Topic 10. Interpolation. Parabolic interpolation. Spline interpolation

When interpolating functions with a large number of nodes, the interpolating polynomial has a high degree that causes its oscillations between nodes. To reduce the degree of polynomial, all interpolation nodes can be divided into groups and construct interpolation polynomials with fewer nodes. But in this case, at the intersections between polynomials, the analytic properties of the interpolating polynomial are violated, and the derivation points appear. One of the outputs of this provision is the use of splines.

The spline on the gap between the interpolation nodes is a low-level polynomial (n = 3,4). The spline over the entire interpolation segment is a function that consists of different parts of the polynomials of a given degree. A striking example of a spline is a pattern.

Spline is a function which, along with several derivatives, is continuous on the entire given interval [], and on each partial segment [,] separately is a certain algebraic polynomial.

The spline's degree is called the maximum of polynomials in all partial segments, and the spline defect is the difference between the degree of spline and the order of the highest continuous on [] of the derivative. For example, a continuous lamina is a spline of degree 1 with a defect 1 (the function itself is continuous, and the first derivative is already discontinuous).

Interpolation in the Maple environment

When interpolating functions with a large number of nodes, the interpolating polynomial has a high degree that causes its oscillations between nodes.