Maple (software)

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Maple
Maple 11 interface
Developed by	Waterloo Maple Inc. (Maplesoft)
Latest release	12 / May, 2008
OS	Cross-platform
Genre	Computer algebra system
License	Proprietary
Website	www.maplesoft.com/products/maple/

Maple is a general-purpose commercial mathematics software package. It was first developed in 1980 by the Symbolic Computation Group at the University of Waterloo in Waterloo, Ontario, Canada.

Since 1988, it has been developed and sold commercially by Waterloo Maple Inc. (also known as Maplesoft), a Canadian company also based in Waterloo, Ontario. The current version is Maple 12 which was released in May 2008. Its main competitor is Mathematica.^[1]^[2]

1 Overview
- 1.1 Core Functionality
- 1.2 Architecture
2 Origin of the Name
3 Examples of Maple code
4 History
5 Past releases
6 Use of the Maple Engine
7 Versions Available
8 See also
9 References
10 External links

[edit] Overview

[edit] Core Functionality

Users can enter mathematics in traditional mathematical notation. Custom user interfaces can also be easily created. There is support for both numeric and symbolic computation, as well as visualization. Examples of symbolic computations are given below.

Maple incorporates a dynamically typed imperative-style programming language. The language permits variables of lexical scope. There are also interfaces to other languages (C, Fortran, Java, Matlab, and Visual Basic). There is also an interface with Excel.

[edit] Architecture

Maple is based around a small kernel written in C which provides the Maple language. Most functionality is provided by libaries from a variety of sources which follow a variety of design conventions. Those written in the Maple language come with source code. Some numerical computations are written in the Maple language and performed in the kernel but many are performed in 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 while the classic interface is written in C.

[edit] Origin of the Name

The name "Maple" is not an abbreviation or acronym, but simply a reference to Maple’s Canadian heritage.

[edit] Examples of Maple code

Find $\int\cos\left(\frac{x}{a}\right)dx$ .

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

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

Compute an exact solution to the linear ordinary differential equation $\frac{d^2y}{dx^2}(x) - 3 y(x) = x$ subject to initial conditions $y(0) = 0 ,\quad \left. \frac{dy}{dx} \right|_{y=0} = 2$

dsolve( {diff(y(x),x,x) - 3*y(x) = x, y(0)=0, D(y)(0)=2}, y(x) );

Answer: $y(x)=\frac{7}{18}e^{\sqrt{3}x}\sqrt{3}-\frac{7}{18}e^{-\sqrt{3}x}\sqrt{3}-\frac{1}{3}x$

Numerically calculate the root of the equation $e^x=x^2+2\,\!$ starting at the point $x=-1\,\!$ ; evaluate the answer to 75 decimal digits.

evalf[75](RootOf(exp(x)=x^2+2,x,-1));

Answer: $1.31907367685736535441789910952084846442196678082549766925608900490512707635$

Compute the determinant of a matrix.

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

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

with(LinearAlgebra):Determinant(M);

Answer: $b z - c y + 3 a y - 2 a z + 2 x c - 3 x b$

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);

Solve the system of partial differential equations

${\frac {\partial }{\partial x}}v \left( x,t \right) =-u \left( x,t \right) v \left( x,t \right)$

${\frac {\partial }{\partial t}}v \left( x,t \right) =-v \left( x,t \right) {\frac {\partial }{\partial x}}u \left( x,t \right) +v \left( x,t \right) \left( u \left( x,t \right) \right) ^{2}$

${\frac {\partial }{\partial t}}u \left( x,t \right) +2\,u \left( x,t \right) {\frac {\partial }{ \partial x}}u \left( x,t \right) -{\frac {\partial ^{2}}{\partial {x}^{2}}}u \left( x,t \right) =0$

with $v(x,t)\neq 0$ .

eqn1:= diff(v(x, t), x) = -u(x,t)*v(x,t):
eqn2:= diff(v(x, t), t) = -v(x,t)*(diff(u(x,t), x))+v(x,t)*u(x,t)^2:
eqn3:= diff(u(x,t), t)+2*u(x,t)*(diff(u(x,t), x))-(diff(diff(u(x,t), x), x)) = 0:
pdsolve({eqn1,eqn2,eqn3,v(x,t)<>0},[u,v]): op(%);

Answer: $v \left( x,t \right) ={e^{\sqrt {{\it \_c}_{{1}}}x}}{\it \_C3 }\,{e^{{\it \_c}_{{1}}t}}{\it \_C1}+{\frac {{\it \_C3}\,{e^{{\it \_c}_ {{1}}t}}{\it \_C2}}{{e^{\sqrt {{\it \_c}_{{1}}}x}}}}, \ \ u \left( x,t \right) =-{\frac {\sqrt {{\it \_c}_{{1}}} \left( {\it \_C1}\, \left( {e^{\sqrt {{\it \_c}_{{1}}}x}} \right) ^{2}-{\it \_C2} \right) }{{\it \_C1}\, \left( {e^{\sqrt {{\it \_c}_{{1}}}x}} \right) ^{2}+{\it \_C2}} }$

Find functions $f$ that satisfy the integral equation $f(x)-3\int_{-1}^1(xy+x^2y^2)f(y)dy = h(x)$ .

eqn:= f(x)-3*Integrate((x*y+x^2*y^2)*f(y), y=-1..1) = h(x):
intsolve(eqn,f(x));

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

Sample imperative programming constructs:

myfac := proc(n)
   local out, i;
   out := 1;
   if n < 0 then 
        error "input must be nonnegative"
   else
        for i from 1 to n do
            out := out * i
        end do;
        out
   end if
end proc;

[edit] 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 more reasonably priced computers. Thus, the project began with the goal of creating a symbolic algebra system accessible to researchers and students.

The initial development of Maple proceeded very quickly, with the first limited version appearing in December 1980. Researchers tried and discarded many different ideas creating a continually evolving system. Maple was demonstrated first at conferences beginning in 1982.

By the end of 1983, over 50 universities had copies of Maple installed on their machines. Due to the large number of support and licensing requests, in 1984, the research group arranged with WATCOM Products Inc to license and distribute Maple.

In 1988, due to the increasing requests for support, 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 much of Maple’s development is done today. Significant development of Maple continues at university research labs including: the Symbolic Computation Laboratory at the University of Waterloo; the Ontario Research Centre for Computer Algebra at the University of Western Ontario; and labs at other universities worldwide.

In 1989, the first graphical user interface for Maple was developed and included with version 4.3 for the Macintosh. Prior versions of Maple included only a command line interface with two dimensional output. X11 and Windows versions of the new interface followed in 1990 with Maple V.

In 1999, with the release of Maple 6, Maple included some of the NAG Numerical Libraries, extended to arbitrary precision.

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^[3]; improvements have been made in later versions, although the Maple 11 documentation^[4] 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^[5]. But 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, so that it appears similar to formulae in a book. 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. ^[6]

[edit] Past releases

Maple 12: May, 2008
Maple 11: February 21, 2007
Maple 10: May 10, 2005
Maple 9.5: April 15, 2004
Maple 9: June 30, 2003
Maple 8: April 16, 2002
Maple 7: July 1, 2001
Maple 6: December 6, 1999
Maple V R5: November 1, 1997
Maple V R4: January, 1996
Maple V R3: March 15, 1994
Maple V R2: November 1992
Maple V: August, 1990
Maple 4.3: March, 1989
Maple 4.2: December, 1987
Maple 4.1: May, 1987
Maple 4.0: April, 1986
Maple 3.3: March, 1985 (first publicly available version)
Maple 3.2: April, 1984
Maple 3.1: October, 1983
Maple 3.0: May, 1983
Maple 2.2: December, 1982
Maple 2.15: August, 1982
Maple 2.1: June, 1982
Maple 2.0: May, 1982
Maple 1.1: January, 1982
Maple 1.0: January, 1982

[edit] Use of the Maple Engine

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 10 also allows users to upload and work with Maple worksheets containing interactive components.
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 contains a portion of the Maple 10 engine.
Older versions of the mathematical editor Scientific Notebook included Maple as a computational engine, though current versions include MuPAD.
Maple Reader, Maplesoft’s platform for DRM controlled electronic books uses the Standard Maple interface.

[edit] Versions Available

Maplesoft sells both student and professional editions of Maple, with a substantial difference in price (US$99.00 compared to US$1,895.00, respectively). Recent student editions (from version 6 onwards) have not placed computational limitations, but rather come with less printed documentation.