Related papers: TaylUR, an arbitrary-order diagonal automatic diff…
We present the singular Euler--Maclaurin expansion, a new method for the efficient computation of large singular sums that appear in long-range interacting systems in condensed matter and quantum physics. In contrast to the traditional…
Automatic differentiation is everywhere, but there exists only minimal documentation of how it works in complex arithmetic beyond stating "derivatives in $\mathbb{C}^d$" $\cong$ "derivatives in $\mathbb{R}^{2d}$" and, at best, shallow…
Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine…
In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from…
In recent years, there has been significant progress in the development and industrial adoption of static analyzers. Such analyzers typically provide a large, if not huge, number of configurable options controlling the precision and…
This short note provides an explicit description of the Fr\'echet derivatives of the principal square root matrix functional at any order. We present an original formulation that allows to compute sequentially the Fr\'echet derivatives of…
Deep reinforcement learning (DRL) has been envisioned to have a competitive edge in quantitative finance. However, there is a steep development curve for quantitative traders to obtain an agent that automatically positions to win in the…
We introduce ToRL (Tool-Integrated Reinforcement Learning), a framework for training large language models (LLMs) to autonomously use computational tools via reinforcement learning. Unlike supervised fine-tuning, ToRL allows models to…
We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations. This algorithm extends the classical Feynman-Kac formula to fully nonlinear partial differential equations, by using random trees that…
We show that the discrete complex, and numerous hypercomplex, Fourier transforms defined and used so far by a number of researchers can be unified into a single framework based on a matrix exponential version of Euler's formula…
The polylogarithm function is one of the constellation of important mathematical functions. It has a long history, and many connections to other special functions and series, and many applications, for instance in statistical physics.…
A recently developed numerical method for the calculation of derivatives of functions of general complex matrices, which can also be combined with implicit matrix function approximations such as Krylov-Ritz type algorithms, is presented. An…
In this brief, we discuss the implementation of a third order semi-implicit differentiator as a complement of the recent work by the author that proposes an interconnected semi-implicit Euler double differentiators algorithm through Taylor…
We present an automated framework for constructing Taylor series expansions of rovibrational kinetic and potential energy operators for arbitrary molecules, internal coordinate systems, and molecular frame embedding conditions. Expressing…
We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the…
A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to…
This paper presents an improved Matlab routine, FO_LE, for the numerical computation of Lyapunov exponents of fractional-order systems modeled by Caputo's derivative. It is conceived as an enhanced version of the former FO_Lyapunov and…
A generalization of the max-plus transformation, which is known as a method to derive cellular automata from integrable equations, is proposed for complex numbers. Operation rules for this transformation is also studied for general number…
We introduce FunKit, a Mathematica package for the derivation and tracing of functional equations from arbitrary master equations. FunKit provides an expression vocabulary and a set of rules that allow for derivations in any given field…
This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular…