The Lorentzian profile on models an absorption line whose sharpness is set by the…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Where a function vanishes, and what lies between the points

Problem

The Lorentzian profile f(x)=1/(1+ax2)f(x) = 1/(1+ax^2) on x∈[−1,1]x \in [-1,1] models an absorption line whose sharpness is set by the parameter aa. It is interpolated at 2121 equally spaced nodes on [−1,1][-1,1], once by the degree-2020 Lagrange polynomial and once by the natural cubic spline through the same nodes. For a broader line, a=9a = 9, the measured maximum error over the whole interval is 2.72.7 for the polynomial and 5.6×10−45.6\times10^{-4} for the spline. For a sharper line, a=25a = 25, measured the same way, it is 59.859.8 for the polynomial and 3.2×10−33.2\times10^{-3} for the spline. (a) By what factor does each interpolant's maximum error grow as the line sharpens from a=9a=9 to a=25a=25? (b) Using the Lagrange error term f(n+1)(ξ)∏j(x−xj)/(n+1)!f^{(n+1)}(\xi)\prod_j(x-x_j)/(n+1)!, explain why sharpening the line inflates the polynomial's error so much more than the spline's.

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