Sharp error bounds for complex floating-point inversion

Abstract : 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.
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Numerical Algorithms, Springer Verlag, 2016, 73 (3), pp.735-760. 〈10.1007/s11075-016-0115-x〉
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Claude-Pierre Jeannerod, Nicolas Louvet, Jean-Michel Muller, Antoine Plet. Sharp error bounds for complex floating-point inversion. Numerical Algorithms, Springer Verlag, 2016, 73 (3), pp.735-760. 〈10.1007/s11075-016-0115-x〉. 〈ensl-01195625v2〉

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