Take , for which and on . (a) Show that , and so obtain the trapezium rule's error on…

PHYSICS · Subjective · 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

Take

\int_0^{\pi}\sin x\,\mathrm{d}x = 2

, for which

|f''| \le 1

and

|f^{(4)}| \le 1

[0,\pi]

. (a) Show that

\int_{x_0}^{x_0+h}(x-x_0)(x-x_0-h)\,\mathrm{d}x = -h^3/6

, and so obtain the trapezium rule's error

-\tfrac{h^3}{12}f''(\xi)

on one panel from the linear interpolation error. (b) Find the smallest number of intervals

N

for which the trapezium rule's error term guarantees an error below

10^{-6}

, and the smallest even

N

for which Simpson's does. (c) The tolerance is tightened to

10^{-8}

. Predict each new

N

from the power of

h

in its error term before recomputing it, then compare the number of function evaluations the two rules need.

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