What is a computer algebra system? Historically, "COMPUTE" means "numerical calculation". Numerical value.

The meaning of calculation is not only the arithmetic calculation of numbers, but also other complicated calculations, such as the calculation of mathematical functions and the calculation of polynomials.

The root of, the calculation of matrix, the calculation of matrix eigenvalue, etc. An essential feature of numerical calculation is that it can't guarantee absoluteness.

Accurate, because in the process of numerical calculation, we use floating-point numbers to calculate. For simple questions, we can

It takes a calculator or computer to manually calculate complex problems with paper and pen. However, for computers,

It is almost impossible to express a floating-point number absolutely and accurately, and errors will inevitably appear in the calculation process.

Besides numerical calculation, mathematical calculation has an important branch, which we call symbolic calculation or algebraic calculation. Jane (female name)

Simply put, it is to calculate symbols representing mathematical objects. These symbols can represent integers, rational numbers, real numbers, complex numbers or

Algebraic numbers can also represent other mathematical objects, such as polynomials, rational functions, matrices, equations or other abstract mathematics.

Such as groups, rings, domains, etc. For these abstract mathematical symbols, we usually calculate them by hand, which is also the tradition of mathematicians.

However, with the development of computer technology and the in-depth study of symbolic algorithm, computers have replaced labor.

Symbolic calculation has become possible.

Since 1960s, the research field of symbolic computation has made great progress. A series of symbolic calculations

This method lays a theoretical foundation for modern computer algebra system. Well-known algorithms include: computing polynomial ideals

Gr÷obner basis algorithm, Berlekamp algorithm for polynomial decomposition, Risch algorithm for calculating rational function integral.

In 1960s, the popular computer programming languages were FORTRAN and ALGOL.

FORTRAN is still one of the standard languages in the field of numerical calculation. But FORTRAN and ALGOL are not suitable for writing symbolic computing software. LISP language appeared in the early 1960s.

No, the computing software provides a suitable language environment, so the early symbolic computing software was written in LISP language. Among them, the most

The famous symbolic computing system REDUCE was developed by Tony Hearn of Stanford University.

The interactive symbolic computing system based on LISP language was originally used for physical computing. In 1970s, an interactive symbolic computing system was developed.

Initially, the MACSYMA system developed by Joel Moses and Willian Martin of MIT was born. That's it.

The most powerful symbolic computing system of this era. Its functions include not only standard algebraic calculation, but also limit calculation and symbol.

Symbolic integration, solving equations, etc. In fact, many standard algorithms of symbolic computation were put forward by the research team of MIT.

SAC/ALDES system developed by G. Collins and R. Loos is another type of symbolic computing system.

Formerly known as PM system, it was written by G. Collins of IBM (it is a symbolic computing system for dealing with polynomials). SAC is a

A non-interactive system, which consists of modules written in ALDES (Algebraic Description) language, and has a

The converter can convert the results into FORTRAN language. By 1990, H. Hong rewrote SAC in C language.

System, forming a new SACLIB system. This system provides a complete source code of C language, which can be freely communicated internationally.

Download it online.

In 1970s, the fourth universal symbolic computing system was muMATH, which was developed by the University of Hawaii.

The first computer algebra system developed by David Stotmeyer and Albert Ritchie can run on IBM's PC.

1

Chapter 1 Introduction of Maple System

The development language it uses is a subset of LISP language called muSIMP.

In the 1980s, with the popularity of personal computers, computer algebra system also developed rapidly.

Most of the computer algebra systems introduced in this era are written in C language. Famous systems include Maple, Mathematica, DERIVE and so on. We will introduce the characteristics of maple later. Here, let's briefly introduce derivative.

And Mathematica.

DERIVE is the subsequent version of muMATH, and it is the first symbolic computing system running on PC.

It has a friendly menu-driven interface and graphical interface, and can display 2D and 3D graphics conveniently. Its only drawback is that it doesn't.

It has programming function, and it didn't provide limited programming function until the third edition of 1994 derivative came out. Now deduce

Most of its functions have been transplanted to graphic calculators made by HP and Texas.

Mathematica is a symbolic computing software developed by stephen wolfram, but the computing power of Mathematica system is not.

Chang Qiang, it has many functions and can be programmed by users themselves. Its greatest advantage is that it can be used on computers with graphical user interfaces.

Mathematica supports a dedicated notebook interface. Through the notebook interface, we can communicate with Mathematica.

The core input command can display the output result of Mathematica, display graphics, animation and play sound. Through the notebook,

We can write reports, papers and even whole books. In fact, most papers, software and magazines about Mathematica

Write it in a notebook and spread it widely on the internet. Another important feature of Mathematica is that

Through Mathlink protocol, we can connect the core of Mathematica with other high-level languages, and we

You can call Mathematica in other languages, and you can also call programs written in other languages with Mathematica.

Up to now, the languages that can be connected by Mathlink are C, Excel, Word and so on. In fact, notebooks are connected through.

Mathlink is connected with Mathematica core.

The software we introduced above is a universal symbolic computing system, and other universal symbolic computing systems include IBM.

AXIOM, developed by Thomas J. Watson Research Center, was formerly called SCRATCHPAD.

In addition to the general symbolic computing system mentioned above, there are some special symbolic computing systems in a certain field, such as

SCHOONSCHIP is used for high-energy physics calculation, and SHEEP and STENSOR are used for general relativity calculation. In the field of mathematics.

Gloria and Gap are group theory, while PARI, Cimas and Kant are number theory. In algebraic geometry and commutative algebra

CoCoA and Macaulay are commonly used systems in this field, as well as Lie which specializes in computing Lie groups.

1.2 Network Resources of Computer Algebra System

Since 1990s, with the rapid development of Internet, more and more symbolic computing systems have been developed.

Open quickly. Information about various symbolic computing systems and other mathematical software can be obtained from the Internet. Some of them.

The new symbolic computing system even provides source code. Some math softwares also have newsgroups or discussion groups, which users can use.

Exchange information with each other and answer questions. Manufacturers can also find software problems in time and make modifications. Here are some common ones.

Network resources of mathematical software, and addresses of major research institutions.

Network resources of Mathematica:

et.mcs.kent.edu

Macau 2

magma

Mupade Bine (4 * cos (x) 3, trig);

cos(3 x) + 3 cos(x)

solve an equation

It is no problem to solve simple equations with Maple. Even very complex equations, Maple can be used for numerical calculation.

Method to handle it.

& gtsolve(x^2-3*x=2,x)；

3 1 3 1

2+

2

p 17，2。 2

p 17

& gtglsys:=f2*x+3*y+z= 1，x-y-z=4，3*x+7*z=3g:

& gt solution (glsys);

..24 97 ..43

fz =

4 1 ; x =

4 1; y =

4 1 }

& gtfsolve(fx^2+y^2= 10,x^y=2g,fx,yg)；

FX = 3: 10244907 1； y = :6 122 170880}

matrix calculation

Maple also has many commands that can handle matrices and vectors, but it needs to call the linear algebra software package linalg. There's another one.

In particular, the multiplication of matrices requires a special operator &; *.

& gt use (linalg):

Warning, a new definition of specification

A new definition of warning and tracking

& gta:= matrix (,]);

& gtinverse(a)，det(a)；

A: =

. ..

23

14

. ..

2 ..

IV. 3

55

.. 12

55

three ..

, 5

& gtb:= matrix()；

8 10

Female: = (..1728019 = 2p2+10800%17+43200%13. 7680 % 13

12

. 3072 % 12 25=2 p2。 32400 15 = 2 p2+3840 23 = 2 p2+28800% 1 9+3072% 13

+23040% 12 2 1 = 2 p2+ 14400% 12 17 = 2 p2。 1 1520 % 1 1 1) .（

（.. 1 1520 1 1 + 1024 13 . 14400 9 . 10800 7) % 13

+ (7680 23=2 p2。 1 1520 19 = 2 p2+2 1600 15 = 2 p2)% 12

+(..7680 12 + 34560 10 + 64800 8) % 1)

% 1 := x。

1 p2 pπ

2

& gtevalf (normal (f))；);

6:(..:4532958 122 109 x 2。 : 1 1253 13 130 109+: 1054 184360 109 x 3+:5353835473 109 x)

((2:x . 2:506628274)

（..: 1097 168700 109 x 2+:8958248690 109 x .: 1356288866 10 10))

chart

The most commonly used drawing commands are plot and plot3d. The following examples illustrate the methods used in these two commands.

& gtplot(sin(x)*exp( 1)^(-x/7),x=0..4 * Pi)；

-0.4-0.200 . 20 . 40 . 60 . 82468 10 12x & gt； plot3d(sin(x)*exp( 1)^y,x=0..2*Pi，y=0..Pi，axes = boxed)；

20- 100 1020

Maple leaf programming

Maple can not only calculate mathematical expressions, but also program. His programming language and other structured programming.

The language is very similar.

10 Chapter 1 Introduction of Maple System

& gtf:=proc(x::nonnegint)

& gt option memory;

& gt if x=0, then 0

& gtelif x= 1 and then 1

& gtelse f(x- 1)+f(x-2) end if

& gt end process:

& gtf(40)；

102334 155

Interactive use of 1.4 Maple system

Maple's window environment provides an advanced workspace interface, and its extended mathematical functions are simple and easy to use, so users can use it in their

Show mathematical ideas, create complex technical reports, and give full play to the functions of Maple.

Figure 1. 1: Window environment of maple trees

Xiang Feng's toolbar

Content toolbar, which also contains an area for entering and editing text.

Title of section c

Enter D Maple, and the prompt is \> ",which is displayed in red.

Interactive use of 1.4 maple leaf system 1 1

The output of E Maple is the result of executing the Maple command, which is usually displayed in blue.

A set of Maple commands and their outputs

G maple leaf's workspace

H part composed of workspace elements

The scope of the first section: indicated by big square brackets \ [".

Default maple leaf input prompt

K symbol template contains many commonly used mathematical symbols.

L expression template

M matrix template

N-vector template

Maple workspace interface

Maple's graphical interface has the common functions of modern application software interface, and supports mouse operations, including cutting and pasting.

Function, if you are used to these usages, you will have the basic knowledge of using Maple workspace interface. Now you can.

Perform some standard operations, such as opening files, saving and printing files.

For Windows platform, just double-click the Maple icon to start Maple. Under Unix system, you can start Maple at the prompt.

Then type the xMaple command to start. When maple is started, a new workspace will be opened.

At the top of the window is a menu bar, including menu items such as File and Edit. Below the menu bar is a toolbar, some of which are used for

Shortcut buttons for frequent operations, such as file opening, saving and printing. Below the toolbar is the content indicator bar, which contains some controls.

The current task has been specified. Further down is a larger workspace, which is your workspace. At the bottom of the window is the status.

Column to display system information.

As an indispensable part of Maple user interface, the workspace is a place where users can solve problems and record their work interactively.

Integrated environment. The so-called interactive problem solving is simply to input the appropriate Maple command and get the result. In the workspace,

You can modify the command, re-execute it and get new results. In addition to the Maple command and its results, you can also add it to the document.

Many other types of information, mainly including:

Text can be added, and users can control text paragraphs character by character.

.

In a text segment, you can add mathematical expressions and Maple commands.

.

You can add hyperlinks, and when you click a specific text area with the mouse, you can jump to other places in the workspace or other documents.

.

Benzhong.

You can specify the structure of the document, including hyperlinks, sections and sections.

.

On the Windows platform, users can embed other objects, and they can embed graphics with the help of OLE 2 (Object Connection and Embedding Standard).

.

Forms and tables.

Add title

In Maple's workspace, we can not only do mathematical calculations, but also write documents. First, we can add a title to the document.

The specific steps are: move the cursor to the first line, and select the area first in the execution group of the insert menu.

Item, a new area will appear at the top. This area contains the prompt of maple leaf input, indicating that it is being input at this time.

The status of the maple headquarters. Click on the toolbar.

Button or select a text entry from the Insert menu, simply put this

This area has become a text input state, and now you can enter text. At this point, a new text selection will appear under the toolbar.

Select the toolbar, from which you can select the font format of the text, etc. If you enter the title of the article, you can enter it in text format.

12 Chapter 1 Introduction of Maple System

Select the title format from the drop-down menu. Enter the title of the book and press enter, and the system will automatically let you enter the author's name and complete the entry.

You can enter text after the name.

Add subtitle

The further processing of the document is to decompose the document into sections. The specific method is to use the mouse to select the relevant area first, and then click.

Click. Key, a small box will appear in front of the selected area, and a brace will be pulled down to enclose the selected area.

Fixed area. And insert a text area before the first command in the area. At this time, you can enter the title of the section and return.

You can also enter other explanatory text at the back of the car. If you need to start a new section, you can choose Section from the Insert menu.

You can create a new partition after this partition.

Embedded mathematical expression

Sometimes it is necessary to insert a mathematical expression into a document, such as the following paragraph:

Look at the integral. X2 sin(x. a) dxPlease note that its integrand function X2sin (x.a) depends on.

About parameter a.

The method of inserting a mathematical formula into it is: first, move the cursor to the corresponding position, and select mathematical input from the insert menu.

Item, and then enter the maple code corresponding to. x2sin (x.a) dx, that is, int (x 2 * sin (x), x). Please note that observation content refers to

The coding area in the column shows the input code, while the workspace shows the integral expression using standard mathematical symbols.

After the mathematical expression input is completed, change the input status to the text input status, and you can continue to input other texts.

Complete our document, which can be saved or printed.

Add hyperlink

In Maple system, users can open multiple workspaces at the same time and establish hyperlinks between different workspaces.

Connection method to establish contact. The way to create a hyperlink is to select a location in the workspace with the mouse, and then insert a plate.

Select a hyperlink item in the list. A dialog box will pop up, asking the user to enter the linked text and another workspace.

File name. Click OK to complete the hyperlink.

Create bookmarks

You can insert bookmarks in the workspace to find content quickly. Click on the hyperlink of the bookmark, and the maple leaf will turn immediately.

Go to the bookmark location. To create a bookmark, first move the cursor to the position where you want to insert the bookmark, and then select it from the View menu.

Edit bookmark items. Type a paragraph of text in the pop-up dialog box, such as \expr command \ as bookmark text, single.

Click the OK button to insert the bookmark. When you move the cursor anywhere in the workspace, select Bookmarks from the View menu, and then

Select expr command from the pop-up menu to jump to the location where the bookmark is inserted.

In addition, bookmarks can also be used in hyperlink mode. The specific method is as follows: first, create a bookmark according to the previous method and put the cursor on it.

Move to the location where the hyperlink is created, and select the hyperlink item from the Insert menu. Enter the linked item in the pop-up dialog box.

Text, and then add the bookmark you have created in the bookmark area, such as \expr command ",and click OK.

Hyperlink complete.

help system

Earlier, we introduced the computing and typesetting capabilities of Maple, but this can only be a brief introduction. In this book, we can't

You can describe all the commands of Maple in detail, because Maple contains thousands of commands. To understand the purpose of these commands,

In this way, you can use a reference manual that comes with Maple software, that is, Maple's help system. With the help system,

1.5 organizational structure of maple 13

In the system, you can query Maple commands and their characteristics by name or subject. In addition, users can also choose their own keywords or terms.

Quickly open a help page that contains these words. Hyperlinks are also provided in each help page so that users can read relevant information.

This page.

In the help system, Maple provides three ways to find information: by directory, by topic and by full text.

Select the directory in the menu, and the help window will become a simple directory of the help system, which users can browse through hyperlinks.

Browse the help system. This is the method of searching by directory. Through this method, we can roughly understand the basic skills of Feng V.

Yes, but it is still difficult to find a specific topic. To search by topic, choose from the help menu.

Topic search. At this time, a dialog box will pop up in the help window, where you can add the topic you need to find, and click OK.

Read the corresponding help document. If you already know the keywords you want to read, you can also visit the page directly from the workspace and read it.

Is the method entered after the Maple prompt? You can read the corresponding page after entering the car.

In most versions of Maple (the only exception is Maple V Realese 4), after entering the help system, Maple

The help browser will open, through which you can easily find the help you need.

Sometimes when solving math problems, I don't know what command Maple should use, but it's up to the math textbook.

It is reasonable to speculate that the help pages of these commands should contain some specific words, so a full-text search is needed at this time.

For example, I want to solve a differential equation, but I don't know what command to use. We can speculate that with the help of this order.

Help should include words such as solution, differential and equation. At this point, you can select the full text in the help menu.

Search, in the pop-up dialog box, enter the key words to find, such as solving the problem of equal space between people and places, and then click.

The search button tells Maple Leaf to start searching. Maple will list the matching topics, indicate the matching degree with numerical values, and use

Users can select the most interesting topic from the list.

In addition, after selecting the balloon help item from the help menu, when the mouse hovers over the button or menu, Maple

Only a short description is displayed. This is also a very useful function.

1.5 organizational structure of maple

Maple is a computer algebra system developed by Symbolic Computing Group of University of Waterloo, Canada. It can be used in various programs.

Run on computers, from supercomputers, such as Cray Y/MP, to microcomputers for desktops, such as IBM PC.

Machine. Maple can be used on a single-user operating system, such as MS-Windows.