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# Improved error bounds for inner products in floating-point arithmetic

Publikationstyp

Journal Article

Publikationsdatum

2013-04-04

Sprache

English

Institut

TORE-URI

Enthalten in

Volume

34

Issue

2

Start Page

338

End Page

344

Citation

SIAM Journal on Matrix Analysis and Applications 2 (34): 338-344 (2013)

Publisher DOI

Scopus ID

Publisher

SIAM

Given two floating-point vectors x, y of dimension n and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for an inner product returns a floating-point number r̂ such that |r̂ - xT y| ≤ nu|x|T|y| with u the unit roundoff. This result, which holds for any radix and with no restriction on n, can be seen as a generalization of a similar bound given in [S. M. Rump, BIT, 52 (2012), pp. 201-220] for recursive summation in radix 2, namely, |r̂ - x T e| ≤ (n - 1)u|x|T e with e = [1, 1, ... , 1] T. As a direct consequence, the error bound for the floating-point approximation Ĉ of classical matrix multiplication with inner dimension n simplifies to |Ĉ - AB| ≤ nu|A||B|. © 2013 Society for Industrial and Applied Mathematics.