Madhavi Prakash Rose Hills
Pseudospectral shattering via stochastic rounding
Eigenvalue problems are central to scientific computing, but solving them efficiently becomes difficult when input matrices are ill-conditioned or non-normal. I began this project by asking whether random perturbations–commonly used in pseudospectral shattering to stabilize these problems–could be replaced by a more structured method. Over the summer, I shifted focus to stochastic rounding, a hardware-friendly, unbiased alternative that modern chips can perform natively. By applying this technique entry-wise, I was able to “shatter” the pseudospectrum and recover clean spectral splits, enabling diagonalization via divide-and-conquer eigensolvers. Experiments across a wide range of challenging matrices showed performance and conditioning on par with Gaussian noise. I’m now developing theoretical guarantees to explain this behavior and exploring how stochastic rounding could serve as a broadly applicable alternative to randomization in numerical linear algebra.
Message To Sponsor
Thank you so much for providing me with the opportunity to pursue this project! I am extremely grateful for your generosity and am beyond excited to work with my mentors to contribute to the field of numerical linear algebra. This experience will allow me to deepen my understanding of algorithm design and grow as a researcher.