Convergence acceleration of through a variational approach

September 30, 2004

Paolo Amore∗, Facultad de Ciencias, Universidad de Colima, Bernal D´ıaz del Castillo 340, Colima, Colima, M´exico Abstract By means of a variational approach we find new series representa- tions both for well known mathematical constants, such as π and the Catalan constant, and for mathematical functions, such as the Rie- mann zeta . The series that we have found are all exponen- tially convergent and provide quite useful analytical approximations. With limited effort our method can be applied to obtain similar expo- nentially convergent series for a large class of mathematical functions.

1 Introduction

This paper deals with the problem of improving the convergence of a series in the case where the series itself is converging very slowly and a large number of terms needs to be evaluated to reach the desired accuracy. This is an interesting and challenging problem, which has been considered by many authors before (see, for example, [FV 1996, Br 1999, Co 2000]). In this paper we wish to attack this problem, although aiming from a different angle: instead of looking for series expansions in some small nat- ural parameter (i.e. a parameter present in the original formula), we in- troduce an artificial dependence in the formulas upon an (not completely)

∗[email protected]

1 arbitrary parameter and then devise an expansion which can be optimized to give faster rates of convergence. The details of how this work will be explained in depth in the next section. This procedure is well known in Physics and it has been exploited in the so-called “Linear Delta Expansion” (LDE) and similar approaches [AFC 1990, Jo 1995, Fe 2000]. The author himself has devised different methods which allow to treat a large class of physical/mathematical problems: the non-perturbative generalization of the Lindstedt-Poincar´e method [AA 2003a, AA 2003b, AM 2004], the solution of the Schr¨odinger equation in quantum mechanics [Am 2004b, Am 2004c], the calculation of a certain class of integrals of physical interest [AS 2004, Am 2004a] and the analytical approximation of WKB expansion in quantum mechanics [AL 2004]. In the following we want to show that such variational techniques can also be used to obtain exponentially convergent series for some mathematical functions. Although we will focus on a few functions, the method that we apply is quite flexible and can be certainly applied to a larger number of examples.

2 Mathematical constants

Many fundamental mathematical constants can be expressed as infinite sums. In many cases such series converge very slowly and a huge number of terms has to be counted before reaching the desired precision. Several examples of this are discussed for example in [FV 1996], where the authors consider a par- ticular rearrangement of the series which transforms them into rapidly con- verging series. In the following we review two of the examples of [FV 1996], π and the Catalan constant, and apply a variational method to further improve the convergence rate of these sums.

2.1 The number π The series ∞ 1 1 S = 4 − (1) 4n − 3 4n − 1 n=1 X   converges very slowly to π and it is known as Gregory’s formula. The sum of the to the first 103 terms yields an approximate value of π, which only has

2 the first 3 decimals correct. As discussed in [FV 1996] it is possible to convert series such as the one in eq. (1) into rapidly converging ones. In the following we generalize the method of Flajolet and Vardi to make room for a variational parameter. The basic idea is to introduce in the series an arbitrary parameter, which is then tuned to accelerate the convergence of the series itself. The principle of minimal sensitivity (PMS) [Ste 1981] is used to fix the value of the parameter. Let us see how this works: we write the series as ∞ 1 1 1 1 S = 3 − 1 . (2) n 1 + λ  4n +λ 4n +λ  n=1 1 − 1 − X 1+λ 1+λ 3 1  4n +λ 4n +λ Provided that 1+λ < 1 and 1+λ < 1, i.e. λ > −5/8, we can expand eq. (2) as:

∞ ∞ m m 1 1 3 + λ 1 + λ S = 4 4n − 4n 4n 1 + λ 1 + λ 1 + λ n=1 m=1      X∞ ∞ X m m k k 1 1 +1 m 3 1 = 4 λm−k − 4n 1 + λ k 4n 4n n=1 m=1   k=1   "    # X∞ X m m X 1 +1 m 3k − 1 = 4 λm−k ζ(k + 1) , (3) 1 + λ k 4k+1 m=1 k X   X=1   where the last line has been obtained by performing the over n. For λ = 0 eq. (3) yields the expression found in [FV 1996], i.e. ∞ 3m − 1 S(F V ) = ζ(m + 1) , (4) 4m m=1 X which converges exponentially to π. Let us go back to eq. (3). This series converges to π for any value of λ > −5/8: as a result, we can regard the dependence upon λ of the partial sums of the first N terms of eq. (3) as artificial, since it is bound to disappear in the limit N → ∞. We therefore minimize such dependence using the PMS and requiring that the of the equation with respect to λ vanish. To leading order, which corresponds to keeping only two terms in the sum, we obtain the optimal value of λ: 3 λ(1) = − ζ(3) ≈ −0.365381 > −5/8 . (5) π2 3 In Fig. 1 we display the partial sums of eq. (3) with 10, 20 and 30 terms as a function of λ: the locations of the optimal value (to first order) and λ = 0 are marked with vertical lines. It turns out that λ(1) is an excellent approximation to the exact minimum of the partial sum even for large values of terms. In Fig. 2 we plot the error obtained by using eq. (3) with λ given by eq. (5) and by using the formula of Flajolet and Vardi, eq. (4). Our series converges exponentially more rapidly than eq. (4).

2.2 The Catalan constant The series ∞ 1 1 S = − (6) (4n − 3)2 (4n − 1)2 n=1 X   is known to slowly converge to the Catalan constant, G ≈ 0.9159656. We rewrite it as d S = lim S˜(a) (7) a→0 da where ∞ 1 1 S˜(a) ≡ − (4n − 3 − a) (4n − 1 − a) n=1   X∞ ∞ m m 1 1 3+a + λ 1+a + λ = 4n − 4n 4n 1 + λ 1 + λ 1 + λ n=1 m=1      X∞ ∞ X m m k k 1 1 +1 m 3 + a 1 + a = λm−k − 4n 1 + λ k 4n 4n n=1 m=1   k=1   "    # X∞ X m m X 1 +1 m (3 + a)k − (1 + a)k = λm−k ζ(k + 1) . (8) 1 + λ k 4k+1 m=1 k X   X=1   In principle we would be tempted to perform the derivation with respect to a and then take the limit a → 0, thus obtaining the expression

∞ m m 1 +1 m 3k−1 − 1 S = λm−k k ζ(k + 1) , (9) 1 + λ k 4k+1 m=1 k X   X=1   4 which indeed for λ = 0 reduces to the formula given in [FV 1996]:

∞ 3m−1 − 1 S = m ζ(m + 1) . (10) 4m+1 m=1 X However in doing so we have neglected that λ can be function of a and therefore we expect that a better approach will consist in first applying the PMS to S˜ and then take the limit. In such a case the PMS to leading order yields the optimal value 3 a λ(1) = − ζ(3) 1 + . (11) π2 2   By using this value we obtain

∞ m m−k 1 1 λ0 m k S = 3+2k m+2 ζ(k + 1) − 3 + 3 k (1 + λ0) 3 2 (1 + λ0) k m=1 k=1   X X k    − 3 −1 + 3 (λ0 − m) (12)

3 where we have defined λ0 ≡ −π2 ζ(3). In Fig. 3 we compare the different approximations, showing that eq. (9) (with λ = λ(1)) and eq. (12) have a greater then the corresponding equation of [FV 1996]. Eq. (12) provides a slightly better approximation. The results derived above can be generalized to sums of the form: ∞ 1 1 S = − (13) n (4n − 3)n (4n − 1)n n=1 X   simply using 1 dn−1 Sn = lim S˜(a). (14) (n − 1)! a→0 dan−1 3 The Riemann zeta function

We can apply the same strategy outlined above to the calculation of the Riemann zeta function. We consider the integral representation (−2)n−1 1 1 logn−1 x ζ(n) = dx (15) 2n−1 − 1 Γ(n) 1 + x Z0 5 and write it as: (−2)n−1 1 1 1 logn−1 x ζ(n) = dx, (16) 2n−1 − 1 Γ(n) 1 + λ 1 + x−λ Z0 1+λ where λ is as usual an arbitrary parameter introduced by hand. x−λ The condition 1+λ < 1 is fullfilled provided that λ > 0; in this case one can expand the denominator in powers of x−λ and obtain: 1+λ

∞ (−2)n−1 1 1 1 ζ(n) = (−x + λ)k logn−1 x dx 2n−1 − 1 Γ(n) (1 + λ)k+1 k=0 Z0 X∞ k (−2)n−1 1 1 k! 1 = λk−j(−1)j xj logn−1 x 2n−1 − 1 Γ(n) (1 + λ)k+1 j!(k − j)! k j=0 0 X=0 X Z ∞ k (−2)n−1 1 1 k! Γ(n) = λk−j(−1)n+j+1 2n−1 − 1 Γ(n) (1 + λ)k+1 j!(k − j)! (1 + j)n k j=0 X=0 X ∞ k (2)n−1 1 k! (−1)j = λk−j . (17) 2n−1 − 1 (1 + λ)k+1 j!(k − j)! (1 + j)n k j=0 X=0 X Although λ appears explicitly in the series (17), the series itself does not depend upon λ, as long as λ > 0. This means that when the sum over k is truncated to a given finite order a residual dependence upon λ will survive: such dependence will be minimized by applying the PMS [Ste 1981], i.e. by asking that the derivative of the partial sum with respect to λ vanish. To (1) −n lowest order one has that λP MS = 2 and the corresponding formula is found:

∞ k (2)n−1 1 k! (−1)j ζ(n) = 2−n(k−j) .(18) 2n−1 − 1 (1 + 2−n)k+1 j!(k − j)! (1 + j)n k j=0 X=0 X We want to stress that eq. (18) is still an exact series representation of the Riemann zeta function. This simple formula yields an excellent approx- imation to the zeta function as it can be appreciated by looking at Fig. 4: there we plot the error (ζ(kmax)(n) − ζ(n))/ζ(n) obtained by using eq. (18) to different orders, for 1 < n ≤ 10. It is remarkable that this simple analyti- cal formula works quite well even in proximity of n = 1 where the ζ function diverges.

6 The rate of convergence of the series is greatly improved by applying the PMS to higher orders†. Although it is possible to find the analytical solution to the PMS equation only to low orders, we observe in Fig. 5 for ζ(3) that λP MS quite rapidly reaches an asymptotic value, which is found numerically (∞) to be λP MS = 0.449408149787716779307327177571409. In Fig. 6 we display the dependence upon λ of the partial sums over the first kmax = 11, 31, 51 terms in the case of ζ(3). The vertical line corresponds (∞) to the location of λP MS. In Fig. 7 we plot the difference |ζapp(3) − ζ(3)| using eq. (17) with λ = (∞) λP MS (solid line), λ = 0 (dashed line) and the series representation

∞ 1 ζ(3) = (19) (n + 1)3 n=0 X which corresponds to the dotted line in the plot. This last series converges quite slowly and a huge number of terms (of the order of 1025) is needed to (∞) 2 obtain the same accuracy that our series with λP MS reaches with just 10 terms. We notice that a special case of eq. (17), corresponding to λ = 1, was already known in the literature [Kn 1996, Ha 1930, So 1994].

3.1 The generalized Hurwitz zeta function We introduce what we call the generalized Hurwitz zeta function‡

∞ 1 ζ(s, u, ξ) = (20) (nu + ξ)s n=0 X which includes as special cases both the Riemann and the Hurwitz zeta func- tions, respectively taking u = 1 and ξ = 1 (Riemann) and u = 1 (Hurwitz). Following the same strategy which we have used before, we want to con- vert eq. (20) into an exponentially convergent series. We write

∞ 1 1 ζ(s, u, ξ) = + (21) ξs (nu + ξ)s n=1 X † A real solution is found only for odd values of kmax. ‡We are not aware of the specific under which such function is known in the literature.

7 and convert it to the form ∞ 1 1 1 1 ζ(s, u, ξ) = + (22) ξs nsu (1 + λ2)s (1 + ∆(n))s n=1 X where we have introduced the definition ξ/nu − λ2 ∆(n) ≡ . (23) 1 + λ2 As usual λ here is an arbitrary parameter which we have introduced “ad 2 ξ−λ 2 ξ−1 hoc”. Provided that 1+λ2 < 1, i.e. that λ > 2 , one can use the expansion:

∞ 1 Γ(k + s) = (−∆(n))k (24) (1 + ∆(n))s Γ(s) k! k X=0 and convert the series (22) to ∞ ∞ 1 1 1 Γ(k + s) ζ(s, u, ξ) = (−∆(n))k ξs nsu (1 + λ2)s Γ(s) k! n=1 k=0 X∞ ∞ Xk 1 Γ(k + s) k λ2(k−j) (−ξ)j = + .(25) ξs Γ(s) k! j (1 + λ2)s+k nu(s+j) n=1 k j=0 X X=0 X   By interchanging the sums and performing the sum over n we obtain the expression

∞ k 1 Γ(k + s) (−ξ)j λ2(k−j) ζ(u, s, ξ) = + ζ(u(s + j))(26). ξs Γ(s) j!(k − j)! (1 + λ2)s+k k j=0 X=0 X 2 ξ−1 We stress that eq. (26) is still exact, provided that λ > 2 and that it converges exponentially. We can obtain the optimal value of λ by applying the PMS, which to leading order provides:

(1) Γ(2 + s)ζ(u(1 + s)) λP MS = ξ . (27) ss(Γ(s) + Γ(1 + s))ζ(su) p (kmax) 3 3 In Fig. 8 we plot the difference |ζ (2, 5 , 1)−ζ(2, 5 , 1)| as a function of the number of terms in the sum of eq. (20). Notice that this series converges very slowly for values su → 1. The horizontal lines are the values obtained by using eq.(26) with the optimal value given in the equation (27).

8 4 Conclusions

The variational approach that we have described in this paper allows in many cases to convert a slowly converging series into a series which converges ex- ponentially. In most cases the results that are obtained by following this approach are completely analytical and therefore extremely useful. Although we have limited ourselves to consider only a few examples, we believe that it should be possible to apply this method to a larger class of series. Indeed the results of [AS 2004, Am 2004a, AL 2004], which however focus more on the physical rather than mathematical aspects, provide a similar series rep- resentation for elliptic functions. In our opinion, the strongest results of the present paper are the equations (17) and (18); it remains to study the pos- sible uses of such equations both in Mathematical and Physical problems (for example, these equations could be used to further improve the rate of convergence of series such as the one in eq. (4)).

The author acknowledges support of Conacyt grant no. C01-40633/A-1.

References

[AA 2003a] Amore, P. and Aranda, A., Presenting a new method for the solution of nonlinear problems, Phys. Lett. A 316 (2003) 218

[AA 2003b] Amore, P. and Aranda, A., Improved Lindstedt-Poincare method for the solution of nonlinear problems, Accepted for publication in Journal of Sound and Vibration, math-ph/0303052

[AL 2004] Amore, P. and Lopez, J., ”The spectrum of a quantum potential”, quant-ph/0405090

[AM 2004] Amore, P. and Montes, H., High order analysis of nonlinear pe- riodic differential equations, Phys. Lett. A 327 (2004) 158-166

[AS 2004] Amore P and S´aenz R., The period of a classical oscillator, Preprint, math-ph/0405030

9 [Am 2004a] Amore, P., Aranda, A., Fernandez, F. and S´aenz, R., Systematic perturbation of integrals with applications to physics, math-ph/0407014 [Am 2004b] Amore, P., Aranda, A. and De Pace, A., A new method for the solution of the Schrodinger equation, Journal of Physics A 37 (2004) 3515-3525 [Am 2004c] Amore, P., Aranda, A., De Pace, A. and Lopez, J., Comparative study of quantum anharmonic potentials, Phys. Lett. A 329 (2004) 451 [AFC 1990] G. A. Arteca, F. M. Fern´andez, and E. A. Castro, Large or- der and summation methods in quantum mechan- ics (Springer, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, Barcelona, 1990). [Br 1999] Bradley D.M., A class of series acceleration formulae for the Cata- lan’s constant, The Ramanujan journal, Vol. 3, Issue 2, June 1999, pp. 159–173 [Co 2000] Cohen, H. , Rodriguez F. and Zagier,D. , Convergence acceleration of , Experimental mathematics 9 (2000) 3 [Fe 2000] Fernandez, F.M., Perturbation theory in quantum mechanics, CRC Press 2000 [FV 1996] Flajolet, P. and Vardi, I. ”Zeta Function Expansions of Classical Constants.” Unpublished manuscript. 1996. [Ha 1930] Hasse, H. ”Ein Summierungsverfahren fr die Riemannsche Zeta- Reihe.” Math. Z. 32, 458-464, 1930. [Jo 1995] Jones, H.F., Nucl. Phys. B 39 (1995) 220 [Kn 1996] Knopp, K. ”4th Example: The Riemann -Function.” Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 51-57, 1996. [So 1994] Sondow, J. ”Analytic Continuation of Riemann’s Zeta Function and Values at Negative Integers via Euler’s Transformation of Series.” Proc. Amer. Math. Soc. 120, 421-424, 1994. [Ste 1981] P. M. Stevenson, Phys. Rev. D 23, 2916 (1981).

10 1

0.01

0.0001

1e-06

Nmax=10

Nmax=20

1e-08 Nmax=30

1e-10 -0.5 0 0.5 1 λ

Figure 1: The error obtained using the partial sum of Eq. (3) over the first 10, 20 and 30 terms respectively as a function of λ.

1

λ=0 (1) λ=λ

0.0001

1e-08

1e-12 0 10 20 30 40 50 Nmax

Figure 2: The error obtained using the partial sum of Eq. (3) with λ = 0 and λ = λ(1) as a function of the number of terms in the sum.

11 1 Eq. 6 Eq. 10 Eq. 9 Eq. 12 0.01

0.0001

1e-06

1e-08

1e-10 0 10 20 30 40 50 Nmax

Figure 3: |S(Nmax)−G| as a function of the number of terms in the sum using eq. (6) (diamonds), eq. (10) (circles), eq. (9) (pluses) and eq. (12) (triangles).

1

kmax = 1

kmax = 11 k = 31 1e-06 max kmax = 51

1e-12

1e-18

1e-24 1 2 3 4 5 6 7 8 9 10 n

Figure 4: The error (ζ(kmax)(n) − ζ(n))/ζ(n) obtained by using eq. (18) to different orders.

12 0.5

0.4

0.3 PMS λ

0.2

0.1 0 10 20 30 40 50 60 70 80 90 100 kmax

Figure 5: The optimal parameter λP MS for ζ(3) calculated to different orders.

1

kmax=11

1e-05 kmax=31

kmax=51

1e-10

1e-15

1e-20

1e-25

1e-30 0 0.2 0.4 0.6 0.8 1 λ

Figure 6: Dependence upon the variational parameter of (kmax) (ζ (s) − ζ(s))/ζ(s) , with kmax = 11, 31, 51.

13 1

1e-10 λ infinity PMS λ = 0 Eq. (19) 1e-20

1e-30

1e-40

1e-50 0 20 40 60 80 100 kmax

Figure 7: Difference |ζapp(3) − ζ(3)| as a function of the number of terms in the sum.

1 kmax = 1

kmax = 3 0.1

kmax = 5

k = 7 0.01 max

kmax = 9

0.001 1 100 10000 1e+06 1e+08 1e+10 1e+12 1e+14 1e+16 kmax

(kmax) 3 3 Figure 8: Difference |ζ (2, 5 , 1) − ζ(2, 5 , 1)| as a function of the number of terms in the sum.

14