Calculation and Analysis of Lagrange, Piecewise Linear, and Cubic Spline Interpolation Methods with MATLAB Implementation

In this course module, we will explore three fundamental interpolation techniques: Lagrange interpolation, piecewise linear interpolation, and cubic spline interpolation. Students will learn to implement these methods using MATLAB programming, including key functions such as polyfit for polynomial fitting, interp1 for one-dimensional interpolation with different method parameters ('linear', 'spline'), and understanding algorithm implementations for node density variation analysis. The curriculum includes practical exercises modifying node quantities to examine how interpolation accuracy changes with data point distribution.

Additionally, we will cover linear least squares implementation using MATLAB's backslash operator (\) for solving overdetermined systems and the lsqlin function for constrained optimization. These mathematical tools extend beyond theoretical concepts to solve practical engineering problems, such as predicting unknown values from sampled data points. Through concrete examples, we will demonstrate application scenarios where interpolation methods excel at exact data point matching versus fitting methods' strength in handling noisy measurements. Special attention will be given to algorithmic differences: interpolation strictly passes through all data points while fitting minimizes overall error, making their respective MATLAB implementations distinct in approach and application. This foundation will significantly enhance your mathematical modeling and computational problem-solving capabilities for engineering applications.