Maple (software)

Maple
Company type	Proprietary software
Industry	Computer software
Founded	1982; 44 years ago
Founder	University of Waterloo developers
Headquarters	Waterloo, Canada
Products	Maple, MapleSim
Owner	Maplesoft (subsidiary of Cybernet Systems since 2009)
Website	www.maplesoft.com

Maple serves as both a symbolic and numeric computation platform and a versatile programming language that supports multiple paradigms. It addresses various technical computing needs such as symbolic mathematics, numerical computations, data analysis, and graphical representation. Its functionality is further enhanced by the MapleSim toolbox, which specializes in multidomain physical system modeling and automated code generation.

Functioning as a comprehensive computer algebra system, Maple is equipped to manipulate complex mathematical expressions and derive symbolic solutions for challenges including ordinary and partial differential equations.

The software is commercially produced by the Canadian company Maplesoft, with its name paying homage to its Canadian roots.

Fundamental Features

Mathematical input can be entered using conventional notation, and users have the option to design personalized interfaces. Maple accommodates high-precision numerical calculations alongside symbolic processing and graphical visualization.

Its programming language is dynamically typed and imperative, bearing similarity to Pascal, with support for lexically scoped variables. Maple provides integration capabilities with several programming environments such as C, C#, Fortran, Java, MATLAB, Visual Basic, and Microsoft Excel.

For web representation of mathematical content, Maple supports MathML 2.0 and can translate classical mathematical notation into LaTeX format for professional typesetting.

System Design

At its core lies a compact kernel developed in C, which delivers the Maple language environment. The majority of its features are implemented through libraries primarily written in Maple itself, with source code accessible for review. Numerical computations utilize established libraries including the NAG Numerical Libraries, ATLAS, and GMP.

Various numerical data types are required by different Maple functions; symbolic data is maintained as directed acyclic graph structures. The standard command-line and calculator-style interfaces are implemented using Java.

History

Maple’s development began in late 1980 at the University of Waterloo, where developers chose to create their own computer algebra system rather than adopt existing Lisp-based solutions like Macsyma. Emphasizing portability, the initial implementation was done in BCPL-related languages before settling on C. A preliminary version was available within three weeks, with more complete releases starting in 1982. By 1983, over fifty academic institutions had adopted Maple.

In 1984, the team licensed version 3.3 to Watcom Products Inc for commercial distribution. The establishment of Waterloo Maple Inc. (later known as Maplesoft) occurred in 1988 to oversee sales and subsequently research and development, which continues to operate alongside university laboratories.

The first GUI appeared with version 4.3 on Macintosh systems in 1989; support for X11 and Windows platforms followed with Maple V in 1990. Maple V Release 2 (1992) introduced the innovative "worksheet" feature that combined text editing, graphics, and formatted input/output seamlessly. In 1994, a dedicated issue of the MapleTech newsletter was published by the developers.

By 1999, Maple 6 integrated components from the NAG Numerical Libraries. The modern standard interface was introduced with Maple 9 in 2003; primarily Java-based, it faced criticism over speed but was subsequently improved. Users constrained by memory were encouraged to use the classic interface instead.

From 1995 through 2005, Maple experienced a decline in market presence due to an inferior user interface. This was addressed with Maple 10 (2005), which launched a new "document mode" interface that has been progressively enhanced in later versions.

In September 2009, Japanese software distributor Cybernet Systems acquired both Maple and Maplesoft.

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 (initial public release)
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 1994
Maple V R4: January 1996
Maple V R5: November 1997
Maple 6: December 1999
Maple 7: July 2001
Maple 8: April 2002
Maple 9: June 2003
Maple 9.5: April 2004
Maple 10: May 2005
Maple 11: February 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 2009
Maple 13.01: July 2009
Maple 13.02: October 2009
Maple 14: April 2010
Maple 14.01: October 2010
Maple 15: April 2011
Maple 15.01: June 2011
Maple 16: March 2012
Maple16.01: May2012
Maple17: March2013
Maple17.01: July2013
Maple18: March2014
Maple18.01a,b,c: May-July-November2014
Maple2015.0,1: March & November2015
Maple2016.0,1,1a: March-April2016
Maple2017.0-3: May-October2017
Maple2018.0: March2018
Maple2019.0-March2019
Maple2020.0-March2020
Maple2021.0-March2021
Maple2022.0-March2022
Maple2023.0-March2023

Features

Maple delivers a broad range of functionalities designed for mathematical and engineering tasks, such as:

Handling both symbolic and numerical computations with arbitrary precision
Extensive collections of basic and specialized mathematical functions
Support for complex numbers, interval arithmetic, and sophisticated polynomial operations across multiple fields
Facilities for evaluating limits, series expansions, asymptotic forms, and Gröbner bases computations
Differential algebra capabilities alongside comprehensive matrix operations, including sparse matrix handling
Graphing utilities for mathematical functions with animation features
Solvers addressing systems of equations, Diophantine problems, ordinary and partial differential equations, differential-algebraic equations, delay differential equations, and recurrence relations
Tools for numeric and symbolic calculus covering integration, summation, automatic differentiation, and integral transforms
Optimization methods for both constrained and unconstrained local and global problems
Statistical tools encompassing model fitting, hypothesis testing, and probability distribution analysis
Data processing functions with visualization options, time-series analytics, and support for unit-based data types
Access to online financial and economic datasets
Financial computation modules for bonds, annuities, derivatives, options pricing, and stochastic process simulations
Text mining features incorporating regular expressions
Signal processing and control system analysis capabilities for linear and nonlinear models
Discrete mathematics utilities including number theory applications and graph visualization/analysis tools
Group theory support covering permutation groups and finitely presented groups
Symbolic tensor algebra functionality
Import/export compatibility with various data formats such as images, audio files, CAD models, and document types
Technical word processing integrated with formula editing tools
A versatile programming environment supporting procedural, functional, and object-oriented approaches
Tools for designing user interfaces to create custom computational applications
Integration options with SQL databases, Java platforms, .NET framework, C++, Fortran languages, and HTTP protocols
Multi-language code generation capabilities including C, C#, Fortran, Java, JavaScript, Julia, Matlab, Perl, Python, R, and Visual Basic
Support for parallel programming paradigms

Together these features make Maple a robust computational environment ideal for enterprises demanding sophisticated tools in finance, engineering design, data science, and software engineering development.

Examples of Maple code

Below is a demonstration of calculating the factorial of a nonnegative integer, showcasing an imperative programming style 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> Alternatively, straightforward functions can be expressed using the "maps to" arrow syntax: <syntaxhighlight lang="maple">

myfac := n -> product(i, i = 1..n);

</syntaxhighlight>

Integration

A sample of symbolic integration in Maple: <syntaxhighlight lang="maple">

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

</syntaxhighlight> Result:

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

Determinant

Calculating the determinant of a matrix example: <syntaxhighlight lang="maple">

M := Matrix([[1,2,3], [a,b,c], [x,y,z]]);
LinearAlgebra:-Determinant(M);

</syntaxhighlight> Output:

Series expansion

Creating a series expansion for the hyperbolic tangent function: <syntaxhighlight lang="maple"> series(tanh(x), x = 0, 15) </syntaxhighlight> The result includes terms up to <math>x^{13}</math>.

Numerical equation solving

Finding numerical solutions for a high-degree polynomial: <syntaxhighlight lang="maple">

f := x^53-88*x^5-3*x-5 = 0
fsolve(f)

</syntaxhighlight> Sample roots:

Solving systems of equations numerically: <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});

</syntaxhighlight> Example solution set:

Plotting functions

Graphing a single-variable function <math>x \sin(x)</math> over the interval [-10,10]: <syntaxhighlight lang="maple">

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

</syntaxhighlight>

Rendering a surface plot for a two-variable function <math>x^2 + y^2</math> with both variables ranging from -1 to 1: <syntaxhighlight lang="maple">

plot3d(x^2 + y^2, x = -1..1, y = -1..1);

</syntaxhighlight>

Function animation

Animating a function depending on two variables: <syntaxhighlight lang="maple"> plots:-animate(subs(k=0.5, (2*k^2)/(cosh(x*k - 4*k^3*t)^2)), x=-30..30, t=-10..10, numpoints=200, frames=50, color=red, thickness=3); </syntaxhighlight>

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

Creating a fly-through animation of 3D plots with customized viewpoints: <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>

Laplace transform

Laplace transform

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

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

</syntaxhighlight>

Inverse Laplace transform example:

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

Fourier transform

Fourier transform

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

</syntaxhighlight>

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

Integral equations

Determine the function that satisfies the following 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>

Application of the Maple engine

Numerous products from Maplesoft operate using the Maple engine:

MapleNet facilitates the development of JSP and Java Applets, with interactive Maple worksheets available from version 12 onwards.
MapleSim, a tool designed for engineering simulations.
The Maple Quantum Chemistry Package by RDMChem, which calculates and displays molecular electronic energies and associated properties.

Previously, several third-party commercial applications incorporated the Maple engine:

Various editions of Mathcad (1994–2006) utilized a Maple-based algebraic engine (MKM), later replaced by MuPAD.
MATLAB's Symbolic Math Toolbox included components of Maple 10 but switched to MuPAD starting with release R2007b.
Earlier versions of Scientific Workplace relied on Maple as their computation core; current releases have transitioned to MuPAD.

External links

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