Abstract. How does GMRES convergence change when the coefficient matrix is perturbed? Using spectral perturbation theory and resolvent estimates, we develop simple, general bounds that quantify the lag in convergence such a perturbation can induce. This analysis is particularly relevant to preconditioned systems, where an ideal preconditioner is only approximately applied in practical computations. To illustrate the utility of this approach, we combine our analysis with Stewart's invariant subspace perturbation theory to develop rigorous bounds on the performance of approximate deflation preconditioning using Ritz vectors.

Abstract. The progressive GMRES algorithm, introduced by Beckermann and Reichel in 2008, is a residual-minimizing short-recurrence Krylov subspace method for solving a linear system in which the coefficient matrix has a low-rank skew-Hermitian part. We analyze this algorithm, observing a critical instability that makes the method unsuitable for some problems. To work around this issue we introduce a different short-term recurrence method based on Krylov subspaces for such matrices, which can be used as either a solver or a preconditioner. Numerical experiments compare this method to alternative algorithms.

We present a simple, accurate method for solving consistent, rank-deficient linear systems, with or without additional rank-completing constraints. Such problems arise in a variety of applications, such as the computation of the eigenvectors of a matrix corresponding to a known eigenvalue. The method is based on elementary linear algebra combined with the observation that if the matrix is rank-k deficient, then a random rank-k perturbation yields a nonsingular matrix with probability 1.

At the beginning of August 2020, the Rio Grande Valley (RGV) of Texas experienced a rapid increase of coronavirus disease 2019 (abbreviated as COVID-19) cases and deaths. This study aims to determine the optimal levels of effective social distancing and testing to slow the virus spread at the outset of the pandemic. We use an age-stratified eight compartment epidemiological model to depict COVID-19 transmission in the community and within households. With a simulated 120-day outbreak period data we obtain a post 180-days period optimal control strategy solution. Our results show that easing social distancing between adults by the end of the 180-day period requires very strict testing a month later and then daily testing rates of 5% followed by isolation of positive cases. Relaxing social distancing rates in adults from 50% to 25% requires both children and seniors to maintain social distancing rates of 50% for nearly the entire period while maintaining maximum testing rates of children and seniors for 150 of the 180 days considered in this model. Children have higher contact rates which leads to transmission based on our model, emphasizing the need for caution when considering school reopenings.

In this paper, first we have defined the uniform distribution on the boundary of a regular hexagon, and then investigate the optimal sets of n-means and the nth quantization errors for all positive integers n. We give an exact formula to determine them if n is of the form n = 6k + 6 for some positive integer k. We further calculate the quantization dimension, and the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number, which supports the well-known result of Bucklew and Wise (1982), which says that for a Borel probability measure P with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number. Then, we define the uniform distribution on the boundary of a semicircular disc, and obtain a sequence and an algorithm, which help us to determine the optimal sets of n-means and the nth quantization errors for all positive integers n with respect to the uniform distribution. Finally, for a uniform distribution defined on an elliptical curve, we investigate the optimal sets of n-means and the nth quantization errors for all positive integers n.