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--- gibson:teaching:fall-2014:iam961:hw5 [2014/12/05 13:48]
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 Can you confirm that the errors scale as stated on page 208 of Trefethen and Bau?
+Note: the suggested line counts above are not requirements. They're meant as guidelines to let you know how roughly how much code each line should require. If you're writing more, you're working too hard.
+----
+**Problem 2:** Implement GMRES and investigate its convergence as a function of the matrix condition number, as follows. Given a value of $M$ and $\kappa$, write code that will
+. Construct a random M x M matrix A with condition number $\kappa$ and exponentially graded singular values, $\sigma_n = \kappa^{n/N}$.
+. Let $x$ be a random M vector with normally-distributed components. Let $b = A x$.
+. Do the GMRES iteration and plot the normalized residual $\| A x_n - b\|/\|b\|$ and the normalized $x$ error $\| x_n - x \|$ as a function of the iteration number $n$. Plot the residual with circles and the error with dots, put them on the same plot, and **use a logarithmic axis for the errors**: e.g. ''semilogy(n,error,'b.',n,residual,'ro')''.
+For testing and developing your code, set $M$ to a small number like 10 and $\kappa$ to $10^8$. Your GMRES code should go all the way to $n=M$ iterations. Your code should have a for-loop to handle iterations $n=1$ to $M-1$ followed by some special-case code to handle the last $n=M$th iterate --the last iteration works a little differently.
+Once your code is working (the residual is $10^{-16}$ at the last iterate), set $M$ to a moderately small number like 30 and produce four residual-and-error plots, for  $\kappa=100$, $\kappa=10^8$, $\kappa=10^{16}$, and $\kappa=10^{32}$.
+Discuss your results. Is anything surprising? What can you explain about the behavior of the plots, given what you know about GMRES and $Ax=b$ problems in general?

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