Fast computation of the exact plasma dispersion functions
The paper is concerned with a computation of the exact relativistic plasma dispersion functions for complex argument z = x + iy in the region Q : y ≥ 0 on the base the theory of the continued fractions of Jacobi. It is first observed that these fractions represent those functions asymptotically...
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| Zitieren: | Fast computation of the exact plasma dispersion functions / S.S. Pavlov, F. Castejón, Á. Cappa, M. Tereshchenko // Вопросы атомной науки и техники. — 2009. — № 1. — С. 69-71. — Бібліогр.: 7 назв. — англ. |
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Pavlov, S.S. Castejón, F. Cappa, Á. Tereshchenko, M. 2015-11-10T20:59:57Z 2015-11-10T20:59:57Z 2009 Fast computation of the exact plasma dispersion functions / S.S. Pavlov, F. Castejón, Á. Cappa, M. Tereshchenko // Вопросы атомной науки и техники. — 2009. — № 1. — С. 69-71. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 52.27.Ny https://nasplib.isofts.kiev.ua/handle/123456789/88226 The paper is concerned with a computation of the exact relativistic plasma dispersion functions for complex argument z = x + iy in the region Q : y ≥ 0 on the base the theory of the continued fractions of Jacobi. It is first observed that these fractions represent those functions asymptotically for z → ∞ in the sector Q. На основі теорії ланцюгових дробів розвивається метод швидкого обчислення точних релятивістських плазмових дисперсійних функцій у комплексній області. На основе теории цепных дробей развивается метод быстрого вычисления точных релятивистских плазменных дисперсионных функций в комплексной области. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Фундаментальная физика плазмы Fast computation of the exact plasma dispersion functions Швидке обчислення точних плазмових дісперсійних функцій Быстрое вычисление точных плазменных дисперсионных функций Article published earlier |
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Fast computation of the exact plasma dispersion functions |
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Fast computation of the exact plasma dispersion functions Pavlov, S.S. Castejón, F. Cappa, Á. Tereshchenko, M. Фундаментальная физика плазмы |
| title_short |
Fast computation of the exact plasma dispersion functions |
| title_full |
Fast computation of the exact plasma dispersion functions |
| title_fullStr |
Fast computation of the exact plasma dispersion functions |
| title_full_unstemmed |
Fast computation of the exact plasma dispersion functions |
| title_sort |
fast computation of the exact plasma dispersion functions |
| author |
Pavlov, S.S. Castejón, F. Cappa, Á. Tereshchenko, M. |
| author_facet |
Pavlov, S.S. Castejón, F. Cappa, Á. Tereshchenko, M. |
| topic |
Фундаментальная физика плазмы |
| topic_facet |
Фундаментальная физика плазмы |
| publishDate |
2009 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Швидке обчислення точних плазмових дісперсійних функцій Быстрое вычисление точных плазменных дисперсионных функций |
| description |
The paper is concerned with a computation of the exact relativistic plasma dispersion functions for complex argument
z = x + iy in the region Q : y ≥ 0 on the base the theory of the continued fractions of Jacobi. It is first observed that
these fractions represent those functions asymptotically for z → ∞ in the sector Q.
На основі теорії ланцюгових дробів розвивається метод швидкого обчислення точних релятивістських
плазмових дисперсійних функцій у комплексній області.
На основе теории цепных дробей развивается метод быстрого вычисления точных релятивистских
плазменных дисперсионных функций в комплексной области.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/88226 |
| citation_txt |
Fast computation of the exact plasma dispersion functions / S.S. Pavlov, F. Castejón, Á. Cappa, M. Tereshchenko // Вопросы атомной науки и техники. — 2009. — № 1. — С. 69-71. — Бібліогр.: 7 назв. — англ. |
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| first_indexed |
2025-11-26T02:45:31Z |
| last_indexed |
2025-11-26T02:45:31Z |
| _version_ |
1850609231483371520 |
| fulltext |
FAST COMPUTATION OF THE EXACT
PLASMA DISPERSION FUNCTIONS
S.S. Pavlov1, F. Castejón2, Á. Cappa2, M. Tereshchenko3
1 Institute of Plasma Physics,
NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
2 Asociación EURATOM-CIEMAT PARA Fusión, Madrid, Spain;
3 General Physics Institute, Russian Academy of Sciences, Moscow, Russia
The paper is concerned with a computation of the exact relativistic plasma dispersion functions for complex argument
iyxz += in the region Q : on the base the theory of the continued fractions of Jacobi. It is first observed that
these fractions represent those functions asymptotically for
0≥y
∞→z in the sectorQ .
PACS: 52.27.Ny
1. INTRODUCTION
An evaluation of plasma dielectric tensor is the basic to
study theoretically phenomena connected with linear
electromagnetic waves in magnetized plasma. In order to
reduce such the evaluation to the evaluation of plasma
dispersion functions (PDFs) one can write this tensor as a
finite Larmor radius expansion in terms of these PDFs. In
many numerical applications those PDFs must be computed
a large number of times. It is therefore important to search
for computation methods which are as fast as possible.
One of such fast methods was developed for calculation
of a special function ×−= )exp()( 2zzW ∫+
z
i dtt
0
22 ))exp(1(
π
,
called Kramp or the complex error function and relating to
the non-relativistic PDF as )()( zWizZ π= , on the base of
the theory of continued fractions [1,2]. The continued
fractions together with their generalization, Padé
approximants, are of great interest in many fields of pure and
applied mathematics and in numerical analysis, where they
are closely connected with the convergence acceleration
techniques and provide, in particulary, efficient computer
approximations to special functions [3]. This method is also
proved suitable for the fast computation of the weakly
relativistic PDFs (Shkarofsky functions) since they are
satisfied up to the second order recursion relation and the
two PDFs, starting the recursive procedure, are expressed
through the non-relativistic PDF.
The exact PDFs, introduced in order to give a recipe
to evaluate the fully relativistic plasma dielectric tensor of
Trubnikov [4] for arbitrary plasma and wave parameters,
were computed on the base of the theory of Cauchy-type
integrals from complex analysis [5,6]. This method, being
exact and general, is not yet as fast as the method [1,2].
The main scope of the present work is generalization of
the last method for the case of the exact PDFs. We review
some analytical properties of these PDFs and relevant
peculiarities of this continued fraction technique and
develop the faster algorithm computing these functions.
The main performance characteristics and data on
algorithm testing are given as well.
2. MATHEMATICAL PRELIMINARIES
The main idea of the works [1, 2], devoted to the fast
computation of the Kramp function ×−= )exp()( 2zzW
)( izerfc − , is utilizing the continued J-fractions instead of
series to accelerate its convergence in the regions, where a
module of the function is monotonous. For instance,
though in the region, where the condition is true,
one can use, in principle, the asymptotic series
z| |>>1
1
0
( ) ~ k
k
k
iW z
z
μ
π
∞
+
=
∑ , 0, 2 1
1 , 2
2
k
k n
k k n
μ
= −⎧
⎪= +⎨ ⎛ ⎞Γ =⎜ ⎟⎪ ⎝ ⎠⎩
, ,...2,1=n (1)
to evaluate the Kramp function, however this series is
divergent and that makes difficult such a usage. Really,
for accuracy it is necessary to control constantly the
number of series terms and, moreover, for not very large
in module z values this number can be proved huge. A
usage of the continued J-fraction on the base equality
1
0
1 2 1 3 2 ... 1 2
1
3 2
...
k
k
k z z z z z z
z
z
z
μ π π∞
+
=
| | | |
= + + + + =
− | − | − | − | −
−
−
−
∑ , (2)
instead the series (1) removes these difficulties, as it was
shown by Gautschi [1]. In consequence of the properties of the
continued fraction in (2), for evaluation of it is enough
only some first terms even for not very large in module
)(zW
z
values. He has shown also that in the case of moderate and
small values one can use the same approach. At that only
coefficients of Teylor series, expanded along positive imagine
axis into inverse direction in early calculated large in module
points, and coefficients of relating continued fraction will
change.
z
z
The exact relativistic PDFs [5,6] are a natural generalization
of non-relativistic and weakly relativistic PDFs to the case of
arbitrary temperature and, as a consequence, their analytical
properties are rather close, though more sophisticated for
higher temperatures. In the case these PDFs have the
next asymptotic expansions:
z| |>>1
∑
∞
=
++
0
123 ~),,(
k
k
q
k
q z
A
zaZ μ , (3)
where , is longitudinal refractive index,
, is thermal speedy of electrons with the rest
mass,
2/2
//Na μ= //N
2)/( TVc=μ TV
)()( 20 μμ KKA q
q = , is Macdonald function of )(xK q
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2009. № 1. 69
Series: Plasma Physics (15), p. 69-71.
order , and reminder coefficients can
be obtained from recursive relation of the second order
[5]. Obviously, this series is a generalization of the series
(1) to the case
q 1
001
+−= qqq AAA q
kA
0>kμ for ( ) and,
consequently, one can hope to use continued fractions
instead of the expression (3) for more fast computation of
functions
12 −= nk ,...2,1=n
),,(23 μzaZ q+
. For this aim it is only necessary to
transform the series of type (3) into continued J-fraction,
i.e. to define in the equality
( )
2 22
0 1
1
0 0 1
1 ... ...k k k
k
k k
m a aa
z z b z b z b
∞
+
=
| ||
− = − − − −
− − | − − | − − |∑ , (4)
the connection between coefficients and coefficients ,
. Using the method of mathematical induction one can
prove the next formulas
km ka
kb
10 1 2
0 1 0 12 2 2 1 2 2 2
0 0 0 1 0 0 1 2 1 2 3
1 2 1 2
2 3 4
, , ,...
m m m
m m m m
a m a a m a a a m m m
m m m m
m m m
−
−= = =
1
0 1 0 11
0 0 1
2 3 1 20
, ,
m m m mmb b b
m m m mm
−
= − + = −
(5)
1
0 1 2 0 1 2
0 1 2 1 2 3 1 2 3
3 4 5 2 3 4
, ...
m m m m m m
b b b m m m m m m
m m m m m m
−
+ + = −
where were used so named determinants of Hankel. These
formulas can be reduced by the change to the formulas
of Heilermann for connection coefficients of Teylor expansion
near zero with coefficients of related continued J-fraction [7].
Now on the base (4), (5) one can obtain the equality (2). Thus,
we can evaluate the functions
xz /1=
),,(23 μzaZ q+
for the case
on the base continued J-fractions also using (4), (5).
For reminder
z| |>>1
z values one can use the technique of Cauchy-
type integrals [6] or the way of Gautschi for small and
moderate in module z values.
3. COMPUTATIONAL PROCEDURE,
PERFORMANCE CHARACTERISTICS
AND TESTING DATA
Our objective is to devise an efficient procedure for
computing the main branch of the function ),,(25 μzaZ to
a given number of correct decimal digits after the
decimal point, i.e., to within an (absolute) error of
. We shall assume
d
d−×105.0 iyxz += to lie in the region
: of the complex plane. This is no restriction of
generality, since the formulas [6]
Q 0≥y
⎪⎩
⎪
⎨
⎧
<>
<<−−
= ∗+
∗+
+
,0,Re),,(
0,Re),,,(2),,(
),,(
yazzaZ
yazzaaifzaZ
zaZ
rq
rrq
q μ
μπμ
μ
1
1 2
( 1) 2
2
1 2
1
22 ( )
0
( , , )
2 ,
2
0, 0.
q
q
t
q
e tt
K a
t
f a t tI a t e
t
μ β
β
πβ
β
μμ μ
μ
β β
μ
−
−
−
−
− −
−
⎧ ⎛ ⎞⎛ ⎞⎪ ⎜ ⎟+ ×⎜ ⎟⎜ ⎟⎪ ⎝ ⎠⎝ ⎠⎪ >⎪= ⎨ ⎛ ⎞⎛ ⎞⎪ ⎜ ⎟+⎜ ⎟⎜ ⎟⎪ ⎝ ⎠⎝ ⎠⎪
≤⎪⎩
can be used to continue ),,(25 μzaZ into the remaining region.
12
10
8,0
6,0
10
4,0
12
2,0
-6 -4 -2 0 2 4 6
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
Y
X
Fig.1 Altitude map of the function ),,(25 μzaZ (Cauchy)-
),,(25 μzaZ (chain fraction, n=9) for 1.1 and
40keV in ion plasma
=//N
=iT
14
12
10
8,0
6,0
4,0
2,0
-6 -4 -2 0 2 4 6
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
Y
A
xi
s
Ti
tle
X Axis Title
Fig. 2 The same as in Fig. 1 excepting n=29
In computational procedure we will use the
asymptotic expansion (3) and the continued J-fraction (4)
for the case z| |>>1 and the exact technique of Cauchy-
type integrals [6] for reminder z values. First let us plot
the line of transition from the method of Cauchy-type
integrals to method of continued fraction to a different
prescribed accuracy for the function ),,(25 μzaZ for two
values of the length of continued fraction (Fig. 1)
and
9=n
29=n (Fig. 2) for longitudinal refractive index
=1.1 and ion temperature =40keV of the ion plasma.
//N iT
This case is very close to the non-relativistic one. From
these pictures it follows, that continued fraction represents
this function asymptotically in essentially wider region
for the case 29=n and for ∞→z in the regionQ :
than for the case
0≥y
9=n and this fact can be used by
different ways (for example, by Gauschi way) for
computation ),,(25 μzaZ for small and moderate in
module values. z
The Fig. 3 presents the function ),,(25 μzaZ for the
relativistic case with longitudinal refractive index =0.6
and
//N
9=n and temperature =40keV of the electron
eT
70
plasma. We can conclude that in the relativistic case
continued fraction as well represents the function
),,(25 μzaZ asymptotically in the same region Q :
for
0≥y
∞→z .
1. If the function can be represented by some
Cauchy-type integral with its continues real density,
defined on the real axis and tending to 0 in the infinity,
then asymptotic series of this function approximates
uniformly this function. in the vicinity of the point
)(zf
∞=z in the sector where (0)( →zf ∞→z along the
ray).
10
8,0
6,0
4,0
8,0
2,0
10
-15 -10 -5 0 5 10
71
15
0
2
4
6
8
10
2. The continued J-fraction, corresponding to this
asymptotic series, extends essentially the present vicinity.
3. The present vicinity is proved to be so wide that the
continued J-fraction describes not only the hermitian part
of but the finite anti-hermitian one as well. This fact
allows one to revivify outside of the vicinity by
different ways.
)(zf
)(zf
Y
X
4. Utilizing of chain fractions for given class of functions
allows one to revivify the whole analytical function on the
base only the asymptotic expansion.
REFERENCES
Fig.3 Altitude map of the function ),,(25 μzaZ (Cauchy)-
),,(25 μzaZ (chain fraction, n=9), for 0.6 and
40keV in electron plasma
=//N
=eT
1. W. Gautschi. Efficient computation of the complex
error function // SIAM J. Numer. Anal. 1970, v. 7, p.187.
2. G.P.M. Poppe, C.M.J. Wijers. More Efficient
Computation of the Complex Error Function // ACM
Transactions on Mathematical Software. 1990, v. 16, N 1,
March, p. 38.
Moreover the calculations show that application of continued
fraction gives acceleration of convergence (and consequently
a gain in computer time) in comparison with an evaluation
on the base of asymptotic expansion approximately with the
factor 2−3. Such an application gives also a gain in time in
comparison with the technique of Cauchy-type integrals
approximately with the factor 100.
3. C. Brezinski. History of continued fractions and Pade
approximants // Springer Series in Computational
Mathematics. 1991, v. 12.
4. B.A. Trubnikov. Plasma Physics and Problems of
Thermonuclear Reactions / ed. M.A. Leontovich. 1959,
v. III, p.122. Thus, the application of continued J-fraction gives
the acceleration in comparison with the technique of
Cauchy-type integrals of the factor 100 and allows
developing essentially faster computational procedure.
5. F. Castejon, S.S. Pavlov. Relativistic plasma
dielectric tensor based on the exact plasma dispersion
functions concept // Phys. Plasmas 2006, v. 13,
p.072105//Phys.Plasmas. 2007, v.14, p.019902 (erratum).
4. CONCLUSIONS
The next conclusions can be drawn from the present
work with the account of results of [1,2].
6. F. Castejon, S.S. Pavlov. The exact plasma dispersion
functions in complex region // Nuclear Fusion. 2008,
v. 48, p. 054003.
7. J.B.H. Heirermann. De transformatione serierum in
fractiones continuas: Dr. Phil. Dissertation, Royal
Academy of Munster, 1845.
Article received 16.10.08
БЫСТРОЕ ВЫЧИСЛЕНИЕ ТОЧНЫХ ПЛАЗМЕННЫХ ДИСПЕРСИОННЫХ ФУНКЦИЙ
C.C. Павлов, F. Castejón, Á. Cappa, М. Терещенко
На основе теории цепных дробей развивается метод быстрого вычисления точных релятивистских
плазменных дисперсионных функций в комплексной области
ШВИДКЕ ОБЧИСЛЕННЯ ТОЧНИХ ПЛАЗМОВИХ ДІСПЕРСІЙНИХ ФУНКЦІЙ
C.C. Павлов, F. Castejón, Á. Cappa, М. Терещенко
На основі теорії ланцюгових дробів розвивається метод швидкого обчислення точних релятивістських
плазмових дисперсійних функцій у комплексній області.
|