Related papers: Quadratic Interval Refinement for Real Roots
We present an algorithm for isolating the roots of an arbitrary complex polynomial $p$ that also works for polynomials with multiple roots provided that the number $k$ of distinct roots is given as part of the input. It outputs $k$ pairwise…
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…
We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented…
The Schur decomposition of a square matrix $A$ is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following…
Iterative refinement (IR) is a popular scheme for solving a linear system of equations based on gradually improving the accuracy of an initial approximation. Originally developed to improve upon the accuracy of Gaussian elimination,…
The isolation intervals of the real roots of the real symbolic monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c\,\,$ are found in terms of simple functions of the coefficients of the polynomial (such as: $-a$, $-a/3$, $-c/b$, $\pm…
In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.
Mixed trigonometric-polynomials (MTPs) are functions of the form $f(x,\sin{x}, \cos{x})$ with $f\in\mathbb{Q}[x_1,x_2,x_3]$. In this paper, an algorithm ``isolating" all the real roots of an MTP is provided and implemented. It automatically…
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 make the interprecision transfers explicit in an algorithmic description of iterative refinement and obtain new insights into the algorithm. One example is the classic variant of iterative refinement where the matrix and the…
In this paper, we present a new method for solving standard quaternion equations. Using this method we reobtain the known formulas for the solution of a quadratic quaternion equation, and provide an explicit solution for the cubic…
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…
We present a proximal augmented Lagrangian based solver for general convex quadratic programs (QPs), relying on semismooth Newton iterations with exact line search to solve the inner subproblems. The exact line search reduces in this case…
We introduce a new iterative rounding technique to round a point in a matroid polytope subject to further matroid constraints. This technique returns an independent set in one matroid with limited violations of the other ones. On top of the…
We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion ($10^9$) so that the observed complexity to find all roots is between $O(d\ln d)$…
The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In…
We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in $B_{\alpha} \in L[y]$, where $L=\QQ(\alpha)$ is a simple algebraic extension of the rational numbers. We…
We present a new rational approximation algorithm based on the empirical interpolation method for interpolating a family of parametrized functions to rational polynomials with invariant poles, leading to efficient numerical algorithms for…
We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…
We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good…