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Our paper "Solving Third Order Linear Difference Equations in Terms of Second Order Equations" gave two algorithms for solving difference equations in terms of lower order equations: an algorithm for absolute factorization, and an algorithm…

Rings and Algebras · Mathematics 2025-12-16 Heba Bou KaedBey , Mark Van Hoeij

For a real function, automatic differentiation is such a standard algorithm used to efficiently compute its gradient, that it is integrated in various neural network frameworks. However, despite the recent advances in using complex…

Machine Learning · Computer Science 2021-01-19 Chu Guo , Dario Poletti

Consider the problem of estimating the median of N items to a precision epsilon, i.e., the estimate should be such that, with a high probability, the number of items, with values both smaller than and larger than this estimate, is less than…

Quantum Physics · Physics 2007-05-23 Lov K. Grover

We show that multiplication can be done in polynomial time on a three counter machine that receives its input as the contents of two counters. The technique is generalized to functions of two variables computable by deterministic Turing…

Computational Complexity · Computer Science 2015-01-12 Holger Petersen

Processors may find some elementary operations to be faster than the others. Although an operation may be conceptually as simple as some other operation, the processing speeds of the two can vary. A clever programmer will always try to…

Data Structures and Algorithms · Computer Science 2012-12-27 Rajat Tandon

We present four quantum algorithms for solving a multidimensional drift-diffusion equation. They rely on a quantum linear system solver, a quantum Hamiltonian simulation, a quantum random walk, and the quantum Fourier transform. We compare…

Quantum Physics · Physics 2025-10-16 Ellen Devereux , Animesh Datta

We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations where n is the number of polynomials (as well as the dimension of the ambient space), D…

Computational Complexity · Computer Science 2010-07-12 Felipe Cucker , Teresa Krick , Gregorio Malajovich , Mario Wschebor

In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of…

Numerical Analysis · Mathematics 2025-10-20 Leszczynski Jacek , Ciesielski Mariusz

We first propose a concise singular value decomposition of dual matrices. Then, the randomized version of the decomposition is presented. It can significantly reduce the computational cost while maintaining the similar accuracy. We analyze…

Numerical Analysis · Mathematics 2024-07-25 Mengyu Wang , Jingchun Zhou , Hanyu Li

Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not…

Algebraic Geometry · Mathematics 2010-08-24 Yong Feng , Wenyuan Wu , Jingzhong Zhang

Quantum algorithms are a very promising field. However, creating and manipulating these kind of algorithms is a very complex task, specially for software engineers used to work at higher abstraction levels. The work presented here is part…

By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are…

Classical Analysis and ODEs · Mathematics 2023-11-16 Hamed Taghavian

Efficient representations of the Hamiltonian such as double factorization drastically reduce circuit depth or number of repetitions in error corrected and noisy intermediate scale quantum (NISQ) algorithms for chemistry. We report a…

A fast Discrete Cosine Transform (DCT) algorithm is introduced that can be of particular interest in image processing. The main features of the algorithm are regularity of the graph and very low arithmetic complexity. The 16-point version…

Information Theory · Computer Science 2022-12-29 Maxim Vashkevich , Alexander Petrovsky

We give algorithms to compute decompositions of a given polynomial, or more generally mixed tensor, as sum of rank one tensors, and to establish whether such a decomposition is unique. In particular, we present methods to compute the…

Algebraic Geometry · Mathematics 2021-07-12 Antonio Laface , Alex Massarenti , Rick Rischter

An efficient numerical algorithm for the computation of linking number is presented. The algorithm keep tracks or rounding error so that it can ensure the correctness of the results.

Algebraic Topology · Mathematics 2020-01-01 Enrico Bertolazzi , Riccardo Ghiloni , Ruben Specogna

We present in this paper two different classes of general $K$-splitting algorithms for solving finite-dimensional convex optimization problems. Under the assumption that the function being minimized has a Lipschitz continuous gradient, we…

Optimization and Control · Mathematics 2015-03-13 Donald Goldfarb , Shiqian Ma

Query complexity is a model of computation in which we have to compute a function $f(x_1, \ldots, x_N)$ of variables $x_i$ which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes.…

Quantum Physics · Physics 2017-12-19 Andris Ambainis

Kernels are key in machine learning for modeling interactions. Unfortunately, brute-force computation of the related kernel sums scales quadratically with the number of samples. Recent Fourier-slicing methods lead to an improved linear…

Numerical Analysis · Mathematics 2025-10-14 Nicolaj Rux , Johannes Hertrich , Sebastian Neumayer

Systems of two ordinary and partial differential equations (ODEs and PDEs) had been obtained from a scalar complex ODE by splitting it into its real and imaginary parts. The procedure was also carried out to obtain a four dimensional system…

Classical Analysis and ODEs · Mathematics 2011-08-29 F M Mahomed , Asghar Qadir