Related papers: An algorithm for dividing quaternions
The problem of representing a given positive integer as a sum of four squares of integers has been widely concerned for a long time, and for a given positive odd $n$ one can find a representation by doing arithmetic in a maximal order of…
We further develop the large $ N $ formalism presented by some of us in earlier works in order to recursively calculate the partition function of a singly pseudoknotted RNA. We demonstrate that this calculation takes time proportional to…
The largest number factored on a quantum device reported until now was 143. That quantum computation, which used only 4 qubits at 300K, actually also factored much larger numbers such as 3599, 11663, and 56153, without the awareness of the…
Matrix multiplication consumes a large fraction of the time taken in many machine-learning algorithms. Thus, accelerator chips that perform matrix multiplication faster than conventional processors or even GPU's are of increasing interest.…
This paper provides the first meaningful documentation and analysis of an established technique which aims to obtain an approximate solution to linear programming problems prior to applying the primal simplex method. The underlying…
A precise estimation of the computational complexity in Shor's factoring algorithm under the condition that the large integer we want to factorize is composed by the product of two prime numbers, is derived by the results related to number…
We investigate the possibilities to calculate vector partition functions by means of iterated partial fraction decomposition, as suggested by Beck (2004). Particularly, for an important type of families of rational functions, we describe an…
The accuracy requirements in many scientific computing workloads result in the use of double-precision floating-point arithmetic in the execution kernels. Nevertheless, emerging real-number representations, such as posit arithmetic, show…
In this paper, an optimized version of classical Bombelli's algorithm for computing integer square roots is presented. In particular, floating-point arithmetic is used to compute the initial guess of each digit of the root, following…
Developing efficient hardware accelerators for mathematical kernels used in scientific applications and machine learning has traditionally been a labor-intensive task. These accelerators typically require low-level programming in Verilog or…
The field of deep learning has seen significant advancement in recent years. However, much of the existing work has been focused on real-valued numbers. Recent work has shown that a deep learning system using the complex numbers can be…
The low-rank quaternion matrix approximation has been successfully applied in many applications involving signal processing and color image processing. However, the cost of quaternion models for generating low-rank quaternion matrix…
In this work, we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum algorithm to solve static, Schr\"odinger-type,…
One of the most significant challenges on cryptography today is the problem of factoring large integers since there are no algorithms that can factor in polynomial time, and factoring large numbers more than some limits(200 digits) remain…
A number of quantum algorithms have been performed on small quantum computers; these include Shor's prime factorization algorithm, error correction, Grover's search algorithm and a number of analog and digital quantum simulations. Because…
Some recent processors are not equipped with an integer division unit. Compilers then implement division by a call to a special function supplied by the processor designers, which implements division by a loop producing one bit of quotient…
Unlike the Hamilton quaternion algebra, the split-quaternions contain nontrivial zero divisors. In general speaking, it is hard to find the solutions of equations in algebras containing zero divisor. In this paper, we manage to derive…
The aim of this paper is to develop novel quantum algorithms for Gaussian process quadrature methods. Gaussian process quadratures are numerical integration methods where Gaussian processes are used as functional priors for the integrands…
In this paper, an exact algorithm in polynomial time is developed to solve unrestricted binary quadratic programs. The computational complexity is $O\left( n^{\frac{15}{2}}\right) $, although very conservative, it is sufficient to prove…
An efficient numerical algorithm is presented for massively parallel simulations of dispersion-managed wavelength-division-multiplexed optical fiber systems. The algorithm is based on a weak nonlinearity approximation and independent…