Complex Roots of a Polynomial and Its Derivative

2007 ◽  
Keyword(s):  
Complex Roots ◽  
Roots Of A Polynomial

scholarly journals Fractional Newton-Raphson Method

2021 ◽  
Vol 8 (1) ◽  
pp. 1-13
Author(s):  
A. Torres-Hernandez ◽  
F. Brambila-Paz
Keyword(s):  
Fractional Calculus ◽  
Iterative Method ◽  
Complex Numbers ◽  
Initial Condition ◽  
The Real ◽  
Complex Roots ◽  
Newton Raphson ◽  
Roots Of A Polynomial ◽  
Raphson Method

The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n, with n ∈ 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 the present work, we explain an iterative method that is created using the fractional calculus, which we will call the Fractional Newton-Raphson (F N-R) Method, which has the ability to enter the space of complex numbers given a real initial condition, which allows us to find both the real and complex roots of a polynomial unlike the classical Newton-Raphson method.

Complex Roots of a Polynomial: 10688

2000 ◽  
Vol 107 (2) ◽  
pp. 181
Author(s):  
Ioan Tomescu ◽  
Kee-Wai Lau ◽  
O. P. Lossers ◽  
K. F. Andersen ◽  
R. J. Chapman ◽  
...  
Keyword(s):  
Complex Roots ◽  
Roots Of A Polynomial

Visualizing the Complex Roots of Quadratic and Cubic Equations

2001 ◽  
Vol 94 (5) ◽  
pp. 410-413
Author(s):  
Alan Lipp
Keyword(s):  
Polynomial Equation ◽  
Complex Roots ◽  
Roots Of A Polynomial

A method for visualizing the complex roots of a polynomial equation, thereby making them a little more “real” and a little less “imaginary.”

scholarly journals Visualising Complex Polynomials: A Parabola Is but a Drop in the Ocean of Quadratics

2018 ◽  
Vol 10 (6) ◽  
pp. 91
Author(s):  
Harry Wiggins ◽  
Ansie Harding ◽  
Johann Engelbrecht
Keyword(s):  
Complex Function ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Complex Polynomials ◽  
Singular Case ◽  
Hyperbolic Paraboloid ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Complex Coefficients ◽  
Roots Of A Polynomial

One of the problems encountered when teaching complex numbers arises from an inability to visualise the complex roots, the so-called "imaginary" roots of a polynomial. Being four dimensional, it is problematic to visualize graphs and roots of polynomials with complex coefficients in spite of many attempts through centuries. An innovative way is described to visualize the graphs and roots of functions, by restricting the domain of the complex function to those complex numbers that map onto real values, leading to the concept of three dimensional sibling curves. Using this approach we see that a parabola is but a singular case of a complex quadratic.  We see that sibling curves of a complex quadratic lie on a three-dimensional hyperbolic paraboloid. Finally, we show that the restriction to a real range causes no loss of generality.

scholarly journals Statistical Distribution of Roots of a Polynomial Modulo Primes III

2017 ◽  
Vol 7 (1) ◽  
pp. 115
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Prime Number ◽  
Linear Relation ◽  
Statistical Distribution ◽  
Ordinary Sense ◽  
Complex Roots ◽  
Distinct Integer ◽  
Distribution Of Roots ◽  
Roots Of A Polynomial ◽  
Expected Density

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ $(a_{n-1},\dots,a_0\in\mathbb Z)$ be a polynomial with complex roots $\alpha_1,\dots,\alpha_n$ and suppose that a linear relation over $\mathbb Q$ among $1,\alpha_1,\dots,\alpha_n$ is a  multiple of $\sum_i\alpha_i+a_{n-1}=0$ only. For a prime number $p$ such that $f(x)\bmod p$ has $n$ distinct integer  roots $0<r_1<\dots<r_n<p$, we proposed in a previous paper a conjecture that the sequence of points $(r_1/p,\dots,r_n/p)$ is equi-distributed in some sense. In this paper, we show that it implies the equi-distribution of the sequence of $r_1/p,\dots,r_n/p$ in the ordinary sense and give the expected density of primes satisfying $r_i/p<a$ for a fixed suffix $i$ and $0<a<1$.

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