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Round Off Errors And Machine Computing

IP.com Disclosure Number: IPCOM000097036D
Original Publication Date: 1962-Apr-01
Included in the Prior Art Database: 2005-Mar-07
Document File: 2 page(s) / 23K

Publishing Venue

IBM

Related People

Thomas, LH: AUTHOR

Abstract

Various techniques have been prepared for concurrently estimating accumulated round-off errors to be expected in a machine computation. Some of these are discussed in Floating-Point Arithmetics by W.C. Wadey, Journal of the Association for Computing Machinery, Volume 7, Number 2, Page 129, 1960. These techniques treat the errors of each machine number as if they were independent and therefore overestimate the accumulated error in many application, including the solution of ordinary or partial differential equations.

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Round Off Errors And Machine Computing

Various techniques have been prepared for concurrently estimating accumulated round-off errors to be expected in a machine computation. Some of these are discussed in Floating-Point Arithmetics by W.C. Wadey, Journal of the Association for Computing Machinery, Volume 7, Number 2, Page 129, 1960. These techniques treat the errors of each machine number as if they were independent and therefore overestimate the accumulated error in many application, including the solution of ordinary or partial differential equations.

One known unbiased technique for rounding utilizes the addition of a pseudo- random number (which may be conveniently produced by one of Lehmer's rules in Mathematic Tables and Other Aides to Computation, published by National Re search Council, Washington, D.C., Number 33, Page 4, 1951, entitled: High Speed Sampling) of as many digits as are to be cut off. This technique is unbiased but the mean square error of each machine number is doubled. It has been found that the use of two of these calculations with different series of pseudo-random digits provides a mean for each machine number with the same mean square error as for the usual method of rounding, and a difference, half of which is a good estimate of the accumulated root mean square error.

Two independent calculations are not necessary. A signed error (alpha,beta) may be retained for each machine number (a,b), where the sum is (a+b)+ (alpha+beta)

the diffe...