Math 6610 • Analysis of Numerical Methods I

\textbf{a)} Give an example to demonstrate the role of pivoting in Gaussian
elimination in solving \(Ax=b\). Define and explain the significance of the condition
number of \(A\). Compare the condition numbers, with respect to the norm of your
choice, of \(A\) and \(PA\), where \(P\) is a permutation matrix.

\textbf{b)} Explain the \(LU\) decomposition of a matrix \(A\), in particular why the
entries of \(L\) are multipliers used in Gaussian elimination, and how to solve
\(Ax=b\) using \(L\) and \(U\). How much work does this take?