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September 17, 1959
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 , STAT - 0 , IINCriii,SSX - -- ' 41 . -...f., Iig I. .-Illitui.,---:;--. to.`xt: ,=1?. - : i i.: ! _ .. ' CLASS1POC#110K:c _ Ata enrauciirica? riroierm- - nitoot, COWEN OE AMA WORT CONCIIIM6 , ugsR . , 1, DATE Of REPORT 17 Sep 1959 . i , i 1 , sumo (Descriptive side. Use iiittividaai reports /yr separate saltier's) , Soviet Document: "Solutions of Engineering Problems on Automatic Computers by B. M. Kagan and T. M. Ter -Mikaelyan 2. The translation consists of two title pages, one page of translator's --, notes, 294 pages of text, five pages of Supplement #1: "List of Operations: of the Computer M-3", four pages of Supplement #2: "List of Operations of the Computer 'Dral'", and four pages of reference, a total of 310, pages. V.- One Copy of orig rpt in Russian (UNCL) ?011 STAT TAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 LASSIYIIDL B. IL KAGAN sad T. M. TER?IIIKAELTAlt SOLUTIONS OF ENGINEERING PROBLEMS ON AUTOMATIC DIGITAL COMPUTERS GoSENERGOIZDAT (STATE POWER ENGINEERING PUBLISHING HOUSE) UNCLASSIFIED STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STATE POWER ENGINEERING PUBLISHING ROUSE MOSKVA 1958 LENINGRAD NCL.A-S:SIPIED Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 ' -14 STAT Pig. 4 of 314 Remarks: Is Rome compopsnts of the .formulas or/and Complex symbols in the original text have been found to be outright illegible. :Such components have been replaced by quest+n markt?, 2. This translation has been made by three trans- latori in succession, which has resulted in a slight difference in the choice of English Terms. So, for example, the Russian word "razryad" has been translated at first as "order", then as "digit" and then as "column". _ URC LA: rnD Declassified in Part - Sanitized Copy Approved for Release STAT 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 ?,,oprfoc,7'' Ifilit4TV601F"-.1155.r. STAT 5 of 314 The book deals with questions concerning the application of automatic digital eeAP4ei'i'(ATsTM) for engineering calculations and , research. It explains the functioning principle of an automatic digital computer as well as the preparation and programming procedures for solving mathematical 1*4s-. Examples of the application of automatic digital computers for solving engineering pAblems-wors examined (Inlivegtition of trans- ition processes in long-distance power linos, calculations of the stability of automatic controlgyitems, investigations concerning the critical speed of turbogeheritor rotors, calculations for a series of electric motors forloweet cost). Although all examples concern themselves with, electric devices the tasks presented have the character of general engineering problems. The book was written for scientific workers, engineers, post- graduate students and senior students of higher educational insti- tutions Authors: Boris Moiseyevich Kagan and 1'c:odor Milchaylovich Ter-Mikaelyan Solution of Engineering Problems by Automatic Digital Computers. Editor:V.M. Kurochkin Technical Editor: (}.Ye. Larionov CRC LAAS npriaRRified in Part - Sanitized COPY Approved for Release STAT ii-,,- 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 ' Declassified in Part- Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED PREFACE Page 6 of 314 Pages The theoretical study of most problems encountered in various branches of engineering is. reduced, to mathematical problems the Vi- gorous solution of whiOh either cannot be found or is so complicated that it is difficult to use it in calculations. For the solution of these problems various approximatemethods have been proposed,. Which make it possible to Obtain the answer in numerical form. However,. the great number of arithmetical operations necessary for applying the methods mentioned,- made them up to recently practically inappli- cable. High-speed electronic computers which have appeared during the last 10 or 15 years considerably expanded the range of solvable problems, and at present have penetrated into the practice of not only scientific but also into engineering-technical investigations. Taking into consideration difficulties arising with engineers who turn to the computers', aid in their activities, the authors made an attempt to compose a book providing basic information on the opera- tional principles of the automatic digital computers and on the pos- sibilities of their employment for engineering investigations and calculations. The book alms to give the reader a sufficient guide for independent programming and formulation of engineering problems for the computers admitting a numerical method of solution. The first four chapters of the book acquaint the reader with the design of automatic digital computers and the methods of pro- gramming mathematical problems. The lectures of Prof. A.A. Lyapunov, delivered by him at the Moscow University in 1954-19551 and also the train of ideas developed by Prof. A.A. Lyapunov and his mathematicians group during the seminar on mechanical mathematics in the IOU (Moscow State University), exerted a great influence on the contents of chapters 3 and 4 and in particilar on paragraph 4-1. These chapters lay down only the foundations of the method and should not be con- sidered as a complete course of the theory and practice of programming. Chapters 5 to 8 describe logical schemes of the programs for a series of practical problems through the example of which the reader can gain an acquaintance with the methoda of solving engineering problems on automatic digital computers. Although the majority of the questions considered here is of the nature of general engineering probleastsuch as the study of transient processes, calculation of the stability of dynamic systems and investigation of resonance phenomena in intricate structures, the exposition of these problems is given UNCLASSIFIED ITharlaccifiPri in Part - Sanitized Copv Approved for Release STAT STAT 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Co .y Approved for Release ? 50-Yr 2014/02/10 CIA-RDP81-01043R003800160006-3 r, .--74,2c44 STAT Page 7 of 314 Pages a-definite electroteOhni7oal trend. It is quite unde;standOble that the authors were not able, even to a small degree* to reflect the diversity of engineering problems for whose solving thet-digi;tal computers are presently employed. The authors hope, hewevert, that the reader will be able to prepare him-. salt independently aid* atter gaining familiarity with the methods of programing described ? in chapters 3 and 4-and with some examples considered in chapters. 5 to 8, to solve on the computer any problem that may arise. At that it is of course assumed that the problem is mathematically formulated and a numerical method for its solution is available. Chapter 9, which is of a reference natures offers brief informa- tion on approximate methods for solving certain mathematical problems. The basic literature on this problem is cited at the and of the book. In 1956, the group of employees of the LUMS of the USSR Academy of Sciences, the XI for the electric engineering industry of the State Planning Committee of the USSR, and Academy of Sciences of the Armenian SSR, under the general supervision of the Corresponding Member of the AS USSR, I.S. BrUk, and Limber of the AS of the Armenian SSR, A.G. Iosiflyans constructed, the M-3 oomputer. The book generalizes the experience gained by the authors who participated in the work of the group of employees of the LUMS AS USSR, the MIX BP of the Gosplan USSR and the AS Armenian SSR on the designing of the M-3 -computer and on conducting a series of engineer- ing investigations on the M-3 computer in the NIX BP Gosplan USSR and on the BMX computer in the ITMIVT AS USSR. Being one of the first publications of this kind, the book cannot be free of drawbacks. The authors will be thankful for all the remarks and suggestions which the readers will make. In conclusion, the authors use this occasion to express their deep gratefulness to Prof. A.A. Lyapunov and Docent V.A. Zurochkin for a nuMber of valuable indications which the authors took into consideration in preparing the manuscript for publication. Chapters 2, 6, 7, 8 and paragraphs 1-1/ 1-2, 1-4 wers'eritten by B.M. Xigan; chapters 3, 4, 5, 9 and paragraphs 1-6, 16.7, 1-8, 1-9 were written by T.M. Ter-Mikamlyan, and paragraphs 1-3 and 1-5 were written by the authors jointly. The Authors UNCLASSIPIED - Cmni+17ari r.nnv Anoroved for Release STAT 50-Yr 2014/02/10 CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED CONTENTS Preface Page 8 of 314 Pages 3 First Chapter. Fundamentals 7 1-1. Introduction 7 1-2. On numerical methods of solving mathematical problems 10 1-3. The block-diagram of "ATsVMH 12 1-4. Systems of counting 14 1-5. Computers with a floating and fixed decimal point 16 1-6. Coding of commands 18 1-7. Certain operations performed by the digital computers 22 1-8. Control operations 23 1-9. Command cods of a conventional computer 25 Second Chapter. Operational principles of the automatic digital computers. 26 2-1. The notions of the subsequent and parallel codes 26 2-2. Basic electronic parts of the "ATM" 27 2-3. Circuits for performing elementary logical operations 32 2-4. The performance of certain operations by means of logical circuits 34 2-5. Peculiarities of performing arithmetical operations on a computer. The concepts of the complementary and re- verse codes 37 2-6. Arithmetical devices 41 2-7. Memory devices 50 2-8. Devioes for input and output 54 2-9. Control units 55 2-10. Main characteristics of digital computers 58 2-11. The universal digital computer M-3 59 2-12. The universal digital computer "Ural" 62 Third Chapter. Programming technique 64 3-1. The simplest example of the program 64 3-2. Conversion of the cell contents 66 3-3. Programs with the automatic choice of the number of cycles 70 3-4. Operation of command adding 73 3-5. Transformation of commands in programs 74 3-6, Examples of more complicated programs 77 3-7. Examples of programa for the M-3 computer 80 3-8. Conversion of numbers from the decimal system into the binary one and vice versa 87 UNCLASSIFIED STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Ap roved for Release ? 50-Yr 2014/02/10 CIA-RDP81-01043R003800160006-3 UNC Page 9 of 314 Pampa 3-9. Separation of the integral and fractional parts in 00a- puters with the floating and fixed point 90 Chapter Four. Programming of mathematical problems 94 4-1. The scheme of a program 94 4-2. The program of solving ordinary differential equations by the Runge-Kutta method 97 4-3. The program of calculating a determinant. The trans- formation of commands in several cycles. 100 4-4. Solution of algebraic and transcendental equations 103 4-5. Storing of functions in the computer 106 Chapter Five. Determination of critical revolution numbers for the rotors of turbogenerators 108 5-1. The definition of the problem 108 5-2. The solution of the problem on a BESM computer 110 Chapter Six. Determination of stability of automatic control eysteme on digital computers 6-1. Basic information 6-2* The scheme of a general program for calculating the regions of stability and equistable lines in the plane of two parameters 6-3. An example of calculating the static stability of a long-distance electric power transmission line 115 115 118 124 Chapter Seven. Calculation and study of transient processes 127 7-1. Preliminary remarks 127 7-2. The logical scheme of the program of integrating a system of ordinary differential equations by the Runge- Kutta method with a constant step 127 7-3. The logical scheme of the program for integrating a system of ordinary differential equations by the Runge- Kutta method with the automatically selected step 132 7-4. The calculation of dynamioal stability of distant electric power lines 135 Chapter Eight. Application of digital computers for calculat- ing electric machines 143 8-1. General remarks 143 8-2. The formulation of the calculating problem 143 8-3. Mathematical treatment of the problem. A remark on the linear and non-linear programming 145 8-4. A method of selecting the optimum variant of a motor with the "ATsVM1'. The logical scheme of the program 146 UNCLASSIFIED STAT STAT 'VA' MA rnnv Anoroved for Release @50-Yr 2014/02/10 CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 UNCLASSIFIED STAT Page 10 of 314 Pages Chapter Nine. Some information on approximate calculations 151 9-1. Theory of errors 151 9-2. Solution of algebraic and transcendental equations 154 9-3. Interpolation of functions 158 9-4. Numerical differentiation and integration of functions 161 9-5. Solution of ordinary differential equations 164 Appendix 1. The list of operations of the M-3 computer 170 Appendix 2. The list of operations of the "Ural" computer 171 Bibliography 174 UNCLASSIFIED STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED FIRST CHAPTER. Fundamentals. 1-1. Introduction SI-AT Page 11 of 314 Pasts The development of technique calls for the increased capacity of individual waits and machines, the intensification of technical processes, the increase of speeds, temperatures, pressures, stresses in the structural materials of machines and apparatuses, the raising of reliability, and speedy and precise operation of various devices. The solution of all these problems is impossible without a deep and all-sided study of processes taking place in the machines, appara- tuses and complicated circuits. In many cases mathematical relations describing the processes in the studied devices turn out to be very complicated duo to the intricacy of a circuit, the presence of elements with distributed parameters, phenomena of saturation and other non-linearities. As a result the theoretical investigation by the conventional methods becomes practically impossible. The engineer-investigator who faces such difficulties can often resort to the physical simulation of the phenomenon under study. At the present time, for instance, physical models of electric trans- mission lines are effectively used for studying processes occurring in these complicated systems. Physics.), simulation is of a special significance when there is no reliable mathematical description of the phenomenon being studied. However, only certain physical phe- nomena can be studied with such a model, and any more or less essen- tial modifications of the original parameters may call for the de- signing of a new model. Modern computers open new possibilities. They can be divided into two large groups: a) electronic simulating analog computers, and b) electronic high-speed digital computers. In electronic simulating analog computers, circuits are devised by means of electronic tubes, capacitors, resistors and some other elements, in which the changes in currents and voltages with time are described by the same differential equations as the phenomenon under investigation. The solution of equations is obtained in the form of osoillograms of corresponding voltages in the circuits of the simulating device. These voltages-analogs of the variable UNCLASSIFIED STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 12 of 314 Pages quantities of the problem being solved vary continuously with time, if this corresponds to the initial equations. Electronic simulating devices are also called computers of continuous action. Their accuracy amounts to 5 to 10%. Electronic simulating computers are not universal devices. They are convenient for solving such problems which in mathematical respect are reduced to ordinary differential equations with constant and variable coefficients(1). Electronic analog computers have been nainly applied for investigating automatic control systems. This is because it is comparatively simple to con- nect an analog device with the operating equipment. During the last years automatic high-speed digital computers (ATent) came into wide use. These computers operate with numerical quantities presented in digital form. The quantities cannot vary continuously in a digital computer. They vary discontinuously, assum- ing individual elementary arithmetical operations (addition, sub- traction, multiplication and division of numbers). The control of the calculating process is carried out automatically according to a program worked out in advance. Main advantages of digital computers consist in their universality and accuracy of operation. High-speed digital computers make it possible to solve a very wide range of problems. It is necessary only that the problem should admit of a numerical method of solution. The accuracy of these computers is high, because the calculations are usually per- formed up to 9th or 10th digits. The calculating accuracy with di- gital computers is not limited by anything and depends only on the number of orders in the numbers with which the computer operates. Digital computers operate at an enormous speed, performing thousands and tons of thousands of operations per second. For ex- ample, the BESU computer designed under the guidance of Academician S.A. Lebedev performs 8,000 to 10,000 operations per second. High-speed digital computers can perform not only arithmetical but also some logical operations. It is possible therefore to auto- mate the process of calculations and to carry out the automatic selection (depending on the satisfaction of certain conditions) of one of several variants in the course of calculations. Although the calculations with these computers call for a preliminary rather labor- consuming work on the compiling of the program of calculations (so- called "programming of a problem") the labor applied is wholly paid off, if a complicated problem is being performed. (1) There exist also analog computers for solving equations with partial derivatives. UNCLASSIFIED STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 .01??? STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 UNCLASSIFIED STAT Page 14 of 314 Pages Studies of losses in the grids, current distribution, and cal- culations of economic lost distribution, are also performed with digital computers. The use of computers for determination of economically optimum regimes of power system operation has a bright prospect. Those cal- culations will determine the grambs of economically reasonable load distribution between individual thermal and hydropower plants of a power system, which take into account the graphs, of the total load of the system, losses in the lines, fuel cost, water levels in re- servoirs, natural discharges, and efficiency factors of individual units. The possibility of determination of economically reasonable load distribution with the ATall makes it possible to automate the dispatcher control of power systems. The ATsTM's are used, in the field of power engineering equip- ment, for designing atomic reactors, for calculating the thermal balance of turbines, for selecting the optimum regime of power units operation, and for studying the strength and vibration of units. It is known, how important it is for high-capacity generators to determine the critical rates of rotor rotation, and to establish the relation between the values of critical speeds and the structural characteristics of the rotor and bearing'. It is practically impossible to carry out any complete investi- gation of this problem by conventional methods in view of the large amount of labor consumption for calculations. Computers, however, make it possible to solve this problem. In a similar way, computers can be applied for calculating resonance frequencies of turbogenerator panels, operating regimes of hydrogenerator footstep bearings, for investigating temperature distribution fields, air velocities in channels, and other problems of heating and ventilation of large electrical machines. Digital computers find application in calculations and investi- gations of stability and various operating regimes in complicated systems of automatic control and regulation, in investigations of dynamic processes in complicated systems of electric drive, in con- trol and regulation circuits containing various non-linear devices, such. as saturation choking coils, mercury rectifiers, etc. They can be used for calculating optimum processes in automatic control systems under certain limitations (limitation of speed, moment, etc.), for synthesis of automatic control systems with prescribed characteristics. Of great interest is the calculation study with the ATsni of UNCLASSIFIED STAT npnlassified in Part - Sanitized COPY Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 13 of 314 Pages The performance of certain logical operations by the automatic digital computers opens_ new possibilitlea in designing the eystema of automatic control and regulation. These possibilities are brought about by means of switching a computer into the system of automatic control. High-speed computers are a powerful means for investigating and calculating complicated engineering problems in various fields of engineering. They make it possible to mathematically (numerically) simulate operating processes in devices of various kinds. While new technique is being designed, many efforts and means are spent for constructing and investigating various physical models and the manufacture of intermediate experimental specimens. However, if differential equations describing the processes in the device being constructed are known, then it is possible, with an "ATsVM", to cal- culate in a short time working processes for a large number of de- signing variants and to choose the best one. With the aid of an ATsVM, it is possible to determine the optimum processes in operation of a complicated machinery by means of calculations. Such a way permits to reduce, in many cases, tlie amount of experimental studies, physical modelling and testing intermediate experimental specimens. In the investigation of working processes it is very important to separate factors essentially affecting the process from secondary ones which may be omitted. Digital computers can be used for cal- culating working processes under various assumptions. The comparison of results obtained permits to determine reasonable limits for simpli- fying assumptions. In a number of cases the detailed investigation of the problem can then be switched to simpler devices, such as electronic computers of continuous action, calculating desks, etc. Calculations and studies on ATsVM will be widely applied in designing electrical engineering equipment within the near future. In connection with the construction and future operation of very large electric power plants which will supply long-distance trans- mission lines with electric power, calculations of static and dynamic stability of systems, and the study of the effects of generator characteristics and excitation circuits of synchronous generators on their operation, will be of great importance. Solution of these problems, connected with considerable calculating difficulties, is comparatively simple for computers. In addition to stability problems, digital computers are used presently for the analysis of a number of other complicated problems in the operation of power systems. UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 15 of 314 Pages Aynamics of automatic control systems subjected to the action of ran- dom signals varying continuously. It is connected with the possibility of using the ATOM for working out random quantities with various lams of distribution. The progress achieved in the field of designing computers put for- ward the problem of using the ATsVM as an element of automatic control systems ("controlling computers") and of designing on this basis the complex automation system of technological processes, as one of the important problems at the recent stage of technical development. In this connection, the application of universal ATsVnts has a great practical importance for engineering and scientific calculations, for designing calculations, or, in other words, for numerical simula- tion of controlling processes in the automation systems with control- ling computers. The universal ATM's can, if equipped with special continuous-discrete converters and their revere*, in a number of cases be connected with actual machines and be used in experimental studies as models of controlling computers. It is expedient to apply computers for calculations of the series of electric machines and transformers. It turns out to be possible to find optimum size and winding characteristics of electric machines and transformers starting from the given nominal data (capacity, ef- ficiency factor, etc.) and certain criteria (minimum weight, minimum cost, etc.) We have listed only a few engineering problems from the field of electrical engineering and some adjacent fields for the solution of which digital computers are applicable. We shall cite also several examples of ATsVM application for engineering calculations in other branches of engineering. In the field of aviation engineering the ATEJVHIs are used for calculating the strength, for studying the problems of vibration and flutter in aircraft construction, for determining the best shape of aircraft wings and reactive engine nozzles, for investigating the problems connected with take-off, landing, and catapulting of aircraft, for determining the take-off speed and trajectory, and for solving other problems. Ballistic tables are calculated with the aid of nen. In the construction field the ATM's may be applied for cal- culating complicated trusses, bridges, buildings, dams, etc. As an example, the shapes of the outline of the steepest canal slopes which do not slip down have been determined by USN computer. UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part- Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 16 of 314 Pages With the aid of ATiVILIs it is. possible to calculate pressures in intricate hydraulic systems and in gas pipelines. In the oil industry the ATM's are applied for solving such problems as outlining the oil strata, determination of the most ef- fective oil refinery processes depending on the properties of initial raw materials, etc. Of a bright prospect is the application of the ATsWits for eco- nomic calculations and studies in the planning of production, analysis of the productional process running, determination of net costs, prices, wages, etc. The possibility of applying universal ATsVM's for solving engin- eering problems does not reduce to any degree the importance of using other computing devices for certain calculations, such as analogs of electric systems, electronic computers of continuous action. For example, calculations of current distribution in intricate power systems are more fitted for models of electric grids. Computers of continuous action, analogs, are preferable for many problems in the field of automatic control and regulation, as they need no programs for a problem solution and admit the direct connection of a computer with the operating equipment. At the present time the calculation methods are being developed which make use of various combinations of ATsVUls and models of electric grids or ATM's and analog computers. 1-2. jOn numerical methods of solving mathematical problems. The investigation of scientific and technical problems is usually reduced in mathematical respect to establishing and analysing solutions of differential equations. Although very many processes in engineer- ing can be sufficiently precisely described by differential equations, however, the solution of these equations in an analytical way (in a "closed form") can be obtained only in very rare cases. There exist numerical (approximate) methods for solving mathematical problems which do not call for analytical solutions. In the using of these methods the solution of complicated mathematical problems is reduced to same sequenes of arithmetical operations performed in a definite order. Numerical methods of solving mathematical problems are de- scribed in Chapter 9. Here we shall confine ourselves to one simple example. Let US consider the non-linear differential equation(1) which is (1)Equation (1-1) describes the motion of the rotor of a synchronous machine, is the angle between the vectors of e.m.f. of the idle run of a generator and the voltage of a receiving system. '\ STAT STAl UNCLASSIFIED nna-Inecifiari in Part - Sanitized Com/ Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 UNCLASSIFIED well known to engineers and electricians: A2 e mo po - Pit sin 9 dt Page 17 of 314 Pages where NO, Po and Po are constants. The solution of equation (1-1) cannot be presented in finite terms by elementary functions. To solve this equation one can use numerical methods, as e.g. the Euler method of finite increments ("method of successive intervals"). Let relative angular speed and acceleration be denoted as follows: A ALE .4 At' and the following initial conditions are given: t = 0 (A-? (n)0 = It is requested to solve equation (1-1), i.e. to determine e= f(t) at OigtoET. The presoribed rang. of t-valuss is dividad into sufficiently small time intervals, steps A t, and it is assumed that acceleration is ccnatent in each interval and equals its value at the beginning of the intervals Indices i and i+1 denote the values of the quanti- ties at the beginning and at the end of the i+1 interval respectively. The magnitude of acceleration which corresponds to the beginning of the i+1 interval is determined by the expression 70 - Fit sin Mo ?Ct " Then the increments of speed 4421+1 and of angle A el., the i+1 interval will be as follows: d AL*, = cti 4t 11 144.1Ltt csai (1-2) during (1-3) At the end of the i+1 interval the relative speed and angle will take the following values: '111,+1 3B 14 + Anit.s - At O + z * (1-4) Finally we have one more relation: ti+1 + UNCLASSIFIED STAT npnlassified in Part - Sanitized COPY Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 18 of 314 Pages The integration of equation (1-1) is carried out in the following way in the hand calculations. Sines the values of St and h in the beginning of the 141 interval are known (for the first intervalik .14, JIL. 0), one determines sink from the tables and calculate accelera- tioncqby formula (1-2). Thehone determinesni+1, 81.41, ti+1 by formulae (1-3) to (X-5) proceeding according to a definite succession. After finishing calculations for one interval one passes over to the next one. When the acceleration value in the beginning of the next is being determined, the value of 61 i+1 obtained for the preceding interval is substituted into (1-2), and so on. Before going over to calculations for the next interval, one has to compare the ti+1 value with the prescribed integration limit T, and if ti+1 T, then the next step of integration must be carried out; otherwise calculations are stopped because the estimation has already been completed. Thus the process of integrating equation (1-1) can be reduced to carrying out a definite sequence of arithmetical operations and one simple logical one, the comparison of the values of q.t.]. end T. Depending on the results of this comparison the calculations either continue or discontinue. The sequence of arithmetical and logical operations which have to be performed over the initial data and over the results of the in- termediate calculations in order to obtain the answer, is called the algorithm of the solution of a mathematical problem (L. 12). 1-3. Block-diagram of the AVAIL computer. In solving the most of mathematical problems by numerical methods, enormous number of arithmetical operations must be carried out. Still recently the performance of these calculations called for such great amount of labor that the solution of many problems was practically unrealizable. During the last one-and-half decades have been designed electronic automatic digital computers (ATsVUls) operating at an enormous speed. The process of calculations in the ATsVld is wholly automated by means of a programming control. Fig. 1-1 shows the simplified block-diagram of the ATsVM. The computer consists of the following basic units: an arithmetical unit, a memory device and an output device, and a controlling unit. The arithmetical unit, AU, performs operations over the numbers introduced into it. The working speed of the arithmetical units in modern computers amounts to thousands of arithmetical operations per second. One can say that the arithmetical unit is like a comptometer UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 19 of 314 Pages STAT operating at a gigantic speed. Nowery Bowie* Warn' Bevis* Aritk- 'Attie leyet Ostrat atit BeWiee Devise Control Unit Figure 1-1. In solving the problems with a comptometer or a desk computer the operator picks up the numbers wanted, switches in the device and than writes down on the paper the result obtained. The applica- tion of a similar method for electronic computers would make complete- ly senseless the high speed of their arithmetical unit. The human being cannot insert numbers into an arithmetical unit at a required speed and read out results of the operations. Therefore these pro- cesses are automated by means of a so-called "memorizing device", ZU, or, briefly expressed, "memory". The memory consists of a series of separate cells in each of which one or several numbers are stored. Cells have numbers which permit to distinguish them from one another. These numbers are called "adresses" of the ZU cells. The numbers are transferred from the memory into the arithmetical unit of the computer ("reading of the number"), and the result ob- tained in the arithmetical unit is put back into the memory ("record- ing of the number"). The time necessary for the transfer of a number from the ZU into AU or back is called the time of addressing to the memory. This time should be commeasurable with the operational speed of the arithmetical unit for the effective utilization of the poten- tialities of the latter. Circuits of the memories are devised in such a way that the con- tents of a cell should not change after reading out a number from it. If this number will be needed in a next calculation, it can be ob- tained from the same cellsp_Rowever, when a new number is transferred into a memory cell, the number previously atorod there is erased and is replaced by the new number. In order to completely automate the calculating process and fully exclude the participation of a human being in the computer operation, the ATsVM's are provided with a control unit, UU, which controls the UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 21 of 514 Pages cannot be brought about on the.conputer directly in this shape. It is necessary for this purpose to code it in such a shape that the computer could "read" it and fulfill. In the following paragraphs we shall consider those operations which an automatic digital computer must carry out in order to make possible the compiling of the program for any calculating process, and shall become acquainted with the methods of coding and storing the numbers and commands in the computer. The program and the numbers are put into the computer memory by means of the "input device". At last, the computer is provided with the "output device V" for typing the results obtained. In modern computers the high speed of calculations (thousands operations per second) is achieved by means of constructing the me- mory of two units (Fig. 1-1): a) the inner high-speed memory ZU for a comparatively small number of cells (usually from 1,000 to 4,000); b) tho external memory VZU which operates comparatively slow but is able to store a great quantity of numbers (several tens or even hundreds of thousands). In this case, all data necessary for the solution of a given problem is inserted into the external memory. In the course of cal- culations, the parts of the program and the constants which corre- spond to individual steps of the problem solving are re-written into the internal memory. Thus the calculating process proper proceeds without addressing the external memory. The next chapter will deal with the operation of elements and units of the automatic digital computer. Although every computer possesses a series of specific (often very essential) peculiarities, the basic properties of all digital computers with the program control are nevertheless mainly the same. Therefore all considerations in the present chapter will be presented for a certain conditional computer. 1-4. Systems of counting. In everyday life the decimal system of counting is used in which ten signs (digits) are employed for presentation of numbers: 0; 1; 2; 3; 4; 5; 6; 7; 8; 9, and any of these digits can be in every place. In other words, in the decimal system every number is repre- sented by the sum of exponents of number 10, and coefficients with tVNAAYI are equal to the number of unities in the corresponding places of the decimal number. For example, the number 37406.15 3 ? 104 + 7 ? 103 + 4 ? 102 + 0 ? 101 + 6 ? 100 + 1 - 10-1 + 5 . 10-2. midiassfmr Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Cop Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 20 of 314 Pages whnl^ calculating process and, in particular, transfers numbers from ZIT into AU, switches in AU for carrying out a required operation and puts the result obtained into W. Is was noted in the preceding paragraph, every numerical method reduces the solution of a mathematical problem to a series of suc- cessive arithmstidal and logical operations over both the numbers given in the problem conditions and those obtained in the process of cal- culations. As the controlling unit of the digital computer controls the whole process of calculations itself, an exact description muat be carried out. Such a description of the whole calculating process is called the program of the solution of the given problem on the automatic digital computer. The automatic programming control is a basic property of the high-speed digital computers. The program consists of separate "commands" (one says also "orders" or "instructions") which indicate which individual operation and over which numbers must be carried out by the computer at the given stage of calculations. These commands incorporate ouccesively all the operations which must b. performedon the computer for solv- ing the given problem by the selected numerical method. The totality of the commands necessary for the solution of a problem, written in a oertain succession forms the program. This can be illustrated by an elementary example. Assume, for instance, that it is necessary to compute a determinant of the second order I: : I ad - bc. To do this, one has to multiply number a by d, then multiply number b by o and subtract the second product from the first. In other words, the following operations are carried out: lx a ad 2x be 3 ? ad be ad - be Here the first column contains the symbol of an operation which has to be carried out, the second and third contain the numbers over which these operations are performed, and finally the fourth column contains the results of the operations. Thus the program for calculating the determinant of the second order consists of three separate commands. However, this program STAT UNCLASSIFIED STA. Dmr1 - Caniti7pd Copy Approved for Release_a?9-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 22 of 314 Pages In hig*spsed digital computers the binary system of counting is often used for presentation of numbers and commands. In this system only digits 0 or 1 can occur in each order of the binary number. In the binary system every number is represented by the sum of the exponents of number 2, and coefficients at the exponents of the num- ber 2 may be either 0 or 1. For example, the decimal number 21.5 4. 1 .24 + 0-23 + 1 .22 + 0 21 + 1.20 + 1.2-1 and it is written as follows 10101.1 in the binary system. Table 1-1 ahows the first 17 decimal and binary numbers. Table 1-1 Decimal numbers - 0 1 2 3 4 5 6 7 8 9 10 11 Binary numbers 0 0 4 - 10 , 11 100 , 101 , 110 111 , .... 1000 1001 1010 1011 Decimal numbers 12 13 _ 14 15 16 17 Binary numbers 1100 1101 1110 1111 10000 10001 1 greater number of orders is needed for representation of binary numbers than for the same numbers in the decimal system. Nevertheless the application of the binary system makes it possible to reduce the total amount of equipment and to provide more convenience in designing digital computers, because any element having only two stable states can be employed for representation in the computer of the order of a binary number. Examples of such elements are relays, trigger circuity, etc. Digit 4 471'4 Button 4 -I Button 0' Relax 4 ??1" Digit 3 or. -I Dutton 2 1 Relay 3 Relay 2 Digit 21 Digit 1 ipj? 'I (ft Figure 1-2. Button 1 Relay 1 10+ It can be assumed that the closed state of a relay represents unity and the disconnected state represents zero. In the circuit of the relay-contact the presence of voltage can represent unity and its absence - zero. For example, having four relays (Fig. 1-2) and UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 011.1. UNCLASSIFIED Page 2 3 of 314 Pages assuming that the state of role:. #1 represents a digit of the first birALTy order, relay #2 - of the second binary order, and so on, any integer from 0 to 15 can be represented in the binary form by means of buttons KN. Figure 1-2 shows relays #1 and e4 switched in, Which corresponds to the binary number 1001 (decimal number 9). The addition of two numbers in the binary system, as well as in the decimal system, can be performed by columns. At that the follow- ing rules are observed in the addition process in each order: 0 + 0 = 0; 0 + 1 . 1 + 0 . 1; 1 + 1 . 0 + unity of transfer into a higher order. For instance, the operation of addition of two numbers 23 + 25 = 48 when represented in the binary system is carried out in the following way 110000 An important advantage of the binary system consists in the extreme simplicity of its multiplication table: As an example, tho operation of multiplication 6 x 5 . 30 looks as follows in the binary system of recording: 110 101 110 000 110 11110 Thus the operation of multiplication is reduced to the operations of shift and addition. At that partial products are obtained by shift- ing the multiplicand to the left by the number of orders which cor- responds to the number of non-zero orders of the multiplier. Besides the binary and decimal systems, are employed also the octal and hexadecimal systems of counting whose bases are the numbers 8 and 16 respectively. In the octal system any digit from 0 to 7 can occur in every order. The decimal number 3011 looks as follows in the octal recording: 5703 r.:5 .83 + 7,82 + 0 ? 81 + 3.80. In the hexadecimal system 15 digits are employed for represen- tation of numbers, and new symbols are introduced to denote figures larger than 9. For example, ;ten can be denoted by 0; eleven by T; UNCLASSIFIED STAT STAT neclassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 UNCLASIFIED Page 24 of 314 Pages twelve by I; thirteen byi. fourteen by 714 and fifteen by 3. The decimal number 3011 will e recorded in the hexadecimal system in the following way:7E3 T? 162 +1%161 + 3.160. The base of any system of counting recorded in the same system is represented by 10 (the number two in the binary system is 10; the number eight in the octal system is 10 and so on). Octal and hexadecimal numbers aro easily converted into binary and vice versa, binary numbers are simply converted into octal or hexadecimal. It is explained by that, that the bases of the octal and hexadeoimal systems are integer exponents of the number 21 8 m 23; 16 . 24. In order to convert an octal number into the binary form it is sufficient to replace each digit of the octal number by the corresponding three-order binary number. In the came way for the conversion from the hexadecimal to the binary system, each digit of the hexadecimal number is replaced by the four-order binary number. For instance, the octal number 5703 looks as follows in the binary systems 101 111 000 011 5 7 0 3 and the hexadecimal number 123 in the binary system is written in the following way: 1011 1100 0011 "--1 ?""""Nr^j L-Ne-"") 1 2 3 In the conversion from the binary to the octal (or hexadecimal) system, the groups by three (or four) binary orders are replaced by the corresponding digits of the octal (or hexadecimal) number, be- ginning consecutively from the lower orders for the integral part of the number and from the higher orders for the fractional part. The considered numbers 101 111 000 011 and 1011 1100 0011 can be looked upon as octal and hexadecimal numbers respectively in which a digit of each order is recorded in the binary system. These forms of number recording are called binary-octal and binary-hexadecimal systems. They are called also binary-coded systems. The binary-decimal system or number presentation is also em- ployed in computers. In this system each digit Of the decimal order is written in the form of the corresponding four-order binary number, as e.g. the number 952 in this system looks as follows* 1001 0101 oplo. At the present time the binary system is the basic system of counting used in most computers. The binary system is employed i UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 25 of 314 Pages computers for representing and storing numbers and commands and for performing arithmetic operations. The octal and hexadecimal systems are employed for compiling the programs of calculations for the shorter and more convenient recording of binary numbers, since these systems need no special operations for being converted to the binary system. The constant quantitiea ("constants") - initial conditions, co- efficients and so on, necessary for the solution of a problem, are inserted into the computer in the octal (hexadecimal) or binary-de- cimal systems. In the latter case the conversion of the binary-de- cimal numbers into binary ones is performed by the computer according to a special program. The conversion from the decimal into binary- decimal system is performed outside of the computer on a tape punch- jig device (perforators, etc.). The results of calculations are obtained from the computer in the octal (hexadecimal) or decimal systems, and as intermediate systems used within the data output unit, the binary coded varieties of these systems are employed. The conversion of data from the binary system into the binary-decimal one is performed by the computer according to a special program. 1.5. Computers with floating and fixed decimal point. Each digit of a binary number is presented on an automatic di- gital computer by some technical device, for example, a relay. The computer contains only a limited number of such devices and therefore it will operate only with numbers of a limited length, i.e. with numbers containing a certain amount of digits. The number of digits is selected once by the accuracy requirements for solving a problem and secondly by technical considerations. Principally, a digital computer may provide any desired accuracy by a corresponding increase of the amount of digits used for repre- senting the number. However, a too great increase of the digit number leads to an increase of the volume of the equipment and to great con- structional difficulties. Usually 30-40 binary digits are used in digital computers for representing numbers. The range of numbers which may be represented in a computer by a given amount of digits depends essentially on the accepted mode of number representation. Two typos of number representations are used in computers: a) with a fixed decimal point; b) with a floating de- cimal point. In computers with a fixed decimal point, which separates the full UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 ?. -???? UNCLASSIFIED Page 26 of 514 Pages number from the fraction, the da04:4n1 point is fixed between certain digits and is kept unehanged during all computing operations. Usually the decimal point is located before the left hand digit, i.e. before the top digit. In this-case, the computer will accept and process only numbers with a modulus smaller than one. For presenting the sign of a number a special digit is used, placed, for example, left of the deeimal point. Thereby the sign "plus" is presented in the sign digit by a zero, while a one is used for the "minus". If, for example, the number of binary digits of a computer is equal to 341 whereby one digit is used for recording the sign, then the digit lattice case the computer from +2433 a modulus smaller will have the form shown by Figure 1-3. In this will accept numbers ranging from -(1 - 2-33) to .to *,(1 - 2-33) and the number zero. A number with than 2-33, but greater than zero may not be pre- sented in the computer and is shown as a zero (the number moves from the digit lattice of the computer to the right). 4k. number having a modulus greater than (1 - 2-33) is also not presented by the computer sinee the number moves out of the digit lattice to the left (-the so- called overflow of digits). 2-' -2- 2.1212-S -6 Sign of Limber 2-Joel-ate 2 . ? ? ? ? ? ? 1112-131*1514 Ii1S 19.11?11111111111*115 I ibl " lisill12?1111211231"12512412T12.13,1313113Z1331 Mantissa Figure 1-3. When such numbers appear during a computation process, then its top digits (digits left of the decimal point) are lost and the result is false. This is avoided in automatic devices in which an automatic device interrupts the work of the computer in case a number greater or equal to nne appears. When programming problems for computers with a fixed decimal point, it is necessary to introduce special scale factors, whereby all basic, intermediate or final magnitudes become smaller than one. The error of the computed result for adding and subtracting operations depends on the absolute accuracy, while multiplying and dividing operation depend on the relative accuracy of number repre- sentation of a computer. The absolute accuracy of number represen- tation is determined by the number of digits used. With a fixed de- cimal point, the relative accuracy is the lower, the smaller the number. Therefore, the scale factor must not be too large, since it _ - UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10 CIA-RDP81-01043R003800160006-3 UNCiASSIYIED Page 29 of 314 Pages necessary to shift the mantissa towards lett by three digits and to reduce the order of the number by three units. As a result, the same number is obtained, but only in a normalized form 211X 0.1011000. When multiplying numbers- n computers with a fixed decimal point, the order of the number is added while the mantissa is multiplied. Analogoualy? the divisor is subtracted from the order of the dividend, the mantissa of the dividend is divided by the mantissa of the divisor and the result is normalized (partially). When adding and subtracting numbers on computers with a floating decimal point, the orders of the numbers are preliminarily equalized. The order of the smallest num- ber is made equal to the order of the largest number of digits, equal to the difference of the orders of the numbers, after which the man- tissa is added (or subtracted). The order of the sum (or the difference) obtained equals to the order of the larger number. All these opera- tions are performed automatically in the arithmetic block of the com- puter. The memory device of the computer will store normalized and not normalized numbers. As we will see in the following chapter, the commands belong to the latter. 1-6. Coding of commands. As it was said before, the digital computers are designed to per- form arithmetic operations with numbers. Thereby, it is necessary that these numbers are stored in the computer itself for an effective use of the speed with which the arithmetic block performs those opera- tions. Consequently, the memory device of the computer must not only store the basic initial numbers of the problem but also the results of intermediate computations used during the course of the problem solution. For example, in case a differential equation has to be solved, Y' = f(x, Y); xo x X; y(x0) yo and Eularts formula is used, Y1+1 a Yi hf(xi? then the computer must store the numbersxo,y,X, step hp and in addition the values xi and yi, obtained during the preceding step of integrating. With the example, listed in paragraph 1-3, it was necessary to store in the computer not only the elements of the determinant,a,b,o UNCLASSIFIED imninceifiori in Part - Sanitized CODV Approved for Release STAT 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STK STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 28 of 314 Pages The smallest, by modulus, normalized number, which is represen- table on our computer, will be 2-31 ? m 2-32. Numbers having a modu- 4- lus smaller than 232, cannot be represented and will be shown as being equal be 2 +31(1 - of -2+31(1 - to zero. The largest, by modulus, normalized number will 2-35)=231. Consequently, only numbers within the range 2-35) to -2-32, of +2-32 to +2+31(1 - 2-35) and the bar zero zero may be represented on the type of computer under consideration. Since 231 and 232 are approximately equal to 109, the operational range of the computer will be the interval from -109 to -109 and from +109 to +109, or in other words, numbers are represented with an accuracy of up to 9 decimal places. Such a range seems to be adequate for the majority of occurring practical problems. The range of numbers with which a computer with a floating de- cimal point will operate is determined by the number of digits placed under the order category, while the accuracy depends on the number of digits under the mantissa category. Actually, with the same num- ber of mantissa digits, the accuracy of a computer with a floating decimal point is higher than the one of a computer with a fixed de- cimal point. This is explained by the fact that the relative accuracy of number representation is decreased with reduced numbers in case a fixed decimal point is used, while it is kept constant for all num- bers within the operating range of a computer with a floating decimal point because of the normalization. In this way, the number representation system witji a floating decimal point permits to obtain a greater working range, a higher accuracy of computations (at an equal number of mantissa digits) and simplifies programming operations compared to the fixed decimal point system. However, the floating decimal point system requires additio- nal computer equipment for representing and processing of orders. Therefore the floating decimal point system is applied in larger com- puters, while the fixed decimal point system is used in smaller com- puters. If not normalized numbers are formed during a computing proccas,? then the computer (with a floating decimal point) will normalize them\ automatically(1). In case the top k digits of the mantissa are equal to zero, then the normalization operation consists of shifting the mantissa towards left by k digits (in order to fulfill requirement 14) ' with a corresponding reduction of the order of the number by k units. For example, for normalizing the binary number 214x 0.0001011, it is In case the normalization operation is not required, then the computer is stopped. UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT ? STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 11/11cliASSIFnD reduces the accuracy of the computation** Page 27 of 314 PAM In computers with a, floating decimal point, all nuabor* are zit, presented in the following way: izP.A,1.1,1-ic 1 ' (1-6) whereby p is a full number (positive, negative or zero), de4ignated by the order of the number X; A - is the mantissa of the number X. The magnitude of the order is determined by the position of the decimal point in the number. Provided the number A satisfies the inequation tit * (177) which says that X is a "normalized number". The afore-mentioned in- equation means that with a normalized number, the first digit after the decimal point is always a one. In computers with a floating deci- mal point, the order - and the mantissa A are presented separately. Thereby, the digital lattice of the computer is divided into two parts: one part of the digits is used for presenting the order p and the rest for presenting the mantissa A. It is necessary to take into consideration that with normalized numbers not only mantissa A but also the order p might have both signs. For example, the number * is represented in the normalized form as 2-2 etc. Consequently it is necessary to reserve one digit for representing the sign of the order and the sign of the mantissa. In the following, some conventional computer was considered which has a floating decimal point and a digital lattice of 42 digits. If it is assumed that the digital lattice computer is of the type as shown by Figure 1-4, then the left six digits are reserved for pre- sentineth:i order (one of them for its sign) and the other 36 for the mantissa (one of them for its sign). With five digits one may write numbers from 0 to 25 - 1 se 31, whereby the order p will comprise full values from -31 to +31. In 35 the normalized form, the mantissa may assume the value of 0.100....0 i.e. of 2-1 to 2-1 + 2-2.+ + 2-'5 . 1 - 2-35. 35 to 0.111....1 ' 5 digits 35 digits A 11131t1s16111119 11110.11-51*Iff 14109 titai 1242111?12fliqvile1413013113211313114361461134001w1g1 Order p \Sim' of Matisse' A Sign of order Mientissa A Figure 1-4. UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Ap roved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 UNCLASSIFIED vo? Page 30 of 314 Pages and d, but also the products ad and bc for having the possibility of subtracting the xecond product from the first one. Therefore, each command must contain information concerning the kind of operation to be performed and on which numbers and where the answer must be placed. the program for solving the problem, i.e1 the sequence of commands which describe completely the entire computation process, must also be stored in the computer. Consequently, commands must be added in such a form that they are easily placed into the computer. For example, it will be necessary to subtract number b from number a. Numbers a and b are stored in certain cells of the memory, which may be called06 and 0 . Further, it will be necessary to place the difference a - b into some cell of the memory, for example into cellY . It is necessary to emphasize theta ,ft r are the numbers of memory device cells, i.e. the numbers by means of which all memory cells are numbered. For the operation under consideration a corresponding command will be used containing the following data: "subtract from the num- ber stored in cella, the number stored in cella and place the re- sult into celly ". Schematically, this command may be written in the following form: 1 B 1 subtract a a -b Here, in the first box, there is the symbol for iie operation which the computer has to perform and the two following boxes con- tain the numbering of the cells in which the humbers are stored, and, finally, in the last box, there is the number of that cell into which the result is to be placed. In case it wore necessary to di- vide number a by number b, then the schematic representation of ouch a command would appear as follows: a divide a a b ie already pointed out that a. ,,64 trare ordinary numbers. Assuming, for example theta,. designates the third 4:311, the binary number of which is 11, 0 is the fifth cell and its binary ?lumber is 101, and finally r is the tenth cell whose' binary number in 1010. Then the command will appear in the following form: : I 11 101 I 1010 We have still to devise a code for this operation which replaces the symbol ":" and Which is convenient to be stored in a coMputer. UNCLASFIFIED STAT STAT npriassifipn in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 1." Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 31 of 314 Pages A binary number, for examplet10$ may serve as such a code. Nov, the command being considered will have the following form: 10 I 11 I 101 J 1010 (1-8) and is represented in the form of the following arrangement of zeros and ones: 10111011010. In this way, when we use some numeral "code of operations" in- stead of the conventional symbols 4., x, : (of course for each operation a different one), then all (=ands appear as ordinary binary numbers. oc, 0 and r were the designations for the seventeenth, thirty second and thirteenth cell, i.e. if their binary numbers were 10001$ 100000 and 1101, then the preceding command in numerical recording would appear in the following way: 10100011000001101. (1-9) The number of digits in the numerical recording of the command was changed with the change of the cell numbers. This circumstance leads to considerable inconvenienoies in deciphering such a record- ing. For example, in the first case, the cell represents the third and fonrth digits, but in the second case it stands for the fourth thru eighth digits. Consequently, it is necessary to indicate anew for each command what the digits of its numerical recording actually represent. Therefore, an equal number of digits is selected for representing commands, in such a way that all commands have a constant length. Thereby, it is necessary to be guided by two con- siderations. First, as mentioned above, the control device steers the entire calculating process automatically and therefore the program for the solution must be stored in the computer. Since each command is al- ready given in the form of a binary number, it is only necessary to provide the control device with a special memory for storing the program of the computer. The cells of this memory device could store all commands of the program. Ho:raver, constructionally it is more , advantageous to place the program into the same memory device which ' is already in the computer for storing numbers. Thus we must take into consideration the lehgth of the memory cella of our conventional computer, which is equal o 42 digits, when composing a numerical command code. Second, any cells (A the memory may be taken as a $ 13 and I ad 7NCL/S3IFIE' STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 32 of 314 Pages consequently for each of these cells a number of digits must be taken which is adequate= to represent the highest number of the cell. For example, if our basic computer contains thousand cells., then not less than ten digits must be used for representing each of the numbers i1 and Y Actually, this number of binary digits is required for recording all numbers from 0 to 1:000 while eL , a and y can, of course, accept all these values. Exactly in the same way we must take a number of digits for the operation code, which permit to represent an adequate number of different operations to be performed by the computer. cob u Based on these considerations, each command of the program will be stored in a separate cell and consequently, each command will be represented by 42 digits. Thereby, those six digits are taken for the operation code which serve for representing the order of the number (the convenience of such a selection becomes evident in Chapter 3). The 36 digits remaining in the cell are separated into three groups of 12 digits each, representing corresponding numbers of cells oL, and 7 . These three groups are called correspondingly the first, second and third address of the command. In this way, we may say that the command consists of a code of operations to be performed by the computer, addresses of numbers with which these operations are performed, and the address at which the result is recorded. The schematic arrangement of the digits in the command is shown by the following table: -, Code of IA - IIA IIIA Operation 12 digits 12 digits 12 digits 6 digits whereby, for example, IA is read as "first address". In case the number of some cell does not require all 12 digits, then in the given case the free digits are filled with zeros. The binary number obtained in this way is called code of command. For example, the codes of the afore mentioned commands (1-8) and (1-9) will have the follewing numbers: 000010 000000000011 000000000101 000000001010 000010 000000010001 000000100000 000000001101 6 digits 12 digits 12 digits 12 digits Now the entire program may be represented in the form of a se- quence of numbers and may be placed into the memory device of the cola- UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 33 of 314 Pages pater. This is one of the most essential properties of computers with program control. In Earagraph 1-8 it will be shown in which way the control device of the computer "distinguisbss" the cells in which the program commands are stored from the cells containing the ordinary numbers. Five digits are usei for representing the modulus of a number order while one digit is used for the sign of the order. Since the presence of a zero or a one corresponds to the different signs in the sign order, the number orders are actually represented by six-digit numbers. With six binary digits it is possible to record 26 64 different numbers - 0, 1, 2, ms, 63. These same six digits are used for representing the operational code. Consequently, the computer will not perform more than 64 dif- ferent operations (each operation must have its individual code). Uodern computers usually perform a smaller number of operations, since the constructional difficulties in the design of the control device will grow with an increase of the number of possible operations. Thus, not any binary number, even if it has the established number of digits (in our example 42), may be considered as the code of some command. The afore-mentioned examples show that the command codes, which appear as numbers, may be non-normalized. Consequently, the memory will store normalized and non-normalized numbers. As it was shown above, each address is the number of that cell in which a number is stored required for performing a given command. Consequently, the possibility must be provided for indicating in each address the numbers of any cell of the memory device. In other words, in the computer under consideration, the operational memory may not have more than 212 . 4,096 cells, because 12 digits are used for re- presenting each address. Besides the operational memory, computers usually have an exter- nal memory (VZU on Figure 1-1) containing tens of thousands of cells. We cannot place the numbers of these cells into the addresses of the aforementioned commands, but the contents of these cells may be tran- scribed into the operational memory by special operations. In this way an increase in the number of memory oells is provided without increasing the length of the command code. Computers built in the afore-mentioned manner are called "com- puters with a three-address control aystem". There are also single-, two- and four-address computers. UNCLASSIFI7D 44::51ei ti STAT STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 34 of 314 Pages In the single-address system, each command contains only the operation code and the address. Three commands are required in the general case for representing an arithmetic operation on two numbers with subsequent recording of the result by the memory. For example, the operation of adding numbers a and bp placed accordingly in cells and with transfer of the result to cell? is recorded in the following way: IA 11 13 Here, Up N, R are the code of operation: U is the transmission of the entry of cel1a. to the register of the arithmetic block; N is the addition of the number located in the register with the number stored in calls ; K is the transfer of the number from the register to cell T of the memory. The single-address programming system is used, for example, in the computer "Ural" (see Paragraph 2-12). In the two-address system, each command contains the code of operation and the addresses of the first and second numbers. The result of the operation is recorded in one of the addresses, for example, in the second. Such a system is used, for example, in the computer ":4-3" (refer to Paragraph 2-11). Another version of the two-address system is also used, in which the first address is used as in the single-addross computers, while the second address indicates the number of the cell in which the sub- sequent command is stored. Finally, in the four-address command system, the first throe addresses are used in the same way as in the three-address commands, while the fourth address indicates in which cell the next command is stored. This problem will not be considered here in further detail. 1-7. Some operations to be RIELormed by digital computers. The four arithmetic operations, addition, subtraction , division and multiplication of numbers are the basic operations performed by digital computers. But by means of these operations the automatic digital computers perform still a number of other operations with which we will get acquainted in this and the subsequent paragraphs. The operation of logical multiplication is very suitable for programming (it is designated by the symbol A ) and which is defined in the following. In one digit the logic operation is performed by the same rules applicable also for the ordinary multiplication, i.e. UNCLASSIFIED narlaccifii=r1 in Part - Sanitized Com/ Approved for Release STAT 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STA1 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 UNCLASSIFIED in Page 35 of 314 Pages 1; 1 A 0 0 A 1 .? 0 A 0 as 0. The n-digit nuabarr. is the result of the logical multipli- cstion of two n-digit binary numbers e?n and p in which each digit ri is the result of the logical multiplication of the figures and Pi standing in one and the same digit of factors. In this way, the logical multiplication is an operation pdr- formed by digits. This means that the result of the operation in each digit does not depend on the value of the other digits of the factors. For example, 1010 A 11.01 1000 If it is necessary for some reason to separate some k or j digiti \ of binary number e6n 0411_1 .... 441, than it is only necessary to muli- tiply logically this number by the n-valued number, whereby the di- gits k and j consist of ones while zeros stand in the remaining digits. a'n an-1c4'k+1 Gk ac1c-1 ". *3+1 aj A o 0 1 0 ... 0 1 0 ? ? ? ? ? ? 1 0 0 ??? 0 ot, 0 0 Frequently, it is necessary to use the operations of shifting numbers to the right or left by some number of digits. In case an n-valued number. . is shifted to the right by digits, .. k 1, than zeros will appear in the first k digits and the following number will be obtained 00 ... 0 et. a. .? . n n-1 . . si k+1 Then shifting the same number by k digits to the loft this number is formed ? c?n-k an-k-1 ...o6100... 0 Ic Frequently, the necessity arises to have a number located in one cell" placed into another cell, for which the operation of trannforrinE a number from one cell into another cell is used. Sometimes it is necessary to transfer only the modulus of this number to another cell; or the number itself is transferred/but with reversed sign. It is possible to transmit a number from any cell of the memory device to the same cell. This, for example, is used for reversing the sign of a number; thereby it is sufficient to perform the transfer of the number with a reversed sign from the cell in which it was originally stored into the same cell. UNCLASSIFIED STAT STAT npnlassified in Part - Sanitized COPY Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 [OIoJOJoloIolol...IoIilo 111 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 36 of 314 Pages For reproducing the numbers from the memory device of the com- puter the printing command is used, aecordi,,e, to which the computer prints by means of the output device the number which is stored in a particular cell of the memory. Finally, let us consider the operation of taking the integral or the fractional part of some number a being located in the cello. The integral part of a non-negative number a is called such an in- tegral number [a] (read "ant/yen a), for which the difference a - (a) in a non-negative number, smaller than one. For example, [3.7] = 3;,[6] = 6; [3/41 -P. The difference a - Ca] is called a fractional part of the number OL. and is designated (a). For example, {3.7} = 0.7; 161 = 0; {3/4) = 3/4. Provided the number a is negative, then under integral part is under- stood such an integral negative number [a] that the difference is a - (a) . (a) is a positive number smaller than 1. The number fal is called the fractional part of a negative number. For example: t- 5.1 = - 5; 1-3?7) --4; (- ;.7.1 . 0.3; (- 0.1] - 1; t- 0.11 0.9. Thereby, the fractional part (a) is recorded in some cell as a non-normalized number 20-{a), i.e. a zero is written into the digits of the order of cell is and the number {al into the digits of tho man- tissa. The integral part of the number is recorded in the last digits of some other cell T . In this way the command will have the follow- ing form: If, for example, the number a . 5i or a . 101.01 in binary re- cording, is located in cello, then after performing this operation, the following numbers will be stored in cells and r sign + 01010101010?40101110101 101 As a conclusion of the subject paragraph it is necessary to re- view briefly still one more extretely important operation, the meaning of which becomes obvious in Chapter 3. As we have shown in Paragraph 1-5, UNCLASSIFIED norinccifian in Part - Sanitized Coov Approved for Release STAT 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STA1 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 ZICLASSIYIED Page 37 of 314 Pao. for the addition of two, normalized numbers 2P-A and 21. B, the com- puter will preliminarily equalize their orders and then add their mantiesas, i.e. consiavring that p> qi it performs the following operations: 2PL + 2418 . 2PA + q+P-15B 2PA + 2P . 2P-cl In case a non-normalized number is obtained in the result (numbers A and B might have different signs), then the computer will automatical- ly normalize the sum obtained. The operation of basic addition is also used in computers, where- by the orders and the mantissas of numbers are added separately. This operation is named addition command and is designated by the symbol SE(OK). In this way, (213.0* SIC (2(18) 21141 (A + 8). 1-8. Control operations. In automatic digital computers those control systems have found the most wide-spread application where, after having performed the command located in the cell with the number k of the memory device, the computer starts to perform automatically the command recorded in cell k+1 and so forth. The change of normal sequential order of commands to be performed ty the computer is achieved by the control operation. If the program for solvin, some problem consists of n commands, then they may be placed into n cells of the memory havinG consecutive numbers, for example, cells numbered 0+1, e+2, c+n. Thereby the first commnnd of the program is placed into the cell having the num- ber e+1, the second one into cell c+2 and so on. The control device(1) has a special address register (counter). The address of a series command is formed in it which must be read from the memory device and which must be performed by the computer(1). The operator at the control panel selects in the address register the number e+1 (the number of that cell in which the first command of the program is stored) and then starts the computer. The control de- vice will select the comm:and from cell c+1, according to the cell number established at th 1 address, register, and after performing ; this command, it will ad a one to the address register and in this 4 (1)Concerning the control device see Paragraph 2-9. UNCUSSIPIED STAT STAT neriassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 -111C148001g1) , way the nAbber 4+2 is-pxoduoid. Then the computer will start:to-per., form the command stored in cell c+2 and so forth. In thin way,all commands of the prograware performed in a consecutive order. However, sometimes it will be necessary that the computer per- forms the command stored in the cell with the number p, after having performed the first k commands of the program, i.e. after having per- formed the commands stored in cells c+1 c+k. In other words, the sequential order of performing commands is interrupted, for example, is it necessary to retorn to a previously performed command. For this purpose the so-called operation of the "unconditional transfer* or the "unconditional transfer ofoontroln is used. This operation (TO is written in the following manner: c+k I TO I c +k+11 I I 4:4 I ??? ???? and is read: "transfer control to the command boated in cell p". Performing the operation of unconditional transfer consists of transmitting the number p into the address register, after which the ordinary addition of a one will not occur. Consequently, after the command of unconditional transfer.of control (the command stored in cell c+k) we do not change over to the command stored in the subsequent cell c + k + 1 as usual, but to the command stored in cell p, the number of which is indicated by the first address of the command of unconditional transfer of control. In this way, an unconditional switching of the computing process is achieved from one section of the program to another one each time when we arrive at the command stored in cell c + k. Besides the aforementioned unconditional transfer of control, there is the so-called conditional transfer of control, whereby the operation of comparing two numbers is used for performing the trans- fer which has the following appearance (we assume that this command is stored in a cell with the number m): m+in 1 01# ? ? S. 1 ? ? ? ? . . . ? ? ? This operation has the following purpose: if a number located in cella, (according to the first address) is smaller than the number stored in cella (according to the second address), then the next one performs the command stored in cell ir , and not the one stored in cell a 4- 111, which is indicated by the third address of the operation of comparison. If this number stored in cella, is greater UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT ?- S TAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 39 of 314 Pages or equal to the number stored in cell then the next one performs the command from cell m + 1. It is said that the operation of com- parison "sends off" to the third address in case the first number is smaller than the second one, and "passes" - in the opposite case. Here, the switching of the count from one section of the program to another one takes place depending upon the performance of a given condition (relation of the magnitude of two numbers). Therefore, this operation is called the conditional transfer of control. As we will see in Chapter 3, the operation of the conditional transfer of control plays a principal role in the automation of the calculating process on digital computers. Practically, the operation of conditional transfer of control is achieved on the basis of determining the sign of the difference of numbers which are stored in cells as (according to the first address) and $ (according to the second address). If the sign of this dif- ference is negative, then the number T is transmitted to the address register of the control device, and if in the opposite case a one, as usual, is added to the same register. The same conditional transfer of control may be achieved by com- paring the moduli of two numbers, which 2sends off" in case the ma,- dulus of the first number is smaller than the modulus of the second one, and it "passes" in the opposite case. We will also introduce the operation of conditional transfer of control which will "pass" in case two numbers are equal and "send off" in the opposite case. Finally, the operation stopping the computer after all calcula- tions have been finishei and after the program has been exhausted, belongs also to the control operations. Acoording to the explanations of this paragraph, it becomes evident in which way the control device of a computer distinguishes the cells in which the numbers (constants) are stored from those cells Containing commands, although both types appear as ordinary numbers and do not show any differences as such. Actually, we ourselves select manually at the control panel tho number of the cell in which the first command is stored. Furthermore, the entire program is either located in consecutive memory cells, or it switches by itself the control to those cells in which the commands are stored until the entire program has been processed, while the last command will stop the computer. UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 40 of 314 Pages 1-9. Command code of a conventional computer. STAT We considered in the preceding paragraphs the basic operations which are to be performed by computers. Different computers may have different ranges of operations depending on the particularities of the respective designs. Below, there is a table of operations for some conventional, three-address computer. The table shows that for some operations, for example the operation of number transfer, not all three addresses are used. The code of such a command contains zeros in the digits of the missing address. If, for examplel the operation of number transfer received the code designation 010001, then the code of the gommand for trans- ferring a number from cell 0 01 to cell 10...01 will have the following form 00 0 6 12 12 12 Command Code of a Conventional Computer Nr Symbol of Operation IA ILL ILIA Contents of Operation 1 2 3 4 5 SX CC. et. a The number from colIct. is added to the number of coil and the sum is recorded in cell T ? The number stored in cell is sub- tracted from the number of cells& and the difference is recorded in cell y The number from cello, is multi- plied by the number from cell fi and the product is recorded in cell dr . The number from cella is divided by the number from sell A and the quotient is recokded in cell y . Adding command. In coll I. a number is formed, the eider of which is equal to the sum of the orders of the numbers stored in celiac...and 0, while the mantissa is the num of their mantissas. UNCLASSIFIED STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 41 of 314 Pages Kr Symbol of Operation IA IIA Contents of Operation 6 06 Take the integral or the fractional part. The fractional part of the number from cella, is recorded in a non-normalized form in cell A , while the integral part goes into the last digits of cell y . 7 ft The number from cell 010 is logically multiplied by the number from cell A and the result is recorded in cell T . 8 k Y Shift to loft. The number from cello, is shifted to the left by k digits and the result is recorded in cell Y . The right k digits of cell y are filled with zeros. 9 Y Shift to right. The number from collet, is shifted by k digits to the right and the result is recorded in celly. The left k digits of cell Y are filled with zeros. 10 PCh Y Transfer number. The number from cella, is transferred to cony- . 11 PCh y Transfer by modulus. The modulus of a number from cell 0J is trans- ferred to cell r . 12 -PCh y Transfer with reverse sign. The num- ber from cell 011 is transferred to cell T with reversed sign. 13 Print eL, ???? The number stored in cell o is printed. 14 PU (TO ??? Transfer of control. The command stored in cellos is performed next. 15 r Comparing numbers. If a number stored in cell od is smaller than the num- ber stored in cell A , the command stored in cell y is perforr.ed next. In the opposite case that command is performed next whose number is by one larger than the number of the given comrand. UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 42 of 314 Pages Ur _ Symbol of Operation IA IIA IIIA Contents of Operation 16 I n, then the machine comes to a stop. Now, the block diagram of the program will look like it is shown in Figure 3-3. STAT Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 NCLASSirirn Tr!.. :-1-5J Preparation of cells STAT ,Page 121, of 314 Pages ...0-1 Forming the argument I + ih and adding 1 in the counter Calculation of sh (I + ih) and oh (I + ih) 4-41ii Calculation of th (I + ih) and comparison of i with n Stop Figure 3-3. III If we attach to every group of commards I - V the name "operator", we shall see, that the operator III fully coincides with the program of computation of sh x and eh x, in our possession. Let us, for sake of brevity, name the program that computes the value of th x "the basic program" and the one that computes sh x and ch x "the subprogram". Including the subprogram into the basic pro- gram, we must foresee the following: Firstly, by plotting the basic program, we must take into con- sideration the way of distribution of points within the subprogram, that is, to take into account in which cells is stored the subprogram itself, in which cells rests the argument x (cell ot,) and where are being formed sh x and eh x (cells $ , , etc.). Secondly, at the end of the operator II, we must introduce a command that switches the count over to the beginning of the subpro- gram, i.e. to cell k+1. At last, namely thirdly, we must foresee the counting switch-over from the end of the subprogram to the be- ginning of the operator IT. Should the subprogram have already been introduced into the memory device of the computor, then the prepara- tion of such switch-over can be performed automatically, leaving it to the basic prograa. Therefore, the command that replaces the last command of the subprogram (operation "Stop" in cell k+16) by a command returning the control to the interrupted place of the basic program, should be placed before the command transferring the control from the 'CLASSIFIEC STAT norincQifipri in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 STAT ;\ CLASSIFIED Page 129 of 314 Pages operator II to the subprogram. Such a command must be constructed beforehand and introduced into the memory device, together with the basic program. Now we have the following picture (the basic program is placed into the cells with numbers p+1, p+2,...., and the sub- program is placed into the cells with numbers k+1, k+2,....): PCh PCh 1 "0" "xn - Y+3 a t "1" 1+3 Y +3 a a 1 The 1r + 3 contains the counter of number i of tabulated points. Preparation of cell OL , in which argument x + ih will be formed. 1 2 h I x1-2h I x I- nh The next command will have to transfer the control to cell k+1 of the subprogram, but, prior to that, the content of cell E ear- marked for accomodation of the command returning the count to the beginning of the operator III, must be transferred into cell k+16. This command can be expressed only after the construction of the basic program. The rest of the program looks as follows: I rt. co.d. 04* ml V P410 PCh 11 PU : Print f , then R(x, yi) must be added to yi, whereupon the computation of R(x, y1+1) must be made. Conversely, when 1 ft I%;k, then the computation can be discontinued. For insuring the univers,1 character of the program, a maximal number capible of being reproduced by the M-3 should be taken as a zero approximation, in order to preclude the relation ? yo formed at the first stage of computation from going out beyond the expanse of the columns. At the same time, should the x be small, such a choice would require a great number of iterations. Practically, in order to save time, the choice of a zero approximation of yo must be harmonized with the quantity of number x. Besides, the requisite that x x at x 1). Now we have a numbertl_ . For separation of 10 10 columns designating the figure et, it is sufficient to have number A' shifted by four columns to the right and multiply it by number p. Inasmuch as the machine 14-3 has no shift operation, it is substituted by multiplicAtion by 1/16 and then we have number 6 . If we divide TFT 10 6 it by --sr we obtain a number. All that is left to be done 161 16' now, is to add it to 6,E1_4 having multiplied the latter beforehand 10 1 by DT Having performed this operation five times, we can get the final number A expressed in the binary notation. This program is so simple, that we do not even bring it up. aritsLASSIREE STAT Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 1 1k 1 A re.trir" Page 141 of 314 pages Program of conversion of numbers from the decimal to the binary System of calculation for the Machine M-3. Cell number Operation IA - lift Remark k+1 k+2 k+3 k+4 k+5 k+6 k+7 k+8 k+9 k+10 . PCh PCh x A, i+ x + 4, I?I, PCh . a+9 a+9 a+5 a+i a+8 a+2 a+6 a+7 a+10 k+3 a+2 a+4 a+2 a+3 - a+2 a+1 a+4 - "Clearing" of cell a+2, whi avcumulates answer "Clearing" of counter Multiplication of ? + 1021-i atL-1 119.+1 + 4. ... 4. 10n-1-1 10 by +6. (at the first stage this operation is unneoessa, Separation of i67 Formation of (ti 10 Accumulation of answer Shift of number subjected t conversion by fourcotvalm5 t the right Addition of 1 into counter Comparison of counter's content with 7 a+1 a+2 a+3 a+4 a+5 a+6 a+7 a+8 1+,1 a+10 1 Binary-decimal code of converted number Formation of binary notation of converted number p.-0, op...wino 24 Counter 1 m.0.000110011001100110011001100110 15 1 mm0,000100,..0 126 7-6. -.0,00100...0 la ...?:412,1900 16 .....0,Q0 Omm0,00...028 13 li -- ....,ri,......ninna%....., ., For realization of this program, the binary-decimal code of num- ber A is perforated in the tape (see Paragraph 2-8) and is p?rt into cell a+1. The address of the cell which after the completion of conversion is supposed to take over the control, must be put into cC)r?Er Y) STAT Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Ap roved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 lENTI Afiqigir/1 Page 142 of 314 Pages the second address of command k+/0. The binary recording of the number takes place in cell a+2. Let us now consider a reverse operation, the way of conversion of positive binary number A a= 0, as,??? ? ? iitn. to decimal designation. In the decimal system of calculation, the number A appears as A 0, 461 n, 0.s al xa ????? ???-? 4.---- 10 102. 11003 where the numbers ec,i are not yet known and must be ascertained. 10 For this we multiply number A by yr and get (a 2 at3 A +.,.+ )51 lm lb ? 4' Orr --2 10 lo ) Inasmuch as the number in parentheses is less than one, the second item isilaced in the column's beginning with the fifth to the n-th, whereas the 4161 is located in the first four columns of number A. Tr Separation of these columns results in definition of the binary ex- pression of figure 061 of the first decimal column of number A. In order to find out column41,2' we must subtract number ?1 from num- 16 ???????? ber A, to obtain a ) 1 AmmG3- + :1 n ,12 +.*.+ -76.7- 0 1010 10 16 l .aVi Tit, ,ultiplied it by yg- , we get: -- 412 t(11.3 "n 10 10 Am.. --7 --2 +4,40,+ 10 --R2) ?2.7 1 16 16 \ow the columns from 5 to 8 are predertly occu-ied by -umber ?4-2 162 and the second item is located to the right from the eighth c lumn. The binary expression (62 of the second decimal group of number A can now be found out by separation of columns 5-8. Repetition of this operat.on n times produces the binary expression of all n columns 411' (1'2 i.e. the binary-decimal code of number A. ' ??? In the 14-3 mncLine, only 28 senior columns can be used by the binary-decimal code, i.e. n 7: But it can occur that from some place all columns of the number undergoing conversion are equal to zero. For elimination of unnecessary cycles, calculation is stopped as soon as the difference between 2-29 + 2-30 . 0, 00.-011 and the 30 number undergoing conversion becomes positive, i.e. as soon as all the remaining columns of the number undergoing conversion with UNCLASSIFIED STAT npriassifipd in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 A cc/clef% Page 143 of 314 Pas exception of its last two columns (29th and 30th) become equal to zero. Program of conversion of numbers from the binary to the decimal system of notation for the machine /4-3 k+1 k+2 k+3 k+4 k+5 k+6 k+7 k+8 k+9 a+1 a+2 a+3 a+4 a+5 a+6 a+7 a+8 a+9 PCh PCh UP a+7 a+3 a+5 a+8 a+2 a+1 a+4 a+1 k+3 a+2 a+8 a+1 a+9 a+2 a+1 a+8 a+6 Clearing of cell a+2 Transfer of number work cell a+8 P Multiplication(761+1 + 10 10 by Tg. Separation of 06i 16i 0, 111100...0 to awn )10n-i to work cell a+9 Accumulating the answer (71+1 + n ) 1 Separation of + 10 TF:r 161 Shift of number p by 4 columns to the right Comparison of number being converted with number 2-29 + 2-30 Binary notation of number subjected to conversion Formation of binary-decimal code of number subjected to con- version p 0,111100....0 1 0,000100....0 10 0,101000....0 2-29 + 2-30 . 0,00....011 0 . 0,00....0 Work cell for storing 0, 0....011110....0 41 Work cell for storing ' 16i \:CL A SSIFIED- ???? STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 Page 144 of 314 Pails 3-9. Separation of integral and fractional parts in machines with floating and fixed commas. The significance of separation of the integral part will be ax- ex plained at examination of the problem of computation of function for large values of x. An we know, the series x2 ex 1 +?+?+ 11 21 fits for every value of xl but when x is sufficiently large, we practically have to sum up a too large number of members before we arrive at a stage where the remainder of the series becomes less than the prescribed error. Let us apraise, for instance, at which n the inequality if x ? e5 146. n has < 0,01 , According to Stirling's formula 111 =:(11)n V-17rTla to concur to the equation , 5_n ) 0,01 (Li)n n e e5n 0,025 (n) n or after logarithming 5n n lgn- n + fr le n + lg 0,025 ; 12n . (2n + 1)1g n - 7.4 . Ass .'in.. than n e6 = 396 we shall in the last left equation have 12e6 w onclusion, that we should apply about 400 members of the series to have a member xn with a modulus ni less than 0,01, when x 146. Consequently, should we try to find out ex using the method of expansion into the at large values of argument x, perform a huge series, we should, number of operations, which would result in a great waste of time and effort. Meanwhile we know, that the plotting of programs has to be done in a most ex- pedient and time-saving way. The longer the machine is put to operation, the more probable would be the presence of errors. Besides, the great cost of contemporaneous electronic machines makes it im- perative, that they should be made use of as efficiently as possible. These considerations force us to look for another, quicker way of computation of ex. For this purpose we may employ the method of separation of the integral part and have the argument x in the form of x [x] + (X), whereinixiis the integral number and {x) is the proper fraction. N a AtgkED STAT npniassified in Part - Sanitized COPY Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 1..::%.CLASS1FIED Now ex . erxi+14 efx.700. Palo 145 of 314 Pages The multiplier eOlis calculated quickly by way of expansion into the series, as (x).(1 and, consequently, ixIn 1 n1 i.e., already at n . 10 ILL < 1 n: 3-628.800 1 1 (it is known that the series 1 +1!+ + ... tallies very quickly). There remains to have the quantity eCA5 multiplied by e (or by 2c if Lxi< 0) exactly f*, times, which is very simple because tx) is an in- teger. In our hypothetical machine we can expand the argument into an integral and fractional part, using one command (see Paragraph 1-7), but this can not be done in the machine M-5. The fact that in here the member xn is taken with the scale coefficient M, no that n1 I211.11.141 4: 1, does not change the situation, because this member, for ascertainment of the end of calculation, is compared not with E but with EA!. Consequently, it is necessary to separate the in- tegral part of the argument x. This can be accomplished by the fol- lowing: let x/M be stored in cell a. and x> 0. Number ? is placed in some cell and then is consecutively subtracted from the content 1 of cell 4061 until in cell 4x, there remains a number less than At every such subtraction the content of cell g is multiplied by e. f By doing so, we get simultaneously the number 3cm in cell elL, and the number efxJ/M in cell S . We shall get the answer after having calculated efx/M in accordance with the progliam described in the preceding paragraph(using it as a subprogram) and multiplied the result by elx]/M. tiNCLASSIPIED STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 U N CLASSIFIED p +1 p + 2 P + 3 P +4 ? + 5 p + 6 P+ 7 p I P + 9 p + 10 p + la p + 12 a .. Pcik t -- DP- 4+, 4 : /1:1FCh )c, 4 g Stop 1 w 8 6 a P + 2 a k -t-i-1 A A I . IM 1 " Is n 1 II P + 5 in uld it, .M if +14;2 i 1" I'M , PUL I p + 9 _ , -- , Page 146 of 314 Pages ? ? ? x -1 x ?2 (x) 14-1 II la U Li kg We shall have to say a few words to explain this program. We had to subtract 1/M from number x/M until the difference became less than 1/M (let us remember that x ?.0). Meanwhile, the control by the first address can be transferred by the operation of conditional conversion of p+4 only then, when in cell ev there arises a negative number. In view of this, at first we transfer into cell& 1/Me (and not 1/M), whereas in the command p+5 we add 1/M to the content of cell 9. (the command of conditional transfer of control p+4 does not change the condition of the register of the arithmetical knot). Separation of the integral part of the positive number x can be easily accomplished when we take the power of number 2 to serve as a scale M: M . 2k. Then the integral part x will be recorded in the senior columns k of the cell and can be separated by way of lo- gical multiplication by number 11...1 00...0 At a long series of calculations, the selection of M . 2k as the scale is less convenient than selection of M 10P, since the division of x by scale II is made prior to introduction of number x into the machine. !NC STAT norinccifipri in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 uNcLASS1HED Pare 147 of 314 ftelle The operation of separation of integral and fractional parts can be applied for calculation of value of function sin x, when x is large. Should we try to accomplish the calculation of sin x by way of ex- pansion into the series, we should have to perform summations of a too large number of members. Therefore, we should better use the cor- relation sin(2kit + x) . sin x and represent the argument in the form of: x m 2kr..+ xo The simplest way of so doing is to have the argument x divided by 2/7 , then take the fractional part of this relation and have it multiplied by 2: x0 Once we have ascertained the value of x, we can perform the cal- culation of sin x.with the use of the program described in Paragraph 3-3. It will be now as follows: p+ 1 p+ 2 P + 3 p + 4 : r 3 X F' U.. I a _ a a k + 1 I 2" " a N ft 21L" ? & a a ? _ s 1 23 In cell 01, is created ( x/2X) , in cell A is created [x/2] 2 TI, Referring to the subprogram of X a calculation of sin x (Par. 3-3) For the expedition of the calculation process it would not be sufficient to have the argument reduced by the modulus to a value less than 2 It . Using some trigonometrical correlations, it might be useful to have the calculation performed so as to have the argu- ment value, in the value of the modulus, less than--' but we shall 4 not dwell on this. Other methods, related to those described in this paragraph can be likewise useful in the calculation of value of function lg x. The ordinary formula of expansion into the series x2 AI lg(1 + x) m x -j-+ 3 - -1 < x is absolutely unusable in this case, since, when the value of x is close to 1, the members of this aeries subside like -- and in order n that this member could become lens than, for example 10 '1 we should have to calculate 104 members. For reduction of the volume of cal- * CLASSIFIED STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Ittrirrt ze 148 of 314 Pen culation work we could substitute x by -x and the have the formula: lg(1 x) - x x3 - Upon subtraction herefrom of the foregoing formula, we should have 1-x x3 X5 g 1+X la (x T+7 ?*') This series is twice as quick because it contains exclusively members with uneven powers. Should we now have to calculate lg y for some value y > 0 and designate y as y ePz wherein p is an integer (positive, negative or zero) and 1 1 assuming that 1-z x, 0 < x< 1+z then 1. 1-s p + + igzp + lgs 1+ 1+z x2n-1)) 1?x 3 + ig x + +...+1 + x 3 2n-1 ?2 (2n+1 1-----+???) ? A ; 2n+1 n n 2n+1 2 2x 2n+1 ,, 2 4 % x2n+1 tl+X +X 1:;:2. A eme2 ----.. +0 ? 2n+1 2n+1 2n +1 x2n+1 Since 22 < 3, then at 2n+1 - the remainder 1-x- 3 in the value of the modulus is less than E . In this case Ix Vi con- sequently at n 6 x2n+1 2n+1 1 ..11C 10-5 213 013 that is, for accomplishing the same a still higher)degree of pre- cision it is now sufficient to take only melpbers instead of 104 members. UNCLASSIRED STAT Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 . CLASSIF1 7.17 -1 ( d ) Cluaring of counter, transfer of e to y?... transfar mf 1to Page 149 of 314 Pages Comparing(i)with 1 Comparing (d) with --e-1 Transfer of to /1 , transfer of -1 to Y2 Dividing (d) by Y, and adding (y4.1 to 0 III IV Forming I - 1 d 12n +1 Calculating zn and su=ing u I2n Comparing ,n+ 1 with3. E VIII V Doubling the series' sum and subtracting from ( ) Ix 12n+1 2n+1 2n+1 3 Figure 3-4 For the reduction of y to oz (in order to determine numbers p and z) we can place y into cell ou and then employ one of the follow- ing operations: if y > 1, we divide the content of cell ou by el until the quo- tient becomes equal to or less than 1, accompanying every division by feeding number 1 into counter ; 1 If y4c , we divide the contents of cell olt, by e-1 until the quotient becomes more than -- accompanying every division by feeding number -1 into counter ft. Doing so, we shall have z in cell oL, and number Mon, by the formula 1-z x a 1+z in counter we determinate the number x, for which we separate the series 4 x3 x5 x +?+?+.. ? until the next calculated member becomes less than L/34? The result is subtracted from p. By this way in doubled and CLASSIFIED STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCI licRIFIM , Fags 150 of 314 Pagea cell is is formed lg y. The logical diagram of this program is shown by Figure 3-4 (cell a. contains number Y). Below is shown the basic part of this program (operators I-NT). ? ? , P43L P at- 7 non it e" _. -- ii /1 Pat- -- "1" '12 4 "1" a k + 8 < 1" a k + 11 "7 Pa,_ 4-1- __ I' P CAA-"-1" __ 1 a a 4. -Y1 PV,. k+4 -- ? ? ? ? ? ? . ? . ? ? ? I UNCLASSMED STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED age 151 of 514 Pages FOURTH CUPTER. The Programaing of Mathematical Problems. 4-1. Program circuit. In the preceding chapter, using a number of examples, we demon- strated the tasic methods of program-plotting. In this chapter we shall expound the programming methods and the formulation of mathe- matical problems with the use of computers. As we have repeatedly noted before, in order to employ the com- puter for solving any desired problem, we should choose some numeral method, through which we might reduce the solving process to a series of arithmetical operations with numbers. Such choice is predicated by considerations of desired precision, operational speed, simplicity and capability of the given machine of coping with the set problem, and so on. In the majority of cases this is a difficult task, con- stituting an analytical approach of approximation. Although these factors do have an influence on the methods of programming, they have no immediate relation thereto and for this reason, we shall not dwell on them. Also, in order to solve a set mathematical problem, we, shall we say have chosen a certain numeral method and now we have to construct a program that would be capable of executing this method in the machine. With this in mind, we must first of all try to conceive a clear picture of the whole process of calculation, i.e. explicitely establish, by which mathematical formulae the computation must be conducted, in which succession and under which circumstances should we employ one or another formula, how many times and for which numbers the given formula should be applied, and so on. Furthermore, we must determine which numbers should be taken out of the machine, and which numbers are needed merely for execution of subsequent calculations. If the latter include numbers needed in subsequent calculations, they must be retained in separate cells. Constructing the calculation chart, we break the whole computation process down into a series of so-called operators of computation, everyone of which makes application of one or more mathematickl for- mulae. As a =le, one such operator may contain such formulae, which are made use of simultaneously in the whole process of computation. Besides, one operator may employ such formulae which deal with the same numeral material, and therefore need the same information. If - 1117r- W N CLASSIFIED Declassified in Part- Sanitized Copy Approved for Release @50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 1\ CLASSiFiED Page 152 of 314 Pastes two different mathematical formulae have, for instance, a compara- tively extensive general part, then it can be separated into one operator. At the same time, such formulae which do not meet one another simultaneously in the course of computation, or are needed now together, now separately, must be contained in separate operators. Likewise, the formulae applied for determination of same quantities, which, however, depending on circumstances are in various situations applied individually, must also be contained in different operators. It is not simple to convey the meaning of the term "operator", it will be elucidated by examples brought up in the coming paragraphs. We shall be designating the computation operators with capital Latin letters A, B, C,... at times assigning to them indices Aj, Bjk, which will signify that these operators change depending on some indices i, j, k, Now we are able to describe the whole computation system with the use of operators, i.e. indicating which operators, in which suc- cession, how often and depending on what conditions should be carried out so as to insure the full realization of the computation process. The preceding chapter provides some of the simplest examples of com- putation. For example, Paragraph 3-7 presents a program of computation of a polynomical by Corner's method. If in that program we designate the operator that feeds the coefficient ai into cell (g and multiply the content of that cell by x, with Ai, then the whole calculation scheme will look as follows: flAiAo.Ai An 9 wherein the sign /1 denotes: the product of operators Ai for i.0 i 0, 1, ..., n. The simplicity of this scheme is explained by our beforehand knowledge of the number of repetitions of the cycle during the calculation process. But, in Paragraph 3-3 we also examined such programs wherein the number of cycle repetitions was determined by the program itself. In this case, it is not sufficient for composition of a calculation scheme to have only calculation operators. It should as well contain the so-called logical operators, which check the ful- fillment of some of the conditions usually brought about with the use of an operation of conditional transfer of control. For example, composing a calculation scheme for determination of ex?, we should have to introduce the n member koi and feeds n1 logical operator X calculation operator A, that calculates the that member into cell A xn comparing Had with f ni UNCLASSIFIED as well as the Then, the computation STAT STAT npnlassified in Part - Sanitized COPY Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 U N CLASSIFIED scheme will be as depicted on Figure 4-1 ro. > Figure 4-1. Pais 153 of 314 rag** Stor These two examples alone make it now evident, that the calcu- lation scheme does not exhaust the meaning of the whole program, for apart from commands needed for the implementation of caloulation Operators and logical operators, the program contains a series of other commands required for the preparation of cells, readdressing the commands, etc. In order to be in a position to describe the program in all details, we have to, on the basis of calculation scheme, construct a program's circuit. To put it bluntly, the cal- culation scheme differs from the program's circuit in that the latter describes the whole program, whereas the former encompasses only the arithmetical commands of that program. he may say, that the program's circuit, apart from strictly arithmetical operations, must deacribe everything alse that pertains to the control over those operations. Consequently, while plotting a program's circuit, we shall have to introduce not bnly the calculation operators, but also a number of other operators. The following operators belong to the catagory of the most frequently encountered operators: hi - operator of reconstruction; it reconstructs the initial shape of those commands which depend on index i, and which undergo transformation in the calculation process. - operator for preparation of counter; it feeds into the counter the initial value of the index i. F(ni) - operator of transformation or readdressing; it transforms the cossands depending on index 1 by the n number of 1. f(1) - operator which adds one to the content of the counter of in- dex i. - logical condition checking the quantity of index i. T - operator of conveyance; out of the whole set of constants, this operator selects the needed group and feeds it into the work cells. It also prepares the cells whose contents undergo transformation during the program run, for example the work cells, the cells accumulating the answer, etc. I & ? s Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 154 of 314 Pages With the use of these operators we are able to construct a logical program's scheme. For example, designating the operator feeding ai into cell ig as T and the operator that increases the in- dex i of operator Ai by one as A F(i), we shall have the program cir- cuit for the calculation of polynomicals, previously examined in Paragraph 3-7, in a different formlas shown on Figure 4-2. -41""A t stop tat Figure 4-2. The block-diagrams we used for description of the program in Paragraph 3 are more illustrative, yet they do not describe the pro- gram just as clearly and exactly as the logical oper4tor schemes do. They will be dealt with in greater detail in the next paragraphs,but now we leave them alone and confine ourselves to making a few general remarks about the methods of problem programming. Firstly, the construction of a logical program's circuit should be performed with the aim possibly shortening the time of calculation to the minimum, which, among other things, can be secured by a re- duction of the number of commands in the program. Particular atten- tion should be paid to the commands contained in the frequently repeated cycles. At times it can be expedient to reconcile oneself to some increase of the total number of commands in the given program, provided that this increase takes place at the cost of such portions of the program, which are to be executed but once or so and for that reduces the number of commands in the cycles. Furthermore, let us note that before we get down to composition of an exact program circuit, we must have apportioned the capacity of the memory device, that is to say, allocate cells to all the con- stants, which either are contained in the conditions of the problem (determinant's elements, for example), or are required for its solution by the machine (the comparison constants, for example). Moreover, some cells must be allocated beforehand for quantities arising in the process of calculation (for example, when the determinant's elements are unknown, but we have formulae by which they will be cal- culated), for use as the work cells, etc. The work cells are those cells which store the rfieults of intermediary calculations. At last we must allocate some 0.0.18 for accommodation of the program itself. Distribution of memory mechanism, particularly important for compo- sition of such operators as the operator of readdressing, calls upon certain experience and cannot be always made up exactly at once, since it is impossible to foretell exactly the total number of commands. UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Vt-WAVIV.C.TVITMT Page 155 of 314 Pages It can be efficiently made, after some practice, though. If it is clear from the beginning, that the whole program (including the con- stants, work cells, etc) cannot be placed within the internal memory unit of the machine, a part thereof is put into the exterior memory device. The time required for the solution of a problem will be re- duced, if we manage to utilize the exterior memory device, as rare as possible. In the Paragraph 3-7 we have already noted, that the reconstruction of commands undergoing transformation in the course of calculation can be accomplished by one of the two methods. Firstly: from the code of the given program we can subtract the number that had been added thereto in the course of the program's run. Secondly: we can keep the initial, the so-called standard structure of the program in a separate cell and feed it into where it is needed in the program. The second method can be particularly recommended, for its application diminishes the chance that either the program-plotter or the machine itself should commit an error. The original structure of the command can be executed at the very beginning of the program. In such a case, the machine can be stopped at any stage of program's run and we can resume the solution of the problem anew, beginning with the first command. The same is true when we have a malfunction. If, however, the standard commands are being fed in after their transformation, or in such a case when the reconstruction of commands is being per- formed by subtraction of added numbers, at every stop of the machine in the middle of the program run and at every malfunction, the whole program must be fed into the machine anew. The programs in which the reconstruction of commands precedes their transformation are usually called self-reconstructing programs. They are very convenient for use in solving of such problems which call for handling of a great number of variants. This method finds detailed illustration in Paragraph 4-3. It is expedient to work out the program circuit 1:* stages. The wnole count is then broken down into enlarged operators and the general program circuit is then made up of those operators, whereupon separate circuits of every such enlarged operator are worked out. Sometimes such introduction of sub-circuits into the principal circuit can be multi-stage and considerably facilitates not only the working out of the circuit, but also provides a better survey of the whole circuit. This method is used in Paragraph 5-2. The working out of the program can be conveniently accomplished in two steps: at first we make up a program of the type described in Chapter 3. That type includes the so to say symbolic programs, wherein the command addresses are shown conditionally as k+1, k+2, , the pgfSSIPIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 age 156 of 314 Pages work cells are substituted with cells 0. ,R,Y, .., and the numbers themselves, in inverted commas (in quotation marks), appear in place of their addresses, and so on. Once we have worked out such a program we can get down to the final allotting of the capacity of the memory device. Such program can be encoded, put on perforated tape and fed into the machine, when required. The majority of mathematical problems include such frequently arising processes as computation of roots, operations with complex numbers, calculation of values of special functions, etc. It would be inexpedient to work out such programs every time anew. Therefore, such processes use to be programmed beforehand and the machine is supplied with a library of such programs, which are usually called "the standard subprograms", on account of their independence from specific qualities of some concrete problem. Making up the program for the given problem, we can easily include into it the already available standard subprograms, just as we did in Paragraph 3-6. The carefully worked out standard subprograms prove to be very economical with respect to the time of the machine's operation. 4-2. The proEram for solution of common differential equations by the Runge-Kutta method. An examination of the ways of finding a sol,Ition of a common differential equation by the Runge-Kutta method, can serve as an example illustrating the methods set forth in the preceding para- graph. Let us find a solution to equation tl? cb (x9 y) 9 satisfactory to the conditions that y(x0) yo; xo X. We know (see Chapter 9) that according to the Runge-Kutta method the segment (x0, X) is broken up into xl, x2, .... xn . X, xi.o? xi h and takes the value of function y(x) at point x1+1 as where yi+1 yi + k' i = 0, 1, 2, ..., n-1 , 1 (k1+2k2 + 2k3 + k4); kim.miNx); I. ki 11 k2 k2-4(xl. 1T, Yi -i); k3-h4(xi Yi h(f)(xj. yi + k3). UNCLASSIFIED STAT STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 TIM^TACOTWITT Page 157 of 314 Pages The preparation of a calculation scheme includes the introduction of the following calculation operators: L - operator, which adds h/2 to the current argument - operator, which adds k1/2 to yi Kr' _ operator, which adds k/2 to yi KIII operator, which adds k3 to yi - operator, which calculates the quantity k - operator, which calculates the value of function R - operator, which calculates k h (x, y) With the aid of these operators we can very easily make up the calculation scheme ricipLoticfmaclNxi 1.1 Prior to working out the program circuit, we must allot the me- mory cells. The values x and y, for which we calculate the function are located respectively into the cells m and a , whereas the va- lue f is formed in the cell 5 . In order to preclude the choking- up of the old value 4 by the new one, numbers kl, k21 k3 must be transferred to cells SA33(the fourth value of , i.e. the k4 may be left alone). Furthermore, inasmuch as to the argument xi is in- variably added the same quantity h/2, this addition may beinvariably made to the same cell 0- . Since, however, the yi sets every time a different quantity added thereto, yi must be stored up in -me more cell T . Having performed such allotment of tne memory and befurehand in- troduced clarity into the functions of the above mentioned operators and having introduced some new operators, we can now work out the program circuit. The new operators will be as follows: P - the operator feeding xo into a. and yo into A ; H the operator calculating h/2; ij the operator cilculating the counter j ; R the operator reconstructing the operator Tj i the operator calculating the value i (x, y) b. x and y resting in cells cf. and A : the value i (x, y) itself is formed in cell E N - the operator formingh (j . 1, 2, 3, 4) in cell g ki the operator transferring kj Sj(j . 1, 2, 3); feeds h/2 into cell old ; K' forms up 1. (divides the contents of cell S in two) 2 ki k2 KII adds the content of cell S (I.e. k3) to y (stored in UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UXCLASSIPIED Page 158 of 314 Pages cell y ) and places the result into cell it ; these two operators Kt and Kt' replace by themselves the afore-introduced operators Kt, K" and Kt"); K - the operator calculating and printing yi+1 ; the value '1+1 V. being fed into A and into T ; - the operator checking the value of index j; F(i) - the operator transforming the operator feeding 1 into counter j; the operator comparing the current argument x1.1.1 of calculated value yi.o. with X. Presently the program circuit will be as shown on Figure 4-3. /..cX x .542 P'H " Q.V." ki .-... Tj ... FO) -.71j r J?2 Figure 4-3. [.al ,,,,, .4 nor J Having worked out the program circuit and knowing exactly all the operators, we can now, without difficulty, make up the program itself. At this, we shall not be interested in the function f and we shall assume, that tne program of calculation of i (x, y) is stored in the cells from t+1 to i+n, and the command returning us to the interrupted portion of the -rogram is stored in cell ?.4-n. uNausSIFIE_ Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 rniffs,10T4V1.11 Page 159 of 314 Pages The program for the solution of common differential equations by the Runge-Kutta Method. ck+1 p +2 k+3 H k+4 Pj t k+5 R {k+6 f(j){k+7 k+8 N k+9 j t-0 Tj { k+11 FORk+12 { k +13 L k+14 PCh PCh PCh PCh PCh + PU x < PCh SK < + : PU < + PU + + PCh Print Stop PCh A _ "xo" "yo" "yo" "0" k+31 "1" +1 S ?3f1 S k+11 "1" p1 S k+7 Po k+16 Es S S1 8 T A 0 S - - - - "2" - - Po - "h" po - "lIIIA" pc, 113" S S 83 " , 1 " 1; S - au B Y P1 Po k+11 Po - 5 k+21 Si k+11 k+18 a, 8 k+15 S 83 83 S S A T k+5 S 2 Clearing of counter j Transfer of control to the calculation program i (x, y). The last command of this pro- gram transfers the control to cell k+9 "2" Kiik+15 ki/2 + K"tk+16 yi + ki/2 or yi + k3 k+17 Apj 0+18 L 0E+19 k+20 + k+21 ki + k4 + I y k+22 (k2 + k3) x u2 is k+23 2(k2 + k3) k+24 k1 + k4 +(k2 + k3)2 k+25 X 8 k+26 Yi+1 k+27 - yi+1 k+28 tk+29 k+30 k+31 Initial appearance of operator Ti In the senswas delineated at the end of preceding paragraph, be called a standard subprogram of solution of a UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP.81-01043R003800160006-3 Declassified in Part - Sanitized Cop Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 VICiASSIFIED Page 160 of 314 Pages common =re:initial equation by the Runge-ratinUthedo having bean worked out and adjusted, it can be incorporated in the perforated tape, which will be then stored in the library of standard sub- programs. Every time when we should have to solve an equation of this type, it would be sufficient to make up its right section, place the first command of this program in cell 2t1 and place at the program's end the transfer of control into cell k+9. This program is then also incorporated in the perforated tape, whereupon both tapes, independent- ly from each other are fed into their respective positions in the memory device. In chapter 7 we shall consider the logical program circuits for the solution of systems of common differential equations by the Runge-Kutta Method, 4-3. The Pruram of Calculation of a Determinant. Transformation of Commands in Several Cycles, General ways of working out a program of mathematical problems can be well illustrated by the example of calculation of the determinant4 D - all a12 a]3...aln a a 21 a ... a 22 23 2n anl an2 a n3... ann For the solution of this problem we can apply Gauss's method, according to which, all determinant elements located below the main diagonal4 turn into zeros and, consequently, the determinant becomes equal to the product of the diagonal elements. The turning of the above-mentioned elements into zero takes place in accordance with the theorem, to the effect that out of all elements of one line of the determinant, it is possible to subtract the elements of another line, multiplied by any chosen coefficient,. In order to turn element a21 into zero, it is necessary to form a coefficientand then a21/ail perform the subtraction from the elements of the second line the ele- ments of the first line and multiplied by this coefficient. -u49ZAC,s1FIED npriassifipri in Part - Sanitized Copy Approved for Release STAT 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 STAT UNCLASSIFIED to 161 of 314 Pages Slemett.iao,..turnsAnto zero at subtraction from the elements of the third lime of respective elements of the first line, multiplied by co- efficient aniall. Thus, it is possible to reAuce to zero all the elements of the first column situated below the diagonal. In order to turn into zero the elements of the second column, we must perform the followingx we make up the coefficient a3 2/i22 and from the elements of the third line subtract the respective elements of the second line,. multiplied by this coefficient. At that* the element a32 turns into zero. In order to turn into zero the element a42, we subtract from the elements of the feurth line the respective elements of the second line multi- plied by the coefficientlz42AMU* and se on, until all other elements of the second column become zero, too (the first element of the second line is a zero already, so that suoh a subtraction will do no harm to the elements of the first column). Having turned into zero all the elements of the second column, we take up the third one, and so forth, As soon as all the elements situated below the diagonal within tho let, 2d,..., iast columns and the first j-i-1 element ai+1,i' ai+ 2 t standing in the i-th column below the diagonal ,11"41 4j-lli element, have been turned into zero, we must turn into zero the element a (The first index denotes the number of the line, the Je second index denotes the number of column, see Fig. 4-4). i+1 ail+1 sji+1 ajh Fig. 4 4. UNCLZS5IFIen STAT A n Can't. ri (Thnv A oro ed for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 TTW011gQIIPYrn STAT Page 162 of 31! Pages For this we calculate the coefficient a aii and from the elements of the j-th line subtract the corresponding elements of the i-th line, multiplied by this coefficient. In order to save the time, it is expedient to subtract only from the elements ah (hami +11 4-2,...,n), since we know in advance, that the elements ail, a12,..., aii_it aji will at this subtraction turn into zero. For convenience we may introduce into the scheme of calculation the following calculation operators: B11-operators calculating the coefficient Aijh-operator calculating the new value of the element a1 a ?jh ..a. aiiaih, jh which is located at the intersection of the j-line and the h-column. With the use of these operators we can express the calculation scheme, upon reduction to triangular mode; as ri i3il fl ikiih. h-i - I Upon the introduction of operator Ci, which multiplies the diagonal elements, we can make up the symbolic calculation scheme It n Bijfl A h.i Now we have to allot the memory cells and, in particular, earmark the cello for the determinant's elements. It is convenient to have the determinant's elements situated in successive series of cells of the memory device, in accordance with their turn on the lines of the determinant, as it is shown at the end of the below-described program. Consequently, at the passage of one element to its neighbor on the line, the number of cell changes by one, whereas at the passage of STAT one element to its neighbor in the column, the number of the cell changes by n ones. t;N:LASSIFIED Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED ?age 163 of 314 Pages STAT Before we set about the making up the program circuit, we must invite the reader's attention to one very important remark. As it is evident from the symbolic scheme of calculation, the operators Bij and Aijh undergo a change not in one cycle, but in several cycles (for each of the indices i,j,h, in the scheme of calculation, there is a corresponding program cycle). It may occur, that every command of these operators depends on only one of the indices ilj,h,. Then, their transformation and reconstruction take place by the usual way. However, such operators often contain commands depending at the same time on several indices. Hence, they undergo changes in several cycles. Hereafter we shall explain the way of transformation and reconstruction of such commands. At the time being, we are going to make a preliminary step and examine, for that purpose, the command k + 15 from the below-described program. Its initial ex- pression is indicated at the end of the program, in cellPS. The commands k+52 k+9 and k+12 convey it, respectively, into cells 03/ 03" and, at last, into its work place in the program, i.e, into cell k+15. Fixing the value 1...10 and taking at first j as ji0 + 1, we shall see that index h will alternately have .ad all values from i0+1 to n. This transformation is made by the command k+17. In order to give index j the value io + 2, it is necessary to have reconstructed in the command k+15 the value h, which is equal to io +1 and increase j by 1. Since j is a line number, te change of j by 1 corresponds to change of addresses depending on j, by n unities This command's code with and j....10 + 1 is stored up in cell k consequently, the index r-3P j in it can be changed by 1 (which is performed by command k+21; and the control can be transferred to command k+121 which performs the transfer of $31 to cell k+15 (actually, command k+23 transfers the control to command k+10) because it is not only the command k+15 that Jeponds on index j). Assuming that indices j and h have become equal to n and it is necessary to impart to i the value io+ 1 (inasmuch as i is a number of the diagonal element, changing of i by 1 corresponds to changing of respective addresses by n+1). Since the cell kilhas already underwent the transformation, the changing of index i takes UN.:?ASSIF:=L Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 CIA-RDP81-01043R003800160006-3 UNC Page 164 of 314 Pages STAT place in the cell 113 (performed by command k+27) and the control is being transferred to the command that sends the contents of this cell onto the work place. Actually, the command k+30 transfers the control to cell k+71 command k+9 transfers the content of cell '13 to cell )13 (index i must be changed once more, but this time from the value i0+2) and, at last, command k+12 transfers the contents of cell 3 to the work place. The command k+5, which transfers the initial ex- Ji pressionacommandic.0.51i.e.thecontentsacod,, to celltt makes the whole program self-reconstructing. From this analysis we can deduce a very useful rule. If a command undergoes transformation in several cycles, then, at the beginning of every cycle it is transferred to some cell, for which it must be taken out of that cell in which it had been stored up at the beginning of the preceding cycle. At this, at the first cycle it is taken out of the cell which stored its initial expression, whereas at the last cycle it is transferred not to reserve cell, but to its work place in the program. Within the cycle, the trans- formation of this command takes place not at the work place (with the exception of the last cycle), but in that cell in which it was stored up at the beginning of the given cycle, Now we can construct a program circuit wherein the counters of indices i? j? j, are respectively designated with Yl, Yj, Th (Fig. 4 - t)tn L , (1 . 0... P (0 -? TO -.Rip) P(M?ii)-? Bin 1....1' P eic? ih).-411,-0.13,.. -IP. A,J?-e. F(h) r. J. n 1 ILO tt2.. fol t aCep b-it ) n This scheme in used for the working out the program, with due consideration of characteristic peculiarities of reconstruction operators Rijh Rjh and R , which were described above. im,---difin,r1 ri Darf - niti7ed Copy Approved for Release 50-Yr 2014/02/10 CIA-RDP81-01043R003800160006-3 STAl Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 TIMPIARSTPT7n T (i-) tk + 1 po-)yo Oct 2 k i 3 Rijh k t4 k *5 k t 6 p(i(1,1p bc+7 k +.8 Rjh kt 9 p(yrty' Lk11-lo k#11 k#12 k I (h) Y k-f-14 k4-15 k +16 x417 k#18 k+19 F(ni) k1-20 t. k +21 111 A) L k -f-22 i k 1-23 Di k +24 STAT Page 165 of 314 Pares a rrogram zor Determination of a Determinant PCh PCh PCh PCh PCh "1" "0" ?If 'am 03 - - - - - it Ti P` ft.) PCh - k+24 .1 PCh Yi - Ifi PCh V'il - k413 PCh' P3 - 1st; 1 PCh `tr t - `ON. PCh el' _ k114 PCh 0131 _ k415 XX X XXX XX XX x xxx xx XX x xxx xx sx k+14 "IIIA" k4-14 SK k4-15 "1.1,IIIA" k#15 + "1" Y 14, n-17 Y k k#14 SK kiw13 "nIA" k+43 SX pt1 "nI , M A" " 4. i y.4 d "n-1" p3 k410 XX X XXX XX 11 UNCLASSIFIED TYa21", "a117, Pi ? .. ... ?? X P1 t"al P2 "a2 2", P2tAn )( I "ai 1; STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED F (n 4.1j k +25 ik +26 k +27 k +28 f( i) k Y1; k+30 q1.1 a11 a+2 a12 al-fl a1n a +r j.i a.21 a t2 a22 a1-2n a2n k 1-31 The first determinant's line Page 166 of 314 Pages STAT I / / SK Pt "(n4-1)I,II n pi SK 111,1_ " (n 4. 1 ) IIA" fl'a. SK p3" (n irl )IIIIIA 1:5 il SK k 71.. 24 "(n i-1) IIA" k 4.24 +" uin y; Nri k -IF 7 Nit Stop : an .t-1 a 4-1 P1 (JO) X P1 a+2 P2 (i,h) - x all. n ,11-2 P2 a -1- 1 atn1-2(iiith) & (i) A _ . a i-n(n-1)ti ani *N. a +n (n-1)+2 an2 a 4-nn ann The seoond 16-counter of index i dot ermi- nant ' s (counter of index j 47 line counfy0 of index h The n-th determinant ' line The initial form of commands. Dependence on an index is indicated in the parantheses UNCLASSIFIED STAT .111=10.... Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 167 of 314 Pages Dotted line shows the limits of the cycles. Alternate commands are not shown (they are replaced by shading), but to the right of the program is shown their initial expressions. The answer accumulates in cell 6 . 4-4. Solution of Algebraic and Transcendental &mations STAT Let it be knomithat an interval (cOo, co) there is a finite number of roots of algebraic or transcendental equation D (oa) zO, and that the distance between the adjacent roots is not less than 2 h. It is de- sired to determine the smallest of those roots. For this, we calculate the values of function D (co) at points wo, att.bu0+h,402 and compare their signs at two successive points. By the strength of our assumptions, the smallest root of equation D (co) =0 is located inside of the first of those intervals (Witali+h), at the ends of which the function D (a)) has different signs. This interval is divided in two and out of the two we eeleot the one at whioh the function changes its sign. Repeating this process a sufficient number of times, we can make, the interval holding the root as small as we please and, consequently, find out the desired root with any degree of accuracy. The program listed below was used for the determination of critical speeds of the rotors of turbogenerators (see chapter 5). Solution of this problem by the machine BESU made high demands upon the scope of the program, which compelled us to make the program somewhat more compli- cated. It will be useful for the reader to get acquainted with such a complicated, although not a long program. Following the customary lino actions(desoribed in Par. 4-1), we shall start off with the construction of a calculation scheme. At the first stage of the calculation we determine the value of the function D at a point (C0o) and retain the value D (COo). Then we add step h to and calculate the value of D at now meaning of argument u: ?cao+h. If the sign of D underwent no change, the newly found value of the function is retained, and to the last value of the argument we add step h and compare the sign of the retained value of function D with the sign of the value found at new value of argument. 7"AtAr= v: STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part- Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED STAT Page 168 of 114 Pnges This process is continued until we get to the first interval (covcoi+h), at whose ends the function D has different signs. After that, the calculation is carried out in accordance with the following rules (see Fig. 4 - 6) (4014 1. At every new stage the step is divided in two. 2. The two consecutive values D(CO) aro compared with each other and the newly found value is discarded when the sign underwent a change, or is retained when it has not. 3. If the sign D (Ca) underwent a change, the next step is subtracted from the last value of the argument, otherwise it is added to the last value of the argument. In other words, in this calculation scheme, the value of function at the newly created point is compared with the last of those values, which has the same sign as its predecessor has. For the working out the program circuit, we may introduce the following operators: T- transfer of initial point wo to tho work cell; reconstruotion of step; Pi- preparation of counter i (counting of number of points in which is calculated function D); P - preparation of counter j (counting of number of divisions); pr- preparation of positive value of controlling parameter r (at r >0 step is undivisible, at r ( 0 divisible); f(r)- imparting to parameter r a negative value; f(t)- working out the controlling parameter t (product of two values of D): at t.:).0 step is added, at t 44. 0 step is subtracted); D- computation of function D (CO); UN:LAS31T7IZZ STAT 1-1,,,ineeifiari in Part - Sanitized CODV Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 trociassiFnD Page 169 f,f A14 Pages STAT Tt- transfer of the found value of function D to the mark cell; L- division of current step in two; BI- addition of current step; B"- subtraction of current step; n (Latin P) - printing the result; f f (j) - increase of indices i and j by 1. Now we can make up the program circuit ( Fig 4 - 7). At this, it is not needed to break up the operator D into separate commands, since the concrete expression of function D (C0) is now immaterial. We assume, that the program of calculation of value of function D at some point, is stored in a group of cells that begins with the cell numbered with p. The last command of this program returns the calculation to the command k+1O. The complexity of this program calls for a few expiatory remarks, that follow below: The Program for the Solution of Equation D (6))1.10 T 0(1.1 R kt2 Pi (k,-3 pi k 4-4 Pr ( k 4-5 A r i. k+6 T' (k+7 B' t k t k-t-9 f(i) tk+10 k-11 f(t) tk+12 At 0(.1.13 f(r) k +14 L 4-15 f(1.) tk +16 Ai tic +17 i\t tk *18 B' {ki-19 k f-20 11 4-21 k4-22 PCh Imow-h PCh PCh PCh PCh PCh -11- PU x < -PCh 2 -IF. 4C.f. _ PU Print Stop _ "h" ne ?i? "1" a ill P a ?0" "1" P uln "n"k P. P 11 A - _ - - - "0" - "2" at yt - "2" Yi? Y;Yt. - P 8* Ir.' Vit, ki-15 a' P" k+7 Yt, k.11-ao T.I. a Y; k421 13 - WilLlts-IsTrp Transfer of control for determination of D STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 170 of 514 Pages a - formation of new value D; at- retention of old value D; p - formation of current argument LO ; at end of calculation in cell $ is formed root. - formation of current step -sr- counter of index i Ii counter of index j yr_ cell for storing the parameter r yt- cell for storing the parameter t. ta 0 P TLL xrr-V ? IV-. 8' D (t)-?kiartv-qL1-1: L-40?A -0- it P r Pr ir Route 1 The system J. unstable Route Formation o 80h-fl 28atinf up the d?.a.1 dt ??=11 Opening up t route 1 I I? Stop 11111. .Emmon. ? ? ..01. ? ? ? ? 1.? ? To the beginning of the operator of the Runge-Xutta method To the calculation operator on the basis of A initial data v ok +1 fc1*---=??';'?''&7---------813.1atingthe transition process - h - Formation of Dok+1=Ook + 5 PriklgY113014 Fig. 7-10 UNCLASSIFIED rlarlaccifiPri in Part - Sanitized Copy Approved for Release Stop STAT 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Page 237 of 314 Pages resistance of dissipation of the generator's stator, including the transmission's inductiVe reactance xp.43-r+ x.4.; xc -- is the full generator inductive reactance of the transverse axle, including the esctsnce transmission's inductiAfxq- x+ x; E -- is the longitudinal qP ot component of the e.m.f. oorrespLinding to the longitudinal component of magnetic flux in the gap; yi --is the resistance of the excitation winding. A vector diagram for this system is shown on Fig, 7 - 11, Oct Fif 7- _UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 238 of 314 Pates All values, including the time value, are expressed in equations (7-13) in relative unitit5 (angle Ostia? t and U0 are expressed in radians). Those (7-13) equations were drawn on the assumption that the transformer e.m.f., induced in the anchor winding along axles d and q during the transition processes, may be ignored. Likewise die- regarded was the rotary moment of currents of the rotor's damper winding. Dependence E am. F (i0 - id), coinciding with the curve of generator's lost motion, takes into account the saturation of the magnetic circuit, in the assumption that it affects nothing else but the value of resistance of mutual inductance along the longitudinal axle. For success in calculation of transition process 6 sawif (t), we must know the parameter values of the block generator-transformer and the parameters of transfer (all reduced to basic quantities), for three work regimes: a) the initial normal regime, b) the short cir- cuit regime and o) the regime subsequent to unehorting the circuit and disconnection of the first section of one of the parallel circuits. Examining dynamic stability of every initial normal regime (04.9 0 ok, Wo..0), we must determine the initial oondition for If,14. Ygo and respective values U70 and E. For the initial normal regime, these values can be determined by parameters (r1)1, (x01, (xp.c.)1' (1)1. In this problem the calculation of the transition process consists of three stages; 1. Integration of the system of differential equations (7-13) at interval 0t ticn_ttz t t. Transfer of End to ce STAT'-'-"4 Page 240 of 314 Pages ans er o Ito cell r Control of con- dition , t k. 3 t tk. 3 tagER.3 Transfer of parameters values(r1)1(k)1(Ipo) (U)1 for the short-circ regime, to the R groui of cells nit Transfer of parameter. valuee(n)2,(I0)21(if (U)2,for the ragime lowing the disoonneoti of short-circuit, to R group of celbs A subprogram for computation of din& and close by the qraument's values in cell C Determination of Et iblia by solving the 7-9 equations from (7-13),when the value El? is taken from oell r and the transmission parameters are taken from the R group of cello and Ids taken from eAll 0 Calculation of in and Mt by the formulae 5 and 6 of (7-13 with the transmiBsion parameters' values taken from the R group of cells Formation of magnitudes f D1,D2, D3, D4 n cells Fts 7-/i. Let us introduce a variable z .415 ?id. After subtraction of the eighth equation from the seventh system, we have a system of equations E E F (z)j where 1 P=P r +Xqp.o x104, r'+x,,,x p.o ; )r cx cos e r sin 19)] a equation in the (7-13) -UNLASS/FIED 2 he Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 241 of 314 Pages The first equation (7-14) is an equation of a straight line, and the second is given by the table of values of the curve of the generator's lost motion (Fig. 7 - 13). N + I value of function (z) for arguments zk k4z (k...0, 1...N) is placed-into the memory device, whereas function F (z) at interval 310 2k+1 is replaced by a straight line, crossing the curve's points oorresponding to abscissas zk and zkia. system LE. ZKt1 F4. 7-13 It is easy to show, that in this case the solution E*, Z* of (7-14), can be had from the expressions (E k+1 k K+1) . k+1 ; (Ek?Ek+1) aiikz b z*.. ------f a (7-15) where Ek+11 Ek, and Ek.o. are the villas of the coordinates of the straight and of the curve, for the respective values of z (Fig. 7 - In order to take advantage of?expreseions (7-15), we must find out the number k of the interval zkt zk+1, which contains the solttion we are after. Performing a succession of examinations, the computer at 1st f k, at whioh USC1A?S4PIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 - -? UNCLASSIFIED Ek+, Bk_41-00? (7-16) Page 242 of 314 Pages then from the memory selects Ek. Elva' calculates Ek+lmmb-- (k+1)4,.z.a and determines E* and z*, by the formulas (7-15). After that are calculated: _ E* and id +z. P.P CHAPTER VIII APPLICATION OF DIGITAL COMPUTING DEVICES FOR CALCULATION OF ELECTRIC MACHINES 8-11 General Remarks Application of small size electronic digital computers for the calculation of a series of electric machines, traneformers and some other unique machines, constitutes an interesting field, especially when a great number of calculations of possible variants must be made for the determination of an optimum solution. The computers can also be successfully applied to obtain data necessary in working out the specification of engineering methods and the calculation of some parameters and characteristics of electric machines (parameters of damper windings of synchronous machines, etc.). The designing of new series of electric machines includes an extensive calculation work for the determination of optimum correlations of structural parameters and calculation of all new electric machines of a given series. Until recently, this hugs labor-consuming effort of calculation stood in the way of the designers of new series of electric machines and prevented them from performing complete caloulatory determinations of optimum variants. The structural parameters used to be chosen after but a limited number of calculation operations on a limited number of variants, with the personal experience and intuition of the designers having a free play. UNCLASSIFIED Declassified in Part- Sanitized Copy Approved for Release @50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIBD Page 245 of 314 Pages The appearance of automatic digital computers opened new possibilities. Beginning with 1956$ the NIIEP conducts the calculation of now series of electric machines with the aid of computer M-5. Performing the proof calculations of electric machines, the computers make use of but a part of their capabilities. The calculation of electric machines is fraught with the calculation of a great deal of initial and final data, so that the printing of the results by the computers takes more time than the calculation itself. Full use of the computer capabilities necessitates such format- ion of the calculation problem, at which the computer's capabilities would be used not only to perform proving calculations, but also for the determination of the optimum geometrical and winding data of the given electric machines. A proving calculation by the computer M-5 lasts about one minute. Therefore, the computer performing a great number of 'calcu- lations envisaged in a given program for various variants of para- meters, can automatically find out and "memorize" the optimum variant and then get the machine's calculation list printed. In the application of computers it is imperative that the calcu- lation lists of electric machines should include all the particular features of calculation performance by the mathematical machine. Magnetization curves must be preset in the form of polynomials. The work of automatic digital computers with fixed commas can be greatly facilitated by utilization of relative unities, whioh makes the choice of scale coeffioients considerably easier. 8-2. Forming of Problem for Calculation This paragraph with some ing of a problem which was used for eimplifications examines the form- the calculation by the computer M-3 of asynchronous motors from 0.6 to 100 kw, in 1957. (This problem, as well as the list of calculation, was worked out by Professor, Doctor of Technical Sciences T. G. Soroker. The program performing the automat- ic) search of the optimum calculation variant was worked out under the 72fCri;i5SIFI ED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 244 of 314 Pages author's direction, with the colaboration of Yu. V. Mordvinov, Ye. V. Plamodiyalo and V0 T. Burmistrov). As the motor's optimum criterion was taken the minimum of the sum C of the cost of a motor's production and the expenditures on electric energy during its service time. At such optimum criterion, the calculation can ' result in the selection of a variant needing a great amount of copper* Therefore, for control and comparison a seoond optimum criterion Cb, the minimal cost of a motor's production was intro- duced, since a motor constructed with the application of this criterion would be most economic with respect to the expenditure of copper. For every motor of the projected series, the computer must find and print (including the quantities C and Cb) tho whole data of opti- mum variants obtained by the criterion C and the criterion C. The projected motors must satisfy certain requirements with re- spect to such basio indices as power coefficient cos tp , efficiency, excess of temperature in the stator's winding, and others. Those requirements narrow down the freedom of choice of the motor's para- meters. At the execution of calculations of asynchronous motors the following factors were subjected to limitation* multiplicity of maximal moment km, excess of temperature of the stator's winding& and power coefficient cosy. Tho magnitude of efficiency was not subjected to limitations, for in a motor that has a minimal total of expenditures C, a high magnitude of efficiency is attained automatically, whereas in a motor chosen by the criterion C-15 the magnitude of efficiency is limited by the admissible excess of temperature. Limitation of the magnitude of the power coefficient is required only in calculations by the criterion Cto since the criterion of nunnery expenditure** simultaneously secures a high value of cosy. Separate limitation of the expenditure of copper was not introduced, because of there being no substantiation for the selection of maximal values on this matter. The expenditure of copper was controlled by way of comparison UNCZASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UNCLASSIFIED Page 246 of 314 Pages with a prescribed outer diameter,,the calculations are made to determine the optimum-inner diameter, the geometrical features of the grooves and the winding data. The inner diameter and the geometrical features of the grooves, obtained as a result of such calculations, are taken for application to motors of primary lengths. The calculation of motors of primary lengths consists in deter- mining the optimum winding data for the inner and outer diameters and the groove geometry obtained by the preceding calculations. 8-3. Mathematical Interpretation of ,the Problem, Remarks on Linear and Non-Linear Px'orammirii? Let us examine a most general case of oaloulation, when all four quantities the outer stator's diameter Da, the inner stator's diameter Di, the inductance in the gap B and the number of effective wires s --axe unknown, and must be defined under mini- mization of the criterial quantities C and C-owhile taking into con- sideration the limitations (8-1). Mathematically, this problem can be formulated in the following manner. It is imperative to find the values of the variables Da, Dit Bband 8, at which the function of these variables or Ca....1? (Da, Dif B8, s) (8-2a) Ct......FD(Da: Di, B8, s) (8-2b) assumes feasibly minimal values with the proviso that the limitations imposed upon the values of some functions of those variables are observed: kir.di (Da, Di, Bt, a...42 (Da, Di, B& cosy..f3 (Da, Di, Be,, ; (8-3) a) (8-4) (8-4) a) clos?f up. (8-5) UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Ap roved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 UNCTIAcTiry Page 245 of 514 Pages with a motor selected,by the criterion C .t, which constitutes the limit of minimal expenditure of copper. The computer must determine the motor parameters, at which the summary expenditure C. (in another instance the cost of production Co) has a feasibly minimal value, provided that the multiplicity of the maximal moment km and the power coefficient are not less and the excess of the stators temperature is not higher than the prescribed maximal values, i.e., that the limiting conditions up 19.1 kid 9. A trniatian i and its transfer to the work cell A ? ? Get tins J.//i111 1-Bifs+. Bi-e? Nb Oa st1 IPrinting 1 of zero Stop 6ot lip It I N0 41i141> Blejtrti3 Ca) COO Control of snP (14 *i 6 pninf te rovits'2 Calculation of cos 1 Control of condition cos y> cos iP cos p.......corp LOpening up the route 31 Calculation of Km < KwL WI Control of condit.Km7; Km icrin Calculation of e oc )4404) rt's Control of condit490 e Opening up tho route4 ICalculation of C xi To x2 x3 c-Pi P )1? Calculation of C +Att. J. on ol of condition IA6 [Transfer of to Ci_14.1 Cai Control of condition Cdi-1 Cat Ac- Cai-4 Cdia'Ca 1 t,a1- Control of condition A to 1C ; Si to Convortion to the deci- mal system Lprinting rre.neFei Da to Pat 2 Di to Di g t.4 [Opening u; th ,.te 6 Toward operatorT, J3--c to a 9 nsio. of ; Das toria ; to Di ; UNCLASSIFIED -; x 6 Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R0038001600n623 STAT STA STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 UJCLA$3I7UD, of 314 lathe course of 'cob, the reepeetiVe quantities of the engine's variant that were found to be of the least cost CI are placed into the memory cells Bap Di, B 6, ;t C,whereas the quantities of another variant found to be of cost Caro placed into the memory = 0 . cells Da, Di, 1115, s, Ct. Clearing the route 1, operator xi, performs the preparation for the printing of zero, a conventional sign signi- fying that no point satisfying the requirements (8-1) is located within the search area. For every calculated engine, the auxiliary operator To determines the boundaries of the search area by calculating the initial value of the outer diameter Duoxa 1 the nominal data of the engine, the initial value of the stator's inner diameter Duoxi and the maximal AP ' values of the diameters DAP'a Di f and then transfers the initial data of DunX and Duox into work cells for Da and D In addition to a (the initial same operator feeds a unity into cells land C tthe initial state of the program). The auxiliary operator RI using the given value of Da anA Di, uox np prepares a search on the Bit r a plane, and calculates s , a and thegrid step 4 B and41 s. Thus, for the number of effeotive wires in the groove and the step Al a, the nearest larger even numbers are taken. Meanwhilifioperator R feeds a unity into cells C and where the cost data for the foregoing point of search are to be uox uox stored and transfers the coordinates B6 and s of the first point of search on the given plane to the respective work cells for 4 and Calculation formulas for an asynchronous engine (at the given Da, Dif B.s. and a), are broken down into several operators 451,021 #3y Ct C 0 Each operator constitutes a completed calculation stage. The first three operators are connected with one another by the logical operations in the form of comparison. In the long runf operator if, determines the power coefficient cosy. Operator 42 calculates torque values and determines the coefficient km. Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 _ Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 P4145,6-41441%," OporktOtli 4;P ariesouthe therzizal calOulation and determines the excess _ of_ stator temperature 9a Operators Ctand. 0, containing in themselves the formulas of economic calculation, fin i out costs 01) and 0. Operators it 74. 21 X.3, verify that the conditions of limitat- np ion (8-i) are complied with. If oos cos Cf , then, contingent ,np upon the condition that the inequality cosy (cos 4e) has not yet been fullfilled during the movement along the given grid vertical line, the search is put to a stop and. operator A a transfers the control to operator 14. A search will then take place on a point with a lesser so Once, the inequality cos y* (cos it)np is fulfilled, the control will-go over to operator x3, which, while opening route 3 will stop the transfer of the control to X4 in case the value of s is on the . nP decrease, thus the condition cos v (cos 11) is not continued any more. In this case, the control will be again turned over to operator 1 5 and the searoh will pass on to a new vertical line. Operator y2 functions then, when the power coefficient is as high as is heeded. Operator A.2 checks the fulfillment of the np n condition k . At km Z. p , the control passes on to operator m m A . When the condition is fulfilled, operator te3 starts working. 4 If the excess of temperature of the stator winding is less than prescribed, the control passes on to operator x4, if it is greater than prescribed, the control passes on to operator .1.4. Operator a 4 compares si at the given point of the round with npo If number ea has not reached its maximal value, the number of wires in the groove is reduced (downward movement on the grid I s - vertical line). Otherwise, the control passes on to operator13 which sees to it that the maximal reading of induction Br is not yet reached. As soon as it is, operator A5 transfers the search onto a new vertical line of the grid, beginning with the uppermost - trialreSIPIED STAT STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Pago2.5 \--Or':;314-715agia -Transfer - to oier,a*4-:feansj.1?, that the current po located within the area of admissible values of variables under the conditions of (8-1). While cutting out the transfer of the control by route 1 from operator operator x4 prepares the printing of the data of th: engine variant which was found by the search to be the beet. As soon as the first point on the given vertical line, at which the conditions (8-1) are fulfilled is found by the computer, routes 4 are opened. Now, should at the further movement on the given vertical line toward a lesser $ the area of admissible values Blo e (under condi- tions of (8-1)) be left out, then the searoh would go over to another vertical line, whereupon the oomputer proceeds to determine the costs Cts and Cs The variants satisfying all limitation conditions are compared with one another, with respect to cost faotors. Operators .16, )L7 compare both costae Di and Ci at the given point with CD.i..1 and Ci_i of the foregoing point on the same vertical line. If the comparison establishes that both costs at the given point are higher than at the foregoing one, then operator A. transfers the control to operator A5, whereby the machine auto- matically relays the calculation process to another vertical line. Operators 3.81 )19, search for points of the lowest cost. OperatorA8 compares total cost Ci at the given point with the minimal cost arrived at the foregoing points. This value is found in cell C. Conditionally upon the cost at the given point Ci being less than the value in cell 6, the coordinates of the current point (Da, Di, B s) are relayed to cells lie Bi, i and a, whereas the value C itself goes into all E. Operator .19 compares the cost of engine C at the given point, with the minimal cost of the engine (in cell CiD), which was arrived at in the course of search at other points. UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 - , maks-matt ? 0 -45.,,,ItTA age 25-7,of 3l4 Pages btj 0 l'then the coordinates of the current point and SW' ? ME Nit values 0% are relayed in cells Da, Di, Bto al OD._ If CC and 00.1.61Z) I then the _exclusion of old coordinates does not take place. Then the control passes on to operator 2.4kand, if it is possible, the movement goes down the vertical line, and the calcu- lation of the engine with the other value of s ensues, eta. (0.? Once the search on the given plane B6 , s is over, i.e. the maximal reading of induction Deis reached, then operator 2. 5 relays the control to an operator which, with a preset step,: changes the value of the inner diameter DI whereupon the search on a new plane B6, st at a changed value of Di, commences. At this, operator Xio sees to it that the value of the inner diameter Di has not yet reqbhed a maximal value. Otherwise, the control goes over to operator Da, with a preset step, changes the value of outer diameter, and to operator D, which in a respective work oelIvieife;ming the initial value of diameter Di. After this search is started in turn in the group of planes B6, so with new values of Da and Di, changing ucx lip with a prescribed stop within limits from Di to Di . The whole process of search for optimum variant ofthe engine is over as soon as operator 2.11 discovers, that the outer diameter has reached the mATimAl value Danp By the end of the search, memory cells DatDi, B8 and tat By 6 8 contain the 000rdi- natee of the beet variants of the engine, corresponding to the lowest costs of 0 and Ob. Once the search is over, operator A. transfers the control 1I to a group of operators performing the printing of the calculation data of the optimum variants. At first,route 5 is cleared (preparat- ion of circutts otoperators performing the printing). The values Of Da, Di, B8 and s of the variant of a minimal cost 0 wbich were fed into respective work cells and the control passes on to operator y it Then calculation of an engine with optimums Da, Di, B6 and s (by criterion 0) in performed .again and prints the data of its INSLAsSIPZED STAT neclassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT ? Then-valnes of Da, Di$ B8 and a of theoptimumvariant found by criterion Cb are fed. into the work cells, the computer performs the calculation of that engine, prints its calculation list and then comes to a stop. The above-described program occupies 1,100 memory cells. 22 program constants undergo transformation at every caloulation of a new engine. A trial calculation of an engine with given Da, Di, B5 and al is being made by the oftputer 14-3 in 45-551. Ascertainment of 40 component data of the calculation hat (conversion tor desima, system and printing) takes 4 minutes. Search of the best variant of engine on plane sis s, with fixed values of Da and Di, lasts, on an aver- age, for about one hour. During that time, the computer performs 60 - 80 calculations of the engine's variants. When the computer searched for the optimum values of Da, Di, 8 and s, the diameters Da, Di assumed 6 various values with a 2.9% step. Thus, the entire calculation of the optimum variant of the engine consisted of searchingrfor the best variant on 36 planes of )3%, s, corresponding to the varied values of Da and D. Such a search was performed by the computer 14-3 in about 36 hours. CHAPTER IX - SOME INFORMATION ON APPROXIMATE CALCULATIONS 971. Theory of Errors In all practical calculation we deal with numbers which are obtained as a result of measuring various magnitudes such as2 e.g. distances, time, weight, eta. Because of limited capabilities of measuring instruments, it is never possible to obtain absolutely exact values of prescribed magnitudes. Conversely, the result is always bound to contain some error, which can be brought to light during a second measuring, when we arrive at a =Tiber differing somewhat from the one we obtained before. Only in very rare cases UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 do w00101tte-,exact values of the m404; STAT 104.4..;? a t,9.174-0.--* Basically this, is true with respect to mathematical conStantai suoh asA,, e and so on. But, even in such cases* these numbers can not be expressed precisely* for they contain an infinite number of decimals. Even when we apply but precise formulae, some errors of initi- ally taken magnitudes will go along through the whole calculation and at the close of the calculation we shall get but an approximate answer. Furthermore, some mathematical problems oan be solved only after an endless process of trial. Integral calculus, the palm. lation of derivatives, etc. can serve as an example of such end- less prooeeses. Once we can not bring the endless process to an end, we have to stop at some final step within its oourse and then we are inescapably liable to have an answer containing an error. At the same time, in the problems of engineering the unknown magnitudes are sought for practical application. Here again, in- evitable'errore arise in the production process, or in sizes of articles, as a result of inexactitude of initial measuring. Con- sequently, the attainment of genuine precision of an unknown mag- nitude has no practical significance and we can be satisfied with an approximate answer. At this, two problems arise; knowing the initial errors of magnitudes we have to determine the error in the anower, and, on the other hand, we have to determine what the ini- tial errors may be in order that we may secure the attainment of the desired precision in the result. In order to be able to answer these questions* we have to define what the concept of exactness of a result actually is. ABSOLUTE AHD RELATIVE ERRORS. If some number a constitutes an approximate value of number A* then the modulus of the differ- ence of IA --. al is called the real absolute error of the approximate number a. In the maiority of cases, the real absolute error is unknimn, WAWESIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT -PA-0f* ft:yrinptcler to, **OW it, we? Shiigd have to know the true v4u0' ef But, as a rule, in conducting a calculation, one can warrant* that in the result of the calculation the value of the committed error will not exceed a certain limit. That certain limit in considered to be a meaiure of precision measurement and is called the absolute error of the approximate number a. Thus, the absolute error of approximate number a is possibly the smallest4 number satisfying the inequality IA?al Ag 416, i.e. the absolute error is amt than, or equal to the true absolute error. It is evident, that the absolute error does not define the measurement quality well. For example, the absolute error of k 17 at weighing railway cars and bricks indicate quite a different accuracy of measurement. Besides, the absolute error as a rule is a concrete quantity and its value is, therefore, changeable along with the change of unit of measurement. Therefore, for the deter? mination the accuracy of approximate numbers, a special term has been introdueed, wooly, the "relative error' Also, the relative error of an approximate number a, is called the relation of the absolute error A to number A: um a The relative error is a dimensionless magnitude, independent of units of measurement. Usually, the relative error in expressed either in per cents (4 ot 0.01) or in thousandth (10/00 4. 0.001). THE NUMBER OF CORRECT CIPHERS: There exists a practical and relatively simple method of determining the absolute and the relative errors by way of counting the number of correct ciphers. It ie known1 that in the decimal scale of notation every number is expressed as a sum of various powers of 10 multiplied by one of the digits 0,1,2,...,9. For instance. 1,023 1.103 + 0.102 2 ? 10i 3 ? 100. Designating an index of the highest power of 10 included in number a as "p", we have the following expressions STAT UNCLASSIFIED ? 254 ? Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 - 40.1CITASSIMO + 4-41+1 41110..n+1.10 +ap_21,1011 n. ..7MVP7rWW--;,11*'"41,;r411.W4i. Page 26?f - -4,1-otg's Other than zero, the first from the left cipher ap is called the first significant or thie senior cipher. For example, in the numbers 1,023 and 0.0023 the first significant cipher itill re- spectively be i and 2, with "p" in these numbers being equal to 3 and -3 respectively. Every digit standing in a certain place in the decimal notation of number "a", has its special value. The first digit is equal to' lop, the second 10124,..., the n-th 10p.-n+1. Rounding the number off, we replace all its ciphers, beginning with some cipher, with zeros. The admissible error does not exceed at this the value of the digit standing in place of the last untouched cipher. Number a is an approximation of quantity A with n being the first correct oipherl provided that the absolute error of the number does not exceed the value of the digit standing in place of the last n-th cipher, i.e. IA -a14.110P -n+1 For example, number 2.718 expresses number 0.2.718281... with four correct ciphers and number 2.7183 expresses e with five correct ciphers. This example illustrates, that the number of correct ciphers may be understood almost literally. Only the last of n "correct" ciphers can be different from the true value, and even then not more than by one. Only some exceptional numbers do not agree with this rule; for example, number 9.999 expresses number 10.000 with four correct ciphers. The resulting rills therefrom is, if an approximate number a is UNGLAtiIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 17. Page 23 of 314 ,gages' exprosain num1er A with n correct ciphers, then the relative error a does not exceed 1 a ?10n-4- e Indeed, by definition the relative error is equal to: < A . 10p-nil 10P-11+1 ? -z- a a.10 =Mt n-1 ? ap.10 Therefrom it follows, that at any value a three correct ciphers guarantee a relative error not exceeding 1%. A number that has n correct ciphers is usually expressed with n signtfioant aphers, even then, when the last ciphers are equal to zero. Por example, if number 3.28 has five correct ciphers, it must be re- presented in the form 3.280.0. ERRORS OP ARITHMETIC RULES. Let positive numbers Ai, A21..., Am be expressed by approximate numbers al, a2,...lam Wring absolute errors Ai, Am and relative errors al, a2,..., amp It is desited to find the error of the sum a affae...+am, which in an approximate value of number A Al+A2+1,?.+41.20 MAgLASSIFIED 4 STAT. Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 1-, 44",;11Nd VASS 1 FTED Since ? ? .:_M7rnnr4r7M7P=M51. -t) Piga ?Wi:of 314 IA --al 3^..t 1(Al+A2 +...+ Am) + a2 + ? ?A + 1301 4?. IA a11 + + , 1A.27,..a1 + +I Am?% II then the absolute error of the sum4 does not exceed the sum of absolute errors. From the correlation a -K- 4 alai+a2a24....+amais MUM Al? a _ *1 mm4lai+ae...+ an a2 4.a2a1+a2+***+4m am +am2+...+e .10/1 it follows, that the relative error of the bum does not exceed the largest relative error of the items. Indeed, if a* is the largest of the numbers from the forgoing inequality, it follows that al at4 al+...tam +*"4. am 10M11*. ar 112001040ant a. Inasmuch as quantity is the multiplier arm2+...+42 of ai' then the principal influence upon the magnitude of the relative error of the sum is exerted by the relative error of CSCLISSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT 1**) 2-87Kaf'31 ages the largest of the items. , 4Thaz.a_fOrai there Will be in the 0112LV* as many correct ciphers as in the largest item. Adding up several numbers having equal numbers of upper ciphers, it is necessary at first to extract the largest item, and in other items to have all columns standing to the right of the last column of the largest item, discarded. For example, in the sum 7728.75 .1. 370.846 + 0.712813 . 8100.308813 only the senior six ciphers can be correct, in view whereof, the summation must be made this ways 7728.75 370.85 0.71 8100.31 Evaluating the error in the difference of two numbers, we must differentiate between two instances. If these numbers have great disparity in value, we may repeat the above-adduoed consideration and be convinced of the correctness of the statement, that is : the error of the largest number, that is exerting the main influence upon the relative error of the difference. Consequently, the un- neoessary columns must be discarded. Let on the other hand, the minudh and the subtrahend having equal numbers of correct ciphers, differ from each other only slightly. If so, then the same absolute error will fall upon the small difference, in view whereof its relative error will be great- ly increased. For example, the absolute error of numbers 150.46 and 150.35 does not exceed 10-4, whereas the relative error of their differ- ence, which is 0.11, can be equal to 10-1, i.e. be increased al- most 1000 times. Consequently, when performing calculations, we have to transform the formulas in such a vny, as to be capable of finding out the difference of numbers with little disparity in value, without knowing those numbers themselves. As regards the multiplication and the division of numbers, it ITECkWIFIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 ..??????/.A.S. gRTV-TrM Z,AS ?rk, Pae 26 of 314 TageSTAT may be stated that:- the relative error of the result of a aUccessiiv series of multiPlications and divisions does not exceed the sum of relative errors of each separate number, We are not going to demonstrate the correctness of this rule, but want to note that when the number of operations is limited (around 10 operations), then from the above-formulated rule it follows, that the result is less by one-two correct ciphers than the smallest number of correct ciphers in the numbers participating in the operations. However, at a large number of operations, the re- sult can be of a considerably lesser accuracy. 9-2. Solution of Algebraic) and Transcendental EqBdjaaL equation Formulating a Problem. Let us have to solve the f (x) 4.0, (9-1) where f(x) is some transcendental, or algebraic function? In other words, we have to find out points xl, x29,..,xn..., at which function f(x) turns into zero (socalled zeroes of function f), Numbers xi can be true as well as complex. There can be an infi- nite number of them, like, for example, an infinite number of roots of equation sin x .0. They may be located in immediate propin- quity from one another, and may even have points of contiguity. At times there arises a problem of finding out the smallest posi- tive root of equation (9-1), or of finding out the equation roots at an interval (alb), and so on. Our problem is to find out approximate values of roots of equation (9-1). If f(x) is a continuous function, then we shall call point x* the approximate value of the root of equation (9-1) an absolute error e, provided that at points x* - 1 and x* + f(x) assumes different signs. A universal numerical 2 method of solution of this problem for arbitrary function f (x) does not exist, so that we havs to consider various particular UN915,pSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 .??? 6 UNCLASSIFIED page 263 of 314 pege?TAT cases. The first and the most difficult step in searohing for material roots of equation (9-1), is the task of separation of roots. We understand this as searching for two such points a and bp between which only one equation root is located. Once all the roots are disengaged, the remaining difficulty will be the great volume of calculation work. Some information on distribution of true zeroes of function f(x) oan be shed by an examination of its derivative, since, accord- ing to Roll's theorem, between two zeroes of f(x) there is situated an uneven number of zeroes of derivative of ft(x). Sometimes, dis- engagement of roots can be managed by moans of physical considerat- ions. For example, in the problem dealt with in-chapter about the determination of frequencies of vibration in turbogenerator rotors, we knew beforehand that all roots of equation DO)--0 were positive and were spaced apart at a distance not less than a fixed number h. Under such conditions, the problem can be solved just the same way as it was solved in paragraph 5-1, that is to say, through finding out such an interval, at whose ends the function has different signs, with subsequent division of that interval in two. This method of division of the interval in two is relatively too laborious even for the highspeed computers. Presuming that we know two numbers a and b between which there is situated only one equation (9-1) root, we are going to present more expeditions ways of definition of its values. Iterational Method. If function 4)(x) in some ambient area of root ol.of equation (9-1) does not turn into zero, and if then equation (x) - (x)f(x), x- (x) (9-2) has in that ambient area only one solution x..a. .7)1U:144%51F I 0 Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 ;..WNCLASSIPIED', Page ?ii3Of 314 l'ISTIVT Assuming that wii,know a rough value xi of root c. and want to have it defined more exactly, we name- xl to be the first approximate and as the second approximate we take: x2 as the third approximate x3 and so on, until we get at the n-th approximate 1 ..'((x_1) eta. When the sequence xn coincides, then its limit is nothing else but the root of equation (9-2). Actually, if we designate the limit of this sequence at, then n-boo n..40?0 mole(lin! Xn.a)milace(a*). 11.4.00 For sequence xn it is sufficiento in order to tally, that in the ambient area of root a/ the condition If t(x):4=:1, was fullfilled, it being known that the tallying is the quicker, the lesser re(x)i. is. Calculation must be stopped as soon as the difference Ilt--xn-li becomes less than the admissible absolute error. In other words, ? it is assumed that the condition I x --x 1 I n n-li MINIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R0038001AnnnR_fl Declassified in Part - Sanitized Co .y Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 engen4ers the me quality age , f ;314STAT4r,. In the simpliest case it is assumed that$p(x)..1 and, conse- quently, solve the equation x-f(x)..xe The iterational methods are particularly convenient for application in programming, on account of similarity of performance of operations. Method of false posture, or method of proportignal parts. Let only one root 4. of equation (9-1) be located at interval (alb). Then let us take b for the first approximation of and determine the iterative process by the formulas af(xl) --xlla) xems af(x2)--x2f(a)t xru.' flx2)--f(a) Geometrically xi+1 is a point of intersection of the axis of abscissas with the chord connecting points (a, f(a)] and ixi f(xi )2 of curve ym.f(x). It in easy to establish, that in t x-a our case funotiorithas the form Vi(x) gar Newton's Method. If function 1 (x) f (x) is taken for lk, then functionYwill have the form f(x) y/(x).mx--- f 1(x) and the itorational process will be expressed by the formula: xemxn-1 -- fl(xn_1). -13NtaSSIFIED rlarlaccifipri in Part - Sanitized Coov Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 rturn Pag 210.,or 31,4 pasTAT Geometrically xi.1 is a point of intersection of the axis of abscissas with a tangent of curve y...f (x) drawn at point [xi, f(xi)). It will be noted, that newton's method secures a quicker tallying than the other above mentioned methods. In para- graph 3-7 we applied this method for the calculation of the value of the square root of x. Algebraic equations. Let function f(x) in equation (9-1) be the polynomial of power soll,with material coefficients, i.e. that (9-1) has the form: Pn(x)=.-7"... a0x21+a1 xn-1+. + a x+a ..0. (9-3) n-1 n According to the principal algrebraic theorem this equation has exactlyKroots, with due regard for their multiplicity. With this, rootais called the k-multiple root of equation (9-3), provid- ed that the conditions (k-1) (a)..0; are met, Theae roots can include complex roots and then the latter are 1.214.416-414 always conjugated in pairs, In other words, numbers a+laare, at the same time, the roots of the equation (9-3). An approximate definition of the roots of equation (9-3) can be accomplished with the use of the above-described methods, but inasmuch as they assume, aa-approximately known arrangement of roots, we are going to start off with the consideration of thin factor. The simplest way would be the consultation of a polynomial graph. Yet, this method is not always within reach and we have to make use of other methods. STAT UNCLASSIFIED 263 - Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10 : CIA-RDP81-01043R003800160006-3 uicii?tftED ?age 2j of 314 PageSTAT Let us, first of all, introduce the following theerem. If the maximum of the moduluses of all coefficients of polynomial P (x) is designated A, then the number constitutes the upper boundary for moduluses of all its roots, be they true or complex. Before we begin the examination of arrangement of real zeroes, let us point out the following. Let: PI (x ) , 1 P2(x)ammd)(--x)1 P3(x)....xnP (-- and let N 0 N2 and N3 be respectively the upper boundaries of their i- Positive roots, Then 1/N1 is the lower boundary of positive roots of polynomial P(x), whereas numbers N2 and -1/N3 respectively ro - present the upper and the lower boundaries, of its negative roots. Thus, we may discuss in advance only the upper boundary of the positive roots of polynomials. Two theorems can thus be formulated =this subject: Langrange's theorem: if a027.,,O, nic(tro 1) is the first negative coefficient and B is the greatest absolute value of negative co- efficients, then number constitutes the upper boundary of positive roots of a multinomial. Newton's theorem: if at x?a-c the polynomial P (x) and all its consecutive derivatives P1(x),P1,(x),..., P(x) are positive, then the upper boundary of the positive-roots is number cc. INeilSSIFIED Declassified in Part - Sanitized Copy Approved for Release 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 71:77:77777777777,77--' '''-',' _q 1,, '-%-fltARR-Tii:?, STAT 1 Page 21 , f 314 Tates Iliiimuoilc,asP (it)( ) niso> 0, then P('n-11(x) is agrowing function and, consequentlyt there exists such a ci, where at at x.,433.4. Consequently, at x?-.c3., derivative (n-2) P (x) is a growing funotion and, therefore, there exists suoh a c2 whereat P(n-2) (x).?...t 0 at x?.02. Continuing this way, we can find the unknown c. Sturm's method provides an exhaustive answer to the problem of determining the real roots, but in View of its cumbersome character it is hardly usable and will not be the subject of our examination. We will now tura to Desoartels theorem, according to which, the number of positive roots of multinomial P(x) (taking into account their multiplicity) is equal to the number of changes of signs in the system of this multinomial, or is less than that number by an even number. For example, polynomial Lobacheveklyts method. For the solution of algebraic equations exists the Lobachevskiy method, which does not call for prelimi- nary separation of roots. Let us for the beginning presume, that all roots of equation (9-4) are material and different. Aligning them in a sequence of decrease of modulus?. we can; on the basis of equation (9:4), make up the equation in accordance with the following rile: efficient at xk minus the double product of the coefficients Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 at:E TK?....71-1Zer' 731 - - STAT product of ho?.Zfioients' i6a At and-ao on, Until the fiat or the last member of the initial eqation. In other Words 2 2 ale..ak--2ak+lake2ak+2ak-2".; Proceding from equation (9-49, another equation oan be constructed, just the same way as equation (9-41) was made from equation (9-4). Continuing this way, at (k-1)-th and at k-th steps we get equations: nCk-1) n-1 x x .(k-1) n-2 (k-1) 7: .012 X +? ? ? +an ?J0001 (9-5) n (k) n-1 (k) n-2 (k) x al x a2 x an 1.01 for whieh the approximation equations 41CL[alk-112; Ea2 a2 k (k-1)/2 ? noran ] ; (9-6) will be met, i.e. in the k-th transformed equation the coeffi- cients will be equal to the square of corresponding coefficients in the preceding equation (with the accepted degree of calculat- ion accuracy). Under these circumstances it may be stated that: - UNCLASSIFIED ? Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 h (k). 41 (k) h a2 (k) h a3 3 (k)' a2 where h.2k (if, for example, km5, then 11..25.32). Prom these equations, it is possible to determine the moduluoes of roots al, the sign of theroots is determined by way of direct substitution in equation (9-4)* Presuming now, that among the roots of equation (9-4) are several pairs of complex-conjugated roots a. ri (cos yi s r. (cos lei ? cps ==.71 a.. r. Cos If ? ? t/? 1Ift gr./) Jti Cf IFI Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 .*1Wowm Itgv-trIMPr vir(4-$?-0-uri and har4lig the roots numbered anewt so that al I > 1 a21 " I ai-11 ri> >. ai+2i> ? ? ? I ai-a I> ai+21 STAT AP1Wi4 we shall again get equations (9-5)p for which the equalities (9-6) will be met for coefficients on all powers of xp with the () exception of powers it jt... Convers k ely, coefficients ai apt) will behave in a most unprediotable manner. They are even capable of changing their signet which is the beat manifest indication of the presence of complex roots. Under these condi- tions, in place of equalities (9-7) there appear equalities (k) ii (k) Ii a1 al I * ? ? ? 0 (k) h aj+1 ry.-77-1 aj -1 (k) h aj+2 j+2 With the use of these equalities we at first determine the material roots a a2,lf OIL. 1.4.2v1v.tait aj+2,..?0111 and moduluses rip rit... of complex roots. If the complex roots are represented by one pair then their argument can be found from --aemal+ a2+"?+111-1+ + 21" con+ai+2+...+a. T.121C1-rIED Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 equations: WCOta are,repreienfid by agea 'thoiT andyi can be found from the system of the two ?B 1a1 cost +alt2f?*?+ai+1+2ri??8 4.4J+2 a n-1 1 , ? fry- 1- 1 an al a3-1 , 2 cos yi+' rj ? 1 1 A 2 -7- 437-- +. ?t "P 008 -4- it2 a r j-1 j The determination of arguments of complex roots when there ard more than two pairs of complex roots, and determination of multiple roots or roots relative to thorn by the modulus, are not included in this work. Pertinent details thereto can be found in the book by A. N. Krylov (L.15). Systems of linear equations. A formally exact solution of a system of linear equations can be made by Kramerld formulas. However, in such cases when the number of a system's equations is suffioiently large, the calculation of determinants in the Kramer formulas becomes practically impossible even for the high-speed contemporaneous computers. Much more practicable is the CAUB8 method by which system UNCLASSIFIrD - .469 - STAT Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 *.; -*=" *ien711; 11211tanxe"e+a2en."132; amei 4-6141Apt...4. pimb ; nn n n is reduced to the triangular form of aux' aieem+4 a22x2+...-411.2nXemib2I a, X mob nn. 11 n at described in paragraph 4-3 dealing with the caloulation of determinant. The solution of the triangular system is not diffi- cult at all. There also exist some very economical iterative methods of approximate solution of systems of equations, but we are not going to mention them. 9-3. Interuolation of Functions. Problem of interpolation. Let us assume that in the n+l-th point xo xit...txn are the prescribed values of 70 --f (x0), Ylamf 65,)"."Yn."(xn) of sone function ymftf(x). The polynomialPn(x) of power. n WISUAing at points so, the values yo,y1,...ant )mrY 11...01 _AVAssinED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 ?P:o I, 11120t0 s'Arr Page 2-7e, of 314 2,4.03.STAL, _ idioalled an interpoletioltpolynomial'tOnneoted with funotiOn f (x) and the aggregate of points xi (the existence pf such nomial has been proven it advanced algebra). The oonstruction of this polynomial and the calculation of its values at points x which do not coincide with points xi, i=0,1,...,n, are called "interpolation", and points xi are called "basic pointtof inter- polation". At times, the calculation of values of Pn(x) at points x located outside of the interval (x01xn) (in this case it is assumed that mo>.x.1}....?.xn) is also called "extrapolation", whereas the name "interpolation" is retained for values of x located within this interval. The neoessity to have such a polynomial constructed, arises in various oases. Let, for example, the values yo, be regarded as obtained by way of experimentation and, cons.- quently4 the expression of function f(x) is not known. Then we take for the value of function gf) at points x differing from xi (i=0,1,!..02) the value of interpolation polynomial Pn(x). Sure enough, in this case function f(x) is subject to imposition of some limitations (it is than regarded as sufficiently smooth). At times, the expressions of function f (x) is known, but it is so complex, that the calculation of its value is quite a labor- consuming process. In the interest of simplicity and labor-saving, the calculation of values of function f (x) in such oases is per- formed at a series of points x0?x1,...,xn, followed by the con- struction of an interpolation polynomial on those points, by taking for the values of function f(x) at points x, different from xi, the values of Pn(x). Case of eaullistamt basic points. Now we shall examine a most frequently encountered case when values of some function 744(x) YitY2,Y31,..?tim (94) are given at points xl,x2. x3....sxm (9-9) equidistant from one,another by equal distances h$ i.e. UNCLASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 In thit daee, the caastruotion of interpolatiam polynomials is made idth the aid of the Opened differences, which have the following definitions.: The nano "difference of the first order" is applied to values ' Alii0maY2'711 A Te?013.-'72, ? ? ? P A Yit-a. Irm-lra-I. ? In turn', -the differences of the first differences are called "differenoe-of the second order". They are, oonsequontly equal: 41 471"."442--47,--2724711 , 2 A y v iesav 2v 4v 2 A YM?.2 ?IR% Yin..3:"". A Ym...2 2 amm7m-- YM -141m -2. In a similar way are determined the differenoes of the third order, and so on. Generally speaking, it offers no difficulty toehow that a difference of the k-th order has the form of: 1, derlD ..0 wherein Ck are binominal coefficients UNeLRAIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT For convenience,thCse differences are gathered into a,table of differonced, which t'apresents the behavior of differences of various orders. For example, a table of differences for function f(x)==x5 is represented herein by table 9 - 1. Table 9-1 Table of Differences of Functions y-x5 X Y I 4 y 2 A 7 , 40 3' Ai 5 7 AI 6 7 0 0 1 30 150 240 120 0 1 1 31 180 390 360 120 0 2 t 32 211 570 750 480 120 0 5-1 243 781 1320 1230 600 120 -- 4 t 1024 2101 2550 1830 720 . -- 5 3125 ,4651 4380 2550 -- -- -- 6 - 77761 9031 6930 -- -- -- -- 7 16807 15961 -- .1.1.1, .111.? A MEM -_ 8 32768 -- -- -- -- -- -- , In this table the differences of the fifth order are oonstant, those of the sixth order are equal to zero. Let now x be an arbitrary point and let u be Wawat where xi is one of the points (9-9). The polynomial power of np assuming at n+lst point xilxi+ip...pxt+np the values yip yi+1,...pynp is defined by the Newton's interpolation formula. ITMASSIFIED Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 (at ximai f (xi )0...P(xi) STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 APAV'rr?Ac ?? - STAT uNr.71) 2 Pn(x.yi+ Trayi yi u(11...1,) (4.7z.? ) ji 3: Age 28), of '314 PtiSeit' ? u(u-I?)(u-=.2) . (u--rn + 1) An .+ Yi. U. Error Rn(x)...4gx)--1.,41(x),, admitted at replacement of function f (x) by interpolation paynomial Pn (x), is expressed by the formula f(n+1)(11)) Rn(x)om. n (x -x *Ix --xisn), (9-10) wherein is a point located between the largest and the smallest of points x, xi, xi+11,..lxilm. The magnitude can be found if the expression of function f(x) is known and the n+lth derivative (n+1) f (x) can be easily appraised. Otherwise, it will be necessary to employ the approximate equality I) A n+lf(xi) hn+ from which such a power n of the interpolation polynomial is taken which can be instrumental in rendering ths n-th difference within the chosen accuracy to be positive (then the n+lth differ- enoes are equal to sero and Rn is equal to zero, too). Considering fOrmula (9-10) we can come to a oonolusion with respect to the chase of the most expedient selection of points xi (contained in the table) at a prescribed value x. Indeed, the formula for Rn includes polynomial (X'""'Xi)cZ"""XiA1.1)Aseo(X...-aifn), whose roots are points xi? UBCLASSIFIM - 274 - Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 STAT Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/02/10: CIA-RDP81-01043R003800160006-3 A , that ite,maiiMuts-are close to the eV-erase root xi+ -- kconsider- ing n as an even number), and less than the maximums close to the end roots xi and Xi+21 ig.3,1). Therefore, xi must be ()heath SO as to have point x as close as possible to the middle of the basic points of interpolation Having made this ohoise, we may be sure, that the interpolation values have the same number of true signs, as there are in the table values of yily2, 1 .41111m?-? 9-I Naturally, when x is located close to the table sides (to xl or x, then the above-mentioned choice of xi is impossible and we have to take the best one from the possible points. 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