Effectively Well-Conditioned Linear Systems

When solving the linear system ${\bf A}x = {\bf b}$, the condition number $K(A) \equiv \| A \| \| A^{ - 1} \|$ is a useful, albeit often overly conservative, measure of the sensitivity of the solution ${\bf x}$ under perturbations $\Delta A$ and $\Delta {\bf b}$ to A and ${\bf b}$. We demonstrate how the projection of ${\bf b}$ onto the range space of A, in addition to $K(A)$, can strongly affect the sensitivity of ${\bf x}$ in specific problem instances. Two practical cases are presented in which the sensitivity of ${\bf x}$ can be substantially smaller than that predicted by $K(A)$ alone. In the first example, we characterize a class of Vandermonde matrices and right-hand sides for which accurate algorithms can exist. For the second example, we show that a (fast Fourier transform-) FFT-based fast Poisson solver can produce very accurate results for smooth right-hand sides. Computational examples on the fast Poisson solver are included to illustrate these concepts.