mathematical modeling, MATLAB serves as a widely-used mathematical software for solving various mathematical problems. This article presents the implementation and underlying principles of several classic algorithms within MATLAB.

First, the Least Squares Method is introduced. This mathematical optimization technique minimizes the sum of squared errors to find the best-fitting line for given data. MATLAB's primary function for this algorithm is lsqcurvefit(), which solves nonlinear least-squares problems through an iterative optimization process. The function requires initial parameter estimates and supports both curve fitting and parameter estimation tasks by minimizing the residual between observed data and model predictions.

Next, Lagrange Interpolation Method is discussed. This polynomial interpolation technique constructs an interpolating polynomial that passes exactly through given data points. MATLAB implements this through the polyfit() function, which calculates polynomial coefficients using a Vandermonde matrix approach. The function accepts data points and desired polynomial degree, returning coefficients that can be evaluated using polyval() for interpolation at any point within the data range.

Finally, Simpson's Rule is presented. This numerical integration method approximates definite integrals by fitting parabolic segments to the function. MATLAB provides the quad() function for adaptive Simpson quadrature, which recursively refines the approximation until achieving the specified tolerance. The algorithm automatically adjusts step sizes based on function behavior, making it efficient for both smooth functions and those with varying curvature.

These algorithm implementations are readily available in MATLAB, providing powerful tools for mathematical modeling applications. Each function includes error handling and supports various options for algorithm customization, enabling users to balance between computational efficiency and numerical accuracy.