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Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
The rounding errors would disappear if computers had infinite capacity to operate with numbers precisely. Round-off errors can affect the final computed result either in accumulating error during a sequence of millions of operations or in catastrophic cancellation. In general, the subject of round-off propagation is poorly understood and its effect on computations depends on many factors. If too many steps are required in the calculation, then eventually round-off error is likely to accumulate to the point that it deteriorates the accuracy of the numerical procedure. Therefore, the round-off error is proportional to the number of computations performed and it is inversely proportional to some power of the step size.
Numerical Analysis
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Loss of accuracy when subtracting numbers of similar magnitudes is called catastrophic cancellation. Consider adding two numbers a = 80.0499 and b = −79.9999 on a computer with 6-digit representation. Even though the result a+b = 0.0500 has no computational error, it has only three significant digits compared to six in the operands. This loss in accuracy is due to cancellation and is due to the large value of 1/(a + b) that multiplies the relative error of the operands.
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
Since the result of any operation between floating-point numbers is subject to rounding, including any intermediate results, additions and multiplications between floating-point numbers are not associative. Going back to our example, , while , an example of the so-called catastrophic cancellation that can happen in floating point. Choosing which operations are done, and in which order, can thus have an enormous impact on the accuracy of the results; in scientific computing, and in computational fluid dynamics in particular, the choice can make the difference between a stable simulation that produces accurate results and unstable simulation that may lead to nonphysical solutions even when the underlying algorithm, in exact arithmetic, would produce the correct result.