Related papers: Root-finding with Implicit Deflation

Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose…

The Iterative Filtering method is a technique developed recently for the decomposition and analysis of non-stationary and non-linear signals. In this work we propose two alternative formulations of the original algorithm which allows to…

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…

Using the option delta systematically, we derive tighter lower and upper bounds of the Black-Scholes implied volatility than those in Tehranchi [SIAM J. Financ. Math. 7 (2016), 893-916]. As an application, we propose a Newton-Raphson…

We consider a popular family of constrained optimization problems arising in machine learning that involve optimizing a non-decomposable evaluation metric with a certain thresholded form, while constraining another metric of interest.…

Univariate polynomial root-finding has been studied for four millennia and very intensively in the last decades. Our new near-optimal root-finders approximate all zeros of a polynomial p almost as fast as one accesses its coefficients with…

We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods…

In this paper, an optimized version of classical Bombelli's algorithm for computing integer square roots is presented. In particular, floating-point arithmetic is used to compute the initial guess of each digit of the root, following…

We present a FFT-based algorithm for the computation of a polynomial's coefficients from its roots, and apply it to obtain the coefficients of interpolation polynomials, to invert Vandermondians and to evaluate the symmetric functions of a…

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and…

The classical division algorithm for polynomials requires $O(n^2)$ operations for inputs of size $n$. Using reversal technique and Newton iteration, it can be improved to $O({M}(n))$, where ${M}$ is a multiplication time. But the method…

In this paper, we present a Newton-like method based on model reduction techniques, which can be used in implicit numerical methods for approximating the solution to ordinary differential equations. In each iteration, the Newton-like method…

We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion…

In our quest for the design, the analysis and the implementation of a subdivision algorithm for finding the complex roots of univariate polynomials given by oracles for their evaluation, we present sub-algorithms allowing substantial…

Superlinear convergence has been an elusive goal for black-box nonsmooth optimization. Even in the convex case, the subgradient method is very slow, and while some cutting plane algorithms, including traditional bundle methods, are popular…

This paper presents a modification of Secant method for finding roots of equations that uses three points for iteration instead of just two. The development of the mathematical formula to be used in the iteration process is provided…

As showed in (Fiedler, 1990), any polynomial can be expressed as a characteristic polynomial of a complex symmetric arrowhead matrix. This expression is not unique. If the polynomial is real with only real distinct roots, the matrix can be…

The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For…

The reciprocal square root is an important computation for which many very sophisticated algorithms exist (see for example \cite{863046,863031} and the references therein). In this paper we develop a simple differential compensation (much…

We propose a new type of multilevel method for solving eigenvalue problems based on Newton iteration. With the proposed iteration method, solving eigenvalue problem on the finest finite element space is replaced by solving a small scale…