Fast linear algebra is stable

Details Fast linear algebra is stable Fast linear algebra is stable

2006-12-10

arxiv.org/abs/math/0612264v3

Numer. Math. 108 (2007), no. 1, 59-91

Mathematics

Numerical Analysis

26 pages; final version; to appear in Numerische Mathematik

Scientific paper

10.1007/s00211-007-0114-x

In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of $n$-by-$n$ matrices can be done by any algorithm in $O(n^{\omega + \eta})$ operations for any $\eta > 0$, then it can be done stably in $O(n^{\omega + \eta})$ operations for any $\eta > 0$. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in $O(n^{\omega + \eta})$ operations.

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