Comparative numerical laboratory

Direct Linear Solvers: Pivoting and Conditioning — Chapter 4

Solve the same \(2\times2\) linear system with and without a stable pivoting idea.

\[ A=\begin{pmatrix}\varepsilon&1\\1&1\end{pmatrix}, Ax=b. \]

Methods compared

Method	Convenience	Inconvenience / warning
No pivoting	Fast and simple in exact arithmetic.	Can divide by a tiny pivot and amplify roundoff.
Partial pivoting	Major stability safeguard in Gaussian elimination.	Still affected by ill-conditioning.
Residual check	Cheap way to verify the computed solution satisfies equations.	Residual alone does not measure forward error when \(A\) is ill-conditioned.

small pivot \(\varepsilon\) \(b_1\) \(b_2\)