Maple (software)

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Maple is a symbolic 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, adds functionality for multidomain physical modeling and code generation.

Maple's capacity for symbolic computing include those of a general-purpose computer algebra system. 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.

OverviewEdit

Core functionalityEdit

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), which permits variables of lexical scope.<ref>Power of two Template:Webarchive Bitwise Magazine</ref> There are also interfaces to other languages (C, C#, Fortran, Java, MATLAB, 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> There is also functionality for converting expressions from traditional mathematical notation to markup suitable for the typesetting system LaTeX.

ArchitectureEdit

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, ATLAS libraries, or GMP libraries.

Different functionality in Maple requires numerical data in different formats. Symbolic expressions are stored in memory as directed acyclic graphs. The standard interface and calculator interface are written in Java.

HistoryEdit

The first concept of Maple arose from a meeting in late 1980 at the University of Waterloo.<ref name="MacCallum2018">Template:Cite journal</ref> 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).<ref name="MacCallum2018"/> A first limited version appeared after three weeks, and fuller versions entered mainstream use beginning in 1982.<ref name="Walz1998">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> By the end of 1983, over 50 universities had copies of Maple installed on their machines.Template:Citation needed

In 1984, the research group arranged with Watcom Products Inc to license and distribute the first commercially available version, Maple 3.3.<ref name="Walz1998"/> 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<ref>Two such university labs are the Symbolic Computation Group at the University of Waterloo and the Ontario Research Centre for Computer Algebra at the University of Western Ontario</ref>).

In 1989, the first graphical user interface for Maple was developed and included with version 4.3 for the Macintosh. 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.<ref>Maple V Release 2 Notes Maplesoft</ref> In 1994 a special issue of a newsletter created by Maple developers called MapleTech was published.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In 1999, with the release of Maple 6, Maple included some of the NAG Numerical Libraries.<ref>Maple 6.0 Template:Webarchive Macworld, Feb 2001</ref> In 2003, the current "standard" interface was introduced with Maple 9. This interface is primarily written in Java (although portions, such as the rules for typesetting mathematical formulae, are written in the Maple language). The Java interface was criticized for being slow;<ref>Capturing knowledge with pure maths, Scientific Computing World.</ref> improvements have been made in later versions, although the Maple 11 documentation<ref>Maple 11 Installation Guide Maplesoft</ref> 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.<ref>Interview with Gaston Gonnet, co-creator of Maple Template:Webarchive, SIAM History of Numerical Analysis and Computing, 16 March 2005</ref> 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Version historyEdit

Template:Div col

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<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

Maple 13.01: July, 2009
Maple 13.02: October, 2009
Maple 14: April 29, 2010<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

Maple 14.01: October 28, 2010
Maple 15: April 13, 2011<ref>{{#invoke:citation/CS1|citation

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Maple 15.01: June 21, 2011
Maple 16: March 28, 2012<ref>{{#invoke:citation/CS1|citation

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Maple 16.01: May 16, 2012
Maple 17: March 13, 2013<ref>{{#invoke:citation/CS1|citation

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Maple 17.01: July, 2013
Maple 18: Mar 5, 2014<ref>{{#invoke:citation/CS1|citation

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Maple 18.01: May, 2014
Maple 18.01a: July, 2014
Maple 18.02: Nov, 2014
Maple 2015.0: Mar 4, 2015<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

Maple 2015.1: Nov, 2015
Maple 2016.0: March 2, 2016<ref>{{#invoke:citation/CS1|citation

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Maple 2016.1: April 20, 2016
Maple 2016.1a: April 27, 2016
Maple 2017.0: May 25, 2017<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

Maple 2017.1: June 28, 2017
Maple 2017.2: August 2, 2017
Maple 2017.3: October 3, 2017
Maple 2018.0: March 21, 2018<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

Maple 2019.0: March 14, 2019<ref>{{#invoke:citation/CS1|citation

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Maple 2020.0: March 12, 2020<ref>{{#invoke:citation/CS1|citation

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Maple 2021.0: March 10, 2021<ref>{{#invoke:citation/CS1|citation

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Maple 2022.0: March 15, 2022<ref>{{#invoke:citation/CS1|citation

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Maple 2023.0: March 9, 2023<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref> Template:Div col end

FeaturesEdit

Features of Maple include:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Support for symbolic and numeric computation with arbitrary precision
Elementary and special mathematical function libraries
Complex numbers and interval arithmetic
Arithmetic, greatest common divisors and factorization for multivariate polynomials over the rationals, finite fields, 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 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 including model fitting, hypothesis testing, and probability distributions
Tools for data manipulation, visualization and analysis
Tools for probability and combinatoric 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 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, and document formats
Technical word processing including formula editing
Programming language supporting procedural, functional and object-oriented constructs
Tools for adding user interfaces to calculations and applications
Tools for connecting to SQL, Java, .NET, C++, Fortran and http
Tools for generating code for C, C#, Fortran, Java, JavaScript, Julia, Matlab, Perl, Python, R, and Visual Basic
Tools for parallel programming

Examples of Maple codeEdit

The following code, which computes the factorial of a nonnegative integer, is an example of an imperative programming construct within Maple:

<syntaxhighlight lang="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; </syntaxhighlight> Simple functions can also be defined using the "maps to" arrow notation: <syntaxhighlight lang="maple">

myfac := n -> product(i, i = 1..n);

</syntaxhighlight>

IntegrationEdit

Find

<math>\int\cos\left(\frac{x}{a}\right)dx</math>.

int(cos(x/a), x);

</syntaxhighlight> Output:

<math>a \sin\left(\frac{x}{a}\right)</math>

DeterminantEdit

Compute the determinant of a matrix. <syntaxhighlight lang="maple">

M := Matrix([[1,2,3], [a,b,c], [x,y,z]]);  # example Matrix

</syntaxhighlight>

<math>

 \begin{bmatrix}
   1 & 2 & 3 \\
   a & b & c \\
   x & y & z
 \end{bmatrix}

</math>

LinearAlgebra:-Determinant(M);

Series expansionEdit

<syntaxhighlight lang="maple"> series(tanh(x), x = 0, 15) </syntaxhighlight>

<math>{}+\frac{62}{2835}\,x^9-\frac{1382}{155925}\,x^{11}+\frac{21844}{6081075}\,x^{13}+\mathcal{O}\left(x^{15}\right)</math>

Solve equations numericallyEdit

The following code numerically calculates the roots of a high-order polynomial: <syntaxhighlight lang="maple">

f := x^53-88*x^5-3*x-5 = 0

fsolve(f)

-1.097486315, -.5226535640, 1.099074017

</syntaxhighlight>

The same command can also solve systems of equations: <syntaxhighlight lang="maple">

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( {f,g,h} );

{x = -2.080507182, y = -5.122547821, z = -0.9408850733}

</syntaxhighlight>

Plotting of function of single variableEdit

Plot <math>x \sin(x)</math> with Template:Mvar ranging from -10 to 10: <syntaxhighlight lang="maple">

plot(x*sin(x), x = -10..10);

</syntaxhighlight>

File:Maple1DPlot.PNG

Plotting of function of two variablesEdit

Plot <math>x^2+y^2</math> with Template:Mvar and Template:Mvar ranging from -1 to 1: <syntaxhighlight lang="maple"> plot3d(x^2+y^2, x = -1..1, y = -1..1); </syntaxhighlight>

File:Maple163DPlot.jpg

Animation of functionsEdit

Animation of function of two variables

<math>f := \frac{2k^2}{\cosh^2\left(x k - 4 k^3 t\right)}</math>

<syntaxhighlight lang="maple"> plots:-animate(subs(k = 0.5, f), x=-30..30, t=-10..10, numpoints=200, frames=50, color=red, thickness=3); </syntaxhighlight>

File:Bellsoliton2.gif

2D bell solution

Animation of functions of three variables

<syntaxhighlight lang="maple"> plots:-animate3d(cos(t*x)*sin(3*t*y), x=-Pi..Pi, y=-Pi..Pi, t=1..2); </syntaxhighlight>

File:3dsincos animation.gif

3D animation of function

Fly-through animation of 3-D plots.<ref>Using the New Fly-through Feature in Maple 13 Maplesoft</ref>

M := Matrix([[400,400,200], [100,100,-400], [1,1,1]], datatype=float[8]):
plot3d(1, x=0..2*Pi, y=0..Pi, axes=none, coords=spherical, viewpoint=[path=M]);

</syntaxhighlight>

File:Maple plot3D flythrough.gif

Maple plot3D fly-through

Laplace transformEdit

Laplace transform

<math> \left(1 + A \, t + B \, t^2\right) e^{c t}</math>

inttrans:-laplace(f, t, s);

</syntaxhighlight>

inverse Laplace transform

<syntaxhighlight lang="maple"> inttrans:-invlaplace(1/(s-a), s, x); </syntaxhighlight>

Fourier transformEdit

Fourier transform

inttrans:-fourier(sin(x), x, w)

</syntaxhighlight>

<math>\mathrm{I}\pi\,(\mathrm{Dirac}(w+1)-\mathrm{Dirac}(w-1))</math>

Integral equationsEdit

Find functions Template:Mvar that satisfy the integral equation

<math>f(x)-3\int_{-1}^1(xy+x^2y^2)f(y)dy = h(x)</math>.

<syntaxhighlight lang="maple"> eqn:= f(x)-3*Int((x*y+x^2*y^2)*f(y), y=-1..1) = h(x): intsolve(eqn,f(x)); </syntaxhighlight>

<math>f \left( x \right) =\int _{-1}^{1}\! \left( -15\,{x}^{2}{y}^{2}-3\,xy \right) h \left( y \right) {dy}+h \left( x \right)

</math>

Use of the Maple engineEdit

The Maple engine is used within several other products from Maplesoft:

MapleNet allows users to create JSP pages and Java Applets. MapleNet 12 and above also allow users to upload and work with Maple worksheets containing interactive components.
MapleSim, an engineering simulation tool.<ref>Template:Cite journal</ref>
Maple Quantum Chemistry Package from RDMChem computes and visualizes the electronic energies and properties of molecules.<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

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 contained a portion of the Maple 10 engine, but now uses MuPAD (starting with MATLAB R2007b+ release).<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

Older versions of the mathematical editor Scientific Workplace included Maple as a computational engine, though current versions include MuPAD.

ReferencesEdit

Template:Reflist

External linksEdit

Template:Sister project Template:Sister project

Maplesoft, division of Waterloo Maple, Inc. – official website

Template:Computer algebra systems Template:Numerical analysis software Template:Statistical software Template:Deep Learning Software Template:Fractal software Template:Graph Analysis Software Template:Authority control

Maple (software)

Contents

OverviewEdit

Core functionalityEdit

ArchitectureEdit

HistoryEdit

Version historyEdit

FeaturesEdit

Examples of Maple codeEdit

IntegrationEdit

DeterminantEdit

Series expansionEdit

Solve equations numericallyEdit

Plotting of function of single variableEdit

Plotting of function of two variablesEdit

Animation of functionsEdit

Laplace transformEdit

Fourier transformEdit

Integral equationsEdit

Use of the Maple engineEdit

See alsoEdit

ReferencesEdit

External linksEdit