Estimating the Backward Error in LSQR
SIAM Journal on Matrix Analysis and Applications
LSMR: An Iterative Algorithm for Sparse Least-Squares Problems
SIAM Journal on Scientific Computing
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A practical estimate for the optimal backward errors of linear least squares problems are studied. Dense, sparse, and iterative methods are developed to evaluate the estimate. Numerical results show that the computed estimate of the optimal backward error is very near the true optimal backward error. Algorithms for calculating sequences of upper and lower bounds for the estimate are also developed, based on Gauss quadrature theory. Numerical results show that the bounds converge quickly and are therefore useful in practice. This solves a twenty-five year old problem suggested by Stewart and Wilkinson. Test-based approaches are explored to construct confidence intervals in clinical trials with survival time as the primary response. Importance resampling techniques are developed to compute tail probabilities of the tests, thereby reducing the variance of the Monte Carlo estimate of an error probability and thus the number of simulations required to compute sample size and power in the design stage of a clinical trial and to construct confidence intervals from the trial's data.