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Maple is a
symbolic Symbolic may refer to: * Symbol, something that represents an idea, a process, or a physical entity Mathematics, logic, and computing * Symbolic computation, a scientific area concerned with computing with mathematical formulas * Symbolic dynam ...
and numeric computing environment as well as a multi-paradigm programming language. It covers several areas of technical computing, such as symbolic mathematics, numerical analysis, data processing, visualization, and others. A toolbox,
MapleSim MapleSim is a Modelica-based, multi-domain modeling and simulation tool developed by Maplesoft. MapleSim generates model equations, runs simulations, and performs analyses using the symbolic and numeric mathematical engine of Maple. Models are c ...
, adds functionality for multidomain physical modeling and code generation. Maple's capacity for symbolic computing include those of a general-purpose
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The ...
. For instance, it can manipulate mathematical expressions and find symbolic solutions to certain problems, such as those arising from ordinary and partial differential equations. Maple is developed commercially by the Canadian software company Maplesoft. The name 'Maple' is a reference to the software's
Canadian heritage The Department of Canadian Heritage, or simply Canadian Heritage (french: Patrimoine canadien), is the department of the Government of Canada that has roles and responsibilities related to initiatives that promote and support "Canadian identity ...
.


Overview


Core functionality

Users can enter mathematics in traditional mathematical notation. Custom user interfaces can also be created. There is support for numeric computations, to arbitrary precision, as well as symbolic computation and visualization. Examples of symbolic computations are given below. Maple incorporates a dynamically typed imperative-style programming language (resembling
Pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, Frenc ...
), which permits variables of lexical
scope Scope or scopes may refer to: People with the surname * Jamie Scope (born 1986), English footballer * John T. Scopes (1900–1970), central figure in the Scopes Trial regarding the teaching of evolution Arts, media, and entertainment * Cinema ...
. There are also interfaces to other languages ( C, C#, Fortran,
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
,
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
, and Visual Basic), as well as to Microsoft Excel. Maple supports MathML 2.0, which is a W3C format for representing and interpreting mathematical expressions, including their display in web pages. There is also functionality for converting expressions from traditional mathematical notation to markup suitable for the typesetting system LaTeX.


Architecture

Maple is based on a small kernel, written in C, which provides the Maple language. Most functionality is provided by libraries, which come from a variety of sources. Most of the libraries are written in the Maple language; these have viewable source code. Many numerical computations are performed by the
NAG Numerical Libraries The NAG Numerical Library is a software product developed and sold by The Numerical Algorithms Group Ltd. It is a software library of numerical analysis routines, containing more than 1,900 mathematical and statistical algorithms. Areas covered by ...
,
ATLAS An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
libraries, or GMP libraries. Different functionality in Maple requires numerical data in different formats. Symbolic expressions are stored in memory as
directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one ...
s. The standard interface and calculator interface are written in
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
.


History

The first concept of Maple arose from a meeting in late 1980 at the University of Waterloo. Researchers at the university wished to purchase a computer powerful enough to run the Lisp-based computer algebra system Macsyma. Instead, they opted to develop their own computer algebra system, named Maple, that would run on lower cost computers. Aiming for portability, they began writing Maple in programming languages from the BCPL family (initially using a subset of B and C, and later on only C). A first limited version appeared after three weeks, and fuller versions entered mainstream use beginning in 1982. By the end of 1983, over 50 universities had copies of Maple installed on their machines. In 1984, the research group arranged with Watcom Products Inc to license and distribute the first commercially available version, Maple 3.3. In 1988 Waterloo Maple Inc. (Maplesoft) was founded. The company’s original goal was to manage the distribution of the software, but eventually it grew to have its own R&D department, where most of Maple's development takes place today (the remainder being done at various university laboratories). In 1989, the first graphical user interface for Maple was developed and included with version 4.3 for the
Macintosh The Mac (known as Macintosh until 1999) is a family of personal computers designed and marketed by Apple Inc., Apple Inc. Macs are known for their ease of use and minimalist designs, and are popular among students, creative professionals, and ...
. X11 and Windows versions of the new interface followed in 1990 with Maple V. In 1992, Maple V Release 2 introduced the Maple "worksheet" that combined text, graphics, and input and typeset output. In 1994 a special issue of a newsletter created by Maple developers called ''MapleTech'' was published. In 1999, with the release of Maple 6, Maple included some of the
NAG Numerical Libraries The NAG Numerical Library is a software product developed and sold by The Numerical Algorithms Group Ltd. It is a software library of numerical analysis routines, containing more than 1,900 mathematical and statistical algorithms. Areas covered by ...
. In 2003, the current "standard" interface was introduced with Maple 9. This interface is primarily written in
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
(although portions, such as the rules for typesetting mathematical formulae, are written in the Maple language). The Java interface was criticized for being slow; improvements have been made in later versions, although the Maple 11 documentation recommends the previous ("classic") interface for users with less than 500 MB of physical memory. Between 1995 and 2005 Maple lost significant market share to competitors due to a weaker user interface. With Maple 10 in 2005, Maple introduced a new "document mode" interface, which has since been further developed across several releases. In September 2009 Maple and Maplesoft were acquired by the Japanese software retailer Cybernet Systems.


Version history

* Maple 1.0: January, 1982 * Maple 1.1: January, 1982 * Maple 2.0: May, 1982 * Maple 2.1: June, 1982 * Maple 2.15: August, 1982 * Maple 2.2: December, 1982 * Maple 3.0: May, 1983 * Maple 3.1: October, 1983 * Maple 3.2: April, 1984 * Maple 3.3: March, 1985 (first public available version) * Maple 4.0: April, 1986 * Maple 4.1: May, 1987 * Maple 4.2: December, 1987 * Maple 4.3: March, 1989 * Maple V: August, 1990 * Maple V R2: November 1992 * Maple V R3: March 15, 1994 * Maple V R4: January, 1996 * Maple V R5: November 1, 1997 * Maple 6: December 6, 1999 * Maple 7: July 1, 2001 * Maple 8: April 16, 2002 * Maple 9: June 30, 2003 * Maple 9.5: April 15, 2004 * Maple 10: May 10, 2005 * Maple 11: February 21, 2007 * Maple 11.01: July, 2007 * Maple 11.02: November, 2007 * Maple 12: May, 2008 * Maple 12.01: October, 2008 * Maple 12.02: December, 2008 * Maple 13: April 28, 2009 * Maple 13.01: July, 2009 * Maple 13.02: October, 2009 * Maple 14: April 29, 2010 * Maple 14.01: October 28, 2010 * Maple 15: April 13, 2011 * Maple 15.01: June 21, 2011 * Maple 16: March 28, 2012 * Maple 16.01: May 16, 2012 * Maple 17: March 13, 2013 * Maple 17.01: July, 2013 * Maple 18: Mar 5, 2014 * Maple 18.01: May, 2014 * Maple 18.01a: July, 2014 * Maple 18.02: Nov, 2014 * Maple 2015.0: Mar 4, 2015 * Maple 2015.1: Nov, 2015 * Maple 2016.0: March 2, 2016 * Maple 2016.1: April 20, 2016 * Maple 2016.1a: April 27, 2016 * Maple 2017.0: May 25, 2017 * Maple 2017.1: June 28, 2017 * Maple 2017.2: August 2, 2017 * Maple 2017.3: October 3, 2017 * Maple 2018.0: March 21, 2018 * Maple 2019.0: March 14, 2019 * Maple 2020.0: March 12, 2020


Features

Features of Maple include: * Support for symbolic and numeric computation with arbitrary precision * Elementary and special mathematical function libraries *
Complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s and interval arithmetic * Arithmetic, greatest common divisors and factorization for multivariate polynomials over the rationals,
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s, algebraic number fields, and algebraic function fields * Limits, series and asymptotic expansions * Gröbner basis * Differential Algebra * Matrix manipulation tools including support for sparse arrays * Mathematical function graphing and animation tools * Solvers for systems of equations, diophantine equations, ODEs, PDEs, DAEs, DDEs and recurrence relations * Numeric and symbolic tools for discrete and continuous
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
including definite and indefinite integration, definite and indefinite summation, automatic differentiation and continuous and discrete integral transforms * Constrained and unconstrained local and global optimization *
Statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
including model fitting, hypothesis testing, and probability distributions * Tools for data manipulation, visualization and analysis * Tools for probability and
combinatoric Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...
problems * Support for time-series and unit based data * Connection to online collection of financial and economic data * Tools for financial calculations including bonds, annuities, derivatives, options etc. * Calculations and simulations on random processes * Tools for text mining including regular expressions * Tools for
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
and linear and non-linear control systems * Discrete math tools including number theory * Tools for visualizing and analysing directed and undirected graphs * Group theory including permutation and finitely presented groups * Symbolic tensor functions * Import and export filters for data, image, sound,
CAD Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...
, and document formats * Technical word processing including formula editing * Programming language supporting procedural,
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
and
object-oriented Object-oriented programming (OOP) is a programming paradigm based on the concept of "objects", which can contain data and code. The data is in the form of fields (often known as attributes or ''properties''), and the code is in the form of ...
constructs * Tools for adding user interfaces to calculations and applications * Tools for connecting to SQL,
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
, .NET, C++, Fortran and http * Tools for generating code for C, C#, Fortran,
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
,
JavaScript JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of websites use JavaScript on the client side for webpage behavior, of ...
, Julia,
Matlab MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
,
Perl Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offic ...
, Python, R, and Visual Basic * Tools for parallel programming


Examples of Maple code

The following code, which computes the factorial of a nonnegative integer, is an example of an imperative programming construct within Maple: myfac := proc(n::nonnegint) local out, i; out := 1; for i from 2 to n do out := out * i end do; out end proc; Simple functions can also be defined using the "maps to" arrow notation: myfac := n -> product(i, i = 1..n);


Integration

Find :\int\cos\left(\frac\right)dx. int(cos(x/a), x); Output: :a \sin\left(\frac\right)


Determinant

Compute the determinant of a matrix. M := Matrix( 1,2,3 ,b,c ,y,z); # example Matrix : \begin 1 & 2 & 3 \\ a & b & c \\ x & y & z \end LinearAlgebra:-Determinant(M); : bz-cy+3ay-2az+2xc-3xb


Series expansion

series(tanh(x), x = 0, 15) :x-\frac\,x^3+\frac\,x^5-\frac\,x^7 :+\frac\,x^9-\frac\,x^+\frac\,x^+\mathcal\left(x^\right)


Solve equations numerically

The following code numerically calculates the roots of a high-order polynomial: f := x^53-88*x^5-3*x-5 = 0 fsolve(f) -1.097486315, -.5226535640, 1.099074017 The same command can also solve systems of equations: f := (cos(x+y))^2 + exp(x)*y+cot(x-y)+cosh(z+x) = 0: g := x^5 - 8*y = 2: h := x+3*y-77*z=55; fsolve( );


Plotting of function of single variable

Plot x \sin(x) with x ranging from -10 to 10: plot(x*sin(x), x = -10..10);


Plotting of function of two variables

Plot x^2+y^2 with x and y ranging from -1 to 1: plot3d(x^2+y^2, x = -1..1, y = -1..1);


Animation of functions

* Animation of function of two variables :f := \frac plots:-animate(subs(k = 0.5, f), x=-30..30, t=-10..10, numpoints=200, frames=50, color=red, thickness=3); * Animation of functions of three variables plots:-animate3d(cos(t*x)*sin(3*t*y), x=-Pi..Pi, y=-Pi..Pi, t=1..2); * Fly-through animation of 3-D plots. M := Matrix( 400,400,200 00,100,-400 ,1,1, datatype=float : plot3d(1, x=0..2*Pi, y=0..Pi, axes=none, coords=spherical, viewpoint= ath=M;


Laplace transform

* Laplace transform f := (1+A*t+B*t^2)*exp(c*t); : \left(1 + A \, t + B \, t^2\right) e^ inttrans:-laplace(f, t, s); :\frac+\frac+\frac * inverse Laplace transform inttrans:-invlaplace(1/(s-a), s, x); : e^


Fourier transform

* Fourier transform inttrans:-fourier(sin(x), x, w) :\mathrm\pi\,(\mathrm(w+1)-\mathrm(w-1))


Integral equations

Find functions f that satisfy the integral equation :f(x)-3\int_^1(xy+x^2y^2)f(y)dy = h(x). eqn:= f(x)-3*Int((x*y+x^2*y^2)*f(y), y=-1..1) = h(x): intsolve(eqn,f(x)); :f \left( x \right) =\int _^\! \left( -15\,^^-3\,xy \right) h \left( y \right) +h \left( x \right)


Use of the Maple engine

The Maple engine is used within several other products from Maplesoft: * Moebius, DigitalEd’s online testing suite, uses Maple to algorithmically generate questions and grade student responses. * MapleNet allows users to create JSP pages and
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
Applets. MapleNet 12 and above also allow users to upload and work with Maple worksheets containing interactive components. *
MapleSim MapleSim is a Modelica-based, multi-domain modeling and simulation tool developed by Maplesoft. MapleSim generates model equations, runs simulations, and performs analyses using the symbolic and numeric mathematical engine of Maple. Models are c ...
, an engineering simulation tool. * Maple Quantum Chemistry Package from RDMChem computes and visualizes the electronic energies and properties of molecules. Listed below are third-party commercial products that no longer use the Maple engine: * Versions of Mathcad released between 1994 and 2006 included a Maple-derived algebra engine (MKM, aka Mathsoft Kernel Maple), though subsequent versions use MuPAD. * Symbolic Math Toolbox in
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
contained a portion of the Maple 10 engine, but now uses MuPAD (starting with MATLAB R2007b+ release). * Older versions of the mathematical editor Scientific Workplace included Maple as a computational engine, though current versions include MuPAD.


See also

*
Comparison of computer algebra systems The following tables provide a comparison of computer algebra systems (CAS). A CAS is a package comprising a set of algorithms for performing symbolic manipulations on algebraic objects, a language to implement them, and an environment in which to ...
* Comparison of numerical-analysis software * Comparison of programming languages * Comparison of statistical packages * List of computer algebra systems *
List of computer simulation software The following is a list of notable computer simulation software. Free or open-source * Advanced Simulation Library - open-source hardware accelerated multiphysics simulation software. * ASCEND - open-source equation-based modelling environmen ...
* List of graphing software *
List of numerical-analysis software Listed here are notable end-user computer applications intended for use with numerical or data analysis: Numerical-software packages General-purpose computer algebra systems Interface-oriented Language-oriented Historically signific ...
* Mathematical software * SageMath (an open source algebra program)


References


External links


Maplesoft, division of Waterloo Maple, Inc.
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