Two intriguing variants of the AAA algorithm for rational approximation

We consider the problem of finding a rational function in barycentric form to approximate a given function or data set in $\mathbb{R}$ or $\mathbb{C}$. The famous AAA algorithm, introduced in 2018, constructs such a rational function: the barycentric weights are the entries of the final right singular vector of a Loewner matrix with more rows than columns. We present two variants of the AAA algorithm, inspired by two intriguing quotations from the original paper. In the first, which we call AAAsmooth, we take the barycentric weights to be a complex linear combination of the last two right singular vectors, which eliminates the problem of spurious poles in real-valued problems and yields smoother convergence curves. In the second, AAAbudget, we incorporate first derivative information. This allows us to use a smaller, square alternative to the Loewner matrix, so the SVDs are cheaper while the resulting approximant is similar to the result of standard AAA. We present numerical tests showing that while both variants behave fairly similarly to standard AAA, AAAsmooth can give somewhat better results and AAAbudget can be much faster.