Math 6610 • Analysis of Numerical Methods I

{Sensitivity}
Consider a \(6\times 6\) symmetric positive definite matrix \(A\) with singular values
\(\sigma_1=1000\), \(\sigma_2=500\), \(\sigma_3=300\), \(\sigma_4=20\), \(\sigma_5=1\),
\(\sigma_6=0.01\).

\textbf{a)} Suppose you use a Cholesky factorization package on a computer with a
machine epsilon \(10^{-14}\) to solve the system \(Ax=b\) for some nonzero vector \(b\).
How many digits of accuracy do you expect in the computed solution? Justify your
answer in terms of condition and stability. You may assume that the entries of \(A\)
and \(b\) are exactly represented in the computer's floating-point system.

\textbf{b)} Suppose that instead you use an iterative method to find an approximate
solution to \(Ax=b\) and you stop iterating and accept iterate \(x^{(k)}\) when the
residual \(r^{(k)}=Ax^{(k)}-b\) has 2-norm less than \(10^{-9}\). Give an estimate of the
maximum size of the relative error in the final iterate? Justify your answer.