On fast multiplication of a matrix by its transpose

We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space of bilinear forms describing 2x2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions.The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions.Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a finite field.To conclude, we show how to use our result in LDLT factorization.Comment: ISSAC 2020, Jul 2020, Kalamata, Greec

Local ensemble transform Kalman filter, a fast non-stationary control law for adaptive optics on ELTs: theoretical aspects and first simulation results

We propose a new algorithm for an adaptive optics system control law, based on the Linear Quadratic Gaussian approach and a Kalman Filter adaptation with localizations. It allows to handle non-stationary behaviors, to obtain performance close to the optimality defined with the residual phase variance minimization criterion, and to reduce the computational burden with an intrinsically parallel implementation on the Extremely Large Telescopes (ELTs).Comment: This paper was published in Optics Express and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://www.opticsinfobase.org/oe/ . Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under la

Evaluating parametric holonomic sequences using rectangular splitting

We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the $n$-th term in a recurrent sequence of suitable type using $O(n^{1/2})$ "expensive" operations at the cost of an increased number of "cheap" operations. Rectangular splitting has little overhead and can perform better than either naive evaluation or asymptotically faster algorithms for ranges of $n$ encountered in applications. As an example, fast numerical evaluation of the gamma function is investigated. Our work generalizes two previous algorithms of Smith.Comment: 8 pages, 2 figure