Classical probabilistic rounding error analysis is particularly well suited to stochastic rounding (SR), and it yields strong results when dealing with floating-point algorithms that rely heavily on summation. For many numerical linear algebra algorithms, one can prove probabilistic error bounds that grow as O(nu), where n is the problem size and u is the unit roundoff. These probabilistic bounds are asymptotically tighter than the worst-case ones, which grow as O(nu). For certain classes of algorithms, SR has been shown to be unbiased. However, all these results were derived under the assumption that SR is implemented exactly, which typically requires a number of random bits that is too large to be suitable for practical implementations. We investigate the effect of the number of random bits on the probabilistic rounding error analysis of SR. To this end, we introduce a new rounding mode, limited-precision SR. By taking into account the number r of random bits used, this new rounding mode matches hardware implementations accurately, unlike the ideal SR operator generally used in the literature. We show that this new rounding mode is biased and that the bias is a function of r. As r approaches infinity, however, the bias disappears, and limited-precision SR converges to the ideal, unbiased SR operator. We develop a novel model for probabilistic error analysis of algorithms employing SR. Several numerical examples corroborate our theoretical findings.

In this paper we generalize the technique of deflation to define two new methods to systematically find many local minima of a nonlinear least squares problem. The methods are based on the Gauss-Newton algorithm, and as such do not require the calculation of a Hessian matrix. They also require fewer deflations than for applying the deflated Newton method on the first order optimality conditions, as the latter finds all stationary points, not just local minima. One application of interest covered in this paper is the inverse eigenvalue problem (IEP) associated with the modelling of spectroscopic data of relevance to the physical and chemical sciences. Open source MATLAB code is provided at<br/>https://github.com/AlbanBloorRiley/DeflatedGaussNewton.