QUADRATIC CONVERGENCE OF THE TANH-SINH ...

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16 JONATHAN M. BORWEIN AND LINGYUN YEAlso, the evaluation of e 2 and e 3 involves the careful estimation of∞∑ ∞∑ e −r2 h 21 − erf(rh)erf(nh)n=a r=bHere we will not give further analysis. For any other quadrature rules with similarproperties, the similar approach can be taken.7. Numerical Convergence and ExamplesWe conclude with a few numerical illustrations and a brief discussion of theoptimality of the coefficient of the infinite trapezoidal rule.7.1. Numerical results. We exhibit results for the following set of test functions,earlier studied in [5]:(1) ∫ 1t log(1 + t) dt = 1/40(2) ∫ 10 t2 arctan t dt = (π − 2 + 2 log 2)/12(3) ∫ π/2(4) ∫ 10e t cos t dt = (e π/2 − 1)/20arctan( √ 2+t 2 )(1+t 2 ) √ dt = 2+t 2 5π2 /96(5) ∫ 1 √0 t log t dt = −4/9(6) ∫ 1 √0 1 − t2 dt = π/4(7) ∫ 10√t√1−t 2 dt = 2√ πΓ(3/4)/Γ(1/4) = β(1/2, 3/4)/2(8) ∫ 10 log2 t dt = 2(9) ∫ π/2(10) ∫ π/20(11) ∫ ∞0(12) ∫ ∞0(13) ∫ ∞log(cos t) dt = −π log(2)/20√ √tan t dt = π 2/211+t 2 dt = π/2e −t√tdt = √ πe −t2 /2 dt = √ π/20(14) ∫ ∞e −t cos t dt = 1/20Note that the first four of these integrals involve well-behaved continuous functionson finite intervals. The integrands in problems 5 and 6 have an infinitederivative at an endpoint. In 7 through 10, there is an integrable singularity atan endpoint. In 11 through 13, the interval of integration is infinite, requiring atransformation such as s = 1/(t + 1). Problem 14 is an oscillatory function on aninfinite interval.Table 1 gives the actual error resulting from using the summation (1.2) as anapproximation to the specified integral, rounded to the nearest power of ten. Theprecision used in these calculations was 1000 digits, so no results are shown witherrors less than 10 −1000 . The “level” shown in the table is an index of the intervalh, according to the rule h = 2 −m , where m is the level. In other words, at eachsuccessive level, the interval h is halved.It can be seen from the results in the table that the tanh-sinh scheme achievesapproximately exponential convergence, in the sense that the number of correctdigits approximately doubles with each level. In some cases, such as Problems 3, 5and 8, the convergence is slightly better than a doubling with each level; in othercases, such as Problems 12, 13 and 14, the rate is slightly less. It is remarkable thatProblems 1, 6, 7, 9 and 10 exhibit virtually identical convergence behavior.
QUADRATIC CONVERGENCE OF THE TANH-SINH QUADRATURE RULE 17Level Prob. 1 Prob. 2 Prob. 3 Prob. 4 Prob. 5 Prob. 6 Prob. 71 10 −4 10 −4 10 −4 10 −4 10 −5 10 −5 10 −62 10 −11 10 −11 10 −9 10 −9 10 −12 10 −12 10 −123 10 −24 10 −19 10 −21 10 −18 10 −28 10 −25 10 −264 10 −51 10 −38 10 −49 10 −36 10 −62 10 −50 10 −495 10 −98 10 −74 10 −106 10 −73 10 −129 10 −99 10 −986 10 −195 10 −147 10 −225 10 −145 10 −265 10 −196 10 −1947 10 −390 10 −293 10 −471 10 −290 10 −539 10 −391 10 −3888 10 −777 10 −584 10 −974 10 −582 10 −779 10 −77791011Level Prob. 8 Prob. 9 Prob. 10 Prob. 11 Prob. 12 Prob. 13 Prob. 141 10 −5 10 −4 10 −6 10 −2 10 −2 10 −1 10 −12 10 −12 10 −11 10 −12 10 −5 10 −4 10 −3 10 −23 10 −29 10 −24 10 −25 10 −11 10 −9 10 −6 10 −54 10 −62 10 −50 10 −48 10 −22 10 −15 10 −9 10 −85 10 −130 10 −97 10 −98 10 −45 10 −28 10 −19 10 −146 10 −266 10 −195 10 −194 10 −91 10 −50 10 −37 10 −267 10 −540 10 −389 10 −388 10 −182 10 −92 10 −66 10 −488 10 −777 10 −777 10 −365 10 −170 10 −126 10 −889 10 −731 10 −315 10 −240 10 −16410 10 −584 10 −457 10 −30411 10 −870 10 −564Table 1. Tanh-sinh errors at level m for the 14 test problems7.2. Non-optimality of the limiting trapezoidal rule. Finally, we examinethe optimality of the coefficient of the infinite trapezoidal rule ̂T h with ψ(x) asdefined in (1.3). Let A := {. . . , a −1 , a 0 , a 1 . . .} be the sequence of coefficients for̂T h such that̂T h;A ̂f =∞ ∑n=−∞a n ̂f(nh)for every ̂f ∈ G 2 , andÊ h;A = Î − ̂T h;AWe may ask whether A h where a n ≡ h is optimal in the sense that‖Êh;A h‖ = inf ‖Êh;A‖for all A’s for which ̂T A is a bounded linear functional. As shown in [1], A h isoptimal if and only if ̂T h integrates F nh exactly for every integer n. But from theanalysis in Section 4.1, we know ÊhF nh ≠ 0; consequently A h is not optimal.References[1] Seymour Haber, “The Tanh Rule for Numerical Integration,” SIAM Journal on NumericalAnalysis, 14 (1977), 668–685.[2] J.M. Borwein, D.H. Bailey D. H. & R. Girgensohn, Experimentation in Mathematics, AKPeters, 2004.
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