Error analysis of some operations involved in the Cooley-Tukey Fast Fourier Transform

Nicolas Brisebarre (1, 2) , Mioara Joldes (3, 2) , Jean-Michel Muller (1, 2) , Ana-Maria Naneş (4) , Joris Picot (1)

1 ARIC - Arithmetic and Computing
2 CNRS - Centre National de la Recherche Scientifique
3 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
4 Technical University of Cluj-Napoca

We are interested in obtaining error bounds for the classical Cooley-Tukey FFT algorithm in floating-point arithmetic, for the 2-norm as well as for the infinity norm. For that purpose we also give some results on the relative error of the complex multiplication by a root of unity, and on the largest value that can take the real or imaginary part of one term of the FFT of a vector $x$, assuming that all terms of $x$ have real and imaginary parts less than some value $b$.

Mots clés: Numeric approximation algorithms; • Theory of computation → Design and analysis of algorithms; Rounding error analysis ACM Reference format:; Floating-point arithmetic; CCS Concepts: • Mathematics of computing → Mathematical software; Rounding error analysis; Additional Key Words and Phrases: Floating-point arithmetic; Fast Fourier Transform

Domaines: Autre [cs.OH]; Arithmétique des ordinateurs

Nicolas Brisebarre, Mioara Joldes, Jean-Michel Muller, Ana-Maria Naneş, Joris Picot. Error analysis of some operations involved in the Cooley-Tukey Fast Fourier Transform. ACM Transactions on Mathematical Software, 2020, 46 (2), pp.1-34. ⟨10.1145/3368619⟩. ⟨hal-01949458v2⟩