A Challenging Test For Convergence Accelerators: Summation Of A Series With A Special Sign Pattern

Transcription

1 Applied Mathematics E-Notes, 6(006), 5-34 c ISSN Available free at mirror sites of ame/ A Challegig Test For Covergece Accelerators: Summatio Of A Series With A Special Sig Patter Avram Sidi Received 5 November 005 Abstract Slowly coverget series that have special sig patters have bee used i testig the efficiecy of covergece acceleratio methods. Ithispaper,we study the series S m, = P /m whe m 1, 0, which has m (k+1) +1 positive terms followed by m egative terms periodically. Usig special fuctios, we first derive its sum i simple terms ivolvig oly the Riema Zeta fuctio ad trigoometric fuctios. With the exact sum available, we ext use this series to test the efficiecy of various oliear covergece acceleratio methods i summig it umerically. We coclude that the Shaks trasformatio ad the Levi Sidi d (m) -trasformatio are two acceleratio methods that produce highly accurate approximatios to the sum of S m,, the latter beig the more effective. 1 Itroductio The ifiite series S 1 = k +1 = = log (1) 4 has bee frequetly used i testig the efficiecy of the various oliear covergece acceleratio methods, such as the Shaks trasformatio [7], the L-trasformatio of Levi [4], ad the θ-algorithm of Breziski [1]. It is well-kow that all three trasformatios (ad several others as well) are very effective o this series, i the sese that they accelerate the covergece of the sequece of the partial sums of the series ad produce excellet approximatios to its limit log. See, e.g., Smith ad Ford [10]. A more geeral versio of this series with a special sig patter, amely, the series S = / k +1 = = π log, () Mathematics Subject Classificatios: 33E0, 40A99, 41A58, 41A99. Computer Sciece Departmet, Techio - Israel Istitute of Techology, Haifa 3000, Israel 5

2 6 Summatio of Series was cosidered by Lubki [6] i the study of his W -trasformatio. Lubki cocluded that the W -trasformatio, either i its simple form or i iterated form, was ot effective i acceleratig the covergece of the partial sums of this series. This series poses a real challege for most of the kow covergece acceleratio methods. It turs out that the Shaks trasformatio produces covergece acceleratio while the Levi trasformatio ad the θ-algorithm are ot effective at all. See Wimp [11, p. 171]. (Icidetally, the special cases of the Levi trasformatio ad of the θ-algorithm, amely, the L -trasformatio ad the θ -algorithm, respectively, are idetical to the W -trasformatio.) Aother oliear method that accelerates the covergece of the series S is the d () -trasformatio of Levi ad Sidi [5]. I view of the fact that the series S, with its special sig patter, poses a real challege for covergece acceleratio methods, we ask whether we ca fid additioal series with kow sums that are more geeral ad challegig tha S, as far as covergece acceleratio methods are cocered. The aim of this paper is, first of all, to provide such ifiite series that are otrivial ad that ca be used as test cases by researchers i the field of covergece acceleratio. The followig geeralizatio of Lubki s series was cosidered i Sidi [9, Sectio 5.9]: It was stated there that S m = S m = 1 m log + π /m, m =1,,.... (3) k +1 cot kπ = 1 m log + π ta kπ. (4) This series too has a special sig patter similar to that of S ;itsfirst m terms are positiveadarefollowedbym egative terms, which are followed by m positive terms, ad so o. Therefore, it is a very appropriate test series for the study of covergece acceleratio methods. The result i (4) was give without proof i [9]; we give its complete proof i Sectio of this ote. I view of the fact that special fuctios are ivolved i the derivatio, that the sum of S m ca be expressed i the simple form give here seems quite surprisig. We ote that, so far, the oly oliear covergece acceleratio methods that are effective o this series are the Shaks trasformatio ad the d (m) -trasformatio of Levi ad Sidi [5]. I Sectio 3, we cosider a further geeralizatio, amely, the series S m, = /m, m =1,,..., =0, 1,.... (5) (k +1) +1 [Thus, S m,0 of (5) is simply S m of (3).] We show that the sum of S m, too ca be expressed i simple terms ivolvig the Riema Zeta fuctio ad trigoometric fuctios: Usig the techiques of Sectio, we show that

3 A. Sidi 7 S m, = π ζ( +1)+ m+1 ()! where ζ(s) is the Riema Zeta fuctio defied by ζ(s) = d ξ dξ ξ= cot, (6) kπ k s, s >1. (7) Our umerical experimets seem to idicate that, with m>1, the covergece of S m, too ca be accelerated by the Shaks trasformatio ad the d (m) -trasformatio of Levi ad Sidi; other oliear methods kow at preset are ot effective. It would be iterestig to kow whether the sum of the compaio series T m, := /m, m, =1,,..., (k +1) ca also be expressed i such simple terms. So far, this does ot seem to be the case. The covergece of T m,, just as that of S m,, ca be accelerated by the Shaks trasformatio ad the Levi Sidi d (m) -trasformatio; agai, other oliear methods kow so far are ot effective. I Sectio 4, we preset brief descriptios of the L-trasformatio, the θ-algorithm, the Shaks trasformatio, ad the Levi Sidi d (m) -trasformatio. Fially, i Sectio 5, we compare umerically these oliear covergece acceleratio methods as they are applied to S m, with m = ad m = 3. Our umerical results idicate that of the two effective methods, the Shaks trasformatio ad the Levi Sidi d (m) - trasformatio, the latter is the more effective i that it requires less terms of the series to produce a required level of accuracy. Derivatio of Eq. (4) The result give i (4) ca be verified very simply for the case m = by rewritig S i the form S = , (8) ad ivokig (1) ad k+1 =arcta1= π 4. This splittig techique ca be exteded to arrive at the result give i (4) for m 3. However, we eed to use some amout of special fuctio theory for this purpose. We start by rewritig (3) i the form m S m = 1 m km + i = km + i = 1 m i β (9) m m

4 8 Summatio of Series where the fuctio β(x) is defied via (see Gradshtey ad Ryzhik [3, p. 947, formula ]) β(x) = x + k. (10) Note that (9) is a rearragemet of the (coditioally coverget) series S m that does ot chage the sum of S m.now, β(1) = log, β(x) = 1 ψ x +1 ψ x, (11) where ψ(z) = d dz log Γ(z) (see [3, p. 947, formula 8.370]). Thus, (9) becomes S m = 1 m log + 1 ψ m + i Rewritig the summatio o the right-had side of (1) i the form ψ i ψ. (1) m + i i s i ψ = ψ ψ s=1 = ψ 1 k k ψ, (13) ad ivokig (see [3, p. 945, formula ]) we arrive at We also have that ψ ψ(1 z) =ψ(z)+π cot πz, (14) m + i i ψ = π cot kπ. (15) cot kπ = cot s=1 (m s)π = s=1 π cot sπ = s=1 ta sπ. (16) Combiig (13) (16) i (1), the result i (4) follows.

5 A. Sidi 9 3 Derivatio of Eq. (6) We ow tur to the derivatio of the result i (6) pertaiig to S m, with =1,,.... Proceedig as i the precedig sectio, we rewrite S m, i the form Deotig S m, = = = m m = 1 m +1 1 (km + i) +1 (km + i) +1 (km + i) +1 + ad makig use of the fact that (km + m) +1 (k + i/m) m +1 η(s) =. (17) (k +1) +1 (k +1) s, (18) ( 1)p = β (p) (x), p =1,,..., (19) (x + k) p+1 p! (see [3, p. 947, formula 8.374]), (17) becomes S m, = 1 m +1 η( +1)+ 1 m +1 ()! Ivokig (11), we obtai i β () = 1 m = 1 +1 = 1 +1 = 1 +1 d dx ψ x +1 m + i ψ () ψ () 1 k d β () i m x ψ x= i m i ψ (). (0) ψ () k ψ(1 z) ψ(z) dz z=. (1) k

6 30 Summatio of Series Note that the third equality i (1) is obtaied i the same way (13) is obtaied. The fourth equality i (1) is made possible by the fact that is a eve iteger. (I case of T m,, this is ot possible, because β () is replaced by β ( 1) ad 1isa odd iteger. As a result, a simple expressio for the sum of T m, does ot seem to be possible.) Ivokig ow (14), we get β () i m = π +1 d dz cot πz z= k Combiig (18) ad () i (0), ad ivokig the result i (6) follows. = +1 π d ξ dξ ξ= cot. kπ () η(s) =(1 1 s )ζ(s), (3) 4 Applicatio of Covergece Acceleratio Methods The covergece of the (slowly covergig) series S m, S m,,adt m, ca be accelerated by applyig to them oliear covergece acceleratio methods. The methods that are kow for their especially good acceleratio properties ad that we have tested here are the Levi L-trasformatio, the Breziski θ-algorithm, the Shaks trasformatio, ad the Levi Sidi d (m) -trasformatio. For the sake of completeess, we recall the defiitios of these methods here. For more iformatio ad recet results o these methods, see [9]. Below, for the L- add (m) -trasformatios, we let a k be the series whose covergece is to be accelerated, ad A = a k, =1,,.... I keepig with covetio, for the Shaks trasformatio ad the θ-algorithm, we let a k be the series whose covergece is to be accelerated, ad A = a k, =0, 1,.... We also defie the forward differece operator such that c j = c j+1 c j for all j. L-trasformatio. Lettig ω r = ra r,wedefie L (j) via the liear equatios A r = L (j) + ω r 1 i=0 β i, J r J + ; J = j +1. ri Here L (j) is the approximatio to the sum of the series ad the β i are auxiliary ukows. (With the preset ω r, this is also kow as the u-trasformatio.) The, L (j) is give by the closed-form expressio L (j) = J 1 A J /ω J i=0 (J 1 = ( 1)i i (J + i) 1 A J+i /ω J+i /ω J ) i=0 ( 1)i ; J = j +1. i (J + i) 1 /ω J+i Note that L (j) is determied by the terms A i, j +1 i j Also, it is kow that L (j) is the approximatio produced by the Lubki W -trasformatio. The diagoal sequeces {L (j) } =1 (j fixed) have the best covergece properties.

7 A. Sidi 31 θ-algorithm. This method is defied via the followig recursive scheme. θ (j) 1 =0, θ(j) 0 = A j, j 0; θ (j) +1 = θ(j+1) 1 + D(j) ; θ (j) + = θ(j+1) D(j) k =1/ θ (j) k for all j, k 0, θ(j+1) D (j) D (j) +1, j, Note that the operator operates oly o the upper idex, amely, o j. Here, the relevat quatities (i.e., the approximatios to the sum of the series) are the θ (j).note that θ (j) is determied by A i, j i j +3. Also, it is kow that θ (j) is the approximatio produced by the Lubki W -trasformatio. The diagoal sequeces {θ (j) } =0 (j fixed) have the best covergece properties. Shaks trasformatio. This method is defied via the liear equatios A r = e (A j )+ ᾱ k A r+k 1, j r j +. Here e (A j ) is the approximatio to the sum of the series ad the ᾱ k are auxiliary ukows. The e (A j ) ca be obtaied recursively with the help of the ε-algorithm of Wy [1] as follows: ε (j) 1 =0, ε(j) 0 = A j, j 0; ε (j) k+1 = ε(j+1) k 1 + ε (j+1) k 1 ε (j) k, j,k 0. The, e (A j )=ε (j) for all j ad. Aother algorithm that is as efficiet as the ε-algorithm is the recet FS/qd-algorithm of the author give i [9, Chapter 1]. Note that e (A j )=ε (j) is determied by A i, j i j +. The diagoal sequeces {ε (j) } =0 (j fixed) have the best covergece properties. d (m) -trasformatio. Pick itegers R l such that 1 R 0 <R 1 <R <, let =( 1,,..., m ), ad defie d (m,j) through A Rl = d (m,j) + m k 1 Rl k ( k 1 a Rl ) i=0 β ki R i l, j l j + N; N = m k. Here, d (m,j) is the approximatio to A, while the β ki are auxiliary ukows. Note that d (m,j) is determied by A i, R j i m + R j+n. The diagoal sequeces {d (m,j) (ν,ν,...,ν) } ν=1 (j fixed) have the best covergece properties. The simplest ad obvious choice for the R l is R l = l +1, l =0, 1,.... It is importat to ote that the R l ca be fixed to achieve best possible covergece acceleratio ad umerical stability i a systematic way. See [9, Sectios 6.5 ad 1.7]. Whe m =1,thed (m,j) ca be computed recursively via the W-algorithm of Sidi [8] (ot to be cofused with the Lubki W-trasformatio). The W-algorithm (with the otatio A (j) d (1,j) ) reads as follows:

8 3 Summatio of Series (i) Set M (j) 0 = A R j R j a Rj, N (j) 1 0 =, j 0. R j a Rj (ii) Compute M (j) M (j) ad N (j) = M (j+1) 1 M (j) Rj+ 1 R 1 j recursively from 1, N (j) = N (j+1) 1 N (j) Rj+ 1 R 1 j 1, j 0, 1. (iii) Set A (j) = M (j), j, 0. N (j) Whe m>1, the d (m,j) (ν,ν,...,ν) ca be computed via the W(m) -algorithm of Ford ad Sidi []; we refer the reader to [] or to [9, Chapter 7] for details. A FORTRAN 77 code that implemets the d (m) -trasformatio with the help of the W (m) -algorithm is give i [9, Appedix I]; it ca also be obtaied from the author via . Whe R l = l+1, l=0, 1,...,ad m =1,thed (m) -trasformatio reduces to the L- trasformatio. This shows that the L-trasformatio ca be implemeted recursively via the W-algorithm. 5 Numerical Example We have applied the trasformatios metioed above to the series S m, S m,,ad T m,. Wehaveusedtheε-algorithm to implemet the Shaks trasformatio, ad the W (m) -algorithm to implemet the d (m) -trasformatio. (Note that the θ-algorithm is already defied via a recursive procedure.) Our umerical results show that the L- trasformatio ad the θ-algorithm are effective accelerators oly whe m = 1;they do ot produce ay acceleratio whe m>1. The Shaks trasformatio ad the d (m) -trasformatio are very effective for all m. The d (m) -trasformatio seems to be more effective i that it uses a smaller umber of sequece elemets to produce a required level of accuracy. Table 1 cotais the results obtaied i quadruple precisio (approximately 35 decimal digits) for the series S m with m =adm = 3. I our computatios with the d (m) -trasformatio, we have chose R l = l +1, l =0, 1,.... Note that, i Table 1, we compare d (,0) (4k,4k) with ε(0) 8k, because they both use about the same umber of terms of S (approximately 8k terms). Similarly, we compare d (3,0) (4k,4k,4k) with ε(0) 1k,because theybothuseaboutthesameumberoftermsofs 3 (approximately 1k terms). It is iterestig to ote that, for each k, the approximatios d (,0) (4k,4k) ad d(3,0) (4k,4k,4k) have comparable accuracies ad so do the approximatios ε (0) 8k ad ε(0) 1k.

9 A. Sidi 33 k d (,0) (4k,4k) S ε (0) 8k S d (3,0) (4k,4k,4k) S 3 ε (0) 1k S ( 5).504( 4) 9.40( 5) 7.459( 4).330( 10).51( 7) 5.561( 10) 7.379( 7) ( 15).33( 10).735( 16) 6.87( 10) ( 0).108( 13) 5.448( 0) 6.164( 13) ( 5) 1.880( 16) 7.413( 5) 5.498( 16) ( 31) 1.665( 19) 3.681( 30) 4.870( 19) ( 34) 1.467( ) 7.704( 34) 4.94( ) ( 5) 3.775( 5) ( 8) 3.310( 8) ( 3).86( 31) Table 1: Results obtaied by applyig the d (m) -trasformatio ad the Shaks trasformatio to the series S m with m =adm =3. Cocludig Remarks Ithiswork,wehavecosideredtheslowlycovergigseriesS m give i (3) ad S m, give i (5). These series, because of their special sig patters, are challegig test cases for covergece acceleratio methods; i fact, most covergece acceleratio methods fail to produce aythig meaigful whe applied to these series. We have derived the exact sums of S m ad S m, i simple terms ad have also cosidered their summatio umerically via covergece acceleratio methods. Based o umerical evidece, we have cocluded that, of the oliear covergece acceleratio methods kow at preset, the Shaks trasformatio ad the Levi Sidi d (m) -trasformatio are the oly effective summatio methods ad that the d (m) -trasformatio is the more effective of the two. Other oliear methods we have tried have ot improved the covergece of S m ad S m,. Refereces [1] C. Breziski, Accélératio de suites à covergece logarithmique, C. R. Acad. Sci. Paris, 73A(1971), [] W. F. Ford ad A. Sidi, A algorithm for a geeralizatio of the Richardso extrapolatio process, SIAM J. Numer. Aal., 4(1987), [3] I. S. Gradshtey ad I. M. Ryzhik, Table of Itegrals, Series, ad Products, Academic Press, New York, [4] D. Levi, Developmet of o-liear trasformatios for improvig covergece of sequeces, Iter. J. Computer Math., B3(1973),

10 34 Summatio of Series [5] D. Levi ad A. Sidi, Two ew classes of oliear trasformatios for acceleratig the covergece of ifiite itegrals ad series, Appl. Math. Comp., 9(1981), [6] S. Lubki, A method of summig ifiite series, J. Res. Nat. Bur. Stadards, 48(195), [7] D. Shaks, Noliear trasformatios of diverget ad slowly coverget sequeces, J. Math. ad Phys., 34(1955), 1 4. [8] A. Sidi, A algorithm for a special case of a geeralizatio of the Richardso extrapolatio process, Numer. Math., 38(198), [9] A. Sidi, Practical Extrapolatio Methods: Theory ad Applicatios, Number 10 i Cambridge Moographs o Applied ad Computatioal Mathematics, Cambridge Uiversity Press, Cambridge, 003. [10] D. A. Smith ad W. F. Ford, Acceleratio of liear ad logarithmic covergece, SIAM J. Numer. Aal., 16(1979), [11] J. Wimp, Sequece Trasformatios ad Their Applicatios, Academic Press, New York, [1] P. Wy, O a device for computig the e m (S ) trasformatio, Mathematical Tables ad Other Aids to Computatio, 10(1956),