Journal of Complexity
Smoothed analysis: an attempt to explain the behavior of algorithms in practice
Communications of the ACM - A View of Parallel Computing
Smooth Analysis of the Condition Number and the Least Singular Value
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Solving polynomial equations in smoothed polynomial time and a near solution to smale's 17th problem
Proceedings of the forty-second ACM symposium on Theory of computing
Randomized preconditioning of the MBA algorithm
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Smoothed Analysis of Moore-Penrose Inversion
SIAM Journal on Matrix Analysis and Applications
Smoothed analysis of algorithms and heuristics
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
Advancing matrix computations with randomized preprocessing
CSR'10 Proceedings of the 5th international conference on Computer Science: theory and Applications
Improved smoothed analysis of multiobjective optimization
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
| Hi-index | 0.00 |
Let Å be an arbitrary matrix and let A be a slight random perturbation of Å. We prove that it is unlikely that A has a large condition number. Using this result, we prove that it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we show that the smoothed precision necessary to solve Ax = b, for any b, using Gaussian elimination without pivoting is logarithmic. Moreover, when Å is an all-zero square matrix, our results significantly improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 18 (1997), pp. 499-517).