Source of Cubic Spline Interpolation: Implementation and Algorithm

Cubic spline interpolation is a classical numerical computation method used to construct smooth curves from a set of discrete data points. Its core concept involves connecting data points using piecewise-defined cubic polynomials while ensuring continuity of both first and second derivatives at the nodes (connection points), thereby guaranteeing curve smoothness.

Implementation of cubic spline interpolation typically follows these algorithmic steps: Data Preparation: Input a set of discrete data points (x_i, y_i) with x_i values sorted in strictly increasing order - this can be implemented using sorting algorithms like quicksort or built-in sort functions. Boundary Condition Specification: Commonly select natural boundary conditions (second derivatives set to zero) or clamped boundary conditions (specified derivatives at endpoints) - in code, this translates to setting appropriate matrix elements in the system equation. Solving Tridiagonal System: Construct and solve a tridiagonal linear system using continuity conditions at nodes to determine second derivatives for each polynomial segment - efficiently implemented using Thomas algorithm (TDMA) with O(n) complexity. Calculating Polynomial Coefficients: Compute cubic polynomial coefficients for each interval using the obtained second derivatives - typically implemented through algebraic formulas derived from Hermite interpolation basis. Interpolation Computation: Calculate interpolation results at given x values using the corresponding polynomial segments - requiring a binary search to locate the correct interval before polynomial evaluation.

The advantage of cubic spline interpolation lies in its ability to produce smoother results than linear interpolation while avoiding oscillation phenomena common in high-degree polynomial interpolation. This method finds extensive applications in engineering, scientific computing, and computer graphics, such as experimental data fitting, smooth trajectory generation, and numerical simulation optimization.

In practical implementations, when the target function exhibits good smoothness properties, cubic spline interpolation typically provides accurate approximations. For large datasets or scenarios requiring higher computational efficiency, variant methods like B-splines or parametric splines may be considered, which offer better numerical stability through localized basis functions.