Smoothed analysis of componentwise condition numbers for sparse matrices
Provides theoretical guarantees for numerical stability of sparse matrix computations, benefiting researchers and practitioners in numerical linear algebra.
The paper performs a smoothed analysis of componentwise condition numbers for determinant computation, matrix inversion, and linear equations solving for sparse n×n matrices, obtaining bounds of order O(log n) for the expectations of the logarithm of these condition numbers. This implies small bounds on the smoothed loss of accuracy for triangular linear systems.
We perform a smoothed analysis of the componentwise condition numbers for determinant computation, matrix inversion, and linear equations solving for sparse n times n matrices. The bounds we obtain for the ex- pectations of the logarithm of these condition numbers are, in all three cases, of the order O(log n). As a consequence, small bounds on the smoothed loss of accuracy for triangular linear systems follow.