Related papers: Exact Computation of the Catalan Number $C(2,050,5…
An efficient method is proposed for computing the structure of Jordan blocks of a matrix of integers or rational numbers by exact computation. We have given a method for computing Jordan chains of a matrix with exact computation. However,…
Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for…
We present factorization and solution phases for a new linear complexity direct solver designed for concurrent batch operations on fine-grained parallel architectures, for matrices amenable to hierarchical representation. We focus on the…
Accurate yet efficient computation of the Voigt and complex error function is a challenge since decades in astrophysics and other areas of physics. Rational approximations have attracted considerable attention and are used in many codes,…
Solving the discrete logarithm problem in a finite prime field is an extremely important computing problem in modern cryptography. The hardness of solving the discrete logarithm problem in a finite prime field is the security foundation of…
We study applications of clustering (in particular, the $k$-center clustering problem) in the design of efficient and practical algorithms for computing an approximate and the exact arithmetic matrix product of two 0-1 rectangular matrices…
For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\sum_{0<k\le n}\frac1{k^m}\ \ (n=0,1,2,\ldots).$$ We discover exact values of some series involving harmonic numbers of order not exceeding four. For…
We refine Catalan numbers and Fu{\ss}-Catalan numbers by introducing colour statistics for triangulations of polygons and $d$-dimensional generalisations there-of which we call Fu{\ss}-Catalan complexes. Our refinements consist in showing…
We study the problem of differentially private continual counting in the unbounded setting where the input size $n$ is not known in advance. Current state-of-the-art algorithms based on optimal instantiations of the matrix mechanism cannot…
We extend the notion of polynomial integration over an arbitrary circle $C$ in the Euclidean geometry over general fields $\mathbb F$ of characteristic zero as a normalized $\mathbb F$-linear functional on $\mathbb{F}\left[\alpha_1,…
Numerical approximate computation can solve large and complex problems fast. It has the advantage of high efficiency. However it only gives approximate results, whereas we need exact results in many fields. There is a gap between…
The closed form for the hyper-Catalan number C[m2,m3,m4,...], which counts the number of subdivisions of a roofed polygon into m2 triangles, m3 quadrilaterals, m4 pentagons, etc., has been known since 1940. In 2025, Wildberger and Rubine…
Decomposing a matrix into a weighted sum of Pauli strings is a common chore of the quantum computer scientist, whom is not easily discouraged by exponential scaling. But beware, a naive decomposition can be cubically more expensive than…
We introduce a backward stable algorithm for computing the CS decomposition of a partitioned $2n \times n$ matrix with orthonormal columns, or a rank-deficient partial isometry. The algorithm computes two $n \times n$ polar decompositions…
We present algorithms for the discrete cosine transform (DCT) and discrete sine transform (DST), of types II and III, that achieve a lower count of real multiplications and additions than previously published algorithms, without sacrificing…
We present a $q$-analog of the super Catalan number $(2m)!(2n)!/2m!n!(m+n)!$, which also generalizes the $q$-Catalan numbers $c_n(\lambda)$, due to F\"urlinger and Hofbauer, for $\lambda=0$ and $\lambda=1$. We give a combinatorial…
Canonical polyadic decomposition (CPD) is at the core of fast matrix multiplication, a computational problem with widespread implications across several seemingly unrelated problems in computer science. Much recent progress in this field…
Ordered search is the task of finding an item in an ordered list using comparison queries. The best exact classical algorithm for this fundamental problem uses $\lceil \log_{2}{n}\rceil$ queries for a list of length $n$. Quantum computers…
Let p be any odd prime. We mainly show that $$\sum_{k=1}^{p-1}binomial(3k,k)*2^k/k=0 (mod p)$$ and $$\sum_{k=1}^{p-1}2^{k-1}C_k^{(2)}=(-1)^{(p-1)/2}-1 (mod p),$$ where $C_k^{(2)}=binomial(3k,k)/(2k+1)$ is the $k$th Catalan number of order…
Programmable Logic Devices (PLDs) continue to grow in size and currently contain several millions of gates. At the same time, research effort is going into higher-level hardware synthesis methodologies for reconfigurable computing that can…