PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics
Mathematical Methods for Physics, Ch Root finding, interpolation, numerical linear algebra and quadrature
Problem
Someone argues: "f(x)=1/(1+25x2) is smooth and bounded by 1 on [−1,1]. Interpolating it at twenty-one equally spaced nodes uses a higher-degree polynomial than eleven equally spaced nodes, and a higher-degree polynomial matches more of the data, so the twenty-one-node interpolant must have the smaller maximum error." Given that eleven equally spaced nodes give a largest error of 1.9 (near x=±0.94) and twenty-one give a largest error of about 60 (near x=±0.975), find the error in the reasoning and explain, using the size of the node product w(x)=∏i(x−xi) near the ends of [−1,1], why adding equally spaced nodes makes the error worse rather than better here.