A detector records the normalised intensity profile across a slit

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Approximating a function: interpolation and differentiation

Problem

A detector records the normalised intensity profile I(x)=11+16x2I(x) = \dfrac{1}{1+16x^2} across a slit, with xx running over [−1,1][-1,1]. A degree-4 polynomial model of II is to be built from samples at the five Chebyshev points of the second kind on [−1,1][-1,1], xj=cos⁡ ⁣(jπ4),j=0,1,2,3,4.x_j = \cos\!\left(\frac{j\pi}{4}\right), \qquad j = 0,1,2,3,4. ((a)) List the five nodes xjx_j, to four decimal places. ((b)) Compute I(xj)I(x_j) at each node, to four significant figures. ((c)) The barycentric weights for these second-kind points are wj=12,−1,1,−1,12w_j=\tfrac12,-1,1,-1,\tfrac12. Evaluate the interpolating polynomial p4(x)p_4(x) at x=0.5x=0.5 by the second barycentric form, giving the denominator sum, the numerator sum and their ratio, and compare with I(0.5)I(0.5).

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