Incremental Condition Estimation

This paper introduces a new technique for estimating the smallest singular value, and hence the condition number, of a dense triangular matrix as it is generated one row or column at a time. It is also shown how this condition estimator can be interpreted as trying to approximate the secular equation with a simpler rational function. While one can construct examples where this estimator fails, numerical experiments demonstrate that despite its small computational cost, it produces reliable estimates. Also given is an example that shows the advantage of incorporating the incremental condition estimation strategy into the QR factorization algorithm with column pivoting to guard against near rank deficiency going unnoticed.