Acceleration of AAA’s linear algebra beyond O(n^4)

Develop numerically stable and effective algorithms to speed up the linear algebra in AAA beyond the current O(n^4) complexity, and establish their accuracy and robustness relative to existing proposals (e.g., Cholesky updates or randomized sketching).

Background

AAA currently incurs an operation count of order O(n^4) due to repeated SVDs of growing tall-skinny Loewner matrices. Although proposals exist (e.g., Cholesky updates and randomized sketching), the authors explicitly state that it is not clear that good methods are available to significantly speed up the linear algebra. A stable acceleration could enable larger-scale rational approximations and broader adoption.