Related papers: Some Numerical Experiments on Round-off Error Grow…
We derive two probabilistic bounds for the relative forward error in the floating point summation of $n$ real numbers, by representing the roundoffs as independent, zero-mean, bounded random variables. The first probabilistic bound is based…
Solving linear systems is a ubiquitous task in science and engineering. Because directly inverting a large-scale linear system can be computationally expensive, iterative algorithms are often used to numerically find the inverse. To…
Compression of floating-point data will play an important role in high-performance computing as data bandwidth and storage become dominant costs. Lossy compression of floating-point data is powerful, but theoretical results are needed to…
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…
Quantum phase estimation is one of the key algorithms in the field of quantum computing, but up until now, only approximate expressions have been derived for the probability of error. We revisit these derivations, and find that by ensuring…
We investigate the possibility of fast, accurate and reliable computation of the Cauchy principal value integrals $\mathrm{P}\!\int_{a}^{b} f(x)(x-\tau)^{-1} dx$ $(a < \tau < b)$ using standard adaptive quadratures. In order to properly…
The idea of computational error correction has been around for over half a century. The motivation has largely been to mitigate unreliable devices, manufacturing defects or harsh environments, primarily as a mandatory measure to preserve…
State-of-the-art static analysis tools for verifying finite-precision code compute worst-case absolute error bounds on numerical errors. These are, however, often not a good estimate of accuracy as they do not take into account the…
A new deterministic floating-point arithmetic called precision arithmetic is developed to track precision for arithmetic calculations. It uses a novel rounding scheme to avoid excessive rounding error propagation of conventional…
Runge-Kutta methods are a popular class of numerical methods for solving ordinary differential equations. Every Runge-Kutta method is characterized by two basic parameters: its order, which measures the accuracy of the solution it produces,…
Higher-order radiative corrections play an important role in precision studies of the electroweak and Higgs sector, as well as for the detailed understanding of large backgrounds to new physics searches. For corrections beyond the one-loop…
This work focuses on the numerical study of a recently published class of Runge-Kutta methods designed for mixed-precision arithmetic. We employ the methods in solving partial differential equations on modern hardware. In particular we…
In order to solve continuous-time optimal control problems, direct methods transcribe the infinite-dimensional problem to a nonlinear program (NLP) using numerical integration methods. In cases where the integration error can be manipulated…
Debugging accumulation of floating-point errors is hard; ideally, computer should track it automatically. Here we consider twofold approximation of an exact real with value + error pair of floating-point numbers. Normally, value + error sum…
Probabilistic models are proposed for bounding the forward error in the numerically computed inner product (dot product, scalar product) between of two real $n$-vectors. We derive probabilistic perturbation bounds, as well as probabilistic…
The quality of numerical computations can be measured through their forward error, for which finding good error bounds is challenging in general. For several algorithms and using stochastic rounding (SR), probabilistic analysis has been…
We describe various issues caused by the lack of round-to-nearest mode in the \textit{gcc} compiler implementation of the fixed-point arithmetic data types and operations. We demonstrate that round-to-nearest is not performed in the…
This paper considers a probabilistic model for floating-point computation in which the roundoff errors are represented by bounded random variables with mean zero. Using this model, a probabilistic bound is derived for the forward error of…
Round-off errors arising from the difference between real numbers and their floating-point representation cause the control flow of conditional floating-point statements to deviate from the ideal flow of the real-number computation. This…
Renewed interest in mixed-precision algorithms has emerged due to growing data capacity and bandwidth concerns, as well as the advancement of GPUs, which enable significant speedup for low precision arithmetic. In light of this, we propose…