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inria-00120352, version 2

Error Bounds on Complex Floating-Point Multiplication

Richard P. Brent, Colin Percival, Paul Zimmermann () 1

Mathematics of Computation 76 (2007) 1469-1481

Abstract: Given floating-point arithmetic with $t$-digit base-$\beta$ significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values $z_0$ and $z_1$ can be computed with maximum absolute error $\abs{z_0} \abs{z_1} \frac{1}{2} \beta^{1 - t} \sqrt{5}$. In particular, this provides relative error bounds of $2^{-24} \sqrt{5}$ and $2^{-53} \sqrt{5}$ for {IEEE 754} single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for {IEEE 754} single and double precision arithmetic.

  • 1:  CACAO (INRIA Lorraine - LORIA)
  • CNRS : UMR7503 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
  • Domain : Computer Science/Data Structures and Algorithms
    Mathematics/Complex Variables
    Mathematics/Numerical Analysis
  • Keywords : IEEE 754 – floating-point number – complex multiplication – roundoff error – error analysis
  • Internal note : RR-6068
  • Available versions :  v1 (2006-12-19) v2 (2006-12-19)
 
  • inria-00120352, version 2
  • http://hal.inria.fr/inria-00120352
  • oai:hal.inria.fr:inria-00120352
  • From: Paul Zimmermann <>
  • Submitted on: Tuesday, 19 December 2006 14:04:38
  • Updated on: Thursday, 15 November 2007 08:43:38