Skip to main content

Exercises 7.5 Exercises

1.
Use Lagrangian polynomials to find the polynomial of least degree that passes through the following set of points:
  1. \(\displaystyle x_0=3,f(x_0)=8,x_1=6,f(x_1)=104\)
  2. \(\displaystyle x_0=1,f(x_0)=2,x_1=3,f(x_1)=12,x_2=7,f(x_2)=146\)
2.
Show that interpolating \(f(x)=\cosh(x)\) with a polynomial of degree 22 in \([-1,1]\) using 23 Chebisehv points gives a maximum error not greater than \(5\times10^{-16}\text{.}\)
3.
Interpolate the function \(f(x)=\frac{1}{1+x^2}\) in the interval \([0,1]\) using:
  1. three equidistant points;
  2. three Chebyshev points.

Estimate the maximum of the distance between the two interpolating polynomials and the function \(f(x)\) (for instance by using MATLAB to plot the difference) and compare it with the upper bound provided by the error formulae of Section 7.3