Algorithms for Various Curve Fitting Techniques in MATLAB Environment

In the MATLAB environment, curve fitting serves as a fundamental technique in data analysis, enabling the discovery of underlying mathematical patterns from discrete data points. MATLAB offers extensive built-in functions and toolboxes that support multiple curve fitting algorithms, catering to various data distributions and fitting requirements.

For simple linear relationships, the polyfit function performs polynomial fitting using least squares method to compute optimal coefficients. This MATLAB function supports polynomials of any degree - users simply specify the polynomial order to obtain corresponding fitted curves. The polyval function complements this by calculating points along the fitted curve for rapid visualization, typically implemented as: fitted_values = polyval(coefficients, x_data).

When dealing with nonlinear data, users can employ either the cftool utility or fit function. The Curve Fitting Tool (cftool) provides an interactive graphical interface supporting multiple preset models (exponential, power functions, Gaussian distributions, etc.), allowing intuitive parameter adjustments with real-time fitting previews. The fit function offers a programmatic interface that accepts custom model equations, ideal for automation scenarios using syntax like: f = fit(x_data, y_data, 'model_type').

For more complex fitting challenges, the lsqcurvefit function from the Optimization Toolbox handles arbitrary nonlinear least-squares problems. This advanced function employs optimization algorithms to find optimal parameter combinations, particularly suitable for custom models with multiple parameters. The Statistics and Machine Learning Toolbox further enhances robustness with functions like robustfit, which minimizes outlier influence on fitting results through weighted least squares algorithms.

MATLAB's curve fitting capabilities include comprehensive quality assessment features, calculating statistical metrics such as R-squared coefficients and residual analysis. The integration of these tools provides flexible solutions for diverse curve fitting needs - from simple linear relationships to complex nonlinear patterns - delivering powerful support for scientific research and engineering applications.