SOVIET DOCUMENT: "SOLUTIONS OF ENGINEERING PROBLEMS ON AUTOMATIC COMPUTERS BY B. M. KAGAN AND T. M. TERMIKAELYAN
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DATE Of REPORT
17 Sep 1959
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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 M3", 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
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LASSIYIIDL
B. IL KAGAN sad T. M. TER?IIIKAELTAlt
SOLUTIONS
OF ENGINEERING PROBLEMS ON AUTOMATIC
DIGITAL COMPUTERS
GoSENERGOIZDAT
(STATE POWER ENGINEERING PUBLISHING HOUSE)
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STATE POWER ENGINEERING PUBLISHING ROUSE
MOSKVA 1958 LENINGRAD
NCL.AS:SIPIED
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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been replaced by quest+n markt?,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".
_
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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 pAblemswors examined (Inlivegtition of trans
ition processes in longdistance 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 TerMikaelyan
Solution of Engineering Problems by Automatic Digital Computers.
Editor:V.M. Kurochkin Technical Editor: (}.Ye. Larionov
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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.
Highspeed 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 engineeringtechnical 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 195419551 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 41. 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
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Page 7 of 314 Pages
adefinite 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 thetdigi;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 4and 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 M3 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 M3 computer and on conducting a series of engineer
ing investigations on the M3 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 11/ 12, 14 wers'eritten
by B.M. Xigan; chapters 3, 4, 5, 9 and paragraphs 16, 16.7, 18, 19
were written by T.M. TerMikamlyan, and paragraphs 13 and 15 were
written by the authors jointly.
The Authors
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CONTENTS
Preface
Page 8 of 314 Pages
3
First Chapter. Fundamentals 7
11. Introduction 7
12. On numerical methods of solving mathematical problems 10
13. The blockdiagram of "ATsVMH 12
14. Systems of counting 14
15. Computers with a floating and fixed decimal point 16
16. Coding of commands 18
17. Certain operations performed by the digital computers 22
18. Control operations 23
19. Command cods of a conventional computer 25
Second Chapter. Operational principles of the automatic digital
computers. 26
21. The notions of the subsequent and parallel codes 26
22. Basic electronic parts of the "ATM" 27
23. Circuits for performing elementary logical operations 32
24. The performance of certain operations by means of
logical circuits 34
25. Peculiarities of performing arithmetical operations on
a computer. The concepts of the complementary and re
verse codes 37
26. Arithmetical devices 41
27. Memory devices 50
28. Devioes for input and output 54
29. Control units 55
210. Main characteristics of digital computers 58
211. The universal digital computer M3 59
212. The universal digital computer "Ural" 62
Third Chapter. Programming technique 64
31. The simplest example of the program 64
32. Conversion of the cell contents 66
33. Programs with the automatic choice of the number of cycles 70
34. Operation of command adding 73
35. Transformation of commands in programs 74
36, Examples of more complicated programs 77
37. Examples of programa for the M3 computer 80
38. Conversion of numbers from the decimal system into the
binary one and vice versa 87
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Page 9 of 314 Pampa
39. Separation of the integral and fractional parts in 00a
puters with the floating and fixed point 90
Chapter Four. Programming of mathematical problems 94
41. The scheme of a program 94
42. The program of solving ordinary differential equations
by the RungeKutta method 97
43. The program of calculating a determinant. The trans
formation of commands in several cycles. 100
44. Solution of algebraic and transcendental equations 103
45. Storing of functions in the computer 106
Chapter Five. Determination of critical revolution numbers
for the rotors of turbogenerators 108
51. The definition of the problem 108
52. The solution of the problem on a BESM computer 110
Chapter Six. Determination of stability of automatic control
eysteme on digital computers
61. Basic information
62* The scheme of a general program for calculating the
regions of stability and equistable lines in the plane
of two parameters
63. An example of calculating the static stability of a
longdistance electric power transmission line
115
115
118
124
Chapter Seven. Calculation and study of transient processes 127
71. Preliminary remarks 127
72. The logical scheme of the program of integrating a
system of ordinary differential equations by the Runge
Kutta method with a constant step 127
73. 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
74. The calculation of dynamioal stability of distant
electric power lines 135
Chapter Eight. Application of digital computers for calculat
ing electric machines
143
81. General remarks 143
82. The formulation of the calculating problem 143
83. Mathematical treatment of the problem. A remark on the
linear and nonlinear programming 145
84. A method of selecting the optimum variant of a motor
with the "ATsVM1'. The logical scheme of the program 146
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Page 10 of 314 Pages
Chapter Nine. Some information on approximate
calculations 151
91. Theory of errors 151
92. Solution of algebraic and transcendental equations 154
93. Interpolation of functions 158
94. Numerical differentiation and integration of functions 161
95. Solution of ordinary differential equations 164
Appendix 1. The list of operations of the M3 computer 170
Appendix 2. The list of operations of the "Ural" computer 171
Bibliography 174
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FIRST CHAPTER.
Fundamentals.
11. Introduction
SIAT
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
allsided 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 nonlinearities. As
a result the theoretical investigation by the conventional methods
becomes practically impossible.
The engineerinvestigator 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 highspeed 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 voltagesanalogs of the variable
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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 highspeed 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.
Highspeed 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.
Highspeed 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.
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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 highcapacity 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 nonlinear 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
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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.
Highspeed 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 longdistance 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.
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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
continuousdiscrete 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 takeoff, landing, and catapulting of aircraft, for
determining the takeoff 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.
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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.
12. 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 nonlinear differential equation(1) which is
(1)Equation (11) 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.
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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 (11) 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 (11), i.e. to determine
e= f(t) at OigtoET.
The presoribed rang. of tvaluss 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
(12)
during
(13)
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 * (14)
Finally we have one more relation:
ti+1 +
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The integration of equation (11) 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 (12). Thehone determinesni+1, 81.41, ti+1 by
formulae (13) to (X5) 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 (12), 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 (11) 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).
13. Blockdiagram 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 oneandhalf 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. 11 shows the simplified blockdiagram 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
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operating
at a gigantic speed.
Nowery Bowie*
Warn' Bevis*
Aritk
'Attie
leyet
Ostrat
atit
BeWiee
Devise
Control Unit
Figure 11.
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 socalled "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
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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. 11): a) the inner highspeed 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 rewritten 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.
14. 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  101 + 5 . 102.
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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 highspeed 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
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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 + 023 + 1 .22 + 0 21 + 1.20 + 1.21 and it is written as
follows 10101.1 in the binary system. Table 11 ahows the first
17 decimal and binary numbers.
Table 11
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 12.
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 relaycontact the presence of voltage can represent unity and
its absence  zero. For example, having four relays (Fig. 12) and
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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 12 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 nonzero 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;
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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 threeorder 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 fourorder 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 LNe"")
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 binaryoctal and binaryhexadecimal systems.
They are called also binarycoded systems.
The binarydecimal 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 fourorder 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
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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 binaryde
cimal systems. In the latter case the conversion of the binaryde
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 binarydecimal 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 3040 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
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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 thiscase, 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 13. In this
will accept numbers ranging from (1  233) to
.to *,(1  233) and the number zero. A number with
than 233, 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  233) 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
2.1212S 6 Sign of Limber 2Joelate 2 . ? ? ? ? ? ?
1112131*1514 Ii1S 19.11?11111111111*115 I ibl " lisill12?1111211231"12512412T12.13,1313113Z1331
Mantissa
Figure 13.
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
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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.
16. 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 13, it was necessary to
store in the computer not only the elements of the determinant,a,b,o
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The smallest, by modulus, normalized number, which is represen
table on our computer, will be 231 ? m 232. 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
235)=231. Consequently, only numbers within the range
235) to 232, of +232 to +2+31(1  235) 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.
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In computers with a, floating decimal point, all nuabor* are zit,
presented in the following way:
izP.A,1.1,1ic 1
' (16)
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 aforementioned 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 22 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 14, 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 21 to 21 + 22.+ + 2'5 . 1  235.
35
to
0.111....1 '
5 digits 35 digits
A
11131t1s16111119 11110.1151*Iff 14109 titai 1242111?12fliqvile1413013113211313114361461134001w1g1
Order p \Sim' of Matisse' A
Sign of order
Mientissa A
Figure 14.
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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.
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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
(18)
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.
(19)
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
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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 (18) and (19)
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
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pater. This is one of the most essential properties of computers with
program control. In Earagraph 18 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 sixdigit
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 aforementioned examples show that the command codes, which
appear as numbers, may be nonnormalized. Consequently, the memory
will store normalized and nonnormalized 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 11) 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 aforementioned manner are called "com
puters with a threeaddress control aystem". There are also single,
two and fouraddress computers.
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In the singleaddress 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 singleaddress programming system is
used, for example, in the computer "Ural" (see Paragraph 212).
In the twoaddress 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 ":43" (refer to Paragraph 211).
Another version of the twoaddress system is also used, in which
the first address is used as in the singleaddross computers, while
the second address indicates the number of the cell in which the sub
sequent command is stored.
Finally, in the fouraddress command system, the first throe
addresses are used in the same way as in the threeaddress commands,
while the fourth address indicates in which cell the next command is
stored. This problem will not be considered here in further detail.
17. 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.
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1; 1 A 0 0 A 1 .? 0 A 0 as 0.
The ndigit nuabarr. is the result of the logical multipli
cstion of two ndigit 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 nvalued number, whereby the di
gits k and j consist of ones while zeros stand in the remaining digits.
a'n an1c4'k+1 Gk ac1c1 ". *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
nvalued 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 n1 . . si k+1
Then shifting the same number by k digits to the loft this number
is formed
?
c?nk ank1 ...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.
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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 nonnegative number a is called such an in
tegral number [a] (read "ant/yen a), for which the difference
a  (a)
in a nonnegative 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; 13?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
nonnormalized 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 15,
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for the addition of two, normalized numbers 2PA 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+P15B 2PA + 2P .
2Pcl
In case a nonnormalized 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).
18. Control operations.
In automatic digital computers those control systems have found
the most widespread 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 29.
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111C148001g1) ,
way the nAbber 4+2 ispxoduoid. Then the computer will start:toper.,
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 socalled 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 socalled 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
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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.
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19. 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,
threeaddress 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.
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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 nonnormalized 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.
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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 33.
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Preparation of cells
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...01 Forming the argument
I + ih and adding 1 in the counter
Calculation of sh (I + ih) and
oh (I + ih)
441ii Calculation of th (I + ih)
and comparison of i with n
Stop
Figure 33.
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 switchover 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 switchover 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
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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 x12h
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 M3 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 143 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.
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Page 141 of 314 pages
Program of conversion of numbers from the decimal to the binary
System of calculation for the Machine M3.
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 ? +
1021i
atL1 119.+1
+ 4. ... 4.
10n11 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
Binarydecimal 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
76. .0,00100...0
la
...?:412,1900
16 .....0,Q0
Omm0,00...028
13 li
 ....,ri,......ninna%.....,
.,
For realization of this program, the binarydecimal code of num
ber A is perforated in the tape (see Paragraph 28) 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)
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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 nth,
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 occuied by umber ?42
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 58. Repetition of
this operat.on n times produces the binary expression of all n columns
411' (1'2 i.e. the binarydecimal code of number A.
' ???
In the 143 mncLine, only 28 senior columns can be used by the
binarydecimal 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 229 + 230 . 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
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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 /43
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 )10ni
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 229 + 230
Binary notation of number subjected to conversion
Formation of binarydecimal code of number subjected to con
version
p 0,111100....0
1
0,000100....0
10
0,101000....0
229 + 230 . 0,00....011
0 . 0,00....0
Work cell for storing 0, 0....011110....0
41
Work cell for storing '
16i
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39. 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 V17rTla
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 timesaving 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.
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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: 3628.800
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 17),
but this can not be done in the machine M5. 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.
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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 ti1
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) 141
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.
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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
33. 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. 33)
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
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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
1x 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
1z
x, 0 < x<
1+z
then
1.
1s
p + + igzp + lgs
1+ 1+z
x2n1))
1?x 3
+ ig x + +...+1 + x 3 2n1
?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
1x 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 105
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.
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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 e1
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 34
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 e1 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
1z
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
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, Fags 150 of 314 Pagea
cell is is formed lg y. The logical diagram of this program is shown
by Figure 34 (cell a. contains number Y).
Below is shown the basic part of this program (operators INT).
? ?
,
P43L
P at
7 non
it e"
_.

ii
/1
Pat

"1"
'12
4
"1"
a
k + 8
<
1"
a
k + 11
"7
Pa,_
41
__
I'
P CAA"1"
__
1
a
a
4.
Y1
PV,.
k+4

? ? ?
? ? ?
. ? .
? ? ?
I
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age 151 of 514 Pages
FOURTH CUPTER.
The Programaing of Mathematical Problems.
41. Program circuit.
In the preceding chapter, using a number of examples, we demon
strated the tasic methods of programplotting. 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 socalled 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
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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 37 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 33 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 socalled 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
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scheme will be as depicted on Figure 41
ro.
>
Figure 41.
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 & ?
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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 37, in a different formlas shown on Figure 42.
41""A t stop
tat
Figure 42.
The blockdiagrams 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.
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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 37 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 socalled 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 programplotter 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 selfreconstructing 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 43.
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 subcircuits into the principal circuit
can be multistage 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 52.
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
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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 36.
The carefully worked out standard subprograms prove to be very
economical with respect to the time of the machine's operation.
42. The proEram for solution of common differential equations by
the RungeKutta method.
An examination of the ways of finding a sol,Ition of a common
differential equation by the RungeKutta 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 RungeKutta 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, ..., n1 ,
1
(k1+2k2 + 2k3 + k4); kim.miNx);
I.
ki 11 k2
k24(xl. 1T, Yi i); k3h4(xi Yi
h(f)(xj. yi + k3).
UNCLASSIFIED
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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
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Page 158 of 314 Pages
cell y ) and places the result into cell it ; these two operators
Kt and Kt' replace by themselves the aforeintroduced 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 43.
/..cX
x .542
P'H " Q.V." ki .... Tj ... FO) .71j
r
J?2
Figure 43.
[.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 ?.4n.
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The program for the solution of common differential equations by the
RungeKutta 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 t0
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
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common =re:initial equation by the RungeratinUthedo 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
RungeKutta Method,
43. 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
abovementioned 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.
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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 ji1 element ai+1,i'
ai+
2 t standing in the ith column below the diagonal ,11"41 4jlli
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. 44).
i+1
ail+1
sji+1
ajh
Fig. 4 4.
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For this we calculate the coefficient a aii and from the elements
of the jth line subtract the corresponding elements of the ith line,
multiplied by this coefficient. In order to save the time, it is
expedient to subtract only from the elements ah (hami +11
42,...,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:
B11operators calculating the coefficient
Aijhoperator calculating the new value of the element
a1
a ?jh ..a.
aiiaih,
jh
which is located at the intersection of the jline and the hcolumn.
With the use of these operators we can express the calculation
scheme, upon reduction to triangular mode; as
ri i3il fl ikiih.
hi  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 belowdescribed 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.
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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 belowdescribed 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
r3P
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
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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 selfreconstructing. 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
bit
)
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.
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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' Lk11lo
k#11
k#12
k
I (h)
Y
kf14
k415
k +16
x417
k#18
k+19
F(ni) k120
t. k +21
111 A) L k f22
i
k 123
Di k +24
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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"
k414
SK
k415
"1.1,IIIA"
k#15
+
"1"
Y 14,
n17
Y k
k#14
SK
kiw13
"nIA"
k+43
SX
pt1
"nI , M A"
"
4.
i
y.4
d
"n1"
p3
k410
XX
X
XXX
XX
11
UNCLASSIFIED
TYa21",
"a117,
Pi
? .. ... ??
X P1 t"al
P2
"a2 2",
P2tAn
)( I "ai 1;
STAT
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F (n 4.1j
k +25
ik +26
k +27
k +28
f( i) k
Y1; k+30
q1.1 a11
a+2 a12
alfl a1n
a +r j.i a.21
a t2 a22
a12n a2n
k 131
The first
determinant's
line
Page 166 of 314 Pages
STAT
I
/
/
SK
Pt
"(n41)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 i1) IIA"
k 4.24
+"
uin
y;
Nri
k IF 7
Nit
Stop
:
an .t1
a 41
P1
(JO)
X
P1
a+2
P2
(i,h)

x
all. n ,112
P2
a 1 1
atn12(iiith)
&
(i)
A
_
.
a in(n1)ti ani *N.
a +n (n1)+2 an2
a 4nn
ann
The seoond 16counter of index i
dot ermi
nant ' s
(counter of index j
47
line counfy0 of index h
The nth determinant '
line
The initial form of commands. Dependence on
an index is indicated in the parantheses
UNCLASSIFIED
STAT
.111=10....
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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 .
44. 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. 41), 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:
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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
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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 44
Pr ( k 45
A r i. k+6
T' (k+7
B' t k t
kt9
f(i) tk+10
k11
f(t) tk+12
At 0(.1.13
f(r) k +14
L 415
f(1.) tk +16
Ai tic +17
i\t tk *18
B' {ki19
k f20
11 421
k422
PCh Imowh
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,
ki15
a'
P"
k+7
Yt,
k.11ao
T.I.
a
Y;
k421
13

WilLltsIsTrp
Transfer of control
for determination
of D
STAT
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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
xrrV ? IV. 8' D (t)?kiartvqL11: L40?A 0 it P r
Pr ir
Route 1
The system J.
unstable
Route
Formation o
80hfl
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
RungeXutta method
To the calculation operator
on the basis of A initial data
v ok +1
fc1*=??';'?''&7813.1atingthe transition process
 h 
Formation of
Dok+1=Ook + 5
PriklgY113014
Fig. 710
UNCLASSIFIED
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Page 237 of 314 Pages
resistance of dissipation of the generator's stator, including the
transmission's inductiVe reactance xp.43r+ 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
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Page 238 of 314 Pates
All values, including the time value, are expressed in equations
(713) in relative unitit5 (angle Ostia? t and U0 are expressed in
radians). Those (713) 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 generatortransformer
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 (713)
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 shortcirc
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 shortcircuit, 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 79 equations
from (713),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 (713
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 (713)
UNLASS/FIED
2
he
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The first equation (714) 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 placedinto 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. 713
It is easy to show, that in this case the solution E*, Z* of
(714), can be had from the expressions
(E
k+1 k K+1) . k+1 ;
(Ek?Ek+1) aiikz
b
z*.. f
a
(715)
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 (715), 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
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Ek+, Bk_4100? (716)
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 (715). After that are
calculated:
_ E*
and id +z.
P.P
CHAPTER VIII
APPLICATION OF DIGITAL COMPUTING DEVICES FOR CALCULATION OF ELECTRIC MACHINES
811 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 laborconsuming 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
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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 M5. 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 M5 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.
82. 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
M3 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
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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 C15 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
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Page 246 of 314 Pages
with a prescribed outer diameter,,the calculations are made to
determine the optimuminner 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.
83. Mathematical Interpretation of ,the Problem,
Remarks on Linear and NonLinear 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 Cowhile taking into con
sideration the limitations (81).
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) (82a)
Ct......FD(Da: Di, B8, s) (82b)
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,,
; (83)
a) (84)
(84)
a) clos?f up. (85)
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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
1Bifs+.
Bie?
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
cPi
P )1?
Calculation of C
+Att. J.
on ol of condition IA6
[Transfer of to Ci_14.1
Cai
Control of condition
Cdi1 Cat Ac Cai4
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,
J3c
to
a 9
nsio. of ; Das toria ; to Di ;
UNCLASSIFIED
;
x 6
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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 (81) 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.
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P4145,641441%,"
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 (8i) 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
willgo 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
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Pago2.5 \Or':;314715agia
Transfer  to oier,a*4:feansj.1?, that the current po
located within the area of admissible values of variables under the
conditions of (81).
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
(81) 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 (81)) 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
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maksmatt
? 0 45.,,,ItTA
age 257,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
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?
Thenvalnes 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 abovedescribed 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 143 in 45551. 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 143 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
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do w00101tte,exact values of the m404;
STAT
104.4..;? a t,9.1740.*
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,
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PA0f*
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
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40.1CITASSIMO
+
441+1
41110..n+1.10 +ap_21,1011 n.
..7MVP7rWW;,11*'"41,;r411.W4i.
Page 26?f 
4,1otg'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 nth
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 nth 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
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Page 23 of 314 ,gages'
exprosain num1er A with n correct ciphers, then the relative
error a does not exceed
1
a ?10n4 e
Indeed, by definition the relative error is equal to:
<
A .
10pnil 10P11+1
?
z
a
a.10
=Mt
n1 ?
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
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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
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1**) 287Kaf'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 aboveadduoed 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 104, whereas the relative error of their differ
ence, which is 0.11, can be equal to 101, 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
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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 aboveformulated rule it
follows, that the result is less by onetwo 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.
92. Solution of Algebraic) and Transcendental EqBdjaaL
equation
Formulating a Problem. Let us have to solve the
f (x) 4.0, (91)
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 (91), 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 (91). If f(x) is a continuous function, then we shall
call point x* the approximate value of the root of equation (91)
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
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page 263 of 314 pege?TAT
cases. The first and the most difficult step in searohing for
material roots of equation (91), 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 inchapter 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 51, 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 (91) 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 (91) does not turn into zero, and if
then equation
(x)  (x)f(x),
x (x) (92)
has in that ambient area only one solution x..a.
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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 nth approximate
1 ..'((x_1)
eta.
When the sequence xn coincides, then its limit is nothing
else but the root of equation (92). Actually, if we designate
the limit of this sequence at, then
nboo
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
Iltxnli
becomes less than the admissible absolute error. In other words,
?
it is assumed that the condition
I x x 1
I n nli
MINIFIED
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age
,
f ;314STAT4r,.
In the simpliest case it is assumed that$p(x)..1 and, conse
quently, solve the equation xf(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 (91) 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
xa
our case funotiorithas the form Vi(x)m.fi(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:
xemxn1 
fl(xn_1).
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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 37 we applied this method for the calculation of the value
of the square root of x.
Algebraic equations. Let function f(x) in equation (91) be
the polynomial of power soll,with material coefficients, i.e. that
(91) has the form:
Pn(x)=.7"... a0x21+a1 xn1+.
+ a x+a ..0. (93)
n1 n
According to the principal algrebraic theorem this equation
has exactlyKroots, with due regard for their multiplicity. With
this, rootais called the kmultiple root of equation (93), provid
ed that the conditions
(k1)
(a)..0;
are met,
Theae roots can include complex roots and then the latter are
1.214.416414
always conjugated in pairs, In other words, numbers a+laare, at
the same time, the roots of the equation (93).
An approximate definition of the roots of equation (93) can
be accomplished with the use of the abovedescribed methods, but
inasmuch as they assume, aaapproximately 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
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?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?ac the polynomial P (x) and all its
consecutive derivatives P1(x),P1,(x),..., P(x) are positive, then
the upper boundary of the positiveroots is number cc.
INeilSSIFIED
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71:77:77777777777,77' '''',' _q
1,,
'%fltARRTii:?, STAT
1
Page 21
, f 314 Tates
Iliiimuoilc,asP (it)( ) niso> 0, then P('n11(x) is agrowing
function and, consequentlyt there exists such a ci, where at
at x.,433.4. Consequently, at x?.c3., derivative
(n2)
P (x) is a growing funotion and, therefore, there exists suoh
a c2 whereat P(n2) (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 (94) 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
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at:E
TK?....711Zer'
731 

STAT
product of ho?.Zfioients'
i6a
At andao on, Until the fiat or the last member of the
initial eqation.
In other Words
2
2
ale..ak2ak+lake2ak+2ak2".;
Proceding from equation (949, another equation oan be
constructed, just the same way as equation (941) was made from
equation (94). Continuing this way, at (k1)th and at kth
steps we get equations:
nCk1) n1
x x
.(k1) n2 (k1)
7: .012 X +? ? ? +an ?J0001 (95)
n (k) n1 (k) n2 (k)
x al x a2 x an 1.01
for whieh the approximation equations
41CL[alk112;
Ea2
a2
k (k1)/2
?
noran ] ; (96)
will be met, i.e. in the kth 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
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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 (94)*
Presuming now, that among the roots of equation (94) are
several pairs of complexconjugated roots
a. ri (cos yi s
r. (cos lei ?
cps
==.71
a.. r. Cos If ? ? t/? 1Ift gr./)
Jti
Cf IFI
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.*1Wowm
ItgvtrIMPr
vir(4$?0uri
and har4lig the roots numbered anewt so that
al I > 1 a21 " I ai11 ri>
>. ai+2i> ? ? ? I aia I>
ai+21
STAT
AP1Wi4
we shall again get equations (95)p for which the equalities
(96) 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 (97) there appear equalities
(k)
ii (k) Ii a1
al I
* ? ? ? 0
(k)
h aj+1
ry.771
aj 1
(k)
h aj+2
j+2
With the use of these equalities we at first determine the
material roots a a2,lf ...pa 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+"?+1111+
+ 21" con+ai+2+...+a.
T.121C1rIED
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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
n1 1 , ? fry
1 1
an al a31
, 2
cos yi+'
rj
? 1 1 A 2
7 437 +. ?t "P 008 4
it2 a r
j1 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 highspeed contemporaneous
computers. Much more practicable is the CAUB8 method by which
system
UNCLASSIFIrD
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*.;
*=" *ien711;
11211tanxe"e+a2en."132;
amei 46141Apt...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 43 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.
93. Interuolation of Functions.
Problem of interpolation.
Let us assume that in the n+lth 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
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?P:o
I, 11120t0
s'Arr
Page 27e, 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 laborsaving,
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 (99)
equidistant from one,another by equal distances h$ i.e.
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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 Yita. IrmlraI. ?
In turn', the differences of the first differences are
called "differenoeof the second order". They are, oonsequontly
equal:
41 471"."44247,2724711
, 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 kth order has the form of:
1,
derlD
..0
wherein Ck are binominal coefficients
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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 91
Table of Differences of Functions yx5
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
51 243
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 (99). 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
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(at ximai f (xi )0...P(xi) STAT
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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(uI?)(u=.2) . (urn + 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), (910)
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 nth 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 (910) 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
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A ,
that ite,maiiMutsare close to the eVerase 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,...am
1
.41111m??
9I
Naturally, when x is located close to the table sides (to xl
or x, then the abovementioned choice of xi is impossible and we
have to take the best one from the possible points. Particularly,
when xrcxitrt cxtseott
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Declassified in Part  Sanitized Copy Approved for Release @ 50Yr 2014/02/10: CIARDP8101043R0038001600063
Declassified in Part Sanitized Co y Ap roved for Release ? 50Yr 2014/02/10: CIARDP8101043R0038001600063
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Declassified in Part  Sanitized Copy Approved for Release
50Yr 2014/02/10 ? CIARDP81 010411RnmsnniAnnnA
Declassified in Part  Sanitized Copy Approved for Release @ 50Yr 2014/02/10: CIARDP8101043R0038001600063
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Declassified in Part  Sanitized Copy Approved for Release @ 50Yr 2014/02/10: CIARDP8101043R0038001600063
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Declassified in Part  Sanitized Copy Approved for Release @ 50Yr 2014/02/10: CIARDP8101043R0038001600063
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Declassified in Part  Sanitized Copy Approved for Release @ 50Yr 2014/02/10: CIARDP8101043R0038001600063
Declassified in Part  Sanitized Copy Approved for Release @ 50Yr 2014/02/10: CIARDP8101043R0038001600063
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Declassified in Part  Sanitized Copy Approved for Release @ 50Yr 2014/02/10: CIARDP8101043R0038001600063
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Declassified in Part  Sanitized Copy Approved for Release ? 50Yr 2014/02/10: CIARDP8101043R0038001600063
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Declassified in Part  Sanitized Copy Approved for Release ? 50Yr 2014/02/10: CIARDP8101043R0038001600063
Declassified in Part  Sanitized Copy Approved for Release 50Yr 2014/02/10: CIARDP8101043R0038001600063
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Declassified in Part  Sanitized Copy Approved for Release @ 50Yr 2014/02/10: CIARDP8101043R0038001600063
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unettinstil anamerp Aocnir npeaeabotorn
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itocrb. acrrapaa olainprautnama WM,
trio rsynt nouropauau nampestint no.ly?
4aerc..: 411C.10. OT.1114310C 01 trpe.aboytne.
to :lamb a ovum pear= czrianx 113
11CC1110 10V1OC 311340101C 13C.13441111, 030
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.12eTC0 yp31111eHHC
z"f a: z*' +,..F +a: = 0
(94')
no caeApouteuy npaallAy:
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licxoati 113 ypaatieulia (941, moiKno
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.
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x"10t,"x" 1441x"2i...1aT=0,
AAR KOTOpb1X 6yAyr 81,1110AnCIILA ripii6m1
Ticeimue paoeticraa ypaaliciniax (95)
lionue nepemeiniue 060311ageuu toAwe
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02 ? juI j
(9.6)
r. e. a lz?Tom npeo6pa3oaaitliosi ypanue
min K044niwieltru 6yAyT paislul !Watt.
param CO0TAC1CTIly101.11.HX X094411UieH100
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3112K implicit onpeAcinierca itenocpeA?
croemioil noAcralionitoilis ypaimicioie (94).
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nap KOMHACKC110 COnp1484C11111AX
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creneinix X, Kriome crcneuell 1, 1,
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