insight - Numerical Linear Algebra

### Improved Lower Bounds for the Maximum Growth Factor in Gaussian Elimination with Complete Pivoting

The maximum growth factor under complete pivoting is larger than 1.0045n for all n > 10, and the lim sup of the ratio with n is greater than or equal to 3.317. This disproves the long-standing conjecture that the growth factor is at most n.

### Randomized Nyström Approximation for Low-Rank Approximation of Non-Negative Self-Adjoint Operators

This paper presents an infinite-dimensional extension of the randomized Nyström approximation to compute low-rank approximations of non-negative self-adjoint trace-class operators. The analysis yields bounds on the expected value and tail bounds for the Nyström approximation error in the operator, trace, and Hilbert-Schmidt norms.

### Convergence Analysis of Quantile Randomized Kaczmarz Method for Linear Systems with Time-Varying Noise and Corruption

The Quantile Randomized Kaczmarz (QRK) method converges at least linearly in expectation up to a convergence horizon even when the linear system is perturbed by time-varying noise and corruption. The rate of convergence depends only on the corruption rate, while the convergence horizon depends on both the corruption rate and the time-varying noise.

### Randomized Matrix Computations: Themes and Variations

This content delves into the application of randomized algorithms in numerical linear algebra, focusing on matrix computations and approximation methods.