A student argues: "An -node Gauss rule is exact to degree

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

Error, Chance and the Machine, Ch Integration: error expansions, Gauss rules and adaptivity

Problem

A student argues: "An nn-node Gauss rule is exact to degree 2n−12n-1, while a Clenshaw-Curtis rule on nn nodes is exact only to degree about nn. The degree of exactness controls the error, so at the same number of function values a Gauss rule must have about twice as many correct digits." For Runge's function 1/(1+25x2)1/(1+25x^2) on [−1,1][-1,1] with 2525 function values, the errors are 9.2×10−59.2\times10^{-5} for Clenshaw-Curtis and 5.0×10−55.0\times10^{-5} for Gauss, and the nearest singularities of the integrand are at ±i/5\pm i/5. Find what is wrong with the argument. Count the correct digits, −log⁡10∣error∣-\log_{10}|\text{error}|, of each rule and compare them with the claim, and say what sets the rate of both rules for this function.

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