scholarly journals A nonlocal transport equation modeling complex roots of polynomials under differentiation

2020 ◽  
pp. 1
Author(s):  
Sean O’Rourke ◽  
Stefan Steinerberger
Keyword(s):  
Transport Equation ◽  
Equation Modeling ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Nonlocal Transport

NUMERICAL TREATMENT OF A NONLINEAR NONLOCAL TRANSPORT EQUATION MODELING CRYSTAL PRECIPITATION

2008 ◽  
Vol 18 (09) ◽  
pp. 1505-1527 ◽  
Author(s):  
ROMINA GOBBI ◽  
SILVIA PALPACELLI ◽  
RENATO SPIGLER
Keyword(s):  
Differential Equations ◽  
Transport Equation ◽  
Communication Theory ◽  
Hyperbolic Partial Differential Equations ◽  
Equation Modeling ◽  
Method Of Solution ◽  
Delta Functions ◽  
Nonlocal Transport ◽  
Precipitation Phenomena ◽  
Initial Crystal

Numerical methods to solve certain nonlinear nonlocal transport equations (hyperbolic partial differential equations with smooth solutions), even singular at the boundary, are developed and analyzed. As a typical case, a model equation used to describe certain crystal precipitation phenomena (a slight variant of the so-called Lifshitz–Slyozov–Wagner model) is considered. Choosing a train of few delta functions as initial crystal size distribution, one can model the technologically important case of having only a modest number of crystal sizes. This leads to the reduction of the transport equation to a system of ordinary differential equations, and suggests a new method of solution for the transport equation, based on Shannon sampling, which is widely used in communication theory.

scholarly journals A Neural Network-Based Approach for Approximating Arbitrary Roots of Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 317
Author(s):  
Diogo Freitas ◽  
Luiz Guerreiro Lopes ◽  
Fernando Morgado-Dias
Keyword(s):  
Iterative Methods ◽  
Fundamental Problem ◽  
Iterative Algorithms ◽  
Rounding Errors ◽  
Suggested Approach ◽  
Simultaneous Iterative Methods ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Comparative Results ◽  
The One

Finding arbitrary roots of polynomials is a fundamental problem in various areas of science and engineering. A myriad of methods was suggested to address this problem, such as the sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute approximations to all the roots of a given polynomial, which can produce inaccurate results due to the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by their capacity to find complex mappings between the dependent and independent variables. In view of this, this paper aims to determine, based on comparative results, whether ANNs can be used to compute approximations to the real and complex roots of a given polynomial, as an alternative to simultaneous iterative algorithms like the D–K method. Although the results are very encouraging and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able to surpass the accuracy of the D–K method. The results indicated, however, that the use of the approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial to the accuracy of this method.

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.

Computing complex zeros of polynomials and power series

2011 ◽  
Author(s):  
Jean-Marc Couveignes
Keyword(s):  
Power Series ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Zeros Of Polynomials ◽  
Complex Data ◽  
Square Roots ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Complex Zeros

The purpose of this chapter is twofold. First, it will prove two theorems (5.3.1 and 5.4.2) about the complexity of computing complex roots of polynomials and zeros of power series. The existence of a deterministic polynomial time algorithm for these purposes plays an important role in this book. More important, it will also explain what it means to compute with real or complex data in polynomial time. The chapter first recalls basic definitions in computational complexity theory, it then deals with the problem of computing square roots. The more general problem of computing complex roots of polynomials is treated thereafter and, finally, the chapter studies the problem of finding zeros of a converging power series.

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