Related papers: GCD Computation of n Integers
This note provides very simple, efficient algorithms for computing the number of distinct longest common subsequences of two input strings and for computing the number of LCS embeddings.
The integer division of a numerator n by a divisor d gives a quotient q and a remainder r. Optimizing compilers accelerate software by replacing the division of n by d with the division of c * n (or c * n + c) by m for convenient integers c…
Many emerging computer applications require the processing of large numbers, larger than what a CPU can handle. In fact, the top of the line PCs can only manipulate numbers not longer than 32 bits or 64 bits. This is due to the size of the…
In this paper we propose a technique to blindly synthesize the generator polynomial of BCH codes. The proposed technique involves finding Greatest Common Divisor (GCD) among different codewords and block lengths. Based on this combinatorial…
A notion of gcd chain has been introduced by the author at ISSAC 2017 for two univariate monic polynomials with coefficients in a ring R = k[x_1, ..., x_n ]/(T) where T is a primary triangular set of dimension zero. A complete algorithm to…
Generalized parton distributions (GPDs) serve as indispensable tools for the exploration of proton structure. In this study, we offer a deep learning-assisted framework for the extraction of GPDs from experimental data and the results of…
We propose a variable decomposition algorithm -greedy block coordinate descent (GBCD)- in order to make dense Gaussian process regression practical for large scale problems. GBCD breaks a large scale optimization into a series of small…
On common processors, integer multiplication is many times faster than integer division. Dividing a numerator n by a divisor d is mathematically equivalent to multiplication by the inverse of the divisor (n / d = n x 1/d). If the divisor is…
A method of determining two factors of an odd integer without need of multiplication or division operation in iterative portion of computation is presented. It is feasible for an implementing algorithm to use only integer addition and…
Upper bounds for GCD sums of the form [\sum_{k,{\ell}=1}^N\frac{(\gcd(n_k,n_{\ell}))^{2\alpha}}{(n_k n_{\ell})^\alpha}] are proved, where $(n_k)_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0<\alpha \le 1$; the…
While studying a characteristic number of manifolds we noticed that the calculation was simply computing a multiple of a multinomial coefficient. We were, at the time, interested in computing the greatest common divisor of these…
Two algorithms for computing $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, are described, and using a combination of these two algorithms, the resulting algorithm is $O(n^{3/2})$. The second algorithm uses a list…
We show asymptotic upper and lower bounds for the greatest common divisor of N and $\sigma(N)$. We also show that there are infinitely many integers N with fairly large g.c.d. of N and $\sigma(N)$.
An input- and output-sensitive GCD algorithm for multi-variate polynomials over finite fields is proposed by combining the modular method with the Ben-Or/Tiwari sparse interpolation. The bit complexity of the algorithm is given and is…
We describe how to compute for two polynomials $f(X), g(X) \in \mathbb{Z}[X]$ with integer coefficients the greatest common divisors of $f(z)$ and $g(z)$ for all integers $z \in \mathbb{Z}$. As an application we determine the structures…
The Euclidean algorithm is the oldest algorithms known to mankind. Given two integral numbers $a_1$ and $a_2$, it computes the greatest common divisor (gcd) of $a_1$ and $a_2$ in a very elegant way. From a lattice perspective, it computes a…
We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and…
A generalized Chinese remainder theorem (CRT) for multiple integers from residue sets has been studied recently, where the correspondence between the remainders and the integers in each residue set modulo several moduli is not known. A…
This contains a new version of the so-called non-commutative Gauss algorithm for polycyclic groups. Its results allow to read off the order and the index of a subgroup in an (possibly infinite) polycyclic group.
To factor an integer N, given that it is equal to the product of two primes, it suffices to find an integer d satisfying a certain simple numerical test. In this approach, the factorization problem equates to the problem of designing an…