(a) A Newton step is applied to a system of nonlinear equilibrium equations. The…

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Error, Chance and the Machine, Ch Iteration: nonlinear systems and eigenvalues

Problem

(a) A Newton step is applied to a system of

n=10

nonlinear equilibrium equations. The Jacobian is built by forward differences, needing one baseline evaluation of the full residual vector

\mathbf F

plus one further evaluation per unknown (

n+1

evaluations in total), each such evaluation costing

c=50

floating-point operations. The resulting

10\times10

linear system is then solved through an LU factorisation, which costs

\tfrac23n^3

flops (Chapter 2); count that, and neglect the

2n^2

flops of the two triangular solves. Estimate the total flop count for this one Newton step. (b) A monic polynomial has a simple root at

r=8

with root condition number

\kappa = 2.0\times10^{10}

(found from the sensitivity formula of Chapter 1). Its roots are computed in binary64 as the eigenvalues of the companion matrix by the (backward stable) QR algorithm, giving a computed root

\tilde r = 8.0000141

. Predict the expected relative error

\kappa u

(with

u=2^{-53}

) and check it against the observed relative error of

\tilde r

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