DOCSLIB.ORG

Algorithms for the Symbolic Integration of Hyperlogarithms with Applications to Feynman Integrals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algorithms for the Symbolic Integration of Hyperlogarithms with Applications to Feynman Integrals Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals Erik Panzer Humboldt-Universität zu Berlin, Institut für Physik, Newton Straße 15, 12489 Berlin, Germany Abstract We provide algorithms for symbolic integration of hyperlogarithms multiplied by rational functions, which also include multiple polylogarithms when their arguments are rational functions. These algorithms are implemented in Maple and we discuss various applications. In particular, many Feynman integrals can be computed by this method. Keywords: Feynman integrals, hyperlogarithms, polylogarithms, computer algebra, symbolic integration, ε-expansions Program summary Unusual features: The complete program works strictly sym- bolically and the obtained results are exact. Whenever a Feyn- Manuscript Title: Algorithms for the symbolic integration of man graph is linearly reducible, its ε-expansion can be com- hyperlogarithms with applications to Feynman integrals puted to arbitrary order (subject only to time and memory Authors: Erik Panzer restrictions) in ε, near any even dimension of space-time and Program Title: HyperInt for arbitrarily ε-dependent powers of propagators with integer Journal Reference: values at ε = 0. Also the method is not restricted to scalar Catalogue identifier: integrals only, but arbitrary tensor integrals can be computed Licensing provisions: GNU General Public License, version 3 directly. Programming language: Maple [1], version 16 or higher Additional comments: Further applications to parametric in- Computer: Any that supports Maple tegrals, outside the application to Feynman integrals in the Operating system: Any that supports Maple Schwinger parametrization, are very likely. RAM: Highly problem dependent; from a few MiB to many Running time: Highly dependent on the particular problem GiB through the number of integrations to be performed (edges of Number of processors used: a graph), the number of remaining variables (kinematic invari- Supplementary material: Example worksheet Manual.mw ex- ants), the order in ε and the complexity of the geometry (topol- plaining most features provided and including plenty of exam- ogy of the graph). Simplest examples finish in seconds, but the ples of Feynman integral computations. time needed increases beyond any bound for sufficiently high Keywords: hyperlogarithms, polylogarithms, symbolic integra- orders in ε or graphs with many edges. tion, computer algebra, Feynman integrals, ε-expansions Classification: 4.4 Feynman diagrams, 5 Computer Algebra External routines/libraries: Subprograms used: References Nature of problem: Feynman integrals and their ε-expansions [1] Maple 161. Maplesoft, a division of Waterloo Maple Inc., Water- in dimensional regularization can be expressed in the Schwinger loo, Ontario. parametrization as multi-dimensional integrals of rational func- arXiv:1403.3385v1 [hep-th] 13 Mar 2014 tions and logarithms. Symbolic integration of such functions therefore serves a tool for the exact and direct evaluation of 1. Introduction Feynman graphs. Solution method: Symbolic integration of rational linear com- An important class of special functions is given by mul- binations of polylogarithms of rational arguments is obtained tiple polylogarithms [1, 2] using a representation in terms of hyperlogarithms. The algo- rithms exploit their iterated integral structure. k X z 1 ··· zkr Restrictions: To compute multi-dimensional integrals with this : 1 r Lin1,...,nr (z1, . , zr) = n1 nr (1.1) k1 ··· kr method, the integrand must be linearly reducible, a criterion 0<k1<···<kr we state in section 4. As a consequence, only a small subset of all Feynman integrals can be addressed. of several complex variables ~z, which generalize the tradi- tional polylogarithms Lin(z) of a single variable (the case Email address: [email protected] (Erik Panzer) 1MapleTM is a trademark of Waterloo Maple Inc. Preprint submitted to Computer Physics Communications March 14, 2014 r = 1) studied for example in [3]. Many properties and Unfortunately, none of these programs was made pub- relations of these multivalued functions can be formulated licly available so far. This might partly be due to the and studied conveniently in terms of combinatorial struc- fact that the exposition in [11] does not provide a sim- tures, which renders them suitable for symbolic algorithms ple method to obtain certain integration constants in a that can be implemented on a computer. crucial intermediate step of the algorithm. In fact, [17] re- This is mainly a consequence of their representation as sorts to numeric evaluations to guess these constants and a special class of iterated integrals [4] and our preferred a similar approach is common to many applications of the basis are the classic hyperlogarithms [5] of symbol- and coproduct-calculus [18–20]. Also within the method of differential equations [21], boundary conditions Definition 1.1. Given a finite set Σ ⊂ C containing occur that must be obtained separately, e.g through phys- 0 ∈ Σ, each word w = ω . ω ∈ Σ× (ω denotes the σ1 σn σ ical reasoning or separate computations of expansions in letter for σ ∈ Σ) defines the hyperlogarithm Lw by setting logn z certain limits. Lωn (z) := and otherwise recursively applying 0 n! We close this gap and provide a complete implementa- Z z 0 tion of the method [11] of symbolic integration using hy- dz 0 L 0 := L 0 (z ). (1.2) perlogarithms in the computer algebra system Maple [22]. ωσ w z0 − σ w 0 This program was used in [23, 24] to compute several non- We also abbreviate L := L and write σ(n) trivial Feynman integrals (including divergences and com- σ1,...,σn ωσ1 ,...,ωσn for a sequence σ, . , σ of n letters σ. plicated kinematics) and we hope that it will prove helpful in further applications by physicists and mathematicians These functions are also referred to as generalized har- alike. monic polylogarithms (with linear weights) or Goncharov Since our foremost goal was to supply a tool for the polylogarithms, since they relate to (1.1) via computation of Feynman integrals, we did not aim for a r most general computer algebra framework to handle hy- (−1) Li~n (~z) = L (nr −1) (n2−1) (n1−1) (z) (1.3) 0 ,σr ,...,0 ,σ2,0 ,σ1 perlogarithms but instead focussed on this particular ap- r plication. Still, the algorithms were implemented for very where ~n = (n1, . , nr) ∈ N and σ1, . , σr 6= 0 are such σ2 σ3 z Qr −1 general situations and may be used for different problems that ~z = , ,..., , equivalently σi = z z . σ1 σ2 σr k=i k as well. Particle physicists observed special classes of hyperloga- For completeness let us mention that while we focus rithms in results of Feynman integral calculations. The on polylogarithms as iterated integrals, the representation most famous example is the case when Σ ⊆ {−1, 0, 1}, (1.1) as nested sums opens the door to completely different called harmonic polylogarithms in [6], and practical tools strategies like [25] with implementations readily available to compute with these are available like [7, 8]. Some algo- [26, 27]. A lot of progress is being made on symbolic ma- rithms for general hyperlogarithms are also implemented nipulation of sums and we like to point out [9] and the in [9]. However, the full power of definition 1.1 can be used numerous references therein. However, we will not pursue not only to express the result of Feynman integrals, but this approach in our work. actually to compute them in the first place. Namely, the study [10] of periods of moduli spaces of 1.1. Plan of the paper curves of genus zero computed multiple integrals In section 2 we present our algorithms to symbolically Z ∞ Z ∞ Z ∞ manipulate hyperlogarithms in sufficient detail so as to fk := fk−1(zk) dzk = dz1 ··· dzk f0 (1.4) 0 0 0 make an implementation straightforward. We follow the ideas of [11] where the reader might find illuminating ex- of certain polylogarithms f0(~z) such that each of the par- amples and details. Our main original contribution is sec- tial integrals fk is a hyperlogarithm in the next integra- tion 2.5 where we solve the problem of determination of tion variable zk+1. This criterion on f0 is called linear integration constants mentioned above. reducibility in [11], where the symbolic integration algo- The Maple implementation HyperInt is presented in rithm of such functions is explained and applied theoret- section 3 and includes examples of its application to inte- ically to some finite scalar single-scale Feynman integrals gration problems and for transformations of arguments of (massless propagators). In [12] it was further shown that polylogarithms. linear reducibility is actually fulfilled for an infinite family To apply these methods to multiple integrals (1.4), we of non-trivial Feynman integrals, but still explicit results review the property of linear reducibility in section 4 and were missing. explain how to exploit the polynomial reduction algorithm This technique has then practically been used in [13] to contained in HyperInt. compute off-shell three-point functions and in [14–16] to Section 5 is devoted to our original motivation and calculate operator insertions into propagator graphs con- main application: the calculation of Feynman integrals. taining a single non-zero mass scale. A further application In HyperInt we supply a couple of commands to facili- to phase-space integrals related to Higgs production can tate the work with Feynman graphs. Detailed examples be found in [17]. 2 and demonstrations are contained in the attached Maple until 1 ¡ w = w ¡ 1 = w where 1 denotes the empty word worksheet Manual.mw. which is the identity of T (Σ). The coproduct ∆ : T (Σ) −→ To ensure correctness of our program, we performed T (Σ) ⊗ T (Σ) of interest is the deconcatenation plenty of tests. Some of them are summarized in Ap- n pendix A and provided in the file HyperTests.mpl. X ∆ (ω . ω ) := ω . ω ⊗ ω . ω . (2.3) Some combinatorial proofs were delegated to Appendix B σ1 σn σ1 σk σk+1 σn k=0 and we supply a short reference of functions and options provided by HyperInt in Appendix C.
Load more

Recommended publications
  • ECE3040: Table of Contents
    ECE3040: Table of Contents Lecture 1: Introduction and Overview Instructor contact information Navigating the course web page What is meant by numerical methods? Example problems requiring numerical methods What is Matlab and why do we need it? Lecture 2: Matlab Basics I The Matlab environment Basic arithmetic calculations Command Window control & formatting Built-in constants & elementary functions Lecture 3: Matlab Basics II The assignment operator “=” for defining variables Creating and manipulating arrays Element-by-element array operations: The “.” Operator Vector generation with linspace function and “:” (colon) operator Graphing data and functions Lecture 4: Matlab Programming I Matlab scripts (programs) Input-output: The input and disp commands The fprintf command User-defined functions Passing functions to M-files: Anonymous functions Global variables Lecture 5: Matlab Programming II Making decisions: The if-else structure The error, return, and nargin commands Loops: for and while structures Interrupting loops: The continue and break commands Lecture 6: Programming Examples Plotting piecewise functions Computing the factorial of a number Beeping Looping vs vectorization speed: tic and toc commands Passing an “anonymous function” to Matlab function Approximation of definite integrals: Riemann sums Computing cos(푥) from its power series Stopping criteria for iterative numerical methods Computing the square root Evaluating polynomials Errors and Significant Digits Lecture 7: Polynomials Polynomials
    [Show full text]
  • Package 'Mosaiccalc'
    Package ‘mosaicCalc’ May 7, 2020 Type Package Title Function-Based Numerical and Symbolic Differentiation and Antidifferentiation Description Part of the Project MOSAIC (<http://mosaic-web.org/>) suite that provides utility functions for doing calculus (differentiation and integration) in R. The main differentiation and antidifferentiation operators are described using formulas and return functions rather than numerical values. Numerical values can be obtained by evaluating these functions. Version 0.5.1 Depends R (>= 3.0.0), mosaicCore Imports methods, stats, MASS, mosaic, ggformula, magrittr, rlang Suggests testthat, knitr, rmarkdown, mosaicData Author Daniel T. Kaplan <[email protected]>, Ran- dall Pruim <[email protected]>, Nicholas J. Horton <[email protected]> Maintainer Daniel Kaplan <[email protected]> VignetteBuilder knitr License GPL (>= 2) LazyLoad yes LazyData yes URL https://github.com/ProjectMOSAIC/mosaicCalc BugReports https://github.com/ProjectMOSAIC/mosaicCalc/issues RoxygenNote 7.0.2 Encoding UTF-8 NeedsCompilation no Repository CRAN Date/Publication 2020-05-07 13:00:13 UTC 1 2 D R topics documented: connector . .2 D..............................................2 findZeros . .4 fitSpline . .5 integrateODE . .5 numD............................................6 plotFun . .7 rfun .............................................7 smoother . .7 spliner . .7 Index 8 connector Create an interpolating function going through a set of points Description This is defined in the mosaic package: See connector. D Derivative and Anti-derivative operators Description Operators for computing derivatives and anti-derivatives as functions. Usage D(formula, ..., .hstep = NULL, add.h.control = FALSE) antiD(formula, ..., lower.bound = 0, force.numeric = FALSE) makeAntiDfun(.function, .wrt, from, .tol = .Machine$double.eps^0.25) numerical_integration(f, wrt, av, args, vi.from, ciName = "C", .tol) D 3 Arguments formula A formula.
    [Show full text]
  • Calculus 141, Section 8.5 Symbolic Integration Notes by Tim Pilachowski
    Calculus 141, section 8.5 Symbolic Integration notes by Tim Pilachowski Back in my day (mid-to-late 1970s for high school and college) we had no handheld calculators, and the only electronic computers were mainframes at universities and government facilities. After learning the integration techniques that you are now learning, we were sent to Tables of Integrals, which listed results for (often) hundreds of simple to complicated integration results. Some could be evaluated using the methods of Calculus 1 and 2, others needed more esoteric methods. It was necessary to scan the Tables to find the form that was needed. If it was there, great! If not… The UMCP Physics Department posts one from the textbook they use (current as of 2017) at www.physics.umd.edu/hep/drew/IntegralTable.pdf . A similar Table of Integrals from http://www.had2know.com/academics/table-of-integrals-antiderivative-formulas.html is appended at the end of this Lecture outline. x 1 x Example F from 8.1: Evaluate e sindxx using a Table of Integrals. Answer : e ()sin x − cos x + C ∫ 2 If you remember, we had to define I, do a series of two integrations by parts, then solve for “ I =”. From the Had2Know Table of Integrals below: Identify a = and b = , then plug those values in, and voila! Now, with the development of handheld calculators and personal computers, much more is available to us, including software that will do all the work. The text mentions Derive, Maple, Mathematica, and MATLAB, and gives examples of Mathematica commands. You’ll be using MATLAB in Math 241 and several other courses.
    [Show full text]
  • Symbolic and Numerical Integration in MATLAB 1 Symbolic Integration in MATLAB
    Symbolic and Numerical Integration in MATLAB 1 Symbolic Integration in MATLAB Certain functions can be symbolically integrated in MATLAB with the int command. Example 1. Find an antiderivative for the function f(x)= x2. We can do this in (at least) three different ways. The shortest is: >>int(’xˆ2’) ans = 1/3*xˆ3 Alternatively, we can define x symbolically first, and then leave off the single quotes in the int statement. >>syms x >>int(xˆ2) ans = 1/3*xˆ3 Finally, we can first define f as an inline function, and then integrate the inline function. >>syms x >>f=inline(’xˆ2’) f = Inline function: >>f(x) = xˆ2 >>int(f(x)) ans = 1/3*xˆ3 In certain calculations, it is useful to define the antiderivative as an inline function. Given that the preceding lines of code have already been typed, we can accomplish this with the following commands: >>intoff=int(f(x)) intoff = 1/3*xˆ3 >>intoff=inline(char(intoff)) intoff = Inline function: intoff(x) = 1/3*xˆ3 1 The inline function intoff(x) has now been defined as the antiderivative of f(x)= x2. The int command can also be used with limits of integration. △ Example 2. Evaluate the integral 2 x cos xdx. Z1 In this case, we will only use the first method from Example 1, though the other two methods will work as well. We have >>int(’x*cos(x)’,1,2) ans = cos(2)+2*sin(2)-cos(1)-sin(1) >>eval(ans) ans = 0.0207 Notice that since MATLAB is working symbolically here the answer it gives is in terms of the sine and cosine of 1 and 2 radians.
    [Show full text]
  • Integration Benchmarks for Computer Algebra Systems
    The Electronic Journal of Mathematics and Technology, Volume 2, Number 3, ISSN 1933-2823 Integration on Computer Algebra Systems Kevin Charlwood e-mail: [email protected] Washburn University Topeka, KS 66621 Abstract In this article, we consider ten indefinite integrals and the ability of three computer algebra systems (CAS) to evaluate them in closed-form, appealing only to the class of real, elementary functions. Although these systems have been widely available for many years and have undergone major enhancements in new versions, it is interesting to note that there are still indefinite integrals that escape the capacity of these systems to provide antiderivatives. When this occurs, we consider what a user may do to find a solution with the aid of a CAS. 1. Introduction We will explore the use of three CAS’s in the evaluation of indefinite integrals: Maple 11, Mathematica 6.0.2 and the Texas Instruments (TI) 89 Titanium graphics calculator. We consider integrals of real elementary functions of a single real variable in the examples that follow. Students often believe that a good CAS will enable them to solve any problem when there is a known solution; these examples are useful in helping instructors show their students that this is not always the case, even in a calculus course. A CAS may provide a solution, but in a form containing special functions unfamiliar to calculus students, or too cumbersome for students to use directly, [1]. Students may ask, “Why do we need to learn integration methods when our CAS will do all the exercises in the homework?” As instructors, we want our students to come away from their mathematics experience with some capacity to make intelligent use of a CAS when needed.
    [Show full text]
  • Sequences, Series and Taylor Approximation (Ma2712b, MA2730)
    Sequences, Series and Taylor Approximation (MA2712b, MA2730) Level 2 Teaching Team Current curator: Simon Shaw November 20, 2015 Contents 0 Introduction, Overview 6 1 Taylor Polynomials 10 1.1 Lecture 1: Taylor Polynomials, Definition . .. 10 1.1.1 Reminder from Level 1 about Differentiable Functions . .. 11 1.1.2 Definition of Taylor Polynomials . 11 1.2 Lectures 2 and 3: Taylor Polynomials, Examples . ... 13 x 1.2.1 Example: Compute and plot Tnf for f(x) = e ............ 13 1.2.2 Example: Find the Maclaurin polynomials of f(x) = sin x ...... 14 2 1.2.3 Find the Maclaurin polynomial T11f for f(x) = sin(x ) ....... 15 1.2.4 QuestionsforChapter6: ErrorEstimates . 15 1.3 Lecture 4 and 5: Calculus of Taylor Polynomials . .. 17 1.3.1 GeneralResults............................... 17 1.4 Lecture 6: Various Applications of Taylor Polynomials . ... 22 1.4.1 RelativeExtrema .............................. 22 1.4.2 Limits .................................... 24 1.4.3 How to Calculate Complicated Taylor Polynomials? . 26 1.5 ExerciseSheet1................................... 29 1.5.1 ExerciseSheet1a .............................. 29 1.5.2 FeedbackforSheet1a ........................... 33 2 Real Sequences 40 2.1 Lecture 7: Definitions, Limit of a Sequence . ... 40 2.1.1 DefinitionofaSequence .......................... 40 2.1.2 LimitofaSequence............................. 41 2.1.3 Graphic Representations of Sequences . .. 43 2.2 Lecture 8: Algebra of Limits, Special Sequences . ..... 44 2.2.1 InfiniteLimits................................ 44 1 2.2.2 AlgebraofLimits.............................. 44 2.2.3 Some Standard Convergent Sequences . .. 46 2.3 Lecture 9: Bounded and Monotone Sequences . ..... 48 2.3.1 BoundedSequences............................. 48 2.3.2 Convergent Sequences and Closed Bounded Intervals . .... 48 2.4 Lecture10:MonotoneSequences .
    [Show full text]
  • Multivariable Calculus with Maxima
    Multivariable Calculus with Maxima G. Jay Kerns December 1, 2009 The following is a short guide to multivariable calculus with Maxima. It loosely follows the treatment of Stewart’s Calculus, Seventh Edition. Refer there for definitions, theorems, proofs, explanations, and exercises. The simple goal of this guide is to demonstrate how to use Maxima to solve problems in that vein. This was originally written for the students in my third semester Calculus class, but once it grew past twenty pages I thought it might be of interest to a wider audience. Here it is. I am releasing this as a FREE document, and other people are free to build on this to make it better. The source for this document is located at http://people.ysu.edu/~gkerns/maxima/ It was inspired by Maxima by Example by Edwin Woollett, A Maxima Guide for Calculus Students by Moses Glasner, and Tutorial on Maxima by unknown. I also received help from the Maxima mailing list archives and volunteer responses to my questions. Thanks to all of those individuals. Contents 1 Getting Maxima 4 1.1 How to Install imaxima for Microsoft Windowsr ................ 4 1.1.1 Install the software ............................ 5 1.1.2 Configure your system .......................... 5 2 Three Dimensional Geometry 6 2.1 Vectors and Linear Algebra ........................... 6 2.2 Lines, Planes, and Quadric Surfaces ....................... 8 2.3 Vector Valued Functions ............................. 14 2.4 Arc Length and Curvature ............................ 19 3 Functions of Several Variables 20 3.1 Partial Derivatives ................................ 23 3.2 Linear Approximation and Differentials ..................... 24 3.3 Chain Rule and Implicit Differentiation ....................
    [Show full text]
  • Highlighting Wxmaxima in Calculus
    mathematics Article Not Another Computer Algebra System: Highlighting wxMaxima in Calculus Natanael Karjanto 1,* and Husty Serviana Husain 2 1 Department of Mathematics, University College, Natural Science Campus, Sungkyunkwan University Suwon 16419, Korea 2 Department of Mathematics Education, Faculty of Mathematics and Natural Science Education, Indonesia University of Education, Bandung 40154, Indonesia; [email protected] * Correspondence: [email protected] Abstract: This article introduces and explains a computer algebra system (CAS) wxMaxima for Calculus teaching and learning at the tertiary level. The didactic reasoning behind this approach is the need to implement an element of technology into classrooms to enhance students’ understanding of Calculus concepts. For many mathematics educators who have been using CAS, this material is of great interest, particularly for secondary teachers and university instructors who plan to introduce an alternative CAS into their classrooms. By highlighting both the strengths and limitations of the software, we hope that it will stimulate further debate not only among mathematics educators and software users but also also among symbolic computation and software developers. Keywords: computer algebra system; wxMaxima; Calculus; symbolic computation Citation: Karjanto, N.; Husain, H.S. 1. Introduction Not Another Computer Algebra A computer algebra system (CAS) is a program that can solve mathematical problems System: Highlighting wxMaxima in by rearranging formulas and finding a formula that solves the problem, as opposed to Calculus. Mathematics 2021, 9, 1317. just outputting the numerical value of the result. Maxima is a full-featured open-source https://doi.org/10.3390/ CAS: the software can serve as a calculator, provide analytical expressions, and perform math9121317 symbolic manipulations.
    [Show full text]
  • Modeling and Simulation of Engineering Systems Laboratory #4
    ENGM 541 Laboratory #4 University of Alberta ENGM 541: Modeling and Simulation of Engineering Systems Laboratory #4 © M.G. Lipsett 2010 MATLAB is capable of solving sets of equations in a number of ways. This lab will explore a few of them for a small linear system. Picard’s Iteration Method and Taylor Series for Short Duration Simulation We’ve looked at two numerical methods for short duration integrations with respect to time. Each has a truncation error because of the simplification of the integration of the governing equations with respect to time by truncating the Taylor series. For a symbolic expression, M ATLAB can evaluate the first few terms of the Taylor Series using the command taylor in the Symbolic Math Toolbox (which is part of the network licence). Typing the following commands, >> syms x >> s = taylor(sin(x)) returns s = x-1/6*x^3+1/20*x^5 which we recognize to be the first three terms of the Taylor Series expansion of sin( x). Actually, the call to the function taylor with no additional arguments (optional parameters) returns the first five terms by default. In the case of sin( x), only three of the first five terms are non-zero. Picard’s Method is not a vendor-supplied routine; but there are third-parties who have developed m-file functions to calculate the integral numerically using Picard’s method. Exercise 1 Use ezplot to plot the symbolic expression s. How accurate does it appear to be near x=0? How accurate does it look farther away from x=0? Find the expression for the first twenty terms of the Taylor Series of sin( x), by typing >> s = taylor(sin(x),20) Use ezplot to plot the symbolic expression s again.
    [Show full text]
  • A Versatile Approach to Calculus and Numerical Methods
    A Versatile Approach to Calculus and Numerical Methods David Tall Mathematics Education Research Centre University of Warwick COVENTRY CV4 7AL U K Traditionally the calculus is the study of the symbolic algorithms for differentiation and integration, the relationship between them, and their use in solving problems. Only at the end of the course, when all else fails, are numerical methods introduced, such as the Newton-Raphson method of solving equations, or Simpson’s rule for calculating areas. The problem with such an approach is that it often produces students who are very well versed in the algorithms and can solve the most fiendish of symbolic problems, yet have no understanding of the meaning of what they are doing. Given the arrival of computer software which can carry out these algorithms mechanically, the question arises as to what parts of calculus need to be studied in the curriculum of the future. It is my contention that such a study can use the computer technology to produce a far more versatile approach to the subject, in which the numerical and graphical representations may be used from the outset to produce insights into the fundamental meanings, in which a wider understanding of the processes of change and growth will be possible than the narrow band of problems that can be solved by traditional symbolic methods of the calculus. The fundamental ideas are those of change, the rate of change, and the accumulation due to change. Symbolically these are represented by the function concept, the derivative and the integral. In the calculus curriculum of the future, we need to broaden our ideas of these three things to give a theory that applies not just to the beautiful abstraction that is traditional calculus, but also a theory that applies to the real world outside.
    [Show full text]
  • The Use of Deep Learning for Symbolic Integration a Review of (Lample and Charton, 2019)
    The Use of Deep Learning for Symbolic Integration A Review of (Lample and Charton, 2019) Ernest Davis Dept. of Computer Science New York University New York, NY 10012 [email protected] December 13, 2019 Abstract Lample and Charton (2019) describe a system that uses deep learning technology to compute symbolic, indefinite integrals, and to find symbolic solutions to first- and second-order ordinary differential equations, when the solutions are elementary functions. They found that, over a particular test set, the system could find solutions more successfully than sophisticated packages for symbolic mathematics such as Mathematica run with a long time-out. This is an impressive accomplishment, as far as it goes. However, the system can handle only a quite limited subset of the problems that Mathematica deals with, and the test set has significant built-in biases. Therefore the claim that this outperforms Mathematica on symbolic integration needs to be very much qualified. Lample and Charton (2019) describe a system (henceforth LC) that uses deep learning technology to compute symbolic, indefinite integrals, and to find symbolic solutions to first- and second-order ordinary differential equations, when the solutions are elementary functions (i.e. compositions of the arithmetic operators with the exponential and trigonometric functions and their inverses). They found that, over a particular test set, LC could find solutions more successfully than sophisticated packages for symbolic mathematics such as Mathematica given a long time-out. This is an impressive accomplishment; however, it is important to understand its scope and limits. We will begin by discussing the case of symbolic integration, which is simpler.
    [Show full text]
  • ON TAYLOR MODEL BASED INTEGRATION of ODES 1. Introduction. the Numerical Solution of Initial Value Problems
    ON TAYLOR MODEL BASED INTEGRATION OF ODES M. NEHER∗, K. R. JACKSON† , AND N. S. NEDIALKOV‡ Abstract. Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods are able to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a solution. Overestimation, however, is a potential drawback of verified methods. For some problems, the computed error bounds become overly pessimistic, or the integration even breaks down. The dependency problem and the wrapping effect are particular sources of overestimations in interval computations. Berz and his co-workers have developed Taylor model methods, which extend interval arithmetic with symbolic compu- tations. The latter is an effective tool for reducing both the dependency problem and the wrapping effect. By construction, Taylor model methods appear particularly suitable for integrating nonlinear ODEs. We analyze Taylor model based integration of ODEs and compare Taylor model methods with traditional enclosure methods for IVPs for ODEs. AMS subject classifications. 65G40, 65L05, 65L70. Key words. Taylor model methods, verified integration, ODEs, IVPs. 1. Introduction. The numerical solution of initial value problems (IVPs) for ODEs is one of the fundamental problems in scientific computation. Today, there are many well-established algorithms for approximate solution of IVPs. However, traditional integration methods usually provide only approxi- mate values for the solution. Precise error bounds are rarely available. The error estimates, which are sometimes delivered, are not guaranteed to be accurate and are sometimes unreliable. In contrast, reliable integration computes guaranteed bounds for the flow of an ODE, including all discretization and roundoff errors in the computation.
    [Show full text]