Fast linear algebra is stable

Authors:
James Demmel;Ioana Dumitriu;Olga Holtz
Affiliations:
University of California, Mathematics Department and CS Division, 94720, Berkeley, CA, USA;University of Washington, Mathematics Department, 98195, Seattle, WA, USA;University of California, Mathematics Department, 94720, Berkeley, CA, USA
Venue:
Numerische Mathematik
Year:
2007

Citing 0
Cited 11

Blendenpik: Supercharging LAPACK's Least-Squares Solver

SIAM Journal on Scientific Computing
The impact of data distribution in accuracy and performance of parallel linear algebra subroutines

VECPAR'10 Proceedings of the 9th international conference on High performance computing for computational science
Communication-optimal parallel 2.5D matrix multiplication and LU factorization algorithms

Euro-Par'11 Proceedings of the 17th international conference on Parallel processing - Volume Part II
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions

SIAM Review
Communication-optimal parallel algorithm for strassen's matrix multiplication

Proceedings of the twenty-fourth annual ACM symposium on Parallelism in algorithms and architectures
Communication-avoiding parallel strassen: implementation and performance

SC '12 Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis
Graph expansion and communication costs of fast matrix multiplication

Journal of the ACM (JACM)
Work-efficient matrix inversion in polylogarithmic time

Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures
Communication efficient gaussian elimination with partial pivoting using a shape morphing data layout

Proceedings of the twenty-fifth annual ACM symposium on Parallelism in algorithms and architectures
Sparsity lower bounds for dimensionality reducing maps

Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Communication costs of Strassen's matrix multiplication

Communications of the ACM

Quantified Score

Hi-index	0.02

Visualization

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(n ω+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.