A root-finding routine working in binary64 ( ) is applied to p(x) = x^3 - 8.001x^2 +…

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

Error, Chance and the Machine, Ch Rounding, backward error and the sensitivity of a computation

Problem

A root-finding routine working in binary64 (u=1.11×10−16u = 1.11\times10^{-16}) is applied to p(x)=x3−8.001x2+21.005x−18.006,p(x) = x^3 - 8.001x^2 + 21.005x - 18.006, whose three roots are exactly 22, 33 and 3.0013.001. Seeking the root near x=3x=3, the routine returns r^=3.0003\hat r = 3.0003. ((a)) Compute the residual p(r^)p(\hat r) and hence the backward error η=∣p(r^)∣∣a3∣r^3+∣a2∣r^2+∣a1∣r^+∣a0∣,\eta = \frac{|p(\hat r)|}{|a_3|\hat r^3 + |a_2|\hat r^2 + |a_1|\hat r + |a_0|}, where a3=1a_3=1, a2=−8.001a_2=-8.001, a1=21.005a_1=21.005, a0=−18.006a_0=-18.006. ((b)) Evaluate p′(r^)=3r^2−16.002r^+21.005p'(\hat r) = 3\hat r^2 - 16.002\hat r + 21.005 and use it to form the coefficient-sum condition number κ=∣a3∣r^2+∣a2∣r^+∣a1∣+∣a0∣/r^∣p′(r^)∣.\kappa = \frac{|a_3|\hat r^2 + |a_2|\hat r + |a_1| + |a_0|/\hat r}{|p'(\hat r)|}. ((c)) Combine (a) and (b) into the first-order forward error estimate κη\kappa\eta for the relative error of r^\hat r, and compare it with the true relative error of r^\hat r from the nearer of the two true roots 33 and 3.0013.001. ((d)) The key result for the sensitivity of a simple root evaluates the condition number at the root itself. Recompute κ\kappa with r=3r=3 in place of r^\hat r, form κη\kappa\eta again, compare it with the true error, and say which condition of the key result fails here.

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