Why Spline Interpolation?

Fitting data with smooth curves avoids sharp jumps that polynomial interpolation may introduce. A natural cubic spline consists of piecewise cubic polynomials joined so the first and second derivatives match at interior points, while the second derivatives vanish at the ends. This produces a flexible yet stable curve.

Input Format

Provide a list of points separated by semicolons, such as 0,0; 1,1; 2,0; 3,2. The calculator sorts them by their $x$-values, constructs the spline, and evaluates it at your chosen coordinate. Because the system involves solving a tridiagonal matrix, at least three points are required.

Behind the Scenes

The method solves for coefficients so that each cubic $S$_i(x) satisfies $S$_i(x_i)=y_i and $S$_i(x_i+1)=y_i+1. Continuity of first and second derivatives leads to a system of linear equations for the second-derivative values at each node.