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# Ultimately fast accurate summation

Publikationstyp

Journal Article

Publikationsdatum

2009-09-04

Sprache

English

Author

Institut

TORE-URI

Enthalten in

Volume

31

Issue

5

Start Page

3466

End Page

3502

Citation

SIAM Journal on Scientific Computing 5 (31): 3466-3502 (2009-12-01)

Publisher DOI

Scopus ID

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.