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The comparison of three advanced novel methods for estimating the boundary trajectory of electron beams propagated in ionized gas, including lower-order interpolation, self-connected interpolation, and extrapolation, as well as higher-order interpolation, is considered and discussed in the article....
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| author | Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo |
| author_facet | Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo |
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| description | The comparison of three advanced novel methods for estimating the boundary trajectory of electron beams propagated in ionized gas, including lower-order interpolation, self-connected interpolation, and extrapolation, as well as higher-order interpolation, is considered and discussed in the article. All estimations of the corresponding errors have been provided relative to numerically solving the set of algebra-differential equations that describe the boundary trajectory of the electron beam. By providing analysis, it is shown and proven that lower-order interpolation usually gives the minimal value of average error, using the method of self-connected interpolation and extrapolation gives the minimal error for estimation of focal beam parameters, and higher-order interpolation is suitable to obtain a uniform error value over the entire interpolation interval. All results of error estimation were obtained using original computer software written in Python. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.4.11 |
| first_indexed | 2025-07-17T10:28:41Z |
| format | Article |
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Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2024
Системні дослідження та інформаційні технології, 2024, № 4 133
UDC 004.942:537.525
DOI: 10.20535/SRIT.2308-8893.2024.4.11
AN ADVANCED METHOD OF INTERPOLATION OF SHORT-
FOCUS ELECTRON BEAMS BOUNDARY TRAJECTORIES
USING HIGHER-ORDER ROOT-POLYNOMIAL FUNCTIONS
AND ITS COMPARATIVE STUDY
I. MELNYK, A. POCHYNOK, M. SKRYPKA
Abstract. The comparison of three advanced novel methods for estimating the
boundary trajectory of electron beams propagated in ionized gas, including lower-
order interpolation, self-connected interpolation, and extrapolation, as well as
higher-order interpolation, is considered and discussed in the article. All estimations
of the corresponding errors have been provided relative to numerically solving the
set of algebra-differential equations that describe the boundary trajectory of the elec-
tron beam. By providing analysis, it is shown and proven that lower-order interpola-
tion usually gives the minimal value of average error, using the method of self-
connected interpolation and extrapolation gives the minimal error for estimation of
focal beam parameters, and higher-order interpolation is suitable to obtain a uniform
error value over the entire interpolation interval. All results of error estimation were
obtained using original computer software written in Python.
Keywords: interpolation, extrapolation, lower-order interpolation, higher-order in-
terpolation, root-polynomial function, ravine function, average error, electron beam,
boundary trajectory, high voltage glow discharge, electron beam technologies.
INTRODUCTION
Interpolation of boundary trajectories of electron beams is very important task
today, taking into account the high level of development electron beam technolo-
gies and its applying in modern industry [1–9].
Really, industrial electron-beam technologies have been developed and
widely applied in industry since 60–70-th years of XX century [6–9], but its
application in modern industry is also continued. Therefore, adaptation of
traditional electron-beam technologies to corresponded advanced technological
processes is successfully provided today [1–5; 10–13].
Today main branches of industry, where electron beam technologies find
high level of application, are follows: metallurgy, mechanical engineering,
electrical engineering, instrument making, microelectronic production,
automotive, aircraft, and space industries [1–5].
For example, in microelectronic production point-focus electron beams with
focal beam radius can be successfully applied for making contacts in precision
cryogenic devices [14; 15]. Corresponded estimation of diameter of welding seam
in electron-beam technologies have been provided recently in the paper [16; 17].
Other advanced application of electron-beam technologies in microelectronic
production is refining of silicon [10–13] and obtaining of chemically-complex
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 134
ceramic films for high quality capacitors and for microwave transmitters and
receivers in advanced communication systems [18–20]. Generally, advanced
possibility of using electron beam technologies in modern microelectronic
production are described in manual book [5].
In metallurgy advanced electron-beam technologies are widely used, since
60-years of XX century, for refining of refractory metals [21–23] and other re-
fractory materials, in particular silicon for the microelectronics industry [11–13].
Among other, advanced application of electron beam heating in metallurgy,
which have been developed in last few years, three-dimensional printing by the
metals, including forming of details with complex spatial shape for aircraft and
space industry, have to be mentioned and considered [24–26].
In energetic industry and electric vehicle production electron beam tech-
nologies are widely used for deposition high-quality ceramic insulator films [27–
30]. Other advanced application of high-quality chemically-complex ceramic
films in automotive, aircraft and space industry is obtaining heat-resistant and
heat-protective thick films for details of engines, which operated under conditions
of high temperature. For deposition such kind of films advanced method of physi-
cal vapor deposition by electron-beam heating is usually applied [27–30].
Special issue is applying of high-energy intensive electron beams, obtained
in the accelerators, for changing the properties of treated materials. Corresponded
technologies are described in the works [4; 31–33].
The advantages of electron beams, which caused its wide application in
modern industrial technologies, are as follows [1–5].
1. High total power and power density in beam focus.
2. High energetic efficiency of electron beam sources.
3. Simplicity of fast control of beam power and spatial position of beam fo-
cus using electric and magnetic fields.
4. Wide range of different technological operations, which can be realized by
electron beam heating and chemical treatment.
Taking into account pointed out advantages of electron beam technologies,
elaboration of advanced improving industrial constructions of electron beam
sources, which are called electron guns, is important scientific and engineering
task today. Usually, this task solving using two different ways, which are, gener-
ally, follows.
1. Improving the constructions of electron guns with heated cathode, oper-
ated in conditions of high vacuum. Such kind of electron guns are traditional and
widely use since 60–70-th years of XX century [6–9].
2. Elaboration of novel types of electron guns, based on auto-electronic
emission of electrons in the string electric fields, photoemission, as well as on
emission in gas discharges. Among this types of guns special place occupied
high-voltage glow discharge electron guns, which particularities of operation will
be considered in next part of this paper.
HIGH VOLTAGE GLOW DISCHARGE ELECTRON GUNS AND
ADVANTAGES ITS APPLYING IN ELECTRON BEAM TECHNOLOGIES
In the last few decades, in some technological processes that are implemented in a
soft vacuum in an environment of air or active gases, instead of the traditionally
used guns with a heated cathode, alternative guns have been successfully applied,
the operation of which is based on the use of a high-voltage glow discharge
(HVGD). From a physical point of view, HVGD is considered a kind of dis-
An advanced method of interpolation of short-focus electron beams boundary trajectories …
Системні дослідження та інформаційні технології, 2024, № 4 135
charge, taking place under voltage between electrodes 5–40 kV and pressure in
the discharge volume range of 0.1–10 Pa [34–36]. In the work [34–36] the basic
principles of simulation HVGD electron guns have been considered, and corre-
sponding mathematical relations were also given and analyzed. In the papers [37;
38] main advantages of applying HVGD electron guns for welding, melting proc-
esses, as well as for deposition of ceramic films have been pointed out. These ad-
vantages are as follows [37; 38].
1. High stability of operation in conditions of soft vacuum.
2. Relative simplicity of gun construction.
3. Relative simplicity of evacuation equipment for obtaining soft vacuum.
4. Since the current density from the cold cathode in HVGD conditions is
not so large, range of 0.01 A/cm2, using of enlarges cathode surface and suitable
self-maintained electron-ion optics allows forming profile electron beams with
linear and ring-like focus [37; 38].
5. Simplicity of control of gun current, both by relatively slow aerodynamic
method using electromagnetic valve [39], and by fast electric method with
lighting of low-voltage additional discharge in anode plasma region [40].
6. Possibility of providing operation of HVGD electron guns in impulse regime
with obtaining advanced technological possibilities of pulsed electron beams [41–43].
The regulation time for slow electrodynamic control systems was estimated
in the paper [39], and for fast electric control systems correspondently, in papers
[41; 43].
However, simulation of HVGD electron guns is realized today mostly by
solving of complex algebra-differential equations, described forming and
interaction of charged particles flows in the soft vacuum conditions. The main
problem in this task is defining of anode plasma boundary form and position. It
caused by the fact, that in HVGD anode plasma is considered as the source of
ions and as electrode with fixed potential, which is transparent to beam electrons
[34–36]. Simplified analytical models for defining of focal beam parameters in
HVGD aren’t existed [34]. But namely such approximative estimations are very
important for defining the technological possibilities of electron beams, especially
on the first stage of gun designing [16; 17]. Absence of such simple approach of
analytical calculations of focal beam parameters significantly hinders
development and implementation in industry of HVGD electron guns, which
advantages have been described above. Also using of sophisticated numerical
calculation methods is lead to increasing the complexity of solving simulation
problems in case of implementing cloud computing. Corresponded estimations
have been given in works [44–46]. Therefore, finding the corresponding
analytical relations for estimation focal parameters of electron beams, formed by
HVGD electron guns, is very important scientific and engineering task today.
This task will be considered in the next section of the article.
GENERAL STATEMENT OF PROBLEM OF INTERPOLATION BOUNDARY
TRAJECTORY OF ELECTRON BEAM, PROPAGATED IN IONIZED GAS,
AND ESTIMATION OF ERRORS
Firstly, the basic approach to interpolation the boundary trajectories of electron
beams have been proposed in the years 2019–2020 in the papers [17; 47–49].
Generally, this approach is based on the following presumptions.
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 136
1. Numerical solving of basic set of algebra-differential equations for the
boundary trajectory of short-focus electron beam, propagated in the soft vacuum
conditions in the medium of ionized gas, which is, in general form, written as fol-
lows [1–6; 34; 49]:
;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
ei
eibi
e
ac
e
b
b
e
rn
U
Um
nM
pnBrn
m
eU
v
r
I
n (1)
;
γβ22
θ
tan ;
γβ2
10
2
θ
tan ;β1γ
2
23
max
2
344
min2 aa ZZ
,,ln
)(8
max
min
2
2
c
v
n
dzZr e
e
ab
where acU is the voltage of HVGD lighting; Ib is the current of electron beam; p
is the residual gas pressure in the volume of HVGD lighting; z is the longitudinal
coordinate; br is the radius of the boundary trajectory of the electron beam; dz is
the length of the electron path in the longitudinal direction at the current iteration;
0in is the concentration of residual gas ions on the beam symmetry axis; en is the
concentration of beam electrons’; ev is the average velocity of the beam
electrons; min and max are the minimum and maximum scattering angles of the
beam electrons, corresponding to Rutherford model [1–6]; min min is the av-
erage scattering angle of the beam electrons; f is the residual level of gas ioniza-
tion; iB is the gas ionization level; em is the electron mass; 0 is the dielectric
constant; c is the light velocity; is the relativistic factor; iM is the molecular
mass of residual gas atoms, and aZ is its’ nuclear charge.
2. Choosing of k basic points ),( zrb on the calculated boundary trajectory.
3. Interpolation of defined function )(zrb using ravine root-polynomial func-
tion [47–49]:
,)( 01
1
1
n n
n
n
nb CzCzCzCzr
(2)
where n = k – 1 is the degree of the polynomial and the order of the root-
polynomial function, and nCC ,,0 are the polynomial coefficients.
4. Defining of interpolation error using relation [47–49]:
%,100
(z)
(z)(z)
(z)ε
num
intnum
b
bb
r
rr
(3)
where (z)
numbr is numerical and (z)
intbr is interpolated values of beam radius br .
An advanced method of interpolation of short-focus electron beams boundary trajectories …
Системні дослідження та інформаційні технології, 2024, № 4 137
Generally, described above method of interpolation of electron beam bound-
ary trajectory is based on the presumption, that dependence )(zr
numb is considered
as ravine one with one global minimum and quasilinear dependence outside the
region of minimum. This presumption is fully corresponded to the conception of
physics of the flows of charged particles, have been described in [1–6]. Provided
theoretical researches shown, that main particularities of root-polynomial function
(2) are usually generally suitable to this presumption. Therefore, the interpolation
error, defined by relation (3), is always very small, range of few percent, and in
some cases is even smaller [47–49]. Typi-
cal dependence of obtained relative error of
interpolation on z coordinate for different
order of polynomial function (2) is pre-
sented in Fig. 1 [49]. In this figure the solid
line corresponded to a second-order func-
tion, the dotted line – to a third-order func-
tion, the dashed line – to a fourth-order
function, and the dash-dotted line – to a
fifth-order function. Model parameters for
the numerical data, presented at Fig. 1, are
follows: 10acU kV, 5.0bI A,
1.0p Pa.
Also provided researches shown, that the interpolation error is strongly de-
pend on position of interpolation points ),( ibi zrP
i
on the interpolated interval. It
has been proven, that the minimum error of interpolation is provided to symmetric
interval of ravine data with location minimum in the medium point. And if the
ravine data is asymmetric, the value of error is significantly increased. Namely
this established rule constituted a theoretical basis for further research, which will
be described later in this article.
By the reason of results of this researches the method of interpolation by
higher-order root-polynomial functions is proposed, which will be considered in
next section of the article.
In the paper [50] was described the method of approximation of the trajecto-
ries of electron beams, propagated in ionized gas, using third-order root-
polynomial function (2). Another approach to simulation of focal beam parame-
ters of HVGD electron guns was given in the paper [51].
ASYMMETRIC RAVINE NUMERICAL DATA AND STATEMENT
THE PROBLEM OF ITS INTERPOLATION AND EXTRAPOLATION
Let’s considering left-hand and right-hand asymmetric ravine root-polynomial
functions, which in general form presented in Fig. 2. Here given the basic pa-
rameters of these functions, such as radiuses of electron beam in the Start Point
SP startr and End Point EP endr , location of this points startz and endz , and posi-
tion of beam focus fz .
It is clear from Fig. 2, that, corresponding to the theory of interpolation, ba-
sic principles of numerical methods, and probability theory [52–58], for interpola-
z,m
Fig. 1. Errors of interpolation of the
boundary trajectory of the electron
beam depend on z coordinate [49]
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 138
tion ravine dependences using root-polynomial functions (2) the additional refer-
ence point with coordinate bpz is considered, and its location is defined by fol-
lowing arithmetic-logic relation:
1
11;10
iz
ii
rir
iz
iii
rir
rriz bp
startstart
startendbp
,1
11;0
iz
ii
rir
iz
iiNi
rir
rr bp
end
P
end
endstart (4)
where pN is the whole number of points in numerical calculation of beam trajec-
tories, which is usually in range 4
10 PN . On the contrary, the value of the basic
points NBP for solving interpolation tasks is significantly smaller: 1 nNBP .
The basic principles of forming arithmetic-logic relations have been consid-
ered in the book chapter [45].
After using iterative relation (4), finding of location of basic points BP on in-
terval ],[ bpstart zz for right-hand asymmetric function or in interval ],[ endbp zz
for left-hand one is defined by following relations:
)),(1()( ))(1( )( )( jkNzzjkNzzjN fffjfffjBP (5)
where )( jk f is the coefficient, which depend on the order of root-polynomial
function and provided the minimal error of interpolation for symmetric ravine
numerical data. For functions of even order usually one basis point is located at
the focus position fz and other symmetrically on the interval of interpolation
],[ bpstart zz or ],[ endbp zz . For example, for fifth-order root polynomial function:
3
1
2 fk and .
20
1
3 fk For six-order function:
3
1
2 fk and .
3
1
3 fk
Correspondent approach to calculation the coefficients )( jk f is related with the
theory of numerical methods [52; 53; 55; 56] and was considered in the paper
[49]. Really, for symmetric ravine function:
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; NSR — numerical simulation result; I — interpolation result; E — extrapola-
tion result
E
ER IR
Base
Points SP
EP BP
rend
r(z)
z zf zbp
zstart zend
NSR
I
E
ERIR
Base
Points
SP
EP
BP
rstart
r(z)
z zf zbp
zstart zend
NSR
I
a b
An advanced method of interpolation of short-focus electron beams boundary trajectories …
Системні дослідження та інформаційні технології, 2024, № 4 139
,
22
pf
p
pf
p Nllk
N
Nllk
N
.even for ,
2
;1
; oddfor ,
2
1
;1
n
n
n
n
l (6)
For asymmetric ravine function problem of minimizing error of interpolation
can be solved by to following ways.
1. Using interpolation method with defining position of basic points by
relations (4)–(6). In this case the points are located evenly on interval
],[ endstart zz , and the focus position pz can’t be considered as a region of
minimal error, because the coefficients fk are calculated for ravine function.
2. Using interpolation method with defining position of basic points by
relations (4)–(6) only for the symmetric region IR (see Fig. 2), and for region ER
solving extrapolation task [53; 54]. In this case the additional basic boundary
point BP is defined by relation (4) and used. Therefore, corresponding to Fig. 2,
for right-hand asymmetric ravine data interpolation provided on the interval
],[ bpstart zz , and extrapolation on the interval ],[ endbp zz . In contrary, for left-
hand asymmetric ravine data interval of extrapolation is ],[ bpstart zz and interval
of interpolation is ],[ endbp zz . For such self-connected interpolation-extrapolation
task maximal error is always observed on the region ER, but in the region of
focus position it is minimized.
Other method of interpolation, which give the average value of error on the
whole interpolation interval, will be considered in the next section of the article.
INTERPOLATION OF ASYMMETRIC RAVINE NUMERICAL DATA USING
ROOT-POLYNOMIAL FUNCTION OF HIGHER ORDER
The main distinguishing feature of proposed method of interpolation is solving of
interpolation task on the whole interval of asymmetric ravine function
][ , endstart zz , but with including into consideration the boundary point BP, which
coordinate, corresponding to Fig. 2, is bpz . In such conditions other basic point
are located in the IR region, but in ER region used interpolation by root-
polynomial function (2) with the same polynomial coefficients. The order of this
function is BPN , where BPN is number of basic points, located in the IR region.
Therefore, all basic points are located evenly in interpolation interval IE using
relations (4)–(6). The arithmetic-logic relation for defining set of coefficients
}{ 0CCn of the root-polynomial function of higher order )(1 zfn by correspond-
ing set of basic points ),,( endbpstart PPP is written as follows:
),,( endbpstart PPP
}),{},,{,},{},,({)(
21
endendbpbp
nj
BPBPstartstartstartstart rzrzrzrzrr
jj
}),{,},{},,{},,({)(
21
endend
nj
BPBPbpbpstartstartendstart rzrzrzrzrr
jj
, (7)
.),()( 1
0
1
0
1
0
nj
jj
nj
jj
nj
jj rzC F .
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 140
Analytical relations for defining the set of coefficients of the root-polynomial
function )( 1
0
nj
jjC through vector-function ),( 1
0
1
0
nj
jj
nj
jj rzF for function
from second to sixth order have been given and analyzed in the paper [49].
Some examples of using relations (4)–(7) for defining the coefficient of root-
polynomial function (2), as well as comparing the error of interpolation using
high-order, low-order functions and combined interpolation-extrapolation
method, will be presented in the next part of the article.
OBTAINED RESULTS OF INTERPOLATION AND EXTRAPOLATION OF
ASYMMETRIC RAVINE NUMERICAL DATA USING ROOT-POLYNOMIAL
FUNCTION OF LOW AND HIGHER ORDER
Comparing study of applying interpolation and combined interpolation-
extrapolation methods, described above, has been provided by comparing such
types of errors: maximal error maxε , average error avε , error of estimation the
focus position Fε , and error of estimation focal beam radius rfε .
Average error is defined by the well-known method of optimization tech-
nique [53; 54] and of mathematical statistics [57; 58] as follows:
p
N
i
simest
av N
rr
p
1ε , (8)
where simr is the radius of the electron beam, calculated numerically by the set of
equations (1) using the fourth-order Runge-Kutt method [55; 56], and estr is the
value of the beam radius, estimated using relation (2). Local error of interpolation
and extrapolation at considered point z has been defined, using relation (3).
All errors have been estimated for different order of root-polynomial func-
tions n and length of extrapolation region addL . Task parameter addL is given in
the tables of obtained testing results in absolute value, in meters, and relatively to
the length of the interpolation region IR, in percents.
Task 1. kV 15 acU ,
A 5.5 bI , Pa 5.4 p , startr
mm 5.8 , ,5.10 θ 0 m 1.0 startz .
End points: ;m 16.0 .1 endz
; 165.0 .2 mzend ;m 17.0 .3 endz
;m 175.0 . 4 endz .18.0 . 5 endz
Additional boundary basic point:
.m 156822.0 bpz
For this example, the depend-
ence )(zr , defined by numerical
solving the set of equations (1), is
presented in Fig. 3.
z, m
r,
m
Fig. 3. Dependence )(zr for kV 15 acU ,
A 5.5 bI , Pa 5.4 P , end point
m 61.0 endz (screen copy)
An advanced method of interpolation of short-focus electron beams boundary trajectories …
Системні дослідження та інформаційні технології, 2024, № 4 141
It is clear that the dependence presented in Fig. 3, corresponding to classifica-
tion, given at Fig. 2, is a right-hand asymmetric ravine function. Errors in solving in-
terpolation and self-connected interpolation-extrapolation tasks for this example are
presented in Table 1. Corresponded polynomial coefficients are given in Table 2.
It is clear that the dependence presented in Fig. 3, corresponding to classifica-
tion, given at Fig. 2, is a right-hand asymmetric ravine function. Errors in solving in-
terpolation and self-connected interpolation-extrapolation tasks for this example are
presented in Table 1. Corresponded polynomial coefficients are given in Table 2.
T a b l e 1 . Errors of estimation for Task 1
Estimation methods
Interpolation
Function Order, n
Extrapolation
Function Order, n
Orders nh of Higher-
Order Method
Type of
Error
4 5 6 4 5 6 5 6
Ladd,
m / %
εmax, % 0.51 0.865 0.118 1.355 2.26 0.76 3.2 0.26
εav, % 0.1586 0.377 3·10–2 0.19 0.338 6·10–2 0.926 0.11
εF, % 3.73·10–2 9.3·10–3 1.87·10–2 0 0 0 0.129 0
εrf, % 1.45·10–4 0.1536 4.07·10–5 4.33·10–12 6.8·10–12 7.12·10–10 0.014 5·10–2
3.17·10–3
/ 5.6
εmax, % 0.98 1.434 0.281 1.355 2.26 0.76 3.35 0.3
εav, % 0.275 0.595 6.6·10–2 0.19 0.338 0.059 0.96 0.16
εF, % 0.1367 2·10–2 7.1·10–2 0 0 0 0.82 0
εrf, % 2·10–3 0.25 5.3·10–4 4.33·10–12 0.068 7.12·10–10 0.069 0.0533
3.82·10–3 /
14.388
εmax, % 1.615 2.263 0.537 2.65 4.45 1.72 4.616 0.843
εav, % 0.47 0.9 0.1376 0.323 0.55 0.147 1.4 0.31
εF, % 0.294 6·10–2 0.17445 0 0 0 0.36 0
εrf, % 8.8·10–3 0.394 3.4·10–3 1.32·10–11 0.0544 1.25·10–9 0.0136 0.2478
1.3175·
10–2 /
23.187
εmax, % 2.4 3.447 0.89 4.13 6.92 2.9 2.7845 0.563
εav, % 0.7465 1.33 0.256 0.53 0.9 0.2925 0.908 0.24
εF, % 0.5257 0.1577 0.345 5.84·10–3 0 0 1.0 0
εrf, % 2.8·10–2 0.6 1.39·10–2 3.89·10–7 0.0454 3·10–11 0.1147 0.0563
1.8175·
10–2 / 32
εmax, % 1.219 5.11 1.32 5.8 9.6 4.367 3.13 0.977
εav, % 3.385 1.87 0.432 0.8166 1.379 0.51725 1.1168 0.37
εF, % 3.078 0.6541 0.6168 0 0 0 1.05 0
εrf, % 1.4 0.8752 4.67·10–2 9.76·10–12 0.0389 4.28·10–12 0.13 0.06
2.1375·
10–2 / 40.8
T a b l e 2 . Coefficients of root-polynomial function (2) for Task 1, zend = 0.16 m
Coefficients of root-polynomial function (2) Estimation
methods
n
C6 C5 C4 C3 C2 C1 C0
4 – – 3.14·10–3 -1.6·10–3 3.13·10–4 -2.72·10–5 8.96·10–7
5 – 4.944·10–5 9.97·10–6 -1.33·10–5 3.1·10–6 -2.6·10–7 9.9·10–9
Lower-order
interpolation
6 2.2·10–4 -1.7·10–4 5.487·10–5 9.45·10–6 9.184·10–7 -4.778·10–8 1.04·10–9
4 – – 3.06·10–3 -1.57·10–3 3.05·10–4 -2.66·10–5 8.774·10–7
5 – -8.96·10–7 3.98·10–5 -2.03·10–5 3.9·10–6 -3.37·10–7 1.1·10–8
Interpolation
and
extrapolation 6 2.11·10–4 -1.626·10–4 5.239·10–5 -9.036·10–6 8.798·10–7 -4.586·10–8 10–9
5 – 4.84·10–4 2.744·10–4 6.074·10–5 -6.474·10–6 -3.245·10–7 -5.8·10–9 Higher-order
interpolation 6 2.63·10–4 -2.03·10–4 6.53·10–5 -1.222·10–5 1.1·10–6 -5.63·10–8 1.22·10–9
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 142
Results of estimations in graphic form for the point 16.0endz m are pre-
sented at Fig. 4.
In the upper graphs straight line correspond to numerical solving the set
of equation (1) and dash line to estimation of numerical solution in depend-
ences on length of propagation of electron beam. On the lower graphs shown
the error of estimation in dependences on length of propagation of electron
beam (screen copy).
z,m
r,
m
a
z,m
r,
m
b
z,m
r,
m
c
Fig. 4. Lower-order interpolation (a), extrapolation (b) and higher-order interpolation (c)
for the Task 1, 6n
An advanced method of interpolation of short-focus electron beams boundary trajectories …
Системні дослідження та інформаційні технології, 2024, № 4 143
Dependence of error of estimation for extrapolation and higher-order inter-
polation tasks on the length of extrapolation region Ladd for different orders of
root-polynomial function presented at Fig. 5.
Task 2. 15acU kV; 5.5bI A, 5.4p Pa, 5.8srartr mm, 05.10 ,
1.0startz m. End points: 155.0endz m, 153.0endz m, 15.0endz m,
147.0endz m, and 145.0endz m.
For this example, the dependence )(zr , defined by numerical solving the set
of equations (1), is presented in Fig 6.
It is clear that the de-
pendence presented in Fig. 6,
corresponding to classifica-
tion, given at Fig. 2, is a left-
hand asymmetric ravine func-
tion. The position of boundary
point for left-hand asymmetric
are always different. Corre-
sponded values for this task
are presented at Table 3. Er-
rors in solving interpolation
and self-connected interpola-
tion-extrapolation tasks for
this example are presented in
Table 4. Obtained polynomial
coefficients for different esti-
mation methods are given in Table 5.
T a b l e 3 . Position of boundary points zbp in depend on position of end point
endz for left-hand asymmetric ravine function, given in Task 2
zend, m 0.155 0.153 0.15 0.147 0.145
zbp, m 0.102 0.1038 0.1044 0.1074 0.10942
Fіg. 5. Dependences of errors of higher-order interpolation (upper) and extrapolation
(lower) tasks on the relative length of extrapolation region addL and order of root-
polynomial function n for Task 1 (screen copy)
Fig. 6. Dependence r(z) for 15acU kV, 5.5bI A,
5,4P Pa, end point 155.0endz m (screen copy)
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 144
T a b l e 4 . Errors of estimation for Task 2
Estimation methods
Interpolation Function
Order, n
Extrapolation Function
Order, n
Orders nh of Higher-
Order Method
Type
of
Error
4 5 6 4 5 6 5 6
Ladd, m /
%
εmax, % 0.344 0.5224 6.6·10–2 0.2357 0.38 0.0735 1.347 0.145
εav, % 0.1 0.24 1.7·10–2 9.147·10–2 0.1887 0.01455 0.4 5.65·10–2
εF, % 3·10–2 4.3·10–3 4.3·10–3 0 0 0 0.6425 0
εrf, % 8.62·10–5 0.09 3.89·10–6 1.24·10–11 0.0634 1.91·10–9 4.426·10–2 0.0297
1.83·10–3 /
3.44
εmax, % 0.344 0.44 7.54·10–2 0.38143 0.63 0.164 1.03 0.084
εav, % 9.7·10–2 0.202 1.58·10–2 0.0736 0.134 0.0137 0.3 0.0343
εF, % 3·10–2 4.1·10–3 8.26·10–3 0 0 0 0.425 0
εrf, % 8.62·10–2 7.07·10–2 1.05·10–5 1.34·10–11 0.0336 1.14·10–9 1.94·10–2 1.5·10–2
3.8·10–3
/ 7.79
εmax, % 0.3725 0.3685 8.97·10–2 0.66 1.17 0.3 0.6386 0.034
εav, % 0.096 0.1628 1.69·10–2 0.069 0.12 0.019 0.172 0.0185
εF, % 4.3·10–2 3.893·10–3 1.16·10–2 0 0 0 0.214 0
εrf, % 1.79·10–4 4.87·10–2 1.3·10–2 1.3·10–11 0.012 3.44·10–10 5·10–3 5.4·10–3
4.415·10–3
/ 9.685
εmax, % 0.3224 0.5224 6.6·10–2 0.2357 0.38 7.35·10–2 1.347 0.145
εav, % 0.1054 0.24 1.7·10–2 9.15·10–2 0.1887 1.45·10–2 0.4 5.65·10–2
εF, % 0.017 4.3·10–3 4.3·10–3 0 0 0 0.6425 0
εrf, % 2.52·10–5 8.9·10–2 3.89·10–2 1.24·10–11 6.34·10–2 1.91·10–9 4.4·10–2 2.97·10–2
7.422·10–3
/ 18.75
εmax, % 0.4 0.398 0.112361 1.107 2.073 0.58 0.22 7.66·10–2
εav, % 0.11 0.129 2.2·10–2 0.11 0.2065 4.81·10–2 0.084 1.66·10–2
εF, % 0.042 3.5·10–3 1.05·10–2 0 0 0 0.06 0
εrf, % 1.95·10–4 0.024 1.31·10–5 9.1·10–12 1.54·10–3 9.76·10–10 4.08·10–4 7.3·10–4
9.42·10–2 /
26.5
T a b l e 5 . Coefficients of root-polynomial function (2) for Task 2, zend = 0.15 m
Coefficients of root-polynomial function (2) Estimation
methods
n
C6 C5 C4 C3 C2 C1 C0
4 – – 2.95·10–3 -1.512·10–3 2.956·10–4 -2.582·10–5 8.538·10–7
5 – -9.963·10–5 9.966·10–5 -3.4734·10–5 5.6475·10–6 -4.408·10–7 1.34·10–8 Lower-order
interpolation
6 2·10–4 -1.549·10–4 5·10–5 -8.64·10–6 8.43·10–7 -4.4046·10–8 9.628·10–10
4 – – 2.8·10–3 -1.435·10–3 2.792·10–4 -2.43·10–5 8.06·10–7
5 – -8.5335·10–7 3.25·10–5 -1.655·10–5 3.198·10–6 -2.766·10–7 9.032·10–9
Interpolation
and
extrapolation 6 1.84·10–4 -1.417·10–4 4.57·10–5 -7.9·10–6 7.7·10–7 -4.03·10–8 8.821·10–10
5 – 4.03·10–4 2.9·10–4 -8.2·10–5 1.15·10–5 -8.018·10–7 2.227·10–8 Higher-order
interpolation 6 2.22·10–4 -1.71·10–4 5.51·10–5 -9.495·10–6 9.235·10–7 -4.81·10–8 1.046·10–9
Results of estimations in graphic form for the point 15.0endz m are pre-
sented at Fig. 7. Dependence of error of estimation for extrapolation and higher-
order interpolation tasks on the length of extrapolation region addL for different
orders of root-polynomial function presented at Fig. 8.
An advanced method of interpolation of short-focus electron beams boundary trajectories …
Системні дослідження та інформаційні технології, 2024, № 4 145
a
z,m
r,
m
z,m
r,
m
b
z,m
r,
m
c
Fig. 7. Lower-order interpolation (a), extrapolation (b) and higher-order interpolation (c)
for the Task 2, n = 6. In the upper graphs straight line correspond to numerical solving
the set of equation (1) and dash line to estimation of numerical solution in dependences
on length of propagation of electron beam. On the lower graphs shown the error of esti-
mation in dependences on length of propagation of electron beam (screen copy)
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 146
PARTICULARITIES OF ELABORATED COMPUTER SOFTWARE
All simulation results presented in this paper have been obtained using original
software, which has been elaborated for simulation and numerical estimation of
the boundary trajectory of an electron beam propagated in ionized gas. The source
program code has been written using the means of programing language Python,
including advanced mathematic and graphic libraries such as tkinter, nympy, and
matplotlib [59; 60]. The distinguishing feature of elaborated software from the
point of view of the means of programming is including additional advanced li-
braries for creating scientific plots from module matplotlib into traditional ele-
ments of the interface window created using the function of module tkinter
[59; 60]. For the correct solution of this sophisticated programming task, specific
system tools have been used, including the definition of virtual variables and cre-
ating on its base the virtual environment for forming a virtual disk in the operative
memory of a local computer [59; 60]. Corresponding graphic interface windows
of elaborated software for the bookmarks “Interpolation” and “Extrapolation” are
presented in Fig. 9. For saving and further analyzing the obtained graphic infor-
mation, the bottom “Save Graph” has been provided in both interface windows.
For automatic creation of root-polynomial functions on both bookmarks, the
bottoms “Import from SDE Task” have been provided. Using this program’s
functionality is possible only after solving the simulation task for the established
electron beam parameters in the corresponded bookmark “Solving of Differential
Equation of Beam Boundary Trajectory”. But the manual creation of the root-
polynomial function by the r and z coordinates, which have to be input in the corre-
sponded textboxes, is also possible by pressing the bottom “Calculate Manually”.
Errors of estimation, presented in Tables 1 and 4, as well as coefficients of
root-polynomial functions, presented in Tables 2 and 5, are written out in the es-
tablished output text windows on the corresponded bookmarks. All described
elements of the graphic user interface are shown in the copy of these bookmarks,
presented in Fig. 9.
Fig. 8. Dependences of errors of higher-order interpolation (upper) and extrapolation
(lower) tasks on the relative length of extrapolation region Ladd and order of root-
polynomial function n for Task 1 (screen copy)
An advanced method of interpolation of short-focus electron beams boundary trajectories …
Системні дослідження та інформаційні технології, 2024, № 4 147
ANALYSIS OF OBTINED RESULTS AND DISCUSSION
The computer simulation results described in this paper showed that higher-order
interpolation for asymmetric ravine functions gives an average error value. No
minimum error value was detected for this novel estimation method. In general,
from a theoretical point of view, this is due to the location of the reference points
for root-polynomial functions of the appropriate order. Indeed, the kf values de-
termined by relations (5), (6) were chosen correctly only for the corresponded
lower-order of the odd or even root polynomial function (2).
For example, for higher-order interpolation with order of function nh = 5, the
basic points are located as for forth order symmetric function, and additional point,
located at the start of interpolated interval for left-hand asymmetric function or on the
end of this interval for right-hand asymmetric function, is artificially added.
Generally, corresponding to Tables 1 and 4, minimal values of maximal and
average interpolation error are corresponded to standard low-order interpolation,
a
b
Fig. 9. Interface windows for bookmarking “Interpolation” (a) and “Extrapolation” (b) in
elaborated computer software (screen copy)
I. Melnyk, A. Pochynok, M. Skrypka
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 148
but self-connected interpolation-extrapolation task usually given the minimal er-
rors in estimation of focal parameters of electron beam. The same conclusion are
follows from graphic dependences, presented at Fig. 4 and Fig. 7.
But, in any case, average integral error of estimation the beam trajectory by
the higher-order root-polynomial function in the whole segment of interpolation
isn’t so large, therefore such estimation can be preferable in some solutions for
practice application. For simplifying the further corresponded analysis in the
digital presentation all estimation errors for the end point 15.0endz are
rewritten from extended Table 4 to smaller Table 6.
T a b l e 6 . Errors of estimation for Task 2 for end point zend = 0.15 m
Methods and
function order
Standard Interpolation Interpolation and
Extrapolation
Higher-Order
Interpolation
N 4 5 6 4 5 6 5 6
εmax, % 0.3725 0.3685 8.97·10–2 0.66 1.17 0.3 0.6386 0.034
εav, % 0.096 0.1628 1.69·10–2 0.069 0.12 0.019 0.172 0.0185
εF, % 4.3·10–2 3.893·10–3 1.16·10–2 0 0 0 0.214 0
Errors
εrf, % 1.79·10–4 4.87·10–2 1.3·10–5 1.3·10–11 0.012 3.44·10–10 5·10–3 5.4·10–3
From the calculation results, presented in Table 6, it is clear, that for higher-
order interpolation the for n = 6 average error ( 0185.0av %) isn’t so small,
than for standard interpolation by the function of same order ( 0169.0av %),
but the difference of these errors isn’t so large. Also, and it is very significant and
important that the estimation using higher-order interpolation for 6n gives the
minimal value of the maximal error, 034.0max %.
It is clear also from numerical data, presented in Table 6, that the best results
for estimation of focal radius of electron beam giving the method of interpolation
and extrapolation by forth and six order functions, the level of error rf is range
of from 10–11 % to 10–10 %. But such precision estimation of focal beam
parameters usually isn’t necessary for the practical applications. Estimation using
higher-order interpolation method give the value of error 3104.5 rf %, which,
certainly, isn’t so small, but usually is suitable for the most of practical applications
[16]. It is also interesting and important, that for self-connected interpolation and
extrapolation method the error of estimation focus position is 0F %, but the same
result is observed for higher-order interpolation function in the case of 6n .
As it is clear from Tables 1 and 4, the particularities of the different methods
of interpolation and extrapolation described above are similar for all positions of
the end point, including left-hand and right-hand ravine functions. But, in any
case, the error in the estimation of electron beam boundary trajectories by using
the root-polynomial function (2) is very small, in the range of a fraction of a per-
cent. This result is confirming the pervious preliminary theoretical estimations,
have been provided in the works [47–49].
All research work 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
Generally, provided research has shown that usually the minimal average error εav
of estimation of the boundary electron beam trajectory using the root-polynomial
An advanced method of interpolation of short-focus electron beams boundary trajectories …
Системні дослідження та інформаційні технології, 2024, № 4 149
function (2) corresponds to the lower-order interpolation method. The best orders
of these functions are even values, such as 2n , 4n , and 6n . The best es-
timations of electron beam focal parameters have been obtained using the self-
connected interpolation-extrapolation method. The level of error in the estimation
of the focal beam radius εrf for this method has been significantly small, ranging
from 10–11 % to 10–10 %, and the estimation by the focus position has been exactly
precise without error. The best results for this method also give the even values of
the order of the root-polynomial function, such as 4n and 6n . It can be gen-
erally explained by the suitable choice of base points position for the symmetric
part of the ravine function, which is evaluated. The proposed method of higher-
order root-polynomial interpolation gives an average value of error both in the focal
region and at the start and end basic points. The larger values of the average error in
this case are explained by the location of the basic points. Unfortunately, solving the
optimization task of defining the basic points position in this case is impossible.
All simulation results presented in this paper have been obtained using origi-
nal computer software elaborated and developed by applying the advanced
mathematical and graphic means of the Python programming language.
Obtained scientific results and practical recommendations can be interesting
to a wide range of experts in the fields of the physics of electron beams and ad-
vanced electron beam technologies, as well as in the computational mathematics
and methods of interpolation and extrapolation of ravine functions.
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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: imel-
nik@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: scien-
tetik@gmail.com
УДОСКОНАЛЕНИЙ МЕТОД ІНТЕРПОЛЯЦІЇ ГРАНИЧНИХ ТРАЄКТОРІЙ
КОРОТКОФОКУСНИХ ЕЛЕКТРОННИХ ПУЧКІВ ЗА ДОПОМОГОЮ
КОРЕНЕВИХ ПОЛІНОМІАЛЬНИХ ФУНКЦІЙ ВИЩОГО ПОРЯДКУ ТА
ЙОГО ПОРІВНЯЛЬНЕ ДОСЛІДЖЕННЯ / І.В. Мельник, А.В. Починок,
М.Ю. Скрипка
Анотація. Розглянуто та обговорено узагальнене порівняння трьох сучасних,
нових методів оцінювання граничної траєкторії електронних пучків, що поши-
рюються в іонізованому газі, включаючи інтерполяцію нижчого порядку, са-
моузгоджену інтерполяцію та екстраполяцію, а також інтерполяцію вищого
порядку. Усі оцінки відповідних похибок були проведені відносно числового
розв’язування системи алгебра-диференціальних рівнянь, що описують грани-
чну траєкторію електронного пучка. Через виконаний аналіз показано та дове-
дено, що інтерполяція нижчого порядку зазвичай дає мінімальне значення се-
редньої похибки, використання методу самоузгодженої інтерполяції та
екстраполяції дає мінімальну похибку щодо оцінки фокальних параметрів еле-
ктронного променя, а інтерполяція вищого порядку може бути використана
для отримання рівномірного значення похибки на всьому інтервалі інтерполя-
ції. Усі результати оцінювання похибок отримано з використанням оригіналь-
ного комп’ютерного програмного забезпечення, створеного засобами мови
програмування Python.
Ключові слова: інтерполяція, екстраполяція, інтерполяція нижчого порядку,
інтерполяція вищого порядку, коренево-поліноміальна функція, яружна функ-
ція, середня похибка, електронний пучок, гранична траєкторія, високовольт-
ний тліючий розряд, електронно-променеві технології.
|
| id | journaliasakpiua-article-322534 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:41Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/82/c0620ad1074a89148b04b461162a6782.pdf |
| spelling | journaliasakpiua-article-3225342025-02-09T21:55:38Z An advanced method of interpolation of short-focus electron beams boundary trajectories using higher-order root-polynomial functions and its comparative study Удосконалений метод інтерполяції граничних траєкторій короткофокусних електронних пучків за допомогою кореневих поліноміальних функцій вищого порядку та його порівняльне дослідження Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo інтерполяція екстраполяція інтерполяція нижчого порядку інтерполяція вищого порядку коренево-поліноміальна функція яружна функція середня похибка електронний пучок гранична траєкторія високовольтний тліючий розряд електронно-променеві технології interpolation extrapolation lower-order interpolation higher-order interpolation root-polynomial function ravine function average error electron beam boundary trajectory high voltage glow discharge electron beam technologies The comparison of three advanced novel methods for estimating the boundary trajectory of electron beams propagated in ionized gas, including lower-order interpolation, self-connected interpolation, and extrapolation, as well as higher-order interpolation, is considered and discussed in the article. All estimations of the corresponding errors have been provided relative to numerically solving the set of algebra-differential equations that describe the boundary trajectory of the electron beam. By providing analysis, it is shown and proven that lower-order interpolation usually gives the minimal value of average error, using the method of self-connected interpolation and extrapolation gives the minimal error for estimation of focal beam parameters, and higher-order interpolation is suitable to obtain a uniform error value over the entire interpolation interval. All results of error estimation were obtained using original computer software written in Python. Розглянуто та обговорено узагальнене порівняння трьох сучасних, нових методів оцінювання граничної траєкторії електронних пучків, що поширюються в іонізованому газі, включаючи інтерполяцію нижчого порядку, самоузгоджену інтерполяцію та екстраполяцію, а також інтерполяцію вищого порядку. Усі оцінки відповідних похибок були проведені відносно числового розв’язування системи алгебра-диференціальних рівнянь, що описують граничну траєкторію електронного пучка. Через виконаний аналіз показано та доведено, що інтерполяція нижчого порядку зазвичай дає мінімальне значення середньої похибки, використання методу самоузгодженої інтерполяції та екстраполяції дає мінімальну похибку щодо оцінки фокальних параметрів електронного променя, а інтерполяція вищого порядку може бути використана для отримання рівномірного значення похибки на всьому інтервалі інтерполяції. Усі результати оцінювання похибок отримано з використанням оригінального комп’ютерного програмного забезпечення, створеного засобами мови програмування Python. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-12-25 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/322534 10.20535/SRIT.2308-8893.2024.4.11 System research and information technologies; No. 4 (2024); 133-153 Системные исследования и информационные технологии; № 4 (2024); 133-153 Системні дослідження та інформаційні технології; № 4 (2024); 133-153 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/322534/312912 |
| spellingShingle | інтерполяція екстраполяція інтерполяція нижчого порядку інтерполяція вищого порядку коренево-поліноміальна функція яружна функція середня похибка електронний пучок гранична траєкторія високовольтний тліючий розряд електронно-променеві технології Melnyk, Igor Pochynok, Alina Skrypka, Mykhailo Удосконалений метод інтерполяції граничних траєкторій короткофокусних електронних пучків за допомогою кореневих поліноміальних функцій вищого порядку та його порівняльне дослідження |
| title | Удосконалений метод інтерполяції граничних траєкторій короткофокусних електронних пучків за допомогою кореневих поліноміальних функцій вищого порядку та його порівняльне дослідження |
| title_alt | An advanced method of interpolation of short-focus electron beams boundary trajectories using higher-order root-polynomial functions and its comparative study |
| title_full | Удосконалений метод інтерполяції граничних траєкторій короткофокусних електронних пучків за допомогою кореневих поліноміальних функцій вищого порядку та його порівняльне дослідження |
| title_fullStr | Удосконалений метод інтерполяції граничних траєкторій короткофокусних електронних пучків за допомогою кореневих поліноміальних функцій вищого порядку та його порівняльне дослідження |
| title_full_unstemmed | Удосконалений метод інтерполяції граничних траєкторій короткофокусних електронних пучків за допомогою кореневих поліноміальних функцій вищого порядку та його порівняльне дослідження |
| title_short | Удосконалений метод інтерполяції граничних траєкторій короткофокусних електронних пучків за допомогою кореневих поліноміальних функцій вищого порядку та його порівняльне дослідження |
| title_sort | удосконалений метод інтерполяції граничних траєкторій короткофокусних електронних пучків за допомогою кореневих поліноміальних функцій вищого порядку та його порівняльне дослідження |
| topic | інтерполяція екстраполяція інтерполяція нижчого порядку інтерполяція вищого порядку коренево-поліноміальна функція яружна функція середня похибка електронний пучок гранична траєкторія високовольтний тліючий розряд електронно-променеві технології |
| topic_facet | інтерполяція екстраполяція інтерполяція нижчого порядку інтерполяція вищого порядку коренево-поліноміальна функція яружна функція середня похибка електронний пучок гранична траєкторія високовольтний тліючий розряд електронно-променеві технології interpolation extrapolation lower-order interpolation higher-order interpolation root-polynomial function ravine function average error electron beam boundary trajectory high voltage glow discharge electron beam technologies |
| url | https://journal.iasa.kpi.ua/article/view/322534 |
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