MATLAB Implementation of Cubic Spline Interpolation

In mathematics and computer science, cubic spline interpolation is a widely-used interpolation method where unknown function values are estimated through known function values. In cubic spline interpolation, a set of piecewise polynomial functions is used to approximate the original function. To compute the coefficients of these polynomials, the three-moment method can be employed. This method involves calculating three bending moment matrices, from which the polynomial coefficients can be derived. In MATLAB implementation, this typically requires solving a tridiagonal system of equations using algorithms like Thomas algorithm for efficient computation. Curve fitting is a similar problem-solving approach that attempts to find an appropriate curve through a set of data points. When calculating function values at interpolation points, these methods can be used to estimate the function's value. Furthermore, the first and second derivatives of the function at interpolation points can also be computed to obtain more information about the function's behavior. In MATLAB code, this involves using spline functions or custom implementations that handle boundary conditions and ensure continuity of derivatives at knots. Key MATLAB functions for implementation may include handling different boundary conditions (natural, clamped, or not-a-knot), constructing the coefficient matrix, and solving the linear system. The implementation typically involves: 1. Data point input and validation 2. Matrix construction for the three-moment equations 3. Efficient solution of the tridiagonal system 4. Polynomial coefficient calculation for each segment 5. Evaluation of function values and derivatives at desired points The algorithm ensures that the spline passes through all data points while maintaining continuous first and second derivatives across intervals, providing smooth and accurate approximations.