Consider polynomial interpolation of the functionf(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolatingpolynomial determined by m equidistant interpolation points, (2) aninterpolating polynomial determined by interpolation at the m zerosof the Chebyshev polynomial T_m(x), and (3) by interpolating bycubic splines instead of by a polynomial. Estimate theapproximation error by evaluation max_i |f(z_i)-p(z_i)| for manypoints z_i on [-1,1]. For instance, you could use 10m points z_i.The cubic spline interpolant can be determined in MATLAB; see \"helpspline\". Use m=10 and m=20. Compute splines that interpolate atequidistant nodes and at Chebyshev nodes. Provide tables of theerrors and plots of the function f and the interpolatingpolynomials and splines.
