Порівняння застосування методів інтерполяції та екстраполяції граничних траєкторій короткофокусних електронних пучків із використанням коренево-поліноміальних функцій
The article considers and discusses the comparison of interpolation and extrapolation methods of estimation of the boundary trajectory of electron beams propagated in ionized gas. All estimations have been computed using root-polynomial functions to numerically solve a differential-algebraic system...
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2024
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System research and information technologies| _version_ | 1867334447288811520 |
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| author | Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo |
| author_facet | Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo |
| author_institution_txt_mv | [
{
"author": "Igor Melnyk",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Alina Pochynok",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Mykhailo Skrypka",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
}
] |
| author_sort | Melnyk, Igor |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2024-11-16T18:06:34Z |
| description | The article considers and discusses the comparison of interpolation and extrapolation methods of estimation of the boundary trajectory of electron beams propagated in ionized gas. All estimations have been computed using root-polynomial functions to numerically solve a differential-algebraic system of equations that describe the boundary trajectory of the electron beam. By providing analysis, it is shown and proven that in the case of solving a self-connected interpolation-extrapolation task, the average error of the beam radius estimation is generally smaller. This approach was especially effective in estimating the focal beam radius. An algorithm for solving self-connected interpolation-extrapolation tasks is given, and its efficiency is explained. Corresponding graphic dependencies are also given and analyzed. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.3.05 |
| first_indexed | 2025-07-17T10:28:36Z |
| format | Article |
| fulltext |
Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2024
74 ISSN 1681–6048 System Research & Information Technologies, 2024, № 3
TIДC
МАТЕМАТИЧНІ МЕТОДИ, МОДЕЛІ,
ПРОБЛЕМИ І ТЕХНОЛОГІЇ ДОСЛІДЖЕННЯ
СКЛАДНИХ СИСТЕМ
UDC 004.942:537.525
DOI: 10.20535/SRIT.2308-8893.2024.3.05
COMPARISON OF METHODS FOR INTERPOLATION
AND EXTRAPOLATION OF BOUNDARY TRAJECTORIES OF
SHORT-FOCUS ELECTRON BEAMS USING ROOT-
POLYNOMIAL FUNCTIONS
I. MELNYK, A. POCHYNOK, M. SKRYPKA
Abstract. The article considers and discusses the comparison of interpolation and
extrapolation methods of estimation of the boundary trajectory of electron beams
propagated in ionized gas. All estimations have been computed using root-
polynomial functions to numerically solve a differential-algebraic system of equa-
tions that describe the boundary trajectory of the electron beam. By providing analy-
sis, it is shown and proven that in the case of solving a self-connected interpolation-
extrapolation task, the average error of the beam radius estimation is generally
smaller. This approach was especially effective in estimating the focal beam radius.
An algorithm for solving self-connected interpolation-extrapolation tasks is given,
and its efficiency is explained. Corresponding graphic dependencies are also given
and analyzed.
Keywords: interpolation, extrapolation, root-polynomial function, ravine function,
average error, electron beam, boundary trajectory.
INTRODUCTION
The development of electron beam technologies is very important today for ad-
vanced branches of industry, including metallurgy, mechanical engineering, in-
strument making, and microelectronic production, as well as automotive, aircraft,
and space industries [1–10]. It is caused by the advantages of the electron beam as
a technological instrument, such as its high total power and power density, the
simplicity of changing and controlling focal beam parameters using electric and
magnetic fields, as well as providing technological operation in the medium of
pure vacuum [1–10].
For example, the use of electron beams today in the electronic industry is
mostly oriented toward welding contacts of electronic devices, including cryo-
genic ones [8–12], deposition of ceramic dielectric films for high-quality capaci-
tors and microwave transmitters and receiver devices [9; 10; 13–15], as well as
toward refining of silicon [16–19]. A special technical problem is obtaining and
applying intensive electron beams in high-energy accelerators [7; 20–22].
Comparison of methods for interpolation and extrapolation of boundary trajectories of …
Системні дослідження та інформаційні технології, 2024, № 3 75
In such circumstances, the development of electron beam technologies and
the sources of electrons are provided in two main directions. The first is improv-
ing the traditional electron guns with heated cathodes, and the second is the elabo-
ration of novel types of electron sources based on other physical principles. For
example, electron sources based on gas discharges are elaborated today and suc-
cessfully applied in industry.
Among this type of electron source, a special place occupied the high-
voltage glow discharge (HVGD) electron guns, which are generally characterized
by the stable operation in the soft vacuum in the medium of different gases, in-
cluding active and noble ones, as well as by simplicity of construction [23–28].
Another well-known advantage of HVGD electron guns is their simplicity in con-
trolling discharge current and beam power. In the paper [29], the slow aerody-
namic control of HVGD current using electromagnetic valves has been consid-
ered, and the corresponding time regimes have been analyzed. Fast electric
control of beam power by lighting the additional low-voltage discharge and
changing the concentration of charged particles in anode plasma, as well as the
corresponding time regimes for such fast regulation, have been considered in the
paper [30].
Generally, the main advantage possibilities for using HVGD electron guns in
industry are as follows:
1. Refining of silicon for the microelectronic industry [16–20; 31].
2. Obtaining high-quality ceramic films for microelectronic production and
for power-energetic insulators. Obtaining defense films for cutting instruments
and heat-protection films for engines in the automotive, aircraft, and space indus-
tries by applying HVGD electron guns is also possible [33–37].
3. Development of 3-D printing technology by metals [38–40].
Therefore, in such circumstances, the development of novel mathematical
approaches for the simulation of short-focus electron beams generated by HVGD
electron guns is generally necessary.
PERVIOUS RESEARCHES AND STATEMENT OF THE SIMULATION
PROBLEM
The problem is that today elaboration of HVGD electron guns is mostly provided
by a combination of sophisticated theoretical approaches [23; 26–30] and com-
plex experimental works [24; 25], because simplified mathematical relations for
estimation of the focal parameters of short-focus electron beams at low pressure
in ionized gas don’t exist [23]. Also, in part of the book [23], general approaches
for the simulation of electric field distribution and the trajectories of charged par-
ticles have been systematized. Corresponding recommendations for the simulation
of HVDG electron guns have also been given in [23]. But in any case, such a so-
phisticated approach to designing this type of guns is significantly hinders the
necessary development and implementation in industry of its novel, advanced
constructions. Therefore, the finding of simplified mathematical relations for in-
terpolation, extrapolation, and approximation of the trajectories of short-focus
electron beams in low-pressure ionized gas is necessary [23]. Corresponding rela-
tions for an intensive long-focus electron beam propagated in high vacuum in ion-
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 76
ized gases are well-known [1–10]. In the paper [41], simple analytical relations
for estimating the depth of welding seam penetration using short-focus electron
beams formed by HVGD electron guns have been proposed. Corresponding simu-
lation results have been analyzed and compared with experimental data [41].
Therefore, it is clear that finding and analyzing the simple analytical relations for
estimation of the focal parameters of short-focus electron beams propagated in
ionized gas is really necessary.
In the years 2019–2020 in the papers [42–44] a simple approach for estimat-
ing the parameters of short-focus electron beams propagated in ionized gas has
been proposed. Generally, this approach is based on the suggestion that, in such
physical conditions, the beam boundary trajectory can be described with high pre-
cision by the ravine function, which has one global minimum, and outside the re-
gion of minimum, this dependence is similar to a linear one. Such an approach
generally corresponds to the main singularities of the physics of electron beams
[1–10]. In the papers [42–44], by the numerical experiments have shown that such
ravine functions with the precision of fractions of a percent can be described by
root-polynomial functions, written as follows [42–45]:
,)( 01
1
1
n n
n
n
nb CzCzCzCzr
(1)
where z is the longitudinal coordinate of beam propagation, zrb is the beam ra-
dius in corresponded cross-section by z coordinate, n is the degree of the poly-
nomial and the order of the root-polynomial function, and nCC 0 are the poly-
nomial coefficients.
The generally obtained results of interpolation have been systematized and
analyzed in the paper [45]. In this work, analytical relations for calculation of the
coefficients of root polynomial functions (1) from second to fifth order n have
been obtained, and the interpolation task was solved relatively to numerical
solution of the corresponded set of algebra-differential equations, which de-
scribed the boundary trajectory of an electron beam propagated in ionized gas
[1–10; 23; 45]:
;θθ;;
2
πε4
)β1(
;
2
2
5.1
0
2
0
s
b
b
b
ac
e
b
ei
e
dz
dr
r
C
dz
rd
U
m
e
fI
C
nn
n
f
;
ε
exp
επ
;
2
;
π 2
0
02
02
be
ac
ace
e
eibi
e
ac
e
b
b
e
rn
U
Um
nM
pnBrn
m
eU
v
r
I
n
;
γβ22
θ
tan ;
γβ2
10
2
θ
tan ;β1γ
2
23
max
2
344
min2 aa ZZ
(2)
, β ,
θ
θ
ln
)(π8
θ
max
min
2
2
c
v
n
dzZr e
e
ab
where acU is the accelerating voltage, bI is the electron beam current, p is the
residual gas pressure, z is the longitudinal coordinate, br is the radius of the
Comparison of methods for interpolation and extrapolation of boundary trajectories of …
Системні дослідження та інформаційні технології, 2024, № 3 77
boundary trajectory of the electron beam, en is the beam electrons’ concentration,
0in is the concentration of residual gas ions on the beam symmetry axis, em is
the electron mass, f is the residual gas ionization level, iB is the gas ionization
level, 0ε is the dielectric constant, ev is the average velocity of the beam elec-
trons, c is the light velocity, γ is the relativistic factor, minθ and maxθ are the
minimum and maximum scattering angles of the beam electrons, corresponding to
Rutherford model [1–10], aZ is the nuclear charge of the residual gas atoms, dz
is the length of the electron path in the longitudinal direction at the current itera-
tion, and θ is the average scattering angle of the beam electrons.
In the work [45], it has also been proven that the ravine function (1) can be
successfully used for interpolation of the results of solving the set of equations (2)
with very small errors in the range of 0.1–8%. Corresponding dependences of
interpolation error on z coordinates are given in Fig. 1 [45]. But the main
conclusion that has been made is that interpolation error strongly depends on the
position of the base points of interpolation, whose number is always equal to
1n . Generally, the ravine function, which is described by a set of relations (2),
is always symmetric relatively to the plane fzz , where fz is the location of
beam focus, which should correspond to the minimum of the ravine root-
polynomial function (1) [1–10]. Therefore, in [45], the symmetric position of the
base point, which gives the minimal error of interpolation, has been considered.
When the numerical solution is asymmetric, the error of interpolation is usually
increasing several times [45]. Such estimations of asymmetric solutions also
increase the error in defining focal beam parameters.
The solid line corresponded to a second-order function, the dotted line — to
a third-order function, the dashed line — to a fourth-order function, and the dash-
dotted line — to a fifth-order function. Model parameters: kV; 10acU A; 5.0bI
Pa 1.0p .
Therefore, the transformation of the asymmetric dataset for solving the set of
relations (2) into a symmetric one has been proposed, and the corresponding
simulation results have been studied. In such conditions, the additional symmetric
point is given to the analyzed data set, and the task of interpolation with defining
the polynomial coefficients nCC 0 is solved between two symmetric points.
And outside this symmetric region, the task of extrapolation for the same root-
Fig. 1. Errors of interpolation of the boundary trajectory of the electron beam depend on
z coordinate [45]
z, m
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 78
polynomial function (1) has been solved, and the corresponding total error of
interpolation and approximation has been defined. The general formulation of this
task and the analysis of the obtained simulation results are given in the next
sections of this article.
In the paper [46], the possibility of using third-order root-polynomial
functions for approximation experimental data about the boundary trajectories of
short-focus electron beams in ionized gases as well as the corresponding
algorithm of approximation has been considered. Another approach for the
simulation of gas discharge electron beam devices with consideration of
experimental data has been proposed in the paper [20].
The provided analysis of relevant sources on methods of interpolation and
extrapolation showed that other known methods for estimation of the ravine
functions either give a large error or are generally not suitable for solving the
problem of estimation boundary trajectories of electron beams [42–50]. For
example, for polynomial interpolation and extrapolation [48; 49], eliminating
unnecessary outliers of the estimated data, especially for high-order polynomials,
is generally impossible. Such an estimate, with the exception of a large error, does
not even provide sufficiently reliable qualitative results about the boundary
trajectory of the electron beam. As a rule, the reason for these outliers is different
values of the derivative of the ravine function in the region of the minimum and
outside it. The main advantages of the proposed approach of interpolation using
root-polynomial functions are as follows:
1. The smoothing of the estimated root-polynomial function in the region of
linear variation of numerical data.
2. Obtaining, by this reason, only one global minimum in corresponded area.
Generally, even for high-order root-polynomial functions, outliers and
unnecessary extremums are not observed [45].
It is for this reason that the interpolation error is significantly smaller, especially
for the symmetrical ravines data sets [45]. Therefore, the main subject of the
presented research is to study the possibilities of transforming asymmetric sets of
numerical data into symmetric ones using appropriate numerical methods of
functional analysis [58–52].
FORMALIZING INTERPOLATION AND EXTRAPOLATION TASKS FOR
ASYMMETRIC RAVINE FUNCTIONS DESCRIBED BOUNDARY
TRAJECTORIES OF ELECTRON BEAM
Generally, the regions of interpolation and approximation for right-hand and left-
hand asymmetric ravine data functions are shown in the Fig. 2. It is clear that the
main idea of this approach is to choose the position of the boundary point bpz ,
which divides the whole range of providing calculations by z coordinate into two
regions: Interpolation Region IR and Extrapolation Region ER.
By this way, the task of calculating the boundary trajectory of an electron
beam comes down to the problem of interpolation of the symmetric ravine func-
tion on the region IE, which has been considered and described in the papers [42–
Comparison of methods for interpolation and extrapolation of boundary trajectories of …
Системні дослідження та інформаційні технології, 2024, № 3 79
45]. After that, the root-polynomial function (1) with the same coefficients is used
for the Extrapolation Region ER. Considering, corresponding to Fig. 2, the values
of the radiuses of Start Point SP startr and End Point EP endr . Also considering
the full set of beam trajectory coordinates ]);1[(),,( Pii Nirz , where PN is the
number of calculated points, has been obtained using a set of equations (2). In
practice, the value of PN is .104PN On the contrary, the value of the basic
points BPN for solving complex interpolation-extrapolation tasks is significantly
smaller: .1 nNBP Also, 0i , the start value of variable i , is necessary to for-
malize the considered task by numerical algorithm. Using the previously de-
scribed assumptions, the position of the boundary point bpz is defined by follow-
ing the recurrent arithmetic-logic relation [47]:
)1(
1
)(
)(
1;1
)(
)()(
0
iz
ii
rir
iz
iii
rir
rriz b
startstart
startendb
.)1(
1
)(
)(
1;
)(
)(
0
iz
ii
rir
iz
iiNi
rir
rr b
end
P
end
endstart (3)
The formalized algorithm for solving the complex self-connected interpola-
tion-extrapolation task using relation (1)–(3) on the different regions of longitudi-
nal coordinate z is described by the flowchart presented in Fig. 3.
SOFTWARE FOR NUMERICAL EXPERIMENTS AND ESTIMATIONS OF
ERRORS
Using the algorithm described in the previous section of the article for finding the
boundary point bz , both interpolation and self-connected interpolation-
extrapolation tasks have been solved. After that, the error of estimation was ana-
lyzed. Both tasks are solved in the region ],[ maxmin zzz . Such types of errors
were considered and analyzed.
Fig. 2. Right-hand (a) and left-hand (b) asymmetric ravine functions: IR — Interpolation
Region; ER — Extrapolation Region; SP — Start Point; EP — End Point; BP — Bound-
ary Point
ERIR
Base
Points
SP
EP
BP
rstart
r(z)
zzf zbp
zstar zend ER IR
Base
PointsSP
EP
BP
rend
r(z)
zzf zbp zstart zend
a b
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 80
1. Maximal error maxε .
2. Average error avε , which is generally defined by the well-known method of
optimization technique [48–52] and of mathematical statistics [53; 54] as follows:
p
N
i
simest
av N
rr
p
1ε ,
where simr is the radius of the electron beam, calculated numerically by the set of
equations (2) using the fourth-order Runge-Kutt method [51; 52], and estr is the
value of the beam radius, estimated using relation (2).
3. Error on focus position Fε .
4. Error on focal beam radius rfε .
Corresponding relations for defining the third and fourth types of errors have
been considered in the papers [43–45].
Corresponding software for the simulation and estimation parameters of
electron beams was realized in the Python programming language, including ad-
vanced libraries for scientific calculations and graphic libraries. For the correct
Start
Reading input data r(z).
Defining values:
rstart, rend, Np
1
Ні
7rend > rstart
2
End
10
5 zb =z(i)
8
5 zb =z(i)
4
10 i = i +1 6 i = i +1
4
2
7 i = Np
r(i) > rend
8
9
Ні
3 i = 1
r(i) > rstart
4
5
Ні
6
Solving of interpolation
task in range [zstart. zb] and
extrapolation task in range
[zb, zend]
11
Solving of interpolation
task in range [zb, zend] and
extrapolation task in range
[zstart. zb]
12
End
Fig. 3. Flowchart of considered algorithm for solving self-connected interpolation-
extrapolation task
Comparison of methods for interpolation and extrapolation of boundary trajectories of …
Системні дослідження та інформаційні технології, 2024, № 3 81
location and including these libraries, a virtual environment, virtual memory, vir-
tual variables, and virtual disk have been created on the local computer [55; 56].
Let’s consider some results of the simulation and errors in estimations.
Task 1. Acceleration voltage acU is 15 kV; beam current bI is 8.5 A; op-
eration pressure in the guiding channel p is 4.5 Pa; the initial radius of the elec-
tron beam startr is 8.5 mm; the initial angle of convergence of the electron beam
θ is 150; the starting point startz is 0.1 m; the first end point endz is 0.15 m; and
symmetrical point bz with the same radius is m. 148.0 That is, the length of the
symmetrical segment for the highest interpolation accuracy: m 1.0 – m 148.0
m. 048.0
For this example, dependence )(zr , defined using the set of equations (2), is
presented in Fig 4.
It is clear that the dependence presented in Fig. 4 is a right-hand asymmetric
ravine function. Errors in solving interpolation and self-connected interpolation-
extrapolation tasks for this example are presented in Table 1. All errors have been
estimated for different order of root-polynomial functions n and length of extrapo-
lation region addL . Task parameter addL is given in Table 1 in absolute value, in
meters, and relatively to the length of the interpolation region IR, in percents.
The dependences of estimation errors for two fourth-order root-polynomial
functions that have been obtained for solving interpolation and self-connected
interpolation-extrapolation tasks are presented in Fig. 5. It is clear that the level of
maximal error is similar, nearly 3%, but in the case of extrapolation error in the
focal region, it is generally smaller. Corresponding data from Table 1 are as
follows; interpolation: % 06.0 ε F , % 1056.7 ε 3–rf ; extrapolation:
% 0041.0 ε F , % 1082.1 ε 6–rf . Data from Table 1 show also that if for the
extrapolation task the maximal error value εmax can be greater, then in the case of
interpolation, the value of the average error for the extrapolation task avε is
generally smaller. The dependences of the value of avε on the relative length of
the extrapolation region ER for the tasks of interpolation and self-connected
interpolation-extrapolation are presented at Fig. 6.
Fig. 4. Dependence r(z) for kV, 15acU A 5.8 bI , Pa 5.4 p , end point m 15.0 end z
(screen copy)
r,
m
z, m
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 82
T a b l e 1 . Interpolation and extrapolation errors for Task 1
maxε , % avε , % Fε , % rfε , %
n Extrapo-
lation
Interepo-
lation
Extrapo-
lation
Interepo-
lation
Extrap-
olation
Interepo-
lation
Extrapo-
lation
Interepo-
lation
% / m
,addL
2 2.29 2.84 1.26 1.4476 0 0.08 5.28·10–13 1.7·10–3
3 5.92 6.58 3.25 3.55 0 0.02 0.6 0.635
4 0.66 0.844 0.2734 0.31 0 0.004 7.7·10–11 2.1·10–4
5 2.08 2.27 0.85 0.925 0 0.004 0.44 0.47
6 0.14 0.203 0.063 0.07 0 0.02 2.4·10–8 8.5·10–5
2·10–3/ 4.2
2 2.3 3.243 1.25 1.546 1.11·10-14 6·10–3 4.57·10-13 6·10–3
3 5.914 7.14 3.2 3.76 0 0.0041 0.629 7.6·10–3
4 0.649 1.0 0.27 0.34 0.0041 0.06 1.82·10–6 7.56·10–3
5 2.2 2.58 0.884 1.03 0.0041 0.02 0.48 0.53
6 0.27 0.256 0.084 0.0654 0 0.041 3.51·10–9 4·10–4
3·10–3 / 6.25
2 2.3 3.7 1.254 1.66 0 0.22 1.23·10-13 0.013
3 5.9 7.71 3.197 3.973 0 4.1·10–3 0.65 0.744
4 0.86 1.17 0.2765 0.38 0.0042 0.1 7.1·10–7 1.878·10–3
5 2.33 2.93 0.93 1.14 0.0042 0.6 0.52 0.03345
6 0.424 0.316 0.07 0.1 0 0.067 2.23·10–8 10–3
4·10–3 / 8.3
2 2.29 4.113 1.27 1.782 2.23·10-14 0.291 2.46·10–13 0.0224
3 5.9 8.3 3.2 4.195 0 8.5·10–3 0.68 0.8
4 1.15 1.35 0.288 0.4264 0 0.14 3.38·10–11 3.86·10–3
5 2.474 3.311 1.0 1.2646 0.0043 0.047 0.5637 0.67
6 0.56 0.383 0.08 0.1166 0 9.376·10–2 2·10–8 2·10–3
5·10–3 / 10.4
2 4.89 6.6 1.4573 2.6 1.11·10-14 0.676 4.39·10-13 0.1263
3 9.52 11.3 3.5 5.4 0 0.02 0.8 1.1443
4 2.97 2.45 0.42 0.76 0 0.44 6.35·10–11 3.56·10–2
5 4.17 5.9 1.39 2.07 4.7·10–3 1.77 0.812 1.1226
6 1.69 0.83 0.17 0.2756 0 0.34 1.12·10–8 0.0273
10–2 / 20.8
Fig. 5. Dependences of electron beam radius (top) and error of estimation (bottom) on z
coordinate for the interpolation task (a) and the self-connected interpolation-
extrapolation task (b). kV 15acU ; A 5.8 bI ; Pa 5.4 p ; 4 n ; m 16.0 end z . On
the top dependence )(zr , the straight line corresponds to the obtained numerical solution,
and the dash line corresponds to estimations (screen copy)
az, m
r,
m
Comparison of methods for interpolation and extrapolation of boundary trajectories of …
Системні дослідження та інформаційні технології, 2024, № 3 83
Task 2. acU is 15 kV; bI is 5.5 A; p is 4.5 Pa; startr is 10.3 mm; θ is 150;
startz is 0.1 m; and endz is 0.15 m. Since in this case endstart rr , symmetrical
point bz with the same radius, as endr , is m. 101743.0bz That is, the length of
the symmetrical segment for the highest interpolation accuracy:
,m 045257.0 m 101743.0 –m 147.0 or, in relative units:
%. 85.3 % 100m 704525.0/m 001743.0
In the next steps of providing the calculations, we will reduce the coordinate
endz of the end point of the considered interval. That is, take .147.0 endz Thus,
we bring it closer to the extremum of the )(zr ravine function, or to the position
of the focus of the electron beam fz . In this way, we will decrease the
interpolation interval IR and increase the extrapolation interval ER. For this task,
dependence )(zr , defined using the set of equations (2), is presented in Fig 7.
z, m
r,
m
b
Contiued Fig. 5
Fig. 6. Dependences of average error εav of interpolation (top) and end interpolation-
extrapolation (bottom) tasks on the value of extrapolation region ER relative to interpola-
tion region IR addL for the root-polynomial functions of different order n . Right-hand
ravine function, Task 1 (screen copy)
L
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 84
It is clear that the dependence presented in Fig. 7 is a left-hand asymmetric
function. Errors in solving interpolation and self-connected interpolation-
extrapolation tasks for this example are presented in Table 2.
T a b l e 2 . Interpolation and extrapolation errors for Task 2
maxε , % εav , % εF , % εrf , %
n Extrapo-
lation
Interepo-
lation
Extrapo-
lation
Interepo-
lation
Extrap-
olation
Interepo-
lation
Extrapo-
lation
Interepo-
lation
,
m / %
addL
2 1.85 2.5215 1.0 1.224 0 0.1 6.68·10–13 2.86·10–3
3 4.8 5.6522 2.6 2.98 0 3.77·10–3 0.46 0.5
4 0.497 0.7 0.2 0.247 0 2.34·10–4 6.37·10–11 0.034
5 1.4255 1.624 0.593 0.6754 0 7.56·10–3 0.29 0.317
6 0.1716 0.1737 4.236·10–2 5.31·10–2 0 2·10–2 1.6·10–8 9.1·10–3
1.75·10–3 /
3.8
2 1.6 2.6 0.88 1.1743 1.11·10-14 0.16 1.4·10–13 6.5·10–3
3 4.2 5.51 2.27 2.81 0 3.7·10–2 0.4 0.45
4 0.583 0.752 0.1722 0.236 0 5·10–2 3.69·10–12 5.16·10–4
5 1.19 1.74 0.5 0.61 0 7·10–3 0.243 0.2764
6 0.26 0.19 0.037 0.05 0 0.026 1.53·10–8 1.55·10–4
2.75·10–3 /
6.35
2 1.686 2.68 0.782 1.1378 0 0.21 2.28·10–13 0.0112
3 3.6427 5.36 1.9842 2.65 0 7.2·10–3 0.346 0.41
4 0.78 0.78 0.152 0.23 0 6.5·10–2 8.73·10–11 8.26·10–4
5 1.13 1.3256 0.43 0.55 0 7.236·10–3 0.2 0.24
6 0.3422 0.2 0.0343 0.05 0 3.26·10–2 1.33·10–9 2.1·10–4
3.75·10–3 /
9.1
2 2.46 2.8 0.64 1.1 1.11·10-14 0.3 6.68·10–13 2.262·10–2
3 4.94 2.36 1.576 5.1 0 10–2 0.256 0.337
4 1.17243 0.8332 0.14 0.2293 3.4·10–3 1.33·10–3 5.62·10–7 8·10–2
5 1.735 1.07 0.3427 0.46 0 10–2 0.134 0.18
6 0.527 0.235 3.77·10–2 5.1·10–2 0 3.46·10–2 3.93·10–13 2.72·10–4
5.75·10–3 /
15.43
2 4.224 3.0 0.65 1.225 3.34·10-14 0.44 2.81·10–13 4.74·10–2
3 8.42 4.34 1.4 1.83 0 0.012 0.105 0.1915
4 2.1 0.934 0.22 0.27 3.1·10–3 8.6·10–2 7.65·10–7 1.45·10–3
5 3.354 0.86 0.39 0.327 0 1.2·10–2 4·10–2 8·10–2
6 1.235 0.31 0.12 6.9·10–2 3.1·10–3 3.1·10–2 3.81·10–6 2.15·10–4
1.075·10–2
/ 39.4
r,
m
z, m
Fig. 7. Dependence )(zr for kV, 15acU A 5.8 bI , and end point m 147.0 end z
(screen copy)
Comparison of methods for interpolation and extrapolation of boundary trajectories of …
Системні дослідження та інформаційні технології, 2024, № 3 85
The dependences for estimation error for fourth-order root-polynomial
functions for interpolation and self-connected interpolation-extrapolation tasks are
presented in Fig. 8. It is clear that the level of maximal error for the task of
interpolation is %, 05.0 εmax and for the self-connected interpolation-
extrapolation task, it is twice as high, %. 1.0 εmax But in this case, as in the
previous example, the extrapolation error in the focal region is generally smaller.
The corresponding data from Table 2 are as follows. Interpolation:
%, 1034.2 ε 4–F %; 034.0 ε rf extrapolation: %, 0 ε F %. 1037.6 ε 11–rf
In this task, the value of the average error for extrapolation avε is also generally
smaller. The dependences of the value of avε on the relative length of the
extrapolation region ER for the interpolation and self-connected interpolation-
extrapolation tasks are presented in Fig. 9.
Another testing numerical experiments for right-hand and left-hand ravine nu-
merical data, which have been provided, give the similar results of error estimation.
For the output of digital and graphic information, advanced libraries of the
Python programming language have been used, such as tkinter, numpy, and mat-
plotlib, which have been located on the virtual disk and included separately [55;
56]. A graphic interface window of elaborated software for the bookmarking “Er-
rors of Estimation” is presented in Fig. 10. For saving and further analyzing the
obtained graphic information, the bottom “Save Graph” has been provided in the
interface window.
r,
m
z, m a
r,
m
z, m b
Fig. 8. Dependences of electron beam radius (top) and error of estimation (bottom) on z
coordinate for the interpolation task (a) and the self-connected interpolation-
extrapolation task (b). kV 15acU ; A 5.5 bI ; Pa 5.5 p ; 4 n ; m. 147.0 end z On
the top dependences, the )(zr straight line corresponds to the obtained numerical solu-
tion, and the dash line corresponds to estimations (screen copy)
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 86
ANALYSIS OF OBTINED RESULTS AND DISCUSSION
The main conclusions about the estimation errors of interpolation and self-
connected interpolation tasks are as follows:
1. Usually, the root-polynomial functions (1) of odd order have a larger error
value than the functions of even order. Generally, the authors have already dis-
cussed this problem in the paper [45], and it is caused by the location of base
Fig. 9. Dependences of average error avε of interpolation (top) and interpolation-
extrapolation (bottom) tasks on value of extrapolation region ER relative to interpolation
region IR addL for the root-polynomial functions of different order .n Left-hand ravine
function. Task 2 (screen copy)
L
Fig. 10. Interface window for bookmarking “Errors of Estimation” in elaborated
computer software
Comparison of methods for interpolation and extrapolation of boundary trajectories of …
Системні дослідження та інформаційні технології, 2024, № 3 87
points. For functions of even order, the number of base points is odd, and in such
a condition, one point is located in the minimum of the ravine function data, and
other points are located evenly symmetrically to it. Such a location is considered
optimal from the point of view of obtaining minimal interpolation error [45].
Since in the case of the odd order of the root-polynomial function and the even
number of base points, this condition can’t be fulfilled, the value of error, espe-
cially in the region of minimum of the ravine function, is usually greater [45]. In
any case, the way to select the base points strongly influences the value of the
maximal and average errors for solving interpolation and self-connected interpo-
lation-extrapolation tasks.
2. The values of maximal and average errors for interpolation and extrapola-
tion tasks in the cases of right-hand and left-hand asymmetric data sets are gener-
ally similar. It is generally clear by comparing the numerical data given in Table 1
and Table 2.
3. The maximal error of extrapolation increases with the enlarging of the in-
terval of extrapolation ER .addL The maximum error of extrapolation usually cor-
responds to the end point of this interval .endz Corresponding dependences
)( addav L are given in Fig. 6 and Fig. 9.
4. In the case of solving self-connected interpolation-extrapolation tasks, the
value of the average error is generally smaller, even in cases of large values of the
interval of extrapolation ER .addL Corresponding digital data are given in Table 1
and Table 2.
5. In the case of solving a self-connected interpolation-extrapolation task, the
values of errors in defining focal beam parameters Fε and rfε are generally
smaller. In most cases, for this task, the error of defining focus position is absent;
.0ε F Corresponding digital data are given in Table 1 and Table 2.
6. For the task of interpolation, the maximal error εmax usually corresponds to
the midpoint of the interval between two selected base points. Corresponding de-
pendences )(zav are given in Fig. 5 and Fig. 8.
7. In any case, using the self-connected interpolation-extrapolation method is
very effective from the point of view of obtaining the minimal error of numerical
data estimation in the region of the position of the focus of an electron beam. Cor-
responding to Table 1 and Table 2, in some cases the error of interpolation-
extrapolation estimations by the focal beam radius rfε is significantly small, re-
sulting in a range of %]. 10 %; 10[ 7–13–
Other features of estimating errors in solving interpolation, extrapolation,
and approximation tasks for the boundary trajectory of electron beams by using
root polynomial functions of the form (1), including comparison with experimen-
tal data, are the subject of separate further studies.
All research described in this paper has been provided in the Scientific and
Educational Laboratory of Electron Beam Technological Devices of the National
Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnical Institute”.
CONCLUSION
The conducted research and the obtained modeling data showed that the use of the
root polynomial function (1) to solve self-connected tasks of the interpolation-
extrapolation problem for ravine digital data obtained as a result of the numerical
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 88
solution of the system of algebraic-differential equations (2) and describing the
boundary trajectory of a short-focus electron beam, propagating in ionized gas are
very effective and provide significantly small estimation errors.
The obtained average error of estimation is in the range of
%, 5.1 – 05.0 ε av and the error of estimation of focal beam radius is in the range
of %. 6.0 –10 ε 13–F An estimations of the value of the average error on the
length of the extrapolation region ER relative to the length of the extrapolation
region IR are also obtained, and the corresponding graphic dependences
)(ε addav L are given and analyzed. It is also significant that the average errors of
solving self-connected problems of interpolation-extrapolation digital data for
right-hand and left-hand asymmetric ravine functions aresimilar.
Obtained simulation results can be interesting for experts in the physics of
electron beams, in the elaboration of HVGD electron guns, and in applying ad-
vanced electron-beam technologies in different branches of modern industry.
REFERENCES
1. J.D. Lawson, The Physics of Charged-Particle Beams. Oxford: Clarendon Press,
1977, 446 p. Available: https://www.semanticscholar.org/paper/The-Physics-of-
Charged-Particle-Beams-Stringer/80b5ee5289d5efd8f480b516ec4bade0aa529ea6
2. M. Reiser, Theory and Design of Charged Particle Beams. John Wiley & Sons, 2008,
634 p. Available: https://www.wiley.com/en-us/Theory+and+Design+of+Charged+
Particle+Beams-p-9783527617630
3. M. Szilagyi, Electron and Ion Optics. Springer Science & Business Media, 2012,
539 p. Available: https://www.amazon.com/Electron-Optics-Microdevices-Miklos-
Szilagyi/dp/1461282470
4. S.J.R. Humphries, Charged Particle Beams. Courier Corporation, 2013, 834 p. Available:
https://library.uoh.edu.iq/admin/ebooks/76728-charged-particle-beams---s.-humphries.pdf
5. S. Schiller, U. Heisig, and S. Panzer, Electron Beam Technology. John Wiley & Sons
Inc, 1995, 508 p. Available: https://books.google.com.ua/books/about/Electron_
Beam_Technology.html?id=QRJTAAAAMAAJ&redir_esc=y
6. R.A. Bakish, Introduction to Electron Beam Technology. Wiley, 1962, 452 p. Avail-
able: https://books.google.com.ua/books?id=GghTAAAAMAAJ&hl=uk&source
=gbs_similarbooks
7. R.C., Davidson, H. Qin, Physics of Intense Charged Particle Beams in High Energy
Accelerators. World Scientific, Singapore, 2001, 604 p. Available:
https://books.google.com.ua/books/about/Physics_Of_Intense_Charged_Particle_Be
am.html?id=5M02DwAAQBAJ&redir_esc=y
8. H. Schultz, Electron Beam Welding. Woodhead Publishing, 1993, 240 p. Available:
https://books.google.com.ua/books?id=I0xMo28DwcIC&hl=uk&source=gbs_book_
similarbooks
9. G. Brewer, Electron-Beam Technology in Microelectronic Fabrication. Elsevier, 2012,
376 р. Available: https://books.google.com.ua/books?idsnU5sOQD6noC&hl=
uk&source=gbs_similarbooks
10. I. Brodie, J.J. Muray, The Physics of Microfabrication. Springer Science & Business
Media, 2013, 504 p. Available: https://books.google.com.ua/books?id=
GQYHCAAAQBAJ&hl=uk&source=gbs_similarbooks
11. A.A. Druzhinin, I.P. Ostrovskii, Y.N. Khoverko, N.S.Liakh-Kaguy, and A.M. Vuytsyk,
“Low temperature characteristics of germanium whiskers,” Functional materials 21,
no. 2, pp. 130–136, 2014. Available: http://dspace.nbuv.gov.ua/bitstream/handle/
123456789/120404/02-Druzhinin.pdf?sequence=1
Comparison of methods for interpolation and extrapolation of boundary trajectories of …
Системні дослідження та інформаційні технології, 2024, № 3 89
12. A.A. Druzhinin, I.A. Bolshakova, I.P. Ostrovskii, Y.N. Khoverko, and N.S. Liakh-
Kaguy, “Low temperature magnetoresistance of InSb whiskers,” Materials Science
in Semiconductor Processing, vol. 40, pp. 550–555, 2015. Available:
https://academic-accelerator.com/search?Journal=Druzhinin
13. A. Zakharov, S. Rozenko, S. Litvintsev, and M. Ilchenko, “Trisection Bandpass Fil-
ter with Mixed Cross-Coupling and Different Paths for Signal Propagation,” IEEE
Microwave Wireless Component Letters, vol. 30, no. 1, pp. 12–15, Jan. 2020.
14. A. Zakharov, S. Litvintsev, and M. Ilchenko, “Trisection Bandpass Filters with All
Mixed Couplings,” IEEE Microwave Wireless Components Letter, vol. 29, no. 9,
pp. 592–594, 2019. Available: https://ieeexplore.ieee.org/abstract/document/8782802
15. A. Zakharov, S. Rozenko, and M. Ilchenko, “Varactor-tuned microstrip bandpass
filter with loop hairpin and combline resonators,” IEEE Transactions on Circuits
Systems. II. Experimental Briefs, vol. 66, no. 6, pp. 953–957, 2019. Available:
https://ieeexplore.ieee.org/document/8477112
16. T. Kemmotsu, T. Nagai, and M. Maeda, “Removal Rate of Phosphorous form Melt-
ing Silicon,” High Temperature Materials and Processes, vol. 30, issue 1-2, pp. 17–
22, 2011. Available: https://www.degruyter.com/journal/key/htmp/30/1-2/html
17. J.C.S. Pires, A.F.B. Barga, and P.R. May, “The purification of metallurgically grade
silicon by electron beam melting,” Journal of Materials Processing Technology,
vol. 169, no. 1, pp. 347–355, 2005. Available: https://www.academia.edu/9442020/
The_purification_of_metallurgical_grade_silicon_by_electron_beam_melting
18. D. Luo, N. Liu, Y. Lu, G.Zhang, and T. Li, “Removal of impurities from metallurgi-
cally grade silicon by electron beam melting,” Journal of Semiconductors, vol. 32,
issue 3, article ID 033003, 2011. Available: http://www.jos.ac.cn/en/ arti-
cle/doi/10.1088/1674-4926/32/3/033003
19. D. Jiang, Y. Tan, S. Shi, W. Dong, Z. Gu, and R. Zou, “Removal of phosphorous in
molten silicon by electron beam candle melting,” Materials Letters, vol. 78, pp. 4–7, 2012.
20. A.F. Tseluyko, V.T. Lazurik, D.L. Ryabchikov, V.I. Maslov, and I.N. Sereda, “Experimental
study of radiation in the wavelength range 12.2-15.8 nm from a pulsed high-current plasma diode,”
Plasma Physics Reports, 34(11), pp. 963–968, 2008. Available: https://link.springer.com/
article/10.1134/S1063780X0811010X
21. V.G. Rudychev, V.T. Lazurik, and Y.V. Rudychev, “Influence of the electron beams inci-
dence angles on the depth-dose distribution of the irradiated object,” Radiation Physics and
Chemistry, 186, 109527, 2021. Available: https://www.sciencedirect.com/science/article/
abs/pii/S0969806X21001778
22. V.M. Lazurik, V.T. Lazurik, G. Popov, and Z. Zimek, “Two-parametric model of elec-
tron beam in computational dosimetry for radiation processing, Radiation Physics and
Chemistry, 124, pp. 230–234, 2016. Available: https://www.sciencedirect.com/science/
article/abs/pii/S0969806X1530133X
23. I. Melnyk, S. Tuhai, and A. Pochynok, “Universal Complex Model for Estimation
the Beam Current Density of High Voltage Glow Discharge Electron Guns,” Lec-
ture Notes in Networks and Systems; Eds: M. Ilchenko, L. Uryvsky, L. Globa,
vol. 152, pp. 319–341, 2021. Available: https://www.springer.com/gp/book/
9783030583583
24. S.V. Denbnovetsky, V.G. Melnyk, and I.V. Melnyk, “High voltage glow discharge
electron sources and possibilities of its application in industry for realizing different
technological operations,” IEEE Transactions on Plasma Science, vol. 31, issue 5,
pp. 987–993, October, 2003. doi: 10.1109/TPS.2003.818444
25. S. Denbnovetskiy et al., “Principles of operation of high voltage glow discharge
electron guns and particularities of its technological application,” Proceedings of
SPIE, The International Society of Optical Engineering, pp. 10445–10455, 2017.
Available: https://spie.org/Publications/Proceedings/Paper/10.1117/12.2280736
26. I.V. Melnyk, “Numerical simulation of distribution of electric field and particle
trajectories in electron sources based on high-voltage glow discharge,” Radio-
electronic and Communication Systems, vol. 48, no. 6, pp. 61–71, 2005. doi:
https://doi.org/10.3103/S0735272705060087
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 90
27. S.V. Denbnovetsky, J. Felba, V.I. Melnik, and I.V. Melnik, “Model of Beam Forma-
tion in a Glow Discharge Electron Gun with a Cold Cathode,” Applied Surface Sci-
ence, 111, pp. 288–294, 1997. Available: https://www.sciencedirect.com/science/ ar-
ticle/pii/S0169433296007611?via%3Dihub
28. J.I. Etcheverry, N. Mingolo, J.J. Rocca, and O.E. Martınez, “A Simple Model of a
Glow Discharge Electron Beam for Materials Processing,” IEEE Transactions on
Plasma Science, vol. 25, no. 3, pp. 427–432, June, 1997.
29. S.V. Denbnovetsky, V.I. Melnyk, I.V. Melnyk, and B.A. Tugay, “Model of control of
glow discharge electron gun current for microelectronics production applications,”
Proceedings of SPIE. Sixth International Conference on “Material Science and Mate-
rial Properties for Infrared Optoelectronics”, vol. 5065, pp. 64–76, 2003. doi:
https://doi.org/10.1117/12.502174
30. I.V. Melnyk, “Estimating of current rise time of glow discharge in triode electrode
system in case of control pulsing,” Radioelectronic and Communication Systems,
vol. 56, no. 12, pp. 51–61, 2017.
31. A. Mitchell, T. Wang, “Electron beam melting technology review,” Proceedings of
the Conference “Electron Beam Melting and Refining State of the Art 2000, Reno,
NV, USA, 2000, ed. R. Bakish, pp. 2–13.
32. D.V. Kovalchuk, N.P. Kondraty, “Electron-beam remelting of titanium – problems
and development prospects,” Titan 2009, no. 1(23), pp. 29–38.
33. V.A. Savenko, N.I. Grechanyuk, and O.V. Churakov, “Electron beam refining in production of
platinum and platinum-base alloys. Information 1. Electron beam refining of platinum,” Ad-
vances in Electrometallurgy, no. 1, pp. 14–16, 2008.
34. T.O. Prikhna et al., “Electron-Beam and Plasma Oxidation-Resistant and Thermal-Barrier Coat-
ings Deposited on Turbine Blades Using Cast and Powder Ni(Co)CrALY(Si) Alloys I. Funda-
mentals of the Production Technology, Structure, and Phase Composition of Cast NiCrAlY Al-
loys,” Powder Metallurgy and Metal Ceramics, vol. 61, issue 1-2, pp. 70–76, 2022. Available:
https://link.springer.com/article/10.1007/s11106-022-00296-8
35. T.O. Prikhna et al., “Electron-Beam and Plasma Oxidation-Resistant and Thermal-Barrier Coat-
ings Deposited on Turbine Blades Using Cast and Powder Ni(Co)CrAlY(Si) Alloys Produced
by Electron-Beam Melting II. Structure and Chemical and Phase Composition of Cast CoCrAlY
Alloys,” Powder Metallurgy and Metal Ceramicsthis, vol. 61, issue 3-4, pp. 230–237, 2022.
36. I.M. Grechanyuk et al., “Electron-Beam and Plasma Oxidation-Resistant and Ther-
mal-Barrier Coatings Deposited on Turbine Blades Using Cast and Powder
Ni(Co)CrAlY(Si) Alloys Produced by Electron Beam Melting IV. Chemical and
Phase Composition and Structure of Cocralysi Powder Alloys and Their Use,” Pow-
der Metallurgy and Metal Ceramics, vol. 61, issue 7-8, pp. 459–464, 2022.
37. M.I. Grechanyuk et al., “Electron-Beam and Plasma Oxidation-Resistant and Thermal-
Barrier Coatings Deposited on Turbine Blades Using Cast and Powder Ni (Co)CrAlY (Si)
Alloys Produced by Electron Beam Melting III. Formation, Structure, and Chemical and
Phase Composition of Thermal-Barrier Ni(Co)CrAlY/ZrO2–Y2O3 Coatings Produced by
Physical Vapor Deposition in One Process Cycle,” Powder Metallurgy and Metal Ceram-
ics, vol. 61, issue 5-6, pp. 328–336, 2022.
38. J. Zhang et al., “Fine equiaxed β grains and superior tensile property in Ti–6Al–4V
alloy deposited by coaxial electron beam wire feeding additive manufacturing,” Acta
Metallurgica Sinica (English Letters), 33(10), pp. 1311–1320, 2020.
39. D. Kovalchuk, O. Ivasishin, “Profile electron beam 3D metal printing,” in Additive
Manufacturing for the Aerospace Industry. Elsevier Inc., 2019, pp. 213–233.
40. M. Wang et al., “Microstructure and mechanical properties of Ti-6Al-4V cruciform
structure fabricated by coaxial electron beam wire-feed additive manufacturing,”
Journal of Alloys and Compounds, vol. 960. article 170943. doi:
https://doi.org/10.1016/j.jallcom.2023.170943
41. I. Melnyk, S. Tuhai, M. Surzhykov, I. Shved, V. Melnyk, and D. Kovalchuk, “Ana-
lytical Estimation of the Deep of Seam Penetration for the Electron-Beam Welding
Technologies with Application of Glow Discharge Electron Guns,” 2022 IEEE 41-st
International Conference on Electronics and Nanotechnology (ELNANO), 2022, pp. 1–5.
Comparison of methods for interpolation and extrapolation of boundary trajectories of …
Системні дослідження та інформаційні технології, 2024, № 3 91
42. I. Melnyk, S. Tuhai, and A. Pochynok, “Calculation of Focal Paramters of Electron
Beam Formed in Soft Vacuum at the Plane which Sloped to Beam Axis,” The Forth IEEE
International Conference on Information-Communication Technologies and Radioelectronics
UkrMiCo’2019. Collections of Proceedings of the Scientific and Technical Conference,
Odesa, Ukraine, September 9-13, 2019. doi: 10.1109/UkrMiCo47782.2019.9165328
43. I. Melnyk, S. Tuhai, and A. Pochynok, “Interpolation of the Boundary Trajectories
of Electron Beams by the Roots from Polynomic Functions of Corresponded Order,”
2020 IEEE 40th International Conference on Electronics and Nanotechnology
(ELNANO), pp. 28–33. doi: 10.1109/ELNANO50318.2020.9088786
44. I. Melnik, S. Tugay, and A. Pochynok, “Interpolation Functions for Describing
the Boundary Trajectories of Electron Beams Propagated in Ionised Gas,” 15-th
International Conference on Advanced Trends in Radioelectronics, Telecommu-
nications and Computer Engineering (TCSET – 2020), pp. 79–83. doi:
10.1109/TCSET49122.2020.235395
45. I.V. Melnyk, A.V. Pochynok, “Study of a Class of Algebraic Functions for In-
terpolation of Boundary Trajectories of Short-Focus Electron Beams,” System
Researches and Information Technologies, no. 3, pp. 23–39, 2020. doi:
https://doi.org/10.20535/SRIT.2308-8893.2020.3.02
46. I. Melnyk, S. Tuhai, M. Skrypka, A. Pochynok, and D. Kovalchuk, “Approximation
of the Boundary Trajectory of a Short-Focus Electron Beam using Third-Order Root-
Polynomial Functions and Recurrent Matrixes Approach,” 2023 International Con-
ference on Information and Digital Technologies (IDT), Zilina, Slovakia, 2023, pp.
133–138, doi: 10.1109/IDT59031.2023.10194399
47. I. Melnyk, A. Luntovskyy, “Estimation of Energy Efficiency and Quality of Service
in Cloud Realizations of Parallel Computing Algorithms for IBN,” in Klymash, M.,
Beshley, M., Luntovskyy, A. (eds) Future Intent-Based Networking. Lecture Notes in
Electrical Engineering, vol. 831, Springer, Cham, pp. 339–379. doi:
https://doi.org/10.1007/978-3-030-92435-5_20
48. G.M. Phillips, Interpolation and Approximation by Polynomials. Springer, 2023, 312
p. Available: http://bayanbox.ir/view/2518803974255898294/George-M.-Phillips-
Interpolation-and-Approximation-by-Polynomials-Springer-2003.pdf
49. N. Draper, H. Smith, Applied Regression Analysis; 3 Edition. Wiley Series, 1998,
706 p. Available: https://www.wiley.com/en-us/Applied+Regression+ Analy-
sis,+3rd+Edition-p-9780471170822
50. C. Mohan, K. Deep, Optimization Techniques. New Age Science, 2009, 628 p. Available:
https://www.amazon.com/Optimization-Techniques-C-Mohan/dp/1906574219
51. M.K. Jain, S.R.K. Iengar, and R.K. Jain, Numerical Methods for Scientific & Engi-
neering Computation. New Age International Pvt. Ltd., 2010, 733 p. Available:
https://www.google.com.ua/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2a
hUKEwippcuT7rX8AhUhlYsKHRfBCG0QFnoECEsQAQ&url=https%3A%2F%2F
www.researchgate.net%2Fprofile%2FAbiodun_Opanuga%2Fpost%2Fhow_can_sol
ve_a_non_linear_PDE_using_numerical_method%2Fattachment%2F59d61f727919
7b807797de30%2FAS%253A284742038638596%25401444899200343%2Fdownlo
ad%2FNumerical%2BMethods.pdf&usg=AOvVaw0MjNl3K877lVWUWw-FPwmV
52. S.C. Chapra, R.P. Canale, Numerical Methods for Engineers; 7th Edition. McGraw
Hill, 2014, 992 p. Available: https://www.amazon.com/Numerical-Methods-
Engineers-Steven-Chapra/dp/007339792X
53. E. Wentzel, L. Ovcharov, Applied Problems of Probability Theory. Mir, 1998, 432 p. Available:
https://mirtitles.org/2022/06/03/applied-problems-in-probability-theory-wentzel-ovcharov/
54. J.A. Gubner, Probability and Random Processes for Electrical and Computer Engineers.
UK, Cambridge: Cambridge University Press, 2006. Available: http://www.amazon.
com/Probability-Processes-Electrical-Computer-Engineers/dp/0521864704
55. M. Lutz, Learning Python; 5th Edition. O’Reilly, 2013, 1643 p.
56. W. McKinney, Python for Data Analysis: Data Wrangling with Pandas, NumPy, and
Jupyter; 3rd Edition. O’Reilly Media, 2023, 579 p.
Received 02.11.2023
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 92
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
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
Mykhailo Yu. Skrypka, ORCID: 0009-0006-7142-5569, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: scientetik@gmail.com
ПОРІВНЯННЯ ЗАСТОСУВАННЯ МЕТОДІВ ІНТЕРПОЛЯЦІЇ ТА
ЕКСТРАПОЛЯЦІЇ ГРАНИЧНИХ ТРАЄКТОРІЙ КОРОТКОФОКУСНИХ
ЕЛЕКТРОННИХ ПУЧКІВ ІЗ ВИКОРИСТАННЯМ КОРЕНЕВО-
ПОЛІНОМІАЛЬНИХ ФУНКЦІЙ / І.В. Мельник, А.В. Починок, М.Ю. Скрипка
Анотація. Розглянуто й обговорено порівняння методів інтерполяційного та
екстраполяційного оцінювання граничної траєкторії електронних пучків, які
поширюються в іонізованому газі. Оцінювання виконано з використанням ко-
ренево-поліноміальних функцій для числового розв’язання системи алгебро-
диференціальних рівнянь, що описують граничну траєкторію електронного
пучка. Показано та доведено, що у разі розв’язання взаємозв’язаної задачі ін-
терполяції-екстраполяції середня похибка оцінювання радіуса променя, зазви-
чай, є меншою. Особливо ефективним виявилося використання цього підходу
в оцінюванні фокального радіуса променя. Наведено алгоритм розв’язання ін-
терполяційно-екстраполяційної задачі та пояснено його ефективність. Наведе-
но та проаналізовано відповідні графічні залежності.
Ключові слова: інтерполяція, екстраполяція, коренево-поліноміальна функ-
ція, яружна функція, електронний пучок, гранична траєкторія.
|
| id | journaliasakpiua-article-315197 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:36Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/83/83e8635256d410ddd903450dc5643783.pdf |
| spelling | journaliasakpiua-article-3151972024-11-16T18:06:34Z Comparison of methods for interpolation and extrapolation of boundary trajectories of short-focus electron beams using root-polynomial functions Порівняння застосування методів інтерполяції та екстраполяції граничних траєкторій короткофокусних електронних пучків із використанням коренево-поліноміальних функцій Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo interpolation extrapolation root-polynomial function ravine function average error electron beam boundary trajectory інтерполяція екстраполяція коренево-поліноміальна функція яружна функція електронний пучок гранична траєкторія The article considers and discusses the comparison of interpolation and extrapolation methods of estimation of the boundary trajectory of electron beams propagated in ionized gas. All estimations have been computed using root-polynomial functions to numerically solve a differential-algebraic system of equations that describe the boundary trajectory of the electron beam. By providing analysis, it is shown and proven that in the case of solving a self-connected interpolation-extrapolation task, the average error of the beam radius estimation is generally smaller. This approach was especially effective in estimating the focal beam radius. An algorithm for solving self-connected interpolation-extrapolation tasks is given, and its efficiency is explained. Corresponding graphic dependencies are also given and analyzed. Розглянуто й обговорено порівняння методів інтерполяційного та екстраполяційного оцінювання граничної траєкторії електронних пучків, які поширюються в іонізованому газі. Оцінювання виконано з використанням коренево-поліноміальних функцій для числового розв’язання системи алгебро-диференціальних рівнянь, що описують граничну траєкторію електронного пучка. Показано та доведено, що у разі розв’язання взаємозв’язаної задачі інтерполяції-екстраполяції середня похибка оцінювання радіуса променя, зазвичай, є меншою. Особливо ефективним виявилося використання цього підходу в оцінюванні фокального радіуса променя. Наведено алгоритм розв’язання інтерполяційно-екстраполяційної задачі та пояснено його ефективність. Наведено та проаналізовано відповідні графічні залежності. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-09-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/315197 10.20535/SRIT.2308-8893.2024.3.05 System research and information technologies; No. 3 (2024); 74-92 Системные исследования и информационные технологии; № 3 (2024); 74-92 Системні дослідження та інформаційні технології; № 3 (2024); 74-92 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/315197/306061 |
| spellingShingle | інтерполяція екстраполяція коренево-поліноміальна функція яружна функція електронний пучок гранична траєкторія Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo Порівняння застосування методів інтерполяції та екстраполяції граничних траєкторій короткофокусних електронних пучків із використанням коренево-поліноміальних функцій |
| title | Порівняння застосування методів інтерполяції та екстраполяції граничних траєкторій короткофокусних електронних пучків із використанням коренево-поліноміальних функцій |
| title_alt | Comparison of methods for interpolation and extrapolation of boundary trajectories of short-focus electron beams using root-polynomial functions |
| title_full | Порівняння застосування методів інтерполяції та екстраполяції граничних траєкторій короткофокусних електронних пучків із використанням коренево-поліноміальних функцій |
| title_fullStr | Порівняння застосування методів інтерполяції та екстраполяції граничних траєкторій короткофокусних електронних пучків із використанням коренево-поліноміальних функцій |
| title_full_unstemmed | Порівняння застосування методів інтерполяції та екстраполяції граничних траєкторій короткофокусних електронних пучків із використанням коренево-поліноміальних функцій |
| title_short | Порівняння застосування методів інтерполяції та екстраполяції граничних траєкторій короткофокусних електронних пучків із використанням коренево-поліноміальних функцій |
| title_sort | порівняння застосування методів інтерполяції та екстраполяції граничних траєкторій короткофокусних електронних пучків із використанням коренево-поліноміальних функцій |
| topic | інтерполяція екстраполяція коренево-поліноміальна функція яружна функція електронний пучок гранична траєкторія |
| topic_facet | interpolation extrapolation root-polynomial function ravine function average error electron beam boundary trajectory інтерполяція екстраполяція коренево-поліноміальна функція яружна функція електронний пучок гранична траєкторія |
| url | https://journal.iasa.kpi.ua/article/view/315197 |
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