Acceleration Methods for Slowly Convergent Sequences and their Applications Naoki Osada Acceleration Methods for Slowly Convergent Sequences and their Applications January 1993 Naoki Osada CONTENTS Introduction ............................. 1 I. Slowly convergent sequences .................... 9 1. Asymptotic preliminaries . 9 1.1 Order symbols and asymptotic expansions . 9 1.2 The Euler-Maclaurin summation formula . 10 2. Slowly convergent sequences . 13 2.1 Order of convergence . 13 2.2 Linearly convergent sequences . 14 2.3 Logarithmically onvergent sequences . 15 2.4 Criterion of linearly or logarithmically convergent sequences . 17 3. In¯nite series . 19 3.1 Alternating series . 19 3.2 Logarithmically convergent series . 23 4. Numerical integration . 28 4.1 Semi-in¯nite integrals with positive monotonically decreasing integrands . 28 4.2 Semi-in¯nite integrals with oscillatory integrands . 29 4.3 Improper integrals with endpoint singularities . 31 II. Acceleration methods for scalar sequences . 33 5. Basic concepts . 33 5.1 A sequence transformation, convergence acceleration, and extrapolation . 33 5.2 A classi¯cation of convergence acceleration methods . 34 6. The E-algorithm . 35 6.1 The derivation of the E-algorithm . 35 6.2 The acceleration theorems of the E-algorithm . 37 7. The Richardson extrapolation . 39 7.1 The birth of the Richardson extrapolation . 39 7.2 The derivation of the Richardson extrapolation . 40 7.3 Generalizations of the Richardson extrapolation . 43 8. The ²-algorithm . 50 8.1 The Shanks transformation . 50 8.2 The ²-algorithm . 51 i 8.3 The asymptotic properties of the ²-algorithm . 52 8.4 Numerical examples of the ²-algorithm . 53 9. Levin's transformations . 57 9.1 The derivation of the Levin T transformation . 57 9.2 The convergence theorem of the Levin transformations . 58 9.3 The d-transformation . 61 10. The Aitken ±2 process and its modi¯cations . 64 10.1 The acceleration theorem of the Aitken ±2 process . 64 10.2 The derivation of the modi¯ed Aitken ±2 formula . 66 10.3 The automatic modi¯ed Aitken ±2 formula . 68 11. Lubkin's W transformation . 72 11.1 The derivation of Lubkin's W transformation . 72 11.2 The exact and acceleration theorems of the W transformation . 74 11.3 The iteration of the W transformation . 75 11.4 Numerical examples of the iterated W transformation . 77 12. The ½-algorithm . 80 12.1 The reciprocal di®erences and the ½-algorithm . 80 12.2 The asymptotic behavior of the ½-algorithm . 82 13. Generalizations of the ½-algorithm . 87 13.1 The generalized ½-algorithm . 87 13.2 The automatic generalized ½-algorithm . 90 14. Comparisons of acceleration methods . 93 14.1 Sets of sequences . 93 14.2 Test series . 94 14.3 Numerical results . 95 14.4 Extraction . 98 14.5 Conclusions . 100 15. Application to numerical integration . 101 15.1 Introduction . 101 15.2 Application to semi-in¯nite integrals . 102 15.3 Application to improper integrals . 106 Conclusions ............................. 108 ii References ............................. 111 Appendix .............................. 115 A. Asymptotic formulae of the Aitken ±2 process . 115 B. An asymptotic formula of Lubkin's W transformation . 117 FORTRAN program . 120 The automatic generalized ½-algorithm The automatic modi¯ed Aitken ±2 formula iii INTRODUCTION Sequence transformations Convergent numerical sequences occur quite often in natural science and engineering. Some of such sequences converge very slowly and their limits are not available without a suitable convergence acceleration method. This is the raison d'^etre of the study of the convergence acceleration method. A convergence acceleration method is usually represented as a sequence transforma- tion. Let S and T be sets of real sequences. A mapping T : S ! T is called a sequence transformation, and we write (tn) = T (sn) for (sn) 2 S . Let T : S ! T be a sequence transformation and (sn) 2 S . T accelerates (sn) if t ¡ s lim n = 0; !1 n sσ(n) ¡ s where σ(n) is the greatest index used in the computation of tn. An illustration : the Aitken ±2 process The most famous sequence transformation is the Aitken ±2 process de¯ned by ¡ 2 2 ¡ (sn+1 sn) ¡ (¢sn) tn = sn = sn 2 ; (1) sn+2 ¡ 2sn+1 + sn ¢ sn 1 2 where (sn) is a scalar sequence. As C. Brezinski pointed out , the ¯rst proposer of the ± process was the greatest Japanese mathematician Takakazu Seki 関 孝 和(or K¹owa Seki, 1642?-1708). Seki used the ±2 process computing ¼ in Katsuy¹oSamp¹ovol. IV 括 要 算 法 巻 四, which was edited by his disciple Murahide Araki 荒 木 村 英in 1712. Let sn be the perimeter of the polygon with 2n sides inscribed in a circle of diameter one. From s15 = 3:14159 26487 76985 6708; s16 = 3:14159 26523 86591 3571; s17 = 3:14159 26532 88992 7759; Seki computed 1C. Brezinski, History of continued fractions and Pad¶eapproximants, Springer-Verlag, Berlin, 1991. p.90. (s16 ¡ s15)(s17 ¡ s16) t15 = s16 + ; (2) (s16 ¡ s15) ¡ (s17 ¡ s16) = 3:14159 26535 89793 2476; and he concluded ¼ = 3:14159 26535 89.2 The formula (2) is nothing but the ±2 process. Seki obtained seventeen-¯gure accuracy from s15; s16 and s17 whose ¯gure of accuracy is less than ten. Seki did not explain the reason for (2), but Yoshisuke Matsunaga 松永良弼 (1692?- 1744), a disciple of Murahide Araki, explained it in Kigenkai 起源解, an annotated edition of Katsuy¹oSamp¹oas follows. Suppose that b = a + ar, c = a + ar + ar2. Then (b ¡ a)(c ¡ b) a b + = ; (b ¡ a) ¡ (c ¡ b) 1 ¡ r the sum of the geometric series a + ar + ar2 + . .3 It still remains a mystery how Seki derived the ±2 process, but Seki's application can be explained as follows. Generally, if a sequence satis¯es » n n sn s + c1¸1 + c2¸2 + . ; where 1 > ¸1 > ¸2 > ¢ ¢ ¢ > 0, then tn in (1) satis¯es ( ¶ ¡ 2 » ¸1 ¸2 n tn s + c2 ¸2 : (3) ¸1 ¡ 1 This result was proved by J. W. Schmidt[48] and P. Wynn[61] in 1966 independently. Since Seki's sequence (sn) satis¯es 1 ¼ X (¡1)j¼2j+1 s = 2n sin = ¼ + (2¡2j)n; n 2n (2j + 1)! j=1 (3) implies that ( ¶ ¼5 1 n+1 t » ¼ + : n 5! 16 2A. Hirayama, K. Shimodaira, and H. Hirose(eds.), Takakazu Seki's collected works, English translation by J. Sudo, (Osaka Kyoiku Tosho, 1974). pp.57-58. 3M. Fujiwara, History of mathematics in Japan before the Meiji era, vol. II (in Japanese), under the auspices of the Japan Academy, (Iwanami, 1956) (= 日本学士院編, 藤原松三郎著, 明治前日本数学史, 岩波書店). p.180. 2 In 1926, A. C. Aitken[1] iteratively applied the ±2 process ¯nding the dominant root of an algebraic equation, and so it is now named after him. He used (1) repeatedly as follows: (n) 2 N T0 = sn; n ; (T (n+1) ¡ T (n))2 T (n) = T (n) ¡ k k k = 0; 1; ...; n 2 N: k+1 k (n+2) ¡ (n+1) (n) Tk 2Tk + Tk This algorithm is called the iterated Aitken ±2 process. Derivation of sequence transformations Many sequence transformations are designed to be exact for sequences of the form sn = s + c1g1(n) + ¢ ¢ ¢ + ckgk(n); 8n; (4) where s; c1; ... ; ck are unknown constants and gj(n)(j = 1; ... k) are known functions of n. Since s is the solution of the system of linear equations sn+i = s + c1g1(n + i) + ¢ ¢ ¢ + ckgk(n + i); i = 0; ... ; k; the sequence transformation (sn) 7! (tn) de¯ned by ¯ ¯ ¯ ¢ ¢ ¢ ¯ ¯ sn sn+1 sn+k ¯ ¯ ¢ ¢ ¢ ¯ ¯ g1(n) g1(n + 1) g1(n + k) ¯ ¯ ¢ ¢ ¢ ¯ ¯ ¯ g (n) g (n + 1) ¢ ¢ ¢ g (n + k) t = E(n) = ¯ k k k ¯ n k ¯ ¢ ¢ ¢ ¯ ¯ 1 1 1 ¯ ¯ ¢ ¢ ¢ ¯ ¯ g1(n) g1(n + 1) g1(n + k) ¯ ¯ ¢ ¢ ¢ ¯ ¯ ¯ gk(n) gk(n + 1) ¢ ¢ ¢ gk(n + k) is exact for the model sequence (4). This sequence transformation includes many famous sequence transformations as follows: 2 (i) The Aitken ± process : k = 1 and g1(n) = ¢sn. j (ii) The Richardson extrapolation : gj(n) = xn, where (xn) is an auxiary sequence. (iii) Shanks' transformation : gj(n) = ¢sn+j¡1. 1¡j (iv) The Levin u-transformation : gj(n) = n ¢sn¡1. In 1979 and 1980, T. Hºavie[20]and C. Brezinski[10] gave independently a recursive (n) algorithm for the computation of Ek , which is called the E-algorithm. 3 By the construction, the E-algorithgm accelerates sequences having the aymptotic expansion of the form X1 sn » s + cjgj(n); (5) j=1 where s; c1; c2; . are unknown constants and (gj(n)) is a known asymptotic scale. More precisely, in 1990, A. Sidi[53] proved that if the E-algorithm is applied to the sequence (5), then for ¯xed k, ( ¶ E(n) ¡ s g (n) k = O k+1 ; as n ! 1: (n) ¡ gk(n) Ek¡1 s Some sequence transformations are designed to accelerate for sequences having a certain asymptotic property. For example, suppose (sn) satis¯es s ¡ s lim n+1 = ¸: (6) !1 n sn ¡ s The Aitken ±2 process is also obtained by solving s ¡ s s ¡ s n+2 = n+1 sn+1 ¡ s sn ¡ s for the unknown s. Such methods for obtaining a sequence transformation from a formula with limit was proposed by C. Kowalewski[25] in 1981 and is designated as thechnique du sous-ensemble, or TSE for short. Recently, the TSE has been formulated by N. Osada[40]. When ¡1 · ¸ < 1 and ¸ =6 0 in (6), the sequence (sn) is said to be linearly convergent sequence. In 1964, P. Henrici[21] proved that the ±2 process accelerates any linearly convergent sequence. When j¸j > 1 in (6), the sequence (sn) diverges but (tn) converges to s. In this case s is called the antilimit of (sn).