A simulation returns for any in

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

Mathematical Methods for Physics, Ch Root finding, interpolation, numerical linear algebra and quadrature

Problem

A simulation returns

f(x) = 1/(1+25x^2)

for any

x

[-1,1]

, but each evaluation takes

1\ \mathrm{s}

. We need

\int_{-1}^{1}f\,\mathrm{d}x

with an error below

10^{-6}

. On

[-1,1]

|f''| \le 50

and

|f^{(4)}| \le 15\,000

, both largest at

x = 0

. ((a)) From the error terms, find how many intervals the trapezium rule and Simpson's rule each need to guarantee the target, and how long each run takes. Which rule would you run? ((b)) A colleague proposes to pass one polynomial of degree ten through eleven equally spaced samples and integrate it exactly, for

11\ \mathrm{s}

in all. For these nodes

|w(x)|

reaches

8.5\times10^{-3}

in the end intervals, and

|f^{(11)}(x)|/11!

reaches

4.4\times10^{7}

near

x = \pm0.02

. Decide from the interpolation error whether the polynomial can be trusted, and so whether the proposal is safe. ((c)) Make the same decision for

g(x) = 1/(1+x^2)

, for which

|g^{(11)}(x)|/11! \le 0.91

[-1,1]

, and name the property of the function, not of the nodes, that decides the matter.

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