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Fast matrix multiplication is stable

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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.

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Correspondence to Olga Holtz.

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J. Demmel acknowledges support of NSF under grants CCF-0444486, ACI-00090127, CNS-0325873 and of DOE under grant DE-FC02-01ER25478.

I. Dumitriu acknowledges support of the Miller Institute for Basic Research in Science.

R. Kleinberg is supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship

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Demmel, J., Dumitriu, I., Holtz, O. et al. Fast matrix multiplication is stable. Numer. Math. 106, 199–224 (2007). https://doi.org/10.1007/s00211-007-0061-6

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  • DOI: https://doi.org/10.1007/s00211-007-0061-6

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