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New vector forms of elemental functions with Taylor series

Xin-Yuan Wu, Jian-Lin Xia
2003 Applied Mathematics and Computation  
In this paper, some new vector forms of elemental functions with Taylor series are developed, with the aid of a new set of vector calculations.  ...  In many practical applications, one often need to use different elementary function in order to approximate the solution of the problem under consideration.  ...  New vector forms of elementary function with Taylor series are derived in Theorem 2 of this paper. Theorem 2 . 2 Let f ðxÞ be any one of the elementary function with Taylor series.  ... 
doi:10.1016/s0096-3003(02)00255-2 fatcat:uo6oq6wdpbgspp3ndorzpbprjm

Convolution number

W.C. Hassenpflug
1993 Computers and Mathematics with Applications  
Generally, the convolution number represents a truncated Taylor series of a function.  ...  convolution number is a new proposed name for the sequence of numbers that constitute the coefficients of polynomials and truncated Taylor expansions of functions.  ...  Taylor Series of Functions All the elements of convolution number analysis were presented separately, which can now be used in combination for a problem whose solution in the form of a Taylor series is  ... 
doi:10.1016/0898-1221(93)90066-5 fatcat:a7wjvz3carguppioixs5tkjwuy

An improved C++ Poisson Series Processor with its applications

José Antonio López Ortí, Vicente Agost Gómez, Miguel Barreda Rochera
2020 Computational and Mathematical Methods  
This package includes the ordinary arithmetic operations with Poisson series-such as sum or product-as well as the most commonly used functions sin, cos, or exp, among others.  ...  In this work we extend the processor in order to obtain the solution to perturbed problems which solution is in the form of a Poisson series. These algorithms have been written in C++.  ...  CONFLICT OF INTEREST The authors declare that they have no conflict of interest with regard to this work. ORCID José Antonio López Ortí https://orcid.org/0000-0002-9620-6454 R E F E R E N C E S  ... 
doi:10.1002/cmm4.1143 fatcat:h5jrshiperck3c7sq4pgjdg7sa

Volume Effects on the Method of Extracting Form Factors at Zero Momentum [article]

Brian C. Tiburzi
2014 arXiv   pre-print
As a test case, we focus on the pion charge radius and show how to ascertain the finite volume effect with the aid of chiral perturbation theory.  ...  The framework developed can easily be generalized to account for modified infrared physics of other low-energy matrix elements extracted at zero momentum.  ...  Instead of computing the Taylor series coefficients directly, it is possible to approximate them by determining the quark propagator as a function of twist angle, and then numerically differentiating with  ... 
arXiv:1409.5769v2 fatcat:i2hsgjya3fhklh24odwtiyllzi

Fast Meshless Reanalysis Using Combined Approximations, Preconditioned Conjugate Gradient, and Taylor Series

Kurt A. Chipperfield, Judy M. Vance, Andrew G. Fischer
2006 AIAA Journal  
Taylor series accuracy depends greatly on the choice of the design variable, the example problem, and the method for computing the sensitivity.  ...  A method is presented that combines the meshless stress analysis method with a reanalysis technique to avoid repeating the time-consuming steps of remeshing and solving for small design changes.  ...  Chen and Sangpil Yoon for their help with the meshless implementation.  ... 
doi:10.2514/1.14170 fatcat:6fzcyv5webhw5kem4tueyavg6m

Application of Taylor series expansion and Least-squares-based lattice Boltzmann method to simulate turbulent flows

C. Shu, Y. Peng, C. F. Zhou, Y. T. Chew
2006 Journal of turbulence  
The Taylor series expansion-and least square-based lattice Boltzmann method (TLLBM) recently developed by us is a flexible lattice Boltzmann approach capable of simulating incompressible viscous flows  ...  with complex geometry.  ...  t δ + , which also has 6 elements, j i s , is the jth element of the vector is the jth element of the vector {V}.  ... 
doi:10.1080/14685240600677624 fatcat:a5urntrwczf7xdr7zb2ilc35hy

Analytic formulation of Cauchy integrals for boundaries with curvilinear geometry

D. R Steward, P. Le Grand, I. Jankovic, O. D.L Strack
2008 Proceedings of the Royal Society A  
Strings formed by the union of adjacent curvilinear elements form a large class of geometries along which Dirichlet and/or Neumann conditions may be applied.  ...  Analytic evaluation of Cauchy integrals along straight elements is presented to establish a context coalescing new developments within the existing body of knowledge.  ...  Management and Waste Water Treatment (RIZA), the Provost Office's Targeted Excellence Program at Kansas State University, the National Science Foundation (grant EPS0553722) and the United States Department of  ... 
doi:10.1098/rspa.2007.0138 fatcat:xiqlhv4n4vhlnefrqjk42kec3e

Taylor Series Based Numerical Integration Method

Petr Veigend, Gabriela Nečasová, Václav Šátek
2020 Open Computer Science  
AbstractThis article deals with a high order integration method based on the Taylor series. The paper shows many positive properties of this method on a set of technical initial value problems.  ...  The MATLAB implementation of the method is compared with state-ofthe-art MATLAB solvers.  ...  Acknowledgement: This research was financially supported by the Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project "IT4Innovations excellence in science  ... 
doi:10.1515/comp-2020-0163 fatcat:l27v7w5rfzhbpo5pqq4dzyu32e

Number System and Errors [chapter]

2018 An Introduction to Numerical Methods  
Acknowledgments We wish to thank the many persons who have helped us in the creation of this book. They are:  ...  To extract the diagonal entries of an existing matrix, the diag function is used: linspace function generates row vectors with equally spaced elements.  ...  Major new features are as follows: • A new chapter on Dynamical systems and Chaos. Preface xv Chapter 1 Introduction The Taylor Series is one of the most important tools in numerical analysis.  ... 
doi:10.1201/9781315107042-8 fatcat:dwmxotj5wrhizfgtsdxtlmjkqu

Closed-form expressions for the finite difference approximations of first and higher derivatives based on Taylor series

Ishtiaq Rasool Khan, Ryoji Ohba
1999 Journal of Computational and Applied Mathematics  
Numerical di erentiation formulas based on interpolating polynomials, operators and lozenge diagrams can be simpliÿed to one of the ÿnite di erence approximations based on Taylor series.  ...  In this paper, we have presented closed-form expressions of these approximations of arbitrary order for ÿrst and higher derivatives.  ...  A set of Taylor series obtained by n + 1 terms backward Taylor series expansion of a function f(t) can be written as F B = A B · D B + O(T n+1 ); (8) where F B is a vector of length n and A B is a n ×  ... 
doi:10.1016/s0377-0427(99)00088-6 fatcat:nwktgc2xmvgpdh7kq47zgv5jbu

Constrained Hermite Interpolation for Mesh-Free Derivative Estimation Near and On Boundaries

Robert A. McDonald, Alejandro Ramos
2011 AIAA Journal  
A Taylor Series Least Squares approach to estimating derivatives on scattered data is extended to include derivative observation or specification.  ...  It also enables a unique method of estimating derivatives in a surface using only surface data.  ...  These results, further unpublished tests, and the full potential equation solutions conducted by the authors 10 were calculated with this implementation of these techniques.  ... 
doi:10.2514/1.j051042 fatcat:crfxgsdvprgablvjalc3z7nowm

Approximating linearizations for nonlinear systems

L. R. Hunt, R. Su, G. Meyer
1986 Circuits, systems, and signal processing  
Of course, one approach is to use the usual Taylor series linearization.  ...  However, the controllability properties of both the nonlinear and linear systems depend on certain Lie brackets of the vector field under consideration.  ...  This allows us to discuss finite Volterra expansions with remainder (analogously to that of Taylor series) as in [23] . The time t will be restricted to the set [0,T].  ... 
doi:10.1007/bf01599618 fatcat:ocgepc4rtveehlc2n4knkcdetm

Aromatic Butcher Series

Hans Munthe-Kaas, Olivier Verdier
2015 Foundations of Computational Mathematics  
We also define a new class of integrators, the class of aromatic Runge-Kutta methods, that extends the class of Runge-Kutta methods, and have aromatic B-series expansion but are not B-series methods.  ...  We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series) which we call aromatic B-series  ...  Such a composition means that the elementary differential consists of two elements of the form f I , one element of the form f I j and one element of the form f I jk .  ... 
doi:10.1007/s10208-015-9245-0 fatcat:avfmqxjozjfmdgkbxmimsalhqq

Page 799 of Mathematical Reviews Vol. , Issue 81B [page]

1981 Mathematical Reviews  
It is shown that every multivalued switching function of these algebras has a Maclaurin series expansion and a Taylor series expansion.  ...  We obtain necessary and sufficient conditions for a func- tion with a given activity vector to belong to certain well-known classes of functions.  ... 

Mathematica program for extracting one-turn Lie generator map. application of TPSA

Dobrin Kaltchev
2008 Physics Procedia  
on truncated multivariate Taylor series.  ...  The Lie Algebra package LieMath, written in the Mathematica language, constructs the one-turn nonlinear map for a given lattice of optical elements. The method used is a BCH-based map concatenation.  ...  In automatic differentiation, or Truncated power series algebra (TPSA) [6] [7] [8] , the array of Taylor coefficients in the expansion of a multivariate function is computed not by symbolic differentiation  ... 
doi:10.1016/j.phpro.2008.07.113 fatcat:lmmrcgz2mvholgoh2cbmxgd7t4
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