Mathematical Tools for Physics

11—Numerical Analysis 349

Show that an equal spaced integration scheme to evaluate such an integral is

P

∫+h

−h

f(x)

x

dx=f(h)−f(−h)−

2

9

h^3 f′′′(0).

Also, an integration scheme of the Gaussian type is

√

3

[

f(h

/√

3)−f(−h

/√

3)

]

+

h^5

675

fv(0).

11.8 Devise a two point Gaussian integration with errors for the class of integrals

∫+∞

−∞

dxe−x

2

f(x).

Find what polynomial has roots at the points wherefis to be evaluated. See problem7.26.

11.9 Same as the previous problem, but make it a three point method.

11.10 Find two and three point Gauss methods for

∫∞

0

dxe−xf(x).

What polynomials are involved here? Look up Laguerre.

11.11 In numerical differentiation it is possible to choose the intervaltoosmall. Every computation is done to
a finite precision. (a) Do the simplest numerical differentiation of some specific function and take smaller and
smaller intervals. What happens when the interval gets very small? (b) To analyze the reason for this behavior,
assume that every number in the two point differentiation formula is kept to a fixed number of significant figures
(perhaps 7 or 8). How does the error vary with the interval? What interval gives the most accurate answer?
Compare this theoretical answer with the experimental value found in the first part of the problem.