Fast Adaptive Condition Estimation

Recursive condition number estimates of matrices are useful in many areas of scientific computing, including recursive least squares computations, optimization, eigenanalysis, and general nonlinear problems solved by linearization methods where matrix modification techniques are used. The purpose of this paper is to propose a fast adaptive condition estimator, called $ACE$, for tracking the condition number of a modified matrix over time, in terms of its triangular factors. Symmetric rank-one modifications are considered, and it is noted how the schemes generalize to higher rank modifications and thus to nonsymmetric rank-one updates. $ACE$ is fast in the sense that only $O(n)$ operations are required for n parameter problems, and is adaptive over time, i.e., estimates at time t are used to produce estimates at time $t + 1$. Traditional condition estimators for triangular factors, such as the LINPACK and LAPACK type schemes, generally require $O(n^2 )$ operations and are not adaptive. The only situation w...