Math packages. Information technologies for mathematical calculations on the example of the Maple package

12.08.2019 Reviews

Currently, software solutions oriented math problems(in this case, under mathematical any problem is understood, the algorithm of which can be described in terms of one or another branch of mathematics), are very extensive and can be conditionally differentiated into five levels:

1. built-in tools of varying degrees of development of a particular programming system; (programming systems like Basic, C, Pascal)

2. special programming languages; (Fortran, Prolog)

3. highly specialized. (Packages MacMath, Phaser, Eureka)

4.special (StatGraf, Macsyma, Dynamics, Derive packages)

5. general packages. ( MathCAD, REDUCE, MatLab)

Finally, the modern development of computer technology focused on the creation of integrated packages multimedia- technologies led to the emergence of a new level of mathematical packages, of which the most famous are packages MAPLE V firms Maple Software Inc. and Mathematica firms Wolfram Research Inc.

MATHCAD package as a tool for solving mathematical problems.

General characteristics of the package

The package has natural an input language for representing mathematical dependencies and tools for their set of the type offered in Microsoft Equation

Mathcad is equipped with a word processor that allows, for example, to format an article without the help of specialized tools.

Input features:

Imaginary unit is written as i or j straightaway after the numeric factor.

· The default angles are in radians.

Latin letters, numbers and signs of operations, including exponentiation

Order-indicating parentheses

Are typed directly from the keyboard.

Pressing after typing a Latin letter converts it to Greek.

Multiplication is typed as *, division is typed by /. During the input process, the multiplication sign is automatically replaced by a dot, and the dividend and divisor are represented as the numerator and denominator of the fraction. The exponentiation sign converts the subsequent expression to an exponent, and the opening square bracket converts it to an index. Return to the main line level (as well as transition to the denominator set) is performed by pressing.

The set \ calls the pattern for the square root, the apostrophe - the appearance of parentheses around the selected subexpression, the vertical bar - the pattern for calculating the absolute value or the determinant of a matrix.

Variable assignment; numerical values are produced by a set of constructs<имя>:<число>(the colon will be replaced with an assignment sign).

For example x: 6 we get x: = 6 on the screen.<

The results are displayed by pressing the [=] key. Equal signs in conditions and equations are typed only by. The dialing is completed by clicking or clicking outside the dialing field.

MATHCAD package interface

MathCAD works with documents... From the user's point of view, a document is a blank sheet of paper on which you can place blocks of three main types: mathematical expressions, text fragments, and graphic areas.

The arrangement of non-text blocks in a document is of fundamental importance - left to right and top to bottom. The insertion point on the worksheet is marked with a red cross, it is called " vizier »

Mathematical expressions

The basic elements of mathematical expressions in MathCAD are data types, operators, functions and control structures.

Operators- MathCAD elements with which you can create mathematical expressions. These include, for example, symbols for arithmetic operations, signs for calculating sums, products, derivative and integral, etc.

Data types include numeric constants, regular and system variables, arrays (vectors and matrices)

Functions

A function is an expression according to which some calculations are performed with arguments and its numerical value is determined.

Of particular note is the difference between arguments and function parameters. The variables specified in parentheses after the name of the function are its arguments and are replaced when the function is evaluated with the values from the parentheses.

Variables on the right side of a function definition that are not listed in parentheses on the left side are parameters and must be set before the function is defined.

The main feature of the function is return value, i.e. the function, in response to calling it by name with an indication of its arguments, must return its value.

Functions in the MathCAD package can be built-in. Ways to insert an inline function

· Select the menu item Insert / Function.

Press the key combination Ctrl + E.

Click on the button

Text Areas

The text area is intended for small pieces of text - signatures, comments, etc. Inserted using the command Insert / Text Area or keyboard shortcuts Shift + "(double quote).

Graphics areas

Graphics areas are divided into three main types - 2D graphics, 3D graphics, and imported graphics. 2D and 3D graphs are built by MathCAD itself based on the processed data.

For creating Cartesian graphics :

1. Install the viewfinder in an empty space of the working document.

2. Select the command Insert / Graph / X-Y graph, or press the key combination Shift + @, or click the button

Charts. A Cartesian chart template appears.

3. Enter the first independent variable in the middle label under the X-axis, the second separated by commas, and so on up to 10, for example x1, x2, ...

4. Enter the first independent variable in the middle label to the left of the vertical Y-axis, the second separated by commas, etc., for example, y1 (x1), y2 (x2), ..., or the corresponding expressions.

5. Click outside the plot area to start plotting.

Three-dimensional, or 3D-graphics, represent functions of two variables of the form Z (X, Y).

Example:

Solving math problems with MATHCAD

Numerical solution of a nonlinear equation

For the simplest equations of the form f (x) = 0 the solution in Mathcad is found using the function root.

root ( f (X 1, x 2, … ), X 1, a, b)

Returns the value X 1 belonging to the segment [a, b ] where the expression or function f (X) becomes 0. Both arguments to this function must be scalars. The function returns a scalar.

Arguments

f (X 1, x 2, … ) - a function, defined elsewhere in the working document, or expression. The expression must return scalar values.

X 1 - the name of the variable that is used in the expression. This variable must be set to a numeric value before using the root function. Mathcad uses it as an initial guess when looking for a root.

a, b - are optional, if used, they must be real numbers, and a < b .

The approximate values of the roots ( initial approximations) may be:

1. Known from the physical meaning of the problem.

2. Found graphically.

Most common graphical way determination of initial approximations. Taking into account that the real roots of the equation f (x) = 0 are the intersection points of the function graph f (x) with the abscissa, it is enough to plot the function f (x) and mark the intersection points f (x) with axis Oh, or mark on the axis Oh segments containing one root.

An example of solving a nonlinear equation:

Finding the roots of a polynomial

To find the roots of an expression of the form

nx n + ... + v2 x 2 + v 1x + v 0,

better use function polyroots than root. Unlike the function root, function polyroots does not require an initial approximation and returns all roots at once, both real and complex.

Polyroots ( v)

returns the roots of a polynomial of degree n... The polynomial coefficients are in the vector v length n+1. Returns a vector of length n consisting of the roots of the polynomial.

Arguments: v is a vector containing the coefficients of the polynomial.

It is convenient to create a vector v using the command Symbols Þ Polynomial coefficients.

An example of finding the roots of a polynomial:

Solving systems of equations

MathCAD also makes it possible to solve systems of equations. The maximum number of equations and variables is 50. The result of solving the system will be the numerical value of the desired root.

To solve the system of equations, you must do the following:

1. Set the initial approximation for all unknowns included in the system of equations. Mathcad solves the system using iterative methods.

2. Print keyword Given... It tells Mathcad that a system of equations follows.

3. Enter equations and inequalities in any order. Use = to print the = character. Any of the symbols can be placed between the left and right sides of the inequalities<, >,

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

differentiation and integration, analytical and numerical;

solution of differential equations;

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

At the same time, the range of tasks 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 computational applications.

Principles of constructing mathematical models. The main stages of modeling.

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

Principles of building mathematical models

The main stages of modeling

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

setting the modeling problem;

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

determination of the optimization criterion or value to be calculated;

highlighting the main variable parameters;

mathematical description of the main parts and processes;

building a solution linking variable parameters and an optimization criterion or calculated value;

investigation of the solution for an extremum or calculation of the required parameter.

Problem statement for modeling

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

Building a model diagram, highlighting the main parts and processes

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

General-purpose packages cannot help here either. Specialized packages usually already contain elements of dividing the model into parts for their subject area.

A quantifiable optimization criterion or quantifiable parameter must be formulated.

A list of all variable parameters and their characteristic quantitative expression should be formulated.

Mathematical description of the main parts and processes

The interaction of the parts of the model must be expressed in mathematical formulas. The section 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 formally describing the interaction of parts and allowing a solution, i.e. dependency derivation: optimization criterion as a function of variable parameters.

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

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

Construction of a solution connecting variable parameters and optimization criterion

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

It is this stage that is the main field of application of the forces 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 the solution is reduced to the construction of a "numerical solver", which, given the values of the variable parameters, can calculate the value of the optimization criterion.

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

There are special subsystems of applied packages of mathematical modeling - systems of analytical (symbolic) calculations - these subsystems can be used to maximize the analyticity of the solution, i.e. replacing numerical methods with finding a functional expression of solutions. Analytical solutions are almost always "better" than numerical ones, because they allow one to express the desired regularities in terms of known functions, which greatly speeds up calculations and increases the accuracy of calculations.

Examining the solution to the extremum

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

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

Parametric Surface Creator

Surfer

package Simulink

gnuplot ImageMagick

Parametric Surface Creator

The program is intended for visual representation of geometric objects described by parametrically defined surfaces, such as a sphere, torus, Mobius 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 the original vector rasterization algorithm, which allows you to get a smooth and natural image even at low monitor resolutions and does not require any hardware support. The image can be exported 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.

package Simulink, focused specifically on the tasks of simulation.

gnuplot 1 is a popular program for creating 2D and 3D charts. gnuplot has its own command system, can run interactively (in command line mode) and execute scripts read from files. Used by gnuplot as a display system for various math packages: GNU Octave, Maxima, and many others. ImageMagick- cross-platform software package for batch processing of 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

In Mathcad documents, it is possible to insert modules (component

) other applications for expanding the capabilities of visualization, data analysis, performing specific calculations.

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

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

There are also free add-ins for integrating Mathcad with Excel programs, AutoCAD.

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

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

Complete set

Mathcad versions may differ in bundle and user license. Versions were delivered at different times Mathcad Professional, Mathcad Premium, Mathcad Enterprise Edition(differ in the package bundle). For academic users, the version is intended Mathcad Academic Professor(has full functionality, but differs in user license and has several times lower cost).

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

However, while the mathematical capabilities of MathCad in the field of computer algebra are much inferior to the systems Maple, Mathematica, MatLab and even the baby Derive. However, many books and training courses have been published under the MathCad program, including in Russia. Today, this system has literally become an international standard for technical computing, and even many schoolchildren master and use MathCad. For a small amount of calculations, MathCad is ideal - here everything can be done very quickly and efficiently, and then formalize the work in a familiar form (MathCad provides ample opportunities for formatting the results, up to publication on the Internet). The package has convenient data import / export capabilities. For example, you can work with Microsoft Excel spreadsheets right inside a 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 versed in mathematics, and especially to those who are just learning the basics.

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

Math 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 2D and 3D graphics and other mathematical objects.

Key features of Matlab

Platform-independent high-level programming language 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 building 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, functions of fast forward and inverse Fourier transform and digital filtering, vector and matrix functions. MatLab "knows" how to 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 C. Inside the package, you can use both MatLab procedures itself and standard C 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;

· Solution of linear equations;

· Decomposition of operators and search for eigenvalues;

· Finding the inverse matrix;

· Search for a determinant;

· Calculation of the 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 a 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, calculating static and dynamic systems and objects. They are the basis for the automatic compilation and solution of equations of state for dynamic objects and systems. It is the universality of the matrix calculus apparatus that significantly increases the interest in the MatLab system, which has incorporated the best achievements in the field of fast solution of matrix problems. Therefore, MatLab has long gone beyond the framework of a specialized matrix system, becoming one of the most powerful universal integrated systems of computer mathematics.

Math package Maple.

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

Pentium III 650 MHz processor;

400 MB of disk space;

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

Maple (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 computation systems. (Figure 2.15,2.16) It provides the user with a convenient intellectual environment for mathematical research of any level and is especially popular in the scientific community.

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

For its sale, a special company was created - Waterloo Maple, Inc., which, unfortunately, is more famous for the mathematical elaboration of its project than for the level of its commercial implementation. As a result, the Maple system was previously available mainly to a narrow circle of professionals. The company is now working in conjunction with the more successful commerce and user interface for math systems firm MathSoft, Inc. - the creator of the very popular and massive systems for numerical calculations MathCad, which have become the international standard for technical calculations.

Maple provides a convenient environment for computer experiments, during which different approaches to the problem are tried, particular solutions are analyzed, and, if programming is necessary, fragments requiring special speed are selected.

The package allows you to create integrated environments with the participation of other systems and high-level universal programming languages. When the calculations are made and you need to format the results, you can use the tools of this package to visualize data and prepare illustrations for publication. To complete the work, it remains to prepare the printed material (report, article, book) directly in the Maple environment, and then you can proceed to the next research. The work takes place interactively - 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 like a traditional programming environment, where a rigid formalization of all variables and actions with them is required. Here, the selection of suitable types of variables is automatically ensured and the correctness of the operations is checked, so that in the general case there is no need for the description of variables and strict formalization of the record.

The Maple package consists of a core (routines written in C and well optimized), a library written in Maple, and a developed front-end. The kernel performs most of the basic operations, and the library contains many commands - procedures that are executed in interpretation mode.

Maple's interface is based on the concept of a worksheet or document containing I / O lines and text and graphics (Figure 2.17).

The package is operated in interpreter mode. In the input line, the user sets a command, presses the Enter key and receives the result - a line (or lines) of output or a message about an erroneously entered command. It immediately prompts you to enter a new command, etc.

Calculations in Maple

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

To work with decimal equivalents, Maple 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, for example, "e "And" pi "). Maple supports hundreds of special functions and numbers found in many areas of mathematics, science, and technology.

Programming in Maple.

Maple 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 the high-level universal languages: C, Fortran, Basic, and Pascal.

Maple can generate code that is compatible with programming languages such as Fortran or C, and with the LaTeX typing language, which is very popular in the 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 specific mathematical model and then use it to generate C code that matches that model. Specially optimized for math development, 4GL can shorten the development process, and customize the user interface with Maplets or Maple documents with embedded graphics.

At the same time, in the Maple environment, you can prepare documentation for the application, since the tools of the package allow you to create professional-looking technical documents containing text, interactive mathematical calculations, graphics, drawings and even sound. You can also create interactive documents and presentations by adding buttons, sliders, and other components, and finally, publish documents to the Internet and deploy interactive computing to 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 Research, Inc., which developed the Mathematica computer mathematics system (Figure 2.27,2.28), is widely regarded as the oldest and most established player in the field. Mathematica (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 can even say that Mathematica has significant functional redundancy (in particular, there is even a possibility for sound synthesis).

Mathematica brings together a numeric and symbolic computing core, a 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. Competitors generally fall into the following groups: numerical packages, computer algebra systems, typing and documentation applications, graphical and statistical systems, traditional programming languages (interface development tools), and spreadsheets. Since Mathematica first appeared, other mathematical packages have significantly expanded their range of capabilities, initially they were intended to solve problems related to only one or two of the above categories.
However, this powerful mathematical system, claiming to be world leadership, is unlikely to be needed by a secretary or even a director of a small commercial firm, let alone ordinary users. But, undoubtedly, any serious scientific laboratory or department of a university should have a similar program if they are seriously interested in automating the performance of mathematical calculations of any degree of complexity. Despite its 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 of mathematical classes of general education and special schools. All of them will find numerous useful applications in such a system.

At the same time, the broadest functions of the program do not overload its interface and do not slow down computations. Mathematica consistently demonstrates the high speed of symbolic transformations and numerical calculations. Of all the systems under consideration, Mathematica is the most complete and versatile program, but 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 systems of computer mathematics should use several programs, because only this guarantees a high level of reliability of complex calculations.

Note that in the development of various versions of the Mathematica system, along with the parent company Wolfram Research, Inc., other firms and hundreds of highly qualified specialists, including mathematicians and programmers, took part. Among them there are also representatives of the Russian mathematical school, which is respected and in demand abroad. The Mathematica system is one of the largest software systems and implements the most efficient computation algorithms. These include, for example, a context mechanism that prevents side effects from appearing in programs.

The Mathematica system is considered today as the world leader among computer systems of symbolic mathematics for PCs, which provide not only the ability to perform complex numerical calculations with the output of their results in the most exquisite graphical form, but also carry out particularly laborious analytical transformations and calculations.

Mathematica has several basic features and is designed to solve a wide variety of problems. Here are some of the classes of problems that Mathematica can solve:

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

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

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

4. Simple and fast application development (RAD) for tech companies and financial institutions.

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

6. Detailed technical documentation, eg for US patents.

7. Conducting special presentations and seminars.

8. Illustrating mathematical or scientific concepts for students from college to graduate school.

Windows versions have a modern user interface and allow you to prepare documents in the form of Notebooks. They combine initial data, descriptions of algorithms for solving problems, 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 automates the work of scientists and mathematicians-analysts as much as possible, so it deserves to be studied 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 researchers, university and university professors, students, engineers, and even schoolchildren.

From the very beginning, much attention was paid to graphics, including dynamic, and even to the possibilities of multimedia - the reproduction of dynamic animation and the synthesis of sounds. The set of graphics functions and options that change their effect is very wide. Graphics have always been a strong point of the various versions of Mathematica and have provided them with a leadership role in computer mathematics systems.

As a result, Mathematica quickly took the lead in the symbolic math systems market. Particularly attractive are the extensive graphical capabilities of the system and the implementation of the Notebook interface. At the same time, the system provided dynamic communication between document cells in the style of spreadsheets even when solving symbolic problems, which fundamentally and favorably distinguished it from other similar systems.

By the way, the central place in systems of the Mathematica class 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, the Front End software interface processor is used. It is he who determines what form the user interface of 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 to call any function from it very quickly. The Library and a set of Add-on Packages are used to expand the set of functions. Extension packages are prepared in the own programming language of Mathematica systems and are the main tool for developing the capabilities of the system and adapting them to solving specific classes of user problems. In addition, the systems have a built-in electronic help system - Help, which contains e-books with real examples.

Thus, Mathematica is, on the one hand, a typical programming system based on one of the most powerful high-level problem-oriented functional programming languages, designed to solve various problems (including mathematical ones), and on the other hand, it is an interactive system for solving most of mathematical tasks in interactive mode without traditional programming. Thus, Mathematica as a programming system has all the possibilities for the development and creation of almost any control structures, the organization of input-output, work with system functions and maintenance of any peripheral devices, and with the help of add-ons, it becomes possible to adapt to the requests of any user, (although an ordinary user may not need these programming tools - he will completely manage with the built-in mathematical functions of the system, which amaze even experienced mathematicians with their abundance and diversity).

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

FlatGraph is a program for graphing functions (regular and parametric) with advanced capabilities (Figure 2.33). Differentiation of any order (simplified). Plotting tangents to the graph. The program is designed for both inexperienced and professional users, because 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 the "live" change of various parameters of functions with the simultaneous display of new graphs, which allows you to determine the influence of the parameters of functions on their appearance;

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

Set and display graphically parametric functions displaying, 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;

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

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

Math packages. Modeling. List the capabilities and main tasks of the packages.

Math packages are an integral part of the world of Computer Aided Engineering (CAE) systems Currently, math packages use the principle of model construction, rather than the traditional "art of programming". That is, the user sets the task, and the system finds the methods and algorithms for the solution 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 a graphics or even sound generator! Nowadays, almost all modern maths have built-in symbolic computation functions. However, the most famous and adapted for mathematical symbolic calculations are Maple, MathCad, Mathematica and MatLab. Mathematical modeling - creating a mathematical description of a real object and studying this description.

Initially, any calculations on the models were done manually. With the development of computing devices, these devices were used to speed up calculations.

The computer makes it possible to use it as a means of automating scientific work, and various specialized programs are used to solve complex computational problems.

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

These simple tasks include, for example, the following:

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

calculation of the results of mathematical operations involving numerical constants, variables and dimensional physical quantities;

operations with vectors and matrices;

solution of 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), analytically

Send your good work in the knowledge base is simple. Use the form below

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/

Introduction

Computers today take on a huge share of the computational and analytical workload of the modern mathematician. Therefore, today's researchers face and, most importantly, seem to be solvable, completely different tasks 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 in the study of space, the search for new sources of energy, 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 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 acquaintance, easily carry out very complex analytical transformations of mathematical expressions, take derivatives, integrals, calculate limits, decompose and simplify expressions, draw graphs. And now you don't need to study programming languages for a long time to master the mathematical capabilities of a computer. Now, in mathematical packages, practically everything necessary for an engineer, economist, sociologist, statistician is realized. Such world-famous packages as Mathematica, Mathcad, MatLAB, Maple have become not only convenient computing, but also amazingly fruitful, flexible educational environment. In my opinion, together with the Internet, these packages can unite 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 the phenomena is clearly demonstrated. They can be modified for your own tasks, supplemented and expanded, resulting in not only numbers, but also new analytical expressions, graphs, tables.

The use of computer mathematical packages allows:

· Expand the range of real applications;

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

· To combine professional orientation, scientific approach, consistency, clarity, interactivity, intersubject communications when solving ODE;

· Instantly exchange information with a person, physical contact with whom is impossible, or difficult to implement;

· Consider more tasks, due to the reduction in 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 their task.

The purpose of this work is to use information technologies 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. Apply the Maple package to math lessons.

4. Make a conclusion on the work done.

1. Modern math packages in education

1.1 Conceptand usemath packagesin education

Methods and forms of application of computer technologies in the educational process is an actual methodological and organizational task of every teacher, every administrator of a school or university.

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

· Development of computer programs for educational purposes, programs specially designed for the study of a particular discipline;

· Use of software developed for professional activities in the relevant field of knowledge; for most natural sciences, these are professional math packages.

Mathematical packages here are 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 ), possessing the means of performing various numerical and analytical (symbolic) mathematical calculations, from simple arithmetic calculations, to solving partial differential equations, solving optimization problems, testing statistical hypotheses, constructing mathematical models and other tools necessary for performing various technical calculations. All of them have advanced scientific graphics, a convenient help system, and reporting tools. The name "professional" or "universal" is used as an alternative to the name "training package".

For many years, teachers of mathematics were quite clearly divided into adherents to 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 have 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 understanding of quality education includes, as a necessary element, fluency in computer technology and, as a result, a computer is perceived as a subject, if not the first, then the second necessity. Most parents cannot imagine raising their own schoolchildren without a computer. An increasing number of students have computers at home and more and more often it is students who initiate the use of computer technologies in the educational process. They are motivated not by "play" interest, as we said and seen before, but by the desire to "make life easier for themselves", the desire to acquire professional skills useful for future careers, and the willingness to learn to work on a computer not only in special computer science classes. It is safe to say that the "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 - it is much easier to learn how to work with ready-made software than to write programs yourself.

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

The struggle for the consumer, the desire to expand the circle of users, led to the fact that, while maintaining individual characteristics, the packages become closer, become so similar that the skills of working with one of them allow you to 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 added, the range of tasks available for research is expanding in accordance with fashion, with the emergence of new applications, new research methods, etc.

The Internet is a new reality in the life of a modern student and specialist. Thanks to global computer networks, the user of any widespread software product gets the opportunity to join the global community of consumers of the same product. He will find information on the network about new products, the latest versions of the program, messages about detected errors, receive expert advice, tell about his findings and get acquainted with the tricks of others, learn about the literature, about the range of problems to be solved, often just find a solution to a similar problem, etc. P.

A separate place is occupied by statistical packages. Today, math statistics is by far the most sought after math course. The data analysis methods studied here are widely used in practice. Consequently, mastering the techniques of working in the environment of a universal statistical package is an element of high-quality vocational education that is in demand on the labor market.

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

Availability of universal math packages and theirs in the professional software market. An essential circumstance that until recently hindered the widespread use of professional packages within the walls of universities is the high cost of professional scientific mathematical support. Recently, however, many companies that develop and distribute programs for science present 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 the packages contain basic computing 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 contributes to the intensification of research activities, allows for wider involvement of students in scientific work, which, as you know, improves the chances of scientific groups in the distribution of grants, and, therefore, allows them to subsequently find funds to purchase more modern licensed software. ...

Availability of documentation and reference literature on math packages. If still 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 would not have published two or three books in Russian.

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

1.2 Comparative analysis of mathematical packages AutoCad, MatLab, Maple, Mathematica

The analysis consists of a table listing the functionality of the programs. It is divided into functional sections of mathematical, graphical, functional capabilities and in the programming environment, a section for import / export of data, the possibility of using it 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 for those programs in which automatic functions are present, a score of 0.9 is given to those applications that need to be installed separately. Programs in which automatic functions are not available receive a score of 0 points. The amount in each column is the total score.

As a result, all grades were assessed as follows:

Mathematical functions 38%;

Graphic functions 10%;

Programming software 9%;

Data import / export 5%;

Operating systems 2%;

Speed comparison 36%.

Common symbols used in various schemes

The function is built into the program

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

$ - The function is supported by an add-on module, which can be downloaded for a 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 no guarantee of service or support. This is a very important point for several types of business (i.e. bank use).

Comparison of math functionality

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

The following table should give a brief overview of the functionality in order to parse data in numerical ways and should indicate which functions are supported by which programs, or these functions are already implemented in the main program, or if you need an add-on module.

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

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 was 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 often used in mathematical models. It is necessary to interpret the timing results in content with whole models as then, small differences in timings of single functions might result in timing differences of minutes up to several hours. However, it is not possible to use complete models for these evaluation tests as a job to make the model work in every mathematical package, and also the duration would be very high.

Functions (version)

Reading data from an ASCII data file
Reading data from a database using the ODBC interface
Extracting 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
Main constituent factorization of 500x500 matrix
Gamma function on a 1500x1500 random matrix
Gaussian error function on 1500x1500 random matrix
Linear regression over 1000x1000 random matrix
Full work

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

The total work was calculated as follows:

The best result of the function performance is estimated as 100%; in order to calculate the results for each function, I will take the best performance and divide it by the timing of the tested program (the formula will look MINUTE (A1; A2;…) / A2) and this is displayed as a percentage. To do the final "Full Job", I will calculate the amount of interest 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 appreciated.

General product information.

Some information like rating, support, newsgroups, books, etc. are essential to users of mathematical or statistical software. Due to the fact that this type of information cannot be characterized objectively, one 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 / Electron. management
Add. books
Frequently Asked Questions
Teleconferences / mailing lists
The program is archived by the software manufacturer
Program archives by external institutions

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

Miscellaneous information: The summary should establish the results of the comparison of speed, functionality of the software environment, data import / export services, and suitability for different platforms with respect to the results of the comparison 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%)
Comparison of graphics functionality (10%)
Functionality of the software environment (9%)
Data circulating (from 5%)
Available platforms (2%)
Speed comparison (36%)
Complete result

Summary: The overall results of some of the tested programs are not the best due to the certain premium of this test report.

2. Development of programming skills among schoolchildren in the environmentMaple

2.1 The concept of programmatically developing a library of procedures 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, specially designed for creating symbolic computation programs. The most interesting features of the Maple system are editing and changing these subroutines, as well as replenishing the libraries with subroutines developed for solving specific problems. They have already appeared in large numbers, and the best of them are included in the Share-library of users, 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.

The creation of a new library is as follows.

First of all, you need to define the name of your library, for example mylib, and create a directory (folder) for it on the disk with the specified 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 enter your library routines. They are specified with a double name - first the name of the library, and then the name of the procedure 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:

It is possible to build graphs of the introduced procedures-functions. They are presented in. Using the with function, you can verify that the mylib library actually contains the procedures you just introduced into it. A list of them should appear when you call with (mylib):

> with (mylib);

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

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

Pay special attention to correctly specifying the fully qualified file name. Commonly used to indicate a path, the \ character in Maple-language strings is used as a line-continuation character. Therefore, either the double \\ character or the / character must be used. In this example, the file is written to the root of the C drive. It is better to place the library file in another folder (for example, in a library already in the system), the full path to it is specified.

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

Using the with command, you can verify that these definitions are no longer there:

> with (mylib):

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

After that, using the read command, you need 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 that your library has a list of fl, f2 and f3 procedures:

> with (mylib):

Finally, you can try again the procedures that are now introduced from the loaded library:

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

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

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

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

The special operator makehelp can be used to provide a standard reference for 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 directory. To register the created help, you need to execute a command of the form:

libname: -libname. "/ mylib":

See the help system for details on how to use these operators. mathematical programming calculating maple

Creation of your own library procedures should be taken with sufficient care. Their use will make your Maple programs incompatible with the standard Maple version. If you use one or two procedures, it is easier to place them in the documents where you really need them. Otherwise, you will be forced to attach a library of procedures to each of your programs. It often turns out to be larger than the file of the document itself. It is not always practical to link a small document file to a large library, most of the procedures of which, most likely, are simply not needed for a given document. It is especially risky to modify the Maple standard library.

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

2.2 Programmatic 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 the 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 science cycle. The idea of integrating these disciplines is undoubtedly very productive, since, on the one hand, it provides a basis for studying these subjects, and on the other hand, it allows the development of information and mathematical culture in the learning process and instilling skills in applied research. At the same time, information technology 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 Technologies and the Maple Package of Symbolic Mathematics".

The program was attended by 10-11 grades of information technology and physical and mathematical profiles. The study of the capabilities of the Maple symbolic mathematics package and its subsequent application was of an applied nature: the students of the physics and mathematics class expanded and deepened their knowledge of mathematics, were able to visualize various mathematical situations, and the information technology classes received useful professional skills as programmers and computer operators ... During the period of implementation of the concept of specialized education at the senior level, it was especially important to introduce into the process of teaching computer science and information technology 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. Also, students needed to instill the skills and abilities of 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 known mathematical packages Maple is optimal for educational purposes. A number of Maple features made it a leading place for the implementation of educational programs: a relatively low cost of the package, a simple and intuitive interface, a programming language closest to the language of mathematical logic, unsurpassed graphical capabilities. All these features make it possible to present a mathematical model of the object or phenomenon under study in a visual interactive graphical form, thereby significantly increasing 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 conducting an elective course "Studying the Maple symbolic mathematics package" (grade 11), 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 the implementation of Maple in 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.

Learning within the framework of this experiment allows you to achieve goals such as self-realization of students and their acquisition of professional competencies, the development of mathematical thinking and scientific creativity of schoolchildren, improving the quality and increasing the effectiveness of the educational process, increasing students' interest in educational activities and interest in its end result, professional orientation students, professional growth of the teaching staff, mastering the methods of information technology, and the creation of computer tools for enhancing the educational process.

In the process of studying the Maple symbolic math package, students practice practical skills in solving math problems using a computer. Maple becomes their learning assistant. Children learn to work on self-control: they solve problems using traditional methods and check the result using Maple. The most interesting and, in the opinion of the students, useful in the elective course program were such topics as "Two-dimensional graphics", "Animation", "Study of function". In the process of studying the Maple application, the students showed a high cognitive interest and a good knowledge of mathematics.

Elective courses are held in various forms: frontal, individual, group. Control and monitoring of knowledge, skills and abilities of students in the study of 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;

Function research and plotting;

Solving geometric problems.

The end result is the project work of each student. Test papers are drawn up 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 goals such as self-realization of students and their acquisition of professional competencies, the development of mathematical thinking and scientific creativity of schoolchildren, improving the quality and efficiency of the educational process, increasing the interest of students to educational activities and interest in its end result, vocational guidance of students, professional growth of the teaching staff, mastery of information technology methods, and the creation of computer tools for enhancing the educational process.

Information support of the educational process is designed to free the student from routine work, allow him to focus on the essence of the material being studied at the moment, consider more examples and solve more problems, and make it easier to understand the material through other ways of presenting the material.

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

The multifunctional package Maple is one of the most powerful math packages. Its capabilities cover quite a few areas of mathematics and can be usefully applied 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 topics of algebra and starting analysis using the Maple package has significantly increased the efficiency of the learning process. By visualizing the material, complex mathematical formulas and transformations become much easier, and the process of mastering the material by high school students is much more efficient.

The possibilities of the Maple package as a means of teaching in the upper grades of secondary school are very extensive and its use in the educational process is a promising direction in modern secondary education.

Bibliography

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

2. Introduction to Maple. A 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 // Informatics and Education.- 1995.- № 1.- P. 3-19.

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

5. Levchenko, I. V. Program and reference materials for teaching practice in informatics: Textbook. manual for ped students. universities and high fur boots [Text] / I.V. Levchenko, O. Yu. Zaslavskaya, L.M. Dergachev, Moscow: MGPU, 2006, 123 p.

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

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

8. Semakin, I. G. Informatics and ICT. Basic course: textbook for grade 9 [Electronic document] / IG Semakin.- (http: www.alleng.ru/edu/comp1.htm). 15.12.08.

9. Ugrinovich, N.D. Informatics and information technology: textbook 10-11 grade [Text] / N.D. Ugrinovich. - M .: Laboratory of basic knowledge, 2002. - 512 p.

10. Ugrinovich, N.D. Workshop on informatics and information technology: textbook 10-11 grade [Text] / N.D. Ugrinovich. - M .: Laboratory of basic knowledge, 2002. - 400 p.

Posted on Allbest.ru

...

10.

Mathcad is a powerful computer mathematics system that combines
in itself a visually oriented input language, a convenient text editor and
formulas, numerical and symbolic processors. The package is simple enough to
study, and the presence of a large number of e-books and "quick
cheat sheets "greatly simplify its use for solving specific
scientific and engineering tasks.

11.

Programs
analytical
calculations

12.

Maxima.
The program is focused on performing calculations and
converting symbolic and numerical expressions, starting from
simplifying algebraic expressions to differentiation,
integration, series expansion, Laplace transform,
solutions of differential equations, tensor problems and
linear algebra

13.

Opportunities
Maxima has the widest range of
funds for
analytical
calculations, numerical
computation and construction
charts. By feature set
the system is close to such
commercial systems,
like Maple and Mathematica. At the same
time she has the highest
degree of tolerance: may
work on all major
modern operating rooms
systems on computers starting
from handheld to the most
powerful.

14.

Programs
constructing
charts

15.

Advanced grapher
Advanced Grapher - Powerful and easy-to-use software for
plotting and analyzing graphs. Supports charting
functions of the form Y (x), X (y), in polar coordinates given
parametric equations, graphs of tables, implicit functions
(equations) and inequalities. Up to 100 charts in one window.

16.

Opportunities
Regression analysis,
finding zeros and extrema
functions,
points of intersection of graphs,
finding derivatives,
equations of tangents and
normals,
numerical integration.
A large number of parameters
graphs and coordinate plane.
Has printing capabilities,
saving and copying charts
in the form of pictures, multi-document
customizable interface.
Supports interface on
Russian and when choosing it
can be used in
non-commercial use for free.

17.

Graph
An open source program designed to build
mathematical graphs. This app supports all
standard functions and allows you to build graphs of sines,
cosines, logarithms, etc.
MagicPlot
A simple application for data analysis, graphing and
nonlinear approximation developed in Russia
ZyukaGraphik
The ZyukaGraphik program is designed for building and researching
graphs set in a tabular way. The program can be useful to everyone
who has to work with datasets presented in the form of two-dimensional
numeric arrays, in particular for registration of measurement results,
registration of laboratory work by students, etc.

18.

Layout programs
mathematical texts.

19.

Latex document preparation system for high-quality typesetting. This
most often used for medium to large technical or
scientific papers, but it can be used for any kind
editions

20.

Opportunities
Hyphenation algorithms, definitions
interword spaces, balancing text
in paragraphs;
automatic generation of content, list
illustrations, tables, etc .;
cross-referencing mechanism
on formulas, tables, illustrations, their numbers
or page;
mechanism for citing bibliographic sources, working with
bibliographic card indexes;
placement of illustrations (illustrations, tables and captions to them automatically
placed on the page and numbered);
design of mathematical formulas, the ability to type multi-line
formulas, a large selection of mathematical symbols;
registration of chemical formulas and structural diagrams
molecules of organic and inorganic chemistry;
design of graphs, schemes, diagrams, syntactic graphs;
design of algorithms, source codes of programs (which may be included in
text directly from your files) with syntax highlighting;
breakdown of the document into separate parts (thematic maps).

Math packages. Information technologies for mathematical calculations on the example of the Maple package

Send your good work in the knowledge base is simple. Use the form below

Similar documents

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Top related articles