Базовий алгоритм апроксимації граничної траєкторії короткофокусного електронного пучка за допомогою коренево-поліноміальних функцій четвертого та п’ятого порядків
The new iterative method of approximating the boundary trajectory of a short-focus electron beam propagating in a free drift mode in a low-pressure ionized gas under the condition of compensation of the space charge of electrons is considered and discussed in the article. To solve the given approxim...
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| description | The new iterative method of approximating the boundary trajectory of a short-focus electron beam propagating in a free drift mode in a low-pressure ionized gas under the condition of compensation of the space charge of electrons is considered and discussed in the article. To solve the given approximation task, the root-polynomial functions of the fourth and fifth order were applied, the main features of which are the ravine character and the presence of one global minimum. As an initial approach to solving the approximation problem, the values of the polynomial coefficients are calculated by solving the interpolation problem. After this, the approximation task is solved iteratively. All necessary polynomial coefficients are calculated multiple times, taking into account the values of the function and its derivative at the reference points. The final values of polynomial coefficients of high-order root-polynomial functions are calculated using the dichotomy method. The article also provides examples of the applying fourth-order and fifth-order root-polynomial functions to approximate sets of numerical data that correspond to the description of ravine functions. The obtained theoretical results are interesting and important for the experts who study the physics of electron beams and design modern industrial electron beam technological equipment. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2023.3.10 |
| first_indexed | 2025-07-17T10:28:23Z |
| format | Article |
| fulltext |
I. Melnyk, A. Pochynok, 2023
Системні дослідження та інформаційні технології, 2023, № 3 127
UDC 004.942:537.525
DOI: 10.20535/SRIT.2308-8893.2023.3.10
BASIC ALGORITHM FOR APPROXIMATION OF THE
BOUNDARY TRAJECTORY OF SHORT-FOCUS ELECTRON
BEAM USING THE ROOT-POLYNOMIAL FUNCTIONS OF THE
FOURTH AND FIFTH ORDER
I. MELNYK, A. POCHYNOK
Abstract. The new iterative method of approximating the boundary trajectory of a
short-focus electron beam propagating in a free drift mode in a low-pressure ionized
gas under the condition of compensation of the space charge of electrons is con-
sidered and discussed in the article. To solve the given approximation task, the root-
polynomial functions of the fourth and fifth order were applied, the main features of
which are the ravine character and the presence of one global minimum. As an ini-
tial approach to solving the approximation problem, the values of the polynomial
coefficients are calculated by solving the interpolation problem. After this, the ap-
proximation task is solved iteratively. All necessary polynomial coefficients are cal-
culated multiple times, taking into account the values of the function and its deriva-
tive at the reference points. The final values of polynomial coefficients of high-order
root-polynomial functions are calculated using the dichotomy method. The article
also provides examples of the applying fourth-order and fifth-order root-polynomial
functions to approximate sets of numerical data that correspond to the description of
ravine functions. The obtained theoretical results are interesting and important for
the experts who study the physics of electron beams and design modern industrial
electron beam technological equipment.
Keywords: approximation, interpolation, root-polynomial function, ravine function,
least-square method, discrepancy, approximation error, electron beam, electron-
beam technologies.
INTRODUCTION
Today, an important task regarding the further development and industrial appli-
cation of electron beam technologies is the preliminary estimate of the boundary
trajectory of the electron beam using different suitable approaches. Therefore, in
addition to development the basic theory of electron beam optics and obtaining
necessary analytical ratios and corresponded numerical methods for solving dif-
ferential equations, methods of interpolation and approximation are widely used
also [1–3].
A separate issue in this aspect is the evaluation of electron beam trajectories
and finding the focal parameters of beams in high-voltage glow discharge
(HVGD) electron guns [1; 4–7]. Main singularities of such kind of beams, at the
physical point of view, is its propagation in the soft vacuum in the medium of re-
sidual gas with compensation the space charge of electrons. In additional, usually
such beams are formed by the cathodes with large emission surface, therefore the
convergence angle of beam is generally large and its focal diameter is not so
I. Melnyk, A. Pochynok
ISSN 1681–6048 System Research & Information Technologies, 2023, № 3 128
small, range of few millimeters. Just today HVGD electron guns widely used in
various branches of industry, in particular in the electronic industry, instrument
building, mechanical engineering, metallurgy, automobile and aerospace industry
[5–9]. The main advantages of these types of guns, regarding the possibility of
their industrial application, are operation in a soft vacuum in the medium of vari-
ous technological gases, including noble and active gases, high stability and reli-
ability of operation of the HVGD electron, the relative simplicity of the design
and the cheap of HVGD electron guns, as well as stability and reliability of its
operation [1–3]. Ease of control the power of the electron beam both by gas dy-
namic lows and changing the operation pressure in discharge chamber and by the
lighting of additional discharges is also possible [8; 9].
Among the advanced application of HVGD electron guns in the modern
electronic production most important are follows.
1. Welding of contacts and casualization of crystals. For example, such ap-
plication is very advanced in the experimental production of cryogenic low-
temperature devices [10; 11].
2. Production of high-quality capacitors with the small value of current losses on
the base of ceramic films [12–14].
3. Production of communication devices as receivers and transmitters of micro-
waves antennas on the base of high-quality ceramic films [12–14].
4. Refining of silicon ingot for obtaining the pure material for electronic in-
dustry [15–18].
Main problems of HVGD optics and energetics are well-known and have
been complexly analyzed in papers [1; 19–21]. The problems of guiding short-
focus electron beam in ionized gas also have been studied carefully both theoreti-
cally and experimentally, corresponded mathematical model was presented in the
paper [1]. However, mathematical methods of interpolation and approximation of
electron beam boundary trajectories in the medium of ionized gas still wasn’t de-
veloped up to the necessary stage, corresponded mathematical function also ha-
ven’t considered complexly. This shortcoming largely hinders the introduction
into the industry of advanced electron-beam technologies.
In the papers [22; 23] the root-polynomial function was considered as the
suitable mathematical tools to interpolation the boundary trajectories of shorty-
focus electron beams in the case of its propagation in the medium of ionized gases
with compensation of the space charge of electrons. Root-polynomial functions
from second to fifth order and corresponded interpolation results were presented
and analyzed in papers [22; 23]. The interpolation results have been compared
with the accurate solution of differential equation of electron beam propagation,
and corresponded interpolation error usually was smaller, than 5% [22; 23].
Therefore, the aim of investigations, which are described in this article, is forming
the algorithm of approximation of boundary trajectories of short-focus electron
beam, propagated in the ionized gas with compensation the space charge of elec-
trons. Testing examples of using such approximation for the root-polynomial
functions of fourth and fifth order are also considered and obtained results of nu-
merical simulation are analyzed.
Basic algorithm for approximation of the boundary trajectory of short-focus electron beam …
Системні дослідження та інформаційні технології, 2023, № 3 129
THE PREVIOUS RESEARCHES AND THEORETICAL FUNDAMENTALS OF
PROPOSED APPROACH
The basic theory of polynomials interpolation and approximation is considered
generally in the manual books [24; 25]. In the papers [22; 23] was considered the
task of interpolation the ravine functions, which corresponded to the boundary
trajectories of electron beam, propagated in the medium of ionized gas, by the
root-polynomial functions, which in the general form are written as:
,)( 01
1
1
n n
n
n
n CzCzCzCzr
(1)
where z is the longitudinal coordinate, r is the radius of the boundary trajectory
of the electron beam, n is the degree of the polynomial, as well as the order of the
root function, 0C – nC are the polynomial coefficients.
The analytical relations for coefficients of forth order root-polynomial func-
tion 3210 ,,, CCCC and 4C , which, in general form, corresponding to relation (1),
is written as follows [22; 23]:
,)( 4
01
2
2
3
3
4
4 CzCzCzCzCzr (2)
are also was obtained and analyzed in the papers [22; 23].
Clear, that for 5 unknown polynomial coefficients of function (2)
3210 ,,, CCCC and 4C , with defined basic values of the spatial coordinates
54321 ,,,, rrrrr , 4321 ,,, zzzz and 5z , corresponded set of 5 linear equation for
calculation the polynomial coefficients is written as follows [22; 23]:
.
;
;
;
;
4
5051
2
52
3
53
4
54
4
4041
2
42
3
43
4
44
4
3031
2
32
3
33
4
34
4
2021
2
22
3
23
4
24
4
1011
2
12
3
13
4
14
rCzCzCzCzC
rCzCzCzCzC
rCzCzCzCzC
rCzCzCzCzC
rCzCzCzCzC
(3)
For solving the set of equations (3) firstly considered the coefficients basic
intermediate variables lka , , where k — number of iterations for solving set of
equation (2) and l — number of equation i in the set (3) [22; 23]. Corresponded
analytical relations are look as follows:
;
12
4
1
4
2
2,1 zz
rr
a
;
13
4
1
4
3
3,1 zz
rr
a
;
14
4
1
4
4
4,1 zz
rr
a
;
15
4
1
4
5
5,1 zz
rr
a
;
23
2,13,1
3,2 zz
aa
a
;
24
2,14,1
4,2 zz
aa
a
.
25
2,15,1
5,2 zz
aa
a
(4)
After that, considering the second set of additional variables lmkb ,, , where
parameter m is the power of variable z in the set of equations (3). Corresponded
analytical relations are written as follows:
;
23
2
2
13
2
11
2
21
2
3
3
2
3
3
3,3,2 zz
zzzzzzzzzz
b
I. Melnyk, A. Pochynok
ISSN 1681–6048 System Research & Information Technologies, 2023, № 3 130
;
23
2131
2
2
2
3
3,3,2 zz
zzzzzz
b
;
24
2
2
14
2
11
2
21
2
4
3
2
3
4
3,3,2 zz
zzzzzzzzzz
b
;
24
2141
2
2
2
4
4,2,2 zz
zzzzzz
b
;
25
2
2
15
2
11
2
21
2
5
3
2
3
5
5,3,2 zz
zzzzzzzzzz
b
;
25
2151
2
2
2
5
5,2,2 zz
zzzzzz
b
(5)
;
3,2,24,2,2
3,3,24,3,2
4,3,3 bb
bb
b
.
3,2,25,2,2
3,3,25,3,2
5,3,3 bb
bb
b
After that, with known values of the coefficients 4,2,2b , 3,2,2b and 5,2,2b ,
five additional variables from the first data set 2,35,34,35,4 ,,, aaaa , and 1,3a as
well as two new coefficients from the second set of variables 1,3b and 2,3b ,
arecalculated by using such analytical relations:
;
3,2,24,2,2
3,24,2
5,4 bb
aa
a
;
3,2,24,2,2
3,24,2
4,3 bb
aa
a
;
3,2,25,2,2
3,25,2
5,3 bb
aa
a
(6)
;
3,2,25,2,2
3,25,2
2,3 bb
aa
a
;
3,2,24,2,2
3,24,2
1,3 bb
aa
a
;
3,2,24,2,2
3,3,24,3,2
1,3 bb
bb
b
.
3,2,25,2,2
3,3,25,3,2
2,3 bb
bb
b
And finally, taking into account relations (4)–(6) and the first equation of the
set (3), all polynomial coefficients of the set of equation (3) are defined with ap-
plying the following relations:
;
1,32,3
1,32,3
4 bb
aa
C
;
3,2,24,2,2
3,3,24,3,2
1,32,3
1,32,3
3,2,24,2,2
3,24,2
3 bb
bb
bb
aa
bb
aa
C
3,2,24,2,2
3,3,24,3,2
1,32,3
1,32,3
3,2,24,2,2
3,24,2
1,32,3
1,32,3
23
2,13,1
2 bb
bb
bb
aa
bb
aa
bb
aa
zz
aa
C (7)
;
24
2
2
14
2
11
2
21
2
4
3
2
3
4
zz
zzzzzzzzzz
;)()()( 212
2
121
2
23
3
12
2
11
2
2
3
242,11 zzCzzzzCzzzzzzCaC
.11
2
12
3
13
4
14
4
10 zCzCzCzCrC
The analytical relations for coefficients of fifth order root-polynomial func-
tion 43210 ,,,, CCCCC and 5C , corresponding to relation (1), is written as fol-
lows [22; 23]:
Basic algorithm for approximation of the boundary trajectory of short-focus electron beam …
Системні дослідження та інформаційні технології, 2023, № 3 131
.)( 5
01
2
2
3
3
4
4
4
5 CzCzCzCzCzCzr (8)
Therefore, the set of equation for defining the polynomial coefficient includ-
ing 6 equations and generally it writing as follows [22; 23]:
.
;
;
;
;
;
5
6061
2
62
3
63
4
64
5
65
5
5051
2
52
3
53
4
54
5
55
5
4041
2
42
3
43
4
44
5
45
5
3031
2
32
3
33
4
34
5
35
5
2021
2
22
3
23
4
24
5
25
5
1011
2
12
3
13
4
14
5
15
rCzCzCzCzCzC
rCzCzCzCzCzC
rCzCzCzCzCzC
rCzCzCzCzCzC
rCzCzCzCzCzC
rCzCzCzCzCzC
(9)
But the advance of proposed method of calculation the polynomial
coefficients is that with using the set of coefficients for four-order function,
defined by relations (4)–(6), some of that relations are also correct for defining
the coefficients of fifth-order polynomial. For example, among the first set of the
coefficient a only the values a1,l are different form relations (4), since they are
including fifth order of beam radius r. Corresponded relations for defining the
coefficients a1,l are written as follows [22; 23]:
;
12
5
1
5
2
2,1 zz
rr
a
;
13
5
1
5
3
3,1 zz
rr
a
;
14
5
1
5
4
4,1 zz
rr
a
.
15
5
1
5
5
5,1 zz
rr
a
(10)
Other two coefficients from the first set a26 and a36 are defined by the fol-
lowing analytical relations:
;
26
2,16,1
6,2 zz
aa
a
.
3,2,26,2,2
3,26,2
6,3 bb
aa
a
(11)
The corresponded coefficients b from the second set of additional variables
are calculated for five order root-polynomial functions by analytical solving the
set of linear equations (9) by the following relations:
;
26
1216
2
2
2
6
6,2,2 zz
zzzzzz
b
;
23
2
3
13
3
1
2
1
2
2
2
1
2
31
3
21
3
3
2
2
4
3
3,4,2 zz
zzzzzzzzzzzzzz
b
;
24
2
3
14
3
1
2
1
2
2
2
1
2
41
3
21
3
4
4
2
4
4
4,4,2 zz
zzzzzzzzzzzzzz
b
;
25
2
3
15
3
1
2
1
2
2
2
1
2
51
3
21
3
5
4
2
4
5
5,4,2 zz
zzzzzzzzzzzzzz
b
;
26
2
2
16
2
11
3
21
2
6
3
2
4
6
6,4,2 zz
zzzzzzzzzz
b
(12)
I. Melnyk, A. Pochynok
ISSN 1681–6048 System Research & Information Technologies, 2023, № 3 132
;
26
2
2
16
2
11
2
21
2
6
3
2
3
6
6,3,2 zz
zzzzzzzzzz
b
;
3,2,24,2,2
3,4,24,4,2
4,3,3 bb
bb
b
;
3,2,26,2,2
3,4,26,4,2
6,4,3 bb
bb
b
.
3,2,26,2,2
3,3,26,3,2
6,3,3 bb
bb
b
With known additional variables a and b, defined by the relations (4), (10)–(12), the
polynomial coefficients of root-polynomial function (8) are calculated with using
following relations:
;
5,4,46,4,4
5,46,4
5 bb
aa
C
;5,4
5,4,46,4,4
5,46,4
5,4,44 a
bb
aa
bC
;5,4
5,4,46,4,4
5,46,4
5,4,44,4,3
5,4,46,4,4
5,46,4
4,4,34,33
a
bb
aa
bb
bb
aa
baC
5,4
5,4,46,4,4
5,46,4
5,4,43,3,2
5,4,46,4,4
5,46,4
3,4,23,22 a
bb
aa
bb
bb
aa
baC
;5,4
5,4,46,4,4
5,46,4
5,4,44,4,3
5,4,46,4,4
5,46,4
4,4,34,33,2,2
a
bb
aa
bb
bb
aa
bab (13)
)( 4
12
3
1
2
1
2
21
3
2
4
2
5,4,46,4,4
5,46,4
3,4,22,11 zzzzzzzz
bb
aa
baC
)( 4
12
3
1
2
1
2
21
3
2
4
25,4
5,4,46,4,4
5,46,4
5,4,4 zzzzzzzza
bb
aa
b
);()( 122
2
112
2
23 zzCzzzzC
.11
2
12
3
13
4
14
5
15
5
10 zCzCzCzCzCrC
Relations (4)–(7) have been used in this work for calculation the coefficients
of forth order root-polynomial function (2), and relations (10)–(13) — for calcula-
tion the corresponded coefficients of fifth order root-polynomial function (8).
Such kind of ravine functions are generally characterized by one minimum, as
well as by quasi-linear dependence outside the region of local minimum. In any
case, such functional dependences are very suitable for approximation the trajec-
tories of electron beam, propagated in the medium of ionized gas with compensa-
tion the space charge of electrons, because, as it was proved theoretically, the be-
havior of electron beams in such physical conditions is exactly the same [1; 20–
23; 26–29]. An effective and simple method of calculation the optimal values of
polynomial coefficients for function (3) and (8), have been used in this work for
solving the task of approximation the suitable numerical data. Describing of this
method, as well as corresponded examples of approximation for some of ravine
functions, will be considered in the next parts of this article.
Basic algorithm for approximation of the boundary trajectory of short-focus electron beam …
Системні дослідження та інформаційні технології, 2023, № 3 133
STATEMENT OF APPROXIMATION PROBLEM
In general, the approximation task is that for given approximation basis points and
a given approximation function r(z), for example, for function (1) with unknown
coefficients Cn, Cn-1, ... C1, C0, write an analytical expression based on the method
of least squares [24; 25; 30; 31].
For example, for fourth order root-polynomial function:
m
i
ii СССССzrrСССССS
1 min
01234
22
01234 )),,,,,((),,,,( (14)
m
i
i СzСzСzСzСr
1 min
01
2
2
3
3
4
4
2 ,
and for fifth-order function, correspondently,
m
i
ii ССССССzrrССССССS
1 min
012345
22
012345 )),,,,,,((),,,,,( (15)
m
i
i СzСzСzСzСzСr
1 min
5 2
01
2
2
3
3
4
4
5
5 ,)(
where n is the degree of the root-polynomial function, m is the number of reference
values.
Applying known methods of solving extremal problems through the search
for partial derivatives of a function of many variables [24; 25], the generalized
relation (14) can be rewritten in the form of a system of algebraic differential
equations as follows [24; 25; 30; 31]:
m
i
i
i
m
i
i
i
m
i
i
i
m
i
i
i
m
i
i
i
C
CCCССzr
СzСzСzСzСr
C
CCCССzr
СzСzСzСzСr
C
CCCССzr
СzСzСzСzСr
C
CCCССzr
СzСzСzСzСr
C
CCCССzr
СzСzСzСzСr
1 4
012344
01
2
2
3
3
4
4
1 3
012344
01
2
2
3
3
4
4
1 2
012344
01
2
2
3
3
4
4
1 1
012344
01
2
2
3
3
4
4
1 0
012344
01
2
2
3
3
4
4
.
),,,,,(
;
),,,,,(
;
),,,,,(
;
),,,,,(
;
),,,,,(
(16)
Correspondently, relation (15) for fifth order root-polynomial function is re-
written as follows:
m
i
i
i
m
i
i
i
m
i
i
i
m
i
i
i
m
i
i
i
C
CCCСССzr
СzСzСzСzСzСr
C
CCCСССzr
СzСzСzСzСzСr
C
CCCСССzr
СzСzСzСzСzСr
C
CCCСССzr
СzСzСzСzСzСr
C
CCCСССzr
СzСzСzСzСzСr
1 4
0123455
01
2
2
3
3
4
4
5
5
1 3
0123455
01
2
2
3
3
4
4
5
5
1 2
0123455
01
2
2
3
3
4
4
5
5
1 1
0123455
01
2
2
3
3
4
4
5
5
1 0
0123455
01
2
2
3
3
4
4
5
5
.
),,,,,,(
;
),,,,,,(
;
),,,,,,(
;
),,,,,,(
;
),,,,,,(
(17)
I. Melnyk, A. Pochynok
ISSN 1681–6048 System Research & Information Technologies, 2023, № 3 134
The problem is that the solution of the set of equations (16), (17) in the case
of a nonlinear function r(z) of many variable parameters, is extremely difficult.
Methods of analytical solution of some simpler approximation problems for
linear, quadratic, polynomial and one-parameter functions f(z), as well as for the
sum of arbitrarily specified functions φ1(z), φ2(z), φn(z) with unknown numerical
coefficients a0, a1, ..., an are described in the textbook [30; 31], and the methods
of numerical solution of systems of nonlinear equations, similar to (4), are
considered in textbooks [24; 25; 32–34]. But generally, in the theory of
approximation is assumed, that with increasing the number of varied variables up
to 5 and more the applying methods of multicriterial analyze aren’t suitable and
lead to obtaining the wrong results. Usually in mathematical software tools the
gradient methods, the Nelder–Mead method, the Broyden–Fletcher–Goldfarb–
Shanno algorithm and others are used for solving multi-criteria optimization tasks
[32–35].
Let’s we will the approximation task for root-polynomial functions (2), (8)
by the other approach. As an initial approximation we will choose the result of
interpolation for four base points using, to calculate the polynomial coefficients of
the root-polynomial function (2), (8) by applying the analytical relations (4–7; 10–13).
Regarding hat the root-polynomial function of the fourth and fifth order
(2), (8) is symmetric about the axis minz z , considering now the linear
approximation for the second and third branches of ravine function and find the
corresponding angles of inclination of the tangents int 2k and int 3k . The solution
of the linear approximation problem is simple and well-known, the corresponding
analytical relations are given in textbooks [30; 31]. For the second branch of
interpolation, they are written as follows:
)(
)(
)()(
)( *
2
2*
2
*
2
*
2
*
22
3
2
3
2
zN
Ni
z
N
Ni
rizi
rB mz
mz
mrmz
mxr
B
B
B
B
;
123
*
2
3
2
BB
N
Ni
i
z NN
z
m
B
B ;
;
123
*
2
3
2
BB
N
Ni
i
r NN
r
m
B
B
3
2
3
2
2*
2
*
2
*
2
2int
)(
)()(
B
B
B
B
N
Ni
z
N
Ni
rizi
mz
mrmz
k , (18)
and for the third branch:
)(
)(
)()(
)( *
3
2*
3
*
3
*
3
*
33
3
3
zN
Ni
z
N
Ni
rizi
rB mz
mz
mrmz
mxr
End
B
End
B
;
13
*
3
3
BEnd
N
Ni
i
z NN
z
m
End
B ;
;
13
*
3
3
BEnd
N
Ni
i
r NN
r
m
End
B
End
B
End
B
N
Ni
z
N
Ni
rizi
mz
mrmz
k
3
3
2*
3
*
3
*
3
3int
)(
)()(
, (19)
Basic algorithm for approximation of the boundary trajectory of short-focus electron beam …
Системні дослідження та інформаційні технології, 2023, № 3 135
where 2BN the starting point of the second approximation branch, 3BN is the
starting point of the third approximation branch, EndN is the end point of the data
set for approximation region.
Taking into account equations (2), (18), (19), let’s we rewrite the set of
equations (16) to find the minimum of the regression function (2) as follows:
.)(
;)(;)(
;
)(
;
)(
55
4433
3int
2
2int
1
rzr
rzrrzr
k
dz
zdr
k
dz
zdr
(20)
Correspondently, to fifth order root polynomial function (8), one can rewrite
the set of equations (17) as follows:
.)(;)(
;)(;)(
;
)(
;
)(
5555
4433
3int
2
2int
1
rzrrzr
rzrrzr
k
dz
zdr
k
dz
zdr
(21)
The separate problem is finding the derivations for root-polynomial func-
tions (2), (8) in the form of suitable polynomials for providing further iterative
calculations. This task was solved in provided researches with applying the tools
of symbolic calculation of the MatLab scientific and technical software [32]. Cor-
responded obtained results for taking a derivative of the function (2) is follows:
6
3
3
1
5
2
3
1
44
1432104 128),,,,,,( zfCfCfzfCfCfzfCfzfkfCfCfCfCfCfR
7
32
2
1
62
2
2
1
343
1
7
4
3
1 722416 zfCfCfCfzfCfCfzfkfCfzfCfCf
9
43
2
1
8
3
2
1
44
2
2
1
8
42
2
1 14454396 zfCfCfCfzfCfCfzfkfCfCfzfCfCfCf
64
4
2
1
102
4
2
1
54
3
2
1 3963 zfkfCfCfzfCfCfzfkfCfCf
8
3
2
21
73
21
242
1 144329124,222 zfCfCfCfzfCfCfzfkfCf (22)
92
321
542
21
9
4
2
21 2163192 zfCfCfCfzfkfCfCfzfCfCfCf
114
221
64
321
10
4321 3846576 zfCfCfCfzfkfCfCfCfzfCfCfCfCf
103
31
34
21
74
421 1088249,4456 zfCfCfzfkfCfCfzfkfCfCfCf
122
431
742
31
11
4
2
31 5763432 zfCfCfCfzfkfCfCfzfCfCfCf
133
41
44
31
84
431 2568249,4456 zfCfCfzfkfCfCfzfkfCfCfCf
zfkfCfzfkfCfCfzfkfCfCf 4
1
454
41
942
41 106563,18249,4453
I. Melnyk, A. Pochynok
ISSN 1681–6048 System Research & Information Technologies, 2023, № 3 136
643
2
10
4
3
2
9
3
3
2
84
2 1289616 zfkfCfzfCfCfzfCfCfzfCf
84
4
2
2
11
43
2
2
102
3
2
2 3576216 zfkfCfCfzfCfCfCfzfCfCf
442
2
44
4
2
2
122
4
2
2 9124,2223384 zfkfCfzfkfCfCfzfCfCf
842
32
12
4
2
32
113
32 3864216 zfkfCfCfzfCfCfCfzfCfCf
54
32
94
432
132
432 8249,44561152 zfkfCfCfzfkfCfCfCfzfCfCfCf
64
42
1042
42
143
42 8249,4453512 zfkfCfCfzfkfCfCfzfCfCf
943
3
13
4
3
3
124
3
24
2
4 43281106563,1 zfkfCfzfCfCfzfCfzfkfCf
642
3
1042
4
2
3
142
4
2
3 9124,2223864 zfkfCfzfkfCfCfzfCfCf
74
43
1142
43
153
43 8249,4453768 zfkfCfCfzfkfCfCfzfCfCf
1243
4
164
4
34
3
4 256106563,1 zfkfCfzfCfzfkfCf
.101024,4106563,19124,222 4544
4
4842
4 kfzfkfCfzfkfCf
For the derivative of fifth order root-polynomial ravine function (8) with us-
ing MatLab symbolic processor such polynomial expression have been obtained:
2
3
4
12
4
1
5
15432105 1510),,,,,,,( zfCfCfzfCfCfCfzfkfCfCfCfCfCfCfR
3
32
3
1
22
2
3
1
4
5
4
1
3
4
4
1 120402520 zfCfCfCfzfCfCfzfCfCfzfCfCf
42
3
3
1
5
52
3
1
4
42
3
1 90200160 zfCfCfzfCfCfCfzfCfCfCf
62
4
3
1
6
53
3
1
5
43
3
1 160300240 zfCfCfzfCfCfCfzfCfCfCf
33
2
2
1
82
5
3
1
7
54
3
1 80250400 zfCfCfzfCfCfzfCfCfCf
6
5
3
2
2
1
5
4
3
2
2
1
4
3
3
2
2
1 600480360 zfCfCfCfzfCfCfCfzfCfCfCf
7
532
2
1
6
432
2
1
52
3
2
2
2
1 18001440540 zfCfCfCfCfzfCfCfCfCfzfCfCfCf
92
52
2
1
8
542
2
1
72
42
2
1 15002400960 zfCfCfCfzfCfCfCfCfzfCfCfCf
8
5
2
3
2
1
7
4
2
3
2
1
63
3
2
1 13501080270 zfCfCfCfzfCfCfCfzfCfCf
102
53
2
1
9
543
2
1
82
43
2
1 225036001440 zfCfCfCfzfCfCfCfCfzfCfCfCf
112
54
2
1
10
5
2
4
2
1
93
4
2
1 30002400640 zfCfCfCfzfCfCfCfzfCfCf
5
3
3
21
44
21
123
5
2
1 480801250 zfCfCfCfzfCfCfzfCfCf
63
2
2
21
7
5
3
21
6
4
3
21 1080800640 zfCfCfCfzfCfCfCfzfCfCfCf (23)
82
4
2
21
8
53
2
21
7
43
2
21 192036002280 zfCfCfCfzfCfCfCfCfzfCfCfCfCf
73
321
102
5
2
21
9
54
2
21 108030004800 zfCfCfCfzfCfCfCfzfCfCfCfCf
Basic algorithm for approximation of the boundary trajectory of short-focus electron beam …
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92
4321
9
5
2
321
8
4
2
321 576054004320 zfCfCfCfCfzfCfCfCfCfzfCfCfCfCf
112
5321
10
54321 900014400 zfCfCfCfCfzfCfCfCfCfCf
122
5421
11
5
2
421
103
421 1200096002560 zfCfCfCfCfzfCfCfCfCfzfCfCfCf
9
4
3
31
84
321
133
5421 21604055000 zfCfCfCfzfCfCfCfzfCfCfCfCf
11
54
2
31
102
4
2
31
10
5
3
31 1080043202700 zfCfCfCfCfzfCfCfCfzfCfCfCf
12
5
2
431
113
431
122
5
2
31 1440038406750 zfCfCfCfCfzfCfCfCfzfCfCfCf
124
431
143
531
132
5431 1280750018000 zfCfCfCfzfCfCfCfzfCfCfCfCf
153
541
5142
5
2
41
413
5
3
41 10102,16400 zfCfCfCfzfCfCfCfzfCfCfCf
6
3
4
2
55
2
5
1
164
51 240323125 zfCfCfzfCfzfkfCfzfCfCf
8
43
3
2
72
3
3
2
8
5
4
2
7
4
4
2 1920720400320 zfCfCfCfzfCfCfzfCfCfzfCfCf
10
54
3
2
92
4
3
2
9
53
3
2 320012802400 zfCfCfCfzfCfCfzfCfCfCf
9
4
2
3
2
2
83
3
2
2
112
5
3
2 432010802000 zfCfCfCfzfCfCfzfCfCf
11
543
2
2
10
43
2
2
10
5
2
3
2
2 1440057605400 zfCfCfCfCfzfCfCfCfzfCfCfCf
12
5
2
4
2
2
113
4
2
2
122
53
2
2 960025609000 zfCfCfCfzfCfCfzfCfCfCf
94
32
143
5
2
2
3132
54
2
2
4 810105102,1 zfCfCfzfCfCfzfCfCfCf
112
4
2
32
11
5
3
32
10
2
3
32 864054004320 zfCfCfCfzfCfCfCfzfCfCfCf
123
432
132
5
2
32
12
54
2
32 76801350021660 zfCfCfCfzfCfCfCfzfCfCfCfCf
142
5432
13
5
2
432 3600028800 zfCfCfCfCfzfCfCfCfCf
14
5
3
42
4134
42
153
532
4 108,122560105,1 zfCfCfCfzfCfCfzfCfCfCf
174
52
163
542
4152
5
2
42
4 6250102104,2 zfCfCfzfCfCfCfzfCfCfCf
12
5
4
3
11
4
4
3
105
3
25
2 20251620243 zfCfCfzfCfCfzfCfzfkfCf
142
5
3
3
13
54
3
3
122
4
3
3 6750108004320 zfCfCfzfCfCfCfzfCfCf
15
54
2
3
414
5
2
4
2
3
4133
4
2
3 107,21016,25760 zfCfCfCfzfCfCfCfzfCfCf
15
5
3
43
4144
43
163
5
2
3 1092,1384011250 zfCfCfCfzfCfCfzfCfCf
184
53
173
543
4162
5
2
43
4 9375103106,3 zfCfCfzfCfCfCfzfCfCfCf
172
5
3
4
416
5
4
4
155
4
35
3 106,164001024 zfCfCfzfCfCfzfCfzfkfCf
45
4
194
54
4183
5
2
4
4 105,12102 zfkfCfzfCfCfzfCfCf
.3125 5
0
55
5
205
5 kfCfzfkfCfzfCf
I. Melnyk, A. Pochynok
ISSN 1681–6048 System Research & Information Technologies, 2023, № 3 138
Obtained polynomial relations (22), (23) have been applied in provided in-
vestigation for iterative calculation the values of polynomial coefficient with us-
ing the method of consistent upper relaxation [33–36]. These relations included
the values of the coefficients root-polynomial function (2), (8), as well as zf co-
ordinate and slope angles of ravine function kf . The particularities of proposed
algorithm, as well as some examples of solving approximation task, will be con-
sidered in the next sections of the article.
PARTICULARITIES OF DEVELOPED ITERATIVE ALGORITHM FOR
SOLVING THE APPROXIMATION TASK
Iterative algorithm for approximation the ravine sets of numerical data by using
root-polynomial dependences four and fifth order (2), (8) generally including the
follow necessary steps.
1. As basic approach the interpolation task is solved and the basic values of
polynomial coefficients are calculated withusing relations (4)–(7) for fourth order
function (2), and by the relations (10)–(13) for fifth order function (8).
2. Solving the relaxation task for the finding values of polynomial coeffi-
cient with using iterative algorithm. Corresponded iterative relations for func-
tion (2), taking into account (22), are the follows:
;),,,,,,(
131
1
4
1
3
1
2
1
1
1
041 C
iiiiii wzfkfCCCCCRC
;),,,,,,(
342
1
4
1
3
1
21
1
043 C
iiiiii wzfkfCCCCCRC
;))(( 4
2
1
021
2
2
1
2
3
23
4
24 4
zfwCzfCzfCzfCrC C
iiiii (24)
;))(( 2
1
1
011
2
1
1
2
4
14
4
12 2
zfwCzfCzfCzfCrC C
iiiii
.)(
0
1
011
2
12
4
14
4
10 C
iiiii wCzfCzfCzfCrC
For fifth order root-polynomial function (8), taking into account (23), the it-
erative relations, in the general form, are rewritten as follows:
;),,,,,,,(
131
1
5
1
4
1
3
1
2
1
1
1
051 C
iiiiiii wzfkfCCCCCCRC
;),,,,,,,(
342
1
5
1
4
1
3
1
21
1
053 C
iiiiiii wzfkfCCCCCCRC
;))(( 5
6
1
061
2
6
1
2
3
63
4
6
1
4
5
65 5
zfwCzfCzfCzfCzfCrC C
iiiiii (25)
;))(( 4
2
1
061
2
2
1
2
3
23
5
25
5
24 4
zfwCzfCzfCzfCzfCrC C
iiiiii
;))(( 2
1
1
011
3
13
4
14
5
15
5
12 2
zfwCzfCzfCzfCzfCrC C
iiiiii
.)(
051
3
52
3
53
4
54
5
55
5
50 C
iiiiii wzfCzfCzfCzfCzfCrC
3. Finding the best function of approximation by calculation the error with
using relations (14), (15) for least-square method.
4. Finding the new optimal values of polynomial coefficients wit using di-
chotomy calculations, namely:
Basic algorithm for approximation of the boundary trajectory of short-focus electron beam …
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,
2
21
i
l
i
li
l
CC
C (26)
where l — the number of polynomial coefficient, defined in the basic relations
(4)–(6), (10)–(13).
As shown the tests calculation experiments, the convergence of proposed
algorithm is provided by 4–6 iterations. It also must be taking into account, that
first iteration for high order root-polynomial function, which have been obtained
by interpolation through n + 1 points, where n is the polynomial order, is really
usually close to optimal. The best solution has been chosen by the smallest value
of sum in relation (14), (15), therefore the well-known least-square method is
considered in this research as the criterium of optimization task [30–39].
Choosing of basic points is also very important, therefore considering the
different sets of its for finding the best approximative root-polynomial function is
also have been provided. Analyzing the interpolation error of the different order
root-polynomial functions in dependence on choosing the set of basic points have
been provided generally in the papers [22; 23].
And finally, choose the correct values of
0Cw –
5Cw relaxation coefficient is
very important problem for providing the stability of convergence of proposed
iteration algorithm, because the value problem of coefficients 50 CC in the
relations (24)–(26) are usually, for the practice engineering tasks, can be both
extra low or extra high.
Some examples of solving approximation for the ravine data sets with using
root-polynomial functions (2), (8) will be considered in the next part of the article.
SOME TESTING EXAMPLES OF SOLVING THE APPROXIMATION TASK
Example 1. Find the coefficients of fourth order approximative root polynomial
function for ravine function data set with one global minimum. Corresponded dig-
ital data are presented at Table 1.
T a b l e 1 . The first set of numerical data of the ravine function, have been used
for testing the software tools for solving the approximation problem
z, mm 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
r, mm 2.5 2.4 2.3 2.1 1.8 1.75 2.11 2.3 2.4
The obtained graphic results of solving the approximation task of this example
with using relations (24), (26) are presented at Fig 1.
Corresponded values of relaxation coefficient for providing the convergence
of iterative relations (24) have been defined as follows:
;101.7995 6
1
Cw ;101.9163 5
3
Cw ;1
4
Cw ;1.131
2
Cw 0.945.
0
Cw
The values of calculated polynomial coefficients are presented at Table 2.
The error of approximation δ, defined by the equation (14) as sum of squares of
differences between basic points and value4 of approximative function (2), also
I. Melnyk, A. Pochynok
ISSN 1681–6048 System Research & Information Technologies, 2023, № 3 140
noted in this table. The relative error of approximation is calculated as
,100/())(]%δ[ 2
max Sr де rmax — the maximum value of electron beam radius.
T a b l e 2 . The results of solving the approximation task for numerical data,
presented in the Table 1
Values of polynomial coefficients Nubmer
of iteration C0 C1 C2 C3 C4
δ, %
Basic Approach –493.16 2750.4 4976.8 3690.6 –961.7 8
First iteration –493.179 2750.41 4976.81 3686.9 –956.2 5
Second iteration –493.172 2750.408 4976.81 3688.7 –958.961 4
Third iteration –493.17 2750.408 4976.7 3689.68 –960.33 3
In this example first iteration by the polynomial coefficients have been real-
ized with applying relations (24), (25), and the second and third iterations, in
which have been obtained the best solution of approximation task, have been pro-
vided with using relation (26).
Example 2. Find the coefficients of fourth order approximative root-
polynomial function for ravine function data set with one global minimum. Corre-
sponded digital data are presented in Table 3.
T a b l e 3 . The second set of numerical data of the ravine function, have been
used for testing the software tools for solving the approximation problem
z, mm 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
r, mm 2.5 2.3 2.1 2.0 1.9 1.8 1.9 2.1 2.3 2.6 2.9
The obtained graphic results of solving the approximation task of this exam-
ple with using one iteration step are presented at Fig 2. Really the task of interpo-
lation by choose 5 basis points among the 11 given have been solved in this case,
and the interpolation error was smaller, than 2%. The calculated values of poly-
nomial coefficients for this test example are presented in Table 4.
Fig. 1. The results of solving the approximation task with using fourth order root-
polynomial function (2) for data set, presented at Table 1. Corresponded values of
polynomial coefficients and the approximation error are given at Table 2
z, mm
r, mm
Basic algorithm for approximation of the boundary trajectory of short-focus electron beam …
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T a b l e 4 . The results of solving the approximation task for numerical data, presented
in the Table 3
Values of polynomial coefficients Nubmer
of iteration C0 C1 C2 C3 C4
δ, %
Basic Approach 328.6 –1251.26 1953.66 –1417.8 398.9 1.5
Example 3. Find the coefficients of fifth order approximative root
polynomial function for ravine function data set with one global minimum.
Corresponded digital data are presented at Table 5.
T a b l e 5 . The third set of numerical data of the ravine function, have been
used for testing the software tools for solving the approximation problem
z, mm 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
r, mm 2.5 2.4 2.31 2.1 1.9 1.7 1.65 1.63
z, mm 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
r, mm 1.6 1.55 1.7 1.8 1.9 2.11 2.32 2.4
In this and in the next example the corresponded values of relaxation coeffi-
cient for providing the consvergence of iterative relations (25) have been defined
as follows:
;105,53 7
1
Cw ;109,5 7
3
Cw ;1
5
Cw ;1
4
Cw ;,390
2
Cw 0,145.
0
Cw
The obtained graphic results of solving the approximation task of this
example with using one iteration step are presented at Fig 3. It is clear from this
example, that considered type of the root-polynomial functions is not very
suitable for approximation the ravine data sets with large area of minimum
numerical values. Corresponded approximation error was greater than 12 %. But
for the left and right branches of considered data set the error of approximation is
much smaller, nearly few percents.
Fig. 2. The results of solving the approximation task with using fourth order root-
polynomial function (2) for data set, presented at Table 3. Corresponded values of poly-
nomial coefficients and the approximation error are given at Table 4
z, mm
r, mm
I. Melnyk, A. Pochynok
ISSN 1681–6048 System Research & Information Technologies, 2023, № 3 142
Example 4. Find the coefficients of fifth order approximative root
polynomial function for ravine function data set with one global minimum.
Corresponded digital data are presented at Table 6.
T a b l e 6 . The fourth set of numerical data of the ravine function, have been
used for testing the software tools for solving the approximation problem
z, mm 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
r, mm 2.5 2.4 2.31 2.1 1.9 1.7 1.65 1.5
z, mm 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
r, mm 1.4 1.5 1.6 1.8 1.9 2.11 2.32 2.4
The obtained graphic results of solving the approximation task of this exam-
ple with using 3 iteration step are presented at Fig 4.
Fig. 4. The results of solving the approximation task with using fifth order root-
polynomial function (8) for data set, presented at Table 6. Corresponded values of the
polynomial coefficients and approximation error are given in Table 7
r, mm
z, mm
Fig. 3. The results of solving the approximation task with using fifth order root-
polynomial function (8) for data set, presented at Table 5. Corresponded approximation
error for the left and right branches of considered ravine function was smaller, than few
percent
r, mm
z, mm
Basic algorithm for approximation of the boundary trajectory of short-focus electron beam …
Системні дослідження та інформаційні технології, 2023, № 3 143
T a b l e 7 . The results of solving the approximation task for numerical data,
presented in the Table 6
Values of polynomial coefficients Nubmer
of iteration C0 C1 C2 C3 C4 C5
δ, %
Basic Approach –231.51 2340.2 –5259.9 5083.15 –2278.7 392.01 5
First iteration –39.32 2340.1 –5260.0 5082.7 –2276.1 392.1 4
Second iteration –151.0 2340.0 –5259.9 5083.0 –2278.2 392.1 3
In this example, the iterative process was carried out in the same way as in
example 1. The first values of polynomial coefficients were obtained by solving
interpolation task. Therefore, for defining the coefficients of the root-polynomial
function (8) relations (10)–(13) was applied. In this step of solving approximation
task 5 basis points have been choose among the 16 given. By the such way the
basic the approach for calculation the polynomial coefficients have been
provided. At the second iteration the values of polynomial coefficient have been
calculated iteratively with using relations (25), and in the third iteration — with
using relation (26). Corresponded results of for the coefficients of fifth order root-
polynomial function, as well as the estimated value of approximation error, have
been presented in the Table 7.
ANALYZING OF OBTAINED RESULTS AND ITS’ DISCUSSION
The provided numerical experiments for solving the approximation task for ravine
data set with using high order root-polynomial functions (2), (8) shown, that
generally by providing simple iterative process with relaxation by using relations
(24), (25) is really possible to find the optimal correct values of polynomial coef-
ficient. Estimated theoretically approximation error in most cases was in range of
few percent, and, taking into account, that usually approximated experimental
data for the trajectories of electron beams are included the large amount instru-
mental errors, such estimations, in the most cases, are generally useful from the
practical point of view.
The main distinguishing feature of proposed algorithm is possibilities of
obtaining the correct results for approximation task just on the first step of
calculations by solving the simpler interpolation task. Such approach is very
effective on the practical point of view for developing corresponded software for
simulation the boundary trajectories of electron beams.
The provided testing experiment also shown, that root-polynomial functions
of high order, like (8), (12), are generally not suitable for approximation the
ravine data sets with the large area near the minimum, but such kind of functions
isn’t corresponded to the trajectories of electron beam in standard physical
conditions. Usually, the region of beam focus is relatively small, and, as shown
the provided numerical experiments, such data sets can be approximated by the
high order root-polynomial functions with small error.
I. Melnyk, A. Pochynok
ISSN 1681–6048 System Research & Information Technologies, 2023, № 3 144
The provided theoretical analyze also shown, that for obtaining the
convergence of proposed iterative approximation algorithm correct choosing of
relaxation coefficients
0Cw –
5Cw is very important numerical factor.
Analyzing of another advanced possibilities of applying high order root-
polynomial functions for accurate approximation of ravine data sets and forming
the corresponded algorithms is the subject of further theoretical researches.
But, generally, the complicity of proposed iterative algorithm is connected
only with solving the strong non-linear task for derivatives of ravine functions (2),
(8) and to numerical solving the complex polynomial relations (22), (23). The
provided numerical experiments shown, that the polynomial coefficient, which
are calculated throw the derivatives by numerical solving the relations (22), (23),
are strongly depended on relaxation parameters and the values of these parameters
are magnificently grater, than for the polynomial coefficients, which are
calculated independently by using simple relations (2), (8). But it is also clear
from the provided testing numerical experiments, that the approximation error is
mostly defined by the polynomial coefficients, which values are calculated
through the derivations. And the values of coefficients, calculated directly by the
relations (2), (8), are generally stable and corresponded relaxation parameter is in
the range of medium value, it is not very large and not so small. But since usually
ravine functions in the region of a local minimum are non-linear, it is often
complex non-linear equations for derivatives, like (22), (23), are solved. Its makes
possible to ensure, that the smallest error and the optimal values of the
polynomial coefficients for solving the approximation problem have been choose.
Choosing the optimal values of relaxation parameters for calculation the
polynomial coefficients throw the derivatives is also the subject of further
theoretical researches.
CONCLUSION
The theoretical researches, have been provided and described in this work, as well
as realized test numerical experiments, given in the corresponded examples, have
shown, that applying of high order root-polynomial functions, which have been
determinate by the analytical relations (2), (8), always leads to solving the task of
approximation the numerical sets of ravine functions data with the small value of
error. Therefore, such functions can be used successfully for approximation the
boundary trajectories of electron beams, propagated in the ionized gas with com-
pensation the space charge of beam electrons. Such approximation can also be
applied to the noisy experimental data, which included experimental errors.
Therefore, studied and proposed in these researches the theoretical approach and
the numerical algorithm of the approximation of digital data are very important
from the practical point of view for further development of modern industrial
electron beam equipment with applying HVGD electron gun as the source of in-
tensive beams. For solving this task proposed iterative algorithm has to be inte-
grated with the software tools for treatment of experimental photographs, where
the beam propagates through ionized gas. Analyzing of the brightness of a burn-
Basic algorithm for approximation of the boundary trajectory of short-focus electron beam …
Системні дослідження та інформаційні технології, 2023, № 3 145
ing discharge on such photographs gives the important experimental information
about the boundary trajectories of propagated electron beam in the real physical
conditions. [1].
Advanced singularities of proposed iterative algorithm, including the
possibilities of approximation the ravine data sets with a large area of the global
minimum, as well as studying of influence of the derivative of root-polynomial
functions on the values of relaxation coefficients, are the subject of further
theoretical researches. But, in any case, the results of provided theoretical
researches, presented in this article, are enough for estimation the boundary
trajectories of electron beams, propagated in ionizing gas. Therefore, theoretical
results, which have already been obtained, are very interesting and important for
experts in the branch of elaboration of modern electron-beam equipment and its
industrial application.
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INFORMATION ON THE ARTICLE
Igor V. Melnyk, ORCID: 0000-0003-0220-0615, National Technical University
of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail:
imelnik@phbme.kpi.ua
I. Melnyk, A. Pochynok
ISSN 1681–6048 System Research & Information Technologies, 2023, № 3 148
Alina V. Pochynok, ORCID: 0000-0001-9531-7593, Research Institute of Electronics
and Microsystem Technology of the National Technical University of Ukraine “Igor Si-
korsky Kyiv Polytechnic Institute”, Ukraine, e-mail: alina_pochynok@yahoo.com
БАЗОВИЙ АЛГОРИТМ АПРОКСИМАЦІЇ ГРАНИЧНОЇ ТРАЄКТОРІЇ
КОРОТКОФОКУСНОГО ЕЛЕКТРОННОГО ПУЧКА ЗА ДОПОМОГОЮ
КОРЕНЕВО-ПОЛІНОМІАЛЬНИХ ФУНКЦІЙ ЧЕТВЕРТОГО ТА П’ЯТОГО
ПОРЯДКІВ / І.В. Мельник, А.В. Починок
Анотація. Розглянуто новий ітераційний метод апроксимації граничної траєк-
торії короткофокусного електронного пучка, який поширюється в режимі
вільного дрейфу в іонізованому газі низького тиску за умови компенсації про-
сторового заряду електронів. Використано коренево-поліноміальні функції че-
твертого та п’ятого порядків, головними особливостями яких є яружний хара-
ктер та наявність одного глобального мінімуму. Як початкове наближення для
розв’язування апроксимаційної задачі розраховано значення поліноміальних
коефіцієнтів через розв’язання задачі інтерполяції. Задачу апроксимації
розв’язано ітераційно. Для цього поліноміальні коефіцієнти обчислено багато-
разово з урахуванням значень функції та її похідної у відлікових точках. Оста-
точні значення поліноміальних коефіцієнтів коренево-поліноміальних функцій
високого порядку розраховано з використанням методу дихотомії. Наведено
приклади використання коренево-поліноміальних функцій четвертого та
п’ятого порядків для апроксимації наборів числових даних, які відповідають
опису яружних функцій. Отримані теоретичні результати є цікавими та корис-
ними для спеціалістів, які вивчають фізику електронних пучків та займаються
проектуванням сучасного промислового електронно-променевого технологіч-
ного обладнання.
Ключові слова: апроксимація, інтерполяція, коренево-поліноміальна функція,
яружна функція, метод найменших квадратів, нев’язка, похибка апроксимації,
електронний пучок, електронно-променеві технології.
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| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:23Z |
| publishDate | 2023 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
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| resource_txt_mv | journaliasakpiua/55/7ec011aeced1210664970e1db8ca8555.pdf |
| spelling | journaliasakpiua-article-2904742023-11-07T22:19:24Z Basic algorithm for approximation of the boundary trajectory of short-focus electron beam using the root-polynomial functions of the fourth and fifth order Базовий алгоритм апроксимації граничної траєкторії короткофокусного електронного пучка за допомогою коренево-поліноміальних функцій четвертого та п’ятого порядків Melnyk, Igor Pochynok, Alina апроксимація інтерполяція коренево-поліноміальна функція яружна функція метод найменших квадратів нев’язка похибка апроксимації електронний пучок електронно-променеві технології approximation interpolation root-polynomial function ravine function least-square method discrepancy approximation error electron beam electron-beam technologies The new iterative method of approximating the boundary trajectory of a short-focus electron beam propagating in a free drift mode in a low-pressure ionized gas under the condition of compensation of the space charge of electrons is considered and discussed in the article. To solve the given approximation task, the root-polynomial functions of the fourth and fifth order were applied, the main features of which are the ravine character and the presence of one global minimum. As an initial approach to solving the approximation problem, the values of the polynomial coefficients are calculated by solving the interpolation problem. After this, the approximation task is solved iteratively. All necessary polynomial coefficients are calculated multiple times, taking into account the values of the function and its derivative at the reference points. The final values of polynomial coefficients of high-order root-polynomial functions are calculated using the dichotomy method. The article also provides examples of the applying fourth-order and fifth-order root-polynomial functions to approximate sets of numerical data that correspond to the description of ravine functions. The obtained theoretical results are interesting and important for the experts who study the physics of electron beams and design modern industrial electron beam technological equipment. Розглянуто новий ітераційний метод апроксимації граничної траєкторії короткофокусного електронного пучка, який поширюється в режимі вільного дрейфу в іонізованому газі низького тиску за умови компенсації просторового заряду електронів. Використано коренево-поліноміальні функції четвертого та п’ятого порядків, головними особливостями яких є яружний характер та наявність одного глобального мінімуму. Як початкове наближення для розв’язування апроксимаційної задачі розраховано значення поліноміальних коефіцієнтів через розв’язання задачі інтерполяції. Задачу апроксимації розв’язано ітераційно. Для цього поліноміальні коефіцієнти обчислено багаторазово з урахуванням значень функції та її похідної у відлікових точках. Остаточні значення поліноміальних коефіцієнтів коренево-поліноміальних функцій високого порядку розраховано з використанням методу дихотомії. Наведено приклади використання коренево-поліноміальних функцій четвертого та п’ятого порядків для апроксимації наборів числових даних, які відповідають опису яружних функцій. Отримані теоретичні результати є цікавими та корисними для спеціалістів, які вивчають фізику електронних пучків та займаються проектуванням сучасного промислового електронно-променевого технологічного обладнання. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2023-09-29 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/290474 10.20535/SRIT.2308-8893.2023.3.10 System research and information technologies; No. 3 (2023); 127-148 Системные исследования и информационные технологии; № 3 (2023); 127-148 Системні дослідження та інформаційні технології; № 3 (2023); 127-148 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/290474/284062 |
| spellingShingle | апроксимація інтерполяція коренево-поліноміальна функція яружна функція метод найменших квадратів нев’язка похибка апроксимації електронний пучок електронно-променеві технології Melnyk, Igor Pochynok, Alina Базовий алгоритм апроксимації граничної траєкторії короткофокусного електронного пучка за допомогою коренево-поліноміальних функцій четвертого та п’ятого порядків |
| title | Базовий алгоритм апроксимації граничної траєкторії короткофокусного електронного пучка за допомогою коренево-поліноміальних функцій четвертого та п’ятого порядків |
| title_alt | Basic algorithm for approximation of the boundary trajectory of short-focus electron beam using the root-polynomial functions of the fourth and fifth order |
| title_full | Базовий алгоритм апроксимації граничної траєкторії короткофокусного електронного пучка за допомогою коренево-поліноміальних функцій четвертого та п’ятого порядків |
| title_fullStr | Базовий алгоритм апроксимації граничної траєкторії короткофокусного електронного пучка за допомогою коренево-поліноміальних функцій четвертого та п’ятого порядків |
| title_full_unstemmed | Базовий алгоритм апроксимації граничної траєкторії короткофокусного електронного пучка за допомогою коренево-поліноміальних функцій четвертого та п’ятого порядків |
| title_short | Базовий алгоритм апроксимації граничної траєкторії короткофокусного електронного пучка за допомогою коренево-поліноміальних функцій четвертого та п’ятого порядків |
| title_sort | базовий алгоритм апроксимації граничної траєкторії короткофокусного електронного пучка за допомогою коренево-поліноміальних функцій четвертого та п’ятого порядків |
| topic | апроксимація інтерполяція коренево-поліноміальна функція яружна функція метод найменших квадратів нев’язка похибка апроксимації електронний пучок електронно-променеві технології |
| topic_facet | апроксимація інтерполяція коренево-поліноміальна функція яружна функція метод найменших квадратів нев’язка похибка апроксимації електронний пучок електронно-променеві технології approximation interpolation root-polynomial function ravine function least-square method discrepancy approximation error electron beam electron-beam technologies |
| url | https://journal.iasa.kpi.ua/article/view/290474 |
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