Fast matrix multiplication is stable

Details Fast matrix multiplication is stable Fast matrix multiplication is stable

2006-03-09

arxiv.org/abs/math/0603207v3

Numer. Math. 106 (2007), no. 2, 199-224

Mathematics

Numerical Analysis

19 pages; final version, expanded and updated to reflect referees' remarks; to appear in Numerische Mathematik

Scientific paper

10.1007/s00211-007-0061-6

We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63--72]. 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 [H. Cohn, and C. Umans, A group-theoretic approach to fast matrix multiplication, FOCS 2003, 438--449] and [H. Cohn, R. Kleinberg, B. Szegedy and C. Umans, Group-theoretic algorithms for matrix multiplication, FOCS 2005, 379--388] 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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