Until the 80s, all computers were designed and used exclusively as stand-alone tools, intended primarily for carrying out complex scientific and engineering calculations. Neither the computer architecture nor their software made it possible to combine individual computers into a multi-machine distributed system with the ability for many users to access it. The following factors contributed to the creation of information computing systems and networks (ICS):

1. The emergence of personal computers and the sharp increase in their number.

2. A sharp expansion of communication capabilities based on digital channels, fiber optics and space technology.

3. The need for collective access to computing resources and databases (knowledge), for the exchange of data between users located at long distances.

These factors have led to the widespread use of information and computing systems in which computers are connected to each other, to data banks and to numerous terminal devices.

By IVS we mean a system for collective use, consisting of one or more processors, computers (computers) and providing independent and simultaneous access to its information and computing resources for many users.

Classification of temporary detention facilities.

Analysis of domestic and foreign information and computing systems for processing and transmitting information and studying their capabilities make it possible to classify IVS according to the following criteria:

Methods for managing temporary detention facilities.

Affiliation.

Operating mode.

Work organization.

Structure.

Type of computing environment IVS.

Number of computers (computers).

Performance.

Let's look at these signs.

By control method Temporary detention facilities are divided into centralized, decentralized and mixed.

Centralized are temporary detention facilities in which all functions of managing the technical means of the temporary detention center are performed by one of the computers. An example of such an IVS is teleprocessing systems.

IN decentralized IVS control functions are distributed between computers. Moreover, each computer operates autonomously and performs everything necessary functions for managing the computing process, data processing and, if necessary, transferring information or tasks to another computer. The machine itself initiates such a transfer and controls it. An example of such an IVS is computer networks.

Mixed are IVS in which some of the control functions are performed by the main computer, and some are distributed among other components of the IVS. This control method is often used in local computer networks, where planning and monitoring of network operation, collection and analysis of statistics on its operation is undertaken by the main computer - the network control center (NCC), and control of information transfer between network nodes, control of transmission errors, control Local data processing is carried out by each computer autonomously.

By affiliation Temporary detention centers are divided into departmental (corporate) and territorial.

Departmental are created for data processing in the interests of an individual enterprise, organization, ministry.

Territorial IVS provide access to many, including remote subscribers of a given area and IVS resource, regardless of their departmental affiliation.

Advantages of territorial temporary detention facilities compared to departmental ones:

Lower (20-40%) cost of information processing.

By operating modes From the user’s point of view, IVS are divided into systems with interactive mode, “request-response” mode, batch and real time. Main modes are the first two modes: interactive and “request-response”.

Work in interactive mode conducted in sessions. The user is allocated certain processor, memory, and other resources for the entire session, and is given the opportunity to continuously influence the task processing process.

IN “request-response” mode the system is configured to work with the user only when receiving a request from him, without maintaining contact with him the rest of the time to issue a response .

Local and remote batch processing, from a computer point of view, is a special case of the “request-response” mode. The computer operating system considers a batch processing task entered into the system as one request with a fairly low priority and a large amount of calculations. Batch mode Use only at night. All tasks arriving at the IVS are grouped into packages and then, as memory and processor resources become available, they are launched into the computer for processing.

Direct interaction of the user with the IVS simultaneously in dialogue and “request-response” modes ensures both high efficiency in the use of IVS equipment and maximum efficiency of the user’s work.

According to the principle of work organization IVS distinguish between local, tele- and distributed processing.

IN IVS of local processing there is no data transmission equipment for communication between individual computers and computers with terminals (LAN).

TO IVS with teleprocessing These include computer systems with a local or remote terminal network via communication channels. All management of the subscriber network is, as a rule, centralized and carried out using the central computer of the system. Teleprocessing systems provide remote collective use of computer resources.

IVS that use network teleprocessing or are built in the form of a computer network are called distributed.

By structural principle IVS are divided into computer centers, hierarchical systems, computer networks and terminal complexes (TC).

Computer center is a temporary detention center consisting of several computers concentrated in one place and united organizationally and methodologically. Methodological unification is understood as a combination of the following factors: a unified principle for managing computing facilities on a computer center, the exchange of information between a computer and a computer center, the possibility of reserving one technical means others (computers, electronic devices, peripheral devices).

Hierarchical IVS is a computer center with a main computer (host machine, mainframe, server, superserver), a developed terminal network (network of personal computers) and means of teleprocessing.

Computer network is an IVS consisting of two or more computers or computing centers remote from each other, interacting through communication channels.

It is customary to divide computer networks into a data processing system (DPS) and a data transmission system (DTS). Data processing system- is a collection of computers, subscriber points, operating system networks, functional software designed to solve information and computing problems of network subscribers. Data transmission system- this is a set of communication channels, hardware (switching centers of teleprocessing processors, data transmission multiplexers, network adapters, repeaters, hubs, bridges, routers, switches, data transmission equipment) and software for establishing and implementing telecommunications (communications).

Terminal complex is an IVS consisting of two or more workstations (user stations) and a central computer (group control device, microcomputer, server). In some cases, an additional intermediate computer (micro-computer) can be used.

By type of computing environment IVS can be divided into homogeneous and heterogeneous . Homogeneous IVS contain computers of the same type, for example, ES computers. Heterogeneous IVS includes a computer various types, series, systems, for example, ES computers and SM computers.

By number of computers There are single-machine and multi-machine IVS. The transition from single-machine to multi-machine IVS is due to the following factors:

The need to increase the capacity of temporary detention facilities;

Increasing requirements for operational reliability;

Specialization of individual computers in execution certain functions as part of the temporary detention facility.

By performance IVS are divided into two subgroups: by speed and by the number of serviced terminals of one IVS.

By speed IVS are divided into small (up to 1 million operations/s), medium (from 1 to 10 million operations/s), large (from 10 to 100 million operations/s) and extra-large (more than 100 million operations ./With).

By number of users served IVS are also divided into small (up to 10 terminals), medium (from 10 to 100 terminals), large (from 100 to 1000 terminals), extra-large (more than 1000 terminals).

Network switching. Routing.

1. Switching methods

The basic data network (BDSN) provides information exchange between subscribers by establishing connections passing through nodes and communication lines (Fig. 1).

The most important characteristic of SPD is data delivery time, which depends on the structure of the data transmission system, the performance of communication nodes and the capacity of communication lines, as well as on the method of organizing communication channels between interacting subscribers and the method of transmitting data over the channels.

Information exchange between subscribers can be carried out different ways, which can be divided into two groups: direct switching And switching with intermediate storage.

Direct switching methods establish direct communication between end users through a sequence of intermediate switching nodes. In this case, a single transmission path is formed, which is assigned to the communication session and is monopolized by it. In this case, not a single resource of this path can be used to organize sessions of other users. To organize the path, it is necessary to carry out a special initial phase of establishing a connection. A representative of this group is the circuit switching method.

With intermediate accumulation user information is packaged into data blocks that are transmitted from node to node, stored on them and then, as resources are released in the direction further movement, move on. In this case, only those resources that are used in this moment for block transmission, the remaining path resources are free for any other transmissions. The essence of methods this group will be discussed using examples message and packet switching.

Circuit switching is a serial-parallel method of data transmission with the organization of parallel paths at the level of transmission of information arrays with zero accumulation of data at switching nodes. Circuit-switched networks are organized on the principle of establishing an entire route for transmitting information from sequentially connected communication channels from the sender to the recipient.

Circuit switching provides the allocation of a physical channel for direct data transmission between subscribers. IN starting moment the sender generates a request (challenge) containing the recipient's address. This request travels through the network and at each switching node finds a free transmission line in the direction of the recipient. If it is present, a new stage of the path is physically connected to an already switched path and retained. This is how the entire transmission path is created step by step.

Switching systems can be fully accessible And incompletely accessible depending on whether the sending node can connect to each subscriber or only to some of them. At switching nodes, one of the disciplines for servicing incoming requests can be implemented:

· discipline with refusals;

· discipline with expectation;

· priority discipline.

First discipline with refusals involves abandoning an attempt to establish a connection if at least one free line in the required direction cannot be found at the next switching node. In this case, the node generates a disconnect signal and sends it in the opposite direction. This signal breaks the already formed path, frees the assigned resources and notifies the sender of this fact. The entire connection procedure must be started again. This property limits the application of failure discipline due to the reduced efficiency of network resource use.

When implementing discipline with expectation A queue of requests is organized in the memory of the switching nodes in anticipation of the release of the required communication channel. During the waiting period, the entire already formed section of the path remains in a fixed state and is inaccessible to other sessions. This discipline cannot be implemented in its pure form, since there are no infinitely large buffer memory capacities. When the storage device is full, the switching system goes into failure mode.

Priority discipline based on ranking users or any network resources by priority. A request from a higher priority user interrupts the already established connection of lower priority users. Due to significant organizational limitations, the application of this discipline is very limited.

The process of channel switching and data transmission between SPD subscribers, shown in Fig. 1, subscriber ai initiates connection with the subscriber aj. Communications center A, reacting to the subscriber's address aj, connects the connection, causing the subscriber's line ai switches with the line connecting the node A with knot IN. Then the connection connection procedure is repeated with nodes IN, WITH And D, as a result of which between subscribers ai And aj the channel is switched.

Upon completion of switching, the node D(or subscriber aj) sends a signal feedback(response), which passes unhindered over an already switched channel. After receiving a response the subscriber aj begins to transmit data in real time (in on-line). Data transfer time depends on the length transmitted message, channel capacity (data transmission rate) and signal propagation time along the channel.

When switching channels, there are different schemes spatial And temporal switching

Spatial switching based on physical connection input and output lines using special devices - switches.

Consider the case of switching any of N inputs and N outputs. In Fig. 2 shows an example with N= 6. In this case, the switching circuit is a square switch with a capacity N N. At each switching point where incoming and outgoing lines intersect, there may be semiconductor switch or metal contact, allowing you to establish a connection between any given input and any given output only possible way. In the switch under consideration, a connection between an input and an output is always possible (provided that the required output has not been connected previously, i.e., is not occupied).

This type of switch is non-blocking. Its complexity is characterized by the number necessary points switching, which is usually equal to N2 and N2-N if the inputs and outputs belong to the same terminals between which a connection is to be established. (IN the latter case terminal connected to the incoming line 1 , also connects to the outgoing line i, . Thus, the terminal can both send and receive a call).

Rice. 2. Square switch with a capacity of 6x6

In more general case the switch can have the form of a matrix of size NK. Obviously, if K more or equal N, the switch will be non-blocking. However, when K less than N blockages are possible. In Fig. Figure 3 shows an example of a switch with N=8 And K=4, in which four connections 1-2, 2-1, 3-3, and 4-4 are installed. From this example it is clear that here the number of outputs differs from the number of inputs. Thus, inputs 5-8 are blocked: connections from these inputs cannot be established to any of the output lines.

Rice. 3. Switch with a capacity of 8x4

As the number of users or connected lines increases, the size and complexity of the switching system increases accordingly. As just noted, the complexity of a spatial switch is usually measured by the number of crosspoints required. For example, if you need to switch 100,000 channels and use a square switch for this purpose, then you will need N2=1010 switching points.

Spatial switching circuits are equally suitable for both analog and digital message transmission.

More modern are time switching systems , which are only suitable for digital transmission. These switches are completely analogous to spatial switches, and the analysis of non-blocking properties or blocking is performed in exactly the same way.

To perform time switching, all connections or messages to be switched must first be sampled in a sequence of time samples, with a group of consecutive samples transmitted one at a time physical line, should be cycle (time frame).

Each cycle, when entering the switching system via an incoming line, is recorded in memory. Switching is then carried out simply by reading individual words in any desired (switched) order. The device that performs the specified operation is called time slot switch(KKI). An example of a CCI is shown in Fig. 4. A cycle consists of five time slots, of which only two, X and Y, are considered active and communicating with each other. On the input side, user data X occupies channel 1 and user data Y occupies channel 3. After each cycle is written to memory, a word on channel Y is read or transmitted on time slot X, and a word on channel X is read on time slot Y. More are also possible. complex work patterns.

Rice. 4. Digital channel switching

The switching node must provide mutual connections between channels of different line bundles.

To ensure that every incoming channel is switched with every outgoing it is necessary to be able to rearrange the time intervals of these channels. Rearranging time intervals can be done using storage devices installed at the inputs and outputs of group blocks. In practice, the number of memory cells is usually taken equal to the number temporary channels in a group block.

Since the memory cells installed at the ends of group blocks are designed to store information arriving through channels, we will agree to call it information memory (IM).

In addition to storage devices that store information, switching requires another group of storage devices to store the addresses of channels and switching points that must be turned on when switching the inputs and outputs of the switching system. We will call this group of storage devices control memory (CM).

The advantages of the circuit switching method should include the ability to transmit data and multimedia traffic in real time. The disadvantages are the low efficiency of using network resources and the difficulty of establishing communication (in some cases, failure or unacceptable big time establishing a physical connection).

Message switching is carried out by transmitting a data block (message), into which all information assigned for transmission is packaged. The message contains a header containing the address (required) and other service information, and the data itself. The message is sent along a route determined by network nodes. The message header indicates the subscriber's address aj- the recipient of the message. Message generated by the sender - subscriber ai, is fully accepted by the node A and is stored in the node memory. Knot A processes the message header and determines the message route leading to the node IN. Knot IN receives the message, placing it in memory, and upon completion of reception, processes the header and outputs the message from memory to the communication line leading to the next node. The process of receiving, processing and transmitting a message is repeated sequentially by all nodes on the route from the subscriber ai to the subscriber aj. Meaning T determines the delivery time of data when switching messages. This time will generally be quite large, since the message cannot be transmitted further until it is completely received and processed by the current node.

Advantages of the message switching method are: increasing the efficiency of using network resources and the absence of monopolization of transmission path resources, since they are immediately released after transmission and processing of the message. Main disadvantage of the method is long transmission time, especially in extended blocks. In addition, switching nodes require large amounts of buffer memory for intermediate storage of all messages arriving at the node.

Packet switching is carried out by breaking the message into packets - message elements equipped with a header and having a fixed maximum length - and then transmitting the packets along a route determined by network nodes. Data transmission during packet switching occurs in the same way as during message switching, but the data is divided into a sequence of packets 1, 2, ......, the length of which is limited by a limit value, for example, 1024 bits.

Packet switching in IVS - main method of data transfer. This is partly due to the fact that packet switching leads to low delays when transmitting data through the data transmission system, as well as the following circumstances.

Firstly, the channel switching method requires that all connecting lines from which the channel is formed have the same throughput, which extremely tightens the requirements for the structure of the data transmission system. Message and packet switching allows you to transfer data over communication lines with any bandwidth.

Secondly, presenting data in batches creates best conditions for multiplexing data streams.

Thirdly, the short length of packets makes it possible to allocate less memory capacity for intermediate storage of transmitted data than is required for messages. In addition, the use of packets simplifies the task of managing data flows, since to receive a stream of packets in communication nodes, less memory needs to be reserved than to receive a stream of messages.

Fourthly, the reliability of data transmission over communication lines is low. A typical communication line provides data transmission with a probability of distortion of 10-4. The longer the length of the transmitted message, the more likely that it will be distorted by interference. The packages, having a small length, are to a greater extent guaranteed against distortion than messages. In addition, distortion is eliminated by re-querying the data (method of automatic request in case of error - ARQ: Automatic ReQuest). Packets are much more consistent with the re-request mechanism than messages and provide the best utilization of link capacity in an interfering environment. These circumstances led to the use of packet switching as the main method of organizing communication channels in the SPD IVS.

Separation of channels by time and frequency

Computing system architectures

Construction principles computer networks. Characteristics of computer networks

Computer network – a network of exchange and distributed information processing, which is formed by many interconnected subscriber systems and communication means. Transmission media are focused on the collective use of network-wide resources - hardware, information and software.

Subscriber system (AS) – a set of computers, software, peripheral equipment, communication equipment, computers that perform application processes, a communication subnetwork (a telecommunication system is a set of physical environment transfer of information, hardware and software that ensure the interaction of the speakers).

Application process – various procedures for processing, storing, and outputting information that are performed in the interests of the user. With the advent of networks, two problems were solved:

1) provision, in principle, unlimited access to the computer

users, regardless of their geographical location;

2) the ability to quickly move large amounts of information over any distance.

The following circumstances are of fundamental importance for networks:

Computers located in different systems of the same network communicate with each other automatically;

Each computer on the network must be adapted to work both in standalone mode under the control of its own OS, and to work as an integral part of the network;

Network computers can operate in various modes: data exchange between speakers, requesting and issuing information, collecting information, batch data processing, etc.

The network hardware consists of: computers of various types; means of communication; AC equipment; equipment of communication centers; communication equipment and coordination of networks of the same level or different levels. The main requirements for computer networks are versatility and modularity. Information Support network is a unified information focused on tasks solved in the network and containing data arrays available to all network users and arrays for individual users.

VS software automates the processes of programming tasks, processing information, planning and organizing collective access to communication and computing resources of the network. The software also dynamically distributes and redistributes these resources.

Types of aircraft software:

General network software, which is formed by a distributed network OS and software included in the complex of maintenance programs;

Special software represented by application software: functional and integrated software packages, libraries of standard programs, as well as programs that reflect the specifics of the subject area;

Basic computer software, including OS, programming automation systems, monitoring and diagnostic test programs.

Classification of computer networks.

The classification of CS is based on the most characteristic, functional and informational features.

By degree territorial distribution network elements. Thus, networks are global, regional and local. The global CS unites ACs concentrated on large territory, covering various countries and continents. The interaction of the AS is carried out on the basis of various territorial communication networks, which use telephone lines, radio, and satellite communications. Regional CS unite AS located at a considerable distance from each other within one country, region, big city. A local CS connects speakers located within a small area. Its length is limited to a few kilometers.

A separate class consists of corporate CS. The corporate network refers to technical base corporations. She plays the leading role in planning, organizing

produced by the corporation.

According to the control method, CSs are divided into networks with centralized, decentralized and mixed control. Based on topology, networks can be divided into two classes: broadcast and serial. For broadcast configurations, at any given time, only one workstation can work to transmit a unit of information, and the rest can receive this frame. Basic types of broadcast configuration:

Ü chain;

Ü a star with an intellectual center;

Data transfer methods

v Wired communication

Ø PSTN telephone network

§ Modem and dial-up

Ø Leased lines

Ø Packet switching

Ø Transmission via fiber optic cable

§ Synchronous optical networking

§ Fiber distributed data interface

v Wireless

Ø Short range

§ Human Area Network

Ø Medium range

§ IEEE 802.16e WiMAX

Ø Long range

§ Satellite connection

§ Data transfer using mobile phones

IEE 802.16e WiMAX

IVS – two or more computers connected via data transmission channels (wired or radio communication lines, optical communication lines) for the purpose of combining resources and exchanging information. Resources refer to hardware and software.

Connecting computers into a network provides the following basic capabilities:

Programs on a computer are downloaded from the network;

There is no need to have a hard drive on your computer;

Saves money and time on purchasing and updating software, because this is done through the network;

Data sharing – the ability to create distributed databases located in the memory of individual computers and manage them from peripheral workstations;

Software sharing – the ability to share software;

Multiplayer mode.

The IVS must be reliable - the failure of any computer should not lead to a stop or malfunction of the system; moreover, the functions of the failed computer must be transferred to another computer on the network.

There is a trend towards connecting computers into networks, this is due to a number of reasons:

1. the need to receive and transmit information at your workplace;

2. the need for rapid exchange of information between users;

3. the ability to quickly obtain a variety of information, depending on its location;

4. there is access to e-mail and Internet resources.

Support for network operation, updating and installation of software, etc. is provided by providers who maintain the network for a subscription fee.

Classification of temporary detention facilities.

IVS can be classified according to various signs, For example:

Around the territory.

· local area networks (LAN) cover small areas with a diameter of 5-10 km. They are created within individual offices, institutions, enterprises, universities, exchanges, banks, etc. With the help common channel LAN connections can be connected from tens to hundreds of PCs.

The combination of several LANs within several buildings (or one) of one corporation is called a corporate (intra) network.

· Regional and global IVS are formed by combining Local LANs in individual territories or throughout the planet. The largest global network– Internet.

According to the control method.

· Networks with centralized management, in which one or more computers are allocated that manage the process of data exchange over the network. These PCs are called servers. Workstations are the remaining computers on the network. Workstations have access to server disks and network printers. Workstations do not contact each other. And to exchange data, users are forced to use server disks. An example of such a network is the Novell NetWare network.

· Decentralized (peer-to-peer) networks do not contain servers. Each workstation can also act as a server. Network management functions are transferred in turn from one workstation to another. Workstations have access to the disks and printers of other workstations. An example network is Windows for Workgroups.

Networks can be divided into public, private and commercial. By recommendation international organization Protocols (for the physical layer) define the following classes of public networks:

Up to 1000 km – medium length;

Up to 10,000 km – long;

Up to 25,000 km - the longest on land;

Up to 80,000 km – trunk routes via satellite;

Up to 160,000 km – international trunk routes via 2 satellites.

System m linear equations c n called unknowns system of linear homogeneous equations if all free members are equal to zero. Such a system looks like:

Where and ij (i = 1, 2, …, m; j = 1, 2, …, n) - given numbers; x i– unknown.

A system of linear homogeneous equations is always consistent, since r(A) = r(). It always has at least zero ( trivial) solution (0; 0; …; 0).

Let us consider under what conditions homogeneous systems have non-zero solutions.

Theorem 1. A system of linear homogeneous equations has nonzero solutions if and only if the rank of its main matrix is r less number unknown n, i.e. r < n.

□ 1). Let a system of linear homogeneous equations have a nonzero solution. Since the rank cannot exceed the size of the matrix, then, obviously, r ≤ n. Let r = n. Then one of the minor sizes n n different from zero. Therefore, the corresponding system of linear equations has only decision: . . . This means that there are no other solutions other than trivial ones. So if there is non-trivial solution, That r < n.

2). Let r < n. Then the homogeneous system, being consistent, is uncertain. So she has infinite set decisions, i.e. has non-zero solutions. ■

Consider a homogeneous system n linear equations c n unknown:

(2)

Theorem 2. Homogeneous system n linear equations c n unknowns (2) has non-zero solutions if and only if its determinant is equal to zero: = 0.

□ If system (2) has a non-zero solution, then = 0. Because when the system has only a single zero solution. If = 0, then the rank r the main matrix of the system is less than the number of unknowns, i.e. r < n. And, therefore, the system has an infinite number of solutions, i.e. has non-zero solutions. ■

Let us denote the solution of system (1) X 1 = k 1 , X 2 = k 2 , …, x n = k n as a string .

Solutions of a system of linear homogeneous equations have the following properties:

1. If the line is a solution to system (1), then the line is a solution to system (1).

2. If the lines And - solutions of system (1), then for any values With 1 and With 2 their linear combination is also a solution to system (1).

The validity of these properties can be verified by directly substituting them into the equations of the system.

From the formulated properties it follows that any linear combination of solutions to a system of linear homogeneous equations is also a solution to this system.

System of linearly independent solutions e 1 , e 2 , …, e r called fundamental, if each solution of system (1) is a linear combination of these solutions e 1 , e 2 , …, e r.

Theorem 3. If rank r coefficient matrices for system variables linear homogeneous equations (1) are less than the number of variables n, then any fundamental system of solutions to system (1) consists of n–r decisions.

That's why common decision system of linear homogeneous equations (1) has the form:

Where e 1 , e 2 , …, e r– any fundamental system of solutions to system (9), With 1 , With 2 , …, with p – arbitrary numbers, R = n–r.

Theorem 4. General solution of the system m linear equations c n unknowns is equal to the sum of the general solution of the corresponding system of linear homogeneous equations (1) and an arbitrary particular solution of this system (1).

Example. Solve the system

Solution. For this system m = n= 3. Determinant

by Theorem 2, the system has only a trivial solution: x = y = z = 0.

Example. 1) Find general and particular solutions of the system

2) Find fundamental system decisions.

Solution. 1) For this system m = n= 3. Determinant

by Theorem 2, the system has nonzero solutions.

Since there is only one independent equation in the system

x + y – 4z = 0,

then from it we will express x =4z- y. Where do we get an infinite number of solutions: (4 z- y, y, z) – this is the general solution of the system.

At z= 1, y= -1, we get one particular solution: (5, -1, 1). Putting z= 3, y= 2, we get the second particular solution: (10, 2, 3), etc.

2) In the general solution (4 z- y, y, z) variables y And z are free, and the variable X- dependent on them. In order to find a fundamental system of solutions, we assign free variable values: at first y = 1, z= 0, then y = 0, z= 1. We obtain partial solutions (-1, 1, 0), (4, 0, 1), which form the fundamental system of solutions.

Illustrations:

Rice. 1 Classification of systems of linear equations

Rice. 2 Study of systems of linear equations

Presentations:

· Solution of SLAU_ matrix method

· Solution SLAE_Cramer method

· Solution SLAE_Gauss method

· Solution packages mathematical problems Mathematica, MathCad: search for analytical and numerical solution systems of linear equations

Control questions :

1. Define a linear equation

2. What type of system does it look like? m linear equations with n unknown?

3. What is called solving systems of linear equations?

4. What systems are called equivalent?

5. Which system is called incompatible?

6. What system is called joint?

7. Which system is called definite?

8. Which system is called indefinite

9. List the elementary transformations of systems of linear equations

10. List the elementary transformations of matrices

11. State the application theorem elementary transformations to a system of linear equations

12. What systems can be solved using the matrix method?

13. What systems can be solved by Cramer's method?

14. What systems can be solved by the Gauss method?

15. List 3 possible cases, arising when solving systems of linear equations using the Gauss method

16. Describe the matrix method for solving systems of linear equations

17. Describe Cramer’s method for solving systems of linear equations

18. Describe Gauss’s method for solving systems of linear equations

19. What systems can be solved using inverse matrix?

20. List 3 possible cases that arise when solving systems of linear equations using the Cramer method

Literature:

1. Higher mathematics for economists: Textbook for universities / N.Sh. Kremer, B.A. Putko, I.M. Trishin, M.N. Friedman. Ed. N.Sh. Kremer. – M.: UNITY, 2005. – 471 p.

2. General course of higher mathematics for economists: Textbook. / Ed. IN AND. Ermakova. –M.: INFRA-M, 2006. – 655 p.

3. Collection of problems in higher mathematics for economists: Tutorial/ Edited by V.I. Ermakova. M.: INFRA-M, 2006. – 574 p.

4. Gmurman V. E. Guide to solving problems in probability theory and magmatic statistics. - M.: graduate School, 2005. – 400 p.

5. Gmurman. V.E Probability theory and math statistics. - M.: Higher School, 2005.

6. Danko P.E., Popov A.G., Kozhevnikova T.Ya. Higher mathematics in exercises and problems. Part 1, 2. – M.: Onyx 21st century: Peace and Education, 2005. – 304 p. Part 1; – 416 p. Part 2.

7. Mathematics in economics: Textbook: In 2 parts / A.S. Solodovnikov, V.A. Babaytsev, A.V. Brailov, I.G. Shandara. – M.: Finance and Statistics, 2006.

8. Shipachev V.S. Higher mathematics: Textbook for students. universities - M.: Higher School, 2007. - 479 p.

Related information.

Let M 0 – many solutions homogeneous system(4) linear equations.

Definition 6.12. Vectors With 1 ,With 2 , …, with p, which are solutions of a homogeneous system of linear equations are called fundamental set of solutions(abbreviated FNR), if

1) vectors With 1 ,With 2 , …, with p linearly independent (i.e., none of them can be expressed in terms of the others);

2) any other solution to a homogeneous system of linear equations can be expressed in terms of solutions With 1 ,With 2 , …, with p.

Note that if With 1 ,With 2 , …, with p– any f.n.r., then the expression k 1× With 1 + k 2× With 2 + … + k p× with p you can describe the whole set M 0 solutions to system (4), so it is called general view of the system solution (4).

Theorem 6.6. Any indeterminate homogeneous system of linear equations has a fundamental set of solutions.

The way to find the fundamental set of solutions is as follows:

Find a general solution to a homogeneous system of linear equations;

Build ( n – r) partial solutions of this system, while the values ​​of the free unknowns must form an identity matrix;

Write out general form solutions included in M 0 .

Example 6.5. Find a fundamental set of solutions next system:

Solution. Let's find a general solution to this system.

~ ~ ~ ~ Þ Þ Þ There are five unknowns in this system ( n= 5), of which there are two main unknowns ( r= 2), there are three free unknowns ( n – r), that is, the fundamental solution set contains three solution vectors. Let's build them. We have x 1 and x 3 – main unknowns, x 2 , x 4 , x 5 – free unknowns

Values ​​of free unknowns x 2 , x 4 , x 5 form the identity matrix E third order. Got that vectors With 1 ,With 2 , With 3 form f.n.r. of this system. Then the set of solutions of this homogeneous system will be M 0 = {k 1× With 1 + k 2× With 2 + k 3× With 3 , k 1 , k 2 , k 3 О R).

Let us now find out the conditions for the existence of nonzero solutions of a homogeneous system of linear equations, in other words, the conditions for the existence of a fundamental set of solutions.

A homogeneous system of linear equations has non-zero solutions, that is, it is uncertain if

1) the rank of the main matrix of the system is less than the number of unknowns;

2) in a homogeneous system of linear equations, the number of equations is less than the number of unknowns;

3) if in a homogeneous system of linear equations the number of equations is equal to the number of unknowns, and the determinant of the main matrix is ​​equal to zero (i.e. | A| = 0).

Example 6.6. At what parameter value a homogeneous system of linear equations has non-zero solutions?

Solution. Let's compose the main matrix of this system and find its determinant: = = 1×(–1) 1+1 × = – A– 4. The determinant of this matrix is ​​equal to zero at a = –4.

Answer: –4.

7. Arithmetic n-dimensional vector space

Basic Concepts

In previous sections we have already encountered the concept of a set of real numbers located in in a certain order. This is a row matrix (or column matrix) and a solution to a system of linear equations with n unknown. This information can be summarized.

Definition 7.1. n-dimensional arithmetic vector called an ordered set of n real numbers.

Means A= (a 1 , a 2 , …, a n), where a iО R, i = 1, 2, …, n– general view of the vector. Number n called dimension vectors, and numbers a i are called his coordinates.

For example: A= (1, –8, 7, 4, ) – five-dimensional vector.

All set n-dimensional vectors are usually denoted as Rn.

Definition 7.2. Two vectors A= (a 1 , a 2 , …, a n) And b= (b 1 , b 2 , …, b n) of the same dimension equal if and only if their corresponding coordinates are equal, i.e. a 1 = b 1 , a 2 = b 2 , …, a n= b n.

Definition 7.3.Amount two n-dimensional vectors A= (a 1 , a 2 , …, a n) And b= (b 1 , b 2 , …, b n) is called a vector a + b= (a 1 + b 1, a 2 + b 2, …, a n+ b n).

Definition 7.4. The work real number k to vector A= (a 1 , a 2 , …, a n) is called a vector k× A = (k×a 1, k×a 2 , …, k×a n)

Definition 7.5. Vector O= (0, 0, …, 0) is called zero(or null vector).

It is easy to check that the actions (operations) of adding vectors and multiplying them by real number have the following properties: " a, b, c Î Rn, " k, lО R:

1) a + b = b + a;

2) a + (b+ c) = (a + b) + c;

3) a + O = a;

4) a+ (–a) = O;

5) 1× a = a, 1 О R;

6) k×( l× a) = l×( k× a) = (l× k)× a;

7) (k + l)× a = k× a + l× a;

8) k×( a + b) = k× a + k× b.

Definition 7.6. A bunch of Rn with the operations of adding vectors and multiplying them by a real number given on it is called arithmetic n-dimensional vector space.