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,, Dec. 2006, 25 pp.

Preprint: dvi (20K), pdf (144K), ps (168K).

Technical report: pdf (296K).


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


In particular, this provides relative error bounds of 2-24sqrt(5) and 2-53sqrt(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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