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Numerical and Computational Issues in Linear Control and System Theory
Published in William S. Levine, Control System Fundamentals, 2019
A.J. Laub, R.V. Patel, P.M. Van Dooren
Finally, let us define a particular number to which we make frequent reference following. The machine epsilon or relative machine precision is defined, roughly speaking, as the smallest positive number ∊ that, when added to 1 on our computing machine, gives a number greater than 1. In other words, any machine representable number δ less than e gets “rounded off” when (floating-point) added to 1 to give exacdy 1 again as the rounded sum. The number ∊, of course, varies depending on the kind of computer being used and the precision of the computations (single precision, double precision, etc.). But the fact that such a positive number ∊ exists is entirely a consequence of finite word length.
Review of Basic Laws and Equations
Published in Pradip Majumdar, Computational Methods for Heat and Mass Transfer, 2005
The value of the machine epsilon (ε) varies from computer to computer depending on the word length, number of words used (i.e., one or two), base (b), and type of number-approximation (rounding or chopping) used.
Accuracy Improvements for Single Precision Implementations of the SPH Method
Published in International Journal of Computational Fluid Dynamics, 2020
Elie Saikali, Giuseppe Bilotta, Alexis Hérault, Vito Zago
The width of the gap between 1 and the next representable floating-point number is called the machine epsilon, and we denote it by , which in the single-precision binary floating-point standard is (other scientists refer to machine epsilon to the upper bound of the relative error that occurs when rounding the exact result of an operation to the nearest representable value, which is exactly half of our definition). Equivalently, this is the relative value of the least significant bit of the representation of a number to the number itself, and its significance can be illustrated by remarking that if a, b are two non-zero representable numbers such that , then (where ⊕ is the result of the addition of the two numbers, rounded to the nearest representable value).