FINDING COMPLEX-VALUED SOLUTIONS TO THE BRENT EQUATIONS BY REDUCING THEM TO A NONLINEAR LEAST SQUARES PROBLEM

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Abstract

Finding nontrivial solutions to the trilinear Brent equations corresponds to the construction of asymptotically fast matrix multiplication algorithms is an important, but in general a very difficult computational task. Methods of parameterization of the Brent equations based on the use of symmetries of the matrix product tensor are proposed, which make it possible to repeatedly reduce the dimension of the problem. The numerical solution of the obtained trilinear or cubic systems of nonlinear equations is carried out by reducing to a nonlinear least squares problem and applying to it a specially developed iterative method that does not require calculation of derivatives. The found solutions of the parameterized Brent equations, as a rule, have a rank no higher (and sometimes even lower) than the known results. Thus, an algorithm for multiplying two 4th-order matrices in 48 active multiplications is obtained.

About the authors

I. E Kaporin

Federal Research Center Computer Science and Control of the Russian Academy of Sciences

Email: igorkaporin@mail.ru
Moscow, Russia

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