21 Jun 2017

Interpolating Runge Function Using Polynomial & Natural Cubic Splines

We could expect that the interpolation error will decrease when we take more points. But in practice, the larger n we take, the wider the deviation of the interpolation polynomial becomes from the function (near the edges). If the number of points goes to infinity, the interpolation error will tend towards infinity too.

interpolating runge function using polynomial & natural cubic splines - 1x1 - Interpolating Runge Function Using Polynomial & Natural Cubic Splines

                      interpolating runge function using polynomial & natural cubic splines - 1x1 - Interpolating Runge Function Using Polynomial & Natural Cubic Splines

a) Using Equidistant points                                                       b) Using Chebyshev points

interpolating runge function using polynomial & natural cubic splines - 1x1 - Interpolating Runge Function Using Polynomial & Natural Cubic Splines

               interpolating runge function using polynomial & natural cubic splines - 1x1 - Interpolating Runge Function Using Polynomial & Natural Cubic Splines

a) Using natural cubic splines                                                   Error Graph of Q3 (a)image2

We conclude that Chebyshev nodes are a very good choice for polynomial interpolation, as the growth in n is

Exponential for equidistant nodes. While in cubic splines the error increases as shown in the figure image2.

Cubic spline is better than the equidistant points using LaGrange’s method.

Leave a Comment