This paper proposes using root moment method to find the roots of arbitrary polynomials. It has been proved that feedforward neural networks (FNN) trained with constrained learning (CL) back propagation (BP) algorithm can be used to estimate the distributions of roots of polynomials. It has been found, however, that using the different priori information implicit in the polynomials for CL-BP al...

We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton-Cotes closed quadrature rules. We prove that when a quadrature rule with n+ 1 nodes is used the resulting iterative method has convergence order at least n+ 2, starting with the case n = 0 (which corresponds to the Newton’s method).

Cardinal’s matrix version of the Sebastiao e Silva polynomial root-finder rapidly approximates the roots as the eigenvalues of the associated Frobenius matrix. We preserve rapid convergence to the roots but amend the algorithm to allow input polynomials with multiple roots and root clusters. As in Cardinal’s algorithm, we repeatedly square the Frobenius matrix in nearly linear arithmetic time p...

In this paper we present a new closed-form solution to a chaotic difference equation, yn+1 = a2y 2 n + a1yn + a0 with coefficient a0 = (a1 − 4)(a1 + 2)/(4a2), and using this solution, show how corresponding exact roots to a special set of related polynomials of order 2, p ∈ N with two independent parameters can be generated, for any p.

In this paper we present some problems and their solutions exploiting lattice based root finding techniques. In CaLC 2001, Howgrave-Graham proposed a method to find the Greatest Common Divisor (GCD) of two large integers when one of the integers is exactly known and the other one is known approximately. In this paper, we present three applications of the technique. The first one is to show dete...

This paper describes a selective root finding method for polynomials based in Complex Analysis results. It can find the poles of the speech signal LPC model that are close to the unit circle, without wasting computations with the others, lesser significant ones. This feature makes our method faster than the standard ones for speech analysis. These poles are in better correspondence with the for...

Constructing r-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree re (where r is a prime) over a given finite field Fq of characteristic p (equivalently, constructing the bigger field Fqr ). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some n...

We introduce the Stochastic Monotone Aggregated Root-Finding (SMART) algorithm, a new randomized operator-splitting scheme for finding roots of finite sums of operators. These algorithms are similar to the growing class of incremental aggregated gradient algorithms, which minimize finite sums of functions; the difference is that we replace gradients of functions with black-boxes called operator...

Reliable risk measurement is a key problem for financial institutions and regulatory authorities. The current industry standard Value-at-Risk has several deficiencies. Improved risk measures have been suggested and analyzed in the recent literature, but their computational implementation has largely been neglected so far. We propose and investigate stochastic approximation algorithms for the co...

In this paper we present a new closed-form solution to a chaotic difference equation, yn+1 = a2y 2 n + a1yn + a0 with coefficient a0 = (a1 − 4)(a1 + 2)/(4a2), and using this solution, show how corresponding exact roots to a special set of related polynomials of order 2, p ∈ N with two independent parameters can be generated, for any p.