scholarly journals Uniform approximations of the first symmetric elliptic integral in terms of elementary functions

2021 ◽  
Vol 116 (1) ◽  
Author(s):  
Blanca Bujanda ◽  
José L. López ◽  
Pedro J. Pagola ◽  
Pablo Palacios
Keyword(s):  
Error Bounds ◽  
Elliptic Integral ◽  
Elementary Functions ◽  
Numerical Examples ◽  
Uniform Approximations

AbstractWe consider the standard symmetric elliptic integral $$R_F(x,y,z)$$ R F ( x , y , z ) for complex x, y, z. We derive convergent expansions of $$R_F(x,y,z)$$ R F ( x , y , z ) in terms of elementary functions that hold uniformly for one of the three variables x, y or z in closed subsets (possibly unbounded) of $$\mathbb {C}{\setminus }(-\infty ,0]$$ C \ ( - ∞ , 0 ] . The expansions are accompanied by error bounds. The accuracy of the expansions and their uniform features are illustrated by means of some numerical examples.

scholarly journals New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices

2019 ◽  
Vol 17 (1) ◽  
pp. 1599-1614
Author(s):  
Zhiwu Hou ◽  
Xia Jing ◽  
Lei Gao
Keyword(s):  
Linear Algebra ◽  
Error Bound ◽  
Linear Complementarity Problem ◽  
Error Bounds ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Numerical Examples ◽  
Numer Algor ◽  
Better Than

Abstract A new error bound for the linear complementarity problem (LCP) of Σ-SDD matrices is given, which depends only on the entries of the involved matrices. Numerical examples are given to show that the new bound is better than that provided by García-Esnaola and Peña [Linear Algebra Appl., 2013, 438, 1339–1446] in some cases. Based on the obtained results, we also give an error bound for the LCP of SB-matrices. It is proved that the new bound is sharper than that provided by Dai et al. [Numer. Algor., 2012, 61, 121–139] under certain assumptions.

scholarly journals Uniform asymptotic expansions for the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) with μ large

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210360
Author(s):  
T. M. Dunster
Keyword(s):  
Analytic Continuation ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Whittaker Functions ◽  
Airy Functions ◽  
Elementary Functions ◽  
Confluent Hypergeometric Functions ◽  
Uniform Asymptotic Expansions

Uniform asymptotic expansions are derived for Whittaker’s confluent hypergeometric functions M κ , μ ( z ) and W κ , μ ( z ) , as well as the numerically satisfactory companion function W − κ , μ ( z   e − π i ) . The expansions are uniformly valid for μ → ∞ , 0 ≤ κ / μ ≤ 1 − δ < 1 and 0 ≤ arg ⁡ ( z ) ≤ π . By using appropriate connection and analytic continuation formulae, these expansions can be extended to all unbounded non-zero complex z . The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.

scholarly journals Simplified error bounds for turning point expansions

2020 ◽  
pp. 1-32
Author(s):  
T. M. Dunster ◽  
A. Gil ◽  
J. Segura
Keyword(s):  
Differential Equations ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Turning Point ◽  
Linear Differential Equations ◽  
Airy Functions ◽  
Elementary Functions ◽  
Varying Coefficient ◽  
Linear Differential ◽  
High Degree

Recently, the present authors derived new asymptotic expansions for linear differential equations having a simple turning point. These involve Airy functions and slowly varying coefficient functions, and were simpler than previous approximations, in particular being computable to a high degree of accuracy. Here we present explicit error bounds for these expansions which only involve elementary functions, and thereby provide a simplification of the bounds associated with the classical expansions of Olver.

scholarly journals Weighted Block Golub-Kahan-Lanczos Algorithms for Linear Response Eigenvalue Problem

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 53 ◽  
Author(s):  
Hongxiu Zhong ◽  
Zhongming Teng ◽  
Guoliang Chen
Keyword(s):  
Eigenvalue Problem ◽  
Error Bounds ◽  
Linear Response ◽  
Numerical Instability ◽  
Lanczos Algorithm ◽  
Numerical Examples ◽  
Block Algorithm ◽  
Restart Strategy ◽  
New Algorithms

In order to solve all or some eigenvalues lied in a cluster, we propose a weighted block Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem. Error bounds of the approximations to an eigenvalue cluster, as well as their corresponding eigenspace, are established and show the advantages. A practical thick-restart strategy is applied to the block algorithm to eliminate the increasing computational and memory costs, and the numerical instability. Numerical examples illustrate the effectiveness of our new algorithms.

scholarly journals Ball convergence theorem for a Steffensen-type third-order method

2017 ◽  
Vol 51 (1) ◽  
pp. 1-14
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Nonlinear Equation ◽  
Error Bounds ◽  
Order Method ◽  
Third Order ◽  
Numerical Examples ◽  
First Derivative ◽  
Fourth Derivative ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

We present a local convergence analysis for a family of Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypothesis up to the first derivative in contrast to earlier studies such as [2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] using hypotheses up to the fourth derivative. This way the applicability of these methods is extended under weaker hypothesis. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

Three-dimensional thermo-electro-elastic field in one-dimensional hexagonal piezoelectric quasi-crystal weakened by an elliptical crack

2021 ◽  
pp. 108128652110592
Author(s):  
Yuwei Liu ◽  
Xuesong Tang ◽  
Peiliang Duan ◽  
Tao Wang ◽  
Peidong Li
Keyword(s):  
Analytical Solution ◽  
Elliptic Integral ◽  
Electric Displacement ◽  
Three Dimensional ◽  
Elastic Field ◽  
Elliptical Crack ◽  
Elementary Functions ◽  
Surface Displacements ◽  
Electric Displacement Intensity Factor ◽  
Generalized Potential Theory

In this paper, an analytical solution is developed for the problem of an infinite 1D hexagonal piezoelectric quasi-crystal medium weakened by an elliptical crack and subjected to mixed loads on the crack surfaces. The mixed loads comprise the phonon pressure, phason pressure, electric displacement, and temperature increment, and the crack surfaces can be electrically permeable or impermeable. Based on a general solution, combined with the generalized potential theory, the steady-state 3D thermo-electro-elastic field variables in the quasi-crystal are obtained in terms of elliptic integral functions and elementary functions. Several important physical quantities on the cracked plane, such as the generalized crack surface displacements, normal stresses, and stress intensity factors, are derived in closed forms. An illustrative numerical calculation verifies the presented analytical solution and shows the distribution of the 3D thermo-electro-elastic field. It is indicated that the influence of the phason field on the result is pronounced, especially for the electric field variables, and the electric permeability of crack surfaces has a significant effect on the electric displacement intensity factor at the crack tip.

scholarly journals Singularity subtraction in the numerical solution of integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 408-418 ◽  
Author(s):  
P. M. Anselone
Keyword(s):  
Numerical Solution ◽  
Numerical Integration ◽  
Integral Equations ◽  
Error Bounds ◽  
Subtraction Technique ◽  
Weakly Singular Kernel ◽  
Numerical Examples ◽  
Weakly Singular ◽  
Convergence Results ◽  
Numerical Integration Procedure

AbstractThe singularity subtraction technique described by Kantorovich and Krylov in [11] is designed to reduce or overcome the effect of a weakly singular kernel in the numerical solution of integral equations. First, the equation is rearranged in such a way that the singularity of the kernel is at least partially cancelled by the smoothness of the solution, and then numerical integration is applied. We present convergence results and error bounds under general conditions on the nature of the singularity and the numerical integration procedure. Numerical examples demonstrate the benefit of the singularity subtraction technique.

High Convergence Order Q-Step Methods for Solving Equations and Systems of Equations

2020 ◽  
pp. 102-109
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Error Bounds ◽  
Local Convergence ◽  
Not Given ◽  
Systems Of Equations ◽  
Numerical Examples ◽  
First Derivative ◽  
Solving Equations ◽  
Local Convergence Analysis

The local convergence analysis of iterative methods is important since it demonstrates the degree of diffculty for choosing initial points. In the present study, we introduce generalized multi-step high order methods for solving nonlinear equations. The local convergence analysis is given using hypotheses only on the first derivative which actually appears in the methods in contrast to earlier works using hypotheses on higher order derivatives. This way we extend the applicability of these methods. The analysis includes computable radius of convergence as well as error bounds based on Lipschitz-type conditions not given in earlier studies. Numerical examples conclude this study.

Error bounds for the Liouville–Green (or WKB) approximation

1961 ◽  
Vol 57 (4) ◽  
pp. 790-810 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Error Bounds ◽  
Bessel Functions ◽  
Approximate Solutions ◽  
Parabolic Cylinder ◽  
Real Interval ◽  
Positive Parameter ◽  
Modified Bessel Functions ◽  
Elementary Functions ◽  
Parabolic Cylinder Functions ◽  
Image Position

ABSTRACTError bounds are derived and examined for approximate solutions in terms of elementary functions of the differential equationsin which u is a positive parameter, the functions f and p are free from singularities and p does not vanish. Bounds are also obtained for the remainder terms in the asymptotic expansions of the solutions in descending powers of u. The variable x ranges over a real interval, finite or infinite or over a region of the complex plane, bounded or unbounded.Applications are made to parabolic cylinder functions of large orders, and modified Bessel functions of large orders.

scholarly journals A Theory of the Trojan Asteroids

1978 ◽  
Vol 41 ◽  
pp. 145-145
Author(s):  
B. Garfinkel
Keyword(s):  
Periodic Solution ◽  
Restricted Problem ◽  
Elliptic Integral ◽  
Mass Parameter ◽  
Small Mass ◽  
Polar Coordinates ◽  
Lagrangian Point ◽  
Elementary Functions ◽  
The Family ◽  
The Mean

AbstractThe paper constructs a long-periodic solution for the case of 1:1 resonance in the restricted problem of three bodies. The polar coordinates r and 0 appear in the formHere λ is the mean synodic longitude, m is the small mass-parameter, k is the integer nearest to the ratio ω2/ω1 of the fundamental angular frequencies of the motion, and ck is a Fourier coefficient of a certain periodic function. Only elementary functions enter r(λ) and θ(λ), while the calculation of λ(t) requires the inversion of a hyper-elliptic integral t(λ).The internal resonant terms, carrying the critical divisor D, impart to the orbit an epicyclic character, in qualitative accord with the results of the numerical integration by Deprit and Henrard (1970). Our solution is valid except in the vicinity of the singularities at D = 0 and λ = 0.The presence of the resonant terms invalidates the Brown conjecture (1911) regarding the termination of the family of the tadpoleshaped orbits at the Lagrangian point L3. However, this conjecture holds for the mean orbits defined by r = r(λ), θ = θ(λ), and it also holds in the limit as m → 0.

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