Sharp error bounds for complex floating-point inversion

International audience; We study the accuracy of the classic algorithm for inverting a complex number given by its real and imaginary parts as floating-point numbers. Our analyses are done in binary floating-point arithmetic, with an unbounded exponent range and in precision $p$; we also assume that the basic arithmetic operations ($+$, $-$, $\times$, $/$) are rounded to nearest, so that the roundoff unit is $u = 2^{-p}$. We bound the largest relative error in the computed inverse either in the componentwise or in the normwise sense. We prove the componentwise relative error bound $3u$ for the complex inversion algorithm (assuming $p \ge 4$), and we show that this bound is asymptotically optimal (as $p\to \infty$) when $p$ is even, and sharp when using one of the basic IEEE 754 binary formats with an odd precision ($p=53,113$). This componentwise bound obviously leads to the same bound $3u$ for the normwise relative error. However, we prove that the smaller bound $2.707131u$ holds (assuming $p \ge 24$) for the normwise relative error, and we illustrate the sharpness of this bound for the basic IEEE 754 binary formats ($p=24, 53, 113$) using numerical examples.