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Acceleration Methods for Slowly Convergent Sequences and Their Applications

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Acceleration Methods for Slowly Convergent Sequences and Their Applications 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).
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