Mathematical modeling programs. Modern mathematical packages in education

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Introduction

Today, computers take on a huge share of the computational and analytical workload of the modern mathematician. Therefore, today's researchers face and, most importantly, seem solvable, completely different problems than half a century ago.

Thanks to the enormous power of computers, it becomes possible to model and study complex and dynamic systems that arise during the study of space, the search for new energy sources, the creation of new technical inventions and many other problems affecting the field of scientific and technological progress. The solution to any problem of this kind can be reduced to performing the following set of actions:

· mathematical modeling of the system;

· construction of a computational algorithm;

· carrying out calculations;

· collection and analysis of the results obtained.

Leading mathematical packages now, with minimal familiarity, easily carry out very complex analytical transformations of mathematical expressions, take derivatives, integrals, calculate limits, expand and simplify expressions, and draw graphs. Moreover, now you don’t need to study programming languages for a long time to master the mathematical capabilities of a computer. Nowadays, almost everything needed by an engineer, economist, sociologist, or statistician is implemented in mathematical packages. Such world-famous packages as Mathematica, Mathcad, MatLAB, Maple have become not only convenient computing, but also an amazingly fruitful, flexible educational environment. In my opinion, together with the Internet, these packages can bring together the efforts of many, many people, providing powerful educational initiatives. Indeed, in computer textbooks and lectures, not ordinary, but directly executable formulas are now inserted into the text, with the help of which the essence of phenomena is clearly demonstrated. They can be modified to suit your own tasks, supplemented and expanded, resulting in not only numbers, but also new analytical expressions, graphs, and tables.

The use of computer mathematical packages allows you to:

· expand the range real applications;

· for visual analysis, build graphs of complex functions and surfaces, with the help of which, for example, solutions of ODEs are evaluated, which greatly facilitates their analysis;

· combine professional orientation, scientific character, systematicity, clarity, interactivity, interdisciplinary connections when solving ODEs;

· instantly exchange information with a person with whom physical contact is impossible or difficult to achieve;

· consider more tasks by reducing the number of routine transformations;

· explore more complex models, since cumbersome calculations can be carried out using appropriate computer systems;

· pay more attention to the qualitative aspects of your task.

The purpose of this work is to use information technology for mathematical calculations using the Maple package as an example.

1. Study the literature on this topic.

2. Conduct a comparative analysis of modern mathematical packages: Mathematica, Maple V, MatLAB, Derive, Mathcad.

3. Use the Maple package in mathematics lessons.

4. Draw a conclusion on the work done.

1. Modern mathematical packages in education

1.1 Conceptand usemath packagesin education

Methods and forms of using computer technologies in the educational process are an urgent methodological and organizational task of every teacher, every school and university administrator.

When organizing computer support for education, two directions can be distinguished:

· development of computer programs for educational purposes, programs specifically designed for the study of a particular discipline;

· use of software developed for professional activities in the relevant field of knowledge; for most natural science disciplines these are professional mathematical packages.

Mathematical packages here refer to systems, environments, languages such as Mathematica, Maple V, MatLAB, Derive, Mathcad, as well as a family of statistical data analysis systems - such as SPSS, Statistica, Statgraphics, Stadia, etc. Modern mathematical packages are programs (software packages ), having the means to perform various numerical and analytical (symbolic) mathematical calculations, from simple arithmetic calculations to solving partial differential equations, solving optimization problems, testing statistical hypotheses, tools for constructing mathematical models and other tools necessary for carrying out a variety of technical calculations. All of them have developed scientific graphics tools, a convenient help system, as well as reporting tools. The name "professional" or "generic" is used as an alternative to the name "training package".

For many years, mathematics teachers were quite clearly divided into supporters of the use of computer programs for educational purposes ("educational packages", training programs) and those who preferred to use universal packages.

Several key points can be identified that determined a radical change in the attitude of teachers and students to the use of universal mathematical packages.

The computer has become an element of "household appliances". The modern idea of quality education includes fluency in computer technology as a necessary element and, as a result, a computer is perceived as a subject, if not the first, then certainly the second necessity. Most parents cannot imagine raising their own schoolchildren without a computer. An increasing number of students have computers at home, and increasingly it is students who initiate the use of computer technology in the educational process. They are driven not by a “gaming” interest, as we said and saw before, but by the desire to “make their life easier,” the desire to acquire professional skills useful for a future career, and the willingness to learn how to use a computer not only through special computer science classes. We can safely say that " home computer" is the most powerful factor that has changed the attitude of teachers towards the use of computers in professional activities. Their position is changing under the influence of public opinion, under the influence of the position of students, and also because many teachers also have computers at home. Hence the interest in universal packages is understandable - learn Working with ready-made software is much easier than writing programs yourself.

In the modern world, standards have been formed and consolidated in the organization of the interface of computer programs. One of the problems that arises when using universal packages is the expenditure of study time on learning the rules of working with the program (learning the interface). However, since developers of scientific mathematical software and developers of "mass consumption" packages adhere to the same standards. Thanks to this, the time spent studying the interface of a specific scientific package is reduced by using skills in working with office programs.

The struggle for the consumer, the desire to expand the circle of users, has led to the fact that, while maintaining individual characteristics, packages are getting closer, becoming so similar that the skills of working with one of them allow you to very quickly get used to working in any other. Developers of mathematical packages very quickly equip their programs with all technological innovations, quickly release versions for new platforms and operating systems, improve command languages, including the latest achievements of algorithmic languages, etc. The intellectual capabilities of packages are developing: new libraries, modules are being added, the range of tasks available for research is expanding in accordance with fashion, with the advent of new applications, new research methods, etc.

Internet is a new reality of life for a modern student and specialist. Thanks to global computer networks, the user of any common software product has the opportunity to join the global community of consumers of the same product. He will find online information about new products, the latest versions of the program, messages about detected errors, receive expert advice, talk about his findings and get acquainted with the tricks of others, learn about the literature, the range of problems being solved, often simply find a solution to a similar problem, etc. p.

Statistical packages occupy a special place. Today, mathematical statistics is by far the most in-demand math course. The data analysis methods studied here are widely used in practice. Consequently, mastery of working techniques in the environment of a universal statistical package is an element of high-quality professional education in demand in the labor market.

Math packages - tool educational activities. A university student works, his work is study. The more perfect the tools a student uses, the better results he achieves. The use of mathematical packages simplifies the preparation of laboratory reports, helps to overcome technical mathematical difficulties in solving engineering problems, expands the range of problems available for solving, and helps present the results of calculations in a visual graphical form. If already in the junior years, while studying mathematics, physics, biology, a student masters the techniques of working with a fairly powerful professional package, then he turns out to be much better prepared for solving mathematical problems in various applications. He will not be afraid of cumbersome calculations, will be ready to solve complex problems, compensating for the lack of his own knowledge by using the intellectual capabilities of the package, has the skills to present research results in a visual graphic form, and knows how to present research results in the form of neat, meaningful reports.

Availability of universal mathematical packages and their availability on the professional software market. A significant circumstance that until recently prevented the widespread use of professional packages within universities is the high cost of professional scientific mathematical software. However, recently many companies developing and distributing programs for science have made it available for free use (including through global networks) previous versions of their programs, widely use a system of discounts for educational institutions, and distribute demo or short-lived versions for free. Publicly available, freely distributed versions of packages contain basic computational and graphical tools and, therefore, are quite suitable for use in the educational process (the modernization of mathematical packages is carried out mainly in the direction of expanding the range of tasks available for professional research by adding more and more subtle computational methods, expanding the capabilities of command languages and adapting to the latest advances in information technology). On the other hand, the use of high-quality software helps to intensify research activities and allows students to be more involved in scientific work, which, as is known, improves the chances of scientific groups when distributing grants, and, therefore, allows them to subsequently find funds for the purchase of more modern licensed software.

Availability of documentation and reference literature on mathematical packages. If relatively recently there was practically no literature on packages in Russian, now new versions, new packages and various user manuals for them appear almost simultaneously. It is difficult to find a package that does not include two or three books in Russian.

It should be noted that developers willingly provide authors with proprietary documentation and the latest versions of packages for their work. In addition, almost all developers maintain servers on which they post descriptions latest news, information about detected errors, extended references for working with the package, descriptions of examples of solving typical problems, and, almost always, information about users in an academic environment with addresses, descriptions of experience and examples of use in education. It can be stated that today reference literature on mathematical packages is publicly available - any user who wants to get acquainted with a particular package and learn how to work with it has the opportunity to receive help that suits his needs. personal requests and qualifications.

1.2 Comparative analysis of Au mathematical packagestoCad, MatLab, Maple, Mathematica

The analysis consists of a table that lists the functionality of the programs. It is divided into functional sections mathematical, graphical, functionality and in the programming environment, section data import/export, possibilities of use in various operating systems, comparison of speed and information in general. To simplify the analysis of all data, we used a simple scoring system.

A score of 1 was given to those programs that contain automatic functions, a score of 0.9 was given to those applications that must be installed separately. Programs in which automatic functions are not available receive a score of 0 points. The total in each column is the total score.

As a result, all assessments were assessed as follows:

Mathematical functions 38%;

Graphics functions 10%;

Programming software 9%;

Data import/export 5%;

Operating systems 2%;

Speed comparison: 36%.

Common symbols used in various circuits

The function is built into the program

m - The function is supported by an additional module that can be downloaded for free.

$ - The function is supported by an additional module, which can be downloaded for an additional fee.

The listed functions are all based on commercial products(except Scilab), which have warranty service and support. Of course there are a huge number of free software applications, modules available, but with no guarantee of service or support. This is a very important point for several types of activities (ie for use in a bank).

Comparison of math functionality

In fact, there are many different math and statistics programs on the market that cover a huge amount of functionality.

The following table should give a brief overview of the functionality for analyzing data in numerical ways and should indicate which functions are supported by which programs, whether these functions are already implemented in the main program or whether you need an additional module.

Algebra and especially linear algebra offer basic functionality for any kind of orientable matrix work. That is, the types of optimizations widely used in the financial sector are also very useful in comparing speed.

The following speed comparison was performed on a Pentium-III with a 550 MHz processor and 384 MB RAM running under Windows XP. Since it could be expected that modern computers could solve these problems within a short time, the maximum duration for each function was limited to 10 minutes.

The speed comparison tests 18 functions that are very commonly used in mathematical models. This is necessary to interpret timing results in content with whole models as then small differences in the timings of single functions might result in timing differences of minutes to several hours. However, it is not possible to use full models for these evaluation tests as it is a job for making the model run in every math package, and also the duration would be very high.

Functions (version)

Reading data from an ASCII data card
Reading data from a database via the ODBC interface
Extracting a descriptive statistic
Loop test 5000 x 5000
3800x3800 random matrix^1000
Sorting 3,000,000 random values
FFT over 1048576 (= 2^20) random values
Triple integration
Determinant 1000x1000 random matrix
Inverse 1000x1000 random matrix
Eigenvalues 600x600 random matrix
Cholesky decomposition 1000x1000 random matrix
1000x1000 crossproduct matrix
Calculating 1,000,000 Fibonacci Numbers
Basic component factorization of a 500x500 matrix
Gamma function on a 1500x1500 random matrix
Gaussian error function on 1500x1500 random matrix
Linear regression on a 1000x1000 random matrix
Full work

* - The maximum duration of 10 minutes has been exceeded.

The total work was calculated as follows:

The best function performance result is rated as 100%; In order to calculate the results for each function I will take the best performance and divide that by the timing of the tested program (the formula will look like MINUTE(A1;A2;...)/A2) and this is displayed as a percentage. To make the final „ Full work", I will calculate the amount of the percentage and divide by the number of programs, which is again displayed as a percentage.

Features that are not supported by the program will not be evaluated.

General information about the product.

Some information such as evaluation, support, newsgroups, books, etc. have significant implications for users of mathematical or statistical software. Due to the fact that this type of information cannot be characterized objectively, you can only mention them without judgment for the final summary of the test report.

Functions (version)

Operation/Programming Processing
User Interface

Programming language (similar)			(Basic, Fortran)
Online help / Electronic. management
Add. books
Lists of frequently asked questions
Teleconferences/mailing lists
The program is archived by the software manufacturer
The program archives by external institutions

The information in this table is rated from 1 to 6 (1 is best, 6 is worst) and represents my own subjective opinion. A score of 6 usually means that something is not supported, meaning that the feature is supported really badly. A score of 1 is given to the feature that is best supported.

Miscellaneous Information: The summary should establish comparison results of speed, software environment functionality, data import/export services, and suitability for various platforms relative to comparison results of mathematical and graphical functionality. The ratio between these four tests is 38:10:9:5:2:36.

Functions (version)

Comparison of math functionality (38%)
Graphics functionality comparison (10%)
Software environment functionality (9%)
Data handling (with 5%)
Available platforms (2%)
Speed comparison (36%)
Full result

Summary: The full results of some tested programs are not the best due to a certain premium of this test message.

2. Development of programming skills among schoolchildren in the environmentMaple

2.1 Concept software development procedure libraries in the environmentMaple

The Maple package consists of a fast core, written in C, containing basic mathematical functions and commands, as well as a large number of libraries that extend its capabilities in various areas of mathematics. The libraries are composed of subroutines written in Maple's own language, specifically designed for creating symbolic computation programs. The most interesting features of the Maple system are the editing and modification of these routines, as well as the addition of libraries with routines designed to solve specific problems. They have already appeared in large numbers, and the best of them have been included in the users' Share library, distributed along with the Maple package.

The program has already turned into a powerful computing system that allows you to perform complex algebraic transformations, including over the field of complex numbers, calculate finite and infinite sums, products, limits and integrals, find the roots of polynomials, solve analytically and numerically algebraic (including transcendental) systems equations and inequalities, as well as systems of ordinary differential equations and partial differential equations. Maple includes specialized packages of routines for solving problems of analytical geometry, linear and tensor algebra, number theory, combinatorics, probability theory and mathematical statistics, group theory, numerical approximation and linear optimization (simplex method), financial mathematics, integral transformations, etc. p.

Creating a new library occurs as follows.

First of all, you need to determine the name of your library, for example mylib, and create a directory (folder) for it on disk with the given name. Procedures in Maple are associated with tables. Therefore, first you need to set a dummy table for future procedures:

> mylib:=tab1e():

mylib:= table()

Now you need to introduce your library procedures. They are specified with a double name - the library name is indicated first, and then the procedure name is indicated in square brackets. For example, let's define three simple procedures named fl, f2 and f3:

> mylib:=proc(x: Anything) sin(x)+cos(x) end:

> mylib:=proc(x:anything) sin(x)^2+cos(x)^2 end:

> mylib:=proc(x::anything) if x=0 then 1 else sin(x)/x fi end:

You can build graphs of the introduced procedures-functions. They are shown in Using the with function, you can verify that the mylib library actually contains the procedures you just added to it. A list of them should appear when calling with (mylib):

> with(mylib);

Now you need to write this library under your name to disk using the save command:

> save(mylib,`c:/ mylib.m);

Pay special attention to specifying the full file name correctly. The \ sign usually used to indicate a path in Maple language lines is used as a line continuation sign. Therefore, you must use either the double sign \\ or the sign /. In this example, the file is written to the root of drive C. It is better to place the library file in another folder (for example, in a library that already exists in the system), the full path to it is indicated.

After all this, you need to make sure that the library file is written. After that, you can immediately count it. To do this, first use the restart command to eliminate the previously entered procedure definitions:

You can use the with command to ensure that these definitions no longer exist:

> with(mylib):

Error, (in pacman:-pexports) mylib is not a package

After this, use the read command to load the library file:

> read("c:/mylib.m");

The file name must be specified according to the rules specified for the save command. If everything is done punctually, then the with command should show the presence in your library of a list of procedures fl, f2 and f3:

> with(mylib):

Finally, you can again try out the procedures that have now been introduced from the loaded library:

sin(x) + cos(x) > simplify(f2(y));

The method described above for creating your own library will suit most users. However, there is a more complex and more “advanced” way to add your own library to an existing one. To implement this, Maple has the following operations for writing to the library of procedures si, s2, ... and reading them from the files filel, file2, ...:

savelib(s1. s2, .... sn, filename)

readlib(f. file1. file2. ...)

Using the special makehelp operator, you can specify a standard help description of new procedures:

makehelp(n.f.b).

where n is the name of the topic, f is the name of the text file containing the help text (the file is prepared as a Maple document), and b is the name of the library. The libname system variable stores the name of the library file directory. To register the created certificate, you need to execute a command like:

libname:-libname. "/mylib":

Details on how to use these operators can be found in the help system. mathematical programming calculation maple

You need to be very careful when creating your own library procedures. Using them will make your Maple programs incompatible with the standard version of Maple. If you use one or two procedures, it's easier to put them in the documents that actually need them. Otherwise, you will be forced to add a library of procedures to each of your programs. It often turns out to be larger in size than the file of the document itself. It is not always practical to link a small document file to a large library, most of whose procedures are likely to of this document simply not needed. It is especially risky to modify the Maple standard library.

However, whether to go for it or not is up to each user. Of course, if you create a serious library of your procedures, then it must be written down and carefully stored. Maple comes with many libraries of useful routines compiled by users around the world, so you can add your own creations to it.

2.2 Software development of a library of procedures in the environmentMaple- as a factor in the development of programming skills

From the experience of some schools, it became known that in recent years there has been a constant reduction in teaching hours in subjects of the physics and mathematics cycle, with a simultaneous expansion of the list of issues studied. In this regard, the need arose for additional and effective study of such basic subjects as mathematics, physics and computer science, as well as other disciplines of the natural sciences. The idea of integrating these disciplines is undoubtedly very productive, since, on the one hand, it provides a basis for the study of these subjects, and on the other hand, it allows us to develop an information and mathematical culture in the learning process and instill skills in applied research. At the same time, information technologies can provide the necessary tools for this integration. In particular, the Maple computer mathematics system is considered as one of such tools.

In practice, one of the schools implemented the program “Integration of physics and mathematics education based on information technology and the Maple symbolic mathematics package.”

The program involved 10-11 grades of information technology and physics and mathematics profiles. The study of the capabilities of the Maple symbolic mathematics package and its subsequent application was of an applied nature: students in the physics and mathematics class expanded and deepened their knowledge of mathematics, were given the opportunity to visually represent various mathematical situations, and the information technology classes acquired useful professional skills as programmers and computer operators . During the period of implementation of the concept of specialized education at the senior level, the introduction into the process of teaching computer science and information technology of such systems and programs that enable students to reveal their mental and creative abilities, acquire basic professional skills and determine the course of their future career. Students also needed to instill skills in computer modeling, which was one of the priority areas in applied sciences.

The experience of using computer mathematics both in universities and at school indicates that, of the well-known mathematical packages, Maple is optimal for educational purposes. A number of features of Maple have put it in a leading position for the implementation of educational programs: the relatively low cost of the package, a simple and understandable interface, a programming language that is closest to the language of mathematical logic, and unsurpassed graphic capabilities. All these features make it possible to present a mathematical model of the object or phenomenon being studied in a visual, interactive graphic form, thereby significantly improving the quality of projects in physical and mathematical disciplines. It is important to note that the results obtained, including animation models of objects and processes, are easily exported to Web pages and text documents.

The introduction of Maple into the education system is carried out in the form of an elective course “Studying the symbolic mathematics package Maple” (11th grade), the main task of which is to create the necessary conditions for the implementation of the experiment program. The main goal of the experimental work on introducing Maple into the learning process is the self-realization of students when introducing new organizational forms of using computers, based on modern packages of symbolic mathematics, into the learning process of computer science and information technology.

Training within the framework of this experiment allows you to achieve such goals as self-realization of students and their acquisition of professional competencies, development of mathematical thinking and scientific creativity of schoolchildren, improving the quality and efficiency of the educational process, increasing students' interest in educational activities and interest in its final result, professional guidance students, professional growth of teaching staff, mastery of information technology methods, and the creation of computer tools to enhance the educational process.

While learning the symbolic mathematics package Maple, students develop practical skills in solving mathematical problems using a computer. Maple becomes their study assistant. Children learn to work under self-control: solve problems using traditional methods and check the results using Maple. The most interesting and, according to students, useful topics in the elective course program were such topics as “Two-Dimensional Graphics”, “Animation”, “Function Research”. In the process of studying the Maple application, students showed high cognitive interest and good knowledge of mathematics.

Elective course classes are conducted in various forms: frontal, individual, group. Control and monitoring of students' knowledge, skills and abilities in studying the Maple symbolic mathematics package is carried out in the form of a credit system. During the academic year, students must pass 4 credits in the main sections of the course:

Solving equations, inequalities and their systems;

2D graphics;

Researching a function and plotting a graph;

Solving geometric problems.

The final result is the project work of each student. Test work is presented in the form of Web documents.

Conclusion

Computer mathematical packages play a very significant role in reforming the teaching of mathematical disciplines in secondary and higher schools, helping to achieve such goals as self-realization of students and their acquisition of professional competencies, development of mathematical thinking and scientific creativity of schoolchildren, improving the quality and efficiency of the educational process, increasing student interest to educational activities and interest in its final result, professional guidance of students, professional growth of teaching staff, mastery of information technology methods, and the creation of computer tools to enhance the educational process.

Information support for the educational process is designed to free the student from routine work, allow him to focus on the essence of the material currently being studied, consider more examples and solve more problems, and facilitate understanding of the material through other ways of presenting the material.

The possibility of computerizing the educational process arises when the functions performed by a person can be formalized and adequately reproduced using technical means. Therefore, before starting to design the educational process, the teacher must determine the relationship between the parts that can be automated and which cannot.

The feature-rich Maple package is one of the most powerful mathematical packages available. Its capabilities cover quite a lot of areas of mathematics and can be usefully used at different levels, from teaching high school students to the level of serious scientific research. Maple is an analytical computing system for mathematical modeling.

The methodology presented in the course work for studying some algebra topics and starting analysis using the Maple package made it possible to significantly increase the efficiency of the learning process. By visually presenting the material, complex mathematical formulas and transformations become much simpler, and the process of mastering the material by high school students is much more effective.

The capabilities of the Maple package as a teaching tool in high school are very extensive and its use in educational process is a promising direction in modern secondary education.

References

1. Bozhovich, L.I. Personality and its formation in childhood. [Text] / L.I. Bozovic. - St. Petersburg: Peter, 2008.- 398 p.

2. Introduction to Maple. Math package for everyone. V.N. Govorukhin, V.G. Tsibulin, Mir, 1997. - 260 p.

3. Ershov, A.P. School informatics (concepts, state, prospects) / A.P. Ershov, G.A. Zvenigorodsky, Yu.A. Pervin // Computer Science and Education. - 1995. - No. 1. - P. 3-19.

4. Lapchik, M.P. Methods of teaching computer science [Text] / M.P. Lapchik, I.G. Semakin, E.K. Hener.- M.: Academy, 2007.- 622 p.

5. Levchenko, I.V. Program and reference materials for teaching practice in computer science: Educational and methodological. manual for pedagogical students. universities and universities [Text] / I.V. Levchenko, O.Yu. Zaslavskaya, L.M. Dergacheva.- M.: MGPU, 2006.- 123 p.

6. Sdvizhkov, O.A. Mathematics on the computer Maple 8: Textbook. manual for students and teachers of universities [Text] / O.A. Sdvizhkov.- M.: SOLON-Press, 2003.- 176 p.

7. Semakin, I.G. Informatics. 11th grade: textbook [Text] / I.G. Semakin.- M.: BINOM, Laboratory of Knowledge, 2005.- 139 p.: ill.

8. Semakin, I.G. Computer Science and ICT. Basic course: textbook for grade 9 [Electronic document] / I.G. Semakin. - (http:www.alleng.ru/edu/comp1.htm). 12/15/08.

9. Ugrinovich, N.D. Computer science and information technology: textbook for grades 10-11 [Text] / N.D. Ugrinovich.- M.: Laboratory of Basic Knowledge, 2002.- 512 p.

10. Ugrinovich, N.D. Workshop on computer science and information technology: textbook for grades 10-11 [Text] / N.D. Ugrinovich.- M.: Laboratory of Basic Knowledge, 2002.- 400 p.

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...

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Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.

Posted on http://www.allbest.ru/

Federal Agency for Education of the Russian Federation

State educational institution of higher professional education

Nakhodka Engineering and Economics Institute (branch)

Far Eastern State Technical University

DVPI named after. V.V. Kuibysheva

Control Job

By subject: "Informatics"

on topic

Mathematical packages (Matlab, Mathcad)

scientific supervisor

Miroshnik E.N.

Nakhodka 2011

Introduction

Language description

Application

Tool sets

Key Features

Comparative characteristics

Expansion of functionality

References

Introduction

One of the factors that determines the level of development of modern society and its intellectual capabilities is its equipment computer technology. The scope of computer use is currently so wide that there is no area where its use would be inappropriate.

The development of computer technology led to the creation and improvement of programming languages, and as a result, software. However, the improvement of software is associated with an increase in its complexity. Therefore, the process of developing programs becomes labor-intensive, and their modification and maintenance is difficult.

Traditional engineering activities are associated with solving a variety of calculation problems, conducting experiments, and preparing documentation. The development of modern methods and computer technology significantly changes the activity of a specialist.

One of the tasks in the area computer technology- automation of intellectual work and increasing the efficiency of scientific research - is successfully solved through the creation of universal packages, in particular mathematical ones. Modern mathematical packages (MPP), developed with the participation of professional mathematicians, use all the achievements accumulated in fundamental and applied science. On the other hand, packages created by programmers /1, 2, 3, 4/ have convenient, flexible interfaces that meet modern standards, provide the user with convenient means of entering conditions and task data, tools for visually presenting calculation results, preparation tools for competent presentation reports.

Currently, the MATLAB system of engineering and scientific calculations is widely used in universities around the world. It is an interactive environment, has a mathematical coprocessor and allows access to programs in Fortran, C and C++.

Areas of application of the MATLAB system:

Mathematics and Computing;

Algorithm development;

Computational experiment, simulation modeling;

Data analysis, research and visualization of results;

Scientific and engineering graphics;

Application development, including graphical user interface, etc.

This system performs all calculations in floating-point arithmetic, in contrast to the DERIVE, Maple, Mathematica systems, where integer representation and symbolic data processing predominate.

The MATLAB system is both an operating environment and a programming language. The user can write specialized functions and programs, which are formatted as M-files. By classifying them by type of tasks, they can be formed into application program packages (APP). Here are several systems and MATLAB software:

MATLAB for Windows - a system for engineering and scientific calculations;

MATLAB C++ Math Library - a library of MATLAB mathematical functions in C++;

The Student Edition - MATLAB version for students;

Statistics Toolbox - statistics;

Optimization Toolbox - optimization;

Partial Differential Equations Toolbox - partial differential equations;

Symbolic Math Toolbox - symbolic mathematics;

Database Toolbox - working with databases, etc.

Thus, depending on the class of problems being solved, the user loads the required operating environment, PPP and creates the necessary MATLAB configuration.

Another most famous and widely used package is MathCAD (Mathematical Computer Aided Design) from Math Soft / 2/. The first version of the MathCAD package for Doc appeared in 1986, the second (2.01) - in 1987; version 2.52 - in 1989. The package is constantly being improved. Starting with version MathCAD Plus 6.0, a built-in programming language appears. Currently, the user has versions of MathCAD 7.0, MathCAD 8.0, MathCAD 2000 for Windows, designed for performing engineering and scientific calculations.

Main advantages of the package:

1) programming in a generally accepted mathematical language allows you to overcome the language barrier between the user and the computer;

2) the package is equipped with Word tools - a similar text editor that allows you to format the text of a document without resorting to special means, and in combination with the graphics processor (drawing graphs and diagrams) allows the user to obtain a finished document during calculations;

3) versatility of the package. MathCAD can be used to solve the most complex and diverse engineering, economic, statistical and other scientific problems, i.e. there is a very wide range of potential users of the package;

4) the package is an open type system. This means that in addition to a certain set of built-in functions designed to solve typical problems, you can create numerous user functions in the package.

The use of all the richest tools and capabilities of MathCAD makes the user’s work more efficient, especially when solving various types engineering problems, including problems of applied mechanics.

1. MATLAB

Story

MATLAB as a programming language was developed by Cleve Moler in the late 1970s, when he was dean of the computer science department at the University of New Mexico. The goal of the development was to give students of the faculty the opportunity to use the Linpack and EISPACK software libraries without the need to study Fortran. Soon new language spread among other universities and was received with great interest by scientists working in the field of applied mathematics. A 1982 version written in Fortran, distributed as open source, can still be found on the Internet. Engineer John N. (Jack) Little was introduced to the language during Cleve Mowler's visit to Stanford University in 1983. Realizing that the new language had great commercial potential, he teamed up with Cleve Mowler and Steve Bangert. Together they rewrote MATLAB in C and founded The MathWorks company in 1984 to further develop it. These libraries, rewritten in C, were known for a long time under the name JACKPAC. MATLAB was originally intended for control system design (John Little's specialty), but quickly gained popularity in many other scientific and engineering fields. It was also widely used in education, particularly for teaching linear algebra and numerical methods.

Descriptionlanguage

The MATLAB language is a high-level interpreted programming language that includes matrix-based data structures, a wide range of functions, an integrated development environment, object-oriented capabilities, and interfaces to programs written in other programming languages.

Programs written in MATLAB come in two types: functions and scripts. Functions have input and output arguments, as well as their own workspace for storing intermediate calculation results and variables. Scripts use a common workspace. Both scripts and functions are not compiled into machine code and are saved as text files. It is also possible to save so-called pre-parsed programs - functions and scripts processed into a form convenient for machine execution. In general, such programs run faster than regular ones, especially if the function contains graphing commands.

The main feature of the MATLAB language is its wide capabilities for working with matrices, which the creators of the language expressed in the slogan “Think vectorized”.

Examples

Example code, part of the magic.m function, generating a magic square M for odd values of side size n:

Meshgrid(1:n);

A = mod(I+J-(n+3)/2,n);

B = mod(I+2*J-2,n);

M = n*A + B + 1;

Example code that loads a one-dimensional array A with the values of array B in reverse order (only if vector A is defined and the number of its elements is the same as the number of elements of vector B):

A(1:end) = B(end:-1:1);

Sinc function graph drawn using MATLAB

Example code plotting a sinc function:

Meshgrid(-8:.5:8);

R = sqrt(X.^2 + Y.^2);

An example of code vectorization. Code

ww = repmat(w, );

runs significantly faster than requiring less memory and arithmetic operations code

for i = 1:size(b,1)

for j = i:size(b,1)

A (i, j) = sum (b (i,:).*b (j,:).*w);

A(i, j) = A(j, i);

which does the same thing.

Application

P 1 . MathematicsAndcalculations

MATLAB provides the user with a large number (several hundred) functions for data analysis, covering almost all areas of mathematics, in particular:

§ Matrices and linear algebra - matrix algebra, linear equations, eigenvalues and vectors, singularities, matrix factorization and others.

§ Polynomials and interpolation - roots of polynomials, operations on polynomials and their differentiation, interpolation and extrapolation of curves and others.

§ Mathematical statistics and data analysis - statistical functions, statistical regression, digital filtering, fast Fourier transform and others.

§ Data processing - a set of special functions, including plotting, optimization, searching for zeros, numerical integration (in quadratures) and others.

§ Differential equations - solving differential and differential-algebraic equations, differential equations with delay, equations with restrictions, partial differential equations and others.

§ Sparse matrices are a special data class of the MATLAB package used in specialized applications.

§ Integer arithmetic - performing integer arithmetic operations in the MATLAB environment.

P. 2 Developmentalgorithms

MATLAB provides convenient tools for developing algorithms, including high-level ones, using object-oriented programming concepts. It has all the necessary tools for an integrated development environment, including a debugger and a profiler. Functions for working with entire data types make it easy to create algorithms for microcontrollers and other applications where needed.

P. 3 Visualizationdata

The MATLAB package has a large number of functions for constructing graphs, including three-dimensional ones, visual data analysis and creating animated videos.

The embedded development environment allows you to create graphical user interfaces with various controls such as buttons, input fields and others. Using the MATLAB Compiler component, these graphical interfaces can be converted into standalone applications that require the MATLAB Component Runtime library to be installed on other computers.

P. 4 Externalinterfaces

MATLAB includes various interfaces for accessing external routines written in other programming languages, data, clients and servers communicating through Component Object Model or Dynamic Data Exchange technologies, and peripheral devices that communicate directly with MATLAB. Many of these capabilities are known as the MATLAB API.

P. 5 COM

MATLAB provides access to functions that allow you to create, manipulate, and delete COM objects (both clients and servers). ActiveX technology is also supported. All COM objects belong to a special COM class of the MATLAB package. All programs that have Automation controller functions can access MATLAB as an Automation server.

P. 6 .NET

MATLAB on Microsoft Windows provides access to the .NET Framework. It is possible to load .NET assemblies and work with .NET class objects from the MATLAB environment. MATLAB version 7.11 (R2010b) supports .NET Framework versions 2.0, 3.0, 3.5, and 4.0.

P. 7 DDE

MATLAB contains functions that allow it to access other Windows applications, and for those applications to access MATLAB data, through Dynamic Data Exchange (DDE) technology. Each application that can be a DDE server has its own unique identification name. For MATLAB this name is Matlab.

P. 8 Web services

In MATLAB, it is possible to call web service methods. A special function creates a class based on the methods of the web service API.

MATLAB interacts with the web service client by accepting messages from it, processing them, and sending a response. The following technologies are supported: Simple Object Access Protocol (SOAP) and Web Services Description Language (WSDL).

P. 9 COM port

MATLAB's serial port interface provides direct access to peripheral devices such as modems, printers, and scientific equipment that connect to a computer through a serial port (COM port). The interface works by creating a special class object for the serial port. The available methods of this class allow you to read and write data to the serial port, use events and event handlers, and also write information to the computer disk in real time. This is necessary when conducting experiments, simulating real-time systems, and for other applications.

P. 10 MEX files

The MATLAB package includes an interface for interacting with external applications written in C and Fortran. This interaction is carried out through MEX files. It is possible to call routines written in C or Fortran from MATLAB as if they were built-in functions of the package. MEX files are dynamic link libraries that can be loaded and executed by the interpreter built into MATLAB. MEX procedures also have the ability to call built-in MATLAB commands.

P. 11 DLL

The MATLAB generic DLL interface allows you to call functions found in common dynamic link libraries directly from MATLAB. These functions must have a C interface.

In addition, MATLAB has the ability to access its built-in functions through a C interface, which allows you to use the package's functions in external applications written in C. This technology in MATLAB is called the C Engine.

Setstools

For MATLAB, it is possible to create special toolboxes that expand its functionality. Toolboxes are collections of functions written in MATLAB to solve a specific class of problems. Mathworks provides toolkits that are used in many areas, including the following:

§ Digital processing of signals, images and data: DSP Toolbox, Image Processing Toolbox, Wavelet Toolbox, Communication Toolbox, Filter Design Toolbox - sets of functions that allow you to solve a wide range of problems of signal processing, images, design of digital filters and communication systems.

§ Control systems: Control Systems Toolbox, µ-Analysis and Synthesis Toolbox, Robust Control Toolbox, System Identification Toolbox, LMI Control Toolbox, Model Predictive Control Toolbox, Model-Based Calibration Toolbox - sets of functions that facilitate analysis and synthesis dynamic systems, design, modeling and identification of control systems, including modern control algorithms, such as robust control, H?-control, LMN synthesis, µ-synthesis and others.

§ Financial analysis: GARCH Toolbox, Fixed-Income Toolbox, Financial Time Series Toolbox, Financial Derivatives Toolbox, Financial Toolbox, Datafeed Toolbox - sets of functions that allow you to quickly and efficiently collect, process and transmit various financial information.

§ Analysis and synthesis of geographic maps, including three-dimensional ones: Mapping Toolbox.

§ Collection and analysis of experimental data: Data Acquisition Toolbox, Image Acquisition Toolbox, Instrument Control Toolbox, Link for Code Composer Studio - sets of functions that allow you to save and process data obtained during experiments, including in real time. A wide range of scientific and engineering measurement equipment is supported.

§ Visualization and presentation of data: Virtual Reality Toolbox - allows you to create interactive worlds and visualize scientific information using technology virtual reality and VRML language.

§ Development tools: MATLAB Builder for COM, MATLAB Builder for Excel, MATLAB Builder for NET, MATLAB Compiler, Filter Design HDL Coder - sets of functions that allow you to create independent applications from the MATLAB environment.

§ Interaction with external software products: MATLAB Report Generator, Excel Link, Database Toolbox, MATLAB Web Server, Link for ModelSim - sets of functions that allow you to save data in various types so that other programs can work with them.

§ Databases: Database Toolbox - tools for working with databases.

§ Scientific and mathematical packages: Bioinformatics Toolbox, Curve Fitting Toolbox, Fixed-Point Toolbox, Fuzzy Logic Toolbox, Genetic Algorithm and Direct Search Toolbox, OPC Toolbox, Optimization Toolbox, Partial Differential Equation Toolbox, Spline Toolbox, Statistic Toolbox, RF Toolbox -- sets of specialized mathematical functions that allow solving a wide range of scientific and engineering problems, including the development of genetic algorithms, solving partial derivative problems, integer problems, system optimization, and others.

§ Neural networks: Neural Network Toolbox - tools for the synthesis and analysis of neural networks.

§ Fuzzy logic: Fuzzy Logic Toolbox - tools for constructing and analyzing fuzzy sets.

§ Symbolic calculations: Symbolic Math Toolbox - tools for symbolic calculations with the ability to interact with the symbolic processor of the Maple program.

In addition to the above, there are thousands of other MATLAB toolkits written by other companies and enthusiasts.

computer package mathcad matlab

2. Mathcad

Screenshot of Mathcad 15 on Windows 7

Type - Computer algebra system

Developer - PTC

OS - Microsoft Windows

Interface language 10 languages

First issue 1986

License Proprietary

Website ptc.com

Basicpossibilities

Three-dimensional graph built in Mathcad

Mathcad contains hundreds of operators and built-in functions for solving various technical problems. The program allows you to perform numerical and symbolic calculations, perform operations with scalar quantities, vectors and matrices, and automatically convert one unit of measurement to another.

Mathcad's capabilities include:

§ Solving differential equations, including numerical methods

§ Construction of two-dimensional and three-dimensional graphs of functions (in different coordinate systems, contour, vector, etc.)

§ Use of the Greek alphabet in both equations and text

§ Perform calculations in symbolic mode

§ Perform operations with vectors and matrices

§ Symbolic solution of systems of equations

§ Curve approximation

§ Execution of subroutines

§ Finding roots of polynomials and functions

§ Carrying out statistical calculations and working with probability distributions

§ Finding eigenvalues and vectors

§ Calculations with units of measurement

§ Integration with CAD systems, use of calculation results as control parameters

With Mathcad, engineers can document all calculations as they are performed.

Comparativecharacteristic

P 1 .Purpose

Mathcad refers to computer algebra systems, that is, tools for automating mathematical calculations. In this class of software there are many analogues of different orientations and principles of construction. Most often, Mathcad is compared with such software systems as Maple, Mathematica, MATLAB, as well as their analogues MuPAD, Scilab, Maxima, etc. However, an objective comparison is complicated due to the different purposes of the programs and the ideology of their use.

The Maple system, for example, is designed primarily for performing analytical (symbolic) calculations and has one of the most powerful arsenals of specialized procedures and functions in its class (more than 3000). For most users who are faced with the need to perform mathematical calculations of an average level of complexity, this configuration is redundant. Maple's capabilities are aimed at users - professional mathematicians; Solving problems in the Maple environment requires not only the ability to operate any function, but also knowledge of the solution methods embedded in it: many built-in Maple functions contain an argument that specifies the solution method.

The same can be said about Mathematica. This is one of the most powerful systems, it has extremely high functionality (there is even sound synthesis). Mathematica has high computational speed, but requires learning a rather unusual programming language.

The Mathcad developers have relied on expanding the system in accordance with the needs of the user. For this purpose, additional libraries and expansion packs are assigned, which can be purchased separately and which have additional functions built into the system during installation, as well as e-books with a description of methods for solving specific problems, with examples of existing algorithms and documents that can be used directly in your own calculations. In addition, if necessary and subject to programming skills in C, it is possible to create your own functions and attach them to the system core through the DLL mechanism.

Mathcad, unlike Maple, was originally created for the numerical solution of mathematical problems; it is focused on solving problems of applied rather than theoretical mathematics, when you need to get a result without delving into the mathematical essence of the problem. However, for those who need symbolic calculations, the integrated Maple kernel (from version 14 - MuPAD) is intended. This is especially useful when it comes to creating documents for educational purposes, when it is necessary to demonstrate the construction of a mathematical model based on the physical picture of a process or phenomenon. The symbolic core of Mathcad, unlike the original Maple (MuPAD), is artificially limited (about 300 functions are available), but in most cases this is quite enough to solve engineering problems.

Moreover, experienced Mathcad users have discovered that in versions up to and including 13, it is possible to use almost the entire functional arsenal of the Maple core (the so-called “undocumented capabilities”) in a not too complicated way, which brings the computing power of Mathcad closer to Maple.

P. 2 Interface

The main difference between Mathcad and similar programs is the graphical rather than textual mode for entering expressions. To set commands, functions, and formulas, you can use both the keyboard and buttons on numerous special toolbars. In any case, the formulas will have a familiar, book-like appearance. That is, no special preparation is needed to set formulas. Calculations with entered formulas are carried out at the user's request or instantly, simultaneously with typing, or by command. Regular formulas are calculated from left to right and top to bottom (similar to reading text). Any variables, formulas, parameters can be changed, observing with your own eyes the corresponding changes in the result. This makes it possible to organize the reality of interactive computing documents.

In other programs (Maple, MuPAD, Mathematica), calculations are carried out in the mode of a software interpreter, which transforms commands entered as text into formulas. Maple's interface is aimed at those users who already have programming skills in traditional languages with the introduction of complex formulas in text mode. To use Mathcad, you don’t have to be familiar with programming in one form or another.

Mathcad was conceived as a programming tool without programming, but if such a need arises, Mathcad has programming tools that are quite easy to learn, allowing, however, to build very complex algorithms, which is resorted to when the built-in means of solving a problem are not enough, and also when necessary perform serial calculations.

Separately, it should be noted that Mathcad can use quantities with dimensions in calculations, and you can choose a system of units: SI, CGS, MKS, English, or build your own. The results of the calculations, of course, also receive the appropriate dimension. The benefit of this feature can hardly be overestimated, since it greatly simplifies the tracking of errors in calculations, especially in physical and engineering ones.

P. 3 Graphics

In the Mathcad environment, there are actually no graphs of functions in the mathematical sense of the term, but there is visualization of data located in vectors and matrices (that is, both lines and surfaces are constructed from points with interpolation), although the user may not know about this, since he has It is possible to use directly functions of one or two variables to construct graphs or surfaces, respectively. One way or another, the Mathcad visualization mechanism is significantly inferior to that of Maple, where it is enough to have only the function view to build a graph or surface of any level of complexity. Compared to Maple, Mathcad graphics also have such disadvantages as: the inability to construct surfaces in non-rectangular areas where two arguments exist, creating and formatting graphs only through the menu, which limits the ability to programmatically control graphics parameters.

However, you should remember about the main area of application of Mathcad - for problems of an engineering nature and the creation of educational interactive documents, the visualization capabilities are quite sufficient. Experienced Mathcad users demonstrate the ability to visualize complex mathematical structures, but objectively this goes beyond the purpose of the package.

Extensionfunctionality

It is possible to add new capabilities to Mathcad using specialized extension packages and libraries that supplement the system with additional functions and constants for solving specialized problems:

§ Data Analysis Extension Pack - provides Mathcad with the necessary tools for data analysis.

§ Signal Processing Extension Pack - contains more than 70 built-in functions for analog and digital signal processing, analysis and presentation of results in graphical form.

§ Image Processing Extension Pack - provides Mathcad with the necessary tools for image processing, analysis and visualization.

§ Wavelets Extension Pack - contains a large set of additional wavelet functions that can be added to the library of built-in functions of the Mathcad Professional base module. The package provides the ability to apply a new approach to signal and image analysis, statistical signal evaluation, data compression analysis, and special numerical methods. Functionality includes one- and two-dimensional wavelets, discrete wavelet transforms, multi-resolution analysis, and more. The package combines more than 60 key wavelet functions. Included are orthogonal and biorthogonal families of wavelets, including Haar wavelet, Daubechies wavelet, simlet, coiflet and B-splines, among others. The package also contains extensive online documentation on the basic principles of wavelets, applications, examples, and reference tables.

§ Civil Engineering Library - includes an English reference book. Roark's Formulas for Stress and Strain, customizable templates for structural design and examples of thermal calculations.

§ Electrical Engineering Library - contains standard computational procedures, formulas and lookup tables used in electrical engineering. Text explanations and examples make the library easy to use—each title is hyperlinked to a table of contents and index, and can be found in the search engine.

§ Mechanical Engineering Library - includes an English reference book. Roark's Formulas for Stress and Strain, containing more than five thousand formulas, computational procedures from the McGraw-Hill reference book and the finite element method. Text explanations, a search engine and examples make your work easier. The library includes David Pintour's e-book Introduction to Finite Element Methods.

Listliterature

1. Dyakonov V.P. Handbook for using the PC MATLAB system. -- M.: "Fizmatlit", 1993. -- P. 112. --ISBN 5-02-015101-7

2. Dyakonov V.P. Computer mathematics. Theory and practice. -- St. Petersburg: "Peter", 1999, 2001. -- P. 1296. -- ISBN 5-89251-065-4

3. Dyakonov V.P. MATLAB 5 is a symbolic mathematics system. -- M.: Knowledge, 1999. -- P. 640. -- ISBN 5-89251-069-7

4. Dyakonov V.P., Abramenkova I.V. MATLAB. Signal and image processing. Special reference book. -- St. Petersburg: "Peter", 2002. -- P. 608. -- ISBN 5-318-00667-608

5. Dyakonov V.P., Kruglov V.V. MATLAB. Analysis, identification and modeling of systems. Special reference book. -- St. Petersburg: "Peter", 2002. -- P. 448. -- ISBN 5-318-00359-1

6. Dyakonov V.P. Simulink 4. Special reference book. -- St. Petersburg: "Peter", 2002. -- P. 528. -- ISBN 5-318-00551-9

7. Dyakonov V.P. MATLAB 6/6.1/6.5 + Simulink 4/5. Application Basics. Complete user manual. -- Moscow: "SOLON-Press", 2002. -- P. 768. -- ISBN 5-98003-007-7

8. Dyakonov V.P. MATLAB 6/6.1/6.5 + Simulink 4/5 in mathematics and modeling. Application Basics. Complete user manual. -- Moscow.: "SOLON-Press", 2003. -- P. 576. -- ISBN 5-93455-177-9

9. Dyakonov V.P. MATLAB 6.0/6.1/6.5/6.5+SP1 + Simulink 4/5. Signal and image processing. Complete user manual. -- Moscow: "SOLON-Press", 2005. -- P. 592. -- ISBN 5-93003-158-8

10. Dyakonov V.P. MATLAB 6.5/7.0 + Simulink 5/6. Application Basics. Professional's library. -- Moscow.: "SOLON-Press", 2005. -- P. 800. -- ISBN 5-98003-181-2

11. Dyakonov V.P. MATLAB 6.5/7.0 + Simulink 5/6 in mathematics and modeling. Professional's library. -- Moscow.: "SOLON-Press", 2005. -- P. 576. -- ISBN 5-98003-209-6

12. Dyakonov V.P. MATLAB 6.5/7.0 + Simulink 5/6. Signal processing and filter design. Professional's library. -- Moscow.: "SOLON-Press", 2005. -- P. 576. -- ISBN 5-98003-206-1

13. Dyakonov V.P. MATLAB 6.5/7.0/7 SP1 + Simulink 5/6. Working with images and video streams. Professional's library. -- Moscow: "SOLON-Press", 2005. -- P. 400. -- ISBN 5-98003-205-3

14. Dyakonov V.P. MATLAB 6.5/7.0/7 SP1/7 SP2 + Simulink 5/6. Artificial intelligence and bioinformatics tools. Professional's library. -- Moscow.: "SOLON-Press", 2005. -- P. 456. -- ISBN 5-98003-255-X

15. Dyakonov V.P. MATLAB R2006/2007/2008 + Simulink 5/6/7. Application Basics. 2nd edition, revised and expanded. Professional's library. -- Moscow.: "SOLON-Press", 2008. -- P. 800. -- ISBN 978-5-91359-042-8

16. Dyakonov V.P. MATLAB 7.*/R2006/2007. Self-instruction manual. -- Moscow: "DMK-Press", 2008. -- P. 768. -- ISBN 978-5-94074-424-5

17. Dyakonov V.P. SIMULINK 5/6/7. Self-instruction manual. -- Moscow: "DMK-Press", 2008. -- P. 784. -- ISBN 978-5-94074-423-8

18. Dyakonov V.P. Wavelets. From theory to practice. Complete user manual. 2nd edition revised and expanded. -- Moscow: SOLON-Press, 2004. -- P. 400. -- ISBN 5-98003-171-5

19. Charles Henry Edwards, David E. Penney Differential Equations and Boundary Value Problems: Computing and Modeling with Mathematica, Maple and MATLAB = Differential Equations and Boundary Value Problems: Computing and Modeling. -- 3rd ed. -- M.: "Williams", 2007. -- ISBN 978-5-8459-1166-7

20. Alekseev E.R., Chesnokova O.V MATLAB 7. Self-instruction manual.. - Press, 2005. - P. 464.

21. Kurbatova Ekaterina Anatolyevna MATLAB 7. Self-instruction manual. -- M.: "Dialectics", 2005. -- P. 256. -- ISBN 5-8459-0904-X

22. John G. Matthews, Curtis D. Fink Numerical methods. Using MATLAB = Numerical Methods: Using MATLAB. -- 3rd ed. -- M.: "Williams", 2001. -- P. 720. -- ISBN 0-13-270042-5

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Before choosing the package you need, you need to evaluate its capabilities in terms of the effectiveness of the problem being solved. Below is a brief description of the most famous mathematical packages:

Derive. This math package is interesting because it allows for symbolic math and two graphics modes. The presence of a graphic cursor allows you to determine the coordinates of characteristic points of curves (extrema, roots, points of intersection with other curves). The Derive package is still attractive for its undemanding requirements for hardware resources. This is the only package that even works on a computer IBM class PC XT without hard drive. Moreover, when solving problems of moderate complexity, it showed higher performance and greater reliability of the solution.

Mathematica. The modern mathematical package Mathematica is a powerful tool for performing and formatting mathematical research in both symbolic and numerical form. It has a built-in programming language and powerful graphics capabilities. The output document can be prepared in conjunction with MS Word, MS Excel, etc. This package is unique in terms of the richness and variety of high-level tools for performing symbolic calculations. The ability to carry out analytical calculations is one of the important advantages of the program. Mathematica can transform and simplify algebraic expressions, differentiate and evaluate definite and indefinite integrals, expand functions into series and find limits, etc. Mathematica contains a large number of effective algorithms for carrying out numerical calculations. The program solves many problems using numerical methods that cannot be solved analytically. The internal algorithms used by the program to operate on mathematical functions are selected to achieve the highest possible accuracy. With all its rich capabilities, the package has non-standard input and inconvenient output mathematical symbolism, inadequate presentation of the results of many operations, poor error diagnostics and requires serious labor to master.

Serious disadvantage The help system of a package, for example, in comparison with other mathematical packages, should be considered a small number of practical examples of the use of a particular operator or function described in it, and without this it is very difficult to work with mathematical packages. A very useful feature is missing - transferring examples from the help database directly into the editing window with the ability to immediately execute them.

MapleV. The range of functionality of MapleV is very wide - the following sections are covered: differential and integral calculus, linear algebra, differential equations, geometry, statistics, number theory, group theory, optimization, numerical calculations, financial functions, combinatorics, graph theory and many other areas of mathematics. MapleV's 2D and 3D graphics provide powerful scientific visualization. MapleV has over 20 types of custom graphs, as well as a large number of options available to customize how each graph is displayed. In addition, it is possible to revive the graphs - animation. The package understands many special functions such as: Delta functions, Dirac functions, etc. MapleV has a powerful help system that includes help files for each command, data type, language construct and library. Another advantage of the system is the huge number of practical examples described in it and the transfer of examples from the help database directly into the editing window with the possibility of their immediate execution. In addition, this software can be used to generate codes in languages such as C, LaTEX, etc.

MathCad. The MathCad package was created as a powerful calculator that makes it easy to cope with routine tasks engineering practice, such as solving algebraic and differential equations with constant and variable parameters, analyzing functions, searching for their extrema, numerical and analytical differentiation and integration, displaying tables and graphs when analyzing the solutions found.

The main advantages of the package are:

Writing complex mathematical expressions in the form in which they are usually written on a piece of paper;

Easy to use;

Carrying out numerical and analytical mathematical calculations;

The ability to create high-quality technical reports with tables, graphs, and text in the form of printed documents using built-in tools; preparation of Web pages and publication of results on the Internet;

Entering source data and outputting results into text files or database files in other formats;

Ease and clarity of task programming; the ability to compose your own program-functions using constructs similar to those used in programming languages (Pascal, Fortran) and use the principles of modular programming to implement user computational algorithms;

Obtaining various background information from the field of mathematics and much more.

MathCad is not intended for professional mathematicians or for programming complex problems.

MatLab. The MatLab package was created by MathWorks more than ten years ago. Its capabilities are constantly expanding, and the algorithms embedded in it are being improved.

The range of problems that can be studied using MatLab includes: matrix analysis, signal and image processing, problems of mathematical physics, optimization problems, data processing and visualization, working with cartographic images, neural networks, fuzzy logic and many others. Specialized tools are collected in packages called ToolBox.

For example, the Simulink package (ToolBox) is designed for interactive modeling of nonlinear dynamic systems consisting of standard blocks.

MatLab implements classical numerical algorithms for solving equations, linear algebra problems, finding the values of definite integrals, interpolation, solving differential equations and systems.

MatLab has well-developed capabilities for visualizing two-dimensional and three-dimensional data.

A simple built-in programming language makes it easy to create your own algorithms. The simplicity of the language is compensated by the huge variety of MatLab and ToolBox functions.

The visual GUIDE environment is designed for writing applications with a graphical user interface.

Modern mathematical packages can be used both as a regular calculator, and as a means to simplify expressions when solving any problems, and as a graphics or even sound generator. Interface with the Internet has also become standard, and HTML pages are now generated as part of the calculation process. Now you can solve a problem and at the same time publish the progress of its solution to your colleagues on your home page.

We can talk about mathematical modeling programs and possible areas of their application for a very long time, but we will limit ourselves to only a brief overview of the leading programs, indicating their common features and differences. Currently, almost all modern CAE programs (Computer Aided Engineering, mathematical modeling packages) have built-in symbolic calculation functions.

So what do these programs do and how do they help mathematicians? Using the described software, you can save a lot of time and avoid many errors in calculations. Note that the range of problems solved by such systems is very wide:

Carrying out mathematical research that requires calculations and analytical calculations;

Development and analysis of algorithms;

Mathematical modeling and computer experiment;

Data analysis and processing;

Visualization, scientific and engineering graphics;

Development of graphic and calculation applications.

The following mathematical packages are considered the most well-known and adapted for mathematical symbolic calculations:

The Mathematica package, presented in Figure 1, is widely used in calculations in modern scientific research and has become widely known in the scientific and educational environment.

Despite their focus on serious mathematical calculations, Mathematica class systems are easy to learn and can be used by a fairly wide category of users - university students and teachers, engineers, graduate students, researchers, and even students in mathematics classes in general education and special schools. At the same time, the program’s extensive functions do not overload its interface and do not slow down calculations. Mathematica consistently demonstrates high speed of symbolic transformations and numerical calculations. Of all the systems under consideration, the Mathematica program is the most complete and universal, however, each program has both its advantages and disadvantages.

Figure 1. Mathematica

Thus, Mathematica is, on the one hand, a typical programming system based on one of the most powerful problem-oriented high-level functional programming languages, designed to solve various problems (including mathematical ones), and on the other, an interactive system for solving most mathematical problems interactively without traditional programming. Mathematica, as a programming system, has all the capabilities to develop and create almost any control structures, organize input-output, work with system functions and service any peripheral devices, and with the help of expansion packs it becomes possible to adapt to the needs of any user.

The disadvantages of the Mathematica system include only a very unusual programming language, which, however, is facilitated by a detailed help system.

The Maple program is a kind of patriarch in the family of symbolic mathematics systems and is still one of the leaders among universal symbolic computing systems. It provides the user with a convenient intellectual environment for mathematical research at any level and is especially popular in the scientific community. Note that the symbolic analyzer of the Maple program is the most powerful part of this software, therefore it was borrowed and included in a number of other CAE packages, such as MathCad and MATLAB, as well as in the packages for preparing scientific publications Scientific WorkPlace and Math Office for Word .

Maple provides a convenient environment for computer experiments, during which different approaches to a problem are tried, particular solutions are analyzed, and, if programming is necessary, fragments that require special speed are selected. The package allows you to create integrated environments with the participation of other systems and universal high-level programming languages. When the calculations have been made and you need to formalize the results, you can use the tools of this package to visualize the data and prepare illustrations for publication. To complete the work, all that remains is to prepare the printed material in the Maple environment, and then you can proceed to the next study. The work is interactive - the user enters commands and immediately sees the result of their execution on the screen (Figure 2). At the same time, the Maple package is not at all similar to a traditional programming environment, which requires strict formalization of all variables and actions with them. Here, the selection of suitable variable types is automatically ensured and the correctness of the operations is checked, so in the general case there is no need to describe variables and strictly formalize the recording.

Figure 2. Maple

Maple is a well-balanced system and the undisputed leader in symbolic computing capabilities for mathematics. At the same time, the original symbolic engine is combined here with an easy-to-remember structured programming language, so that Maple can be used for both small tasks and large projects.

The only disadvantages of the Maple system include its somewhat “thoughtful” nature, which is not always justified, as well as the very high cost of this program.

The MATLAB system, presented in Figure 3, belongs to the middle level of products intended for symbolic mathematics, but is designed for widespread use in the field of CAE.

MATLAB is one of the oldest, carefully developed and time-tested systems for automating mathematical calculations, built on an advanced representation and application of matrix operations. This is reflected in the very name of the system - MATrix LABoratory, that is, matrix laboratory. However, the syntax of the system's programming language is thought out so carefully that this orientation is almost not felt by those users who are not directly interested in matrix calculations.

MATLAB libraries are characterized by high speed of numerical calculations. However, matrices are widely used not only in such mathematical calculations as solving problems of linear algebra and mathematical modeling, calculation of static and dynamic systems and objects. They are the basis for the automatic compilation and solution of equations of state of dynamic objects and systems. It is the universality of the matrix calculus apparatus that significantly increases interest in the MATLAB system, which has absorbed the best achievements in the field of quickly solving matrix problems. Therefore, MATLAB has long gone beyond the scope of a specialized matrix system, becoming one of the most powerful universal integrated systems of computer mathematics.

Figure 3. MATLAB

Among the disadvantages of the MATLAB system, we can note the low integration of the environment (a lot of windows that are better to work with on two monitors), a not very clear help system (the volume of proprietary documentation reaches almost 5 thousand pages, which makes it difficult to review) and a specific MATLAB code editor -programs (Figure 4). Today, the MATLAB system is widely used in technology, science and education, but still it is more suitable for data analysis and organizing calculations than for purely mathematical calculations.

Unlike the powerful MATLAB package, which is focused on highly efficient calculations in data analysis, the MathCad program is rather a simple but advanced mathematical text editor with extensive symbolic calculation capabilities and an excellent interface. MathCad does not have a programming language as such, and the symbolic calculation engine is borrowed from the Maple package. But the interface of the MathCad program is very simple, and the visualization capabilities are rich. All calculations here are carried out at the level of visual recording of expressions in commonly used mathematical form. The package has good tips, detailed documentation, training function, a number of additional modules and decent technical support from the manufacturer. However, so far the mathematical capabilities of MathCad in the field of computer algebra are much inferior to the systems Maple, Mathematica, MATLAB. However, many books and training courses have been published on the MathCad program. Today, this system has become an international standard for technical computing, and even many schoolchildren are learning and using MathCad.

Figure 4. MathCad

For a small amount of calculations, MathCad is ideal - here everything can be done very quickly and efficiently, and then the work can be formatted in the usual form (MathCad provides ample opportunities for formatting the results, even publishing them on the Internet). The package has convenient data import/export capabilities. For example, you can work with electronic Microsoft tables MS Excel right inside the MathCad document.

In general, MathCad is a very simple and convenient program that can be recommended to a wide range of users, including those who are not very knowledgeable in mathematics, and especially those who are just learning its basics.

Cheaper, simpler packages include UMS and Microsoft MS Excel.

Once upon a time, symbolic mathematics systems were aimed exclusively at a narrow circle of professionals and worked for large computers. But with the advent of PCs, these systems were redesigned for them and brought to the level of mass production software systems. Nowadays, symbolic mathematics systems of various calibers coexist on the market - from the MathCad system designed for a wide range of consumers to the computer monsters Mathematica, MATLAB and Maple, which have thousands of built-in and library functions, extensive capabilities for graphical visualization of calculations and developed tools for preparing documentation.

Note that almost all of these systems work not only on personal computers equipped with the popular Windows operating systems, but also under the control of operating systems Linux systems, UNIX, Mac OS, and also on PDAs.

Let's move on to the packages most often used in schools when conducting mathematics lessons in high school. These include: Universal Math Solver (UMS), Microsoft MS Excel.

The UMS program - "Universal Mathematical Solver" allows you to solve problems from many sections of algebra and analysis. The knowledge of the "Universal Solver" covers almost the entire course in algebra and analysis in high school and first years of higher education.

Unlike a number of powerful mathematical packages, UMS is accessible for quick learning thanks to a simple interface and deals with the proposed problems exclusively using “school” methods, formalizing all stages of the solution as a teacher would do it (Figure 5).

If we look at the practical value of Universal Math Solver more broadly, then the application will successfully serve parents who are accustomed to monitoring their child’s homework, and mathematics teachers. The latter can use the interactive capabilities of the program in the educational process, placing the explanation of problem solutions on the “shoulders” of the electronic teacher.

Universal Math Solver comes in two editions - desktop and online. The cost of an annual license for one installation of the first version is 3000 tenge, the price of the online edition is three times higher.

Figure 5. Universal Math Solve

Unfortunately, in school practice it is not possible to use such powerful mathematical packages as Mathematica, Mathcad, MathLab, Maple due to the high cost of their licensed copies. However office applications MS Office is available in every school. The use of the mathematical shell of the office spreadsheet processor MS Excel allows you to solve mathematical problems of high complexity.

identical transformations of expressions (including simplification), analytical solution of equations and systems;

differentiation and integration, analytical and numerical;

solving differential equations;

Carrying out a series of calculations with different values of initial conditions and other parameters.

At the same time, the range of problems solved by such systems is very wide:

conducting mathematical research requiring calculations and analytical calculations;
development and analysis of algorithms;
mathematical modeling and computer experiment;
data analysis and processing;
visualization, scientific and engineering graphics;
development of graphic and calculation applications.

Principles of constructing mathematical models. Main stages of modeling.

Mathematical modeling is the creation of a mathematical description of a real object and the study of this description.

Principles of constructing mathematical models

Main stages of modeling

The entire modeling process can be divided into the following stages:

formulation of the modeling problem;

constructing a model diagram, highlighting the main parts and processes;

determination of the optimization criterion or value to be calculated;

highlighting the main changeable parameters;

mathematical description of the main parts and processes;

constructing a solution linking the variable parameters and the optimization criterion or calculated value;

studying the solution for an extremum or calculating the required parameter.

Statement of the modeling problem

The problem statement is usually formulated in the form of a verbal description. At the formulation stage, the modeling object, the goals of building the model and optimization criteria should be described.

Building a model diagram, highlighting the main parts and processes

At this stage, based on the problem statement, the modeling object is divided into main parts and a list of processes of interaction between these parts is determined.

Here are the packages general purpose They also can't help. Specialized packages usually already contain elements for dividing the model into parts for their subject area.

Must be formulated amenable quantification optimization criterion or the desired quantitative parameter.

A list of all changeable parameters and their characteristic quantitative expression must be formulated.

Mathematical description of the main parts and processes

The interaction of parts of the model must be expressed by mathematical formulas. The branch of mathematics that will be used for the description is chosen for reasons of convenience. Those. First of all, this section should be able to quantitatively describe this type of interaction.

The result of this stage is a system of equations or other mathematical expressions that formally describes the interaction of parts and allows for a solution, i.e. obtaining a dependency: optimization criterion as a function of variable parameters.

In particular, it is desirable that the system of equations be closed and that there be a formal proof of the existence of a solution.

Here, general purpose packages are provided with only the apparatus. Specialized packages usually have a predefined mathematical apparatus and are based on a ready-made mathematical description of the problem.

Construction of a solution linking variable parameters and optimization criterion

A SOLUTION is being built, i.e. an explicit functional connection is determined: an optimization criterion or a calculated parameter as a function of the variable parameters.

It is this stage that is the main field of application of applied mathematical modeling packages. This is due to the fact that analytical solutions for the mathematical description of complex objects are usually impossible. And the construction of a solution comes down to the construction of a “numerical solver”, which, based on the given values of variable parameters, can calculate the value of the optimization criterion.

In rare cases of the existence of an analytical solution to the model, the role of applied mathematical modeling packages is reduced to determining the solution function.

There are special subsystems of applied mathematical modeling packages - systems of analytical (symbolic) calculations - these subsystems can be used to maximize the analyticity of the solution, i.e. replacing numerical methods with searching for functional expressions of solutions. Analytical solutions are almost always “better” than numerical ones, because they allow the desired patterns to be expressed through known functions, which greatly speeds up calculations and increases the accuracy of calculations.

Study of the solution to the extremum

The complexity of studying a solution to an extremum is most often associated with a significant amount of time spent on calculating the optimization criterion for given values of variable parameters and/or the large number of permissible combinations of variable parameters, which leads to a huge number of calculations and, again, significant time consumption.

This stage is another field for applying forces to packages. Methods for studying functions for extrema are well developed in mathematics and can be formally applied to any given function.

Parametric Surface Creator

Surfer

plastic bag Simulink

gnuplot ImageMagick

Parametric Surface Creator

The program is designed for a visual representation of geometric objects described by parametrically defined surfaces, such as a sphere, torus, Möbius strip and others. To describe objects, a Pascal-like language is used with support for all standard mathematical functions of the Pascal language and several additional ones. The resulting object is displayed in vector form using an original vector rasterization algorithm, which allows you to get a smooth and natural image even on a low monitor resolution and does not require any hardware support. It is possible to export the image to a BMP file.

Surfer- a program for creating three-dimensional surfaces. Commercial simulation programs for tasks with a predominance of “logical aspects”: AutoMod, Process Model, SIMFACTORY, etc.

plastic bag Simulink, focused specifically on simulation modeling tasks.

gnuplot 1 is a popular program for creating two- and three-dimensional graphs. gnuplot has its own command system and can work interactively (in command line) and execute scripts read from files. Used gnuplot as an image output system in various mathematical packages: GNU Octave, Maxima and many others. ImageMagick– cross-platform software package for batch processing graphic files. Supports a huge number of graphic formats. Can be used with Perl, C, C++, Python, Ruby, PHP, Pascal, Java, in shell scripts, or on its own.

Using Components

Mathcad program documents have the ability to insert modules (component

) other applications to expand the capabilities of visualization, data analysis, and perform specific calculations.

The Axum Graph component is designed for advanced data visualization. To work with tabular data - Microsoft Excel.

Data Acquisition Components, ODBC Input allow you to use external databases.

Free modules (add-in) for integrating Mathcad with Excel programs are also offered, AutoCAD.

The Axum S-PLUS Script component is designed for statistical analysis.

Significant expansion of the package's capabilities is achieved when integrated with the super-powerful MATLAB application.

Options

Versions of Mathcad may differ in configuration and user license. Versions were supplied at different times Mathcad Professional, Mathcad Premium, Mathcad Enterprise Edition(differ in configuration). The version is intended for academic users Mathcad Academic Professor(has full functionality, but differs in the user license and has several times lower cost).

For some time, simplified and noticeably “cut down” student versions of the program were also released.

However, so far the mathematical capabilities of MathCad in the field of computer algebra are much inferior to the systems Maple, Mathematica, MatLab and even the little Derive. However, many books and training courses have been published using the MathCad program, including in Russia. Today, this system has literally become an international standard for technical computing, and even many schoolchildren are learning and using MathCad. For a small amount of calculations, MathCad is ideal - here everything can be done very quickly and efficiently, and then the work can be formatted in the usual form (MathCad provides ample opportunities for formatting the results, even publishing them on the Internet). The package has convenient data import/export capabilities. For example, you can work with Microsoft Excel spreadsheets directly inside a MathCad document.

As cheaper, simpler, but ideologically similar alternatives to the MathCad program, one can note such packages as the already mentioned YaCaS, the commercial MuPAD system ( http://www.mupad.de/) and the free KmPlot program

Mathematical package Mupad

As for the MuPAD program (Figure 2.6), it is a modern integrated system of mathematical calculations, with which you can perform numerical and symbolic transformations, as well as draw two-dimensional and three-dimensional graphs of geometric objects. However, in terms of its capabilities, MuPAD is significantly inferior to its venerable competitors and is, rather, an entry-level system designed for training.

MuPAD Pro 3 is a relatively new computer algebra system with an extensive set of tools, including mathematical algorithms for symbolic and numerical calculations, and tools for visualization, animation and interactive manipulation of two- and three-dimensional graphs and other mathematical objects.

Key Features Matlab

· Platform independent high level language programming focused on matrix calculations and algorithm development

· Interactive environment for code development, file and data management

· Functions of linear algebra, statistics, Fourier analysis, solving differential equations, etc.

· Rich visualization tools, 2-D and 3-D graphics.

· Built-in user interface development tools for creating complete MATLAB applications

· Integration tools with C/C++, code inheritance, ActiveX technologies

The basic set of MatLab includes arithmetic, algebraic, trigonometric and some special functions, fast forward and inverse Fourier transform functions and digital filtering, vector and matrix functions. MatLab “can” perform operations with polynomials and complex numbers, build graphs in Cartesian and polar coordinate systems, and generate images of three-dimensional surfaces. MatLab has tools for calculating and designing analog and digital filters, constructing their frequency, impulse and transient characteristics and the same characteristics for linear electrical circuits, tools for spectral analysis and synthesis.

The C Math library (MatLab compiler) is object-based and contains over 300 data processing procedures in the C language. Inside the package, you can use both MatLab procedures and standard C language procedures, which makes this tool a powerful tool for developing applications (using the C Math compiler , you can embed any MatLab procedures into ready-made applications).

The C Math library allows you to use the following categories of functions:

· operations with matrices;

· comparison of matrices;

· solving linear equations;

· expansion of operators and search for eigenvalues;

· finding the inverse matrix;

· search for a determinant;

· calculation of matrix exponential;

· elementary mathematics;

· functions beta, gamma, erf and elliptic functions;

· basics of statistics and data analysis;

· search for roots of polynomials;

· filtering, convolution;

· fast Fourier transform (FFT);

· interpolation;

· operations with strings;

· file I/O operations, etc.

Moreover, all MatLab libraries are distinguished by high speed of numerical calculations. However, matrices are widely used not only in such mathematical calculations as solving problems of linear algebra and mathematical modeling, calculation of static and dynamic systems and objects. They are the basis for the automatic compilation and solution of equations of state of dynamic objects and systems. It is the universality of the matrix calculus apparatus that significantly increases interest in the MatLab system, which has incorporated the best achievements in the field of quickly solving matrix problems. Therefore, MatLab has long gone beyond the scope of a specialized matrix system, becoming one of the most powerful universal integrated systems of computer mathematics.

Maple math package.

Maple ( http://www.maplesoft.com/)

Processor Pentium III 650 MHz;

400 MB of disk space;

Operating systems: Windows NT 4 (SP5)/98/ME/2000/2003 Server/XP Pro/XP Home.

The Maple program (latest version 10.02) is a kind of patriarch in the family of symbolic mathematics systems and is still one of the leaders among universal symbolic computing systems. (Figure 2.15,2.16) It provides the user with a convenient intellectual environment for mathematical research at any level and is especially popular in the scientific community.

Note that the symbolic analyzer of the Maple program is the most powerful part of this software, which is why it was borrowed and included in a number of other CAE packages, such as MathCad and MatLab, as well as in the Scientific WorkPlace and Math Office for Word packages for preparing scientific publications . The Maple package is a joint development of the University of Waterloo (Ontario, Canada) and the ETHZ, Zurich, Switzerland.

A special company was created for its sale - Waterloo Maple, Inc., which, unfortunately, became more famous for the mathematical study of its project than for the level of its commercial implementation. As a result, the Maple system was previously available primarily to a narrow circle of professionals. Now this company works together with the company MathSoft, Inc., which is more successful in commerce and in developing the user interface of mathematical systems. - the creator of the very popular and widespread systems for numerical calculations MathCad, which have become the international standard for technical calculations.

The package allows you to create integrated environments with the participation of other systems and universal high-level programming languages. When the calculations have been made and you need to formalize the results, you can use the tools of this package to visualize the data and prepare illustrations for publication. To complete the work, all that remains is to prepare printed material (report, article, book) directly in the Maple environment, and then you can proceed to the next study. The work is interactive - the user enters commands and immediately sees the result of their execution on the screen. At the same time, the Maple package is not at all similar to a traditional programming environment, which requires strict formalization of all variables and actions with them. Here, the selection of suitable variable types is automatically ensured and the correctness of the operations is checked, so in the general case there is no need to describe variables and strictly formalize the recording.

The Maple package consists of a core (procedures written in C and well optimized), a library written in the Maple language, and a developed external interface. The kernel performs most of the basic operations, and the library contains many commands - procedures executed in interpretive mode.

The Maple interface is based on the concept of a worksheet, or document, containing input/output lines and text, as well as graphics (Figure 2.17).

The package is processed in interpreter mode. In the input line, the user specifies a command, presses the Enter key, and receives the result - an output line (or lines) or a message about an erroneously entered command. An invitation is immediately issued to enter a new command, etc.

Computing in Maple

The Maple system can be used at the most basic level of its capabilities - as a very powerful calculator for calculations using given formulas, but its main advantage is the ability to perform arithmetic operations in symbolic form, that is, the way a person does it. When working with fractions and roots, the program does not convert them to decimal form during the calculations, but makes the necessary reductions and transformations into a column, which allows you to avoid rounding errors.

To work with decimal equivalents, the Maple system has a special command that approximates the value of an expression in floating point format. The Maple system calculates finite and infinite sums and products, performs computational operations with complex numbers, easily converts a complex number to a number in polar coordinates, calculates the numerical values of elementary functions, and also knows many special functions and mathematical constants (such as "e" " and "pi"). Maple supports hundreds of special functions and numbers found in many areas of mathematics, science, and engineering.

Programming in Maple.

The Maple system uses a 4th generation procedural language (4GL). This language is specifically designed for the rapid development of mathematical routines and custom applications. The syntax of this language is similar to the syntax of universal high-level languages: C, Fortran, Basic and Pascal.

Maple can generate code compatible with programming languages such as Fortran or C, and with the LaTeX typing language, which is very popular in scientific world and is used to design publications. One of the advantages of this property is the ability to provide access to specialized numerical programs that maximize the speed of solving complex problems. For example, using the Maple system, you can develop a certain mathematical model, and then use it to generate C code that matches that model. The 4GL language, specially optimized for the development of mathematical applications, allows you to shorten the development process, and Maplets elements or Maple documents with built-in graphics components help you customize the user interface.

At the same time, in the Maple environment you can prepare documentation for the application, since the package tools allow you to create technical documents professional looking, containing text, interactive math calculations, graphs, pictures, and even sound. You can also create interactive documents and presentations by adding buttons, sliders and other components, and finally publish documents on the Internet and deploy interactive computing on the Web using the MapleNet server.

Mathematica package.

Mathematica ( http://www.wolfram.com/)

Minimum system requirements:

Pentium II processor or higher;

400-550 MB of disk space;

operating systems: Windows 98/Me/NT 4.0/2000/2003 Server/2003x64/XP/XP x64.

Wolfram Reseach, Inc., which developed the Mathematica computer mathematics system (Figure 2.27,2.28), is rightfully considered the oldest and most reputable player in this field. The Mathematica package (current version 5.2) is widely used in calculations in modern scientific research and has become widely known in the scientific and educational environment. You could even say that Mathematica has significant functional redundancy (in particular, there is even the ability to synthesize sound).

Mathematica combines a numerical and symbolic computing core into a single whole, graphics system, a programming language, a documentation system, and the ability to interact with other applications. There is no single competitor for the entire Mathematica environment. Broadly speaking, competitors fall into the following groups: numerical packages, computer algebra systems, typing and documentation applications, graphics and statistical systems, traditional programming languages (interface development tools), and spreadsheets. Since Mathematica first appeared, other mathematics packages have significantly expanded their range of capabilities, originally intended to solve problems falling into only one or two of the above categories.
However, it is unlikely that this powerful mathematical system, which claims to be a world leader, is needed by a secretary or even the director of a small commercial company, not to mention ordinary users. But, undoubtedly, any serious scientific laboratory or university department should have a similar program if they are seriously interested in automating the performance of mathematical calculations of any degree of complexity. Despite their focus on serious mathematical calculations, Mathematica class systems are easy to learn and can be used by a fairly wide category of users - university students and teachers, engineers, graduate students, researchers, and even students in mathematics classes in general education and special schools. All of them will find numerous useful possibilities for application in such a system.

At the same time, the program’s extensive functions do not overload its interface and do not slow down calculations. Mathematica consistently demonstrates high speed of symbolic transformations and numerical calculations. Of all the systems under consideration, the Mathematica program is the most complete and universal, however, each program has both its advantages and disadvantages. And most importantly, they have their own adherents, whom it is useless to convince of the superiority of another system. But those who seriously work with computer mathematics systems should use several programs, because only this guarantees a high level of reliability of complex calculations.

Note that in development different versions The Mathematica system, along with the parent company Wolfram Research, Inc., involved other companies and hundreds of highly qualified specialists, including mathematicians and programmers. Among them there are also representatives of the Russian mathematical school, which is respected and in demand abroad. Mathematica is one of the largest software systems and implements the most efficient calculation algorithms. These include, for example, the context mechanism, which eliminates the appearance of side effects in programs.

The Mathematica system is today considered as the world leader among computer symbolic mathematics systems for the PC, providing not only the ability to perform complex numerical calculations with the output of their results in the most sophisticated graphical form, but also carrying out particularly labor-intensive analytical transformations and calculations.

Mathematica has several main features and is designed to solve a wide range of problems. Here are some classes of problems solved using Mathematica:

1. Working with symbolic complex calculations using hundreds of thousands or millions of terms.
loading, analysis and visualization of data.

2. Solving ordinary and differential equations, as well as numerical or symbolic minimization problems.

3. Numerical modeling and simulation, construction of control systems, ranging from the simplest to galactic collisions, financial losses, complex biological systems, chemical reactions, studying the impact on the environment and magnetic fields in particle accelerators.

4. Easy and Fast Application Development (RAD) for tech companies and financial institutions.

5. Create professional, interactive, technical reports and documents for distribution electronically or on paper.

6. Detailed technical documentation eg for US patents.

7. Conducting special presentations and seminars.

8. Illustrate math or science concepts for students ranging from college to graduate school.

Versions of the system for Windows have a modern user interface and allow you to prepare documents in the form of Notebooks (notebooks). They combine source data, descriptions of problem solving algorithms, programs and solution results in a wide variety of forms (mathematical formulas, numbers, vectors, matrices, tables and graphs).

Mathematica was conceived as a system that would automate the work of scientists and analytical mathematicians as much as possible, so it deserves study even as a typical representative of elite and highly intelligent software products of the highest degree of complexity. However, it is of much greater interest as a powerful and flexible mathematical toolkit that can provide invaluable assistance to most scientists, university teachers, students, engineers and even schoolchildren.

From the very beginning, much attention was paid to graphics, including dynamic, and even multimedia capabilities - the reproduction of dynamic animation and sound synthesis. The range of graphics functions and options that change their effect is very wide. Graphics have always been the strength of various versions of the Mathematica system and provided them with leadership among computer mathematics systems.

As a result, Mathematica quickly took a leading position in the market for symbolic mathematical systems. Particularly attractive are the system’s extensive graphical capabilities and the implementation of a Notebook-type interface. At the same time, the system provided a dynamic connection between document cells in the style of spreadsheets, even when solving symbolic problems, which fundamentally and advantageously distinguished it from other similar systems.

By the way, the central place in Mathematica-class systems is occupied by a machine-independent core of mathematical operations, which allows the system to be transferred to various computer platforms. To transfer the system to another computer platform, a Front End software interface processor is used. It is he who determines what type of user interface the system has, that is, the interface processors of Mathematica systems for other platforms may have their own nuances. The kernel is made compact enough so that any function can be called from it very quickly. To expand the set of functions, use the Library and a set of Add-on Packages. Extension packages are prepared in the Mathematica systems' own programming language and are the main means for developing system capabilities and adapting them to solve specific classes of user problems. In addition, the systems have a built-in electronic help system - Help, which contains electronic books with real examples.

Thus, Mathematica is, on the one hand, a typical programming system based on one of the most powerful problem-oriented high-level functional programming languages, designed to solve various problems (including mathematical ones), and on the other hand, an interactive system for solving most mathematical problems. tasks online without traditional programming. Thus, Mathematica as a programming system has all the capabilities to develop and create almost any control structures, organize input-output, work with system functions and service any peripheral devices, and with the help of extension packages (Add-ons) it becomes possible to adapt to the needs of any user (although the average user may not need these programming tools - he will get by with the built-in mathematical functions of the system, which amaze even experienced mathematicians with their abundance and variety).

The disadvantages of the Mathematica system include only a very unusual programming language, which, however, is facilitated by a detailed help system.

FlatGraph is a program for constructing graphs of functions (regular and parametric) with advanced capabilities (Figure 2.33). Differentiation of any order (with simplification). Construction of tangents to the graph. The program is designed for both inexperienced and professional users, as it combines an intuitive interface with professional functions.

FlatGraph allows you to:

Enter one or more functional expressions of any complexity to display and (or) differentiate them;

Perform symbolic differentiation for the specified order of the derivative, as well as simplify the resulting derivative;

Explore “live” changes in various function parameters with simultaneous display of new graphs, which allows you to determine the influence of function parameters on their appearance;

Use automatic or manual scaling of function graphs for linear scales;

Set and display graphically parametric functions that display, for example, ellipsoids, cardioids, Bernoulli lemniscates and other similar graphs (where the abscissa and ordinate depend on one parameter “t”);

Solve equations, systems of equations and inequalities graphically;

Receive and display the tangent to the graph of the function at point x0 (defined by the user).

FlatGraph has a simple and intuitive interface, equipped with detailed documentation on use and examples of work.

Math packages. Modeling. List the capabilities and main tasks solved by the packages.

Mathematical packages are an integral part of the world of CAE systems. (Computer Aided Engeneering) Currently, mathematical packages apply the principle of model construction, rather than the traditional “art of programming.” That is, the user poses a problem, and the system finds the methods and algorithms for solving it itself. Modern mathematical packages can be used both as a regular calculator and as a means to simplify expressions when solving any problems, as well as as a graphics or even sound generator! Currently, almost all modern mathematical systems have built-in symbolic calculation functions. However, Maple, MathCad, Mathematica and MatLab are considered the most well-known and suitable for mathematical symbolic calculations. Mathematical modeling – creating a mathematical description of a real object and studying this description.

Initially, any calculations using models were done manually. As computing devices developed, these devices were used to speed up calculations.

The computer allows it to be used as a means of automating scientific work and various specialized programs are used to solve complex calculation problems.

At the same time, in scientific work there is a wide range of simple mathematical problems, for solving which universal professional tools can be used.

Such simple tasks include, for example, the following:

preparation of scientific and technical documents containing text and formulas written in a form familiar to specialists;

calculating the results of mathematical operations that involve numerical constants, variables and dimensional physical quantities;

operations with vectors and matrices;

solving equations and systems of equations (inequalities);

statistical calculations and data analysis;

construction of two-dimensional and three-dimensional graphs;

identical transformations of expressions (including simplification), analytical