Related papers: Efficient computation of the Zassenhaus formula
In this paper, we give a recursive algorithm to compute the multivariable Zassenhaus formula $$e^{X_1+X_2+\cdots +X_n}=e^{X_1}e^{X_2}\cdots e^{X_n}\prod_{k=2}^{\infty}e^{W_k}$$ and derive an effective recursion formula of $W_k$.
We propose and analyze a symmetric version of the Zassenhaus formula for disentangling the exponential of two non-commuting operators. A recursive procedure for generating the expansion up to any order is presented which also allows one to…
We present the algebraic foundations of the symmetric Zassenhaus algorithm and some of its variants. These algorithms have proven effective in devising higher-order methods for solving the time-dependent Schr\"{o}dinger equation in the…
In this paper, we contribute operator-splitting methods improved by the Zassenhaus product for the numerical solution of linear partial differential equations. We address iterative splitting methods, that can be improved by means of the…
We provide a simple method for the calculation of the terms c_n in the Zassenhaus product $e^{a+b}=e^a e^b \prod_{n=2}^{\infty} e^{c_n}$ for non-commuting a and b. This method has been implemented in a computer program. Furthermore, we…
This paper introduces a new Lie-theoretic approach to the computation of counterterms in perturbative renormalization. Contrary to the usual approach, the devised version of the Bogoliubov recursion does not follow a linear induction on the…
In this paper, we introduce a novel algorithm for calculating arbitrary order cumulants of multidimensional data. Since the $d^\text{th}$ order cumulant can be presented in the form of an $d$-dimensional tensor, the algorithm is presented…
We show that the Zassenhaus decomposition for the exponential of the sum of two non-commuting operators, simplifies drastically when these operators satisfy a simple condition, called the no-mixed adjoint property. An important application…
The complexity of matrix multiplication (hereafter MM) has been intensively studied since 1969, when Strassen surprisingly decreased the exponent 3 in the cubic cost of the straightforward classical MM to log 2 (7) $\approx$ 2.8074.…
We establish a polynomial recursion formula for linear Hodge integrals. It is obtained as the Laplace transform of the cut-and-join equation for the simple Hurwitz numbers. We show that the recursion recovers the Witten-Kontsevich theorem…
Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of…
In this paper we give efficient algorithms for computing second-, third-, and fourth-order linear recurrences. We also present an algorithm scheme for computing terms with the indices $N,\ldots,N+n-1$ of an $n$th-order linear recurrence.…
In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and recurrent formalism, incorporating Strassen's fast matrix multiplication. Our research places particular emphasis on triangular…
Efficient matrix determinant calculations have been studied since the 19th century. Computers expand the range of determinants that are practically calculable to include matrices with symbolic entries. However, the fastest determinant…
A numerical approach to compute tensor integrals in one-loop calculations is presented. The algorithm is based on a recursion relation which allows to express high rank tensor integrals as a function of lower rank ones. At each level of…
The Collatz conjecture, which posits that any positive integer will eventually reach 1 through a specific iterative process, is a classic unsolved problem in mathematics. This research focuses on designing an efficient algorithm to compute…
We present a recursive minimal polynomial theorem for finite sequences over a commutative integral domain $D$. This theorem is relative to any element of $D$. The ingredients are: the arithmetic of Laurent polynomials over $D$, a recursive…
This paper studies the exponential of the sum of two non-commuting operators as an infinite product of exponential operators involving repeated commutators of increasing order. It will be shown how to determine two coefficients in front of…
After revisiting Cantor-Zassenhaus polynomial factorization algorithm, we describe a new simplified version of it, which requires less computational cost. Moreover we show that it is able to find a factor of a fully splitting polynomial of…
The purpose of the paper is to introduce two new algorithms. The first one computes a linear recursion for proper hypergeometric multisums, by treating one summation variable at a time, and provides rational certificates along the way. A…