Related papers: Solution of Real Cubic Equations without Cardano's…
Omar Khayyam's studies on cubic equations inspired the 12th century Persian mathematician Sharaf al-Din Tusi to investigate the number of positive roots. According to the noted mathematical historian Rashed, Tusi analyzed the problem for…
According to the Abel-Ruffini theorem [1] and Galois theory [2], there is no solution in finite radicals to the general quintic equation. This article takes a different approach and proposes a new method to solve the quintic by iteration of…
In this paper we show that Cardanos formula for the solution of cubic equations can be reduced to expressions involving only square roots if the real root is rational.
We derive the Cardano formula of cubic equations by completing the cube, and provide radical solutions to some algebraic equations of higher degree by completing powers. The main idea of completing powers arises from Harrison's center…
The solution of the cubic equation has a century-long history; however, the usual presentation is geared towards applications in algebra and is somewhat inconvenient to use in optimization where frequently the main interest lies in real…
One can hardly believe that there is still something to be said about cubic equations. To dodge this doubt, we will instead try and say something about Sylvester. He doubtless found a way of solving cubic equations. As mentioned by Rota, it…
Let $p(z)$ be a monic cubic complex polynomial with distinct roots and distinct critical points. We say a critical point has the {\it Voronoi property} if it lies in the Voronoi cell of a root $\theta$, $V(\theta)$, i.e. the set of points…
This article provides a simple trigonometric method for determining how many roots of a quintic equation are real and how many are complex, without solving the equation. The approach transforms a depressed quintic $t^5 + mt^3 + nt^2 + pt +…
We show that every cubic form with coefficients in an imaginary quadratic number field $K/\mathbb{Q}$ in at least $14$ variables represents zero non-trivially. This builds on the corresponding seminal result by Heath-Brown for rational…
The cubic regularization method (CR) is a popular algorithm for unconstrained non-convex optimization. At each iteration, CR solves a cubically regularized quadratic problem, called the cubic regularization subproblem (CRS). One way to…
There are thousands of papers on rootfinding for nonlinear scalar equations. Here is one more, to talk about an apparently new method, which I call ``Inverse Cubic Iteration'' (ICI) in analogy to the Inverse Quadratic Iteration in Richard…
In this paper we prove that all irrational numbers from totally real cubic number fields are well approximable by rationals (i.e. the partial quotients in the continued fraction expansion of such a number are unbounded). This settles the…
This article introduces an intuitive function MY that simplifies solving cubic equations without venturing into the complex space. To many, it's quite strange that cubic root(s) are expressed using trigonometric functions in the…
The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck--Marden--Northshield equilateral triangle and regular tetrahedron. The Vi\`ete trigonometric formulae for the roots of…
This article provides a simple trigonometric method for determining how many roots of a quartic equation are real and how many are complex, without solving the equation. The approach replaces the quartic's classical discriminant -- a…
The exact solutions of both the cubic Duffing equation and cubic-quintic Duffing equation are presented by using only leaf functions. In previous studies, exact solutions of the cubic Duffing equation have been proposed using functions that…
We propose an efficient algorithm to compute the real roots of a sparse polynomial $f\in\mathbb{R}[x]$ having $k$ non-zero real-valued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given…
In this paper, we consider an unconstrained optimization model where the objective is a sum of a large number of possibly nonconvex functions, though overall the objective is assumed to be smooth and convex. Our bid to solving such model…
We present a new method to calculate analytically the roots of the general complex polynomial of degree three. Thismethod is based on the approach of appropriated changes of variable involving an arbitrary parameter. The advantageof this…
This work introduces a new cubic regularization method for nonconvex unconstrained multiobjective optimization problems. At each iteration of the method, a model associated with the cubic regularization of each component of the objective…