4 papers
A Variational Equation and Lower Bound for the Linear Least-Squares Backward Error
Eric Hallman
This paper derives a new variational equation for the linear least-squares backward error by expressing the backward error in terms of a generalized eigenvalue problem and using re…
Two Variations on the XTrace Algorithm
Eric Hallman
This paper studies two potential modifications of XTrace (Epperly et al., SIMAX 45(1):1-23, 2024), a randomized algorithm for estimating the trace of a matrix. The first is a varia…
Extremal bounds for Gaussian trace estimation
Eric Hallman
This work derives extremal tail bounds for the Gaussian trace estimator applied to a real symmetric matrix. We define a partial ordering on the eigenvalues, so that when a matrix h…
Deterministic and Probabilistic Error Bounds for Floating Point Summation Algorithms
Eric Hallman, Ilse C. F. Ipsen
We analyse the forward error in the floating point summation of real numbers, from algorithms that do not require recourse to higher precision or better hardware. We derive informa…