ASTR615 Fall 2015 Problem Set #3

Due Oct 21, 2015

Topics for this problem set include round-off error and linear algebra.

1. As an example of an unstable algorithm, consider integer powers of the Golden Mean'' . It can be shown that , i.e. successively higher powers of can be computed from a single subtraction rather than a more expensive multiply. Write a single-precision program to compute a table consisting of the columns , computed from the recursion relation, and computed directly (i.e. ), for ranging from 1 to 20. Is the round-off error random? What happens in double precision?

2. Write a program to compute the instantaneous spin period of a rigid body made up of identical, discrete, point particles. Use the fact that the angular momentum is
 (1)

where is the mass of particle , and are its position and velocity vectors with respect to the centre of mass, is the spin vector, and is the inertia tensor

where is the unit matrix. [For continuous bodies the summations are replaced by volume integrations and the particle masses become a mass density. In the present case the 's can be omitted entirely since the particles are identical.] Write a program to solve Eq. () for (feel free to use the routines in Numerical Recipes). The spin period is then .