Topics in matrix analysis
Incremental condition estimation
SIAM Journal on Matrix Analysis and Applications
Incremental condition estimation for sparse matrices
SIAM Journal on Matrix Analysis and Applications
Estimating the largest eigenvalues by the power and Lanczos algorithms with a random start
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
The symmetric eigenvalue problem
The symmetric eigenvalue problem
A Jacobi--Davidson Type SVD Method
SIAM Journal on Scientific Computing
Computing Probabilistic Bounds for Extreme Eigenvalues of Symmetric Matrices with the Lanczos Method
SIAM Journal on Matrix Analysis and Applications
Augmented Implicitly Restarted Lanczos Bidiagonalization Methods
SIAM Journal on Scientific Computing
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We develop probabilistic upper bounds for the matrix two-norm, the largest singular value. These bounds, which are true upper bounds with a user-chosen high probability, are derived with a number of different polynomials that implicitly arise in the Lanczos bidiagonalization process. Since these polynomials are adaptively generated, the bounds typically give very good results. They can be computed efficiently. Together with an approximation that is a guaranteed lower bound, this may result in a small probabilistic interval for the matrix norm of large matrices within a fraction of a second.