Computer Physics Communications 125 (2000) 21–59 www.elsevier.nl/locate/cpc
An accurate eighth order exponentially-fitted method for the efficient solution of the Schrödinger equation T.E. Simos 1 Section of Mathematics, Department of Civil Engineering, School of Engineering, Democritus University of Thrace, GR 671 00 Xanthi, Greece Received 9 March 1999; received in revised form 28 April 1999
Abstract An accurate eighth algebraic order exponentially-fitted method is developed for the numerical solution of radial Schrödinger equation and of the coupled differential equations of the Schrödinger type. The free parameters of the new scheme are defined in order to integrate exactly exponential functions. Numerical and theoretical results indicate that the new method is much more efficient than other classical and exponentially fitted methods. 2000 Elsevier Science B.V. All rights reserved. PACS: 65L05 Keywords: Exponentially-fitted methods; Hybrid methods; Scattering problems; Coupled differential equations; Schrödinger equation; Finite differences
1. Introduction There is a great activity in last decade for the numerical solution of the radial Schrödinger equation and coupled differential equations of the Schrödinger type. The aim of this activity is the construction of an efficient and reliable algorithm that approximates the solution (see [1] and references therein). The radial Schrödinger equation can be written as (1) y 00 (r) = f (r)y(r) = l(l + 1)/r 2 + V (r) − k 2 y(r). Differential equations of the above type occur very frequently in many problems in theoretical physics and chemistry, in chemical physics, in physical chemistry, in astrophysics, in electronics and elsewhere (see, for example, [2]). For the above reason it is needed the construction of an efficient and reliable numerical method. In (1) the function W (r) = l(l + 1)/r 2 + V (r) 1 Please use the following address for all correspondence: Dr. T.E. Simos, 26 Menelaou Street, Amfithea – Paleon Faliron, GR-175 64 Athens, Greece. E-mail:
[email protected].
0010-4655/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 9 9 ) 0 0 4 5 9 - 2
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T.E. Simos / Computer Physics Communications 125 (2000) 21–59
denotes the effective potential, which satisfies W (r) → 0 as r → ∞, k 2 is a real number denoting the energy, l is a given integer and V is a given function which denotes the potential. The boundary conditions are y(0) = 0
(2)
and a second boundary condition, for large values of r, determined by physical considerations. An explanation about the not efficiency of boundary and initial value methods is given in [3]. One of the most popular and well known methods for the numerical solution of (1) is Numerov’s method. The reason is given in [3]. In [4,5] high order numerical methods for the eigenvalue problem of the radial Schrödinger equation are developed for some special potentials V (x) which are even functions. In [6] the Runge–Kutta type or hybrid methods, which are an alternative approach for deriving higher order methods, are introduced. In [7] (and references therein) another approach for developing efficient methods for the numerical solution of (1), which is the exponential fitting (see [7] and references therein), is introduced. This approach is appropriate because for large values of r and positive k 2 the solution of (1) is periodic. Many authors have investigated the idea of exponential fitting, since Raptis and Allison [8]. Ixaru and Rizea [7] have produced a method which integrates functions of the form (3) 1, x, x 2, . . . , x p , exp(±vx), x exp(±vx), . . . , x m exp(±vx) , where v is the frequency of the problem. For the method obtained by Ixaru and Rizea [7] we have m = 1 and p = 1. Raptis [9] develop an exponentially-fitted method with m = 2 and p = 0. Simos [10] has derived a four-step method of this type which integrates more exponential functions and gives much more accurate results than the four-step methods of Raptis [11,12]. For this method we have m = 3 and p = 0. Simos [13] has derived a family of four-step methods which give more efficient results than other four-step methods. In particular, he has derived methods with m = 0 and p = 5, m = 1 and p = 3, m = 2 and p = 1 and finally m = 3 and p = 0. Also Raptis and Cash [14] have derived a two-step method fitted to (3) with m = 0 and p = 5 based on the well known Runge– Kutta-type sixth-order formula of Cash and Raptis [15]. The method of Cash, Raptis and Simos [16] is also based on the formula proposed in [15] and is fitted to (3) with m = 1 and p = 3. All the above methods are of algebraic order four and six. The purpose of this paper is to derive an eighth algebraic order method fitted to (3) and in particular to derive an exponentially-fitted method with m = 4 and p = 0. Based on the new developed method and the sixth-order exponentially-fitted method of Simos [17] a variable-step procedure is introduced. We have applied the new methods to the solution of coupled differential equations arising from the Schrödinger equation. The results indicate that the approach is more efficient than the well known iterative Numerov method of Allison [18], the exponentially-fitted variable-step method of Raptis and Cash [14] and the classical (with constant coefficients) variable-step method of Simos [19].
2. Exponential multistep methods For the numerical solution of the initial value problem y (r) = f (x, y),
y (j )(A) = 0,
j = 0, 1, . . . , r − 1
(4)
the multistep methods of the form k X i=0
ai yn+i = hr
k X
bi f (xn+i , yn+i )
i=0
over the equally spaced intervals {xi }ki=0 in [A, B] can be used.
(5)
T.E. Simos / Computer Physics Communications 125 (2000) 21–59
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The method (5) is associated with the operator L(x) =
k X
ai z(x + ih) − hr bi z(r)(x + ih) ,
(6)
i=0
where z is a continuously differentiable function. Definition 1. The multistep method (5) called algebraic (or exponential) of order p if the associated linear operator L vanishes for any linear combination of the linearly independent functions 1, x, x 2, . . . , x p+r−1 (or exp(v0 x), exp(v1 x), . . . , exp(vp+r−1 x) where vi , i = 0, 1, . . . , p + r − 1 are real or complex numbers). Remark 1 (see [20]). If vi = v for i = 0, 1, . . . , n, n 6 p + r − 1 then the operator L vanishes for any linear combination of (7) exp(vx), x exp(vx), x 2 exp(vx), . . . , x n exp(vx), exp(vn+1 x), . . . , exp(vp+r−1 x) . Remark 2 (see [20]). Every exponential multistep method corresponds in a unique way, to an algebraic multistep method (by setting vi = 0 for all i). Lemma 1 (For proof see [20] and [21]). Consider an operator L of the form (6). With v ∈ C, h ∈ R, n > r if v = 0, and n > 1 otherwise, then we have L x n exp(vx) 6= 0 (8) L x m exp(vx) = 0, m = 0, 1, . . . , n − 1, if and only if the function ϕ has a zero of exact Pmultiplicity s at exp(vh), P where s = n if v 6= 0, and s = n − r if v = 0, ϕ(w) = ρ(w)/ logr w − σ (w), ρ(w) = ki=0 ai wi and σ (w) = ki=0 bi wi . Proposition 1 (For proof see [22] and [23]). Consider an operator L with L exp(±vi x) = 0,
p+r −1 2 there is a unique set of bi such that bi = bk−i .
j = 0, 1, . . . , k 6
then for given ai and p with ai = (−1)r ak−i
(9)
In the present paper we investigate the case r = 2.
3. The new exponentially-fitted method Consider the following method: (10) yn+1 + q1 yn + yn−1 = h2 t0 (fn+1 + fn−1 ) + t1 fn + t2 (fn+s + fn−s ) + t3 (fn+q + fn−q ) . √ This method for appropriate values of ti , i = 0(1)3, for s = 21/3 and arbitrary values of q is of algebraic order eight (see for details [19]). We require that the methods (10) should integrate exactly any linear combination of the functions: (11) exp(±vx), x exp(±vx), x 2 exp(±vx), x 3 exp(±vx), x 4 exp(±vx) . To construct a method of the form (10) which integrates exactly the functions (11), we require that the method (10) integrates exactly (see [22] and [23]): (12) exp(±v0 x), exp(±v1 x), exp(±v2 x), exp(±v3 x), exp(±v4 x)
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and then put: v0 = v1 = v2 = v3 = v4 = v.
(13)
Demanding that (10) integrates (12) exactly, we obtain the following system of equations for q1 and for ti , i = 0(1)3 2t2 wj2 cosh(wj s) + 2 cosh(wj )t0 wj2 − q1 + t1 wj2 + 2t3 wj2 cosh(wj q) = 2 cosh(wj )
(14)
where wj = vj h, j = 0(1)4. √ Solving for q1 and for ti , i = 0(1)3 and putting s = 21/3 and q = 12 , we obtain the coefficients which are given in the Appendix A. The above formulae for q1 and ti , i = 0(1)3 are subject to heavy cancellations for small values of w = vh. In this case it is much more convenient to use the series expansions for the coefficients q1 and ti , i = 0(1)3 of the method given in the Appendix B. We now seek computable approximations to yn±s and yn±q . For these approximations we still require to be exact to any linear combination of the functions of the form (15) exp(±vx), x exp(±vx), . . . , x m exp(±vx) . Following [19] we look for approximations of the form yn+s + yn−s = a0 (yn+1 + yn−1 ) + a1 yn + h2 a2 (fn+1 + fn−1 ) + a3 fn , yn+s − yn−s = b0 (yn+1 − yn−1 ) + h2 b1 (fn+1 − fn−1 )
(16) (17)
and of the form
yn+q + yn−q = c0 (yn+1 + yn−1 ) + c1 yn + h2 c2 (fn+1 + fn−1 ) + c3 fn + c4 (fn+s + fn−s ) , yn+q − yn−q = d0 (yn+1 − yn−1 ) + h2 d1 (fn+1 − fn−1 ) + d2 (fn+s − fn−s ) .
(18) (19)
We require that the method (16) should integrate exactly any linear combination of the functions (15) (for m = 3). We also require that the method (17) should integrate exactly any linear combination of the functions (15) (for m = 1). To construct a method of the form (16) which integrates exactly the functions (15) (with m = 3), we require (see [22] and [23]) that the method (16) integrates exactly (20) exp(±v0 x), exp(±v1 x), exp(±v2 x), exp(±v3 x) and then put v0 = v1 = v2 = v3 = v.
(21)
To construct a method of the form (17) which integrates exactly the functions (15) (with m = 1), we require that the method (17) integrates exactly (see [22] and [23]): (22) exp(±v0 x), exp(±v1 x) and then put: v0 = v1 = v.
(23)
Demanding that (16) integrates (20) exactly, we obtain the following system of equations for ai , i = 0(1)3. Demanding, also, that (17) integrates (22) exactly, we obtain the following system of equations for bi , i = 0, 1. 2a0 cosh(wj ) + 2a2 wj2 cosh(wj ) + a1 + a3 wj2 = 2 cosh(wj s),
(24)
2b0 sinh(wj ) + 2b1 wj2 sinh(wj ) = 2 sinh(wj s)
(25)
where wj = vj h, j = 0, 1, 2, 3 for (24) and j = 0, 1 for (25). √ Solving for ai , i = 0(1)3 and for bi , i = 0, 1 and putting s = 21/3 and q = 12 , we obtain the coefficients which are given in the Appendix C.
T.E. Simos / Computer Physics Communications 125 (2000) 21–59
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The above formulae for a1 and b1 are subject to heavy cancellations for small values of w = vh. In this case it is much more convenient to use the series expansions for the coefficient ai , i = 0(1)3 and bi , i = 0, 1 of the method given in the Appendix D. Following the same procedure as above, we require that the method (18) should integrate exactly any linear combination of the functions (15) (for m = 4). We also require that the method (19) should integrate exactly any linear combination of the functions (15) (for m = 2). To construct a method of the form (18) which integrates exactly the functions (15) (with m = 4), we require that the method (17) integrates exactly (12) (see [22] and [23]) and then put (13). To construct a method of the form (19) which integrates exactly the functions (15) (with m = 2), we require that the method (19) integrates exactly (see [22] and [23]): (26) exp(±v0 x), exp(±v1 x), exp(±v2 x) and then put: v0 = v1 = v2 = v.
(27)
Demanding that (18) integrates (12) exactly, we obtain the following system of equations for ci , i = 0(1)4. Demanding, also, that (19) integrates (26) exactly, we obtain the following system of equations for di , i = 0(1)2. 2c4 w2 cosh(w s) + 2c2 w2 cosh(w) + 2c0 cosh(w) + c3 w2 + c1 = 2 cosh(w q),
(28)
(29) 2d2 w sinh(w s) + 2d0 sinh(w) + 2d1w sinh(w) = 2 sinh(w q). √ 1 Solving for ci , i = 0(1)4 and for di , i = 0(1)2 and putting s = 21/3 and q = 2 , we obtain the coefficients which are given in the Appendix E. The above formulae for ci , i = 0(1)4 and di , i = 0(1)2 are subject to heavy cancellations for small values of w = vh. In this case it is much more convenient to use the series expansions for these coefficients of the method which are given in the Appendix F. Since the Taylor series expansions are very accurate (we have given series expansions of 14th order), one can use them in all cases. Finally, in the Appendix G a description of the computer algebra program, which is used for the derivation of the coefficients presented in Appendices A–F, is presented. It is easy for one to see that the approximations of yn±s and yn±q are given by 2
2
y n+s = 12 (a0 + b0 )yn+1 + 12 a1 yn + 12 (a0 − b0 )yn−1 + h2
1 2 (a2
00 00 + b1 )yn+1 + 12 a3 yn00 + 12 (a2 − b1 )yn−1 ,
y n−s = 12 (a0 − b0 )yn+1 + 12 a1 yn + 12 (a0 + b0 )yn−1 + h2 y n+q =
y n−q =
1 2 (a2
00 00 − b1 )yn+1 + 12 a3 yn00 + 12 (a2 + b1 )yn−1 ,
1 1 2 (c0 + d0 )yn+1 + 2 c1 yn 00 + h2 12 (c2 + d1 )yn+1 + + h2 12 (c4 + d2 )y 00n+s + 1 1 2 (c0 − d0 )yn+1 + 2 c1 yn 00 + h2 12 (c2 − d1 )yn+1 + 2 1 00 + h 2 (c4 − d2 )y n+s +
(30)
+ 12 (c0 − d0 )yn−1 00 00 1 1 2 c3 yn + 2 (c2 − d1 )yn−1 , 00 1 2 (c4 − d2 )y n−s , + 12 (c0 + d0 )yn−1 1 00 1 00 2 c3 yn + 2 (c2 + d1 )yn−1 , 00 1 2 (c4 + d2 )y n−s .
If w = iφ, then the method (10), (30) with coefficients given by (58), (61) and (63) is exact for any linear combination of the functions: sin(φx), cos(φx), x sin(φx), x cos(φx), x 2 sin(φx), x 2 cos(φx), (31) x 3 sin(φx), x 3 cos(φx), x 4 sin(φx), x 4 cos(φx) .
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4. Error analysis In this section we will examine theoretically the method developed in [19], the exponentially-fitted method developed in [24] and the new exponentially-fitted method. We will also find a quantitative estimation for the extend of the accuracy gain to be expected from the exponentially-fitted versions. Definition 2. A method is called classical if it has constant coefficients. Remark 3. An exponentially-fitted method is not a classical one because it has coefficients which are depended from the quantity vh, where v is the frequency of the problem and h is the step length. We first write f (x) of (1) in the form f (x) = g(x) + d,
(32)
where g(x) = U (x) − Uc = g, where Uc is the constant approximation of the potential and d = w2 = Uc − k 2 . So, g(x) depends on the potential and its constant approximation and d shows √ the energy dependence. The local truncation error of the method developed in [19] (with s = 21/3 and arbitrary q) is given by L.T .E.clas. =
(10) h10 3yn T1 + 56yn(8)Fn (13 − 42q 2) + O(h12 ), 2 152409600(7 − 3q )
(33)
where T1 = 765q 4 − 2007q 2 + 518. √ The local truncation error of the exponentially-fitted method developed in [24] is given by (with s = 21/3 and arbitrary q) L.T .E.exp.[24] =
h10 152409600(7 − 3q 2 ) × 3(yn(10) − dyn(8))T1 + 56(yn(8) − dyn(6))Fn (13 − 42q 2) + O(h12 ).
(34) √ The local truncation error of the exponentially- fitted method developed in this paper is given by (with s = 21/3 and arbitrary q) 41 41 41 yn(10) − dyn(8) − yn d 5 L.T .E.new = h10 203212800 40642560 203212800 41 41 41 d 3 yn(4) + d 2 yn(6) + d 4 yn(2) + O(h12 ). − (35) 20321280 20321280 40642560 We express, now, the derivatives y 00 (x), y (4) , y (6) , y (8) and y (10) in terms of Eq. (1), i.e. y 00 = f (x)y(x), 2 ∂ ∂ ∂ U (x) y(x) + y(x)U (x)2 − 2y(x)U (x)Uc U (x) y(x) + 2 y (4) = ∂x ∂x ∂x 2 + 2y(x)U (x) d + y(x) Uc2 − 2y(x)Uc d + y(x) d 2,
(36)
etc. We note that g (n) (x) = U (n) (x) for nth order derivative with respect to x. We also express the terms as polynomials of d. If we introduce, now, these expressions to the local truncation error formulae (33) and (34), then we obtain the following expressions (as polynomials of d) for the local truncation error. We note that in the following expressions q = 12 .
T.E. Simos / Computer Physics Communications 125 (2000) 21–59
Classical method [19]. 10 L.T .E.clas. = h d 5
41 4 y(x) + d T2 + · · · + O(h12 ), 203212800
27
(37)
where T2 = −
1 −448y(x) − 3075y(x)U (x) + 3075y(x)Uc . 3048192000
Exponentially-fitted method [24]. 1 10 4 (−615y(x)U (x) + 615y(x)Uc) + · · · + O(h12 ). L.T .E.exp.[24] = h d − 3048192000 New exponentially-fitted method. 3 3 41 ∂ ∂ ∂ ∂ 41 y(x) U (x) − y(x) Uc U (x) U (x) L.T .E.new = h10 d 5080320 ∂x 3 ∂x 5080320 ∂x 3 ∂x 41 41 41 X1 y(x)U (x)2 − X1 U (x)y(x)U c + X1 y(x)U c2 + 5080320 2540160 5080320 2 ∂ 41 41 ∂ ∂ U (x) y(x) X1 + U (x) y(x)U (x) + 2540160 ∂x ∂x 3386880 ∂x 2 3 ∂ 533 ∂ ∂ 41 U (x) y(x)U c + U (x) y(x) U (x) − 3386880 ∂x 12700800 ∂x ∂x 3 4 ∂ 943 U (x) y(x)U (x) + 50803200 ∂x 4 4 ∂ 697 943 X2 y(x) U (x) y(x)U c + − 4 50803200 ∂x 25401600 1 6 5 ∂ ∂ 41 ∂ 41 y(x) + · · · , U (x) y(x) + U (x) + 8467200 ∂x 6 6350400 ∂x 5 ∂x
(38)
(39)
∂2 U (x). ∂x 2 From the relations (37) and (38) we obtain the asymptotic expansions of the errors (in the case where d 0 or d 0). X1 =
Classical method [19]. L.T .E.clas. = h10 d 5
41 y(x). 203212800
Exponentially-fitted method [24]. 1 10 4 −615y(x)U (x) + 615y(x)Uc . L.T .E.exp.[24] = h d − 3048192000 New exponentially-fitted method. 3 3 ∂ 41 ∂ ∂ ∂ 41 y(x) U (x) − y(x) Uc U (x) U (x) L.T .E.new = h10 d 5080320 ∂x 3 ∂x 5080320 ∂x 3 ∂x
(40)
(41)
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T.E. Simos / Computer Physics Communications 125 (2000) 21–59
+ + − + − +
41 41 41 X1 y(x)U (x)2 − X1 U (x)y(x)U c + X1 y(x)U c2 5080320 2540160 5080320 2 ∂ ∂ ∂ 41 41 U (x) y(x) X1 + U (x) y(x)U (x) 2540160 ∂x ∂x 3386880 ∂x 2 3 ∂ ∂ ∂ 533 41 U (x) y(x)U c + U (x) y(x) U (x) 3386880 ∂x 12700800 ∂x ∂x 3 4 ∂ 943 U (x) y(x)U (x) 50803200 ∂x 4 4 697 ∂ 943 X2 y(x) U (x) y(x)U c + 50803200 ∂x 4 25401600 1 6 5 41 ∂ ∂ ∂ 41 y(x) , U (x) y(x) + U (x) 8467200 ∂x 6 6350400 ∂x 5 ∂x
(42)
∂2 U (x). ∂x 2 From the above equations we have that in the classical eighth-order method developed in [19] the error increases as a fifth power of d. In the exponentially-fitted eighth-order method developed in [24] the error increases as a fourth power of d. In the exponentially-fitted eighth-order method developed in this paper the error increases as a first power of d. Based on the above analysis we have the following theorem X1 =
Theorem 1. For the exponentially-fitted method developed in this paper and for problems of the form (1) with f (x) given by (32), the error increases as a first power of d, while for the exponentially-fitted method developed in [24] the error increases as a fourth power of d and for the the eighth algebraic order method developed in [19] the error increases as a fifth power of d. So the new exponentially-fitted method is more accurate than the eighth-order exponentially-fitted method of [24] and the eighth-order method developed in [19] for the problems mentioned above especially for large values of |d| = |Uc − E|. 5. Numerical illustration 5.1. Error estimation It is known from the literature (see, for example, [25] and references therein) that there are many methods for the estimation of the local truncation error (LTE) in the integration of systems of initial-value problems. We note that the LTE is based on the algebraic order of the method. The local error estimation algorithm used in this paper is based on the fact that when we have a method with local error of higher order then the approximation of the solution for the problems which have a periodic or oscillating solution is better. H and the solution obtained with lower Denoting the solution obtained with higher algebraic order method as yn+1 L , we have the following definition algebraic order method as yn+1 L by the quantity Definition 3. We define the local error estimate in the lower order solution yn+1 H L L.T .E. = yn+1 − yn+1 .
(43)
H can be neglected compared with Under the assumption that when h is sufficiently small, the local error in yn+1 L that in yn+1 .
T.E. Simos / Computer Physics Communications 125 (2000) 21–59
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H is obtained using the exponentially-fitted method developed in this paper and We assume that the solution yn+1 L the solution yn+1 is obtained using the exponentially-fitted method of Simos [17]. If the local phase-lag error is bounded by acc and the step size of the integration used for the nth step length is hn , the estimated step size for the (n + 1)st step, which will give a local error bounded by acc, must be 1/q acc , (44) hn+1 = hn L.T .E. where q is the order of the local error. Following [14], we have considered all step changes to halving and doubling. Thus, based on the procedure developed in [14], for the Local Truncation Error, the step control procedure which we use for the Local Error is
hn+1 = 2hn ,
If L.T .E. < acc,
If 100 acc > L.T .E. > acc, hn+1 = hn , (45) hn and repeat the step. If L.T .E. > 100 acc, hn+1 = 2 It is known that the local error estimate is obtained to the lower order solution. This is applied, also, in our L . However, if this error estimate is case of the local error estimate, i.e. the local error estimate is obtained for yn+1 acceptable, i.e. less than the bound acc, we consider the widely used local extrapolation technique. Thus, although L , we use the higher order solution we are controlling an estimation of the local error in the lower order solution yn+1 H yn+1 at each accepted step. 5.2. Coupled differential equations There are many problems in theoretical physics, atomic physics, physical chemistry, quantum chemistry and chemical physics which can be transformed to the solution of coupled differential equations of the Schrödinger type. The close-coupling differential equations of the Schrödinger type may be written in the form 2 N X li (li + 1) d 2 + ki − − Vii yij = Vim ymj (46) dx 2 x2 m=1
for 1 6 i 6 N and m 6= i. We have investigated the case in which all channels are open. So we have the following boundary conditions (see for details [18]): yij = 0
at x = 0,
yij ∼ ki xjli (ki x)δij +
ki kj
(47)
1/2 Kij ki xnli (ki x),
(48)
where jl (x) and nl (x) are the spherical Bessel and Neumann functions, respectively. We note here that since the method presented in this paper have much larger intervals of periodicity (the property which must have a method to avoid instabilities) than the Numerov’s method, the method of Cash and Raptis [15] and other finite difference methods, we can use the present methods to problems involve close channels. Based on the detailed analysis developed in [18] and defining a matrix K 0 and diagonal matrices M, N by: 1/2 ki Kij , Kij0 = kj Mij = ki xjli (ki x)δij , Nij = ki xnli (ki x)δij ,
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T.E. Simos / Computer Physics Communications 125 (2000) 21–59
we find that the asymptotic condition (48) may be written as: y ∼ M + NK0 .
(49)
One of the most well-known methods for the numerical solution of the coupled differential equations arising from the Schrödinger equation is the Iterative Numerov method of Allison [18]. A real problem in theoretical physics, atomic physics, quantum chemistry and molecular physics which can be transformed to close-coupling differential equations of the Schrödinger type is the rotational excitation of a diatomic molecule by neutral particle impact. Denoting, as in [18], the entrance channel by the quantum numbers (j, l), the exit channels by (j 0 , l 0 ), and the total angular momentum by J = j + l = j 0 + l 0 , we find that 2 Jj l l 0 (l 0 + 1) Jj l 2µ X X 0 0 d 2 + k − (50) j l ; J |V |j 00 l 00 ; J yj 00 l 00 (x), yj 0 l 0 (x) = 2 0 jj 2 2 dx x h¯ j 00 l 00 where k
=
2µ
h¯ 2 0 0 j (j + 1) − j (j + 1) , E+ 2I
(51) h¯ 2 E is the kinetic energy of the incident particle in the center-of-mass system, I is the moment of inertia of the rotator, and µ is the reduced mass of the system. Following the analysis of [18], the potential V may be written as (52) V (x, kˆ j 0 j kˆ jj ) = V0 (x)P0 (kˆ j 0 j kˆ jj ) + V2 (x)P2 (kˆ j 0 j kˆ jj ), j 0j
and the coupling matrix element is given by
00 j l ; J |V |j 00 l 00 ; J = δj 0 j 00 δl 0 l 00 V0 (x) + f2 (j 0 l 0 , j 00 l 00 ; J )V2(x),
(53)
where the f2 coefficients can be obtained from formulas given by Bernstein et al. [26] and kˆ j 0 j is a unit vector parallel to the wave vector kj 0 j and Pi , i = 0, 2 are Legendre polynomials (see for details [26]). The boundary conditions may then be written as (see [18]) Jj l
yj 0 l 0 (x) = 0 at x = 0, Jj l yj 0 l 0 (x) ∼ δjj 0 δll 0 exp −i kjj x − 12 lπ −
ki kj
1/2
(54) S J (j l; j 0 l 0 ) exp i kj 0 j x − 12 l 0 π ,
(55)
where the scattering S matrix is related to the K matrix of (48) by the relation S = (I + iK)(I − iK)−1 .
(56)
The calculation of the cross sections for rotational excitation of molecular hydrogen by impact of various heavy particles requires the existence of the numerical method for step-by-step integration from the initial value to matching points. In our numerical test we choose the S matrix which is calculated using the following parameters µ = 2.351, E = 1.1, I h¯ 1 1 V0 (x) = 12 − 2 6 , V2 (x) = 0.2283V0(x). x x As is described in [18], we take J = 6 and consider excitation of the rotator from the j = 0 state to levels up to j 0 = 2, 4 and 6 giving sets of four, nine and sixteen coupled differential equations, respectively. Following Bernstein [27] and Allison [18] the reduction of the interval [0, ∞) to [0, x0] is obtained. The wavefunctions are then vanished in this region and consequently the boundary condition (54) may be written as 2µ 2
Jj l
= 1000.0,
yj 0 l 0 (x0 ) = 0.
(57)
T.E. Simos / Computer Physics Communications 125 (2000) 21–59
31
Table 1 RTC (real time of computation (in seconds)) to calculate |S|2 for the variable-step methods (1)–(4). acc = 10−6 . hmax is the maximum stepsize Method Iterative Numerov [18]
Variable-step method of Raptis and Cash [14]
Variable-step method of Simos [19]
Variable-step method of Simos [24]
Variable-step method of Simos [28]
New variable-step method
N
hmax
RTC
4
0.014
3.25
9
0.014
23.51
16
0.014
99.15
4
0.056
1.55
9
0.056
8.43
16
0.056
43.32
4
0.056
1.05
9
0.056
5.25
16
0.056
27.15
4
0.448
0.37
9
0.448
3.22
16
0.448
14.22
4
0.448
0.35
9
0.448
1.38
16
0.448
6.51
4
0.896
0.08
9
0.896
0.54
16
0.896
3.11
For the numerical solution of this problem we have used (1) the well-known Iterative Numerov method of Allison [18], (2) the variable-step method of Raptis and Cash [14], (3) the variable-step method of Simos [19] and (4) the variable-step exponentially-fitted method of Simos [24], (5) the embedded variable-step method of Simos [28] and (6) the new embedded variable-step exponentially-fitted method. In Table 1 we present the real time of computation required by the methods mentioned above to calculate the square of the modulus of the S matrix for sets of 4, 9 and 16 coupled differential equations. In Table 1 N indicates the number of equations of the set of coupled differential equations. All computations were carried out on an Pentium PC using double precision arithmetic of 16 digits accuracy. 6. Conclusions In this paper an eighth algebraic order exponentially-fitted method is developed. Based on this method and on the sixth algebraic order exponentially-fitted method of Simos [17] a new variable-step method is obtained. The application of the new variable-step algorithm to the systems of 4, 9 and 16 coupled differential equations arising from the Schrödinger equation indicate that this new algorithm is much more computationally efficient than other known and efficient methods.
32
T.E. Simos / Computer Physics Communications 125 (2000) 21–59
Acknowledgements The author wishes to thank the anonymous referee and the Principal Editor Professor Dr. P.R. Taylor for their careful reading of the manuscript and their fruitful comments and suggestions.
References [1] G. Avdelas, T.E. Simos, Embedded methods for the numerical solution of the Schrödinger equation, Comput. Math. Appl. 31 (1996) 85–102. [2] L.D. Landau, F.M. Lifshitz, Quantum Mechanics, Pergamon, New York, 1965. [3] T.E. Simos, Eighth-order method for accurate computations for the elastic scattering phase-shift problem, Int. J. Quant. Chem. 68 (1998) 191–200. [4] V. Fack, G. Vanden Berghe, (Extended) Numerov method for computing eigenvalues of specific Schrödinger equations, J. Phys. A: Math. Gen. 20 (1987) 4153. [5] V. Fack, G. Vanden Berghe, A finite difference approach for the calculation of perturbed oscillator energies, J. Phys. A: Math. Gen. 18 (1987) 3355. [6] J.R. Cash, A.D. Raptis, A high order method for the numerical solution of the one-dimensional Schrödinger equation. Comput. Phys. Comm. 33 (1984) 299–304. [7] R.M. Thomas, T.E. Simos, A family of hybrid exponentially fitted predictor – corrector methods for the numerical integration of the radial Schrödinger equation, J. Comput. Appl. Math. 87 (1997) 215–226. [8] A.D. Raptis, A.C. Allison, Exponential-fitting methods for the numerical solution of the Schrödinger equation, Comput. Phys. Comm. 14 (1978) 1–5. [9] A.D. Raptis, Two-step methods for the numerical solution of the Schrödinger equation, Computing 28 (1982) 373–378. [10] T.E. Simos, A four-step method for the numerical solution of the Schrödinger equation, J. Comput. Appl. Math. 30 (1990) 251–255. [11] A.D. Raptis, On the numerical solution of the Schrödinger equation, Comput. Phys. Commun. 24 (1981) 1–4. [12] A.D. Raptis, Exponentially-fitted solutions of the eigenvalue Schrödinger equation with automatic error control, Comput. Phys. Comm. 28 (1983) 427–431. [13] T.E. Simos: Some new four-step exponential fitting methods for the numerical solution of the radial Schrödinger equation, IMA J. Numerical Anal. 11 (1991) 347–356. [14] A.D. Raptis, J.R. Cash, Exponential and Bessel fitting methods for the numerical solution of the Schrödinger equation, Comput. Phys. Comm. 44 (1987) 95–103. [15] J.R. Cash, A.D. Raptis, A high order method for the numerical solution of the one-dimensional Schrödinger equation, Comput. Phys. Comm. 33 (1984) 299–304. [16] J.R. Cash, A.D. Raptis, T.E. Simos, A sixth order exponentially-fitted method for the numerical solution of the radial Schrödinger equation, J. Comput. Phys. 91 (1990) 413–423. [17] T.E. Simos, Exponential fitted methods for the numerical solution of the Schrödinger equation, J. Comput. Math. 14 (1996) 120–134. [18] A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation, J. Comput. Phys. 6 (1970) 378–391. [19] T.E. Simos, An eighth order method with minimal phase-lag for accurate computations for the elastic scattering phase-shift problem, Internat. J. Modern Phys. C 7 (1996) 825–835. [20] T. Lyche, Chebyshevian multistep methods for ordinary differential equations, Numer. Math. 10 (1972) 65–75. [21] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York, 1962. [22] A.D. Raptis, Exponential multistep methods for ordinary differential equations, Bull. Greek Math. Soc. 25 (1984) 113–126. [23] T.E. Simos, Numerical solution of ordinary differential equations with periodical solution, Doctoral Dissertation, National Technical University of Athens, 1990. [24] T.E. Simos, An eighth order exponentially-fitted method for the numerical solution of the Schrödinger equation, Internat. J. Modern Phys. C 9 (1998) 271–288. [25] L.F. Shampine, H.A. Watts, S.M. Davenport, Solving non-stiff ordinary differential equations-the state of the art, SIAM Rev. 18 (1975) 376–411. [26] R.B. Bernstein, A. Dalgarno, H. Massey, I.C. Percival, Thermal scattering of atoms by homonuclear diatomic molecules, Proc. Royal Soc. London Ser. A 274 (1963) 427–442. [27] R.B. Bernstrein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams, J. Chemical Phys. 33 (1960) 795–804. [28] T.E. Simos, Eighth order methods for accurate computations for the Schrödinger equation, Comput. Phys. Comm. 105 (1997) 127–138.
T.E. Simos / Computer Physics Communications 125 (2000) 21–59
Appendix A
q1 = −486 w3 sinh(T1 ) + 18000 cosh(T3 ) + 1179 w2 cosh(T6 ) + 3960 w sinh(T3 ) √ − 18000 cosh(T4 ) − 6120 cosh(T5 ) + 4572 w sinh(T5 ) − 1080 21 cosh(T2 ) − 3321 w2 cosh(T2 ) + 114 w3 sinh(T6 ) − 11880 cosh(T2 ) − 1179 w2 cosh(T5 ) √ + 3321 w2 cosh(T1 ) + 1080 21 cosh(T6 ) − 9828 w sinh(T1 ) + 4572 w sinh(T6 ) √ − 9828 w sinh(T2 ) + 300 w3 sinh(T3 ) + 300 w3 sinh(T4 ) + 1080 21 cosh(T5 ) + 114 w3 sinh(T5 ) − 486 w3 sinh(T2 ) + 4500 w2 cosh(T3 ) − 4500 w2 cosh(T4 ) √ + 3960 w sinh(T4 ) − 1080 21 cosh(T1 ) + 6120 cosh(T6 ) + 11880 cosh(T1 ) √ √ √ − 1548 w 21 sinh(T2 ) − 960 w 21 sinh(T3 ) − 711 w2 21 cosh(T2 ) √ √ √ + 450 w2 21 cosh(T4 ) + 450 w2 21 cosh(T3 ) + 1548 w 21 sinh(T1 ) √ √ √ − 948 w 21 sinh(T5 ) + 948 w 21 sinh(T6 ) + 960 w 21 sinh(T4 ) √ √ √ − 200 w3 21 sinh(T4 ) + 200 w3 21 sinh(T3 ) + 26 w3 21 sinh(T6 ) √ √ √ − 126 w3 21 sinh(T2 ) + 261 w2 21 cosh(T5 ) + 126 w3 21 sinh(T1 ) √ √ √ − 711 w2 21 cosh(T1 ) + 261 w2 21 cosh(T6 ) − 26 w3 21 sinh(T5 ) / −63 w3 sinh(T1 ) + 11880 cosh(T3 ) + 324 w sinh(T3 ) − 11880 cosh(T4 ) √ √ √ − 1080 21 cosh(T2 ) + 1080 21 cosh(T3 ) + 1080 21 cosh(T4 ) + 414 w2 cosh(T2 ) − 6120 cosh(T2 ) − 414 w2 cosh(T1 ) + 324 w sinh(T1 ) + 324 w sinh(T2 ) − 63 w3 sinh(T3 ) − 63 w3 sinh(T4 ) − 63 w3 sinh(T2 ) − 486 w2 cosh(T3 ) + 486 w2 cosh(T4 ) √ √ + 324 w sinh(T4 ) − 1080 21 cosh(T1 ) + 6120 cosh(T1 ) − 84 w 21 sinh(T2 ) √ √ √ + 684 w 21 sinh(T3 ) + 126 w2 21 cosh(T2 ) − 126 w2 21 cosh(T4 ) √ √ √ − 126 w2 21 cosh(T3 ) + 84 w 21 sinh(T1 ) − 684 w 21 sinh(T4 ) √ √ √ √ + 33 w3 21 sinh(T4 ) − 33 w3 21 sinh(T3 ) − 17 w3 21 sinh(T2 ) + 17 w3 21 sinh(T1 ) √ + 126 w2 21 cosh(T1 ) , √ T1 = 16 w (9 + 2 21), √ T2 = 16 w (−9 + 2 21), √ T3 = 16 w (2 21 − 3), √ T4 = 16 w (2 21 + 3), √ T5 = 16 w (15 + 2 21), √ T6 = 16 w (−15 + 2 21), t0 = 1062 w3 sinh(T7 ) + 2592 cosh(T9 ) − 4608 w sinh(T9 ) − 2592 cosh(T10 ) √ √ √ + 1728 21 cosh(T8 ) − 1728 21 cosh(T9 ) − 1728 21 cosh(T10 ) + 5526 w2 cosh(T8 ) + 2592 cosh(T8 ) − 5526 w2 cosh(T7 ) + 9792 w sinh(T7 ) + 9792 w sinh(T8 ) − 6138 w3 sinh(T9 ) − 6138 w3 sinh(T10 ) + 1062 w3 sinh(T8 ) − 8874 w2 cosh(T9 ) √ + 8874 w2 cosh(T10 ) − 4608 w sinh(T10 ) + 1728 21 cosh(T7 ) − 2592 cosh(T7 )
33
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√ √ √ − 33 w5 21 sinh(T9 ) + 17 w5 21 sinh(T7 ) + 33 w5 21 sinh(T10 ) √ √ √ − 284 w4 21 cosh(T10 ) − 284 w4 21 cosh(T9 ) − 16 w4 21 cosh(T8 ) √ √ − 16 w4 21 cosh(T7 ) − 17 w5 21 sinh(T8 ) − 1074 w4 cosh(T9 ) + 1074 w4 cosh(T10 ) − 63 w5 sinh(T7 ) − 63 w5 sinh(T8 ) + 174 w4 cosh(T7 ) − 63 w5 sinh(T9 ) − 63 w5 sinh(T10 ) √ √ − 174 w4 cosh(T8 ) + 1728 w 21 sinh(T8 ) − 1728 w 21 sinh(T9 ) √ √ √ + 984 w2 21 cosh(T8 ) − 984 w2 21 cosh(T10 ) − 984 w2 21 cosh(T9 ) √ √ √ − 1728 w 21 sinh(T7 ) + 1728 w 21 sinh(T10) + 558 w3 21 sinh(T10 ) √ √ √ − 558 w3 21 sinh(T9 ) + 258 w3 21 sinh(T8 ) − 258 w3 21 sinh(T7 ) √ + 984 w2 21 cosh(T7 ) / w4 (−63 w3 sinh(T7 ) + 11880 cosh(T9 ) + 324 w sinh(T9 ) √ √ √ − 11880 cosh(T10 ) − 1080 21 cosh(T8 ) + 1080 21 cosh(T9 ) + 1080 21 cosh(T10 ) + 414 w2 cosh(T8 ) − 6120 cosh(T8 ) − 414 w2 cosh(T7 ) + 324 w sinh(T7 ) + 324 w sinh(T8 ) − 63 w3 sinh(T9 ) − 63 w3 sinh(T10 ) − 63 w3 sinh(T8 ) − 486 w2 cosh(T9 ) √ + 486 w2 cosh(T10 ) + 324 w sinh(T10 ) − 1080 21 cosh(T7 ) + 6120 cosh(T7 ) √ √ √ − 84 w 21 sinh(T8 ) + 684 w 21 sinh(T9 ) + 126 w2 21 cosh(T8 ) √ √ √ − 126 w2 21 cosh(T10 ) − 126 w2 21 cosh(T9 ) + 84 w 21 sinh(T7 ) √ √ √ √ − 684 w 21 sinh(T10 ) + 33 w3 21 sinh(T10 ) − 33 w3 21 sinh(T9 ) − 17 w3 21 sinh(T8 ) √ √ + 17 w3 21 sinh(T7 ) + 126 w2 21 cosh(T7 )) , √ T7 = 16 w (9 + 2 21), √ T8 = 16 w (−9 + 2 21), √ T9 = 16 w (2 21 − 3), √ T10 = 16 w (2 21 + 3), √ t1 = − 6894 w3 sinh(T11 ) − 43200 cosh(T13 ) + 38 w5 21 sinh(T15 ) √ √ − 141 w4 21 cosh(T15 ) + 141 w4 21 cosh(T16 ) + 182 w5 sinh(T16 ) + 599 w4 cosh(T16 ) √ + 599 w4 cosh(T15 ) + 38 w5 21 sinh(T16 ) − 182 w5 sinh(T15 ) − 2454 w2 cosh(T16 ) −8640 w sinh(T13 ) − 43200 cosh(T14 ) + 14688 cosh(T15 ) − 12096 w sinh(T15 ) √ + 2592 21 cosh(T12 ) + 11754 w2 cosh(T12 ) − 1806 w3 sinh(T16 ) + 28512 cosh(T12 ) √ − 2454 w2 cosh(T15 ) + 11754 w2 cosh(T11) + 2592 21 cosh(T16 ) − 31104 w sinh(T11 ) + 12096 w sinh(T16 ) + 31104 w sinh(T12 ) − 13920 w3 sinh(T13 ) + 13920 w3 sinh(T14 ) √ − 2592 21 cosh(T15 ) + 1806 w3 sinh(T15 ) − 6894 w3 sinh(T12 ) − 9300 w2 cosh(T13 ) √ − 9300 w2 cosh(T14 ) + 8640 w sinh(T14 ) − 2592 21 cosh(T11 ) + 14688 cosh(T16 ) √ √ √ + 28512 cosh(T11 ) + 100 w5 21 sinh(T13 ) − 162 w5 21 sinh(T11) + 100 w5 21 sinh(T14 ) √ √ √ − 1500 w4 21 cosh(T14 ) + 1500 w4 21 cosh(T13 ) − 1359 w4 21 cosh(T12 ) √ √ + 1359 w4 21 cosh(T11 ) − 162 w5 21 sinh(T12 ) − 1850 w4 cosh(T13 ) − 1850 w4 cosh(T14 ) + 882 w5 sinh(T11 ) − 882 w5 sinh(T12 ) − 5949 w4 cosh(T11 ) + 1400 w5 sinh(T13 ) √ √ − 1400 w5 sinh(T14 ) − 5949 w4 cosh(T12) + 8064 w 21 sinh(T12 ) − 960 w 21 sinh(T13 )
T.E. Simos / Computer Physics Communications 125 (2000) 21–59
√ √ √ 21 cosh(T12 ) + 4800 w2 21 cosh(T14 ) − 4800 w2 21 cosh(T13 ) √ √ √ + 8064 w 21 sinh(T11 ) + 3264 w 21 sinh(T15 ) + 3264 w 21 sinh(T16 ) √ √ √ − 960 w 21 sinh(T14 ) − 6180 w3 21 sinh(T14 ) − 6180 w3 21 sinh(T13 ) √ √ √ − 354 w3 21 sinh(T16 ) − 2754 w3 21 sinh(T12 ) + 786 w2 21 cosh(T15 ) √ √ √ − 2754 w3 21 sinh(T11) − 4014 w2 21 cosh(T11 ) − 786 w2 21 cosh(T16 ) √ − 354 w3 21 sinh(T15 ) / w4 (−119 w3 sinh(T11 ) − 7560 cosh(T13) − 4788 w sinh(T13 ) √ √ √ − 7560 cosh(T14 ) + 2040 21 cosh(T12) − 3960 21 cosh(T13 ) + 3960 21 cosh(T14 ) + 4014 w2
− 882 w2 cosh(T12 ) + 7560 cosh(T12 ) − 882 w2 cosh(T11 ) − 588 w sinh(T11 ) + 588 w sinh(T12 ) + 231 w3 sinh(T13 ) − 231 w3 sinh(T14 ) + 119 w3 sinh(T12 ) √ + 882 w2 cosh(T13 ) + 882 w2 cosh(T14 ) + 4788 w sinh(T14 ) − 2040 21 cosh(T11 ) √ √ √ + 7560 cosh(T11 ) − 108 w 21 sinh(T12) − 108 w 21 sinh(T13 ) − 138 w2 21 cosh(T12 ) √ √ √ − 162 w2 21 cosh(T14) + 162 w2 21 cosh(T13 ) − 108 w 21 sinh(T11 ) √ √ √ √ − 108 w 21 sinh(T14 ) + 21 w3 21 sinh(T14 ) + 21 w3 21 sinh(T13 ) + 21 w3 21 sinh(T12 ) √ √ + 21 w3 21 sinh(T11 ) + 138 w2 21 cosh(T11 )) , √ T11 = 16 w (9 + 2 21), √ T12 = 16 w (−9 + 2 21), √ T13 = 16 w (2 21 − 3), √ T14 = 16 w (2 21 + 3), √ T15 = 16 w (15 + 2 21), √ T16 = 16 w (−15 + 2 21), t2 = −18 432 cosh( 32 w) − 144 cosh( 52 w) + 216 w3 sinh( 12 w) − 288 cosh( 12 w) − 288 w sinh( 32 w) − w4 cosh( 52 w) + 72 w3 sinh( 32 w) + 15 w2 cosh( 52 w) − 9 w2 cosh( 32 w) − 9 w4 cosh( 32 w) − 6 cosh( 12 w) w2 + 864 sinh( 12 w) w + 46 w4 cosh( 12 w) / w4 (−119 w3 sinh(T17 ) − 7560 cosh(T19 ) − 4788 w sinh(T19 ) − 7560 cosh(T20 ) √ √ √ + 2040 21 cosh(T18 ) − 3960 21 cosh(T19 ) + 3960 21 cosh(T20 ) − 882 w2 cosh(T18 ) + 7560 cosh(T18 ) − 882 w2 cosh(T17 ) − 588 w sinh(T17 ) + 588 w sinh(T18 ) + 231 w3 sinh(T19 ) − 231 w3 sinh(T20 ) + 119 w3 sinh(T18 ) + 882 w2 cosh(T19 ) √ + 882 w2 cosh(T20 ) + 4788 w sinh(T20 ) − 2040 21 cosh(T17 ) + 7560 cosh(T17 ) √ √ √ − 108 w 21 sinh(T18) − 108 w 21 sinh(T19 ) − 138 w2 21 cosh(T18 ) √ √ √ − 162 w2 21 cosh(T20) + 162 w2 21 cosh(T19 ) − 108 w 21 sinh(T17 ) √ √ √ √ − 108 w 21 sinh(T20 ) + 21 w3 21 sinh(T20 ) + 21 w3 21 sinh(T19 ) + 21 w3 21 sinh(T18 ) √ √ + 21 w3 21 sinh(T17 ) + 138 w2 21 cosh(T17 )) , √ T17 = 16 w (9 + 2 21), √ T18 = 16 w (−9 + 2 21), √ T19 = 16 w (2 21 − 3),
35
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√ T20 = 16 w (2 21 + 3), √ √ √ √ t3 = 32 324 cosh(T26 ) + 216 cosh( 13 w 21) 21 + 28 w4 cosh( 13 w 21) 21 √ √ √ √ + 42 cosh( 13 w 21) 21 w2 + 882 w3 sinh( 13 w 21) − 72 sinh( 13 w 21) w √ √ √ − 21 21 w2 cosh(T26 ) − 21 21 w2 cosh(T25 ) + 108 w 21 sinh(T26 ) √ √ √ − 108 w 21 sinh(T25 ) + 10 w4 21 cosh(T26 ) + 10 w4 21 cosh(T25 ) √ √ − 27 w3 21 sinh(T26 ) + 27 w3 21 sinh(T25 ) + 99 w2 cosh(T26 ) − 99 w2 cosh(T25 ) √ √ − 108 21 cosh(T26 ) − 108 21 cosh(T25 ) + 153 w3 sinh(T26 ) − 42 w4 cosh(T26 ) + 42 w4 cosh(T25 ) + 153 w3 sinh(T25 ) − 612 w sinh(T26 ) − 612 w sinh(T25 ) − 324 cosh(T25 ) / w4 (−63 w3 sinh(T21 ) + 11880 cosh(T23 ) + 324 w sinh(T23 ) − 11880 cosh(T24 ) √ √ √ − 1080 21 cosh(T22 ) + 1080 21 cosh(T23 ) + 1080 21 cosh(T24 ) + 414 w2 cosh(T22 ) − 6120 cosh(T22 ) − 414 w2 cosh(T21 ) + 324 w sinh(T21 ) + 324 w sinh(T22 ) − 63 w3 sinh(T23 ) − 63 w3 sinh(T24 ) − 63 w3 sinh(T22 ) − 486 w2 cosh(T23 ) + 486 w2 cosh(T24 ) √ √ + 324 w sinh(T24 ) − 1080 21 cosh(T21 ) + 6120 cosh(T21 ) − 84 w 21 sinh(T22 ) √ √ √ + 684 w 21 sinh(T23 ) + 126 w2 21 cosh(T22 ) − 126 w2 21 cosh(T24 ) √ √ √ − 126 w2 21 cosh(T23) + 84 w 21 sinh(T21 ) − 684 w 21 sinh(T24 ) √ √ √ √ + 33 w3 21 sinh(T24 ) − 33 w3 21 sinh(T23 ) − 17 w3 21 sinh(T22 ) + 17 w3 21 sinh(T21 ) √ + 126 w2 21 cosh(T21 )) , √ T21 = 16 w (9 + 2 21), √ T22 = 16 w (−9 + 2 21), √ T23 = 16 w (2 21 − 3), √ T24 = 16 w (2 21 + 3), √ T25 = 13 w (−6 + 21), √ T26 = 13 w (6 + 21). Appendix B 41 29693 25036829 w10 + w12 − w14 + · · · , 203212800 2703868231680 39368321453260800 41 11 52681 1018207991 9589567003 − w2 − w4 + w6 − w8 t0 = 560 112896 12517908480 984208036331520 122479222299033600 39035969395583 168903180855390896711 w10 − w12 + 8550682772769865728000 716297946853922960880697344000 4832018927488114120127 w14 + · · · , + 417171924247724732416918133145600 41 326 355753 1549241257 136765543891 − w2 − w4 − w6 − w8 t1 = 735 32928 3651056640 287060677263360 321507958534963200 15775682947969709 343004047144453960103 w10 − w12 + 606029641520064233472000 208920234499060863590203392000 q1 = −2 −
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T.E. Simos / Computer Physics Communications 125 (2000) 21–59
1600153234280837955899 w14 + · · · , 18435627965492885902262796288000 123 9 70283 1736345993 + w2 − w4 + w6 t2 = − 19600 2195200 16226918400 6379126161408000 65306657723 32313479393985497 w8 + w10 − 4286772780466176000 40401976101337615564800000 2240625777206516309 86306291395759665110351 w12 + w14 + · · · , − 54619669150081271526850560000 40558381524084348984978151833600000 41 136 55997 48641363 5688784027 + w2 + w4 + w6 − w8 t3 = 525 44100 977961600 34950569472000 28706067726336000 3904144288624633 46734209227376971373 w10 − w12 + 270548947107171532800000 55960777097962731318804480000 35155349956079433349517 + w14 + · · · . 814788914546337368001793228800000
37
+
Appendix C √ √ √ 1 24 21 w2 cosh(T2 ) − 24 21 w2 cosh(T1 ) + 18 w 21 sinh(T2 ) a0 = − 18 √ + 18 w 21 sinh(T1 ) − 21 w3 sinh(T2 ) + 21 w3 sinh(T1 ) − 126 w2 cosh(T2 ) √ √ − 126 w2 cosh(T1 ) + 7 21 w3 sinh(T2 ) + 7 21 w3 sinh(T1 ) − 126 w sinh(T2 ) √ √ + 126 w sinh(T1 ) + 18 21 cosh(T2 ) − 18 21 cosh(T1 ) / 5 w2 + w2 cosh(2 w) + 3 − 3 cosh(2 w) , √ T1 = 13 w (−3 + 21 , √ T2 = 13 w (3 + 21 , √ √ √ √ 1 −18 21 cosh(T4 ) + 36 w sinh( 13 w 21) 21 − 15 21 w2 cosh(T4 ) a1 = 18 √ √ √ √ + 18 w 21 sinh(T4 ) + 18 w 21 sinh(T3 ) − 10 w3 21 sinh(T3 ) − 10 w3 21 sinh(T4 ) √ √ √ + 15 21 w2 cosh(T3 ) + 42 w3 sinh(T3 ) + 12 w3 sinh( 13 w 21) 21 − 42 w3 sinh(T4 ) √ √ − 126 w sinh(T3 ) + 126 w sinh(T4 ) + 18 21 cosh(T3 ) + 108 cosh( 13 w 21) √ − 54 cosh(T3 ) − 54 cosh(T4 ) − 198 cosh( 13 w 21) w2 − 45 w2 cosh(T4 ) − 45 w2 cosh(T3 ) / 5 w2 + w2 cosh(2 w) + 3 − 3 cosh(2 w) , √ T3 = 13 w (6 + 21 , √ T4 = 13 w (−6 + 21 , √ √ 1 −21 w sinh( 13 w (3 + 21)) + 21 w sinh( 13 w (−3 + 21)) a2 = 18 √ √ √ √ √ √ − 4 21 cosh( 13 w (3 + 21)) + 4 21 cosh( 13 w (−3 + 21)) + 7 w 21 sinh( 13 w (3 + 21)) √ √ + 7 w 21 sinh( 13 w (−3 + 21)) / 5 w2 + w2 cosh(2 w) + 3 − 3 cosh(2 w) , √ √ √ √ 1 −10 w 21 sinh(T6 ) − 10 w 21 sinh(T5 ) + 12 w sinh( 13 w 21) 21 a3 = − 18 √ √ − 13 21 cosh(T6 ) + 13 21 cosh(T5 ) + 42 w sinh(T6 ) − 42 w sinh(T5 ) √ − 126 cosh( 13 w 21) + 63 cosh(T6 ) + 63 cosh(T5 ) / 5 w2 + w2 cosh(2 w) + 3 − 3 cosh(2 w) ,
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√ 21), √ 1 T6 = 3 w (6 + 21), √ √ √ b0 = 16 3 w sinh( 13 w (3 + 21)) + 3 w sinh( 13 w (−3 + 21)) + 6 cosh( 13 w (3 + 21)) √ √ √ √ √ − 6 cosh( 13 w (−3 + 21)) − w 21 sinh( 13 w (3 + 21)) + w 21 sinh( 13 w (−3 + 21)) / −1 + cosh(2 w) , √ √ √ √ b1 = − 16 3 sinh( 13 w (3 + 21)) + 3 sinh( 13 w (−3 + 21)) − 21 sinh( 13 w (3 + 21)) √ √ + 21 sinh( 13 w (−3 + 21)) / w (−1 + cosh(2 w)) . T5 = 13 w (−6 +
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Appendix D 49 224 2 20 4 5704 93838 10784 − w − w − w6 − w8 − w10 81 729 729 22733865 18619035435 451153550925 403496356 178405115344 w12 − w14 + · · · , + 69095519784816525 413536685912126902125 64 448 2 40 4 11408 2308711 13709449 + w + w + w6 − w8 − w10 a1 = 81 729 729 22733865 7447614174 902307101850 19619622719 6934169136149 w12 − w14 + · · · , − 44221132662282576 827073371824253804250 56 98 4 8 44816 + w2 + w4 + w6 + w8 a2 = 243 2187 10935 13640319 279285531525 143368 368000824 15304688096 w10 + w12 − w14 + · · · , − 17594988486075 1036432796772247875 1240610057736380706375 560 2 224 292 4 20870 53514947 + w + w + w6 + w8 a3 = 243 2187 10935 13640319 1117142126100 30201449 39208407053 36142895623 w10 + w12 + w14 + · · · , + 28151981577720 2551219192054764000 244304749831164200640 √ √ √ 1√ 8 2 32 w6 21 + w8 21 − w10 21 21 − b0 = 3 25515 45927 5412825 √ √ 3632 2788 w12 21 − w14 21 + · · · , + 3723807087 39897933075 √ √ √ 2 √ 4 8 8 4 w 21 − w6 21 + w8 21 21 + b1 = 27 8505 137781 1082565 √ √ √ 1816 1472 5576 10 12 w w w14 21 + · · · . 21 + 21 − − 6206345145 17099114175 119693799225 a0 =
Appendix E 1 2124 w3 sinh(S3 ) + 5184 cosh(S5 ) + 2016 w sinh(S5 ) − 5184 cosh(S6 ) c0 = 32 √ √ √ − 1728 21 cosh(S4 ) + 1728 21 cosh(S5 ) + 1728 21 cosh(S6 ) + 2664 w2 cosh(S4 )
− 5184 cosh(S4 ) − 2664 w2 cosh(S3 ) − 7200 w sinh(S3 ) − 7200 w sinh(S4 ) + 3276 w3 sinh(S5 ) + 3276 w3 sinh(S6 ) + 2124 w3 sinh(S4 ) − 7272 w2 cosh(S5 )
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√ + 7272 w2 cosh(S6 ) + 2016 w sinh(S6 ) − 1728 21 cosh(S3 ) + 5184 cosh(S3 ) √ √ √ − 33 w5 21 sinh(S5 ) + 17 w5 21 sinh(S3 ) + 33 w5 21 sinh(S6 ) √ √ √ + 146 w4 21 cosh(S6 ) + 146 w4 21 cosh(S5 ) + 46 w4 21 cosh(S4 ) √ √ + 46 w4 21 cosh(S3 ) − 17 w5 21 sinh(S4 ) − 150 w4 cosh(S5 ) + 150 w4 cosh(S6 ) − 63 w5 sinh(S3 ) − 63 w5 sinh(S4 ) − 78 w4 cosh(S3 ) − 63 w5 sinh(S5 ) − 63 w5 sinh(S6 ) √ √ √ + 78 w4 cosh(S4 ) − 864 w 21 sinh(S4 ) − 864 w 21 sinh(S5 ) + 312 w2 21 cosh(S4 ) √ √ √ − 312 w2 21 cosh(S6 ) − 312 w2 21 cosh(S5 ) + 864 w 21 sinh(S3 ) √ √ √ + 864 w 21 sinh(S6 ) − 36 w3 21 sinh(S6 ) + 36 w3 21 sinh(S5 ) √ √ √ + 420 w3 21 sinh(S4 ) − 420 w3 21 sinh(S3 ) + 312 w2 21 cosh(S3 ) / 324 cosh(S2 ) √ √ √ √ √ √ + 216 cosh( 13 w 21) 21 + 28 w4 cosh( 13 w 21) 21 + 42 cosh( 13 w 21) 21 w2 √ √ √ + 882 w3 sinh( 13 w 21) − 72 sinh( 13 w 21) w − 21 21 w2 cosh(S2 ) √ √ √ − 21 21 w2 cosh(S1 ) + 108 w 21 sinh(S2 ) − 108 w 21 sinh(S1 ) √ √ √ √ + 10 w4 21 cosh(S2 ) + 10 w4 21 cosh(S1 ) − 27 w3 21 sinh(S2 ) + 27 w3 21 sinh(S1 ) √ √ + 99 w2 cosh(S2 ) − 99 w2 cosh(S1 ) − 108 21 cosh(S2 ) − 108 21 cosh(S1 ) + 153 w3 sinh(S2 ) − 42 w4 cosh(S2 ) + 42 w4 cosh(S1 ) + 153 w3 sinh(S1 ) − 612 w sinh(S2 ) − 612 w sinh(S1 ) − 324 cosh(S1 ) , √ S1 = 13 w (−6 + 21), √ S2 = 13 w (6 + 21), √ S3 = 16 w (9 + 2 21), √ S4 = 16 w (−9 + 2 21), √ S5 = 16 w (2 21 − 3), √ S6 = 16 w (2 21 + 3), √ 1 c1 = − 32 −6372 w3 sinh(S11 ) − 10368 cosh(S13 ) + 26 w5 21 sinh(S9 ) √ √ − 181 w4 21 cosh(S9 ) − 181 w4 21 cosh(S10 ) − 114 w5 sinh(S10 ) − 843 w4 cosh(S10 ) √ + 843 w4 cosh(S9 ) − 26 w5 21 sinh(S10 ) − 114 w5 sinh(S9 ) + 7128 w2 cosh(S10 ) − 2880 w sinh(S13 ) + 10368 cosh(S14 ) − 5184 cosh(S9 ) + 12384 w sinh(S9 ) √ √ √ + 5184 21 cosh(S12 ) − 6912 21 cosh(S13 ) − 6912 21 cosh(S14 ) − 7992 w2 cosh(S12 ) − 324 w3 sinh(S10 ) + 15552 cosh(S12 ) − 7128 w2 cosh(S9 ) + 7992 w2 cosh(S11 ) √ + 1728 21 cosh(S10 ) + 21600 w sinh(S11 ) + 12384 w sinh(S10 ) + 21600 w sinh(S12 ) √ − 20520 w3 sinh(S13 ) − 20520 w3 sinh(S14 ) + 1728 21 cosh(S9 ) − 324 w3 sinh(S9 ) − 6372 w3 sinh(S12 ) + 15120 w2 cosh(S13 ) − 15120 w2 cosh(S14 ) − 2880 w sinh(S14 ) √ √ + 5184 21 cosh(S11 ) + 5184 cosh(S10 ) − 15552 cosh(S11 ) − 200 w5 21 sinh(S13 ) √ √ √ − 126 w5 21 sinh(S11 ) + 200 w5 21 sinh(S14 ) − 1058 w4 21 cosh(S14 ) √ √ √ − 1058 w4 21 cosh(S13 ) + 87 w4 21 cosh(S12 ) + 87 w4 21 cosh(S11 ) √ + 126 w5 21 sinh(S12 ) − 6612 w4 cosh(S13 ) + 6612 w4 cosh(S14 ) + 486 w5 sinh(S11 )
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+ 486 w5 sinh(S12 ) − 1161 w4 cosh(S11 ) − 300 w5 sinh(S13 ) − 300 w5 sinh(S14 ) √ √ + 1161 w4 cosh(S12 ) + 2592 w 21 sinh(S12 ) − 936 w2 21 cosh(S12 ) √ √ √ − 480 w2 21 cosh(S14 ) − 480 w2 21 cosh(S13 ) − 2592 w 21 sinh(S11 ) √ √ √ − 2592 w 21 sinh(S9 ) + 2592 w 21 sinh(S10 ) + 480 w3 21 sinh(S14 ) √ √ √ − 480 w3 21 sinh(S13 ) − 12 w3 21 sinh(S10 ) − 1260 w3 21 sinh(S12 ) √ √ √ + 1416 w2 21 cosh(S9 ) + 1260 w3 21 sinh(S11 ) − 936 w2 21 cosh(S11 ) √ √ + 1416 w2 21 cosh(S10 ) + 12 w3 21 sinh(S9 ) / 324 cosh(S8 ) √ √ √ √ √ √ + 216 cosh( 13 w 21) 21 + 28 w4 cosh( 13 w 21) 21 + 42 cosh( 13 w 21) 21 w2 √ √ √ + 882 w3 sinh( 13 w 21) − 72 sinh( 13 w 21) w − 21 21 w2 cosh(S8 ) √ √ √ − 21 21 w2 cosh(S7 ) + 108 w 21 sinh(S8 ) − 108 w 21 sinh(S7 ) √ √ √ √ + 10 w4 21 cosh(S8 ) + 10 w4 21 cosh(S7 ) − 27 w3 21 sinh(S8 ) + 27 w3 21 sinh(S7 ) √ √ + 99 w2 cosh(S8 ) − 99 w2 cosh(S7 ) − 108 21 cosh(S8 ) − 108 21 cosh(S7 ) + 153 w3 sinh(S8 ) − 42 w4 cosh(S8 ) + 42 w4 cosh(S7 ) + 153 w3 sinh(S7 ) − 612 w sinh(S8 ) − 612 w sinh(S7 ) − 324 cosh(S7 ) , √ S7 = 13 w (−6 + 21), √ S8 = 13 w (6 + 21), √ S9 = 16 w (15 + 2 21), √ S10 = 16 w (−15 + 2 21), √ S11 = 16 w (9 + 2 21), √ S12 = 16 w (−9 + 2 21), √ S13 = 16 w (2 21 − 3), √ S14 = 16 w (2 21 + 3), 1 −63 w3 sinh(S17 ) + 486 cosh(S19 ) + 846 w sinh(S19 ) − 486 cosh(S20 ) c2 = − 32 √ √ √ − 216 21 cosh(S18 ) + 216 21 cosh(S19 ) + 216 21 cosh(S20 ) − 510 w2 cosh(S18 )
− 378 cosh(S18 ) + 510 w2 cosh(S17 ) − 1170 w sinh(S17 ) − 1170 w sinh(S18 ) − 63 w3 sinh(S19 ) − 63 w3 sinh(S20 ) − 63 w3 sinh(S18 ) − 738 w2 cosh(S19 ) √ + 738 w2 cosh(S20 ) + 846 w sinh(S20 ) − 216 21 cosh(S17 ) + 378 cosh(S17 ) √ √ √ √ − 198 w 21 sinh(S18 ) − 54 w 21 sinh(S19 ) − 96 w2 21 cosh(S18 ) − 12 w2 21 cosh(S20 ) √ √ √ √ − 12 w2 21 cosh(S19 ) + 198 w 21 sinh(S17 ) + 54 w 21 sinh(S20 ) + 33 w3 21 sinh(S20 ) √ √ √ √ − 33 w3 21 sinh(S19 ) − 17 w3 21 sinh(S18 ) + 17 w3 21 sinh(S17 ) − 96 w2 21 cosh(S17 ) / √ √ √ √ 324 cosh(S16 ) + 216 cosh( 13 w 21) 21 + 28 w4 cosh( 13 w 21) 21 √ √ √ √ + 42 cosh( 13 w 21) 21 w2 + 882 w3 sinh( 13 w 21) − 72 sinh( 13 w 21) w √ √ √ − 21 21 w2 cosh(S16 ) − 21 21 w2 cosh(S15 ) + 108 w 21 sinh(S16 ) √ √ √ − 108 w 21 sinh(S15 ) + 10 w4 21 cosh(S16 ) + 10 w4 21 cosh(S15 )
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√ √ 21 sinh(S16 ) + 27 w3 21 sinh(S15 ) + 99 w2 cosh(S16 ) − 99 w2 cosh(S15 ) √ √ − 108 21 cosh(S16 ) − 108 21 cosh(S15 ) + 153 w3 sinh(S16 ) − 42 w4 cosh(S16 ) + 42 w4 cosh(S15 ) + 153 w3 sinh(S15 ) − 612 w sinh(S16 ) − 612 w sinh(S15 ) − 324 cosh(S15 ) , √ S15 = 13 w (−6 + 21), √ S16 = 13 w (6 + 21), √ S17 = 16 w (9 + 2 21), √ S18 = 16 w (−9 + 2 21), √ S19 = 16 w (2 21 − 3), √ S20 = 16 w (2 21 + 3), − 27 w3
1 882 w3 sinh(S25 ) − 5940 cosh(S27 ) + 39 w2 cosh(S24 ) + 5760 w sinh(S27 ) c3 = − 32 √ − 5940 cosh(S28 ) − 1782 cosh(S23 ) + 1638 w sinh(S23 ) + 1134 21 cosh(S26 ) √ √ − 1440 21 cosh(S27 ) + 1440 21 cosh(S28 ) − 1581 w2 cosh(S26 ) + 182 w3 sinh(S24 ) √ + 7722 cosh(S26 ) + 39 w2 cosh(S23 ) − 1581 w2 cosh(S25 ) − 306 21 cosh(S24 )
− 2970 w sinh(S25 ) − 1638 w sinh(S24 ) + 2970 w sinh(S26 ) + 1400 w3 sinh(S27 ) √ − 1400 w3 sinh(S28 ) + 306 21 cosh(S23 ) − 182 w3 sinh(S23 ) − 882 w3 sinh(S26 ) √ + 2406 w2 cosh(S27 ) + 2406 w2 cosh(S28 ) − 5760 w sinh(S28 ) − 1134 21 cosh(S25 ) √ √ − 1782 cosh(S24 ) + 7722 cosh(S25 ) + 1302 w 21 sinh(S26 ) − 660 w 21 sinh(S27 ) √ √ √ − 639 w2 21 cosh(S26 ) − 2204 w2 21 cosh(S28 ) + 2204 w2 21 cosh(S27 ) √ √ √ + 1302 w 21 sinh(S25 ) − 426 w 21 sinh(S23 ) − 426 w 21 sinh(S24 ) √ √ √ − 660 w 21 sinh(S28 ) + 100 w3 21 sinh(S28 ) + 100 w3 21 sinh(S27 ) √ √ √ + 38 w3 21 sinh(S24 ) − 162 w3 21 sinh(S26 ) − 29 w2 21 cosh(S23 ) √ √ √ − 162 w3 21 sinh(S25 ) + 639 w2 21 cosh(S25 ) + 29 w2 21 cosh(S24 ) √ √ √ √ √ + 38 w3 21 sinh(S23 ) / −24 w sinh( 13 w 21) 21 + 294 w3 sinh( 13 w 21) 21 √ √ √ − 756 cosh(S22 ) + 1512 cosh( 13 w 21) + 196 w4 cosh( 13 w 21) + 294 cosh( 13 w 21) w2 √ √ √ + 33 21 w2 cosh(S22 ) − 33 21 w2 cosh(S21 ) − 204 w 21 sinh(S22 ) √ √ √ − 204 w 21 sinh(S21 ) − 14 w4 21 cosh(S22 ) + 14 w4 21 cosh(S21 ) √ √ + 51 w3 21 sinh(S22 ) + 51 w3 21 sinh(S21 ) − 147 w2 cosh(S22 ) − 147 w2 cosh(S21 ) √ √ + 108 21 cosh(S22 ) − 108 21 cosh(S21 ) − 189 w3 sinh(S22 ) + 70 w4 cosh(S22 ) + 70 w4 cosh(S21 ) + 189 w3 sinh(S21 ) + 756 w sinh(S22 ) − 756 w sinh(S21 ) − 756 cosh(S21 ) , √ S21 = 13 w (−6 + 21), √ S22 = 13 w (6 + 21), √ S23 = 16 w (15 + 2 21), √ S24 = 16 w (−15 + 2 21), √ S25 = 16 w (9 + 2 21),
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√ S26 = 16 w (−9 + 2 21), √ S27 = 16 w (2 21 − 3), √ S28 = 16 w (2 21 + 3), 9 −w2 cosh( 52 w) − 9 w2 cosh( 32 w) − 24 w sinh( 32 w) + 15 cosh( 52 w) c4 = − 16 − 72 sinh( 12 w) w − 9 cosh( 32 w) − 6 cosh( 12 w) + 46 cosh( 12 w) w2 / √ √ √ √ − 24 w sinh( 13 w 21) 21 + 294 w3 sinh( 13 w 21) 21 − 756 cosh(S30 ) √ √ √ + 1512 cosh( 13 w 21) + 196 w4 cosh( 13 w 21) + 294 cosh( 13 w 21) w2 √ √ √ + 33 21 w2 cosh(S30 ) − 33 21 w2 cosh(S29 ) − 204 w 21 sinh(S30 ) √ √ √ − 204 w 21 sinh(S29 ) − 14 w4 21 cosh(S30 ) + 14 w4 21 cosh(S29 ) √ √ + 51 w3 21 sinh(S30 ) + 51 w3 21 sinh(S29 ) − 147 w2 cosh(S30 ) − 147 w2 cosh(S29 ) √ √ + 108 21 cosh(S30 ) − 108 21 cosh(S29 ) − 189 w3 sinh(S30 ) + 70 w4 cosh(S30 ) + 70 w4 cosh(S29 ) + 189 w3 sinh(S29 ) + 756 w sinh(S30 ) − 756 w sinh(S29 ) − 756 cosh(S29 ) , √ S29 = 13 w (−6 + 21), √ S30 = 13 w (6 + 21), √ √ √ d0 = − 18 −3 w2 21 cosh(S33 ) + 3 w2 21 cosh(S36 ) − 3 w2 21 cosh(S35 ) √ √ √ √ − 8 w 21 sinh(S36 ) + 8 w 21 sinh(S34 ) + 8 w 21 sinh(S33 ) − 8 w 21 sinh(S35 ) √ − 24 21 cosh(S34 ) + 72 cosh(S34 ) − 72 cosh(S33 ) + 72 cosh(S35 ) − 72 cosh(S36 ) √ √ + 3 w2 21 cosh(S34 ) + 8 w sinh(S35 ) − 56 w sinh(S36 ) − 24 21 cosh(S36 )
+ 33 w2 cosh(S33 ) − 33 w2 cosh(S35 ) + 17 w2 cosh(S36 ) − 17 w2 cosh(S34 ) √ √ + 24 21 cosh(S33 ) + 24 21 cosh(S35 ) − 56 w sinh(S34 ) + 8 w sinh(S33 ) / √ 4 sinh( 13 w 21) w + 10 w sinh(S32 ) + 10 w sinh(S31 ) − 9 cosh(S32 ) + 9 cosh(S31 ) √ √ √ √ √ − 6 cosh( 13 w 21) 21 + 3 21 cosh(S32 ) + 3 21 cosh(S31 ) − 2 w 21 sinh(S32 ) √ + 2 w 21 sinh(S31 ) , √ S31 = 13 w (−6 + 21), √ S32 = 13 w (6 + 21), √ S33 = 16 w (2 21 + 3), √ S34 = 16 w (9 + 2 21), √ S35 = 16 w (2 21 − 3), √ S36 = 16 w (−9 + 2 21), √ √ √ d1 = 18 −3 w2 21 cosh(S39 ) + 3 w2 21 cosh(S41 ) − 3 w2 21 cosh(S42 ) √ √ √ √ + 8 w 21 sinh(S41 ) − 8 w 21 sinh(S40 ) + 24 w 21 sinh(S39 ) − 24 w 21 sinh(S42 ) √ − 12 cosh(S40 ) + 36 cosh(S39 ) − 36 cosh(S42 ) + 12 cosh(S41 ) + 3 w2 21 cosh(S40 ) − 18 w sinh(S42 ) + 18 w sinh(S41 ) + 33 w2 cosh(S39 ) − 33 w2 cosh(S42 ) + 17 w2 cosh(S41 ) √ − 17 w2 cosh(S40 ) + 18 w sinh(S40 ) − 18 w sinh(S39 ) / w2 (4 sinh( 13 w 21) w
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d2 =
3 4
√ √ + 10 w sinh(S38 ) + 10 w sinh(S37 ) − 9 cosh(S38 ) + 9 cosh(S37 ) − 6 cosh( 13 w 21) 21 √ √ √ √ + 3 21 cosh(S38 ) + 3 21 cosh(S37 ) − 2 w 21 sinh(S38 ) + 2 w 21 sinh(S37 )) , √ S37 = 13 w (−6 + 21), √ S38 = 13 w (6 + 21), √ S39 = 16 w (2 21 + 3), √ S40 = 16 w (9 + 2 21), √ S41 = 16 w (−9 + 2 21), √ S42 = 16 w (2 21 − 3), 22 sinh( 12 w) w + w sinh( 52 w) − 9 w sinh( 32 w) + 4 cosh( 12 w) + 2 cosh( 52 w) − 6 cosh( 32 w) / √ w2 (4 sinh( 13 w 21) w + 10 w sinh(S44 ) + 10 w sinh(S43 ) − 9 cosh(S44 ) + 9 cosh(S43 ) √ √ √ √ √ − 6 cosh( 13 w 21) 21 + 3 21 cosh(S44 ) + 3 21 cosh(S43 ) − 2 w 21 sinh(S44 ) √ + 2 w 21 sinh(S43 )) , √ S43 = 13 w (−6 + 21), √ S44 = 13 w (6 + 21).
Appendix F
3211 451 2445823 4001907731 − w2 − w4 + w6 4352 1775616 47813787648 6145027988520960 597166167641357 15449745768882434917 w8 − w10 + 1434102051849067560960 307184659506070271557632000 613771437522347047212407 w12 + 157165501712409745177574375424000 4393658945657064254276762749 w14 + · · · , − 17505722242735047056858944232226816000 3211 2 3901 2445823 4001907731 + w + w4 − w6 c1 = 2176 887808 23906893824 3072513994260480 597166167641357 164748346268375963 w8 − w10 − 717051025924533780480 153592329753035135778816000 245350378618576357297033 w12 78582750856204872588787187712000 2688115359648583042429394051 w14 + · · · , − 8752861121367523528429472116113408000 93119 3967 238457299 100832529217 + w2 − w4 − w6 c2 = − 1044480 426147840 80327163248640 210686673892147200 26473419115079029 w8 + 481858289421286700482560 5950569274099655743 w10 − 1504577924111364595384320000 c0 =
43
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63211306505625208566244139 w12 264038042876848371898324950712320000 8132174607689063497190745361 w14 + · · · , − 600196191179487327663735230819205120000 266513 70211 1788502847 524971765007 + w2 + w4 + w6 c3 = 456960 186439680 35143133921280 645227938794700800 2262875218912381 46528191032221262831 w8 + w10 − 42162600324362586292224 32254389248137378513551360000 5600312588614821667047833 w12 + 115516643758621162705517165936640000 15484931228512156056520038289 w14 + · · · , − 1838100835487179940970189144383815680000 387 21513797 32352872623 9633 − w2 + w4 − w6 c4 = 2437120 331448320 6941853614080 127452432354508800 2229629828948647 w8 + 124926223183296551976960 21914159576560202597 w10 − 19113712147044372452474880000 4801338409124497334900089 w12 + 68454307412516244566232394629120000 1526460980961731847559418719 w14 + · · · , − 363081646515986161179296621112852480000 61 1 283 137287 863953 w6 + w8 − w10 + w12 d0 = − 2 92160 1769472 6688604160 706316599296 43731142691 w14 + · · · , + 385648863215616000 183 2 47 179 6498257 1113076207 + w + w4 − w6 + w8 d1 = − 512 20480 327680 22295347200 23543886643200 1033277723411 13597048809341 w10 − w12 − 257099242143744000 333200617818292224000 32427119523995353 w14 + · · · , + 432554983858655723520000 √ √ √ 15 √ 183 283 3010543 w2 21 + w4 21 − w6 21 21 − d2 = 3584 143360 1376256 156067430400 √ √ 477793048211 3038527 w8 21 + w10 21 + 10987147100160 1799694695006208000 √ √ 61509757733996239 143327944745 12 w w14 21 + · · · . 21 + − 2665604942546337792 11102244585705496903680000 +
(64)
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Appendix G
PROGRAM SUMMARY
Distribution format: default ASCII else uuencoded compressed tar file
Title of program (32 characters maximum): MAPLESIM
Keywords (descriptive of problem and method of solution): Maple programming, construction of exponentially-fitted methods
Catalogue identifier: ADLI Program Summary URL: http://cpc.cs.qub.ac.uk/summaries/ADLI Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland Computer for which the program is designed and others on which it has been tested: IBM Compatible Pentium Operating systems or monitors under which the program has been tested: DOS, Windows Programming language used: Maple Language
Please enclose a summary (300 words maximum), under the following headings: Nature of physical problem With the present program the derivation of the coefficients produced by Eq. (14) is obtain. The first part of the proposed program consists of the calculation of the matrix elements which form the coefficients of the system of equations. The second part of the proposed program, as this has been explained in [20,22,23], consists of the iterative application of the L’Hospital’s rule (to avoid coefficients of the form 00 ) for the computation of the solution of these equations that make up the coefficients of the method (14). We note that the system of equations produced by Eq. (14) is solved by an application of Cramer’s rule. The above procedure is repeated for the calculation of the coefficients of the methods (24)–(25) and for the methods (28)–(29).
Memory required to execute with typical data: words: 20 MBytes Method of solution: Symbolic Computation using Maple No. of bits in a word: 16 Restrictions on the complexity of the problem: None No. of processors used: 1 Typical running time: 1800 seconds Has the code been vectorized or parallelized?: No Unusual features of the program: None No. of bytes in distributed program, including test data, etc.: 16 462
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