scholarly journals Solution of nonlinear equations via Pade approximation. A Computer Algebra approach

2021 ◽  
Vol 66 (2) ◽  
pp. 321-328
Author(s):  
Radu T. Trimbitas
Keyword(s):  
Numerical Methods ◽  
Nonlinear Equations ◽  
Computer Algebra ◽  
Padé Approximation ◽  
High Order ◽  
Pade Approximation ◽  
Algebra Approach

"We generate automatically several high order numerical methods for the solution of nonlinear equations using Pad e approximation and Maple CAS."

High-order recurrence relations, Hermite-Padé approximation and Nikishin systems

2018 ◽  
Vol 209 (3) ◽  
pp. 385-420 ◽  
Author(s):  
D. Barrios Rolanía ◽  
J. S. Geronimo ◽  
G. López Lagomasino
Keyword(s):  
Recurrence Relations ◽  
Padé Approximation ◽  
High Order ◽  
Pade Approximation

scholarly journals A Few Iterative Methods by Using [1,n]-Order Padé Approximation of Function and the Improvements

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 55 ◽  
Author(s):  
Shengfeng Li ◽  
Xiaobin Liu ◽  
Xiaofang Zhang
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Padé Approximation ◽  
Single Step ◽  
Order Convergence ◽  
Pade Approximation ◽  
Derivatives Of ◽  
Third Derivative ◽  
Approximation Of Function

In this paper, a few single-step iterative methods, including classical Newton’s method and Halley’s method, are suggested by applying [ 1 , n ] -order Padé approximation of function for finding the roots of nonlinear equations at first. In order to avoid the operation of high-order derivatives of function, we modify the presented methods with fourth-order convergence by using the approximants of the second derivative and third derivative, respectively. Thus, several modified two-step iterative methods are obtained for solving nonlinear equations, and the convergence of the variants is then analyzed that they are of the fourth-order convergence. Finally, numerical experiments are given to illustrate the practicability of the suggested variants. Henceforth, the variants with fourth-order convergence have been considered as the imperative improvements to find the roots of nonlinear equations.

scholarly journals Numerical Methods for a Class of Differential Algebraic Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Lei Ren ◽  
Yuan-Ming Wang
Keyword(s):  
Numerical Methods ◽  
Constant Coefficient ◽  
Numerical Experiments ◽  
Inverse Method ◽  
Padé Approximation ◽  
Differential Algebraic Equations ◽  
Drazin Inverse ◽  
Time Varying ◽  
Algebraic Equations ◽  
Pade Approximation

This paper is devoted to the study of some efficient numerical methods for the differential algebraic equations (DAEs). At first, we propose a finite algorithm to compute the Drazin inverse of the time varying DAEs. Numerical experiments are presented by Drazin inverse and Radau IIA method, which illustrate that the precision of the Drazin inverse method is higher than the Radau IIA method. Then, Drazin inverse, Radau IIA, and Padé approximation are applied to the constant coefficient DAEs, respectively. Numerical results demonstrate that the Padé approximation is powerful for solving constant coefficient DAEs.

scholarly journals Fractional Riccati Equation and Its Applications to Rough Heston Model Using Numerical Methods

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 959 ◽  
Author(s):  
Siow W. Jeng ◽  
Adem Kilicman
Keyword(s):  
Numerical Methods ◽  
Characteristic Function ◽  
Riccati Equation ◽  
Financial Market ◽  
Approximation Method ◽  
Padé Approximation ◽  
Heston Model ◽  
Pade Approximation ◽  
Multipoint Padé Approximation ◽  
Rough Volatility

Rough volatility models are recently popularized by the need of a consistent model for the observed empirical volatility in the financial market. In this case, it has been shown that the empirical volatility in the financial market is extremely consistent with the rough volatility. Currently, fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form and therefore, we must rely on numerical methods to obtain a solution. In this paper, we will be giving a short introduction to option pricing theory (Black–Scholes model, classical Heston model and its characteristic function), an overview of the current advancements on the rough Heston model and numerical methods (fractional Adams–Bashforth–Moulton method and multipoint Padé approximation method) for solving the fractional Riccati equation. In addition, we will investigate on the performance of multipoint Padé approximation method for the small u values in D α h ( u − i / 2 , x ) as it plays a huge role in the computation for the option prices. We further confirm that the solution generated by multipoint Padé (3,3) method for the fractional Riccati equation is incredibly consistent with the solution generated by fractional Adams–Bashforth–Moulton method.

Modeling of Rails Expressed using Pade Approximation Considering Frequency Dependent Effect

2017 ◽  
Vol 137 (2) ◽  
pp. 147-153
Author(s):  
Akinori Hori ◽  
Hiroki Tanaka ◽  
Yuichiro Hayakawa ◽  
Hiroshi Shida ◽  
Keiji Kawahara ◽  
...  
Keyword(s):  
Padé Approximation ◽  
Pade Approximation ◽  
Frequency Dependent ◽  
Dependent Effect

Optimization of the Navy’s three-dimensional mine impact burial prediction simulation model, Impact35, using high-order numerical methods

2021 ◽  
pp. 154851292110286
Author(s):  
Athanasios Donas ◽  
Ioannis Famelis ◽  
Peter C Chu ◽  
George Galanis
Keyword(s):  
Numerical Analysis ◽  
Numerical Methods ◽  
Prediction Model ◽  
Simulation Model ◽  
Three Dimensional ◽  
Simulation System ◽  
High Order ◽  
Alternative Methodology ◽  
Runge Kutta ◽  
Fifth Order

The aim of this paper is to present an application of high-order numerical analysis methods to a simulation system that models the movement of a cylindrical-shaped object (mine, projectile, etc.) in a marine environment and in general in fluids with important applications in Naval operations. More specifically, an alternative methodology is proposed for the dynamics of the Navy’s three-dimensional mine impact burial prediction model, Impact35/vortex, based on the Dormand–Prince Runge–Kutta fifth-order and the singly diagonally implicit Runge–Kutta fifth-order methods. The main aim is to improve the time efficiency of the system, while keeping the deviation levels of the final results, derived from the standard and the proposed methodology, low.

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