Related papers: On Newton-Raphson iteration for multiplicative inv…
We present a deterministic algorithm that, given a prime $p$ and a solution $x \in \mathbb Z$ to the discrete logarithm problem $a^x \equiv b \pmod p$ with $p\nmid a$, efficiently lifts it to a solution modulo $p^k$, i.e., $a^x \equiv b…
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
To approximate a simple root of an equation we construct families of iterative maps of higher order of convergence. These maps are based on model functions which can be written as an inner product. The main family of maps discussed is…
We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…
We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…
Given an odd prime $p$, we provide formulas for the Hensel lifts of polynomial roots modulo $p$, and give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose…
We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the…
Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic…
We present a new algorithm for fast matrix multiplication using tensor decompositions which have special features. Thanks to these features we obtain exponents lower than what the rank of the tensor decomposition suggests. In particular for…
In this note we give concise formulas, which lead to a simple and fast computer program that computes a powerful knot invariant. This invariant $\rho_1$ is not new, yet our formulas are by far the simplest and fastest: given a knot we write…
The "self-power" map $x \mapsto x^x$ modulo $m$ and its generalized form $x \mapsto x^{x^n}$ modulo $m$ are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use $p$-adic…
The capacity for solving eigenstates with a quantum computer is key for ultimately simulating physical systems. Here we propose inverse iteration quantum eigensolvers, which exploit the power of quantum computing for the classical inverse…
In the following document, we present a way to obtain the order of convergence of the Fractional Newton-Raphson (F N-R) method, which seems to have an order of convergence at least linearly for the case in which the order $\alpha$ of the…
Recent technological developments have led to big data processing, which resulted in significant computational difficulties when solving large-scale linear systems or inverting matrices. As a result, fast approximate iterative matrix…
Modular composition is the problem of computing the composition of two univariate polynomials modulo a third one. For a long time, the fastest algebraic algorithm for this problem was that of Brent and Kung (1978). Recently, we improved…
We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$. In particular, in the integer case, we improve a recent bound…
We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power…
Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use…
In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured…
In our recent publication we have proposed a new methodology for determination of the two-term Machin-like formula for pi with small arguments of the arctangent function of kind $$ \frac{\pi }{4} = {2^{k - 1}}\arctan \left(…