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Maple (software)

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Template:Short description Template:Other uses Template:Infobox software

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.

Overview

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Core functionality

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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>Template:Cite web</ref> There is also functionality for converting expressions from traditional mathematical notation to markup suitable for the typesetting system LaTeX.

Architecture

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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.

History

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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">Template:Cite 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>Template:Cite 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>Template:Cite web</ref>

Version history

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  • 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>Template:Cite web</ref>
  • Maple 13.01: July, 2009
  • Maple 13.02: October, 2009
  • Maple 14: April 29, 2010<ref>Template:Cite web</ref>
  • Maple 14.01: October 28, 2010
  • Maple 15: April 13, 2011<ref>Template:Cite web</ref>
  • Maple 15.01: June 21, 2011
  • Maple 16: March 28, 2012<ref>Template:Cite web</ref>
  • Maple 16.01: May 16, 2012
  • Maple 17: March 13, 2013<ref>Template:Cite web</ref>
  • Maple 17.01: July, 2013
  • Maple 18: Mar 5, 2014<ref>Template:Cite web</ref>
  • Maple 18.01: May, 2014
  • Maple 18.01a: July, 2014
  • Maple 18.02: Nov, 2014
  • Maple 2015.0: Mar 4, 2015<ref>Template:Cite web</ref>
  • Maple 2015.1: Nov, 2015
  • Maple 2016.0: March 2, 2016<ref>Template:Cite web</ref>
  • Maple 2016.1: April 20, 2016
  • Maple 2016.1a: April 27, 2016
  • Maple 2017.0: May 25, 2017<ref>Template:Cite 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>Template:Cite web</ref>
  • Maple 2019.0: March 14, 2019<ref>Template:Cite web</ref>
  • Maple 2020.0: March 12, 2020<ref>Template:Cite web</ref>
  • Maple 2021.0: March 10, 2021<ref>Template:Cite web</ref>
  • Maple 2022.0: March 15, 2022<ref>Template:Cite web</ref>
  • Maple 2023.0: March 9, 2023<ref>Template:Cite web</ref>

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Features

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Features of Maple include:<ref>Template:Cite web</ref>

Examples of Maple code

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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>

Integration

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Find

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

<syntaxhighlight lang="maple"> 

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

</syntaxhighlight> Output:

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

Determinant

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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);
<math>bz-cy+3ay-2az+2xc-3xb</math>

Series expansion

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

<math>x-\frac{1}{3}\,x^3+\frac{2}{15}\,x^5-\frac{17}{315}\,x^7</math>
<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 numerically

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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 variable

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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 variables

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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 functions

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  • 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

<syntaxhighlight lang="maple"> 

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 transform

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<syntaxhighlight lang="maple"> f := (1+A*t+B*t^2)*exp(c*t); </syntaxhighlight>

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

<syntaxhighlight lang="maple"> 

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

</syntaxhighlight>

<math>\frac{1}{s-c}+\frac{A}{(s-c)^2}+\frac{2B}{(s-c)^3}</math>
  • inverse Laplace transform

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

<math>e^{ax}</math>

Fourier transform

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<syntaxhighlight lang="maple"> 

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

</syntaxhighlight>

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

Integral equations

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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 engine

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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>Template:Cite 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>Template:Cite web</ref>
  • Older versions of the mathematical editor Scientific Workplace included Maple as a computational engine, though current versions include MuPAD.

See also

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References

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