# Maple (software)

(Redirected from Maple computer algebra system)
Developer(s) Maple interface Waterloo Maple (Maplesoft) 1982 2016 / March 2, 2016 C, Java, Maple Microsoft Windows (7, 8 and 10), Apple OS X, Linux English, Japanese, and limited support in additional languages[1] Computer algebra system, Numeric computation Proprietary commercial software www.maplesoft.com/products/maple/

Maple is a symbolic and numeric computing environment, and multi-paradigm programming language.

Developed by Maplesoft, Maple also covers other aspects of technical computing, including visualization, data analysis, matrix computation, and connectivity.

A toolbox, MapleSim, adds functionality for multidomain physical modeling and code generation

## 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 which resembles Pascal.[2] The language permits variables of lexical scope. There are also interfaces to other languages (C, C#, Fortran, Java, MATLAB, and Visual Basic). There is also an interface to Excel.

Maple supports MathML 2.0, a W3C format for representing and interpreting mathematical expressions, including their display in Web pages.[3]

### 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, 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.

### Features

• 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 function fields
• Limits, series and asymptotic expansions
• Groebner bases
• 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, image 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 interface to calculations and applications
• Tools for connecting to DLL, SQL, Java, .NET, C++, Fortran and http
• Tools for generating code for C, C#, Fortran, Java, JavaScript, Julia, Matlab, Perl, Python, R, and VisualBasic
• Tools for parallel programming

## History

The first concept of Maple arose from a meeting in November 1980 at the University of Waterloo. Researchers at the university wished to purchase a computer powerful enough to run Macsyma. Instead, it was decided that they would develop their own computer algebra system that would be able to run on lower cost computers. The first limited version appearing in December 1980 with Maple demonstrated first at conferences beginning in 1982. The name is a reference to Maple's Canadian heritage. 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 Maple. In 1988 Waterloo Maple Inc. was founded. The company’s original goal was to manage the distribution of the software. Eventually, the company evolved to have an R&D department where most of Maple's development is done today with the rest done at university research labs worldwide including: the Symbolic Computation Laboratory at the University of Waterloo and the Ontario Research Centre for Computer Algebra at the University of Western Ontario[who?].

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 1994 a special issue of a newsletter created by Maple developers called MapleTech was published.[4]

In 1999, with the release of Maple 6, Maple included some of the NAG Numerical Libraries.[5] 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;[6] improvements have been made in later versions, although the Maple 11 documentation[7] recommends the previous (“classic”) interface for users with less than 500 MB of physical memory. This classic interface is no longer being maintained.

Between the mid 1995 and 2005 Maple lost significant market share to competitors due to a weaker user interface.[8] In 2005, Maple 10 introduced a new “document mode”, as part of the standard interface. The main feature of this mode is that math is entered using two dimensional input. In 2008, Maple 12 added additional user interface features found in Mathematica, including special purpose style sheets, control of headers and footers, bracket matching, auto execution regions, command completion templates, syntax checking and auto-initialization regions. Additional features were added for making Maple easier to use as a MATLAB toolbox.[9]

Maple 13 introduced a fly-through feature for animating 3-D plots.[10]

In September 2009 Maple and Maplesoft were acquired by the Japanese software retailer Cybernet Systems.

In 2016 the Maple Workbook was introduced, which is a container format for storing Maple documents and data into a single file.

## 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, 2009
• Maple 13.01: July, 2009
• Maple 13.02: October, 2009
• Maple 14: April, 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, 2013
• Maple 17.01: July, 2013
• Maple 18: Mar, 2014
• Maple 18.01: May, 2014
• Maple 18.01a: July, 2014
• Maple 18.02: Nov, 2014
• Maple 2015: Mar, 2015
• Maple 2015.1: Nov, 2015
• Maple 2016: March 2, 2016

## Examples of Maple code

Sample imperative programming constructs:

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

${\displaystyle \int \cos \left({\frac {x}{a}}\right)dx}$.
int(cos(x/a), x);

${\displaystyle a\sin \left({\frac {x}{a}}\right)}$

### Determinant

Compute the determinant of a matrix.

M:= Matrix([[1,2,3], [a,b,c], [x,y,z]]);  # example Matrix
${\displaystyle {\begin{bmatrix}1&2&3\\a&b&c\\x&y&z\end{bmatrix}}}$
LinearAlgebra:-Determinant(M);
${\displaystyle bz-cy+3ay-2az+2xc-3xb}$

### Series expansion

series(tanh(x),x=0,15)
${\displaystyle x-{\frac {1}{3}}\,x^{3}+{\frac {2}{15}}\,x^{5}-{\frac {17}{315}}\,x^{7}}$
${\displaystyle +{\frac {62}{2835}}\,x^{9}-{\frac {1382}{155925}}\,x^{11}+{\frac {21844}{6081075}}\,x^{13}+O(x^{15})}$

### Solve equation numerically

High order polynomial equation

>f := x^53-88*x^5-3*x-5 = 0

>fsolve(f)

-1.097486315, -.5226535640, 1.099074017

### Solve equation set

>f := (sin(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 = -1.543352313, y = -1.344549481, z = -.7867142955}

### Plotting of function of single variable

Plot ${\displaystyle x\cdot \sin(x)}$ with ${\displaystyle x}$ ranging from -10 to 10
plot(x*sin(x),x=-10..10);

### Plotting of function of two variables

Plot ${\displaystyle x^{2}+y^{2}}$ with ${\displaystyle x}$ and ${\displaystyle y}$ 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
${\displaystyle f:=2\cdot k^{2}/\cosh(k\cdot (x-4\cdot k^{2}\cdot t))^{2}}$
with(plots);
animate(subs(k = .5, f), x = -30 .. 30, t = -10 .. 10, numpoints = 200, frames = 50, color = red, thickness = 3);
 2D bell soliton 3D animation of function
• animation of functions of three variables
with(plots)
animate3d(cos(t*x)*sin(3*t*y), x = -Pi .. Pi, y = -Pi .. Pi, t = 1 .. 2)

### Laplace transform

with(inttrans);
• Laplace transform
f := (1+A*t+B*t^2)*exp(c*t);
${\displaystyle (1+A\cdot t+B\cdot t^{2})\cdot e^{c\cdot t}}$
laplace(f, t, s);
${\displaystyle {\frac {1}{s-c}}+{\frac {A}{(s-c)^{2}}}+{\frac {2B}{(s-c)^{3}}}}$
• inverse Laplace transform
invlaplace(1/(s-a),s,x)
${\displaystyle e^{ax}}$

### Fourier transform

with(inttrans):
fourier(sin(x),x,w)
${\displaystyle \mathrm {I} \pi \,(\mathrm {Dirac} (w+1)-\mathrm {Dirac} (w-1))}$

### Integral equations

Find functions ${\displaystyle f}$ that satisfy the integral equation

${\displaystyle f(x)-3\int _{-1}^{1}(xy+x^{2}y^{2})f(y)dy=h(x)}$.
eqn:= f(x)-3*Int((x*y+x^2*y^2)*f(y), y=-1..1) = h(x):
intsolve(eqn,f(x));
${\displaystyle f\left(x\right)=\int _{-1}^{1}\!\left(-15\,{x}^{2}{y}^{2}-3\,xy\right)h\left(y\right){dy}+h\left(x\right)}$

## Use of the Maple engine

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

• Maple T.A., Maplesoft’s online testing suite, uses Maple to algorithmically generate questions and grade student responses.
• 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.

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).[11]
• Older versions of the mathematical editor Scientific Workplace included Maple as a computational engine, though current versions include MuPAD.