Correction d'une somme en arithmetique a virgule flottante

Pichat, Michaël

doi:10.1007/bf01404922

Cited by 57 publications

(19 citation statements)

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“…Since sums of floating-point numbers are ubiquitous in scientific computations, there is a vast amount of literature to that, among them [2,3,7,10,13,14,18,21,22,23,24,25,31,32,33,34,35,36,37,39,40,41,42,46,47,48,49,50], all aiming on some improved accuracy of the result. Higham [19] devotes an entire chapter to summation.…”

mentioning

confidence: 99%

“…Many algorithms [23,24,25,36,39,46,49,48] including those by Kahan, Babuška and Neumaier and others use compensated summation, i.e. the error of the individual additions is somehow corrected.…”

mentioning

confidence: 99%

“…There are few methods [34,39,7,40,41,46,13,14,50] to compute an accurate approximation of a sum independent of the condition number, with the ultimate goal of the faithfully rounded or rounded-to-nearest exact sum. The aim of the present papers (Parts I and II) are such algorithms, and we shortly overview known approaches.…”

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confidence: 99%

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Accurate Floating-Point Summation Part I: Faithful Rounding

Rump¹,

Ogita²,

Oishi³

2008

SIAM J. Sci. Comput.

164

199

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Abstract. Given a vector of floating-point numbers with exact sum s, we present an algorithm for calculating a faithful rounding of s, i.e. the result is one of the immediate floating-point neighbors of s. If the sum s is a floating-point number, we prove that this is the result of our algorithm. The algorithm adapts to the condition number of the sum, i.e. it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number. All statements are also true in the presence of underflow. The algorithm does not depend on the exponent range. Our algorithm is fast in terms of measured computing time because it allows good instruction-level parallelism, it neither requires special operations such as access to mantissa or exponent, it contains no branch in the inner loop, nor does it require some extra precision: The only operations used are standard floating-point addition, subtraction and multiplication in one working precision, for example double precision. Certain constants used in the algorithm are proved to be optimal.

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confidence: 99%

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Accurate Floating-Point Summation Part I: Faithful Rounding

Rump¹,

Ogita²,

Oishi³

2008

SIAM J. Sci. Comput.

164

199

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“…Many summation algorithms published in the literature (see for instance [1], [18], [19], [17], [5], [22]) are based on this property and implicitly or explicitly use basic blocks such as Dekker's Fast2Sum and Knuth's 2Sum algorithms (Algorithms 1 and 2 below) to compute the rounding error generated by a floating-point addition.…”

Section: Introductionmentioning

confidence: 99%

On the Computation of Correctly Rounded Sums

Kornerup

Lefèvre

Louvet

et al. 2012

IEEE Trans. Comput.

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Abstract-This paper presents a study of some basic blocks needed in the design of floating-point summation algorithms. In particular, in radix-2 floating-point arithmetic, we show that among the set of the algorithms with no comparisons performing only floatingpoint additions/subtractions, the 2Sum algorithm introduced by Knuth is minimal, both in terms of number of operations and depth of the dependency graph. We investigate the possible use of another algorithm, Dekker's Fast2Sum algorithm, in radix-10 arithmetic. We give methods for computing, in radix 10, the floating-point number nearest the average value of two floating-point numbers. We also prove that under reasonable conditions, an algorithm performing only round-to-nearest additions/subtractions cannot compute the round-to-nearest sum of at least three floating-point numbers. Starting from an algorithm due to Boldo and Melquiond, we also present new results about the computation of the correctly-rounded sum of three floating-point numbers. For a few of our algorithms, we assume new operations defined by the recent IEEE 754-2008 Standard are available.

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“…Pichat [33] and Neumaier [26] independently and apparently without knowing FastTwoSum use this EFT to add the p i . This approach was called "compensated summation": Priest [34,35] first sorts the input data by absolute value and then applies a scheme similar to that in (1.3) to add the errors y i .…”

Section: Introductionmentioning

confidence: 99%

Ultimately Fast Accurate Summation

Rump¹

2009

SIAM J. Sci. Comput.

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Abstract. We present two new algorithms FastAccSum and FastPrecSum, one to compute a faithful rounding of the sum of floating-point numbers and the other for a result "as if" computed in K-fold precision. Faithful rounding means the computed result either is one of the immediate floating-point neighbors of the exact result or is equal to the exact sum if this is a floating-point number. The algorithms are based on our previous algorithms AccSum and PrecSum and improve them by up to 25%. The first algorithm adapts to the condition number of the sum; i.e., the computing time is proportional to the difficulty of the problem. The second algorithm does not need extra memory, and the computing time depends only on the number of summands and K. Both algorithms are the fastest known in terms of flops. They allow good instruction-level parallelism so that they are also fast in terms of measured computing time. The algorithms require only standard floating-point addition, subtraction, and multiplication in one working precision, for example, double precision.

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Correction d'une somme en arithmetique a virgule flottante

Cited by 57 publications

References 0 publications

Accurate Floating-Point Summation Part I: Faithful Rounding

Accurate Floating-Point Summation Part I: Faithful Rounding

On the Computation of Correctly Rounded Sums

Ultimately Fast Accurate Summation

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