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.
a) Using Equidistant points b) Using Chebyshev points
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.