Realistic Graphics Lab

Exercise session - Extra questions (1)

$$ \newcommand{\vb}{\mathbf{b}} \newcommand{\vc}{\mathbf{c}} \newcommand{\vx}{\mathbf{x}} \newcommand{\mA}{\mathbf{A}} \newcommand{\mL}{\mathbf{L}} \newcommand{\mU}{\mathbf{U}} \newcommand{\mP}{\mathbf{P}} \newcommand{\mI}{\mathbf{I}} $$
  1. Given least squares problem $\mA\vx\approx\vb$, suppose that the right hand side $\vb$ matches one of the columns of $\mA$. Does this problem have a unique solution? Can the problem be solved with residual $\mA=0$? Do these answers change when $\vb\notin\mathrm{span}\mA$?

  2. Suppose that we just performed a LU factorization of a matrix $\mA\in\mathbb{R}^{n\times n}$ with partial pivoting, resulting in the factorization $\mA=\mP\mL\mU$. Is it possible to efficiently compute the determinant of $\mA$ given this information? (You may need to review rules for computing the determinant of products and orthogonal and triangular matrices). Approximately how many FLOPs will this computation consume?

    Assuming we find that $\det{\mA}=\varepsilon$ (where $\varepsilon$ is a tiny number, e.g. $10^{-300}$), should this make us nervous about potential numerical difficulties when solving linear systems involving $\mA$?