相关论文: An algorithm for multipication of Kaluza numbers

One of the most common, but at the same time expensive operations in linear algebra, is multiplying two matrices $A$ and $B$. With the rapid development of machine learning and increases in data volume, performing fast matrix intensive…

Hurwitz (1887) defined a continued fraction algorithm for complex numbers which is better behaved in many respects than a more "natural" extension of the classical continued fraction algorithm to the complex plane would be. Although the…

In this paper, we apply results on number systems based on continued fraction expansions to modular arithmetic. We provide two new algorithms in order to compute modular multiplication and modular division. The presented algorithms are…

This work establishes the existence of addition theorems and double-angle formulas for Ck real scalar functions. Moreover, we determine necessary and sufficient conditions for a bivariate function to be an addition formula for a Ck real…

In this paper, the problem of multiplicative anomaly of zeta regularization is solved for polynomials. For a regularizable sequence $\Lambda$, we explicitly calculate the zeta regularized product of $(\Lambda-z_1)\dots(\Lambda-z_n)$ for…

In this paper, we present a probabilistic algorithm to multiply two sparse polynomials almost as efficiently as two dense univariate polynomials with a result of approximately the same size. The algorithm depends on unproven heuristics that…

Factorization of polynomials arises in numerous areas in symbolic computation. It is an important capability in many symbolic and algebraic computation. There are two type of factorization of polynomials. One is convention polynomial…

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…

An algorithm for matrix factorization of polynomials was proposed in \cite{fomatati2022tensor} and it was shown that this algorithm produces better results than the standard method for factoring polynomials on the class of summand-reducible…

We describe an incremental algorithm to enumerate the isomorphism classes of double pseudoline arrangements. The correction of our algorithm is based on the connectedness under mutations of the spaces of one-extensions of double pseudoline…

This paper proposes new factorizations for computing the Neumann series. The factorizations are based on fast algorithms for small prime sizes series and the splitting of large sizes into several smaller ones. We propose a different basis…

We present a novel class of methods to compute functions of matrices or their action on vectors that are suitable for parallel programming. Solving appropriate simple linear systems of equations in parallel (or computing the inverse of…

Matrix multiplication is a foundational operation in scientific computing and machine learning, yet its computational complexity makes it a significant bottleneck for large-scale applications. The shift to parallel architectures, primarily…

Matrix multiplication is a fundamental building block for large scale computations arising in various applications, including machine learning. There has been significant recent interest in using coding to speed up distributed matrix…

We improve the space complexity of Karatsuba multiplication on a quantum computer from $O(n^{1.427})$ to $O(n)$ while maintaining $O(n^{\lg 3})$ gate complexity. We achieve this by ensuring recursive calls can add their outputs directly…

We introduce and study the recursive divisor function, a recursive analog of the usual divisor function: $\kappa_x(n) = n^x + \sum_{d\lfloor n} \kappa_x(d)$, where the sum is over the proper divisors of $n$. We give a geometrical…

We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs…

The existing doubling algorithms have been proven efficient for several important nonlinear matrix equations arising from real-world engineering applications. In a nutshell, the algorithms iteratively compute a basis matrix, in one of the…

In this article we will be dedicated some algorithms of addition, subtraction, multiplication and division of two positive integers using Zeckendorf form. Such results find application in coding theory.

The complexity of matrix multiplication is measured in terms of $\omega$, the smallest real number such that two $n\times n$ matrices can be multiplied using $O(n^{\omega+\epsilon})$ field operations for all $\epsilon>0$; the best bound…