Fast matrix multiplication is stable

Authors:
James Demmel;Ioana Dumitriu;Olga Holtz;Robert Kleinberg
Affiliations:
University of California, Mathematics Department, 94720, Berkeley, CA, USA and University of California, Computer Science Division, 94720, Berkeley, CA, USA;University of California, Mathematics Department, 94720, Berkeley, CA, USA;University of California, Mathematics Department, 94720, Berkeley, CA, USA;University of California, Computer Science Division, 94720, Berkeley, CA, USA
Venue:
Numerische Mathematik
Year:
2007

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Abstract

We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of Bini and Lotti [Numer. Math. 36:63–72, 1980]. As a consequence of our analysis, we show that the exponent of matrix multiplication (the optimal running time) can be achieved by numerically stable algorithms. We also show that new group-theoretic algorithms proposed in Cohn and Umans [Foundations of Computer Science, 44th Annual IEEE Symposium, pp. 438–449, 2003] and Cohn et al. [Foundations of Computer Science, 46th Annual IEEE Symposium, pp. 379–388, 2005] are all included in the class of algorithms to which our analysis applies, and are therefore numerically stable. We perform detailed error analysis for three specific fast group-theoretic algorithms.