Numerische Mathematik

, Volume 108, Issue 1, pp 59–91

Fast linear algebra is stable

Article

DOI: 10.1007/s00211-007-0114-x

Cite this article as:
Demmel, J., Dumitriu, I. & Holtz, O. Numer. Math. (2007) 108: 59. doi:10.1007/s00211-007-0114-x

Abstract

In Demmel et al. (Numer. Math. 106(2), 199–224, 2007) 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(nω+η) operations for any η >  0, then it can be done stably in O(nω+η) operations for any η >  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(nω+η) operations.

Mathematics Subject Classification (2000)

65F0565F1565F2565F3065F3565F4065G3065G5065Y2068Q2568W2068W4015A52

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematics Department and CS DivisionUniversity of CaliforniaBerkeleyUSA
  2. 2.Mathematics DepartmentUniversity of WashingtonSeattleUSA
  3. 3.Mathematics DepartmentUniversity of CaliforniaBerkeleyUSA