Related papers: An iterative algorithm for approximating roots of …
We extend the algorithms of Robinson, Smyth, and McKee--Smyth to enumerate all real-rooted integer polynomials of a fixed degree, where the first few (at least three) leading coefficients are specified. Additionally, we introduce new linear…
The problem of root mean square approximation of a square integrable function by finite linear combinations of exponential functions is considered. It is subdivided into linear and nonlinear parts. The linear approximation problem is…
We present a new algorithm for reconstructing an exact algebraic number from its approximate value using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
We generalize Ramanujan method of approximating the smallest root of an equation which is found in Ramanujan Note books, Part-I. We provide simple analytical proof to study convergence of this method. Moreover, we study iterative approach…
We present a novel iterative algorithm for approximating the linear least squares solution with low complexity. After a motivation of the algorithm we discuss the algorithm's properties including its complexity, and we present theoretical…
We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field $\mathbb{K}$. More precisely, given a precision $d$, and a polynomial $Q$ whose coefficients are power series in $x$, the…
We construct a family of iterations for computing the principal square root of a square matrix $A$ using Zolotarev's rational minimax approximants of the square root function. We show that these rational functions obey a recursion, allowing…
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 consider the problem of numerically identifying roots of a target function - under the constraint that we can only measure the derivatives of the function at a given point, not the function itself. We describe and characterize two…
Among many existing algorithms, convergence methods are the most popular means of computing square root and the reciprocal of square root of numbers. An initial approximation is required in these methods. Look up tables (LUT) are employed…
Integer data sets frequently appear in many applications in sciences and technology. To analyze these, integer low rank approximation has received much attention due to its capacity of representing the results in integers preserving the…
Many problems in applied mathematics require root finding algorithms. Unfortunately, root finding methods have limitations. Firstly, regarding the convergence, there is a trade-off between the size of it's domain and it's rate. Secondly the…
We study how well a real number can be approximated by sums of two or more rational numbers with denominators up to a certain size.
We use simple equations in order to compare the basins of attraction on the complex plane, corresponding to a large collection of numerical methods, of several order. Two cases are considered, regarding the total number of the roots, which…
It is well known that the repeated square and multiply algorithm is an efficient way of modular exponentiation. The obvious question to ask is if this algorithm has an inverse which would calculate the discrete logarithm efficiently. The…
Potential theory for rational approximation is reviewed by means of examples computed with the AAA algorithm.
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…
In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…