On the precision of a certain procedure of numerical integration

Huskey, Harry D.

doi:10.6028/jres.042.005

Cited by 13 publications

(7 citation statements)

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“…Backward error and its bounds for the matrix--vector product y = Ax, with a matrix A and vector x with random uniform entries. Here, N\itt \ite \its \itt = 10 and \lambda = 1. than cancelling was observed and analyzed by Huskey and Hartree [18] in the numerical solution of differential equations on the ENIAC . 4.3.…”

Section: Numerical Experimentsmentioning

confidence: 96%

A New Approach to Probabilistic Rounding Error Analysis

Higham¹,

Mary²

2019

SIAM J. Sci. Comput.

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Traditional rounding error analysis in numerical linear algebra leads to backward error bounds involving the constant \gamma n = nu/(1 -nu), for a problem size n and unit roundoff u. In light of large-scale and possibly low-precision computations, such bounds can struggle to provide any useful information. We develop a new probabilistic rounding error analysis that can be applied to a wide range of algorithms. By using a concentration inequality and making probabilistic assumptions about the rounding errors, we show that in several core linear algebra computations \gamma n can be replaced by a relaxed constant \widetil \gamma n proportional to \surd n log n u with a probability bounded below by a quantity independent of n. The new constant \widetil \gamma n grows much more slowly with n than \gamma n. Our results have three key features: they are backward error bounds; they are exact, not first order; and they are valid for any n, unlike results obtained by applying the central limit theorem, which apply only as n \rightar \infty . We provide numerical experiments that show that for both random and real-life matrices the bounds can be much smaller than the standard deterministic bounds and can have the correct asymptotic growth with n. We also identify two special situations in which the assumptions underlying the analysis are not valid and the bounds do not hold. Our analysis provides, for the first time, a rigorous foundation for the rule of thumb that``one can take the square root of an error constant because of statistical effects in rounding error propagation.""

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Section: Numerical Experimentsmentioning

confidence: 96%

A New Approach to Probabilistic Rounding Error Analysis

Higham¹,

Mary²

2019

SIAM J. Sci. Comput.

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“…In Huskey's integrations of the system y, -x on the ENIAC [2], the successive values of e were found to be far from independent. In most intervals of the integration the e's were so regularly distributed that the accumulated error was considerably less than that for independent random variables.…”

Section: Reprint Of a Note On Rounding-off Errors George E Forsythementioning

confidence: 93%

Reprint of a Note on Rounding-Off Errors

Forsythe¹

1959

SIAM Rev.

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WHEN ORDINARY (CAuSSIAN) ROUNDING OFF is used in the numerical integration of ordinary differential equations, the individual rounding-off errors e are really not random variableswthey are unknown constants. This fact is probably generally accepted in principle, but many persons have assumed that with an appropriate scaling in a practical computation the constants e behave nearly like identical random variables independently and uniformly distributed on (-.5, .5). If this assumption were correct, the e's could be treated formally as independent random variables, and estimates like those of Rademacher [i] would apply to the ordinary rounding-off procedure.In Huskey's integrations of the system y, -x on the ENIAC [2], the successive values of e were found to be far from independent. In most intervals of the integration the e's were so regularly distributed that the accumulated error was considerably less than that for independent random variables. In certain intervals, however, the e's had a biased distribution which caused unexpectedly large accumulations of the rounding-off error. Following the lead of Hartree, the present writer (in an unpublished paper) has explained these phenomena and has shown how to predict the rounding-off errors for the system y, -x (and to a certain extent for other systems). It seems clear that in the integration of smooth functions the ordinary rounding-off errors will frequently not be distributed like independent random variables.To circumvent this difficulty the present writer [3] has proposed a random rounding-off procedure which make e a true random variable. Suppose, for example, that a real number u is to be rounded off to an integer. Let [u] be the greatest integer not exceeding u, and let u [u] v. In the proposed procedure u is "rounded up" to [u] W 1 with probability v, and "rounded down" to [u] with probability 1

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“…To the best of our knowledge, the earliest proposal of SR was in a one-paragraph abstract of a communication presented by Forsythe in 1949 at the 52nd meeting of the American Mathematical Society [ 3 ]. The abstract claims that SR can be used to reduce the accumulation of round-off errors observed by Huskey [ 4 ] in solving a simple system of ordinary differential equations (ODEs). The numerical integration that Forsythe and Huskey consider entails a sum of real values which is further reduced to a sum of integers, most likely intended as fixed-point representations of reals.…”

Section: Early History Of Stochastic Roundingmentioning

confidence: 99%

Stochastic rounding: implementation, error analysis and applications

et al. 2022

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Stochastic rounding (SR) randomly maps a real number x to one of the two nearest values in a finite precision number system. The probability of choosing either of these two numbers is 1 minus their relative distance to x . This rounding mode was first proposed for use in computer arithmetic in the 1950s and it is currently experiencing a resurgence of interest. If used to compute the inner product of two vectors of length n in floating-point arithmetic, it yields an error bound with constant n u with high probability, where u is the unit round-off. This is not necessarily the case for round to nearest (RN), for which the worst-case error bound has constant nu . A particular attraction of SR is that, unlike RN, it is immune to the phenomenon of stagnation, whereby a sequence of tiny updates to a relatively large quantity is lost. We survey SR by discussing its mathematical properties and probabilistic error analysis, its implementation, and its use in applications, with a focus on machine learning and the numerical solution of differential equations.

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On the precision of a certain procedure of numerical integration

Cited by 13 publications

References 0 publications

A New Approach to Probabilistic Rounding Error Analysis

A New Approach to Probabilistic Rounding Error Analysis

Reprint of a Note on Rounding-Off Errors

Stochastic rounding: implementation, error analysis and applications

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