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Section: New Results

On Matrices with Displacement Structure: Generalized Operators and Faster Algorithms

For matrices with displacement structure, basic operations like multiplication, inversion, and linear-system solving can all be expressed in terms of a single task: evaluating the product $𝖠𝖡$, where $𝖠$ is a structured $n\times n$ matrix of displacement rank $\alpha$, and $𝖡$ is an arbitrary $n\times \alpha$ matrix. Given $𝖡$ and a so-called generator of $𝖠$, this product is classically computed with a cost ranging from $O\left({\alpha }^2𝖬\left(n\right)\right)$ to $O({\alpha }^2𝖬\left(n\right)log\left(n\right))$ arithmetic operations, depending on the specific structure of $𝖠$. (Here, $𝖬$ is a cost function for polynomial multiplication.) In [5], Alin Bostan, jointly with Claude-Pierre Jeannerod (AriC), Christophe Mouilleron (ENSIIE), and Éric Schost (University of Waterloo), has generalized classical displacement operators, based on block diagonal matrices with companion diagonal blocks, and has also designed fast algorithms to perform the task above for this extended class of structured matrices. The cost of these algorithms ranges from $O\left({\alpha }^{\omega -1}𝖬\left(n\right)\right)$ to $O({\alpha }^{\omega -1}𝖬\left(n\right)log\left(n\right))$, with $\omega$ such that two $n\times n$ matrices over a field can be multiplied using $O\left(n^{\omega }\right)$ field operations. By combining this result with classical randomized regularization techniques, he has obtained faster Las Vegas algorithms for structured inversion and linear system solving.