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1.
Two basic problems of dynamics, one of which was tackled in the extensive work of Z. Kopal (see e.g. Kopal, 1978, Dynamics of Close Binary Systems, D. Reidel Publication, Dordrecht, Holland.), are presented with their approximate general solutions. The ‘penetration’ into
the space of solution of these non-integrable autonomous and conservative systems is achieved by application of ‘The Last Geometric Theorem of Poincaré’ (Birkhoff, 1913, Am. Math. Soc. (rev. edn. 1966)) and the calculation of sub-sets of ‘solutions précieuses’ that are covering densely the spaces of all solutions
(non-periodic and periodic) of these problems. The treated problems are: 1. The two-dimensional Duffing problem, 2. The restricted
problem around the Roche limit. The approximate general solutions are developed by applying known techniques by means of which
all solutions re-entering after one, two, three, etc, revolutions are, first, located and then calculated with precision.
The properties of these general solutions, such as the morphology of their constituent periodic solutions and their stability
for both problems are discussed. Calculations of Poincaré sections verify the presence of chaos, but this does not bear on
the computability of the general solutions of the problems treated. The procedure applied seems efficient and sufficient for
developing approximate general solutions of conservative and autonomous dynamical systems that fulfil the PoincaréBirkhoff
theorems. The same procedure does not apply to the sub-set of unbounded solutions of these problems. 相似文献
2.
I. Stellmacher 《Celestial Mechanics and Dynamical Astronomy》1999,75(3):185-200
The motion of Hyperion is an almost perfect application of second kind and second genius orbit, according to Poincaré’s classification.
In order to construct such an orbit, we suppose that Titan’s motion is an elliptical one and that the observed frequencies
are such that 4n
H−3n
T+3n
ω=0, where n
H, n
T are the mean motions of Hyperion and Titan, n
ω is the rate of rotation of Hyperion’s pericenter. We admit that the observed motion of Hyperion is a
periodic motion
such as
. Then,
.N
H, N
T, k∈ N
+. With that hypothesis we show that Hyperion’s orbit tends to a particular periodic solution among the periodic solutions
of the Keplerian problem, when Titan’s mass tends to zero. The condition of periodicity allows us to construct this orbit
which represents the real motion with a very good approximation.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
3.
This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x
0 ∊ [−2,2],C ∊ [−2,5]) of the (x
0, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x
0 ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order
n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x
0, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D
i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families. 相似文献
4.
The general solution of the Henon–Heiles system is approximated inside a domain of the (x, C) of initial conditions (C is the energy constant). The method applied is that described by Poincaré as ‘the only “crack” permitting penetration into
the non-integrable problems’ and involves calculation of a dense set of families of periodic solutions that covers the solution
space of the problem. In the case of the Henon–Heiles potential we calculated the families of periodic solutions that re-enter
after 1–108 oscillations. The density of the set of such families is defined by a pre-assigned parameter ε (Poincaré parameter),
which ascertains that at least one periodic solution is computed and available within a distance ε from any point of the domain
(x, C) for which the approximate general solution computed. The approximate general solution presented here corresponds to ε =
0.07. The same solution is further improved by “zooming” into four square sub-domain of (x, C), i.e. by computing sufficient number of families that reduce the density parameter to ε = 0.003. Further zooming to reduce
the density parameter, say to ε = 10−6, or even smaller, although easily performable in both areas occupied by stable as well as unstable solutions, was found unnecessary.
The stability of all members of each and all families computed was calculated and presented in this paper for both the large
solution domain and for the sub-domains. The correspondence between areas of the approximate general solution occupied by
stable periodic solutions and Poincaré sections with well-aligned section points and also correspondence between areas occupied
by unstable solutions and Poincaré sections with randomly scattered section points is shown by calculating such sections.
All calculations were performed using the Runge-Kutta (R-K) 8th order direct integration method and the large output received,
consisting of many thousands of families is saved as “Atlas of the General Solution of the Henon–Heiles Problem,” including
their stability and is available at request. It is concluded that approximation of the general solution of this system is
straightforward and that the chaotic character of its Poincaré sections imposes no limitations or difficulties. 相似文献
5.
In this paper, we have investigated that tilted Bianchi Type I cosmological models for stiff perfect fluid under a supplementary
condition A = B
n
between metric potentials, is not possible. The tilted solution is also not possible when we assume A = t
ℓ, B = t
m
, C = t
n
; ℓ, m and n are constants for ε = p. Thus to preserve tilted nature of model, we assume p = γε, 0 ≤ γ ≤ 1 (barotropic equation of state) for the case A = t
ℓ
B = t
m
and C = t
n
. The physical and geometrical aspects of the models are also discussed.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
6.
Poincaré designed the méthode nouvelle in order to build approximate integrals of Hamiltonians developed as series of a small
parameter. Due to several critical deficiencies, however, the method has fallen into disuse in favor of techniques based on
Lie transformations. The paper shows how to repair these shortcomings in order to give Poincaré’s méthode nouvelle the same
functionality as the Lie transformations. This is done notably with two new operations over power series: a skew composition
to expand series whose coefficients are themselves series, and a skew reversion to solve implicit vector equations involving
power series. These operations generalize both Arbogast’s technique and Lagrange’s inversion formula to the fullest extent
possible.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
7.
String cosmological models with bulk viscosity are investigated in Kantowski-Sachs space-time. To obtain a determinate solution,
it is assumed that the coefficient of bulk viscosity is a power function of the scalar of expansion ζ = kθm and the scalar of expansion is proportional to the shear scalar θ ∝ σ, which leads to a relation between metric potentials
R = AS
n
. The physical and geometrical aspects of the model are also discussed. It is shown that the bulk viscosity has significant
influence on the evolution of the universe. There is a ‘big bang’ start in the model when m ≤ 1 but there is no ‘big bang’
start when m > 1. 相似文献
8.
Poincaré surface of section technique is used to study the evolution of a family ‘f’ of simply symmetric retrograde periodic orbits around the smaller primary in the framework of restricted three-body problem for a number of systems, actual and hypothetical, with mass ratio varying from 10−7 to 0.015. It is found that as the mass ratio decreases the region of phase space containing the two separatrices shrinks in size and moves closer to the smaller primary. Also the corresponding value of Jacobi constant tends towards 3. 相似文献
9.
The energy density of Vaidya-Tikekar isentropic superdense star is found to be decreasing away from the center, only if the
parameter K is negative. The most general exact solution for the star is derived for all negative values of K in terms of circular and inverse circular functions. Which can further be expressed in terms of algebraic functions for K = 2-(n/δ)2 < 0 (n being integer andδ = 1,2,3 4). The energy conditions 0 ≤ p ≤ αρc
2, (α = 1 or 1/3) and adiabatic sound speed conditiondp dρ ≤ c
2, when applied at the center and at the boundary, restricted the parameters K and α such that .18 < −K −2287 and.004 ≤ α ≤ .86. The maximum mass of the star satisfying the strong energy condition (SEC),
(α = 1/3) is found to be3.82 Mq· at K=−2/3, while the same for the weak energy condition (WEC), (α =1) is 4.57 M_
⊙ atK=−>5/2. In each case the surface density is assumed to be 2 × 1014 gm cm-3. The solutions corresponding to K>0 (in fact K>1) are also made meaningful by considering the hypersurfaces t= constant as 3-hyperboloid by replacing the parameter R
2 by −R2 in Vaidya-Tikekar formalism. The solutions for the later case are also expressible in terms of algebraic functions for K=2-(n/δ2 > 1 (n being integer or zero and δ =1,2,3 4). The cases for which 0 < K < 1 do not possess negative energy density gradient and therefore are incapable of representing any physically plausible
star model. In totality the article provides all the physically plausible exact solutions for the Buchdahl static perfect
fluid spheres.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
10.
Masaya Masayoshi Saito Kiyotaka Tanikawa 《Celestial Mechanics and Dynamical Astronomy》2009,103(3):191-207
We study the change of phase space structure of the rectilinear three-body problem when the mass combination is changed. Generally,
periodic orbits bifurcate from the stable Schubart periodic orbit and move radially outward. Among these periodic orbits there
are dominant periodic orbits having rotation number (n − 2)/n with n ≥ 3. We find that the number of dominant periodic orbits is two when n is odd and four when n is even. Dominant periodic orbits have large stable regions in and out of the stability region of the Schubart orbit (Schubart
region), and so they determine the size of the Schubart region and influence the structure of the Poincaré section out of
the Schubart region. Indeed, with the movement of the dominant periodic orbits, part of complicated structure of the Poincaré
section follows these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit. 相似文献
11.
Otávio de Oliveira Costa Filho Wagner Sessin 《Celestial Mechanics and Dynamical Astronomy》1999,74(1):1-17
This paper is concerned with the extended Delaunay method as well as the method of integration of the equations, applied to
first order resonance. The equations of the transformation of the extended Delaunay method are analyzed in the (p + 1)/p type resonance in order to build formal, analytical solutions for the resonant problem with more than one degree of freedom.
With this it is possible to gain a better insight into the method, opening the possibility for more generalized applications.
A first order resonance in the first approximation is carried out, giving a better comprehension of the method, including
showing how to eliminate the ‘Poincaré singularity’ in the higher orders.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
12.
The work presented in paper I (Papadakis, K.E., Goudas, C.L.: Astrophys. Space Sci. (2006)) is expanded here to cover the evolution of the approximate general solution of the restricted problem covering symmetric and escape solutions for values of μ in the interval [0, 0.5]. The work is purely numerical, although the available rich theoretical background permits the assertions that most of the theoretical issues related to the numerical treatment of the problem are known. The prime objective of this work is to apply the ‘Last Geometric Theorem of Poincaré’ (Birkhoff, G.D.: Trans. Amer. Math. Soc. 14, 14 (1913); Poincaré, H.: Rend. Cir. Mat. Palermo 33, 375 (1912)) and compute dense sets of axisymmetric periodic family curves covering the initial conditions space of bounded motions for a discrete set of values of the basic parameter μ spread along the entire interval of permissible values. The results obtained for each value of μ, tested for completeness, constitute an approximation of the general solution of the problem related to symmetric motions. The approximate general solution of the same problem related to asymmetric solutions, also computable by application of the same theorem (Poincaré-Birkhoff) is left for a future paper. A secondary objective is identification-computation of the compact space of escape motions of the problem also for selected values of the mass parameter μ. We first present the approximate general solution for the integrable case μ = 0 and then the approximate solution for the nonintegrable case μ = 10−3. We then proceed to presenting the approximate general solutions for the cases μ = 0.1, 0.2, 0.3, 0.4, and 0.5, in all cases building them in four phases, namely, presenting for each value of μ, first all family curves of symmetric periodic solutions that re-enter after 1 oscillation, then adding to it successively, the family curves that re-enter after 2 to 10 oscillations, after 11 to 30 oscillations, after 31 to 50 oscillations and, finally, after 51 to 100 oscillations. We identify in these solutions, considered as functions of the mass parameter μ, and at μ = 0 two failures of continuity, namely: 1. Integrals of motion, exempting the energy one, cease to exist for any infinitesimal positive value of μ. 2. Appearance of a split into two separate sub-domains in the originally (for μ = 0) unique space of bounded motions. The computed approximations of the general solution for all values of μ appear to fulfill the ‘completeness’ criterion inside properly selected sub-domains of the domain of bounded motions in the (x, C) plane, which means that these sub-domains are filled countably densely by periodic family curves, which form a laminar flow-line pattern. The family curves in this pattern may, or may not, be intersected by a ‘basic’ family curve segment of order from 1 up to 3. The isolated points generating asymptotic solutions resemble ‘sink’ points toward which dense sets of periodic family curves spiral. The points in the compact domain in the (x, C) plane resting outside the domain of bounded motions (μ = 0), including the gap between the two large sub-domains (μ > 0) created by the aforementioned split, generate escape motions. The gap between the two large sub-domains of bounded motions grows wider for growing μ. Also, a number of compact gaps that generate escape motions exist within the body of the two sub-domains of bounded motions. The approximate general solutions computed include symmetric, heteroclinic, asymptotic, collision and escape solutions, thus constituting one component of the full approximate general solution of the problem, the second and final component being that of asymmetric solutions. 相似文献
13.
Naveen Bijalwan 《Astrophysics and Space Science》2011,336(2):413-418
Recently, Bijalwan (Astrophys. Space Sci., doi:, 2011a) discussed charged fluid spheres with pressure while Bijalwan and Gupta (Astrophys. Space Sci. 317, 251–260, 2008) suggested using a monotonically decreasing function f to generate all possible physically viable charged analogues of Schwarzschild interior solutions analytically. They discussed
some previously known and new solutions for Schwarzschild parameter
u( = \fracGMc2a ) ￡ 0.142u( =\frac{GM}{c^{2}a} ) \le 0.142, a being radius of star. In this paper we investigate wide range of u by generating a class of solutions that are well behaved and suitable for modeling Neutron star charge matter. We have exploited
the range u≤0.142 by considering pressure p=p(ω) and
f = ( f0(1 - \fracR2(1 - w)a2) +fa\fracR2(1 - w)a2 )f = ( f_{0}(1 - \frac{R^{2}(1 - \omega )}{a^{2}}) +f_{a}\frac{R^{2}(1 - \omega )}{a^{2}} ), where
w = 1 -\fracr2R2\omega = 1 -\frac{r^{2}}{R^{2}} to explore new class of solutions. Hence, class of charged analogues of Schwarzschild interior is found for barotropic equation
of state relating the radial pressure to the energy density. The analytical models thus found are well behaved with surface
red shift z
s
≤0.181, central red shift z
c
≤0.282, mass to radius ratio M/a≤0.149, total charge to total mass ratio e/M≤0.807 and satisfy Andreasson’s (Commun. Math. Phys. 288, 715–730, 2009) stability condition. Red-shift, velocity of sound and p/c
2
ρ are monotonically decreasing towards the surface while adiabatic index is monotonically increasing. The maximum mass found
to be 1.512 M
Θ with linear dimension 14.964 km. Class of charged analogues of Schwarzschild interior discussed in this paper doesn’t have
neutral counter part. These solutions completely describe interior of a stable Neutron star charge matter since at centre
the charge distribution is zero, e/M≤0.807 and a typical neutral Neutron star has mass between 1.35 and about 2.1 solar mass, with a corresponding radius of about
12 km (Kiziltan et al., [astro-ph.GA], 2010). 相似文献
14.
For the n-centre problem of one particle moving in the potential of attracting centres of small mass fixed in an arbitrary smooth potential
and magnetic field, we prove the existence of periodic and chaotic trajectories shadowing sequences of collision orbits. In
particular, we obtain large subshifts of solutions of this type for the circular restricted 3-body problem of celestial mechanics.
Poincaré had conjectured existence of the periodic ones and given them the name ‘second species solutions’.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
15.
Based on the data on a spectral dependence of the geometric albedo of giant planet discs, we obtained depth variations in
the optical thickness τ
a
of the aerosol component and relative concentration γ of methane (Uranium, Neptune) lnτ
a
= −0.720 + 1.507Δlnp (for −2.2085 ≤ lnp ≤ −1.0018), lnτ
a
= +1.224 + 1.160Δlnp (for −1.0018 ≤ lnp ≤ −0.0595), lnτ
a
= +2.318 + 0.192Δlnp (for −0.0595 ≤ lnp), γ = 0.0027 for Jupiter; lnτ
a
= −0.846 + 1.598Δlnp (for −3.3619 ≤ lnp ≤ −2.0575), lnτ
a
= +1.238 + 1.342Δlnp (for −2.0575 ≤ lnp ≤ −1.2074), lnτ
a
= +2.379 + 0.722 (for −1.2074 ≤ lnp ≤ −0.6501), lnτ
a
= +2.781 + 0.326Δlnp (for 0.6501 ≤ lnp), γ = 0.0027 for Saturn; lnτ
a
= −2.694 + 0.087Δlnp (for +0.3685 ≤ lnp ≤ +1.2314), lnτ
a
= −2.619 + 7.341Δlnp (for +1.2314 ≤ lnp ≤ +1.7556), lnτ
a
= +1.229 + 0.956Δlnp (for +1.7556 ≤ lnp) for Uranium; lnτ
a
= −1.861 + 1.248Δlnp (for +0.3204 ≤ lnp ≤ +0.9051), lnτ
a
= −1.131 + 0.347Δlnp (for +0.9051 ≤ lnp) for Neptune; depth-averaged relative methane concentration lnγ = −9.982 + 2.676Δlnp(0.3584 ≤ lnp ≤ 1.5445); ln γ = −9.738 + 2.561Δlnp(0.3237 ≤ lnp ≤ 1.6156) and γ = 0.00382(lnp ≥ 1.6156); 0.00554(lnp ≥ 1.6156) for Uranium and Neptune, respectively (p is in bar). 相似文献
16.
E.A. Dorfi 《Astrophysics and Space Science》2000,272(1-3):227-238
Supernova Remnants (SNRs) are the most likely sources of the galactic cosmic rays up to energies of about 1015 eV/nuc. The large scale shock waves of SNRs are almost ideal sites to accelerate particles up to these highly non-thermal
energies by a first order Fermi mechanism which operates through scattering of the particles at magnetic irregularities. In
order to get an estimate on the total amount of the explosion energy E
SNconverted into high energy particles the evolution of a SNR has to be followed up to the final merging with the interstellar
medium. This can only be done by numerical simulations since the non-linear modifications of the shock wave due to particle
acceleration as well as radiative cooling processes at later SNR stages have to be considered in such investigations. Based
on a large sample of numerical evolution calculations performed for different ambient densities n
ext, SN explosion energies, magnetic fields etc. we discuss the final ‘yields’ of cosmic rays at the final SNR stage where the
Mach number of the shock waves drops below 2. At these times the cosmic rays start to diffuse out of the remnant. In the range
of external densities of10-2 ≤ n
ext/[cm-3] ≤ 30 we find a the total acceleration efficiency of about 0.15 E
SN with an increase up to 0.24 E
SN at maximum for an external density of n
ext = 10 cm-3. Since for the larger ambient densities radiative cooling can reduce significantly the total thermal energy content of the
remnant dissipation of Alfvén waves can provide an important heating mechanism for the gas at these later stages. From the
collisions of the cosmic rays with the thermal plasma neutral pions are generated which decay subsequently into observable
γ-rays above 100 MeV. Hence, we calculate these γ-ray luminosities of SNRs and compare them with current upper limits of ground
based γ-raytelescopes. The development of dense shells due to cooling of the thermal plasma increases the γ-ray luminosities
and e.g. an external density of n
ext = 10 cm-3 with E
SN = 1051 erg can lead to a γ-ray flux above 10-6 ph cm-2 s-1 for a remnant located at a distance of 1 kpc.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
17.
For the case of Tycho’s supernova remnant (SNR) we present the relation between the blast wave and contact discontinuity radii
calculated within the nonlinear kinetic theory of cosmic ray (CR) acceleration in SNRs. It is demonstrated that these radii
are confirmed by recently published Chandra measurements which show that the observed contact discontinuity radius is so close
to the shock radius that it can only be explained by efficient CR acceleration which in turn makes the medium more compressible.
Together with the recently determined new value E
sn=1.2×1051 erg of the SN explosion energy this also confirms our previous conclusion that a TeV γ-ray flux of (2–5)×10−13 erg/(cm2 s) is to be expected from Tycho’s SNR. Chandra measurements and the HEGRA upper limit of the TeV γ-ray flux together limit the source distance d to 3.3≤d≤4 kpc. 相似文献
18.
A new class of charged super-dense star models is obtained by using an electric intensity, which involves a parameter, K. The metric describing the model shares its metric potential g 44 with that of Durgapal’s fourth solution (J. Phys. A, Math. Gen. 15:2637, 1982). The pressure-free surface is kept at the density ρ b =2×1014 g/cm3 and joins smoothly with the Reissner-Nordstrom solution. The charge analogues are well-behaved for a wide range, 0≤K≤59, with the optimum value of X=0.264 i.e. the pressure, density, pressure–density ratio and velocity of sound are monotonically decreasing and the electric intensity is monotonically increasing in nature for the given range of the parameter K. The maximum mass and the corresponding radius occupied by the neutral solution are 4.22M Θ and 20 km, respectively for X=0.264. For the charged solution, the maximum mass and radius are defined by the expressions M≈(0.0059K+4.22)M Θ and r b ≈−0.021464K+20 km respectively. 相似文献
19.
We obtain a well behaved class of charge analogues of neutral superdense star model due to Kuchowicz, by using a particular
electric field, which involves a parameter K and vanishes when K=0. The members of this class are seen to satisfy the various physical conditions e.g. c
2
ρ≥3p≥0, dp/dr<0, dρ/dr<0, along with the velocity of sound, dp/c
2
dρ<1 and the adiabatic index ((p+c
2
ρ)/p)(dp/(c
2
dρ))>1, for the interval 0<K<1 with the maximum mass 6.8374M
Θ and the radius 23.4679 km with the central red shift Z
c
=0.75364. In the interval, 0<K≤0.1179, the velocity of sound and the ratio p/c
2
ρ are found monotonically decreasing towards the pressure free interface, which presents a relevant model for massive star
like Neutron star or pulsar with the maximum mass as 4.1474M
Θ and the radius 20.5481 km with the central red shift Z
c
=0.6654. 相似文献
20.
Toshio Fukushima 《Celestial Mechanics and Dynamical Astronomy》2009,105(4):305-328
As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete
elliptic integral of the first and second kinds, K(m) and E(m), for the standard domain of the elliptic parameter, 0 < m < 1. For the case 0 < m < 0.9, the method utilizes 10 pairs of approximate polynomials of the order of 9–19 obtained by truncating Taylor series
expansions of the integrals. Otherwise, the associate integrals, K(1 − m) and E(1 − m), are first computed by a pair of the approximate polynomials and then transformed to K(m) and E(m) by means of Jacobi’s nome, q, and Legendre’s identity relation. In average, the new method runs more-than-twice faster than the existing methods including
Cody’s Chebyshev polynomial approximation of Hastings type and Innes’ formulation based on q-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, sn(u|m), cn(u|m), and dn(u|m), by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic
argument, u, after its domain is reduced to the standard range, 0 ≤ u < K(m)/4, with the help of the new method to compute K(m). The new procedure is 25–70% faster than the methods based on the Gauss transformation such as Bulirsch’s algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of K(m) is not taken into account. 相似文献