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Improved error bounds for floating-point products and Horner’s scheme

Abstract : Let $u$ denote the relative rounding error of some floating-point format. Recently it has been shown that for a number of standard Wilkinson-type bounds the typical factors $\gamma_k:=k u / (1-k u)$ can be improved into $k u$, and that the bounds are valid without restriction on $k$. Problems include summation, dot products and thus matrix multiplication, residual bounds for $LU$- and Cholesky-decomposition, and triangular system solving by substitution. In this note we show a similar result for the product $\prod_{i=0}^k x_i$ of real and/or floating-point numbers $x_i$, for computation in any order, and for any base $\beta \ge 2$. The derived error bounds are valid under a mandatory restriction of $k$. Moreover, we prove a similar bound for Horner's polynomial evaluation scheme.
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https://hal.inria.fr/hal-01137652
Contributor : Claude-Pierre Jeannerod <>
Submitted on : Tuesday, March 31, 2015 - 10:26:44 AM
Last modification on : Thursday, November 21, 2019 - 2:25:37 AM
Long-term archiving on: : Wednesday, July 1, 2015 - 11:42:59 AM

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Siegfried M. Rump, Florian Bünger, Claude-Pierre Jeannerod. Improved error bounds for floating-point products and Horner’s scheme. BIT Numerical Mathematics, Springer Verlag, 2016, 56 (1), pp.293 - 307. ⟨10.1007/s10543-015-0555-z⟩. ⟨hal-01137652⟩

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