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Alexandre Sedoglavic

Publications and source records attributed to Alexandre Sedoglavic.

15 recordsLinked to original sources
arXiv2026

A more accurate rational non-commutative algorithm for multiplying 4x4 matrices using 48 multiplications

We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2. This algorithm achieves an error bound exponent of only log4($\gamma$$\infty$,2) $\approx$2.335. In practice, it also reaches a better accuracy w.r.t. max-norm, when compared to previously known such fast algorithms. Furthermore, we propose a straight line program of this algorithm, giving a leading constant in its complexity bound of $316/32 n^{2+log\_4(3)} + o(n^{2+log\_4(3))$ operations over any ring containing an inverse of 2.

cs.DS↗
arXiv2025

Towards automated generation of fast and accurate algorithms for recursive matrix multiplication

We propose a strategy for the generation of fast and accurate versions of non-commutative recursive matrix multiplication algorithms. To generate these algorithms, we consider matrix and tensor norm bounds governing the stability and accuracy of numerical matrix multiplication. We start by a unification on known max-norm bounds on matrix multiplication stability and then extend them to further norms and more generally to recursive bilinear algorithms and the alternative basis matrix multiplication algorithms. Then our strategy has three phases. First, we reduce those bounds by minimizing a growth factor along the orbits of the associated matrix multiplication tensor decomposition. Second, we develop heuristics that minimize the number of operations required to realize a bilinear formula, while further improving its accuracy. Third, we perform an alternative basis sparsification that improves on the time complexity constant and mostly preserves the overall accuracy. For instance this strategy allows us to propose a non-commutative algorithm for multiplying 2x2-matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast ___2x2x2:7___ Strassen-like algorithms and a time complexity bound with the best currently known leading term (obtained via alternative basis sparsification). We also present detailed results of our technique on other recursive matrix multiplication algorithms, such as Smirnov's ___3x3x6:40___ family of algorithms.

math.NA↗
arXiv2025

A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications

The quest for non-commutative matrix multiplication algorithms over non-commutative rings in small dimensions has recently seen significant progress. Specifically, the number of scalar multiplications required to multiply two 4x4 matrices was reduced in \cite{Fawzi:2022aa} from 49 (using two recursion levels of Strassen's algorithm) to 47 in characteristic 2, and more recently to 48 in \cite{alphaevolve} over the complex numbers. We propose an algorithm requiring 48 multiplications that uses only rational coefficients, thereby removing the requirement for complex-number arithmetic, and making this algorithm valid over any ring except those of characteristic 2. We also produce a straight-line program of this algorithm reducing the number of additions and scalar multiplications, reaching a running time of $\frac{347}{32}n^{2+\log_4{3}}+o(n^{2+\log_4{3}})$ operations, as well as an alternative basis variant of it, leading to an algorithm running in $7n^{2+\log_4{3}} +o(n^{2+\log_4{3}})$ operations over any ring containing an inverse of 2. Similarly, the number of scalar multiplications required to multiply a 3x4 matrix by a 4x7 matrix was reduced from 66 in \cite{Smirnov:2021aa} to 63 in \cite{alphaevolve} by an algorithm over complex numbers. Using the same techniques, we propose an equivalent algorithm in 63 multiplications using only rational coefficients. In both cases the rational algorithm is obtained by identifying an isotropy that projects the previously known complex-valued decomposition onto the field of rational numbers.

cs.SC↗
arXiv2024

Strassen's algorithm is not optimally accurate

We propose a non-commutative algorithm for multiplying 2x2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity bound with the best currently known leading term (obtained via alternate basis sparsification). To build this algorithm, we consider matrix and tensor norms bounds governing the stability and accuracy of numerical matrix multiplication. First, we reduce those bounds by minimizing a growth factor along the unique orbit of Strassen's 2x2-matrix multiplication tensor decomposition. Second, we develop heuristics for minimizing the number of operations required to realize a given bilinear formula, while further improving its accuracy. Third, we perform an alternate basis sparsification that improves on the time complexity constant and mostly preserves the overall accuracy.

math.NA↗
arXiv2021

The tensor rank of 5x5 matrices multiplication is bounded by 98 and its border rank by 89

We present a non-commutative algorithm for the product of 3x5 by 5x5 matrices using 58 multiplications. This algorithm allows to construct a non-commutative algorithm for multiplying 5x5 (resp. 10x10, 15x15) matrices using 98 (resp. 686, 2088) multiplications. Furthermore, we describe an approximate algorithm that requires 89 multiplications and computes this product with an arbitrary small error.

cs.CC↗
arXiv2021

Some fast algorithms multiplying a matrix by its adjoint

We present a non-commutative algorithm for the multiplication of a 2 x 2 block-matrix by its adjoint, defined by a matrix ring anti-homomorphism. This algorithm uses 5 block products (3 recursive calls and 2 general products)over C or in positive characteristic. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its adjoint to general matrix product, improving by a constant factor previously known reductions. We prove also that there is no algorithm derived from bilinear forms using only four products and the adjoint of one of them. Second we give novel dedicated algorithms for the complex field and the quaternions to alternatively compute the multiplication taking advantage of the structure of the matrix-polynomial arithmetic involved. We then analyze the respective ranges of predominance of the two strategies. Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a prime field.

cs.SC↗
arXiv2020

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.

cs.SC↗
arXiv2017

Laderman matrix multiplication algorithm can be constructed using Strassen algorithm and related tensor's isotropies

In 1969, V. Strassen improves the classical~2x2 matrix multiplication algorithm. The current upper bound for 3x3 matrix multiplication was reached by J.B. Laderman in 1976. This note presents a geometric relationship between Strassen and Laderman algorithms. By doing so, we retrieve a geometric formulation of results very similar to those presented by O. Sykora in 1977.

cs.SC↗
arXiv2010

A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations

This paper deals with the index reduction problem for the class of quasi-regular DAE systems. It is shown that any of these systems can be transformed to a generically equivalent first order DAE system consisting of a single purely algebraic (polynomial) equation plus an under-determined ODE (that is, a semi-explicit DAE system of differentiation index 1) in as many variables as the order of the input system. This can be done by means of a Kronecker-type algorithm with bounded complexity.

math.CA↗
arXiv2006

Reduction of Algebraic Parametric Systems by Rectification of their Affine Expanded Lie Symmetries

Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equations. We apply this principle by finding some \emph{affine derivations} that induces \emph{expanded} Lie point symmetries of considered system. By rewriting original problem in an invariant coordinates set for these symmetries, we \emph{reduce} the number of involved parameters. We present an algorithm based on this standpoint whose arithmetic complexity is \emph{quasi-polynomial} in input's size.

cs.SC↗
arXiv2006

Fast computation of power series solutions of systems of differential equations

We propose new algorithms for the computation of the first N terms of a vector (resp. a basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations which is quasi-linear with respect to N. Similar results are also given in the non-linear case. This extends previous results obtained by Brent and Kung for scalar differential equations of order one and two.

cs.SC↗
arXiv2006

Polynomial Time Nondimensionalisation of Ordinary Differential Equations via their Lie Point Symmetries

Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equation. We apply this principle by finding dilatations and translations that are Lie point symmetries of considered ordinary differential system. By rewriting original problem in an invariant coordinates set for these symmetries, one can reduce the involved number of parameters. This process is classically call nondimensionalisation in dimensional analysis. We present an algorithm based on this standpoint and show that its arithmetic complexity is polynomial in input's size.

cs.SC↗
arXiv2000

A probabilistic algorithm to test local algebraic observability in polynomial time

The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input-output behavior of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of the \emph{local algebraic observability} problem which is concerned with the existence of a non trivial Lie subalgebra of the symmetries of the model letting the inputs and the outputs invariant. We present a \emph{probabilistic seminumerical} algorithm that proposes a solution to this problem in \emph{polynomial time}. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the \emph{complexity of evaluation} of the model and in the number of variables. Furthermore, we show that the \emph{size} of the integers involved in the computations is polynomial in the number of variables and in the degree of the differential system. Last, we estimate the probability of success of our algorithm and we present some benchmarks from our Maple implementation.

math.OC↗