scholarly journals Polynomial Solutions of the Heun Equation

10.14311/1416 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
B. Shapiro ◽  
M. Tater
Keyword(s):  
Root Distribution ◽  
Heun Equation ◽  
Real Roots ◽  
Polynomial Solutions ◽  
Complex Roots ◽  
Interlacing Property

We review properties of certain types of polynomial solutions of the Heun equation. Two aspects are particularly concerned, the interlacing property of spectral and Stieltjes polynomials in the case of real roots of these polynomials and asymptotic root distribution when complex roots are present.

scholarly journals Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 169
Author(s):  
Avram Sidi
Keyword(s):  
Nonlinear Equations ◽  
Local Convergence ◽  
Real Root ◽  
Numerical Procedure ◽  
Secant Method ◽  
Convergence Theory ◽  
Simple Complex ◽  
Real Roots ◽  
Complex Roots ◽  
Appl Math

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

On Graeffe's Method for Complex Roots of Algebraic Equations

1924 ◽  
Vol 22 (2) ◽  
pp. 83-87 ◽  
Author(s):  
S. Brodetsky ◽  
G. Smeal
Keyword(s):  
Nineteenth Century ◽  
Real Axis ◽  
Practical Method ◽  
Algebraic Equations ◽  
Argand Diagram ◽  
Considerable Difficulty ◽  
The Real ◽  
Real Roots ◽  
Bipolar Coordinates ◽  
Complex Roots

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

scholarly journals The New New-Nacci Method for Calculating the Roots of a Univariate Polynomial and Solution of Quintic Equation in Radicals

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 746
Author(s):  
Ilija Tanackov ◽  
Ivan Pavkov ◽  
Željko Stević
Keyword(s):  
Impossibility Theorem ◽  
Second Phase ◽  
Polynomial Roots ◽  
Real Roots ◽  
Quintic Equation ◽  
Complex Roots ◽  
Abel Polynomial ◽  
Quintic Polynomial ◽  
Univariate Polynomial ◽  
Newton’S Identities

An arbitrary univariate polynomial of nth degree has n sequences. The sequences are systematized into classes. All the values of the first class sequence are obtained by Newton’s polynomial of nth degree. Furthermore, the values of all sequences for each class are calculated by Newton’s identities. In other words, the sequences are formed without calculation of polynomial roots. The New-nacci method is used for the calculation of the roots of an nth-degree univariate polynomial using radicals and limits of successive members of sequences. In such an approach as is presented in this paper, limit play a catalytic–theoretical role. Moreover, only four basic algebraic operations are sufficient to calculate real roots. Radicals are necessary for calculating conjugated complex roots. The partial limitations of the New-nacci method may appear from the decadal polynomial. In the case that an arbitrary univariate polynomial of nth degree (n ≥ 10) has five or more conjugated complex roots, the roots of the polynomial cannot be calculated due to Abel’s impossibility theorem. The second phase of the New-nacci method solves this problem as well. This paper is focused on solving the roots of the quintic equation. The method is verified by applying it to the quintic polynomial with all real roots and the Degen–Abel polynomial, dating from 1821.

scholarly journals Polynomial solutions of q-Heun equation and ultradiscrete limit

2019 ◽  
Vol 25 (5) ◽  
pp. 647-664 ◽  
Author(s):  
Kentaro Kojima ◽  
Tsukasa Sato ◽  
Kouichi Takemura
Keyword(s):  
Heun Equation ◽  
Polynomial Solutions

Singularities at the Tip of a Crack Terminating Normally at an Interface Between Two Orthotropic Media

1996 ◽  
Vol 63 (2) ◽  
pp. 264-270 ◽  
Author(s):  
J. C. Sung ◽  
J. Y. Liou
Keyword(s):  
Characteristic Equation ◽  
Complex Form ◽  
Stress Singularities ◽  
Real Form ◽  
Antisymmetric Mode ◽  
Real Roots ◽  
Complex Roots ◽  
Principal Axes ◽  
Orthotropic Media ◽  
Set Up

The order of stress singularities at the tip of a crack terminating normally at an interface between two orthotropic media is analyzed. Characteristic equation in complex form for the power of singularity s, where 0 < Re{s} < 1, is first set up for two general anisotropic materials. Attention is then focused on the problem that is composed by two orthotropic media where one of them (say, material #2 ) the material principal axes are aligned while the other one (say, material #1) the principal axes can have an angle γ relative to the interface. For such a problem, a real form of the characteristic equation is obtained. The roots are functions of γ in general. Two real roots exist for most values of γ; however, there are possible ranges of γ that the complex roots will occur. The roots s are found to be independent of γ when material #1 has the property that δ(1) = 1. When γ = 0, two roots are always real. Furthermore, each of these two roots is associated with symmetric or antisymmetric mode and they become equal when Δ = 1. Many other features of the effects of the material parameters on the behaviors of the roots s are further investigated in the present work, where the six generalized Dundurs’ constants, expressed in terms of Krenk’s parameters, play an important role in the analysis.

scholarly journals Increasing the interval of convergence for a generalized Newton's method of solving nonlinear equations

2016 ◽  
pp. 7-12
Author(s):  
А.Н. Громов
Keyword(s):  
Continuous Function ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
The Real ◽  
Real Roots ◽  
Complex Roots ◽  
Transcendental Equations ◽  
Generalized Newton’S Method

Рассмотрен подход к построению расширения промежутка сходимости ранее предложенного обобщения метода Ньютона для решения нелинейных уравнений одного переменного. Подход основан на использовании свойства ограниченности непрерывной функции, определенной на отрезке. Доказано, что для поиска действительных корней вещественнозначного многочлена с комплексными корнями предложенный подход дает итерации с нелокальной сходимостью. Результат обобщен на случай трансцендентных уравнений. An approach to the construction of an extended interval of convergence for a previously proposed generalization of Newton's method to solve nonlinear equations of one variable. This approach is based on the boundedness of a continuous function defined on a segment. It is proved that, for the search for the real roots of a real-valued polynomial with complex roots, the proposed approach provides iterations with nonlocal convergence. This result is generalized to the case transcendental equations.

Notes on the Theory of Equations

1939 ◽  
Vol 23 (256) ◽  
pp. 376-379
Author(s):  
E. P. Lewis
Keyword(s):  
Real Root ◽  
Real Roots ◽  
The Third ◽  
Complex Roots ◽  
Image Position

Multiply throughout by a 2 and write y for ax+ b ; the equation becomes where H ≡ ac − b 2, G ≡ a2d − 3abc + 2b3. Since in an equation with real coefficients complex roots occur in conjugate pairs, (i) must have at least one real root; so if α is this root, (i) may be written Accordingly the two remaining roots are also real if But since α satisfies (i), and so Hence if (i) has three real roots, G2 +4H 3 ≤ 0; and clearly, when G2 +4H 3 = 0, two roots are numerically equal to and the third to .

Lower bounds for discriminants of polynomials

2017 ◽  
Vol 13 (10) ◽  
pp. 2769-2776
Author(s):  
Kwang-Seob Kim
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Irreducible Polynomial ◽  
Real Roots ◽  
Integer Coefficients ◽  
Complex Roots ◽  
Polynomial Discriminant

Let [Formula: see text] be a monic irreducible polynomial of degree [Formula: see text] having exactly [Formula: see text] real roots and [Formula: see text] complex roots with integer coefficients and [Formula: see text] be its polynomial discriminant. Suppose that [Formula: see text] is square-free. In this paper, we will give a lower bound for polynomial discriminants [Formula: see text] for given [Formula: see text].

Complex Roots of a Quadratic Equation Graphically

1970 ◽  
Vol 63 (3) ◽  
pp. 229
Author(s):  
W. G. Brady
Keyword(s):  
Graphical Method ◽  
Quadratic Equation ◽  
Geometric Method ◽  
Real Roots ◽  
Complex Roots

The graphical method for solving a quadratic equation as it is usually presented indicates real roots and coincident roots as intersections of a certain parabola with the x-axis and signals, by failure of intersection, that the roots are either complex or imaginary. In this note, we shall show a geometric method for finding the complex roots of the quadratic.

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