## Error Bounds on Complex Floating-Point Multiplication

221. R. P. Brent, Colin Percival and P. Zimmermann,
Error bounds on complex floating-point multiplication,
*Mathematics of Computation* 76 (2007), 1469-1481.
Also (extended version) Report RR-6068, INRIA, France,
http://hal.inria.fr/inria-00120352/en/, Dec. 2006, 25 pp.
Preprint:
dvi (20K),
pdf (144K),
ps (168K).

Technical report:
pdf (296K).

## Abstract

Given floating-point arithmetic with t-digit
base-b 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
u and v can be computed with maximum absolute error

|uv|b^{1-t}sqrt(5)/2.
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.

The Technical Report also considers generic worst cases and Karatsuba
multiplication, and comments on a paper by Olver in which a slightly
weaker bound is proved.

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