## Error bounds on complex floating-point multiplication

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- by Richard Brent, Colin Percival and Paul Zimmermann PDF
- Math. Comp.
**76**(2007), 1469-1481 Request permission

## 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 $|z_0\|z_1| \frac {1}{2} \beta ^{1 - t} \sqrt {5}$. 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.## References

- Nicholas J. Higham,
*Accuracy and stability of numerical algorithms*, 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. MR**1927606**, DOI 10.1137/1.9780898718027 - Colin Percival,
*Rapid multiplication modulo the sum and difference of highly composite numbers*, Math. Comp.**72**(2003), no. 241, 387–395. MR**1933827**, DOI 10.1090/S0025-5718-02-01419-9

## Additional Information

**Richard Brent**- Affiliation: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
- Email: complex@rpbrent.com
**Colin Percival**- Affiliation: IRMACS Centre, Simon Fraser University, Burnaby, BC, Canada
- Email: cperciva@irmacs.sfu.ca
**Paul Zimmermann**- Affiliation: INRIA Lorraine/LORIA, 615 rue du Jardin Botanique, F-54602 Villers-lès-Nancy Cedex, France
- MR Author ID: 273776
- Email: zimmerma@loria.fr
- Received by editor(s): November 21, 2005
- Received by editor(s) in revised form: February 21, 2006
- Published electronically: January 24, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**76**(2007), 1469-1481 - MSC (2000): Primary 65G50
- DOI: https://doi.org/10.1090/S0025-5718-07-01931-X
- MathSciNet review: 2299783

Dedicated: In memory of Erin Brent (1947–2005)