Related papers: A Simple Randomized Algorithm to Compute Harmonic …
The harmonic numbers are the sequence 1, 1+1/2, 1+1/2+1/3, ... Their asymptotic difference from the sequence of the natural logarithm of the positive integers is Euler's constant gamma. We define a family of natural generalizations of the…
In various areas of applied numerics, the problem of calculating the logarithm of a matrix A emerges. Since series expansions of the logarithm usually do not converge well for matrices far away from the identity, the standard numerical…
We provide approximations to the prime counting function by various discretized versions of the logarithmic integral function, expressed solely in terms of the harmonic numbers. We demonstrate with explicit error bounds that these…
An algorithm for computing /pi(N) is presented.It is shown that using a symmetry of natural numbers we can easily compute /pi(N).This method relies on the fact that counting the number of odd composites not exceeding N suffices to calculate…
Harmonic numbers arise from the truncation of the harmonic series. The $n^\text{th}$ harmonic number is the sum of the reciprocals of each positive integer up to $n$. In addition to briefly introducing the properties of harmonic numbers, we…
We explore an algorithm for approximating roots of integers, discuss its motivation and derivation, and analyze its convergence rates with varying parameters and inputs. We also perform comparisons with established methods for approximating…
Harmonic polylogarithms $\H(\vec{a};x)$, a generalization of Nielsen's polylogarithms ${S}_{n,p}(x)$, appear frequently in analytic calculations of radiative corrections in quantum field theory. We present an algorithm for the numerical…
Given real numbers whose sum is an integer, we study the problem of finding integers which match these real numbers as closely as possible, in the sense of L^p norm, while preserving the sum. We describe the structure of solutions for this…
We introduce and analyse a new, extremely simple, randomised sorting algorithm: - choose a pair of indices $\{i, j\}$ according to some distribution $q$; - sort the elements in positions $i$ and $j$ of the array in ascending order. Choosing…
We present a randomized polynomial-time algorithm to generate a random integer according to the distribution of norms of ideals at most N in any given number field, along with the factorization of the integer. Using this algorithm, we can…
Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that exploit low-rank structure. However, estimating the rank has been done largely in an…
A simple yet efficient computational algorithm for computing the continuous optimal experimental design for linear models is proposed. An alternative proof the monotonic convergence for $D$-optimal criterion on continuous design spaces are…
We consider the harmonic balance method for finding approximate periodic solutions of the Lorenz system. When developing software that implements the described method, the math package Maxima was chosen. The drawbacks of symbolic…
Given $n$ colored balls, we want to detect if more than $\lfloor n/2\rfloor$ of them have the same color, and if so find one ball with such majority color. We are only allowed to choose two balls and compare their colors, and the goal is to…
We report on the results of a computer search for primes $p$ which divide an Harmonic number $H_{\lfloor p/N \rfloor}$ with small $N > 1$.
We study the algorithmic complexity of fair division problems with a focus on minimizing the number of queries needed to find an approximate solution with desired accuracy. We show for several classes of fair division problems that under…
We consider the following "partition and sum" operation on a natural number: Treating the number as a long string of digits insert several plus signs in between some of the digits and carry out the indicated sum. This results in a smaller…
In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers.…
We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In…
We offer multiplication method for factoring big natural numbers which extends the group of the Fermat's and Lehman's factorization algorithms and has run-time complexity $O(n^{1/3})$. This paper is argued the finiteness of proposed…