What is Cubic, natural and monotone splines, Runge phenomenon?

A cubic spline through $(x_i, y_i)$ is a piecewise cubic with continuous first and second derivatives; a natural cubic spline has zero second derivative at both ends, and its second derivatives at the nodes solve a tridiagonal system. Monotone cubic interpolation chooses the node slopes of a piecewise cubic Hermite interpolant so that it is monotone wherever the data are. The Runge phenomenon (Runge, 1901) is the growth, near the ends of the interval, of the error of polynomial interpolation at equally spaced nodes as the degree increases.