Homework 10.

Assigned: Mon Nov 15. Due: Mon Nov 22, at midnight
  1. Any two of the four exercises in Lecture 24 (page 188-189) of the text. (For Exercise 25.3, type "help expm" in Matlab.)
  2. Modify my code inv_power.m to implement the Rayleigh quotient iteration (see Lecture 27). This requires defining the shift mu inside the loop, instead of outside it, and also solving the system of equations inside the loop, since now the matrix is different each time. There is no longer any reason to use L and U; you can just use the \ syntax. An important part of this question is to investigate the rate of convergence. The book speaks of "cubic convergence" when A is symmetric: this means the error (difference between approximate and exact eigenvalue) is cubed each iteration (e.g. 10^-1; 10^-3; 10^-9). Equivalently, the number of correct decimal digits approximately triples each iteration. Do you observe this? What if A is not symmetric? Include some judiciously selected printed output to justify your observations.