Series expansions and direct inversion for the Heston model
SSERIES EXPANSIONS AND DIRECT INVERSION FOR THEHESTON MODEL ∗ SIMON J. A. MALHAM † , JIAQI SHEN † , AND
ANKE WIESE † Abstract.
Efficient sampling for the conditional time integrated variance process in the Hestonstochastic volatility model is key to the simulation of the stock price based on its exact distribution.We construct a new series expansion for this integral in terms of double infinite weighted sums of par-ticular independent random variables through a change of measure and the decomposition of squaredBessel bridges. When approximated by series truncations, this representation has exponentially de-caying truncation errors. We propose feasible strategies to largely reduce the implementation of thenew series to simulations of simple random variables that are independent of any model parameters.We further develop direct inversion algorithms to generate samples for such random variables basedon Chebyshev polynomial approximations for their inverse distribution functions. These approxima-tions can be used under any market conditions. Thus, we establish a strong, efficient and almostexact sampling scheme for the Heston model.
Key words.
Series expansion, Direct inversion, Chebyshev approximation, Stochastic volatility
AMS subject classifications.
1. Introduction.
Stochastic volatility models involving a pair of stochastic dif-ferential equations, with the diffusion term of the first one governed by the evolutionof the second equation, are immensely popular in the pricing of derivatives. Amongthe existing stochastic volatility models, the Heston model plays an important roleand is used widely. It can be expressed in the form of a two-dimensional system dS t S t = µ dt + (cid:112) V t (cid:16) ρ dW t + (cid:112) − ρ dW t (cid:17) , (1.1) dV t = κ ( θ − V t ) dt + σ (cid:112) V t dW t , (1.2)where W and W are two independent standard Brownian motions, and κ , θ , σ andtypically also µ are positive constants with ρ ∈ [ − , S characterisesthe dynamics of the stock price while the component V specifies the variances of itsreturns. The introduction of randomness to the volatility has been used to explain thelong-observed features of the implied volatility surface in a self-consistent way. Thevariance process follows a mean-reverting square-root or Cox-Ingersoll-Ross (CIR)process (Cox, Ingersoll and Ross [13]).Closed form solutions for standard vanilla option prices under the Heston modelare available; see Heston [19] and Kahl and J¨ackel [23]. However for exotic options,especially path-dependent options, such closed form solutions are not known in gen-eral and Monte Carlo simulation is often employed. Typically, continuous stochasticprocesses are approximated by paths simulated on discrete time grids. It is normallynatural to consider the Euler-Maruyama scheme which converges weakly with con-vergence rate one under certain regularity conditions; see Section 14 . ∗ Submitted to the editors DATE.
Funding:
James Watt Scholarship. † Maxwell Institute for Mathematical Sciences and School of Mathematical and Computer Sci-ences, Heriot-Watt University, Edinburgh, EH14 4AS, UK ([email protected], [email protected],[email protected]). 1 a r X i v : . [ q -f i n . P R ] A ug laten [24]; see Section 6 . (cid:16)(cid:82) t V s ds (cid:12)(cid:12)(cid:12) V , V t (cid:17) . Theybuild on the results (2.m) and (6.d) in Pitman and Yor [30] to derive the explicit formfor the corresponding characteristic function. Fourier inversion techniques in con-junction with the trapezoidal rule are applied to numerically evaluate the probabilitydistribution function. This is followed by inverse transform sampling to simulate thevalue of the above integral. Their numerical results imply that the proposed methodhas a faster convergence rate compared to the Euler scheme with bias-free simulation.Because of the dependence on V and V t , Broadie and Kaya [12] compute thecharacteristic function for each step and path in the Monte Carlo simulation. At the xpense of a small bias, Smith [34] presents an approximation to the characteristicfunction, which makes it possible to precalculate and store the values of the character-istic function for all the points required in advance. Glasserman and Kim [17] provideanother sampling method for the time integrated conditional variance, which relies onan explicit representation as infinite sums and mixtures of gamma random variables.When combined with the exact simulation method suggested by Broadie and Kaya[12], their method is highly effective in terms of both accuracy and computationalspeed for pricing non-path-dependent options across a full range of model parametervalues.Motivated by the decomposition in Glasserman and Kim [17] (Theorem 2 .
2. Main results.
The method we propose closely follows the lead of Broadie andKaya [12] and Glasserman and Kim [17] with the key difference for the simulation ofthe conditional integral of the variance process. To complete the understanding of themotivation for sampling from the conditional integral, we first quote some propertieswith regard to the Heston model.We start with the variance process governed by (1.2), which is a CIR process(Cox, Ingersoll and Ross [13]) with transition probability given explicitly as a scalednoncentral chi-squared distribution. With the degrees of freedom for this processdefined to be δ := 4 κθ/σ , we have V t ∼ σ (1 − exp ( − κt ))4 κ χ δ (cid:18) κ exp ( − κt ) σ (1 − exp ( − κt )) V (cid:19) , t > , (2.1)where V > χ δ ( λ ) denotes a noncentral chi-squared ran-dom variable with degrees of freedom δ and noncentrality parameter λ . This meansthat conditional on V , V t is distributed as σ (1 − exp ( − κt )) / (4 κ ) multiplied bya noncentral chi-squared distribution with degrees of freedom δ and noncentralityparameter λ := 4 κ exp ( − κt ) σ (1 − exp ( − κt )) V . he above law provides a way of exactly simulating V t from V , see Broadie and Kaya[12], Scott [32] and Malham and Wiese [26] for details.By employing the explicit solution of the stock price process (1.1) and Itˆo’s for-mula, we obtainlog S t = log S + µt − (cid:90) t V s ds + ρ (cid:90) t (cid:112) V s dW s + (cid:112) − ρ (cid:90) t (cid:112) V s dW s . Integrating the variance process (1.2) also gives (cid:90) t (cid:112) V s dW s = (cid:90) t σ ( dV s − κ ( θ − V s ) ds ) = V t − V − κθtσ + κσ (cid:90) t V s ds. Combining these two results, Broadie and Kaya [12] observe that given V , V t and (cid:82) t V s ds , the distribution of log ( S t /S ) is Gaussian with known moments since theprocess V is independent of the Brownian motion W , i.e.log S t S ∼ N (cid:18) µt + ρσ ( V t − V − κθt ) + (cid:18) ρκσ − (cid:19) (cid:90) t V s ds, (cid:0) − ρ (cid:1) (cid:90) t V s ds (cid:19) . Hence, an exact simulation for the stock price S t given the initial conditions S and V is now reduced to sampling a conditional normal random variable given aboveprovided there is a way to sampling from the joint distribution (cid:16) V t , (cid:82) t V s ds (cid:17) . As V t can be simulated using the transition law in (2.1), the main challenge is now todevelop a tractable method for sampling from the time integral of the variance process V s over [0 , t ] given its endpoints V and V t , i.e. (cid:18)(cid:90) t V s ds (cid:12)(cid:12)(cid:12)(cid:12) V , V t (cid:19) . In what follows we focus on developing a new representation for the above integralbuilding upon the decomposition suggested by Glasserman and Kim [17], which appliesthe decomposition of the squared Bessel bridges from Pitman and Yor [30].Before proceeding, we reduce the model to a special case by time-rescaling anda measure transformation. First, define ˜ A t = V t/σ . Then, ˜ A t satisfies the followingstochastic differential equation d ˜ A t = (cid:0) δ − q ˜ A t (cid:1) dt + 2 (cid:113) ˜ A t d ˜ W t , where q := 2 κ/σ and ˜ W t := σW t/σ / Q , meaning that our target now is to simulate (cid:20)(cid:90) t V s ds (cid:12)(cid:12)(cid:12)(cid:12) V = v , V t = v t (cid:21) d == (cid:20) σ (cid:90) τ ˜ A s ds (cid:12)(cid:12)(cid:12)(cid:12) ˜ A = a , ˜ A τ = a τ (cid:21) under Q with τ = σ t/ a = v and a τ = v t .Second, we further simplify the model by introducing a new probability measure P such that d P d Q = exp (cid:18) q (cid:90) τ (cid:113) ˜ A s d ˜ W s − q (cid:90) τ ˜ A s ds (cid:19) . (2.2) y the Girsanov theorem, W P τ := ˜ W τ − (cid:82) τ q (cid:112) ˜ A s ds is a standard Brownian motionunder P . With this replacement, the rescaled process ˜ A satisfies d ˜ A t = δ dt + 2 (cid:113) ˜ A t dW P t , (2.3)which is a δ -dimensional squared Bessel process under P . Hence, our objective is tosample from the time integral of a squared Bessel process ˜ A given its values at theendpoints, denoted by I , under the new probability measure P , i.e. I = (cid:18)(cid:90) τ ˜ A s ds (cid:12)(cid:12)(cid:12)(cid:12) ˜ A = a , ˜ A τ = a τ (cid:19) , (2.4)and to find a connection between the distributions for the conditional integral under P and Q .Next, we state our main result which relies on decomposing the conditional inte-gral as introduced by Glasserman and Kim [17]. Theorem
Under the new probability measure P , the conditional integral ofthe rescaled variance process ˜ A is equivalent in distribution to the sum of three infiniteseries of random variables I = (cid:18)(cid:90) τ ˜ A s ds (cid:12)(cid:12)(cid:12)(cid:12) ˜ A = a , ˜ A τ = a τ (cid:19) d == X + X + η (cid:88) j =1 Z j , where X , X , η , Z , Z , . . . are mutually independent, and η is a Bessel randomvariable with parameters ν = δ/ − and z = √ a a τ /τ , i.e. η ∼ Bessel ( ν, z ) .Moreover, X , X , Z , Z , . . . admit the following representations: ( a ) We have X d == ∞ (cid:88) n =0 τ n P n (cid:88) k =1 S n,k , where for n = 0 , , . . . , the P n are independent Poisson random variables with mean ( a + a τ ) 2 n − /τ and for k = 1 , , . . . , P n , the S n,k are independent copies of therandom variable S := (cid:0) /π (cid:1) (cid:80) ∞ l =1 (cid:15) l /l and (cid:15) l ∼ Exp (1) are independent exponentialrandom variables for l = 1 , , . . . ; ( b ) Further we have X d == ∞ (cid:88) n =1 τ n C δ/ n , where for n = 1 , , . . . , the C δ/ n are independent copies of the random variable C δ/ := (cid:0) /π (cid:1) (cid:80) ∞ l =1 Γ δ/ ,l / ( l − / and Γ δ/ ,l ∼ Gamma ( δ/ , are independent gammarandom variables for l = 1 , , . . . ; ( c ) And also we have the Z j , j = 1 , , . . . , η , which are independent copies of therandom variable Z such that Z d == ∞ (cid:88) n =1 τ n C (cid:48) n , where for n = 1 , , . . . , the C (cid:48) n are independent copies of the random variable C := (cid:0) /π (cid:1) (cid:80) ∞ l =1 Γ ,l / ( l − / and Γ ,l ∼ Gamma (2 , are independent gamma randomvariables for l = 1 , , . . . . roof. We work on the probability measure P throughout this proof. We notethat for a fixed τ > (cid:18)(cid:90) τ ˜ A s ds (cid:12)(cid:12)(cid:12)(cid:12) ˜ A = a , ˜ A τ = a τ (cid:19) = (cid:18) τ (cid:90) A s ds (cid:12)(cid:12)(cid:12)(cid:12) A = x, A = y (cid:19) , (2.5)where A s is defined by setting A s = ˜ A sτ /τ for 0 ≤ s ≤ x = a /τ, y = a τ /τ .Then using equation (2.3), the process A satisfies dA s = δ ds + 2 (cid:112) A s dW s , where W s := W P sτ / √ τ is a standard Brownian motion. We observe that { A s } ≤ s ≤ isa δ -dimensional squared Bessel process. Conditional on the end points, the process (cid:0) A s , ≤ s ≤ (cid:12)(cid:12) A = x, A = y (cid:1) is then a squared Bessel bridge, denoted by A δ, x,y = (cid:8) A δ, x,y ( s ) (cid:9) ≤ s ≤ . Then the right hand side of (2.5) has the same distribution as (cid:18) τ (cid:90) A δ, x,y ( s ) ds (cid:19) . (2.6)We prove the result in three steps.First, the integral (2.6) can be decomposed into the sum of three independentparts as follows: τ (cid:90) A δ, x,y ( s ) ds d == X (cid:48) + X (cid:48) + η (cid:88) j =1 Z (cid:48) j , where X (cid:48) = τ (cid:90) A , x + y, ( s ) ds,X (cid:48) = τ (cid:90) A δ, , ( s ) ds, and for j = 1 , , . . . , η , Z (cid:48) j are independent copies of Z (cid:48) = τ (cid:90) A , , ( s ) ds, and η is an independent Bessel random variable with parameters ν = δ/ − z = √ xy = √ a a τ /τ , i.e. η ∼ Bessel ( ν, z ). This is a direct result from Glassermanand Kim [17], who apply the decomposition of squared Bessel bridges proposed byPitman and Yor [30] to the transformed variance process.Second, it follows from Glasserman and Kim [17] that the Laplace transforms of X (cid:48) , X (cid:48) and Z (cid:48) for b ≥ (cid:48) ( b ) = exp (cid:18) a + a τ τ (cid:16) − √ bτ coth (cid:0) √ bτ (cid:1)(cid:17)(cid:19) , (2.7) Φ (cid:48) ( b ) = (cid:32) √ b τ sinh (cid:0) √ b τ (cid:1) (cid:33) δ/ , (2.8) Φ (cid:48) ( b ) = (cid:32) √ b τ sinh (cid:0) √ b τ (cid:1) (cid:33) . (2.9) hird, to verify the random variables X (cid:48) , X (cid:48) and Z (cid:48) have the same distributionas the series expansions which define X , X and Z respectively, it is sufficient toshow that they have identical Laplace transforms. To do this, let us first rewrite Φ (cid:48) i , i = 1 , , ζ ≡ coth ζ − ζ , sinh ζ ≡ ζ ζ . Iterating N times gives uscoth ζ ≡ coth ζ N +1 − N (cid:88) n =0 ζ n , (2.10) sinh ζ ≡ N sinh ζ N N (cid:89) n =1 cosh ζ n . (2.11)Substituting (2.10) into (2.7) and rearranging the terms, we getexp (cid:18) a + a τ τ (1 − ζ coth ζ ) (cid:19) = N (cid:89) n =0 exp (cid:32) a + a τ τ n ζ n sinh ζ n (cid:33) I N ( ζ, τ, a , a τ ) , where I N ( ζ, τ, a , a τ ) := exp (cid:0) − ( a + a τ ) (cid:0) ζ coth( ζ/ N +1 ) − (cid:1) / (2 τ ) (cid:1) . On the otherhand, it follows from (cid:80) Nn =0 n = 2 N +1 − I N ( ζ, τ, a , a τ ) = N (cid:89) n =0 exp (cid:18) − a + a τ τ n (cid:19) exp ( ε ,N ( ζ, τ, a , a τ )) , where ε ,N ( ζ, τ, a , a τ ) := − ( a + a τ ) (cid:0) ζ coth (cid:0) ζ/ N +1 (cid:1) − N +1 (cid:1) / (2 τ ) → N →∞ . Thus, for ζ = √ b τ , we have an alternative form for Φ (cid:48) given byΦ (cid:48) ( b ) = ∞ (cid:89) n =0 exp (cid:32) a + a τ τ n (cid:32) √ b τ n sinh √ b τ n − (cid:33)(cid:33) . Similarly, for Φ (cid:48) after substitution and rearrangement, we have (cid:18) ζ sinh ζ (cid:19) δ/ = N (cid:89) n =1 (cid:18) cosh ζ n (cid:19) − δ/ ε ,N ( ζ, δ ) , where ε ,N ( ζ, δ ) := (cid:0)(cid:0) ζ/ N (cid:1) / sinh (cid:0) ζ/ N (cid:1)(cid:1) δ/ → N → ∞ . As a result, plugging ζ = √ b τ into this expression yieldsΦ (cid:48) ( b ) = ∞ (cid:89) n =1 (cid:32) cosh √ b τ n (cid:33) − δ/ . Next, we derive the Laplace transforms of X , X and Z , denoted by Φ , Φ and Φ respectively. For any b ≥
0, we haveΦ ( b ) = E [exp ( − bX )] ∞ (cid:89) n =0 E (cid:34) exp (cid:32) − b τ n P n (cid:88) k =1 S n,k (cid:33)(cid:35) = ∞ (cid:89) n =0 E (cid:34) P n (cid:89) k =1 E (cid:18) exp (cid:18) − b τ n S n,k (cid:19)(cid:19)(cid:35) = ∞ (cid:89) n =0 E (cid:113) bτ n sinh (cid:113) bτ n P n = ∞ (cid:89) n =0 exp (cid:32) a + a τ τ n (cid:32) √ b τ n sinh √ b τ n − (cid:33)(cid:33) , where the second equality comes from the interchange of expectation and limit by theBounded Convergence Theorem and the fourth equality holds due to the fact that E [exp ( − bS n,k )] = √ b/ sinh √ b for all n ≥ k ≥ for X .Indeed, from E (cid:104) exp (cid:16) − bC δ/ n (cid:17)(cid:105) = (cid:16) cosh √ b (cid:17) − δ/ for any n ≥ ( b ) = E [exp ( − bX )]= ∞ (cid:89) n =1 E (cid:20) exp (cid:18) − b τ n C δ/ n (cid:19)(cid:21) = ∞ (cid:89) n =1 (cid:32) cosh √ b τ n (cid:33) − δ/ . Hence, we can now deduce that X (cid:48) i d == X i as Φ (cid:48) i = Φ i for i = 1 ,
2. In line withthe steps explained above, Z (cid:48) d == Z follows since this is a special case when δ = 4,completing the proof. Remark τ , the depen-dence of X on model parameters is only though the Poisson random variable P n and X depends only on one parameter δ . This feature provides us with a possibilitythat the task of sampling the conditional integral I can be largely reduced to thesimulations of simple random variables, whose distributions remain unchanged as wechange the values for the model parameters; see section 3.1 and section 3.5.We have represented the conditional time integral I by double infinite weightedsums and mixtures of simple independent random variables under the new probabilitymeasure P , which serves as a theoretical basis for the exact simulation from thedistribution of (2.4) under P . However, our goal is set up under the probabilitymeasure Q . We now focus on the task of generating a sample from the distributionof the conditional integral I under Q once we have generated a sample under P . Inparticular, we explore the relationship between the probability density functions of theintegral under these two probability measures. We specify the details in the followingtheorem. Theorem
Suppose that f P and f Q are the probability density functions of I nder the probability measures P and Q , respectively. Then, we have f Q ( x ) = L ( q, ν, τ, a , a τ ) exp (cid:18) − q x (cid:19) f P ( x ) , where L ( q, ν, τ, a , a τ ) = sinh ( qτ ) qτ exp (cid:18) a + a τ τ ( qτ coth ( qτ ) − (cid:19) I ν (cid:16) √ a a τ τ (cid:17) I ν (cid:16) q √ a a τ sinh ( qτ ) (cid:17) with I ν ( · ) denoting the modified Bessel function of the first kind.Proof. We will make use of the shift property of the Laplace transform to justifythe theorem. We first establish a connection between their respective Laplace trans-forms. For any b ≥
0, consider the Laplace transform
L { f Q } ( b ) of f Q at b , which isthe Q -expectation of exp ( − bI ). Thus, we get L { f Q } ( b ) = E Q (cid:20) exp (cid:18) − b (cid:90) τ ˜ A s ds (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ˜ A = a , ˜ A τ = a τ (cid:21) = E P (cid:104) exp (cid:16) − (cid:16) b + q (cid:17) (cid:82) τ ˜ A s ds (cid:17)(cid:12)(cid:12)(cid:12) ˜ A = a , ˜ A τ = a τ (cid:105) E P (cid:104) exp (cid:16) − q (cid:82) τ ˜ A s ds (cid:17)(cid:12)(cid:12)(cid:12) ˜ A = a , ˜ A τ = a τ (cid:105) = L { f P } (cid:16) b + q (cid:17) L { f P } (cid:16) q (cid:17) = L f P L { f P } (cid:16) q (cid:17) (cid:18) b + q (cid:19) . The second equality is a result of the change of law formula (6 . d) from Pitman andYor [30], see the Appendix of Broadie and Kaya [12] as well. Now by the applicationof the shift property, we can write f Q ( x ) = f P ( x ) L { f P } (cid:16) q (cid:17) exp (cid:18) − q x (cid:19) . Using the formula (2 . m) in Pitman and Yor [30] for the Laplace transform L { f P } of f P at q / L { f P } (cid:18) q (cid:19) = qτ sinh ( qτ ) exp (cid:18) a + a τ τ (1 − qτ coth ( qτ )) (cid:19) I ν (cid:16) q √ a a τ sinh ( qτ ) (cid:17) I ν (cid:16) √ a a τ τ (cid:17) and setting L ( q, ν, τ, a , a τ ) := (cid:0) L { f P } (cid:0) q / (cid:1)(cid:1) − establishes the stated result.The above theorem relates the density f P of the distribution explicitly given byTheorem 2.1 in terms of infinite sums to the density f Q of the distribution we areinterested in. This means we can simulate the random variable I under the measure Q provided we have an observation from its distribution under the measure P . In eneral, we construct the acceptance-rejection algorithm outlined in Algorithm 2.1 togenerate samples from f Q .On average, the probability of accepting a proposed sample is P (cid:18) U ≤ exp (cid:18) − q Y (cid:19)(cid:19) = 1 L ( q, ν, τ, a , a τ ) , where U ∼ Unif (0 ,
1) and independently Y follows the distribution of I under P .Consequently, we require L ( q, ν, τ, a , a τ ) ≥ L closer to one asit indicates higher acceptance probability on average, and thus fewer iteration stepsneeded. Algorithm 2.1
Acceptance-Rejection Simulate a realisation Y of the random variable I under P using Theorem 2.1. Obtain a sample U independently from the uniform distribution Unif (0 ,
1) overunit interval. If U ≤ exp (cid:0) − q Y / (cid:1) , accept Y as a sample drawn from the distribution of I under Q ; otherwise reject the value of Y and return to the first step. Remark L ( q, ν, τ, a , a τ ) ≥ I is (0 , + ∞ ), wehave (cid:82) ∞ L ( q, ν, τ, a , a τ ) exp (cid:0) − q x/ (cid:1) f P ( x ) dx = (cid:82) ∞ f Q ( x ) dx = 1. Noticing thatexp (cid:0) − q x/ (cid:1) ≤ x ≥ < (cid:82) ∞ exp (cid:0) − q x/ (cid:1) f P ( x ) dx ≤ (cid:82) ∞ f P ( x ) dx = 1then leads to L ( q, ν, τ, a , a τ ) ≥
3. Simulation.
In this section, we outline how to generate an exact sample for I under P by Theorem 2.1 introduced earlier. In particular, we discuss the samplingtechniques corresponding to X and X . We note that Z is a special case of X with δ = 4. In order to apply the decomposition theorem to sample the conditionalintegral, we need to determine a point at which the infinite summation is terminated.We consider the truncation for the outer summation now, leaving the inner one tobe discussed further in the following contexts. Let us denote the truncation levelby K and the resulting remainder random variables of X and X by R K and R K espectively, i.e. R K := ∞ (cid:88) n = K +1 τ n P n (cid:88) k =1 S n,k ,R K := ∞ (cid:88) n = K +1 τ n C δ/ n . We evaluate the effect of truncation by summarising the means and variances of theremainder terms in the next lemma; see Appendix A for a detailed proof.
Lemma
Given the truncation level
K > , we have E (cid:2) R K (cid:3) = ( a + a τ ) τ K , Var (cid:2) R K (cid:3) = ( a + a τ ) τ
90 18 K , E (cid:2) R K (cid:3) = δτ K , Var (cid:2) R K (cid:3) = δτ
45 116 K . Remark K . Hence, including the terms at lower levels will beenough to produce an accurate approximation. This is supported by our numericalsimulations in section 4. X . Recall that by dropping the remainders, we approxi-mate X by X K where X K = K (cid:88) n =0 τ n P n (cid:88) k =1 S n,k . Notice that the S n,k are independently and identically distributed as S = (cid:0) /π (cid:1) (cid:80) ∞ l =1 (cid:15) l /l . To reduce the truncation error further, we simulate the tail sum R K as well.Glasserman and Kim [17] use the central limit theorem to show the validity of a normalapproximation for the remainders. They also point out that a gamma approximationis feasible and better in the sense that its cumulant generating function is closer tothat of the remainder random variable compared with that of a normal approximation.Therefore, inspired by this, the approximation to X with tail simulation for a giventruncation level K is X ≈ X K + Γ K , where Γ K is a gamma random variable such that its first two moments match thoseof the remainder R K from Lemma 3.1.We now detail our sampling strategy for X K . The series which defines X K sug-gests two potential problems. First, the random variables S n,k d == S are representedby an infinite weighted sum of independent exponential random variables, which re-quires an efficient simulation method. Second, given a Poisson sample P n = P for afixed level n = 0 , . . . , K , sampling the sum of P independent random variables S be-comes increasingly computationally demanding when the sample P tends to be larger. hus, we now incorporate these two tasks with each other and consider simulatingthe sum of P independent random variables S directly, denoted by S P , i.e. S P = P (cid:88) k =1 S k , where S k are independent copies of S . Using the Laplace transform for S given inBiane, Pitman and Yor [8], S P has the following Laplace transform:Φ S P ( b ) = E (cid:2) exp (cid:0) − bS P (cid:1)(cid:3) = (cid:32) √ b sinh √ b (cid:33) P , (3.1)for b > P can be expressed in the form P = p + 10 p + 50 p + 5000 p + 10 p + 10 p + 10 p . Here p is the multiples of 10 present in the integer P , i.e. p = (cid:98) P/ (cid:99) , p is the multiples of 10 present in the remainder of the division of P by 10 , i.e. p = (cid:98) (cid:0) P − p (cid:1) / (cid:99) , and so forth. As the law of S P is infinitely divisible forany P > . S P admits therepresentation S P d == (cid:88) k ∈ S p k (cid:88) i =1 S ki , where S = (cid:8) , , , , , , (cid:9) and for i = 1 , . . . , p k , S ki are independentcopies of S k , i.e. the sum of k independent random variables S , with k ∈ S . Then,the above representation can be intended as a basis for an efficient sampling schemefor S P for all P > S k effectively for k ∈ S . Indeed, we applythe direct inversion method to simulate S k with their inverse distribution functionsapproximated by predetermined Chebyshev polynomials for each k ∈ S . In general,the direct inversion algorithm for generating the samples of S P for any P >
Algorithm 3.1
Direct inversion for S P For each k ∈ S , sample p k independent random variables S ki , i = 1 , . . . , p k fromthe distribution of S k using the inverse distribution functions based on the corre-sponding Chebyshev polynomial approximations. Compute the accumulated sum, i.e. (cid:80) k (cid:80) p k i =1 S ki ∼ S P .The advantage of this algorithm is that we only need to construct the Cheby-shev polynomial approximations for the inverse distribution function of S k for k ∈ S .With this replacement, the complicated inverse distribution function becomes veryeasy to compute at arbitrary points. Moreover, since S k does not depend on anymodel parameters, the coefficients of the polynomials can be computed and tabulatedin advance. As such, when a sample for X is needed, we truncate the series rep-resentation to include the terms at n ≤ K with the tail approximated by a gammadistribution. For each n = 0 , . . . , K , we generate Poisson samples P n and simulate the ums S P n directly by Algorithm 3.1, which requires evaluating some prescribed poly-nomials with coefficients drawn directly from the cached table; see the supplementarymaterials. To make the above process fast for implementation, we take advantage ofthe direct inversion to obtain Poisson samples when the mean is less than 10. Forlarger means, the PTRD transformed rejection method suggested by H¨ormann [21]will be used.To obtain the Chebyshev coefficients, it is crucial to determine the values ofthe inverse distribution functions at several points efficiently and accurately. Forlarge P , we derive an asymptotic series expansion for the distribution function of S P when P → + ∞ through the inverse Fourier transform of its characteristic function.While for small P , we utilise the explicit expression for the density function givenby Biane and Yor [9], which involves the parabolic cylinder functions. To derive therepresentation for the distribution function, we use a routine consisting of the powerseries and asymptotic expansions for the parabolic cylinder functions to evaluate thedensity function followed by term-wise integration. With these expansions computed,we apply root-finding algorithms to calculate the required values. S P for large P . Before proceeding, it is worth noticing that in the limit P → ∞ , the expectation E (cid:2) S P (cid:3) = P/ (cid:2) S P (cid:3) = 2 P/
45 of S P will diverge. Thus, we stan-dardise the random variable S P by Z P = (cid:0) S P − P/ (cid:1) / (cid:112) P /
45, so that the newrandom variable Z P has mean zero and variance one. As S P is non-negative, thesupport of Z P is (cid:104) −√ P / √ , + ∞ (cid:17) . Then, taking the inverse Fourier transform ofthe characteristic function, the probability density function f Z P of Z P has the form f Z P ( x ) = (cid:114) P f S P (cid:32) P x (cid:114) P (cid:33) = (cid:114) P
45 12 π (cid:90) + ∞−∞ exp (cid:32) − i z (cid:32) P x (cid:114) P (cid:33)(cid:33) (cid:18) √− z isinh √− z i (cid:19) P dz, where f S P denotes the probability density function of S P and the first equality followsfrom the classical theorem on transforming density functions. By introducing β = x (cid:112) / / √ P , the above equation can be written as f Z P ( x ) = 14 π (cid:114) P (cid:90) + ∞−∞ exp ( P ρ ( z ; β )) dz, (3.2)where ρ ( z ; β ) satisfies ρ ( z ; β ) = log (cid:32) √ z isinh √ z i (cid:33) + z i (cid:18)
16 + 12 β (cid:19) . (3.3)We apply the standard technique of the steepest descent method to develop theasymptotic approximation for f Z P , where all the higher order terms are given inreciprocal powers of P , see Bender and Orszag [6], Bleistein and Handelsman [10]and Ablowitz and Fokas [1]. The expansion is then integrated term-wise to generatethe asymptotic representation for the distribution function. The general procedureis given below. We first identify the critical points including saddle points z of ρ ( z ; β ) such that ρ (cid:48) ( z ; β ) = 0. Note that since ρ ( z ; β ) depends on the parameter , the saddle point z will also depend on β . Due to the fact that β is quite smallas P → + ∞ , we can establish a useful expression for z as an asymptotic series in β . Afterwards, we demonstrate that the original contour of integration, i.e. the realline, can be deformed onto the steepest descent paths, obtained by considering thecontour defined by Im ( ρ ( z ; β )) = Im ( ρ ( z ; β )) and Re ( ρ ( z ; β )) < Re ( ρ ( z ; β )), inthe domain where the integrand is analytical. In this way, the rapid oscillations ofthe integrand can be removed when P is large, whence the asymptotic behaviour ofthe integral can be determined locally depending only on a small neighbourhood ofthe critical points. We present the results in the next theorem, which is proved inAppendix B. Theorem As P → + ∞ for fixed x with | x | (cid:28) √ P / (cid:112) / , i.e. β (cid:28) , wehave f Z P ( x ) ∼ π (cid:114)
245 exp (cid:32) P ∞ (cid:88) l =2 ˆ ρ l β l (cid:33) ∞ (cid:88) j =0 ∞ (cid:88) l =0 (cid:98) j (cid:99) (cid:88) n =0 ˆ α n,l,j Γ (cid:18) j + 12 (cid:19) β l P n − j , (3.4) where ˆ ρ l and ˆ α n,l,j are constants with explicit form derived in the proof and Γ ( c ) isthe gamma function.Remark ρ l is defined in the proof in equation (B.16) usingconstants ˆ r k , ˆ ξ k and ˆ υ l,j defined in equations (B.3), (B.5) and (B.9), respectively.The constant ˆ α n,l,j is defined in equation (B.30), which depends on constants ˆ ω n,j ,ˆ E l,k,n , ˆ K k , ˆ γ l,k , ˆ C l,k ,k , ··· ,k n , ˆ (cid:36) l , ˆ φ l,n , ˆ ϕ k,n , ˆ υ l,j , ˆ r k and ˆ ξ k defined in equations (B.28)-(B.29), (B.20)-(B.23), (B.27), (B.18), (B.19), (B.26), (B.11), (B.7), (B.9), (B.3) and(B.5), respectively.Having developed the large P asymptotic approximation for the probability den-sity function f Z P with all the higher order terms given in reciprocal powers of P , thenext stage is to derive an asymptotic representation for the corresponding distributionfunction. Before that, we first consider the asymptotic expansion for the probability P (cid:0) z < Z P ≤ z (cid:1) for some | z | , | z | (cid:28) √ P / (cid:112) /
45, which can be accomplished bytaking the integration of (3.4) on the finite interval ( z , z ]. We then explain howthis expression can be used to approximate the distribution function. The results aresummarised in the next theorem with the proof given in Appendix C. Theorem
For | z | , | z | (cid:28) √ P / (cid:112) / , the following asymptotic series ex-pansion holds as P → + ∞ . For z < z < , we have (cid:90) z z f Z P ( x ) dx ∼ π (cid:114) ∞ (cid:88) j =0 P − j j (cid:88) r =0 r (cid:88) n =0 j − r (cid:88) l =0 ˆ η n,r ˆ λ l,j − r ( − r + l (cid:16) √ (cid:17) n + r + l − · (cid:32) γ (cid:32) n + r + l + 12 , ( z ) (cid:33) − γ (cid:32) n + r + l + 12 , ( z ) (cid:33)(cid:33) . For z < ≤ z , we have (cid:90) z z f Z P ( x ) dx ∼ π (cid:114) ∞ (cid:88) j =0 P − j j (cid:88) r =0 r (cid:88) n =0 j − r (cid:88) l =0 ˆ η n,r ˆ λ l,j − r (cid:16) √ (cid:17) n + r + l − · (cid:32) ( − r + l γ (cid:32) n + r + l + 12 , ( z ) (cid:33) + γ (cid:32) n + r + l + 12 , ( z ) (cid:33)(cid:33) . ere, ˆ η n,r and ˆ λ l,j − r are constants explicitly given in (C.5) - (C.6) and (C.7) - (C.8) inthe proof and γ ( s, z ) is the lower incomplete gamma function.Remark P asymptotic series expansion forthe probability that the random variable Z P takes values in ( z , z ]. Notice thatthis representation is valid when | z i | (cid:28) √ P / (cid:112) /
45 for i = 1 ,
2. This restrictioncan be traced back to Theorem 3.3, where the saddle point is given in an asymptoticexpansion in β for β (cid:28)
1. Hence for practical applications, we truncate the asymptoticexpansion to generate an accurate approximation for the saddle point when β issufficiently small. More precisely, there is a region centred around zero with width˜ β , throughout which the error of the approximation is below a given threshold. Therange of validity can be determined by numerical comparisons using for exampleMaple in practice. This range of validity turns out to be large enough so that theerror in computing the distribution function for large P is negligible. More precisely,we take the error below 10 − for all the cases considered here and the correspondingvalues for ˜ β are 0 . . . . P = 5000 , , and 10 ,respectively.In summary, we have so far developed a tractable method to evaluate the distri-bution function F Z P ( z ). This is approximated by integration of the correspondingdensity function on some restricted interval ( − ˜ z, z ] with ˜ z carefully chosen for each P .We derive an asymptotic expansion for the integral in reciprocal powers of P for allorders following the steepest descent method. In practice, we compute enough termsfor the expansion to achieve the desirable accuracy in Maple with 50 digit accuracyfor P = 5000 , , and 10 , along with the root-finding for F − Z P values at nodalpoints required by Chebyshev polynomial approximations. S P for small P . In this section, we turn to the specifics of the series expansion for the distributionfunction of S P for small P . Recall from (3.1) that S P has the Laplace transformΦ S P ( b ) = (cid:16) √ b/ sinh √ b (cid:17) P for b >
0. Biane and Yor [9, formula (3x)] have given anexplicit expression for the probability density function with such a Laplace transform.Namely, for arbitrary
P >
0, the probability density function f S P is of the form f S P ( y ) = 1 √ π P Γ ( P ) y − ( P +2) ∞ (cid:88) n =0 Γ ( n + P )Γ ( n + 1) exp (cid:32) − (2 n + P ) y (cid:33) D P +1 (cid:18) n + P √ y (cid:19) , where D P +1 ( z ) is the parabolic cylinder function with order P + 1. We use differentstrategies to calculate these functions according to different ranges of z . For small z , the power series is preferable while for large z an asymptotic expansion will beapplied. We summarise these properties here. First, the series expansion for theparabolic cylinder function can be written as D P +1 ( z ) = ∞ (cid:88) k =0 ˆ d k ( P ) z k , (3.5)with the coefficients ˆ d k ( P ) satisfying some recurrence relations, that can be found inGil, Segura and Temme [15] or Abramowitz and Stegun [2]. We use Maple for theirpractical implementation. Second, in the limit z → + ∞ , D P +1 ( z ) has the following symptotic behaviour (Gil, Segura and Temme [15, formula (23) , (24) , (25)]): D P +1 ( z ) ∼ exp (cid:18) − z (cid:19) z P +1 ∞ (cid:88) k =0 ( − k ( − ( P + 1)) k k ! (2 z ) k , (3.6)where ( a ) k denotes the Pochhammer symbol ( a ) k = Γ ( a + k ) / Γ ( a ). For comparisonsof different computational methods, see Temme [35] and Gil, Segura and Temme [15].Finally, integrating the density function f S P term-wise yields the series representationfor the distribution function F S P of S P stated below. See Appendix D for a detailedproof. Theorem
For any ≤ x < ∞ and P ∈ (0 , ∪ N , the distribution function F S P ( x ) can be written as the following convergent series F S P ( x ) = 1 √ π P +1 Γ ( P ) ∞ (cid:88) n =0 Γ ( n + P )Γ ( n + 1) (2 n + P ) − P G (cid:18) n + P √ x (cid:19) , (3.7) where the function G ( y ) for y > is given by G ( y ) = (cid:90) + ∞ y z P − exp (cid:18) − z (cid:19) D P +1 ( z ) dz. Further, G satisfies G ( y ) = G ( y, y ∗ ) + G ( y ∗ ) for all y ∗ ≥ y that are sufficiently large, where G can be expressed as the convergentseries G ( y, y ∗ ) = ∞ (cid:88) k =0 ˆ d k ( P ) 2 P + k − (cid:32) Γ (cid:18) P + k , y (cid:19) − Γ (cid:32) P + k , ( y ∗ ) (cid:33)(cid:33) , and G has the asymptotic expansion G ( y ∗ ) ∼ ∞ (cid:88) k =0 ( − k ( − ( P + 1)) k k ! 2 P − k − Γ (cid:32) P − k + 12 , ( y ∗ ) (cid:33) , as y ∗ → + ∞ . Here,
Γ ( s, z ) is the upper incomplete gamma function. The previous theorem provides an effective approach to calculate the distributionfunction F S P for small P across its support with high precision. In practice, wechoose to use the asymptotic expansion for the parabolic cylinder function D P +1 ( z )whenever z ≥ ∆ ( P + 3 /
2) for some positive constant ∆ (cid:29)
1, suggested by Gil,Segura and Temme [15, Section 5]. Accordingly, we set y ∗ = max { ∆ ( P + 3 / , y } when computing the function G ( y ) for fixed y >
0. This means only asymptoticseries is involved in the computation of G ( y ) = G ( y ) for sufficiently large y suchthat y ≥ ∆ ( P + 3 / P , can be determined by numerical trials of comparing the accuracy and efficiency ofevaluating both the power series and asymptotic representations at particular points.As in the case for large P , we compute the above series representation for F S P andperform the root-finding for F − S P in Maple for P = 1 ,
10 and 50. Notice that the seriesexpansion developed here is valid for any P ∈ (0 , P . This willbe useful in the simulation of X later on. .4. Chebyshev polynomial approximation for the inverse distributionfunction of S P . As presented above, for any positive integer P , the simulation of S P is based on generating a series of random variables S k for k ∈ S by direct inversion.This method takes a uniform sample u ∼ Unif (0 ,
1) and returns the quantile functionevaluated at u as a sample for the associated distribution, which requires computingthe inverse of the distribution function. However, it is often the case that the invertingprocess is computationally inefficient due to many factors such as poor initial guessand the lack of an analytical expression for the corresponding quantile function. Sincea large number of samples is needed for the Monte Carlo simulation when the samenumber of inversions of the distribution will be performed, we now look for a moretractable technique to complete this step.Indeed, we employ the method of Chebyshev polynomials explained below toapproximate the inverse distribution function F − S P for P ∈ S . Despite the fact thatthe polynomial is just an approximation, we can still obtain highly accurate resultsby restricting the error, which is controlled by the degrees of the polynomials weconstruct. In practice, we require the uniform error to be far smaller than the MonteCarlo error, e.g. of order 10 − .Recall that a degree n Chebyshev polynomial approximation has the form c T ( z ) + c T ( z ) + · · · + c n T n ( z ) − c , where T k ( z ) = cos ( k arccos z ) for k = 1 , . . . , n are the Chebyshev polynomials ofdegree k defined on [ − ,
1] and c k for k = 1 , . . . , n are the Chebyshev coefficientscomputed in the standard way following Press et al. [31]. Since polynomials of-ten exhibit more rapid changes than the distribution functions, approximations bypolynomials might not be able to fully capture the behaviour of the inverse function F − S P ( u ). Hence, identifying appropriate scaling schemes of the argument u is of greatimportance to allow the application of the Chebyshev polynomial approximation. Thechoices of the scales are mainly characterised by the behaviour of the function depend-ing on the range of P . We briefly state the scaling and its rationale behind for large P and small P separately. Large P . Instead of the sum S P , we take the normalised random variable Z P with zero mean and unit variance into consideration. For the approximation of theinverse distribution function F − Z P , we focus on the sub-interval [ F Z P (0) ,
1) of itssupport [0 ,
1] first, corresponding to the region where the random variable Z P takespositive values. In the limit of large P , the distribution function of Z P resemblesthat of a standard normal distribution. Thus, we generalise and apply the ideasunderlying the Beasley-Springer-Moro direct inversion method for standard normalrandom variables; see Moro [29], Joy, Boyle and Tan [22] and Malham and Wiese[27]. The normal distribution function has three regions exhibiting different charac-teristic behaviours on the positive real line. Accordingly, we roughly split the interval[ F Z P (0) ,
1) into three regimes: the central [ F Z P (0) , u ], the middle ( u , u ] and thetail (cid:0) u , − − (cid:3) regimes. In general, the central regime roughly represents the areawhere the decreasing density function has a increasing slope while the middle regimerepresents the area where the decreasing density function has a declining slope withthe tail regime representing the region where the density function is flat taking valuesclose to zero. We neglect the regimes from 1 − − to 1. Remark u and u by a small umber of trials in Maple to ensure that the resulting Chebyshev polynomial approx-imations have moderate degrees while retaining the accuracy for all three regimes.We may come across the circumstance that the approximations which achieves thedesired accuracy have degrees of say 15 for both the central and middle regimes buta higher degree of say 50 for the tail regime for some given u and u . Such a caseshould be avoided from the perspective of efficiency as higher degree often comes withhigher computational cost. Hence, it is necessary to set the values u and u againthrough further investigations so that the degrees of the approximation for all regionsare balanced with each other. If both of the degrees of the Chebyshev polynomialsconstructed for two neighbouring regions are at relative lower level, we may combinethose two regimes to one and produce a unified approximation.In the central regime, we follow Malham and Wiese [27] to scale and shift thevariable. Define U ( u ) := √ π ( u − F Z P (0)) and z ( u ) := k U ( u ) + k , where theparameters k and k are chosen to make sure z ( F Z P (0)) = − z ( u ) = 1.Then, we approximate the inverse distribution function by F − Z P ( u ) ≈ U ( u ) · (cid:18) c T ( z ( u )) + c T ( z ( u )) + · · · + c n T n ( z ( u )) − c (cid:19) . In the middle and tail regimes, we approximate F − Z P ( u ) ≈ c T ( z ( u )) + c T ( z ( u )) + · · · + c n T n ( z ( u )) − c , where U ( u ) := log ( − log (1 − u )) and z ( u ) := k U ( u ) + k with the parameters k and k chosen such that z ( u ) = − z ( u ) = 1 at the rightendpoint. The ansatz for U follows from inverting the asymptotic tail approximationfor the standard normal, which is equivalent in distribution to Z P when P → + ∞ bythe central limit theorem; see Moro [29].The above serves as a general discussion for choosing the scaled variables andapproximations in the region of (cid:2) F Z P (0) , − − (cid:3) for large P . We apply thisprocedure to the cases P = 10 , , , , and 10 , the inverse distributionfunctions of which are roughly anti-symmetric. For the remaining half sub-interval (cid:2) − , F Z P (0) (cid:1) of its support, we can apply similar results to the scaling and ap-proximation following the arguments mentioned above. Small P . In the Chebyshev polynomial approximation for small P , the idea re-mains the same as above. Notice that the random variable S P takes positive valuesonly. Since the distribution has a heavy right tail, we break down the support of F − S P into four regimes: the left (cid:2) − , u (cid:3) , the central ( u , u ], the middle ( u , u ] andthe right tail (cid:0) u , − − (cid:3) regimes. We neglect the regimes at a distance of 10 − from its endpoints. In theory, these boundary points are determined in accordancewith the behaviour of the distribution function, but again it is better to set them viaempirical studies in practice.The central limit theorem tells us the asymptotic distribution of the sum S P when P is large. However, for small P we have to analyse the limiting behaviour ofthe distribution function F S P and its inverse F − S P in order to help us find the properscaled variables when we construct Chebyshev polynomial approximations. We buildon the series representation for F S P given in Theorem 3.7 to derive the results below;see Appendix E for a detailed proof. Corollary
For any P ∈ (0 , ∪ N , the distribution function F S P has the ollowing asymptotic expansion F S P ( x ) ∼ √ π P + P P − x − P + exp (cid:18) − P x (cid:19) , as x → + . (3.8)The above expression describes the behaviour of the distribution function F S P ( x )as x → + . Now, our goal is to invert this relation to obtain the asymptotic approx-imation for the inverse distribution function F − S P ( u ) as u → + . Let u = F S P ( y ),then from (3.8) it is clear u √ π P + P P − ∼ y − P + exp (cid:18) − P y (cid:19) , as y → + . (3.9)Introducing the new variable v := u √ π/ (cid:16) P + P P − (cid:17) and taking logarithm on bothsides, we can rewrite (3.9) aslog v ∼ (cid:18) P − (cid:19) log 1 y − P y , as y → + . After rearrangement, the above expression becomes1 y ∼ P (cid:18) P − (cid:19) log 1 y − P log v, as y → + . Since log y − /y − → y → + , we have − v/P ∼ y − in the limit y → + .This relation further giveslog (cid:18) − P log v (cid:19) = o (cid:18) y (cid:19) , as y → + . Hence, taking advantage of order relations we can write y in terms of v as1 y ∼ − P log v + 2 P (cid:18) P − (cid:19) log (cid:18) − P log v (cid:19) (3.10) = − P log v + 2 P (cid:18) P − (cid:19) log (cid:18) P (cid:19) + 2 P (cid:18) P − (cid:19) log ( − log v )when y → + . In particular by the fact that log ( − log v ) = o (cid:0) y − (cid:1) as y → + , itsleading order behaviour yields y ∼ (cid:18) P (cid:18) P − (cid:19) log 2 P − P log v (cid:19) − = (cid:18) P (cid:18) P − (cid:19) log 2 P − P log (cid:18) u √ π P + P P − (cid:19)(cid:19) − , as y → + , i.e. u → + , where the last equation comes from the transformation v = u √ π/ (cid:16) P + P P − (cid:17) .As u →
1, i.e. y → + ∞ , we adopt a gamma approximation for the tail. This isimplied by empirical tests which show that the distribution is positively skewed witha longer right tail. Hence, by matching the mean and variance of S P with those of agamma random variable, the shape and rate parameters take the form s = 5 P/ = 15 /
2. Then, the distribution function F S P is approximated by the distributionfunction F X of a gamma random variable X with parameters s and r given as follows: F X ( y ) = 1 − (cid:0) P (cid:1) Γ (cid:18) P, y (cid:19) , Making use of the asymptotic relationship given below for the incomplete gammafunction (Abramowitz and Stegun [2, formula (6 . . s, z ) ∼ z s − exp ( − z ) ∞ (cid:88) k =0 Γ ( s )Γ ( s − k ) z − k , as z → + ∞ establishes as y → + ∞ , F X ( y ) ∼ − (cid:0) P (cid:1) (cid:18) y (cid:19) P − exp (cid:18) − y (cid:19) . (3.11)Set u = F X ( y ). After rewriting (3.11), we obtain(1 − u ) Γ (cid:18) P (cid:19) ∼ (cid:18) y (cid:19) P − exp (cid:18) − y (cid:19) , as y → + ∞ . To generate an asymptotic expression for y , we start by taking logarithm and definingthe new variable v := (1 − u ) Γ (5 P/ v ∼ (cid:18) P − (cid:19) log (cid:18) y (cid:19) − y, as y → + ∞ . (3.12)Rearranging (3.12) leads to y ∼ −
215 log v + 215 (cid:18) P − (cid:19) log (cid:18) y (cid:19) , as y → + ∞ . By a short calculation analogous to (3.10), we conclude y ∼ −
215 log v + 215 (cid:18) P − (cid:19) log ( − log v ) , as y → + ∞ . On substituting v = (1 − u ) Γ (5 P/ y → + ∞ , i.e. u →
1, its leading order is ofthe form y ∼ −
215 log (cid:18) (1 − u ) Γ (cid:18) P (cid:19)(cid:19) . The analysis above outlines the asymptotic approximation for F − S P ( u ) as u → u →
1, and provides the ansatz behind the choices of reasonable scaling variablesfor Chebyshev polynomial approximations for small P , i.e. P = 1. Accordingly, wereport the routines for approximations of the inverse distribution function F − S P ( u )through Chebyshev polynomials for the four regimes identified in detail. Note againthe parameters k and k given below restrict the ranges of the transformed variable z to the interval [ − , or all the four regimes, we approximate F − S P ( u ) by a degree n Chebyshev poly-nomial approximation of a scaled and shifted variable z ( u ) := k U ( u ) + k as below: F − S P ( u ) ≈ c T ( z ( u )) + c T ( z ( u )) + · · · + c n T n ( z ( u )) − c , where U ( u ) := (cid:0)(cid:0) /P (cid:1) ( P − /
2) log (cid:0) /P (cid:1) − (cid:0) /P (cid:1) log (cid:0) u √ π/ (cid:0) P +1 / P P − (cid:1)(cid:1)(cid:1) − in the left regime (cid:2) − , u (cid:3) , U ( u ) := ( u − u ) (cid:112) P/
45 in the central regime ( u , u ]and U ( u ) := ( − /
15) log ((1 − u ) Γ (5 P/ u , u ] and righttail regime (cid:0) u , − − (cid:3) . Fig. 1 . We plot the errors in the Chebyshev polynomial approximations to the inverse distri-bution functions F − Z P ( u ) with P = 10 ( top panel ) and F − S P ( u ) with P = 1 ( bottom panel ) acrossall regimes, respectively. Note that to highlight the tail we use a log-log scale with − u on theabscissa. We have summarised the approximation techniques for the inverse distributionfunction F − Z P ( u ) or F − S P ( u ) taking into account the various values that P and u might take. Following this, we compute the coefficients for the Chebyshev polyno-mial approximations in the standard fashion (see Press et al. [31]) for the cases = 1 , , , , , , 10 using Maple. With all these accurate and reliablecoefficients, quoted in the supplementary materials, then imported to Matlab, sub-sequent Chebyshev approximations are evaluated by Clenshaw’s recurrence formula,which can be found in Press et al. [31]. Therefore, for any P > S P can be sampledrepeatedly with high efficiency using Algorithm 3.1.We end this section by showing the respective errors in the Chebyshev polynomialapproximations to F − Z P ( u ) and F − S P ( u ) with u ∈ (cid:2) − , − − (cid:3) when P is 10 and 1 in Figure 1. For each u , the error of the approximation is c − ˆ c , where c isobtained by a high precision root-finding procedure applied to the expansions for thedistribution functions and ˆ c is evaluated by the prescribed Chebyshev polynomials.To highlight the tail we plot the errors on a log-log scale with 1 − u on the abscissa.For P = 10 , we split the interval (cid:2) − , − − (cid:3) into two regimes: (cid:2) − , . (cid:1) and (cid:2) . , − − (cid:3) , where both of the Chebyshev polynomials have degrees 16.For P = 1, we generate approximations for five regimes as described above where theright tail region is further split into two, the degrees for which are 25 in the left with u ∈ (cid:2) − , . (cid:1) , 18 in the central with u ∈ [0 . , . u ∈ [0 . , . . , . (cid:2) . , − − (cid:1) ,respectively. Notice the error in all cases is of order 10 − . Results for the othervalues, reported in Shen [33], have similar accuracy as those in Figure 1. X and Z . Let us first introduce the notation h = δ/ δ = 4 κθ/σ is the degrees of freedom. Note that for Heston models calibratedto real market data, the zero boundary of the variance process is typically attainableand strongly reflecting; see Haastrecht and Pelsser [36] and Lord, Koekkoek and VanDijk [25]. By the Feller condition, this requires δ <
2, i.e. h <
1. Hence, afterseparating the time parameter, X can be written in the form X = ∞ (cid:88) n =1 τ n C hn = τ ∞ (cid:88) n =1 n C hn = τ Y h , with Y h := (cid:80) ∞ n =1 C hn / n depending only on the parameter h . The structure of therandom variable Y h , along with its dependence on the model parameters, providesus with another possibility to sample Y h and thus X , apart from the truncationmethod. In fact, the Laplace transform of Y h is identical to that for S h . We cantherefore try to extend the direct inversion of S h for any h ∈ N developed above forthe simulation of Y h .Recall that Y h has the Laplace transformΦ Y h ( b ) = E (cid:2) exp (cid:0) − bY h (cid:1)(cid:3) = (cid:32) √ b sinh √ b (cid:33) h , for b > , which has the same expression as that of S h given by (3.1) after replacing P by h .The only difference is that the parameter h now is restricted to (0 ,
1) rather thanpositive integers. This suggests the decomposition proposed in section 3.1 for integer h and the resulting S h is no longer reasonable. However, motivated by Malham andWiese [26], we have the following alternative formula for 0 < h <
1, which is given tothe first three decimal places h = h h
10 + h
20 + h
50 + h
100 + h
200 + h
500 + h h , here h k ∈ { , , } for k ∈ H = { , , , , , , , , } . The abovedecomposition works for the case when h is rounded to three decimal digits, but itcan be generalised to h ∈ (0 ,
1) given to any decimal places in principle. Next, wegive the direct inversion algorithm for generating Y h for any h ∈ (0 ,
1) given to thefirst three decimal places.
Algorithm 3.2
Direct inversion for Y h For each k ∈ H , sample h k independent random variables Y /k ,i , i = 1 , . . . , h k fromthe distribution of Y /k by inverse transform sampling based on the correspondingChebyshev polynomial approximations. Compute the accumulated sum, i.e. (cid:80) k (cid:80) h k i =1 Y /k ,i ∼ Y h . Fig. 2 . We plot the errors in the Chebyshev polynomial approximations to the inverse distri-bution functions F − Y h ( u ) with h = 0 . across all regimes. Note as above we use a log-log scalewith − u on the abscissa. Given Algorithm 3.2, for a general h , the simulation of Y h is reduced to simu-lating several particular random variables such as Y / , Y / , . . . using their inversedistribution functions, which are approximated by the associated Chebyshev polyno-mials. To design these Chebyshev polynomial approximations, the approach for S P with small integer P introduced in section 3.3 and section 3.4 can be fully used here.This is because the series expansion and the asymptotic approximation for the distri-bution function of S P remain valid for small non-integer P . Therefore, we apply thesame strategy to calculate the Chebyshev polynomial approximations for the inversedistribution function of Y h with fixed h = 1 /k , k ∈ H , the values for the coefficients ofwhich are presented in the supplementary materials. Figure 2 shows the errors in theapproximation for F − Y h by Chebyshev polynomials across all regimes when h = 1 / (cid:2) − , . (cid:1) , [0 . , . . , . . , . (cid:2) . , − − (cid:3) with degrees 24, 19, 18, 19 ig. 3 . We plot the errors in the Chebyshev polynomial approximations to the inverse distribu-tion functions F − Y h ( u ) with h = 2 , i.e. F − Z (cid:48) ( u ) across all regimes. Note as above we use a log-log scale with − u on the abscissa. and 31, respectively. All errors are fluctuating at the level of 10 − . See Shen [33] formore results of the other cases.As Z is a special case of X when h = 2, the approach to generating samples of X discussed above is fully applicable here. In fact, we directly construct the Chebyshevpolynomial approximations for the inverse distribution function F − Z (cid:48) with Z (cid:48) = Z/τ since Z (cid:48) is independent of any parameters. We plot the resulting errors in Figure 3.The polynomials have degrees between 22 and 27 in the four regions with errors oforder 10 − . Table 1
Parameters for the Heston model.
Parameters European Path-dependentCase 1 Case 2 Case 3 Case 4 Asian Barrier κ . . .
21 1 . . θ .
04 0 .
04 0 .
09 0 .
019 0 . . σ . .
61 0 . ρ − . − . − . − . − . t
10 15 5 1 4 1 v .
04 0 .
04 0 .
09 0 . . . S
100 100 100 100 100 100 r .
05 0 .
4. Numerical analysis.
In this section, we compare our direct inversion methodwith the gamma expansion of Glasserman and Kim [17] by pricing four challengingEuropean call options in the Heston model. The sets of parameters considered aregiven in Table 1. These four sets are taken from Glasserman and Kim [17], which arefound to be in the typical range of parameter values of the Heston model in practice.Two path-dependent options including an Asian option with yearly fixings (see Smith Before giving simulation resultsfor option prices, we first illustrate the performance of the above direct inversionmethod in terms of accuracy based upon the series expansion given in Theorem 2.1.Recall that our objective is to sample from the distribution of the random variable (cid:82) t V s ds given its endpoints V and V t , denoted by ¯ I , i.e.¯ I = (cid:18)(cid:90) t V s ds (cid:12)(cid:12)(cid:12)(cid:12) V = v , V t = v t (cid:19) = 4 σ (cid:18)(cid:90) τ ˜ A s ds (cid:12)(cid:12)(cid:12)(cid:12) ˜ A = a , ˜ A τ = a τ (cid:19) = 4 σ I, under the probability measure Q . We have decomposed the integral into the sum ofthree independent series after measure transformation. Among the realisation of thosethree series, the first one is truncated with tail approximated by a gamma randomvariable and the remaining two series are simulated exactly by direct inversion. (a) Case 1: v = v t = 0 . Fig. 4 . We indicate the absolute errors in the first four moments of the conditional integral ¯ I simulated by direct inversion and gamma expansion versus the truncation levels for Case withdifferent values for v t . Both methods are implemented with tail simulation. We perform · simulations for each case. Below the dashed line, the errors are not statistically significant at thelevel of three standard deviations. In Figure 4 and Figure 5 the absolute errors in the first four moments are displayedfor simulating the conditional integral ¯ I with different values of v t using our method.For comparison, we include the results by employing the gamma expansion fromGlasserman and Kim [17] as well. For both methods, we apply tail approximationswith truncation level increasing in integers. The number of samples generated in eachcase is 5 · . The three panels shown in Figure 4 from top to bottom correspondto the three representative values v t = 0 . , , . v t = 0 . , . , .
05 (top tobottom) for Case 4. Note that the true moments can be computed by evaluating he respective derivatives of the moment generating functions derived by Broadie andKaya [12] at the origin. (b) Case 1: v = 0 . , v t = 4(c) Case 1: v = 0 . , v t = 0 . Fig. 4 . (cont.) We indicate the absolute errors in the first four moments of the conditionalintegral ¯ I simulated by direct inversion and gamma expansion versus the truncation levels for Case with different values for v t . Both methods are implemented with tail simulation. We perform · simulations for each case. Below the dashed line, the errors are not statistically significantat the level of three standard deviations. We observe that most errors for the first two moments across different values of v t and truncation levels for both Case 1 and 4 are not significantly different fromzero at the level of three standard deviations, suggesting both methods achieve highaccuracy for these two moments as expected. This is consistent with the theory as tail imulation in each method is designed such that the first two moments are matched.In other words, the simulations should lead to the exact first and second moments inprinciple, whence only Monte Carlo noise with a scaling as the inverse of the squareroot of the sample size, i.e. (cid:0) · (cid:1) − / , is observed.For the higher moments, the errors of the direct inversion are fluctuating at somelevel below the statistical significance for all circumstances considered. These errorsare so small that a decreasing trend is not visible when increasing the truncation level.In contrast, with the increment of the truncation levels, the errors of the gammaexpansion first exhibit a decaying pattern until the curves become horizontal. Forexample, the behaviour of the decreasing errors of the third and fourth moments isobvious when the truncation level is increased from one to two. The falling tendencyappears to be more significant when we increase the sample size, thus, reduce theMonte Carlo effect, which will be discussed later. This suggests that there exists somebias in the gamma expansion with small truncation levels while the direct inversionwith lower truncation levels has the same accuracy as that with higher ones. (a) Case 4: v = v t = 0 . Fig. 5 . We indicate the absolute errors in the first four moments of the conditional integral ¯ I simulated by direct inversion and gamma expansion versus the truncation levels for Case withdifferent values for v t . Both methods are implemented with tail simulation. We perform · simulations for each case. Below the dashed line, the errors are not statistically significant at thelevel of three standard deviations. While the figures for Case 1 and 4 have many details in common, they also revealnoteworthy differences in the first two moments. As illustrated in the upper panels inFigure 5a, Figure 5b and Figure 5c for Case 4, most of the first and second momenterrors in the direct inversion are slightly higher compared to those in the gammaexpansion at the same truncation level. Errors of the two schemes considered in thefirst two moments for Case 1 on the other hand seem to be of the same order to someextent with the same truncation level, which can be seen from the upper panels inFigure 4a, Figure 4b and Figure 4c. In order to find a plausible explanation for thisdifference, we increase the sample size by a factor of 10 and plot the resulting errorsversus the truncation levels in Figure 6 and Figure 7. v = 0 . , v t = 0 . v = 0 . , v t = 0 . Fig. 5 . (cont.) We indicate the absolute errors in the first four moments of the conditionalintegral ¯ I simulated by direct inversion and gamma expansion versus the truncation levels for Case with different values for v t . Both methods are implemented with tail simulation. We perform · simulations for each case. Below the dashed line, the errors are not statistically significantat the level of three standard deviations. For Case 1, Figure 6 demonstrates the errors in all four moments based on thedirect inversion are decreased as expected, i.e. proportional to the reciprocal of thesquare root of the sample size across all the values of v t and truncation levels. Thisconfirms that the moment errors observed in Figure 4 using the direct inversion aredominated by the Monte Carlo error. On the other hand, for the gamma expansionwe note in the upper panels of each subplots that the first two moments of the simu- v = v t = 0 . v = 0 . , v t = 4 Fig. 6 . We indicate the absolute errors in the first four moments of the conditional integral ¯ I simulated by direct inversion and gamma expansion versus the truncation levels for Case withdifferent values for v t . Both methods are implemented with tail simulation. We perform · simulations for each case. Below the dashed line, the errors are not statistically significant at thelevel of three standard deviations. lations for all five truncation levels are indeed matched with errors improving roughlyaccording to the expected scaling when increasing the sample size. However, we seein the lower panels that the errors in the third and fourth moments hardly show anychanges for lower truncation levels such as one and two while the accuracy for theother truncation levels is improved with the increase of the sample size. In fact, afterreducing the Monte Carlo noise, there exists an even more clear decreasing trend for v = 0 . , v t = 0 . Fig. 6 . (cont.) We indicate the absolute errors in the first four moments of the conditionalintegral ¯ I simulated by direct inversion and gamma expansion versus the truncation levels for Case with different values for v t . Both methods are implemented with tail simulation. We perform · simulations for each case. Below the dashed line, the errors are not statistically significantat the level of three standard deviations. the higher order moment errors with the gamma expansion as the truncation levelincreases. This seems to corroborate the observations from Figure 4 for Case 1, in-dicating that the gamma expansion with small truncation levels exhibits some biaswhile the direct inversion achieves the same accuracy, restricted by the Monte Carloerror, for all truncation levels.In comparison, Figure 7 shows different behaviour for the errors related to thedirect inversion for Case 4 while similar conclusion can be reached for the gammaexpansion as Case 1. More specifically, we notice that all moment errors in directinversion sampling for Case 4 are invariant to increasing the sample size when thetruncation levels are fixed. Further we observe that the errors, all remaining steadyacross a set of different truncation levels, become statistically significant when thenumber of samples is increased, especially for the first and second moments. Thus,this implies in Case 4 the direct inversion performs equally well for all truncationlevels, nevertheless, the accuracy of which is overridden by some bias. We should notfail to mention that the bias is roughly of the same order as the Monte Carlo errorwith 5 · samples, whence it is not reflected in Figure 5. This accounts for thefinding for Case 4 that the first and second moment errors for the direct inversion arealways slightly larger than those for gamma expansion, where only Monte Carlo erroris in presence. We give a possible explanation for this bias as follows.The reason for the bias with the direct inversion method for Case 4 lies in thearithmetic precision we use for the parameter h , which is related to the randomvariable X . Recall that the proposed decomposition requires the rational parameter h is given as a decimal with three significant figures. Let ˜ h stand for the roundednumber and ˜ X denote the approximation to X by replacing h with ˜ h . Next, we givethe exact errors in the first and second moments of X . Directly computing the first v = v t = 0 . Fig. 7 . We indicate the absolute errors in the first four moments of the conditional integral ¯ I simulated by direct inversion and gamma expansion versus the truncation levels for Case withdifferent values for v t . Both methods are implemented with tail simulation. We perform · simulations for each case. Below the dashed line, the errors are not statistically significant at thelevel of three standard deviations. two moments using the series which defines X , we can write E [ X ] = 13 τ h, E (cid:2) X (cid:3) = 245 τ h + 19 τ h . Then, the corresponding relative errors are (cid:12)(cid:12)(cid:12) E [ X ] − E (cid:104) ˜ X (cid:105)(cid:12)(cid:12)(cid:12) E [ X ] = (cid:12)(cid:12)(cid:12) h − ˜ h (cid:12)(cid:12)(cid:12) h , (cid:12)(cid:12)(cid:12) E (cid:2) X (cid:3) − E (cid:104) ˜ X (cid:105)(cid:12)(cid:12)(cid:12) E [ X ] = (cid:12)(cid:12)(cid:12) (cid:16) h − ˜ h (cid:17) + 5 (cid:16) h − ˜ h (cid:17)(cid:12)(cid:12)(cid:12) h + 5 h . The above equations shows a linear scaling of the moment errors of X in terms ofthe discrepancy between the true value h and the approximated value ˜ h . Table 2quotes the values for h and ˜ h for all the four European cases. Note that for Case 1and 3 accurate values of h are used while the relative errors for Case 2 and 4 are oforder 10 − and 10 − , respectively. In Figure 8 the panels show the relative errorsin the first four moments of X for Case 1 to Case 4 using 10 and 10 simulations.For Case 1 and Case 3, by successively increasing the sample size the high accuracyfor the first four moments of X sampled by direct inversion Algorithm 3.2 is indeedlimited by the Monte Carlo error, which improves roughly according to the expectedscale. However, the errors are invariant for Case 2 and Case 4 when increasing thesample size. For these two cases, the systematic Monte Carlo error is lower than thebias caused by replacing the true value h with the approximated value ˜ h . Hence, theerrors reflected in Figure 8, dominated by the bias, fail to show improvement whenthe sample size is increased by a factor of 10. v = 0 . , v t = 0 . v = 0 . , v t = 0 . Fig. 7 . (cont.) We indicate the absolute errors in the first four moments of the conditionalintegral ¯ I simulated by direct inversion and gamma expansion versus the truncation levels for Case with different values for v t . Both methods are implemented with tail simulation. We perform · simulations for each case. Below the dashed line, the errors are not statistically significantat the level of three standard deviations. Table 2
True value h and rounded value ˜ h . Case 1 Case 2 Case 3 Case 4 h . . . . h . . . . ig. 8 . We plot the relative errors in the first four moments of X simulated by direct inversionAlgorithm for Case to Case . By increasing the sample size by a factor of , we note thatthe accuracy in the moment errors is improved as expected for Case and Case . The four momenterrors are invariant for Case and Case when increasing the sample size, suggesting possible biasin the direct inversion for these two cases. Based on the above analysis, we conclude that the direct inversion method for X exhibits some small bias when approximation of the parameter h is adopted.This can conceivably lead to the bias of the general direct inversion scheme for theconditional integral ¯ I . However, this bias has nothing to do with the development ofthe method, but is associated with the decomposition technique and the arithmeticprecision involved. Without loss of generality, this method can be extended to allowfor a finer decomposition of the parameter h given to any number of decimal places.In this sense, we expect that the accuracy available for this method will become moreapparent. In this section, we apply the direct inversion method andthe gamma expansion to pricing four European call options and two path-dependentoptions including an Asian option and a barrier option. These two schemes are bothbased on the known conditional non-central chi-square distribution for the varianceprocess and the conditional lognormal distribution for the asset price. We furthercompare the above two methods with the full truncation scheme of Lord, Koekkoekand Van Dijk [25], which is a time stepping method with asset price and variancesimulated on discrete time grids. This type of equidistant discretization scheme for theone-dimensional CIR process has been shown to have an arbitrarily slow convergencerate in the strong sense in general; see Hefter and Jentzen [18]. Hence, developingother simulation methods becomes essential for practical purposes.We first demonstrate the tradeoff between the truncation level and accuracy forthe direct inversion scheme and the gamma expansion. Figure 9 plots the root meansquare error for the price of an at the money European call option against the trunca-tion level for Case 1 and Case 4. For both methods, we use a sample size of 5 · andtruncate after M terms, increasing M in integers from 1 to 10. For Case 1, the directinversion exhibits a faster convergence rate, revealed by the steeper slope in Figure 9a, Fig. 9 . We show the root mean square error in the option price (strike ) versus the trunca-tion level for Case and Case . We use a sample size of · with truncation levels increasingin integers from to . in contrast with the gamma expansion. Indeed, truncation after three terms alreadyprovides a satisfactory estimator with error curve eventually becoming noisy in thelarger M regime. To obtain the same accuracy, many more terms up to M = 10 arerequired for the gamma expansion. For Case 4, increasing M from one to two indeedhelps to reduce the error. However, further increase in M does not seem to bring mprovement to the error for both methods, as seen from the horizontal error curveswith small fluctuations in Figure 9b. This implies that approximations with small M are sufficient to achieve acceptable accuracy. Remark
Fig. 10 . We show the convergence of the root mean square error in the option price (strike ) for Case to Case of gamma expansion and direction inversion, both at a truncation level M = 5 , and full truncation Euler scheme, with number of time steps equal to the square root of thesample size. Next, we give the comparisons between the direct inversion method, the gammaexpansion and the full truncation Euler scheme. In Figure 10, we plot the root meansquare error in the option price as a function of the CPU time required on a log-log scale for all schemes. For fair comparisons, we implement these three methods asefficiently as possible and generate the CPU time using compiled Matlab code. Forthe two non-discretization methods, we choose to use truncation level M = 5. Forthe full truncation Euler method, we set the number of time steps equal to the squareroot of the sample size. This is motivated by the optimal allocation for the numberof time steps from Duffie and Glynn [ ? ], which is proportional to the square root ofthe number of trials for methods with weak order of convergence being equal to one;see Broadie and Kaya [12] and Lord, Koekkoek and Van Dijk [25].We can see from the upper panels in Figure 10 that the bias in the gamma ex-pansion with M = 5 for Case 1 and Case 2 eventually dominates the root meansquare error when the number of sample trails increases. By comparison, the rootmean square errors for the direct inversion and full truncation Euler scheme are de-clining monotonically, with the former presenting a more rapid rate with reducedcomputational cost. For Case 3 and Case 4, the two non-discretization methods bothoutperform the full truncation Euler scheme, which has a slower convergence rate eflected by the less steeper slope in the graph.With regard to the computing time, the gamma expansion is at least two to threetimes slower than the direct inversion with similar accuracy for Case 1, Case 2 andCase 3. For Case 4, the new method takes much more time compared with Case 1.This is because more effort is needed for the acceptance-rejection sampling of Case4 due to the slightly unfavourable values for the model parameters. Although thetime needed for the direct inversion is marginally more than the gamma expansionfor Case 4 with M = 1, as the desired accuracy is increased the new method requiresless computational budget. In summary, we conclude that the performance of thedirect inversion is the best among the three schemes considered here.Now we turn to the pricing of options with payoffs depending on sample paths.We first consider an at the money Asian option with yearly fixings, the payoff ofwhich is determined by the average of asset prices at the end of each year. We showin Figure 11 the root mean square error of the price versus the CPU time on a log-log scale. For the direct inversion and the gamma expansion, we truncate the series after M = 1 and simulate the asset prices for each year. Within one year, the terminal valueis obtained directly using a single step. For the time discretization scheme, multiplesteps are needed for each year. In this test, the number of time steps is taken to bethe square root of the sample size in a similar manner to Broadie and Kaya [12] andSmith [34]. Fig. 11 . We show the convergence of the root mean square error in the option price (strike ) for the Asian option of gamma expansion and direction inversion, both at a truncation level M = 1 , and full truncation Euler scheme, with number of time steps equal to the square root of thesample size. We observe that both the gamma expansion and the direct inversion, even with alower truncation level, deliver similar accuracy compared to the full truncation schemefor small sample sizes. However when the number of simulations increases, bias of theestimated price starts to dominate the root mean square error for all three methodswith the standard deviation decreasing according to the expected scaling, i.e. the able 3 Estimated prices with standard errors and CPU time for the barrier option.
Stepsize Direct inversion Gamma expansion . . . . .
33 179 . / . . . . .
52 328 . / . . . . .
68 593 . / . . . . .
35 1152 . /
16 Estimated price 0 . . . . .
06 2278 . /
32 Estimated price 0 . . . . .
38 3952 . /
64 Estimated price 0 . . . . .
79 9904 . /
128 Estimated price 0 . . . . .
93 24521 . M = 1 for a double no touch barrier option with barriers90 and 110. We sample a total of 10 paths for each case. We increase the numberof time steps per year from 1 to 128 and monitor at each time step if the asset pricehas hit one of the two barriers.We see from Table 3 that as we decrease the stepsize, the estimated price of boththe direct inversion and the gamma expansion is decreasing monotonically. This is inaccordance with our expectation since when more dates are being monitored, there re more chances for the asset price to cross the barriers. Because of the nature ofthese two methods, we expect their estimated price will eventually be almost exactwith negligible truncation errors when the asset price is monitored on a more frequentbasis, for instance, every trading day. The results here are also consistent with thoseof the four schemes tested in Malham and Wiese [26, Table 5] and the PT, FT andABR scheme in Lord, Koekkoek and Van Dijk [25, Table 7] in terms of accuracy.Similar conclusions can be reached as the cases for European and Asian options interms of the computational time. The time required for the gamma expansion is 1 .
5. Conclusion.
In this paper, we have designed a new series expansion for thetime integrated variance process under the Heston stochastic volatility model. Ourexpansion is built on a change of measure argument and the decomposition techniquesfor the integral of squared Bessel bridges by Pitman and Yor [30] and Glasserman andKim [17]. Acceptance-rejection and direct inversion methods are developed to realisethe conditional integral. On combining this result with the method of Broadie andKaya [12], almost exact simulations of the stock price and variance can be generatedon the basis of their exact distributions. We compare our approach with Glassermanand Kim [17] through pricing four practical and challenging options. Apart fromthat, two path-dependent options including an Asian option and a barrier option arealso tested using the above two methods. Further comparisons with a standard timediscretization method, i.e. the full truncation Euler scheme, are performed as well.Evidence implies faster computational speed with comparable error in our method.The series representation and sampling techniques above can also be transferredto the generalised squared Ornstein-Uhlenbeck process x t with parameters b ∈ R and δ > dx t = ( δ + 2 bx t ) dt + 2 √ x t dW t , where W t denotes a standard Brownian motion. Although in this paper we focusonly on the case 0 < δ <
2, the present result can be applied to other cases δ ≥ δ and hence establishefficient Chebyshev polynomial approximations required for the resulting direct in-version algorithm. The expansions derived in section 3 will be helpful in determiningthe coefficients.Lastly, we recommend a direction for future research. Our method entails anacceptance-rejection algorithm with acceptance probability depending on model pa-rameters. Thus, it is difficult to measure its general computational complexity, i.e. theaverage number of iterations needed. Besides, in the application of risk managementand trading, the acceptance-rejection scheme is less favourable as it will introduceconsiderable Monte Carlo noise in sensitivity analysis. For these reasons, an alterna-tive should be considered. One realistic way to avoid the use of acceptance-rejection isto sample the Radon-Nikod´ym derivative directly under the new probability measureinstead. Appendix A. Proof of Lemma 3.1.
For the remainder R K , as stated inTheorem 2.1, the S n,k are independent and identically distributed random variablesand the P n are independent Poisson random variables with mean ( a + a τ ) 2 n − /τ .Taking the expectation of R K directly, we have E (cid:2) R K (cid:3) = ∞ (cid:88) n = K +1 τ n E [ P n ] E [ S n,k ] ∞ (cid:88) n = K +1 τ n (cid:18) a + a τ τ n − (cid:19) (cid:32) π ∞ (cid:88) l =1 l (cid:33) = ( a + a τ ) τ K , where the last identity holds since (cid:80) ∞ l =1 l − = π / (cid:80) ∞ n = K +1 − ( n +1) = 2 − ( K +1) .Similarly, we can computeVar (cid:2) R K (cid:3) = ∞ (cid:88) n = K +1 τ n (cid:16) Var [ P n ] ( E [ S n,k ]) + E [ P n ] Var [ S n,k ] (cid:17) = ∞ (cid:88) n = K +1 τ n (cid:18) a + a τ τ n − (cid:19) (cid:32) π ∞ (cid:88) l =1 l (cid:33) + 4 π ∞ (cid:88) l =1 l = ( a + a τ ) τ
90 18 K , where we use the formulae (cid:80) ∞ l =1 l − = π /
90 and (cid:80) ∞ n = K +1 − n = 8 − K / R K , similar to the calculations for the moments of R K , we find E (cid:2) R K (cid:3) = ∞ (cid:88) n = K +1 τ n E (cid:104) C δ/ n (cid:105) = ∞ (cid:88) n = K +1 τ n (cid:32) π ∞ (cid:88) l =1 δ (cid:0) l − (cid:1) (cid:33) = δτ K . Note that the last step is a direct result of (cid:80) ∞ l =1 (2 l − − = π / (cid:80) ∞ n = K +1 − n =4 − K /
3. Further we can proceed with the computation of the variance:Var (cid:2) R K (cid:3) = ∞ (cid:88) n = K +1 τ n Var (cid:104) C δ/ n (cid:105) = ∞ (cid:88) n = K +1 τ n (cid:32) π ∞ (cid:88) l =1 δ (cid:0) l − (cid:1) (cid:33) = δτ
45 116 K , in which we apply (cid:80) ∞ l =1 (2 l − − = π /
96 and (cid:80) ∞ n = K +1 − n = 16 − K / Appendix B. Proof of Theorem 3.3.
We use the method of steepest descentsto find the limiting behaviour as P → + ∞ of the probability density function f Z P ofthe standardised random variable Z P given by f Z P ( x ) = 14 π (cid:114) P (cid:90) + ∞−∞ exp ( P ρ ( z ; β )) dz, (B.1)where β = x (cid:112) / / √ P and ρ ( z ; β ) = log (cid:32) √ z isinh √ z i (cid:33) + z i (cid:18)
16 + 12 β (cid:19) . e are interested in the saddle point z such that ρ (cid:48) ( z ; β ) = 0. Before that, let ς ( z ) := √ z isinh √ z i , which gives ρ ( z ; β ) = log ( ς ( z )) + z i (cid:18)
16 + 12 β (cid:19) . We observe ς ( z ) and ρ ( z ; β ) are analytic when Im ( z ) < π after defining ς (0) := 1and ρ (0; β ) := 0.Bleistein and Handelsman [10, Chapter 7 .
6] suggest that we should seek a saddlepoint near z = 0, which will be the dominant one. To obtain an explicit form for thesaddle point, we take advantage of the series expansions of ρ ( z ; β ) and its differentia-tion ρ (cid:48) ( z ; β ). Let us first consider the series expansion of ρ ( z ; β ) about z = 0, whichis of the form ρ ( z ; β ) = 12 β i z + ∞ (cid:88) k =2 ˆ r k z k , (B.2)where ˆ r = − , ˆ r = i2835(B.3)and so forth. Although we can compute the coefficients ˆ r k analytically through Taylorexpansion of ρ ( z ; β ) up to any order, we compute them using Maple in practice.Hence, its differentiation can be expressed as ρ (cid:48) ( z ; β ) = 12 β i + ∞ (cid:88) k =2 k ˆ r k z k − . The above two series converge pointwise in the domain where | z | < π . It seemsthat a precise form for the saddle point z such that ρ (cid:48) ( z ; β ) = 0 is not obtainable.However, we can get a good approximation by making use of the smallness of β . For β (cid:28)
1, i.e. | x | (cid:28) √ P / (cid:112) /
45, we solve the above equation by iteration. In fact, aftertwo iterations, we have the following approximate form for the desired saddle point z = 45i β − β + O (cid:0) β (cid:1) . In this sense by successive iterations, we can approximate the saddle point z by anasymptotic expansion in β to any order, i.e. z ∼ β ∞ (cid:88) k =0 ˆ ξ k β k , (B.4)in which ˆ ξ = 45i , ˆ ξ = − nd so on. Again, all these coefficients ˆ ξ k are calculated via Maple in practice. Noticethat the saddle point z is near the origin and along the imaginary axis.In order to deform the original contour into the steepest descent path passingthrough the saddle point in the domain of analyticity of ρ ( z ; β ), we need to show that z will not hit its singularity, i.e. Im ( z ) < π . Indeed, we know that z satisfies ρ (cid:48) ( z ; β ) = ς (cid:48) ( z ) ς ( z ) + i (cid:18)
16 + 12 β (cid:19) = 0 . This is equivalent to Im (cid:18) ς (cid:48) (Im ( z ) i) ς (Im ( z ) i) (cid:19) + (cid:18)
16 + 12 β (cid:19) = 0 . If we introduce the function κ ( z ) := Im ( ς (cid:48) ( z i) /ς ( z i)), then we see κ ( z ) is a mono-tonically decreasing function for real z < π and κ ( z ) → −∞ as z ↑ π . Adding apositive constant to κ ( z ) moves its graph upwards, whence the intersection with thereal axis is shifted to the right. Due to the limiting behaviour as z ↑ π , the zero of κ ( z )+(1 / β/
2) that we are interested in is always below π , yielding Im ( z ) < π .Now, we can expand ρ ( z ; β ) as a Taylor series near the saddle point z ρ ( z ; β ) = ρ ( z ; β ) + 12! ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) + ( z − z ) (cid:88) k ≥ ρ ( k +3) ( z ; β )( k + 3)! ( z − z ) k , (B.6)which converges in a neighbourhood of z . For preparations, we must evaluate ρ ( k ) ( z ; β ) for k ≥ z = z . Differentiating the series expansion for ρ in equa-tion (B.2) leads to ρ ( n ) ( z ; β ) = ∞ (cid:88) k = n k ( k − · · · ( k − n + 1) ˆ r k z k − n = ∞ (cid:88) k = n ˆ ϕ k,n z k − n for n ≥
2, where ˆ ϕ k,n := k ( k − · · · ( k − n + 1) ˆ r k (B.7)for k ≥ n . This series converges in the same domain as the expansion for ρ ; seeequation (B.2). By substituting the asymptotic approximation (B.4) regarding thesaddle point z into the above equation and noting that z j ∼ β j ∞ (cid:88) l =0 ˆ υ l,j β l (B.8)for j ≥
0, where ˆ υ ,j = ˆ ξ j , ˆ υ l,j = 1 l ˆ ξ l (cid:88) k =1 ( kj − l + k ) ˆ ξ k ˆ υ l − k,j , for l ≥ , (B.9)we get that for n ≥ ρ ( n ) ( z ; β ) ∼ ∞ (cid:88) l =0 ˆ φ l,n β l (B.10) ith β (cid:28)
1, where ˆ φ l,n = l (cid:88) k =0 ˆ ϕ n + l − k,n ˆ υ k,l − k (B.11)for l ≥ z given byIm ( ρ ( z ; β )) − Im ( ρ ( z ; β )) = Im ( ρ ( z ; β ) − ρ ( z ; β )) = 0 . When expanding ρ ( z ; β ) close to the saddle point z , see (B.6), we consider the leadingorder and setIm ( ρ ( z ; β ) − ρ ( z ; β )) = Im (cid:18) ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) + O (cid:16) ( z − z ) (cid:17)(cid:19) = 0 . (B.12)On the other hand, from (B.10) we have ρ (cid:48)(cid:48) ( z ; β ) ∼ ˆ φ , + O ( β ) = ˆ ϕ , ˆ υ , + O ( β ) = 2ˆ r + O ( β ) = −
190 + O ( β ) . If we set z := u +i v for u, v ∈ R , then (B.12) implies that the paths of steepest descentand ascent from z lie along the curves2 u ( v − Im ( z )) + O (cid:16) | z − z | (cid:17) = 0as z is purely imaginary. These paths, close enough to the saddle point z , thatis when | z − z | is small, consist of the straight lines u = 0 and v = Im ( z ). Todistinguish between the ascent and descent paths, we consider Re ( ρ ( z ; β )) along thetwo lines near z = z . Along u = 0, we haveRe ( ρ ( z ; β )) = Re ( ρ ( z ; β )) − ρ (cid:48)(cid:48) ( z ; β ) ( v − Im ( z )) + O (cid:16) | z − z | (cid:17) ≥ Re ( ρ ( z ; β ))when z is near z . Along v = Im ( z ), we haveRe ( ρ ( z ; β )) = Re ( ρ ( z ; β )) + 12! ρ (cid:48)(cid:48) ( z ; β ) u + O (cid:16) | z − z | (cid:17) ≤ Re ( ρ ( z ; β ))for z close enough to z . Thus, the path of steepest descents from z is v = Im ( z ),parallel to the real axis.As z is in the domain of analyticity of ρ ( z ; β ), we can deform the original contourof the integration (B.1) onto the steepest descent paths through the saddle point z ,denoted by C l for u < C r for u >
0, both pointing a direction away from z . Itfollows from Cauchy’s theorem that f Z P ( x ) = 14 π (cid:114) P (cid:90) + ∞−∞ exp ( P ρ ( z ; β )) dz = 14 π (cid:114) P (cid:90) C r −C l exp ( P ρ ( z ; β )) dz, whence the main contributions to the asymptotic expansion of the integral now comesfrom a small neighbourhood of z for large P . We use Laplace’s method to evaluatethis integral. For some (cid:15) >
0, we have the following asymptotic relation: f Z P ( x ) ∼ π (cid:114) P (cid:90) Im( z )i+ (cid:15) Im( z )i − (cid:15) exp ( P ρ ( z ; β )) dz, as P → + ∞ , (B.13) here by replacing the contour of integration C r − C l with a narrow interval centredaround z , only exponentially small errors are introduced for large P . Now, (cid:15) canbe chosen so small that we can replace ρ ( z ; β ) by its Taylor expansion (B.6), whichconverges on the interval (Im ( z ) i − (cid:15), Im ( z ) i + (cid:15) ). Then, separating the quadraticterm from all the higher-order terms of the series expansion (B.6) in exp ( P ρ ( z ; β ))and setting g ( z ; β ) := exp P ( z − z ) (cid:88) k ≥ ρ ( k +3) ( z ; β )( k + 3)! ( z − z ) k , (B.14)the integral (B.13) becomes f Z P ( x ) ∼ π (cid:114) P
45 exp (
P ρ ( z ; β )) (cid:90) Im( z )i+ (cid:15) Im( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) g ( z ; β ) dz, (B.15)as P → + ∞ .To find ρ ( z ; β ), we use (B.2), (B.4) and (B.8) to write ρ ( z ; β ) = 12 β i z + ∞ (cid:88) k =2 ˆ r k z k ∼ ∞ (cid:88) k =2 ˆ ρ k β k , where for k ≥
2, ˆ ρ k := i ˆ ξ k − k (cid:88) m =2 ˆ r m ˆ υ k − m,m . (B.16)Since the series in the argument of the exponential function which defines g ( z ; β )in (B.14) is convergent near z , we can write as z → z , g ( z ; β ) = exp P ( z − z ) (cid:88) k ≥ ˆ σ k ( β ) ( z − z ) k ∼ ∞ (cid:88) n =0 n ! P n ( z − z ) n (cid:88) k ≥ ˆ σ k ( β ) ( z − z ) k n , (B.17)where ˆ σ k ( β ) := ρ ( k +3) ( z ; β ) / ( k + 3)! for k ≥
0. Further, the asymptotic approxima-tion (B.10) for ρ ( k +3) ( z ; β ) gives us ˆ σ k ( β ) ∼ (cid:80) ∞ l =0 ˆ γ l,k β l withˆ γ l,k := ˆ φ l,k +3 ( k + 3)!(B.18)for l, k ≥ β is small. As an immediate consequence of the properties forasymptotic series, we have for n ≥ (cid:88) k ≥ ˆ σ k ( β ) ( z − z ) k n ∼ ∞ (cid:88) k =0 k (cid:88) k =0 · · · k n − (cid:88) k n =0 ˆ σ k n ( β ) ˆ σ k n − − k n ( β ) · · · ˆ σ k − k ( β ) ( z − z ) k , s z → z . In addition, we observe for n ≥ ≤ k n ≤ k n − ≤ · · · ≤ k ,ˆ σ k n ( β ) ˆ σ k n − − k n ( β ) · · · ˆ σ k − k ( β ) ∼ (cid:32) ∞ (cid:88) l =0 ˆ γ l ,k n β l (cid:33) (cid:32) ∞ (cid:88) l =0 ˆ γ l ,k n − − k n β l (cid:33) · · · (cid:32) ∞ (cid:88) l n =0 ˆ γ l n ,k − k β l n (cid:33) ∼ ∞ (cid:88) l =0 l (cid:88) l =0 · · · l n − (cid:88) l n =0 ˆ γ l n ,k n ˆ γ l n − − l n ,k n − − k n · · · ˆ γ l − l ,k − k β l ∼ ∞ (cid:88) l =0 ˆ C l ,k ,k , ··· ,k n β l , where ˆ C l ,k ,k , ··· ,k n := l (cid:88) l =0 · · · l n − (cid:88) l n =0 ˆ γ l n ,k n ˆ γ l n − − l n ,k n − − k n · · · ˆ γ l − l ,k − k (B.19)for l ≥
0. Generally, for n ≥ (cid:88) k ≥ ˆ σ k ( β ) ( z − z ) k n ∼ ∞ (cid:88) k =0 ˆ θ k,n ( β ) ( z − z ) k , as z → z . Here ˆ θ k,n ( β ) are functions of β satisfying ˆ θ k,n ( β ) ∼ (cid:80) ∞ l =0 ˆ E l,k,n β l for k ≥ E l,k,n as stated below: for n = 0,ˆ E l,k,n = ˆ E l,k, = (cid:40) , for k = l = 0 , , otherwise,(B.20)for n = 1, ˆ E l,k,n = ˆ E l,k, = ˆ γ l,k , for k, l ≥ , (B.21)for n = 2, ˆ E l,k,n = ˆ E l,k, = k (cid:88) k =0 ˆ C l,k,k , for k, l ≥ , (B.22)for n ≥
3, ˆ E l,k,n = k (cid:88) k =0 k (cid:88) k =0 · · · k n − (cid:88) k n =0 ˆ C l,k,k , ··· ,k n , for k, l ≥ . (B.23)Using these factors, we can rewrite g ( z ; β ) in (B.17) as g ( z ; β ) ∼ ∞ (cid:88) n =0 n ! P n ( z − z ) n (cid:32) ∞ (cid:88) k =0 ˆ θ k,n ( β ) ( z − z ) k (cid:33) ∼ ∞ (cid:88) n =0 ∞ (cid:88) k =0 n ! P n ˆ θ k,n ( β ) ( z − z ) n + k ∞ (cid:88) j =0 (cid:88) n + k = j n ! P n ˆ θ k,n ( β ) ( z − z ) j ∼ ∞ (cid:88) j =0 (cid:98) j (cid:99) (cid:88) n =0 n ! P n ˆ θ j − n,n ( β ) ( z − z ) j ∼ ∞ (cid:88) j =0 ˆ g j ( β ) ( z − z ) j in the limit z → z , where ˆ g j ( β ) := (cid:80) (cid:98) j/ (cid:99) n =0 P n ˆ θ j − n,n ( β ) /n ! for j ≥
0. Hence by thedefinition for asymptotic expansions, we have g ( z ; β ) − J (cid:88) j =0 ˆ g j ( β ) ( z − z ) j = o (cid:16) ( z − z ) J (cid:17) , as z → z for any J ≥
0. From this it follows that for any (cid:15) ∗ > | z − z | < L for some L >
0, in which (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( z ; β ) − J (cid:88) j =0 ˆ g j ( β ) ( z − z ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) ∗ (cid:12)(cid:12)(cid:12) ( z − z ) J (cid:12)(cid:12)(cid:12) . Therefore for any 0 < (cid:15) < L , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)
Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) g ( z ; β ) − J (cid:88) j =0 ˆ g j ( β ) ( z − z ) j dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( z ; β ) − J (cid:88) j =0 ˆ g j ( β ) ( z − z ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz ≤ (cid:15) ∗ (cid:90) Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) (cid:12)(cid:12)(cid:12) ( z − z ) J (cid:12)(cid:12)(cid:12) dz = (cid:15) ∗ ( − J (cid:90) Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) ( z − z ) J dz. Then as (cid:15) → + , we can write (cid:90) Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) g ( z ; β ) dz = J (cid:88) j =0 ˆ g j ( β ) (cid:90) Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) ( z − z ) j dz + o (cid:32)(cid:90) Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) ( z − z ) J dz (cid:33) , hich gives (cid:90) Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) g ( z ; β ) dz ∼ ∞ (cid:88) j =0 ˆ g j ( β ) (cid:90) Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) ( z − z ) j dz for small (cid:15) . Now the above integrals can be evaluated by change of variables. Forarbitrary j ≥
0, the substitution z = Im ( z ) i + x yields (cid:90) Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) ( z − z ) j dz = (cid:90) − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) x (cid:19) x j dx = 12 ( − j (cid:18) − ρ (cid:48)(cid:48) ( z ; β ) (cid:19) ( j +1) · (cid:90) − ρ (cid:48)(cid:48) ( z ; β ) (cid:15) ζ ( j − exp ( − P ζ ) dζ, where the last step is a result of the change of variable ρ (cid:48)(cid:48) ( z ; β ) x / − ζ . Thus as (cid:15) → + , we have (cid:90) Im( z )iIm( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) g ( z ; β ) dz ∼ ∞ (cid:88) j =0 ˆ g j ( β ) 12 ( − j (cid:18) − ρ (cid:48)(cid:48) ( z ; β ) (cid:19) ( j +1) (cid:90) − ρ (cid:48)(cid:48) ( z ; β ) (cid:15) ζ ( j − exp ( − P ζ ) dζ. Similar arguments give us that (cid:90)
Im( z )i+ (cid:15) Im( z )i exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) g ( z ; β ) dz ∼ ∞ (cid:88) j =0 ˆ g j ( β ) 12 (cid:18) − ρ (cid:48)(cid:48) ( z ; β ) (cid:19) ( j +1) (cid:90) − ρ (cid:48)(cid:48) ( z ; β ) (cid:15) ζ ( j − exp ( − P ζ ) dζ, as (cid:15) → + .Hence, the integration in (B.15) can be expanded in an asymptotic series for small (cid:15) as follows: (cid:90) Im( z )i+ (cid:15) Im( z )i − (cid:15) exp (cid:18) P ρ (cid:48)(cid:48) ( z ; β ) ( z − z ) (cid:19) g ( z ; β ) dz ∼ ∞ (cid:88) j =0 (cid:16) − j (cid:17) ˆ g j ( β ) (cid:18) − ρ (cid:48)(cid:48) ( z ; β ) (cid:19) ( j +1) (cid:90) − ρ (cid:48)(cid:48) ( z ; β ) (cid:15) ζ ( j − exp ( − P ζ ) dζ, where terms with odd j vanish. For large P , we can extend the integration regionin each integral to infinity. With this replacement, we introduce only exponentiallysmall errors for large P , whence we have as P → + ∞ , (cid:90) − ρ (cid:48)(cid:48) ( z ; β ) (cid:15) ζ ( j − exp ( − P ζ ) dζ ∼ (cid:90) + ∞ ζ ( j − exp ( − P ζ ) dζ = P − ( j +1) Γ (cid:18) j (cid:19) or j ≥
0. Assembling the above results, we have the following asymptotic series f Z P ( x ) ∼ π (cid:114) P
45 exp (cid:32) P ∞ (cid:88) l =2 ˆ ρ l β l (cid:33) ∞ (cid:88) j =0 ˆ g j ( β ) (cid:18) − ρ (cid:48)(cid:48) ( z ; β ) (cid:19) j + P − ( j + )Γ (cid:18) j + 12 (cid:19) (B.24)in the limit P → + ∞ for fixed x such that | x | (cid:28) √ P / (cid:112) / β , i.e. ˆ g j ( β ) ( − /ρ (cid:48)(cid:48) ( z ; β )) j +1 / as an asymptotic series in β . This can be achieved by collecting the coefficients fromthe product of their individual series. Assume that (cid:115) − ρ (cid:48)(cid:48) ( z ; β ) ∼ ∞ (cid:88) n =0 ˆ K n β n (B.25)for some constants ˆ K n for n ≥
0. Then taking the square on both sides yields − ρ (cid:48)(cid:48) ( z ; β ) ∼ ∞ (cid:88) l =0 l (cid:88) k =0 ˆ K k ˆ K l − k β l ∼ ∞ (cid:88) l =0 ˆ µ l β l , where ˆ µ l := (cid:80) lk =0 ˆ K k ˆ K l − k for l ≥
0. On the other hand, by performing simplearithmetical operations on the asymptotic series (B.10) with n = 2 for ρ (cid:48)(cid:48) ( z ; β ), wesee − ρ (cid:48)(cid:48) ( z ; β ) ∼ ∞ (cid:88) l =0 ˆ (cid:36) l β l , where ˆ (cid:36) = − φ , , ˆ (cid:36) l = − φ , l − (cid:88) m =0 ˆ (cid:36) m ˆ φ l − m, , for l ≥ . (B.26)Hence, by equating the coefficients following the uniqueness of asymptotic expansions(Bender and Orszag [6, Chapter 3 .
8, p. 125]), we findˆ (cid:36) l = ˆ µ l = l (cid:88) k =0 ˆ K k ˆ K l − k , for l ≥ , (B.27)providing the values for the constants ˆ K k with k ≥ − /ρ (cid:48)(cid:48) ( z ; β )) j +1 / for β (cid:28)
1. Indeed, from (B.25) we have for j ≥ (cid:18) − ρ (cid:48)(cid:48) ( z ; β ) (cid:19) j + ∼ (cid:32) ∞ (cid:88) n =0 ˆ K n β n (cid:33) j +1 ∼ ∞ (cid:88) n =0 ˆ ω n,j β n , where for j = 0, ˆ ω n,j = ˆ ω n, = ˆ K n , for n ≥ , (B.28) or j ≥ ω n,j = n (cid:88) n =0 n (cid:88) n =0 · · · n j (cid:88) n j +1 =0 ˆ K n j +1 ˆ K n j − n j +1 · · · ˆ K n − n ˆ K n − n , for n ≥ . (B.29)If we combine the series which is asymptotic to ˆ g j ( β ) with the explicit expansiongiven above, we obtain for j ≥ g j ( β ) (cid:18) − ρ (cid:48)(cid:48) ( z ; β ) (cid:19) j + ∼ (cid:98) j (cid:99) (cid:88) n =0 n ! P n ∞ (cid:88) l =0 ˆ E l, j − n,n β l (cid:32) ∞ (cid:88) n =0 ˆ ω n,j β n (cid:33) ∼ ∞ (cid:88) l =0 (cid:98) j (cid:99) (cid:88) n =0 n ! P n ˆ E l, j − n,n β l (cid:32) ∞ (cid:88) k =0 ˆ ω k,j β k (cid:33) ∼ ∞ (cid:88) l =0 l (cid:88) k =0 (cid:98) j (cid:99) (cid:88) n =0 n ! P n ˆ E k, j − n,n ˆ ω l − k,j β l ∼ ∞ (cid:88) l =0 (cid:98) j (cid:99) (cid:88) n =0 l (cid:88) k =0 n ! ˆ ω l − k,j ˆ E k, j − n,n P n β l ∼ ∞ (cid:88) l =0 (cid:98) j (cid:99) (cid:88) n =0 ˆ α n,l,j P n β l , where ˆ α n,l,j := 1 n ! l (cid:88) k =0 ˆ ω l − k,j ˆ E k, j − n,n (B.30)for 0 ≤ n ≤ (cid:98) j/ (cid:99) and l ≥
0. Then (B.24) becomes f Z P ( x ) ∼ π (cid:114) P
45 exp (cid:32) P ∞ (cid:88) l =2 ˆ ρ l β l (cid:33) ∞ (cid:88) j =0 ∞ (cid:88) l =0 (cid:98) j (cid:99) (cid:88) n =0 ˆ α n,l,j P n β l P − ( j + )Γ (cid:18) j + 12 (cid:19) , (B.31)as P → + ∞ with β (cid:28)
1, which completes the proof.
Appendix C. Proof of Theorem 3.5.
Before integrating the density function,we first rewrite its asymptotic expansion in Theorem 3.3 in terms of the originalvariable x by using the identity β = x (cid:112) / / √ P . Accordingly in the limit P → + ∞ with | x | (cid:28) √ P / (cid:112) /
45, the probability density function has the following asymptoticexpansion f Z P ( x ) ∼ π (cid:114)
245 exp (cid:32) ∞ (cid:88) l =2 ˆ ρ l (cid:18) (cid:19) l P − l x l (cid:33) (C.1) · ∞ (cid:88) j =0 ∞ (cid:88) l =0 (cid:98) j (cid:99) (cid:88) n =0 ˆ α n,l,j (cid:18) (cid:19) l Γ (cid:18) j + 12 (cid:19) P n − l − j x l . o justify that the integrated series is indeed asymptotic to the distribution func-tion, we adjust the terms in (C.1) to form a more appropriate expression for easiercomputation.Specifically, we separate the quadratic term ˆ ρ (2 / x from the argument (cid:80) ∞ l =2 ˆ ρ l (2 / l/ P − l/ x l of the exponential function. As the integration is taken withrespect to x , we expand the remaining term in an asymptotic approximation in P withall the coefficients given as polynomials of x . Note that ˆ ρ = i ˆ ξ / r (cid:16) ˆ ξ (cid:17) = − / (cid:32) ∞ (cid:88) l =2 ˆ ρ l (cid:18) (cid:19) l P − l x l (cid:33) ∼ exp (cid:18) − x (cid:19) exp (cid:32) ∞ (cid:88) l =3 ˆ ρ l (cid:18) (cid:19) l P − l x l (cid:33) , (C.2)as P → + ∞ , whereexp (cid:32) ∞ (cid:88) l =3 ˆ ρ l (cid:18) (cid:19) l P − l x l (cid:33) ∼ ∞ (cid:88) n =0 n ! (cid:32) ∞ (cid:88) l =3 ˆ ρ l (cid:18) (cid:19) l P − l x l (cid:33) n ∼ ∞ (cid:88) n =0 n ! (cid:18) (cid:19) n P − n x n (cid:32) ∞ (cid:88) k =0 ˆ ρ k +3 (cid:18) (cid:19) k P − k x k (cid:33) n . (C.3)Analogous to the previous computations, the generalisation of multiplication of asymp-totic expansions tells us for n ≥ (cid:32) ∞ (cid:88) k =0 ˆ ρ k +3 (cid:18) (cid:19) k P − k x k (cid:33) n ∼ ∞ (cid:88) k =0 ˆ ϑ k,n (cid:18) (cid:19) k P − k x k , (C.4)where ˆ ϑ ,n = (ˆ ρ ) n , ˆ ϑ k,n = 1 k ˆ ρ k (cid:88) m =1 ( mn − k + m ) ˆ ρ k +3 ˆ ϑ k − m,n , for k ≥ . (C.5)Using (C.2) − (C.4) amounts toexp (cid:32) ∞ (cid:88) l =2 ˆ ρ l (cid:18) (cid:19) l P − l x l (cid:33) ∼ exp (cid:18) − x (cid:19) ∞ (cid:88) n =0 n ! (cid:18) (cid:19) n P − n x n · ∞ (cid:88) k =0 ˆ ϑ k,n (cid:18) (cid:19) k P − k x k ∼ exp (cid:18) − x (cid:19) ∞ (cid:88) n =0 ∞ (cid:88) k =0 n ! (cid:18) (cid:19) n + k ˆ ϑ k,n P − n + k x n + k ∼ exp (cid:18) − x (cid:19) ∞ (cid:88) j =0 (cid:32) j (cid:88) n =0 n ! (cid:18) (cid:19) n + j ˆ ϑ j − n,n x n + j (cid:33) P − j ∼ exp (cid:18) − x (cid:19) ∞ (cid:88) j =0 ˆ A j ( x ) P − j , s P → + ∞ , whereˆ A j ( x ) := j (cid:88) n =0 n ! (cid:18) (cid:19) n + j ˆ ϑ j − n,n x n + j = j (cid:88) n =0 ˆ η n,j x n + j , for j ≥ η n,j := (cid:18) (cid:19) n + j ˆ ϑ j − n,n n !(C.6)for 0 ≤ n ≤ j . Further, we see that ∞ (cid:88) j =0 ∞ (cid:88) l =0 (cid:98) j (cid:99) (cid:88) n =0 ˆ α n,l,j (cid:18) (cid:19) l Γ (cid:18) j + 12 (cid:19) x l P n − l − j ∼ ∞ (cid:88) m =0 ∞ (cid:88) l =0 3 m (cid:88) j = m ˆ α j − m,l,j (cid:18) (cid:19) l Γ (cid:18) j + 12 (cid:19) x l P − ( m + l ) ∼ ∞ (cid:88) r =0 r even r (cid:88) l =0 l even r − l )2 (cid:88) j = r − l ˆ α j − r − l ,l,j (cid:18) (cid:19) l Γ (cid:18) j + 12 (cid:19) x l P − r + ∞ (cid:88) r =1 r odd r (cid:88) l =1 l odd r − l )2 (cid:88) j = r − l ˆ α j − r − l ,l,j (cid:18) (cid:19) l Γ (cid:18) j + 12 (cid:19) x l P − r ∼ ∞ (cid:88) r =0 ˆ B r ( x ) P − r when P → + ∞ with ˆ B r ( x ) := r (cid:88) l =0 ˆ λ l,r x l , for r ≥ , where for even r ,ˆ λ l,r := , for odd l, r − l )2 (cid:80) j = r − l ˆ α j − r − l ,l,j (cid:0) (cid:1) l Γ (cid:0) j + (cid:1) , for even l, (C.7)and for odd r , ˆ λ l,r := , for even l, r − l )2 (cid:80) j = r − l ˆ α j − r − l ,l,j (cid:0) (cid:1) l Γ (cid:0) j + (cid:1) , for odd l. (C.8) ollowing the above discussion, (C.1) can be rearranged as f Z P ( x ) ∼ π (cid:114)
245 exp (cid:18) − x (cid:19) ∞ (cid:88) j =0 ˆ A j ( x ) P − j (cid:32) ∞ (cid:88) r =0 ˆ B r ( x ) P − r (cid:33) ∼ π (cid:114)
245 exp (cid:18) − x (cid:19) ∞ (cid:88) j =0 (cid:32) j (cid:88) r =0 ˆ A r ( x ) ˆ B j − r ( x ) (cid:33) P − j ∼ π (cid:114)
245 exp (cid:18) − x (cid:19) ∞ (cid:88) j =0 ˆ ψ j ( x ) P − j , where for j ≥ ψ j ( x ) := j (cid:88) r =0 ˆ A r ( x ) ˆ B j − r ( x ) = j (cid:88) r =0 r (cid:88) n =0 j − r (cid:88) l =0 ˆ η n,r ˆ λ l,j − r x n + r + l . Then, by the definition for asymptotic expansions, we have the order relation givenbelow: for any J ≥ f Z P ( x ) − π (cid:114)
245 exp (cid:18) − x (cid:19) J (cid:88) j =0 ˆ ψ j ( x ) P − j = o (cid:16) P − J (cid:17) , as P → + ∞ with | x | (cid:28) √ P / (cid:112) /
45. Integrating on finite interval ( z , z ] such that | z i | (cid:28) √ P / (cid:112) /
45 for i = 1 ,
2, we have (cid:90) z z f Z P ( x ) dx = 14 π (cid:114) J (cid:88) j =0 P − j (cid:90) z z exp (cid:18) − x (cid:19) ˆ ψ j ( x ) dx + o (cid:16) P − J (cid:17) . Next, we show the integrals on the right hand side are finite. In fact, for 0 ≤ j ≤ J ,we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) z z exp (cid:18) − x (cid:19) ˆ ψ j ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) z z exp (cid:18) − x (cid:19) (cid:12)(cid:12)(cid:12) ˆ ψ j ( x ) (cid:12)(cid:12)(cid:12) dx ≤ j (cid:88) r =0 r (cid:88) n =0 j − r (cid:88) l =0 (cid:12)(cid:12)(cid:12) ˆ η n,r ˆ λ l,j − r (cid:12)(cid:12)(cid:12) (cid:90) z z exp (cid:18) − x (cid:19) (cid:12)(cid:12) x n + r + l (cid:12)(cid:12) dx< + ∞ . Notice that the constants ˆ η n,r and ˆ λ l,j − r are finite. Hence, we have the followingasymptotic expansion: when P → + ∞ , (cid:90) z z f Z P ( x ) dx ∼ π (cid:114) ∞ (cid:88) j =0 P − j (cid:90) z z exp (cid:18) − x (cid:19) ˆ ψ j ( x ) dx ∼ π (cid:114) ∞ (cid:88) j =0 P − j (cid:90) z z exp (cid:18) − x (cid:19) j (cid:88) r =0 r (cid:88) n =0 j − r (cid:88) l =0 ˆ η n,r ˆ λ l,j − r x n + r + l dx ∼ π (cid:114) ∞ (cid:88) j =0 P − j j (cid:88) r =0 r (cid:88) n =0 j − r (cid:88) l =0 ˆ η n,r ˆ λ l,j − r (cid:90) z z exp (cid:18) − x (cid:19) x n + r + l dx. (C.9) e apply the change of variable v = x / z < z < q ≥
0, we have (cid:90) z z exp (cid:18) − x (cid:19) x q dx = ( − q (cid:16) √ (cid:17) q − (cid:90) z z exp ( − v ) v q +12 − dv = ( − q (cid:16) √ (cid:17) q − (cid:32) γ (cid:32) q + 12 , ( z ) (cid:33) − γ (cid:32) q + 12 , ( z ) (cid:33)(cid:33) , where γ ( s, z ) is the lower incomplete gamma function. For z < ≤ z and q ≥ z ,
0) and [0 , z ] separately. Byadditivity, we get (cid:90) z z exp (cid:18) − x (cid:19) x q dx = ( − q (cid:16) √ (cid:17) q − (cid:90) z exp ( − v ) v q +12 − dv + (cid:16) √ (cid:17) q − (cid:90) z exp ( − v ) v q +12 − dv = (cid:16) √ (cid:17) q − (cid:32) ( − q γ (cid:32) q + 12 , ( z ) (cid:33) + γ (cid:32) q + 12 , ( z ) (cid:33)(cid:33) . Substituting the explicit form for the integrals back into (C.9) yields the stated rep-resentation, completing the proof.
Appendix D. Proof of Theorem 3.7.
Recall that the probability densityfunction f S P of S P has the form f S P ( y ) = 1 √ π P Γ ( P ) y − ( P +2) ∞ (cid:88) n =0 Γ ( n + P )Γ ( n + 1) exp (cid:32) − (2 n + P ) y (cid:33) D P +1 (cid:18) n + P √ y (cid:19) , in which D P +1 ( z ) is a parabolic cylinder function of order P + 1. The distributionfunction F S P is derived by term-wise integration of the above series. First we showthat (cid:80) ∞ n =0 (cid:82) x | f n ( y ) | dy < ∞ for any finite x ≥
0, where f n ( y ) := Γ ( n + P )Γ ( n + 1) exp (cid:32) − (2 n + P ) y (cid:33) D P +1 (cid:18) n + P √ y (cid:19) y − ( P +2) . For fixed n ≥
0, application of the variable transformation z = (2 n + P ) / √ y gives (cid:90) x | f n ( y ) | dy = 2 Γ ( n + P )Γ ( n + 1) (2 n + P ) − P (cid:90) + ∞ n + P √ x exp (cid:18) − z (cid:19) | D P +1 ( z ) | z P − dz. Notice that the parabolic cylinder function D P +1 ( z ) is square integrable on [0 , ∞ )(Gradshteyn and Ryzhik [ ? , Chapter 7 . (cid:107) D P +1 (cid:107) := (cid:18)(cid:90) + ∞ | D P +1 ( z ) | dz (cid:19) < ∞ . y H¨older’s inequality, we have (cid:90) + ∞ y exp (cid:18) − z (cid:19) | D P +1 ( z ) | z P − dz ≤ (cid:18)(cid:90) + ∞ y z P − exp (cid:18) − z (cid:19) dz (cid:19) (cid:107) D P +1 (cid:107) (D.1)for y ≥
0. Next we consider the following two cases for P separately: P ∈ (0 ,
1) and P ∈ N .For any P ∈ (0 , z P − is monotonically decreasing, which yields (cid:90) + ∞ y exp (cid:18) − z (cid:19) | D P +1 ( z ) | z P − dz ≤ y P − (cid:18)(cid:90) + ∞ y exp (cid:18) − z (cid:19) dz (cid:19) (cid:107) D P +1 (cid:107) = (2 π ) (cid:107) D P +1 (cid:107) y P − (1 − Φ ( y )) for y >
0, where Φ ( y ) is the distribution function of a standard normal randomvariable. Then, it follows that the sequence (cid:82) x | f n ( y ) | dy with finite x is bounded by (cid:90) x | f n ( y ) | dy ≤ b n , where b n := 2 (2 π ) (cid:107) D P +1 (cid:107) x (1 − P ) Γ ( n + P )Γ ( n + 1) 12 n + P (cid:18) − Φ (cid:18) n + P √ x (cid:19)(cid:19) . By the ratio test, we can deduce that the series (cid:80) ∞ n =0 b n is convergent. In fact, wehave (cid:12)(cid:12)(cid:12)(cid:12) b n +1 b n (cid:12)(cid:12)(cid:12)(cid:12) = n + Pn + 1 2 n + P n + 1) + P − Φ (cid:16) n +1)+ P √ x (cid:17) − Φ (cid:16) n + P √ x (cid:17) → , as n → ∞ . The comparison test implies that the series (cid:80) ∞ n =0 (cid:82) x | f n ( y ) | dy is also convergent forany finite x .For any P ∈ N , the integral on the right hand side of (D.1) can be regarded asthe moment of some transformation of a standard normal random variable Z , i.e. (cid:90) + ∞ y z P − exp (cid:18) − z (cid:19) dz = √ π E (cid:104) Z P − { Z ≥ y } (cid:105) ≤ √ π (cid:16) E (cid:104) Z P − (cid:105)(cid:17) (cid:16) E (cid:104)(cid:0) { Z ≥ y } (cid:1) (cid:105)(cid:17) = √ π (cid:16) E (cid:104) Z P − (cid:105)(cid:17) (1 − Φ ( y )) for y ≥
0, where { z ≥ y } is the indicator function and the inequality follows fromH¨older’s inequality. Hence, the above argument gives the bounds for (cid:82) x | f n ( y ) | dy with x < ∞ as (cid:90) x | f n ( y ) | dy ≤ b n , here b n := 2 (2 π ) (cid:107) D P +1 (cid:107) (cid:16) E (cid:104) Z P − (cid:105)(cid:17) Γ ( n + P )Γ ( n + 1) (2 n + P ) − P (cid:18) − Φ (cid:18) n + P √ x (cid:19)(cid:19) . Similarly, the series (cid:80) ∞ n =0 b n converges by the fact that (cid:12)(cid:12)(cid:12)(cid:12) b n +1 b n (cid:12)(cid:12)(cid:12)(cid:12) = n + Pn + 1 (cid:18) n + P n + 1) + P (cid:19) P − Φ (cid:16) n +1)+ P √ x (cid:17) − Φ (cid:16) n + P √ x (cid:17) → , as n → ∞ , implying the convergence of the series (cid:80) ∞ n =0 (cid:82) x | f n ( y ) | dy for finite x as well.Thus, for fixed 0 ≤ x < ∞ and P ∈ (0 , ∪ N , we have (cid:80) ∞ n =0 (cid:82) x | f n ( y ) | dy < ∞ . Then, applying a corollary of the Dominated Convergence Theorem (Rudin [ ? ,Theorem 1 . ∞ (cid:88) n =0 (cid:90) x f n ( y ) dy = (cid:90) x ∞ (cid:88) n =0 f n ( y ) dy, where the integration and summation can be interchanged. This leads to the followingconvergent series expansion for the distribution function F S P : for any x < ∞ , F S P ( x ) = 1 √ π P Γ ( P ) ∞ (cid:88) n =0 (cid:90) x f n ( y ) dy = 1 √ π P Γ ( P ) ∞ (cid:88) n =0 Γ ( n + P )Γ ( n + 1) (cid:90) x y − ( P +2) exp (cid:32) − (2 n + P ) y (cid:33) D P +1 (cid:18) n + P √ y (cid:19) dy = 1 √ π P +1 Γ ( P ) ∞ (cid:88) n =0 Γ ( n + P )Γ ( n + 1) (2 n + P ) − P G (cid:18) n + P √ x (cid:19) , where the function G ( y ) is defined as G ( y ) = (cid:90) + ∞ y z P − exp (cid:18) − z (cid:19) D P +1 ( z ) dz. (D.2)To evaluate the function G , we first follow the methods mentioned earlier tocalculate the parabolic cylinder functions D P +1 and hence its integrand. We mayreplace D P +1 ( z ) by its convergent power series on the entire interval of integration toderive the corresponding series expansion for G . However, the power series convergestoo slowly to be of practical use for large z . Instead, we split the interval of integration[ y, + ∞ ) into two small elements, say [ y, y ∗ ) and [ y ∗ , + ∞ ) for some sufficiently large y ∗ ≥ y , where we apply different representations for D P +1 ( z ) depending on the valueof z . Then, we have G ( y ) = G ( y, y ∗ ) + G ( y ∗ ) , where G ( y, y ∗ ) := (cid:90) y ∗ y z P − exp (cid:18) − z (cid:19) D P +1 ( z ) dz,G ( y ∗ ) := (cid:90) + ∞ y ∗ z P − exp (cid:18) − z (cid:19) D P +1 ( z ) dz. n [ y ∗ , + ∞ ), the asymptotic expansion (3.6) is a convenient way to compute D P +1 ( z )and hence G . On [ y, y ∗ ), the power series (3.5) will be useful for G . Next, weconsider the integral on the two sub-intervals case by case.On [ y ∗ , + ∞ ), we approximate the parabolic cylinder function D p +1 ( z ) by itsasymptotic series (3.6) on the entire interval under consideration. The series is thenmultiplied by z P − exp (cid:0) − z / (cid:1) and integrated term by term to generate a seriesapproximation for the integral G . To confirm that the resulting series is the correctasymptotic expansion for the integral with large y ∗ , we note that z P − exp (cid:18) − z (cid:19) D P +1 ( z ) ∼ z P exp (cid:18) − z (cid:19) ∞ (cid:88) k =0 ˆ a k z − k , as z → + ∞ , where ˆ a k = ( − k − k ( − ( P + 1)) k /k ! for k ≥
0. By the definition for asymptoticexpansions, we have for each K , z P − exp (cid:18) − z (cid:19) D P +1 ( z ) − z P exp (cid:18) − z (cid:19) K (cid:88) k =0 ˆ a k z − k = o (cid:18) z P exp (cid:18) − z (cid:19) z − K (cid:19) , as z → + ∞ , meaning that for any (cid:15) >
0, there exists a z > z > z , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z P − exp (cid:18) − z (cid:19) D P +1 ( z ) − z P exp (cid:18) − z (cid:19) K (cid:88) k =0 ˆ a k z − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) z P exp (cid:18) − z (cid:19) z − K (cid:12)(cid:12)(cid:12)(cid:12) . Then, properties of integration yield for any y ∗ > z , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ( y ∗ ) − K (cid:88) k =0 ˆ a k (cid:90) + ∞ y ∗ z P exp (cid:18) − z (cid:19) z − k dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) + ∞ y ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z P − exp (cid:18) − z (cid:19) D P +1 ( z ) − z P exp (cid:18) − z (cid:19) K (cid:88) k =0 ˆ a k z − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz ≤ (cid:15) (cid:90) + ∞ y ∗ z P exp (cid:18) − z (cid:19) z − K dz. Hence, we have the following asymptotic relation: as y ∗ → + ∞ , G ( y ∗ ) − K (cid:88) k =0 ˆ a k (cid:90) + ∞ y ∗ z P exp (cid:18) − z (cid:19) z − k dz = o (cid:18)(cid:90) + ∞ y ∗ z P exp (cid:18) − z (cid:19) z − K dz (cid:19) , which further gives the asymptotic expansion G ( y ∗ ) ∼ ∞ (cid:88) k =0 ˆ a k (cid:90) + ∞ y ∗ z P exp (cid:18) − z (cid:19) z − k dz, as y ∗ → + ∞ . Introducing a new variable ζ = z /
2, we find that G ( y ∗ ) ∼ ∞ (cid:88) k =0 ˆ a k P − k − (cid:90) + ∞ ( y ∗ )22 ζ P − k − exp ( − ζ ) dζ ∼ ∞ (cid:88) k =0 ˆ a k P − k − Γ (cid:32) P − k + 12 , ( y ∗ ) (cid:33) n the limit y ∗ → + ∞ , where Γ ( s, z ) is the upper incomplete gamma function. Re-placing ˆ a k by the explicit form given above generates the stated asymptotic expansionfor G .On [ y, y ∗ ), approximating the parabolic cylinder function D P +1 ( z ) using thepower series (3.5), we can write G ( y, y ∗ ) = (cid:90) y ∗ y z P − exp (cid:18) − z (cid:19) (cid:32) ∞ (cid:88) k =0 ˆ d k ( P ) z k (cid:33) dz = ∞ (cid:88) k =0 ˆ d k ( P ) (cid:90) y ∗ y z P + k − exp (cid:18) − z (cid:19) dz = ∞ (cid:88) k =0 ˆ d k ( P ) 2 P + k − (cid:90) ( y ∗ ) y ζ P + k − exp ( − ζ ) dζ = ∞ (cid:88) k =0 ˆ d k ( P ) 2 P + k − (cid:32) Γ (cid:18) P + k , y (cid:19) − Γ (cid:32) P + k , ( y ∗ ) (cid:33)(cid:33) , where the interchange of integration and summation in the second step follows fromthe fact that the power series is uniformly convergent over the interval of integrationand a change of variable ζ = z / < y ≤ y ∗ . Appendix E. Proof of Corollary 3.9.
Recall from Theorem 3.7 that thedistribution function F S P for P ∈ (0 , ∪ N takes the form F S P ( x ) = 1 √ π P +1 Γ ( P ) ∞ (cid:88) n =0 Γ ( n + P )Γ ( n + 1) (2 n + P ) − P G (cid:18) n + P √ x (cid:19) (E.1)for any 0 ≤ x < ∞ , where the function G has the asymptotic approximation G ( y ) ∼ ∞ (cid:88) k =0 ( − k ( − ( P + 1)) k k ! 2 P − k − Γ (cid:18) P − k + 12 , y (cid:19) , as y → + ∞ . Then it follows from the definition for asymptotic expansions that G ( y ) = 2 P − Γ (cid:18) P + 12 , y (cid:19) + o (cid:18) Γ (cid:18) P + 12 , y (cid:19)(cid:19) , as y → + ∞ . Further, by the asymptotic expansion for the incomplete gamma function (Abramowitzand Stegun [2, formula (6 . . s, z ) ∼ z s − exp ( − z ) ∞ (cid:88) k =0 Γ ( s )Γ ( s − k ) z − k , as z → + ∞ , we haveΓ (cid:18) P + 12 , y (cid:19) = (cid:18) y (cid:19) P − exp (cid:18) − y (cid:19) + o (cid:18) y P − exp (cid:18) − y (cid:19)(cid:19) , as y → + ∞ . The above analysis yields G ( y ) = y P − exp (cid:18) − y (cid:19) + o (cid:18) y P − exp (cid:18) − y (cid:19)(cid:19) , as y → + ∞ . ence, we can write G (cid:18) n + P √ x (cid:19) = (2 n + P ) P − x − P exp (cid:32) − (2 n + P ) x (cid:33) + o (cid:32) (2 n + P ) P − x − P exp (cid:32) − (2 n + P ) x (cid:33)(cid:33) in the limit x → + . The observation(2 n + P ) P − exp (cid:16) − (2 n + P ) x (cid:17) exp (cid:0) − P x (cid:1) = (2 n + P ) P − exp (cid:18) − n + 4 nP x (cid:19) → , as x → + for any n ≥ G (cid:18) n + P √ x (cid:19) = o (cid:18) G (cid:18) P √ x (cid:19)(cid:19) , as x → + . Therefore, (E.1) becomes F S P ( x ) = 1 √ π P +1 Γ ( P ) Γ ( P )Γ (1) P − P G (cid:18) P √ x (cid:19) + o (cid:18) G (cid:18) P √ x (cid:19)(cid:19) = 1 √ π P +1 P P − x − P exp (cid:18) − P x (cid:19) + o (cid:18) x − P exp (cid:18) − P x (cid:19)(cid:19) ∼ √ π P +1 P P − x − P exp (cid:18) − P x (cid:19) , as x → + . REFERENCES[1] M. J. ABLOWITZ AND A. S. FOKAS,
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In the tables below, we quote theChebyshev coefficients of the approximations to the inverse distribution functions forthe (standardised) sum ( Z P ) S P . Note that the u denotes the right boundary pointof each regime. Table 4
Chebyshev coefficients c n for P = 1 . n left central middle0 1.870164486816790e-01 4.979491420716220e-01 9.481879998153620e-011 7.543713026654420e-02 8.479024376573340e-02 1.331260262296900e-012 -8.902496689813970e-04 5.515061682001420e-03 -2.432806629044970e-043 5.081110592347190e-04 1.234615565280710e-03 7.536358417854170e-054 -6.749946441432770e-05 7.826573846201580e-05 -1.832873207509700e-055 1.869595919328090e-05 3.190249135857450e-05 3.735655017085910e-066 -4.954381513667230e-06 6.911355172727800e-07 -6.763222944593870e-077 1.546665396688050e-06 1.101914889591240e-06 1.152436476933680e-078 -5.158806635369980e-07 -4.732064488662130e-08 -1.948453255574870e-089 1.840226824599990e-07 4.708456855473810e-08 3.380721765672400e-0910 -6.893572202548840e-08 -5.304437974915840e-09 -6.060338021312360e-1011 2.688851615671340e-08 2.373716110861710e-09 1.110982538990550e-1012 -1.083817122960260e-08 -4.052654986799920e-10 -2.059320821371500e-1113 4.489823790157280e-09 1.347795701927130e-10 3.835128401417810e-1214 -1.903315339772780e-09 -2.837136223549530e-11 -6.941256697800590e-1315 8.228619256107180e-10 8.275614752183520e-1216 -3.618241198589310e-10 -1.932522965168290e-1217 1.614597119724730e-10 4.988201361092360e-1318 -7.298521657921080e-1119 3.336786065994890e-1120 -1.540524697535400e-1121 7.165207365475960e-1222 -3.335368012294930e-1223 1.513427174673910e-1224 -5.858440022503530e-13 u k k k k k k -1.186186343393790e+00 1.000000000000000e+00 -2.084692506698210e+0059 able 4 (cont.) Chebyshev coefficients c n for P = 1 . n right tail right tail0 2.147319302658020e+00 7.301291887179110e+001 4.666085467273610e-01 2.110381075693420e+002 -5.745397085771070e-06 -3.401385019474830e-123 3.926281274449900e-06 3.135451774873670e-124 -2.335817034793870e-06 -2.797671851935420e-125 1.221893569112290e-06 2.415971255636180e-126 -5.676317622953180e-07 -2.018406573717770e-127 2.363679020058830e-07 1.629565101375640e-128 -8.898571009069280e-08 -1.267921039843530e-129 3.052532824574800e-08 9.444179410663350e-1310 -9.610414759815690e-09 -6.622577811449470e-1311 2.796013299028150e-09 4.176644424605940e-1312 -7.568881452656740e-10 -2.012958534474450e-1313 1.920738140207490e-1014 -4.610765872702790e-1115 1.059477675547800e-1116 -2.363411529421280e-1217 4.992754651101120e-1318192021222324 9.990000000000000e-01 k k k k -1.124938736608290e+00 -1.635958636579130e+0060 able 5 Chebyshev coefficients c n for P = 10 . n left tail left central0 -4.186236205146250e+00 -1.966613010380010e+001 1.605190906754510e+00 -1.483681112879300e-022 1.059708264495660e-02 -5.446775372602540e-033 -2.569138906164220e-02 2.105452172418740e-064 2.657630896552880e-03 -3.250251439573260e-055 5.038422707739470e-04 6.250462841383680e-076 -1.420477191393670e-04 -2.522741048643350e-077 -2.983752657616170e-06 1.026418500454660e-088 5.644770851826920e-06 -2.286307882545840e-099 -4.127971330420580e-07 1.386590908961820e-1010 -2.164910928216110e-07 -2.303940103504890e-1111 3.970382219542570e-08 1.773158449711400e-1212 8.721275989102100e-09 -2.474347778199780e-1313 -3.382084859676620e-0914 -2.332160758995600e-1015 2.833835970869750e-1016 -2.159881036758510e-1117 -1.980176647976920e-1118 5.076230137995960e-1219 8.012161542244290e-1320 -6.172636726201170e-13 u k k -6.189978932685550e-01 -3.895807869740060e+00 k k n right central right tail0 2.105468260211710e+00 1.013961870007640e+011 5.526660297932240e-02 5.881348912778960e+002 6.562580263096990e-03 1.613214220004240e+003 4.554851569107730e-04 3.583662354665820e-014 4.959544883969950e-05 6.562610713857870e-025 4.554114153141270e-06 1.013276567637800e-026 4.895102446478050e-07 1.335612973263590e-037 5.011014704681740e-08 1.508499455059680e-048 5.446620224111430e-09 1.461564610943890e-059 5.859569797393820e-10 1.229129368330510e-0610 6.466760957427150e-11 9.368122509902420e-0811 7.143472968048850e-12 6.969255008216930e-0912 7.892055904355890e-13 5.168922908502960e-1013 3.395515411597460e-1114 1.469649165527180e-1215 -3.355536462420540e-141617181920 7.168371818502030e-01 k k k k -1.000000000000000e+00 -1.147724394557630e+0061 able 6 Chebyshev coefficients c n for P = 50 . n left tail left central0 -5.288942152307260e+00 -2.000965923235480e+001 2.354558187240890e+00 5.027272329447840e-032 -2.148300929212860e-01 -5.388241960386400e-033 1.167801704830350e-03 1.321210600565510e-044 4.271918579401850e-03 -3.328258515654870e-055 -7.367600678980360e-04 1.691117475054500e-066 3.190352612884990e-05 -2.758671231347400e-077 1.049570860070370e-05 2.011261792238410e-088 -2.661096411008500e-06 -2.690233651951720e-099 2.849363340335400e-07 2.380979895569440e-1010 6.003686941040730e-09 -2.896342891338410e-1111 -9.314474427911970e-09 2.851594174388340e-1212 2.078536143889170e-09 -3.279640552952750e-1313 -2.199997312217100e-1014 -1.383145073419240e-1115 1.142880443224370e-1116 -2.619066923423380e-1217 3.160327088396940e-13 u k k -6.266377357307220e-01 -4.044263478442090e+00 k k n right central right tail0 2.062558090116340e+00 8.002513934544340e+001 3.614884716178920e-02 4.310172977793670e+002 5.905537215326120e-03 9.680044235587960e-013 3.302424143714490e-04 1.854270427762370e-014 4.120186343509380e-05 3.016976801851500e-025 3.385543742585720e-06 4.333887776280000e-036 3.843164000781120e-07 5.620910828054070e-047 3.717029485793840e-08 6.549643858862510e-058 4.112363385448630e-09 6.704595648415990e-069 4.289267466426690e-10 5.921631388292800e-0710 4.741128697087930e-11 4.529653831824030e-0811 5.137505874838590e-12 3.042863599628060e-0912 5.647548074791590e-13 1.746343337679330e-1013 9.109476838228590e-1214 8.873841036286460e-13151617 7.027660432349340e-01 k k k k -1.000000000000000e+00 -1.122310983565170e+0062 able 7 Chebyshev coefficients c n for P = 5000 . n left tail left middle0 -9.007526636815000e+00 -2.320006146407520e+001 2.127535755063470e+00 1.266023709561750e+002 -1.873229942604270e-01 -1.152857342660610e-013 1.220927893353500e-02 9.871525791146420e-034 -5.143519783570370e-04 -6.330801524297140e-045 1.053201736042580e-05 2.721250039684760e-056 2.597870360642370e-07 -5.511437046510560e-077 -3.977373581792380e-08 -9.780799116408280e-098 2.731783965620670e-09 7.487832322276510e-109 -1.221138438089840e-10 -6.846285108906070e-1110 3.067277829880080e-12 1.419621555779360e-1111 -1.657314975397310e-13 -1.036284107015480e-1212 -3.034958826524900e-1413141516171819 u k k -1.210717168704150e+00 -9.809650284289560e-01 k k n right middle right tail0 2.350506283752280e+00 9.391788080584070e+001 1.288612184352350e+00 2.304276718789270e+002 1.238550415075650e-01 2.244838261893660e-013 1.126367631985240e-02 1.723724750804200e-024 8.113776712522790e-04 1.029865654753350e-035 4.574695148394010e-05 5.284626922206710e-056 2.146343317877790e-06 2.632282130705390e-067 1.059607255591610e-07 1.297109856324030e-078 6.522023447796370e-09 6.005448267256560e-099 3.433932966389800e-10 3.273100583196690e-1010 7.415030905551360e-12 5.306924125563730e-1111 -5.333234673760680e-14 2.645356635940220e-1112 1.613648499072340e-1113 9.700093322305410e-1214 5.663793575999130e-1215 3.213742114800220e-1216 1.771549706162470e-1217 9.437229274020540e-1318 4.731397672671720e-1319 1.949505078193680e-139.950000000000000e-01 k k k k -6.435551648966840e-01 -3.019180921864380e+0063 able 8 Chebyshev coefficients c n for P = 10 . n left tail left middle0 -8.763208308698120e+00 -2.117708885805710e+001 2.272856177544270e+00 1.148679380728190e+002 -2.195485611789300e-01 -9.719454925386660e-023 1.600603323734860e-02 7.829228309183660e-034 -7.913864835076260e-04 -4.795725327622320e-045 2.398092020649720e-05 2.045360216276630e-056 -1.924215235351760e-07 -5.013317907576720e-077 -3.279659242674000e-08 3.503726442772780e-098 3.433751594624110e-09 -1.253562317413470e-109 -1.944135457019680e-10 -1.311511811147320e-1210 5.718823252641270e-12 5.140621840511810e-1211 -3.006387310778820e-13 -4.635014571234970e-13121314151617 u k k -1.115994743243930e+00 -1.054206693922630e+00 k k n right middle right tail0 2.134535481408690e+00 9.022503475215680e+001 1.161232186820710e+00 2.401317040413510e+002 1.019807405937260e-01 2.487767815506420e-013 8.555012569847670e-03 2.028231950132690e-024 5.664908014626390e-04 1.266220882300120e-035 2.890763831163810e-05 6.623334821975800e-056 1.181852005087120e-06 3.326141775983180e-067 4.960639757165790e-08 1.667239146689460e-078 2.822677760359780e-09 7.705454473071610e-099 1.409816724283650e-10 3.768066578229160e-1010 1.871694274577610e-12 3.560180300524010e-1111 -1.571409536591980e-13 1.210420538657680e-1112 7.304190736146010e-1213 4.532327470602990e-1214 2.727718438162450e-1215 1.578085621557310e-1216 8.497824077877610e-1317 3.683334903311600e-139.900000000000000e-01 k k k k -6.158684948176150e-01 -2.704324434390850e+0064 able 9 Chebyshev coefficients c n for P = 10 . n left tail right tail0 -5.706926051399320e+00 5.760300411129710e+001 3.395553188579180e+00 3.436321449713240e+002 -6.300054999277720e-01 6.483188840749760e-013 9.838818354304640e-02 1.035993426134920e-014 -1.136750936655030e-02 1.252833208873680e-025 9.513883577249630e-04 1.161417162277510e-036 -6.015836221231600e-05 9.191380802217640e-057 3.618126012554300e-06 7.737590464616790e-068 -1.697979831324770e-07 6.400595433239310e-079 -1.897253631189280e-08 2.904203939446630e-0810 3.326071860640010e-09 1.050861883541800e-0911 2.442263784022650e-10 6.008685289953490e-1012 -3.833832227759890e-11 6.651321295963480e-1113 -1.297154428240060e-11 -1.031199486940060e-1114 1.412983301319550e-12 -9.752972166933010e-1315 4.089569727968860e-13 5.588332180241710e-13 u k k -5.424593574874610e-01 5.427821460191420e-01 k k Table 10
Chebyshev coefficients c n for P = 10 . n left tail right tail0 -5.946145460337920e+00 5.964470382846980e+001 3.556004911984500e+00 3.570041770711440e+002 -6.801894131410160e-01 6.865586194977590e-013 1.092510150797910e-01 1.110985275994380e-014 -1.307725205702110e-02 1.349685679309470e-025 1.160328523663860e-03 1.237761453545620e-036 -8.240107479627320e-05 9.434859113712600e-057 5.926003936265940e-06 7.508635778142890e-068 -3.827985039853400e-07 5.673276390309780e-079 -4.939610258605130e-09 1.429761666233160e-0810 2.191431812357590e-09 -4.002728561697650e-1011 4.611932833470640e-10 6.107231293159660e-1012 -4.959733545874770e-11 6.167427914388110e-1113 -1.701165301087770e-11 -1.590293060974530e-1114 1.492500677577930e-12 -1.334659663440940e-1215 5.860252785293660e-13 6.358368604978550e-13 u k k -5.310898090226540e-01 5.311876095528130e-01 k k able 11 Chebyshev coefficients c n for h = 2 . n left left0 3.746732186396810e-01 1.246285546643250e+001 1.195997564921540e-01 3.358246308532060e-012 -8.070601474351940e-03 2.386732322681120e-023 1.339603001568510e-03 1.294898023275990e-024 -2.506080830464610e-04 2.637423242240120e-035 5.637743633696060e-05 1.091486949977880e-036 -1.413866007641720e-05 3.083088185622240e-047 3.869664142567620e-06 1.183343193111800e-048 -1.132969831139660e-06 3.838648490201040e-059 3.501617974093610e-07 1.440406811311990e-0510 -1.130976314574740e-07 5.002488355310480e-0611 3.788372056248530e-08 1.869987422785930e-0612 -1.308238855480900e-08 6.739900491698620e-0713 4.635781671551490e-09 2.526970560301390e-0714 -1.679323548784110e-09 9.311688370921460e-0815 6.200204087490220e-10 3.508754493020290e-0816 -2.327351020956820e-10 1.311638643142900e-0817 8.863586972126330e-11 4.968660909382380e-0918 -3.418842923801670e-11 1.875958145331950e-0919 1.333107561719920e-11 7.141781072249230e-1020 -5.234957827868200e-12 2.716166269988070e-1021 2.035050527498960e-12 1.038650818883500e-1022 -7.009107859412300e-13 3.972098592976600e-1123 1.524237223546780e-1124 5.838842289005870e-1225 2.206347821363880e-1226 7.400818875262400e-13 u k k k k -1.301445639726460e+00 -1.662175466937440e+0066 able 11 (cont.) Chebyshev coefficients c n for h = 2 . n middle right tail0 3.527966142386780e+00 1.003328309377590e+011 7.472142334253430e-01 2.496706503355630e+002 -1.333644417363060e-02 -1.589949448670880e-023 2.622018509548740e-03 3.113590178926820e-034 -6.027577883155430e-04 -6.963073964368400e-045 1.515242268297150e-04 1.678769292503920e-046 -4.036373098988620e-05 -4.250114645375870e-057 1.119644049107180e-05 1.113756440553480e-058 -3.199289309119440e-06 -2.994727316140500e-069 9.349240809409510e-07 8.214913272814210e-0710 -2.779917579878980e-07 -2.289772183834370e-0711 8.378973634959820e-08 6.466492239429990e-0812 -2.552869970129400e-08 -1.846256500319380e-0813 7.845488794437930e-09 5.320336235290850e-0914 -2.428118880696940e-09 -1.545402456085360e-0915 7.559005461053680e-10 4.520062644698680e-1016 -2.364994613022200e-10 -1.330078378584710e-1017 7.431800136657720e-11 3.934888781070520e-1118 -2.344374813365950e-11 -1.169587411803260e-1119 7.415598838082110e-12 3.489114825460500e-1220 -2.333558791113770e-12 -1.037689978383220e-1221 6.745346171083730e-13 2.859557405213800e-132223242526 9.997923134230980e-01 k k k k -6.514171570857550e-01 -1.450272673332570e+0067 able 12 Chebyshev coefficients c n for h = 1 / . n left central middle0 1.551440631887020e-02 5.965851817537560e-02 2.365922039769380e-011 7.154056831584140e-03 1.701537131814850e-02 7.509528018047980e-022 1.120793580526900e-04 2.597262497204220e-03 6.749060031298230e-033 -2.767589127786680e-05 4.709183006272860e-04 -9.756034732487190e-054 5.310776527089690e-06 7.145518311775060e-05 -4.546713797013440e-055 -1.242484981459680e-06 1.330260630541950e-05 2.265934249688400e-066 3.545760586969740e-07 1.967551836645490e-06 3.137406427278980e-077 -1.173714662110980e-07 3.949016650674680e-07 1.344476627067420e-088 4.313888580276150e-08 5.420695609123570e-08 -8.390259114557710e-099 -1.709037630074550e-08 1.243550067642890e-08 7.380356092377570e-1010 7.159028761307770e-09 1.439656776199120e-09 -2.253776759627590e-1011 -3.130641321239270e-09 4.213671249996360e-10 4.735083288679870e-1112 1.416650107438240e-09 3.320038533665110e-11 -6.226333936166760e-1213 -6.591639233672150e-10 1.566391114883240e-11 2.369177040131130e-1214 3.138939220131410e-10 4.052720676134970e-13 -5.354588204982440e-1315 -1.524291404102210e-1016 7.526974215608190e-1117 -3.770866706212000e-1118 1.912726184698180e-1119 -9.801629544246880e-1220 5.055418302750250e-1221 -2.598617692870460e-1222 1.287201224090770e-1223 -5.308318388467400e-1324 u k k k k k k -1.121868009800680e+00 1.000000000000000e+00 -1.516097941825860e+0068 able 12 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.766531067045780e-01 5.548556192934720e+001 3.018335097399800e-01 2.006312378503930e+002 1.137764493492600e-02 2.401553462574660e-023 -1.796551902064940e-03 -5.827238250950370e-034 2.961890615175630e-04 1.564108444981830e-035 -3.906140490786650e-05 -4.425250341488140e-046 3.228136981746220e-07 1.292806696330280e-047 1.990381470117290e-06 -3.860207865611520e-058 -7.180791153931000e-07 1.171917601240410e-059 1.237304075899040e-07 -3.609533592707940e-0610 4.797867870237380e-09 1.128193509794600e-0611 -1.022283804766720e-08 -3.587762176720570e-0712 3.242766128356910e-09 1.166622015545870e-0713 -3.934089138513020e-10 -3.905857435831940e-0814 -1.071903231878940e-10 1.356481292566670e-0815 7.087360088039800e-11 -4.912140911093550e-0916 -1.797818306057740e-11 1.854288283705740e-0917 1.002649821269050e-12 -7.243408051055730e-1018 1.130725736108390e-12 2.888927287597270e-1019 -5.251498162236730e-13 -1.156809097791070e-1020 4.568355593550080e-1121 -1.746849139303900e-1122 6.337718113837970e-1223 -2.123287366692640e-1224 6.058538275109010e-139.985870051144820e-01 k k k k -2.078549151656860e+00 -1.568565612274160e+0069 able 13 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.130510092436490e-03 2.226333345761950e-02 1.988982349173310e-011 1.417002017122940e-03 1.019151935447450e-02 8.748062377822970e-022 2.360674641499860e-05 2.673461828675060e-03 1.322178512127460e-023 -6.764554780577100e-06 7.541005867012190e-04 1.783172989082840e-054 1.073524111115770e-06 1.825724014495570e-04 -1.742061549930610e-045 -2.165240827636890e-07 5.070406265568330e-05 1.493474602053700e-066 5.714805445297300e-08 1.156695182629020e-05 3.359055560341550e-067 -1.802784817620440e-08 3.413930484845420e-06 5.333580684343860e-088 6.351907282247530e-09 6.943759577207300e-07 -5.521667146922240e-089 -2.410001034673690e-09 2.411908456774400e-07 -2.276235131191460e-0910 9.649353406509410e-10 3.679790048414060e-08 -2.893556379100810e-1011 -4.026554373758090e-10 1.888991234749750e-08 -6.436850207940650e-1112 1.736498040397040e-10 1.106003127127480e-09 2.350168269709410e-1113 -7.693121262340320e-11 1.752401085721140e-09 2.321358068727610e-1114 3.485269197769610e-11 -1.364762944049400e-10 2.636593526158320e-1215 -1.608506662519110e-11 2.000879783288790e-10 -1.139207052258690e-1216 7.530478144378150e-12 -4.354349576970780e-11 -7.314416583435220e-1317 -3.545863047530400e-12 2.738074123398660e-1118 1.631081082539700e-12 -8.687479490879170e-1219 -6.387767991337960e-13 4.196188738890250e-1220 -1.521565690726940e-1221 5.748616824737580e-13 u k k k k k k -1.150665798108190e+00 1.000000000000000e+00 -9.422665062905070e-0170 able 13 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 1.167088563196570e+00 5.557949689428240e+001 4.028821977072450e-01 1.804403214930680e+002 1.589726864878320e-02 2.001076313509200e-023 -2.776444180529080e-03 -4.157908839745580e-034 5.440877740378380e-04 9.533639401005070e-045 -1.067963570631740e-04 -2.298471294547880e-046 1.602535986540860e-05 5.706424934572420e-057 3.639164819106400e-07 -1.443342632426860e-058 -1.502345300041770e-06 3.696661627610680e-069 6.684757371238390e-07 -9.553281498410910e-0710 -1.716295249441850e-07 2.486726588153440e-0711 1.635394935133220e-08 -6.518766056569640e-0812 8.892938254607350e-09 1.724037249294330e-0813 -5.718412804481970e-09 -4.621243803810750e-0914 1.784898899321040e-09 1.266095041880910e-0915 -2.594370983148820e-10 -3.591505441133870e-1016 -6.261300169685750e-11 1.071702367595880e-1017 5.827210983507840e-11 -3.409479146600350e-1118 -2.189105354284160e-11 1.160098934787360e-1119 4.514597736601780e-12 -4.170286799495250e-1220 -6.083212355274480e-14 1.531989721713420e-1221 -5.089272134222670e-139.998315799958800e-01 k k k k -1.885294461971110e+00 -1.777456775656590e+0071 able 14 Chebyshev coefficients c n for h = 1 / . n left central middle0 8.320108386176220e-04 1.028289222803410e-02 1.796716788945920e-011 3.797922844843720e-04 6.051387649842750e-03 9.215513210293030e-022 6.119152516315670e-06 2.386515701734710e-03 1.798443115218630e-023 -2.216499535843730e-06 9.149383182711970e-04 3.536164396647120e-044 3.549796569970470e-07 3.260804312989750e-04 -3.290230257519100e-045 -6.909777679284210e-08 1.174302244391960e-04 -9.190331115977200e-066 1.800406773387920e-08 4.046558068563540e-05 8.941392960865100e-067 -5.727807832664590e-09 1.430511004864880e-05 4.569100427021660e-078 2.048306829365090e-09 4.845772394724280e-06 -1.898690252517320e-079 -7.892933375352240e-10 1.708561223033980e-06 -1.470693084569350e-0810 3.208272429161230e-10 5.709919634096540e-07 9.061517318686090e-1011 -1.358581479450610e-10 2.027725977295490e-07 -4.996134827280970e-1012 5.944136353935260e-11 6.667913363885630e-08 -3.085715272976990e-1113 -2.671169627505600e-11 2.408641753810920e-08 8.541875621658270e-1114 1.227327977365980e-11 7.729248497752890e-09 3.049756612877180e-1115 -5.743936308401000e-12 2.879366774439790e-09 7.727805117204190e-1316 2.726212185552570e-12 8.872469780543650e-10 -3.534457536346370e-1217 -1.300587961514960e-12 3.484429145625870e-10 -1.357036283678540e-1218 6.052241498351530e-13 1.001335286917030e-10 4.056653202382400e-1419 -2.390157663283510e-13 4.301663419431590e-11 2.166251245948460e-1320 1.093494057671210e-1121 5.462280485697060e-1222 1.109419945826560e-1223 6.264905101784660e-13 u k k k k k k -1.142970216093080e+00 9.999999999999990e-01 -6.895558541579670e-0172 able 14 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.761057883039990e-01 5.196073467821080e+001 3.017284864244540e-01 1.831141920349900e+002 1.166946937938660e-02 2.543065037748800e-023 -1.687052247930830e-03 -5.741796704905520e-034 2.745955374164150e-04 1.425454072455170e-035 -4.255962891388790e-05 -3.709730129757980e-046 3.654738784807000e-06 9.914921759807340e-057 9.678759239832630e-07 -2.692978838795350e-058 -5.962323527180660e-07 7.390951335687150e-069 1.619073566570090e-07 -2.044323535428510e-0610 -2.044848058779350e-08 5.700139242154520e-0711 -3.557663286168380e-09 -1.608555974873110e-0712 2.792106872070250e-09 4.637422326004230e-0813 -8.014791797406380e-10 -1.388223005657000e-0814 9.886383182987470e-11 4.408914855308610e-0915 2.338331909817720e-11 -1.513613329915950e-0916 -1.720362865683670e-11 5.636677579981080e-1017 5.096915629190340e-12 -2.238514411774360e-1018 -7.701281537440130e-13 9.204119152754320e-1119 -3.802089545459430e-1120 1.539622910176960e-1121 -5.993091125355200e-1222 2.193183001782910e-1223 -6.910989846330220e-139.997667653179200e-01 k k k k -2.168616210929900e+00 -1.650521806358000e+0073 able 15 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.342404855864130e-04 5.287504606339480e-03 1.679802218836370e-011 1.582119045977740e-04 3.609167042053250e-03 9.425159027828390e-022 -4.034459826869400e-06 1.936145076792520e-03 2.146641276528250e-023 -1.781674777546940e-06 9.633501098384940e-04 7.761771414171220e-044 6.111894067922420e-07 4.586062099584780e-04 -4.715700923865780e-045 -1.616413643230570e-07 2.128565367871240e-04 -2.911833910065400e-056 4.535507820139480e-08 9.711724060740940e-05 1.543681974168120e-057 -1.504566368490170e-08 4.380458926575550e-05 1.397441335389570e-068 5.916977370301490e-09 1.959490913685950e-05 -3.947158787904670e-079 -2.631104077272530e-09 8.713044455946110e-06 -4.582398590716490e-0810 1.265674285428750e-09 3.857017778878410e-06 4.537121001350180e-0911 -6.412984582939380e-10 1.701669482580570e-06 -8.587058106556170e-1012 3.373488093146030e-10 7.488328990975590e-07 -3.395277343066420e-1013 -1.827196103322590e-10 3.288850425725570e-07 1.497940638456500e-1014 1.013577890763340e-10 1.442278857066740e-07 9.047367570431180e-1115 -5.736165706600700e-11 6.317620038821080e-08 1.532712463574530e-1116 3.301900838213170e-11 2.764873413720030e-08 -5.899234983630290e-1217 -1.928175236669560e-11 1.209228320347630e-08 -4.759599480764640e-1218 1.139207390775470e-11 5.285996098982320e-09 -9.845099825540210e-1319 -6.784952996564550e-12 2.309885154214620e-09 3.353301735326350e-1320 4.046041780101790e-12 1.009129182529960e-0921 -2.377799356621700e-12 4.407929681530880e-1022 1.318077536426880e-12 1.925241011297400e-1023 -5.870490384521500e-13 8.408559765252580e-1124 3.672433365477270e-1125 1.603764366665520e-1126 6.998237196315440e-1227 3.040158479208750e-1228 1.288881492848470e-1229 4.729004482220720e-13 u k k k k k k -1.055332320056090e+00 1.000000000000000e+00 -5.434844889094470e-0174 able 15 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.761146326247330e-01 5.440971360286990e+001 3.016928648060250e-01 1.956911680269630e+002 1.167231718705550e-02 2.774260925258900e-023 -1.651276452041110e-03 -6.436890817427110e-034 2.666875570608340e-04 1.641299885337880e-035 -4.247191460497490e-05 -4.384480040071970e-046 4.247919767558590e-06 1.201938574075200e-047 7.240192379129270e-07 -3.345172414930230e-058 -5.468504974426770e-07 9.394648359883000e-069 1.621838761986780e-07 -2.653557174889410e-0610 -2.469682517941960e-08 7.531228230971810e-0711 -1.846812979021160e-09 -2.152447292666580e-0712 2.453530993347940e-09 6.237985768142160e-0813 -8.104795070564450e-10 -1.859198071101320e-0814 1.328212021134190e-10 5.827250350019090e-0915 9.746012118699450e-12 -1.970501685911350e-0916 -1.445305942484290e-11 7.299803884539640e-1017 5.164008080182300e-12 -2.939243341028580e-1018 -1.029938970952770e-12 1.250293085781470e-1019 -5.430549502597070e-1120 2.340535807434650e-1121 -9.802247791887960e-1222 3.924623772975130e-1223 -1.472095272217630e-1224 4.733000270115350e-132526272829 9.999145446729710e-01 k k k k -2.179745438075220e+00 -1.614273003898510e+0075 able 16 Chebyshev coefficients c n for h = 1 / . n left central middle0 1.064576523939530e-04 7.757528496943130e-04 6.583911279090610e-021 4.967483886634590e-05 3.963182148041290e-04 4.630386572794740e-022 -2.574907330657650e-06 1.452820177102820e-04 1.757544141274230e-023 -5.303715403034560e-07 4.855954814287430e-05 3.315284363718880e-034 2.663871929167150e-07 1.531863558980870e-05 1.968319378136490e-055 -8.401958707782960e-08 4.676444520070540e-06 -1.223428005218250e-046 2.541840517940180e-08 1.394003778702580e-06 -1.557379115932900e-057 -8.458536138015860e-09 4.088490384098540e-07 3.231990249141240e-068 3.265291063082850e-09 1.184092050832930e-07 9.273005743030430e-079 -1.446267628365560e-09 3.397158988245310e-08 -3.854154049131550e-0810 7.070186386866000e-10 9.672491318170320e-09 -3.568282952834910e-0811 -3.686990455737450e-10 2.737461712291000e-09 -8.276509465005390e-1012 2.008214150700500e-10 7.708833517653440e-10 1.206706537754180e-0913 -1.129090416703240e-10 2.161982864910220e-10 9.225936702119910e-1114 6.508194177274510e-11 6.042419184248020e-11 -3.617701898545360e-1115 -3.828935422940620e-11 1.683809589516920e-11 -4.590404543093130e-1216 2.291651606295020e-11 4.678693771234680e-12 9.605943640102260e-1317 -1.391339883329400e-11 1.290808850907550e-12 1.614097426563680e-1318 8.543046330402440e-12 3.320491288954080e-1319 -5.282416801082780e-1220 3.263505722460030e-1221 -1.979537858186990e-1222 1.125706972359250e-1223 -5.098805064619270e-1324252627282930 u k k k k k k -1.042176040321580e+00 1.000000000000000e+00 2.136086687920960e-0176 able 16 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 5.721713263001810e-01 5.012707101453440e+001 2.017820038440820e-01 2.062096651831810e+002 1.377375013917080e-02 3.999684930491530e-023 -1.806370775567440e-03 -1.094915283125730e-024 1.523593047177300e-04 3.280050342478840e-035 2.295752543436930e-05 -1.026359592164560e-036 -1.211645711839800e-05 3.290839108300630e-047 1.558915013492110e-06 -1.072510048733340e-048 3.379002506728430e-07 3.549425811224860e-059 -1.589627733030860e-07 -1.199878274669100e-0510 8.281053242094140e-09 4.199796533313170e-0611 9.754591213299930e-09 -1.552499344038120e-0612 -2.587101485419130e-09 6.174229919308370e-0713 -2.654922061326020e-10 -2.654025845860760e-0714 2.854986006586110e-10 1.213436337756580e-0715 -4.288462498279570e-11 -5.724458644126490e-0816 -1.614286040010390e-11 2.696385391883720e-0817 8.227664210952470e-12 -1.231399370142220e-0818 -8.469562416364850e-13 5.303258910110760e-0919 -2.080169945416400e-0920 6.955732723896000e-1021 -1.600540556356640e-1022 -1.256262769207190e-1123 4.730496708984270e-1124 -3.958827080866930e-1125 2.486066180938770e-1126 -1.342780201739020e-1127 6.510864817550600e-1228 -2.877910389935850e-1229 1.155822468250220e-1230 -3.885406981704210e-139.997327929473050e-01 k k k k -1.873568041871660e+00 -1.410192678438050e+0077 able 17 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.518991274756410e-05 8.392520820142320e-04 3.802327783487690e-021 1.590990829653510e-05 6.431136826076690e-04 2.293406644160830e-022 -1.491507257739220e-06 4.085596936397020e-04 6.950796412254640e-033 -9.066540515692110e-08 2.399296017984750e-04 1.080933113927390e-034 1.118185077651380e-07 1.344982201802300e-04 3.968276530040410e-055 -4.689629675507760e-08 7.316918708694770e-05 -1.510396050891140e-056 1.672578243190620e-08 3.898020499880180e-05 -2.164545910662930e-067 -5.962483120983330e-09 2.045075653316820e-05 1.031238628209580e-078 2.299928220409480e-09 1.060530105647130e-05 4.715822375760130e-089 -9.951311785026230e-10 5.449951689909780e-06 1.338195202962630e-0910 4.813973704272410e-10 2.780434074253660e-06 -7.162524074055580e-1011 -2.538921373245920e-10 1.410169685110300e-06 -5.838646258223070e-1112 1.421639472958470e-10 7.117308974598160e-07 9.035922879320330e-1213 -8.292852778673330e-11 3.577606972109210e-07 1.324669429761400e-1214 4.981742953446680e-11 1.792155094219710e-07 -9.136939392295850e-1415 -3.060881416945410e-11 8.951262809172500e-0816 1.915173995253600e-11 4.459614066323360e-0817 -1.216412493952420e-11 2.216977555006020e-0818 7.820244746397910e-12 1.100014354771840e-0819 -5.072097961571200e-12 5.448923967042200e-0920 3.302395224132890e-12 2.695162687572190e-0921 -2.138908948989580e-12 1.331357683924750e-0922 1.351588122026080e-12 6.569059660914260e-1023 -7.937121438141370e-13 3.237909926903940e-1024 3.670269446605750e-13 1.594506248721030e-1025 7.845610579728340e-1126 3.857391556723120e-1127 1.895058050737680e-1128 9.300136827782300e-1229 4.552865562504610e-1230 2.209843826831820e-1231 1.035453103819430e-1232 4.092881791896850e-13 u k k k k k k -1.029129856396840e+00 1.000000000000000e+00 4.634771910249610e-0178 able 17 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 4.996811103033990e-01 4.914258356400550e+001 2.283166265702920e-01 2.011738127244230e+002 2.514423513031150e-02 3.937356800757610e-023 -3.474198991631110e-03 -1.065266526129480e-024 5.077220214696990e-05 3.152291195579000e-035 1.613616422231280e-04 -9.738895553647930e-046 -3.880889586523480e-05 3.081786621477630e-047 -3.634159932107070e-06 -9.909792676876680e-058 3.510602114428830e-06 3.236113469325020e-059 -1.965557011348590e-07 -1.080367954217740e-0510 -2.893679425553000e-07 3.742488200776480e-0611 4.919338342933830e-08 -1.374128809584770e-0612 2.766224437511820e-08 5.450001386646280e-0713 -9.150633315570560e-09 -2.341737771120620e-0714 -2.629338358854130e-09 1.069346503420290e-0715 1.702910382225310e-09 -5.021967626980490e-0816 1.193636414090970e-10 2.344515094335100e-0817 -2.861147205442400e-10 -1.056108154875420e-0818 3.711031444447810e-11 4.460714134159530e-0919 3.822894605668730e-11 -1.700913808398100e-0920 -1.452954868328450e-11 5.418507670597160e-1021 -2.818928436440390e-12 -1.082511209350420e-1022 3.052195946657310e-12 -2.332308071729190e-1123 -4.763636711850320e-13 4.458678049756930e-1124 -3.438522185303330e-1125 2.066800196185070e-1126 -1.078966568122660e-1127 5.070919968888780e-1228 -2.172493412014030e-1229 8.448217334599220e-1330 -2.756966775927990e-133132 9.998681694019440e-01 k k k k -1.323148927258810e+00 -1.420354956151160e+0079 able 18 Chebyshev coefficients c n for h = 1 / . n left central middle0 5.483749876010110e-06 6.081052138201630e-05 5.473769059145160e-021 2.486444309865320e-06 3.802911084933040e-05 4.351538755149600e-022 -2.246254082499010e-07 1.823368632895480e-05 2.198780054496340e-023 -1.578587968151560e-08 7.946990081244110e-06 6.662984075703530e-034 1.732923293003170e-08 3.273356396625710e-06 7.907173467155090e-045 -7.081399642607440e-09 1.301100356483330e-06 -2.243067264619820e-046 2.482198403770580e-09 5.043448241124090e-07 -1.032261735179890e-047 -8.749831841180770e-10 1.919418161993120e-07 -5.350827374548080e-068 3.358446096487070e-10 7.202571800642060e-08 6.032870474106270e-069 -1.452715547964040e-10 2.672894811858390e-08 1.389775608451330e-0610 7.036053028305950e-11 9.830459878805920e-09 -1.667794341081260e-0711 -3.712078515936340e-11 3.588823452512350e-09 -1.098049445185060e-0712 2.075613750341240e-11 1.302082454421860e-09 -3.990118701505280e-0913 -1.206646596058050e-11 4.699379616814380e-10 6.210083081288950e-0914 7.203733700544880e-12 1.688418171221930e-10 9.633344895225680e-1015 -4.375271295344380e-12 6.042392709176170e-11 -2.609783367202760e-1016 2.674170121231750e-12 2.154618744858790e-11 -8.315421351488390e-1117 -1.612760826306620e-12 7.648543523947050e-12 6.884100040797630e-1218 9.148126116574220e-13 2.675596311035590e-12 5.334393632012140e-1219 -4.140244318492860e-13 8.418748326551260e-13 8.096094478676190e-14202122232425262728 u k k k k k k -1.030074288714920e+00 1.000000000000000e+00 4.184273889203900e-0180 able 18 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 5.722628305041320e-01 4.619812351486580e+001 2.017761232436850e-01 1.860589481022910e+002 1.372549559855640e-02 3.705233566867140e-023 -1.799486944230340e-03 -9.679912077752170e-034 1.548179312896890e-04 2.763008630922080e-035 2.188831960478410e-05 -8.227338295193510e-046 -1.205509962337280e-05 2.508114545966230e-047 1.626588892457300e-06 -7.771817988229730e-058 3.184920827863750e-07 2.450094311357910e-059 -1.604199428018640e-07 -7.933144007790290e-0610 1.026914924420960e-08 2.688223932510620e-0611 9.459364972251830e-09 -9.765416694004470e-0712 -2.712392941794810e-09 3.865875322747190e-0713 -2.064497813254320e-10 -1.657778704727940e-0714 2.839303432413100e-10 7.480889417663450e-0815 -4.871222002206250e-11 -3.419915104148750e-0816 -1.444200355589120e-11 1.528919983568170e-0817 8.372113940734360e-12 -6.479029251360280e-0918 -8.215034741114160e-13 2.513840140749050e-0919 -5.718274703852940e-13 -8.420176316921800e-1020 2.056434736961990e-1021 -1.906441397640450e-1222 -4.118010901086650e-1123 3.573866647613250e-1124 -2.210862584205230e-1125 1.157843167001790e-1126 -5.392753508375620e-1227 2.255758034817050e-1228 -7.825720526313930e-139.999476900342120e-01 k k k2
Chebyshev coefficients c n for P = 10 . n left tail right tail0 -5.946145460337920e+00 5.964470382846980e+001 3.556004911984500e+00 3.570041770711440e+002 -6.801894131410160e-01 6.865586194977590e-013 1.092510150797910e-01 1.110985275994380e-014 -1.307725205702110e-02 1.349685679309470e-025 1.160328523663860e-03 1.237761453545620e-036 -8.240107479627320e-05 9.434859113712600e-057 5.926003936265940e-06 7.508635778142890e-068 -3.827985039853400e-07 5.673276390309780e-079 -4.939610258605130e-09 1.429761666233160e-0810 2.191431812357590e-09 -4.002728561697650e-1011 4.611932833470640e-10 6.107231293159660e-1012 -4.959733545874770e-11 6.167427914388110e-1113 -1.701165301087770e-11 -1.590293060974530e-1114 1.492500677577930e-12 -1.334659663440940e-1215 5.860252785293660e-13 6.358368604978550e-13 u k k -5.310898090226540e-01 5.311876095528130e-01 k k able 11 Chebyshev coefficients c n for h = 2 . n left left0 3.746732186396810e-01 1.246285546643250e+001 1.195997564921540e-01 3.358246308532060e-012 -8.070601474351940e-03 2.386732322681120e-023 1.339603001568510e-03 1.294898023275990e-024 -2.506080830464610e-04 2.637423242240120e-035 5.637743633696060e-05 1.091486949977880e-036 -1.413866007641720e-05 3.083088185622240e-047 3.869664142567620e-06 1.183343193111800e-048 -1.132969831139660e-06 3.838648490201040e-059 3.501617974093610e-07 1.440406811311990e-0510 -1.130976314574740e-07 5.002488355310480e-0611 3.788372056248530e-08 1.869987422785930e-0612 -1.308238855480900e-08 6.739900491698620e-0713 4.635781671551490e-09 2.526970560301390e-0714 -1.679323548784110e-09 9.311688370921460e-0815 6.200204087490220e-10 3.508754493020290e-0816 -2.327351020956820e-10 1.311638643142900e-0817 8.863586972126330e-11 4.968660909382380e-0918 -3.418842923801670e-11 1.875958145331950e-0919 1.333107561719920e-11 7.141781072249230e-1020 -5.234957827868200e-12 2.716166269988070e-1021 2.035050527498960e-12 1.038650818883500e-1022 -7.009107859412300e-13 3.972098592976600e-1123 1.524237223546780e-1124 5.838842289005870e-1225 2.206347821363880e-1226 7.400818875262400e-13 u k k k k -1.301445639726460e+00 -1.662175466937440e+0066 able 11 (cont.) Chebyshev coefficients c n for h = 2 . n middle right tail0 3.527966142386780e+00 1.003328309377590e+011 7.472142334253430e-01 2.496706503355630e+002 -1.333644417363060e-02 -1.589949448670880e-023 2.622018509548740e-03 3.113590178926820e-034 -6.027577883155430e-04 -6.963073964368400e-045 1.515242268297150e-04 1.678769292503920e-046 -4.036373098988620e-05 -4.250114645375870e-057 1.119644049107180e-05 1.113756440553480e-058 -3.199289309119440e-06 -2.994727316140500e-069 9.349240809409510e-07 8.214913272814210e-0710 -2.779917579878980e-07 -2.289772183834370e-0711 8.378973634959820e-08 6.466492239429990e-0812 -2.552869970129400e-08 -1.846256500319380e-0813 7.845488794437930e-09 5.320336235290850e-0914 -2.428118880696940e-09 -1.545402456085360e-0915 7.559005461053680e-10 4.520062644698680e-1016 -2.364994613022200e-10 -1.330078378584710e-1017 7.431800136657720e-11 3.934888781070520e-1118 -2.344374813365950e-11 -1.169587411803260e-1119 7.415598838082110e-12 3.489114825460500e-1220 -2.333558791113770e-12 -1.037689978383220e-1221 6.745346171083730e-13 2.859557405213800e-132223242526 9.997923134230980e-01 k k k k -6.514171570857550e-01 -1.450272673332570e+0067 able 12 Chebyshev coefficients c n for h = 1 / . n left central middle0 1.551440631887020e-02 5.965851817537560e-02 2.365922039769380e-011 7.154056831584140e-03 1.701537131814850e-02 7.509528018047980e-022 1.120793580526900e-04 2.597262497204220e-03 6.749060031298230e-033 -2.767589127786680e-05 4.709183006272860e-04 -9.756034732487190e-054 5.310776527089690e-06 7.145518311775060e-05 -4.546713797013440e-055 -1.242484981459680e-06 1.330260630541950e-05 2.265934249688400e-066 3.545760586969740e-07 1.967551836645490e-06 3.137406427278980e-077 -1.173714662110980e-07 3.949016650674680e-07 1.344476627067420e-088 4.313888580276150e-08 5.420695609123570e-08 -8.390259114557710e-099 -1.709037630074550e-08 1.243550067642890e-08 7.380356092377570e-1010 7.159028761307770e-09 1.439656776199120e-09 -2.253776759627590e-1011 -3.130641321239270e-09 4.213671249996360e-10 4.735083288679870e-1112 1.416650107438240e-09 3.320038533665110e-11 -6.226333936166760e-1213 -6.591639233672150e-10 1.566391114883240e-11 2.369177040131130e-1214 3.138939220131410e-10 4.052720676134970e-13 -5.354588204982440e-1315 -1.524291404102210e-1016 7.526974215608190e-1117 -3.770866706212000e-1118 1.912726184698180e-1119 -9.801629544246880e-1220 5.055418302750250e-1221 -2.598617692870460e-1222 1.287201224090770e-1223 -5.308318388467400e-1324 u k k k k k k -1.121868009800680e+00 1.000000000000000e+00 -1.516097941825860e+0068 able 12 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.766531067045780e-01 5.548556192934720e+001 3.018335097399800e-01 2.006312378503930e+002 1.137764493492600e-02 2.401553462574660e-023 -1.796551902064940e-03 -5.827238250950370e-034 2.961890615175630e-04 1.564108444981830e-035 -3.906140490786650e-05 -4.425250341488140e-046 3.228136981746220e-07 1.292806696330280e-047 1.990381470117290e-06 -3.860207865611520e-058 -7.180791153931000e-07 1.171917601240410e-059 1.237304075899040e-07 -3.609533592707940e-0610 4.797867870237380e-09 1.128193509794600e-0611 -1.022283804766720e-08 -3.587762176720570e-0712 3.242766128356910e-09 1.166622015545870e-0713 -3.934089138513020e-10 -3.905857435831940e-0814 -1.071903231878940e-10 1.356481292566670e-0815 7.087360088039800e-11 -4.912140911093550e-0916 -1.797818306057740e-11 1.854288283705740e-0917 1.002649821269050e-12 -7.243408051055730e-1018 1.130725736108390e-12 2.888927287597270e-1019 -5.251498162236730e-13 -1.156809097791070e-1020 4.568355593550080e-1121 -1.746849139303900e-1122 6.337718113837970e-1223 -2.123287366692640e-1224 6.058538275109010e-139.985870051144820e-01 k k k k -2.078549151656860e+00 -1.568565612274160e+0069 able 13 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.130510092436490e-03 2.226333345761950e-02 1.988982349173310e-011 1.417002017122940e-03 1.019151935447450e-02 8.748062377822970e-022 2.360674641499860e-05 2.673461828675060e-03 1.322178512127460e-023 -6.764554780577100e-06 7.541005867012190e-04 1.783172989082840e-054 1.073524111115770e-06 1.825724014495570e-04 -1.742061549930610e-045 -2.165240827636890e-07 5.070406265568330e-05 1.493474602053700e-066 5.714805445297300e-08 1.156695182629020e-05 3.359055560341550e-067 -1.802784817620440e-08 3.413930484845420e-06 5.333580684343860e-088 6.351907282247530e-09 6.943759577207300e-07 -5.521667146922240e-089 -2.410001034673690e-09 2.411908456774400e-07 -2.276235131191460e-0910 9.649353406509410e-10 3.679790048414060e-08 -2.893556379100810e-1011 -4.026554373758090e-10 1.888991234749750e-08 -6.436850207940650e-1112 1.736498040397040e-10 1.106003127127480e-09 2.350168269709410e-1113 -7.693121262340320e-11 1.752401085721140e-09 2.321358068727610e-1114 3.485269197769610e-11 -1.364762944049400e-10 2.636593526158320e-1215 -1.608506662519110e-11 2.000879783288790e-10 -1.139207052258690e-1216 7.530478144378150e-12 -4.354349576970780e-11 -7.314416583435220e-1317 -3.545863047530400e-12 2.738074123398660e-1118 1.631081082539700e-12 -8.687479490879170e-1219 -6.387767991337960e-13 4.196188738890250e-1220 -1.521565690726940e-1221 5.748616824737580e-13 u k k k k k k -1.150665798108190e+00 1.000000000000000e+00 -9.422665062905070e-0170 able 13 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 1.167088563196570e+00 5.557949689428240e+001 4.028821977072450e-01 1.804403214930680e+002 1.589726864878320e-02 2.001076313509200e-023 -2.776444180529080e-03 -4.157908839745580e-034 5.440877740378380e-04 9.533639401005070e-045 -1.067963570631740e-04 -2.298471294547880e-046 1.602535986540860e-05 5.706424934572420e-057 3.639164819106400e-07 -1.443342632426860e-058 -1.502345300041770e-06 3.696661627610680e-069 6.684757371238390e-07 -9.553281498410910e-0710 -1.716295249441850e-07 2.486726588153440e-0711 1.635394935133220e-08 -6.518766056569640e-0812 8.892938254607350e-09 1.724037249294330e-0813 -5.718412804481970e-09 -4.621243803810750e-0914 1.784898899321040e-09 1.266095041880910e-0915 -2.594370983148820e-10 -3.591505441133870e-1016 -6.261300169685750e-11 1.071702367595880e-1017 5.827210983507840e-11 -3.409479146600350e-1118 -2.189105354284160e-11 1.160098934787360e-1119 4.514597736601780e-12 -4.170286799495250e-1220 -6.083212355274480e-14 1.531989721713420e-1221 -5.089272134222670e-139.998315799958800e-01 k k k k -1.885294461971110e+00 -1.777456775656590e+0071 able 14 Chebyshev coefficients c n for h = 1 / . n left central middle0 8.320108386176220e-04 1.028289222803410e-02 1.796716788945920e-011 3.797922844843720e-04 6.051387649842750e-03 9.215513210293030e-022 6.119152516315670e-06 2.386515701734710e-03 1.798443115218630e-023 -2.216499535843730e-06 9.149383182711970e-04 3.536164396647120e-044 3.549796569970470e-07 3.260804312989750e-04 -3.290230257519100e-045 -6.909777679284210e-08 1.174302244391960e-04 -9.190331115977200e-066 1.800406773387920e-08 4.046558068563540e-05 8.941392960865100e-067 -5.727807832664590e-09 1.430511004864880e-05 4.569100427021660e-078 2.048306829365090e-09 4.845772394724280e-06 -1.898690252517320e-079 -7.892933375352240e-10 1.708561223033980e-06 -1.470693084569350e-0810 3.208272429161230e-10 5.709919634096540e-07 9.061517318686090e-1011 -1.358581479450610e-10 2.027725977295490e-07 -4.996134827280970e-1012 5.944136353935260e-11 6.667913363885630e-08 -3.085715272976990e-1113 -2.671169627505600e-11 2.408641753810920e-08 8.541875621658270e-1114 1.227327977365980e-11 7.729248497752890e-09 3.049756612877180e-1115 -5.743936308401000e-12 2.879366774439790e-09 7.727805117204190e-1316 2.726212185552570e-12 8.872469780543650e-10 -3.534457536346370e-1217 -1.300587961514960e-12 3.484429145625870e-10 -1.357036283678540e-1218 6.052241498351530e-13 1.001335286917030e-10 4.056653202382400e-1419 -2.390157663283510e-13 4.301663419431590e-11 2.166251245948460e-1320 1.093494057671210e-1121 5.462280485697060e-1222 1.109419945826560e-1223 6.264905101784660e-13 u k k k k k k -1.142970216093080e+00 9.999999999999990e-01 -6.895558541579670e-0172 able 14 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.761057883039990e-01 5.196073467821080e+001 3.017284864244540e-01 1.831141920349900e+002 1.166946937938660e-02 2.543065037748800e-023 -1.687052247930830e-03 -5.741796704905520e-034 2.745955374164150e-04 1.425454072455170e-035 -4.255962891388790e-05 -3.709730129757980e-046 3.654738784807000e-06 9.914921759807340e-057 9.678759239832630e-07 -2.692978838795350e-058 -5.962323527180660e-07 7.390951335687150e-069 1.619073566570090e-07 -2.044323535428510e-0610 -2.044848058779350e-08 5.700139242154520e-0711 -3.557663286168380e-09 -1.608555974873110e-0712 2.792106872070250e-09 4.637422326004230e-0813 -8.014791797406380e-10 -1.388223005657000e-0814 9.886383182987470e-11 4.408914855308610e-0915 2.338331909817720e-11 -1.513613329915950e-0916 -1.720362865683670e-11 5.636677579981080e-1017 5.096915629190340e-12 -2.238514411774360e-1018 -7.701281537440130e-13 9.204119152754320e-1119 -3.802089545459430e-1120 1.539622910176960e-1121 -5.993091125355200e-1222 2.193183001782910e-1223 -6.910989846330220e-139.997667653179200e-01 k k k k -2.168616210929900e+00 -1.650521806358000e+0073 able 15 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.342404855864130e-04 5.287504606339480e-03 1.679802218836370e-011 1.582119045977740e-04 3.609167042053250e-03 9.425159027828390e-022 -4.034459826869400e-06 1.936145076792520e-03 2.146641276528250e-023 -1.781674777546940e-06 9.633501098384940e-04 7.761771414171220e-044 6.111894067922420e-07 4.586062099584780e-04 -4.715700923865780e-045 -1.616413643230570e-07 2.128565367871240e-04 -2.911833910065400e-056 4.535507820139480e-08 9.711724060740940e-05 1.543681974168120e-057 -1.504566368490170e-08 4.380458926575550e-05 1.397441335389570e-068 5.916977370301490e-09 1.959490913685950e-05 -3.947158787904670e-079 -2.631104077272530e-09 8.713044455946110e-06 -4.582398590716490e-0810 1.265674285428750e-09 3.857017778878410e-06 4.537121001350180e-0911 -6.412984582939380e-10 1.701669482580570e-06 -8.587058106556170e-1012 3.373488093146030e-10 7.488328990975590e-07 -3.395277343066420e-1013 -1.827196103322590e-10 3.288850425725570e-07 1.497940638456500e-1014 1.013577890763340e-10 1.442278857066740e-07 9.047367570431180e-1115 -5.736165706600700e-11 6.317620038821080e-08 1.532712463574530e-1116 3.301900838213170e-11 2.764873413720030e-08 -5.899234983630290e-1217 -1.928175236669560e-11 1.209228320347630e-08 -4.759599480764640e-1218 1.139207390775470e-11 5.285996098982320e-09 -9.845099825540210e-1319 -6.784952996564550e-12 2.309885154214620e-09 3.353301735326350e-1320 4.046041780101790e-12 1.009129182529960e-0921 -2.377799356621700e-12 4.407929681530880e-1022 1.318077536426880e-12 1.925241011297400e-1023 -5.870490384521500e-13 8.408559765252580e-1124 3.672433365477270e-1125 1.603764366665520e-1126 6.998237196315440e-1227 3.040158479208750e-1228 1.288881492848470e-1229 4.729004482220720e-13 u k k k k k k -1.055332320056090e+00 1.000000000000000e+00 -5.434844889094470e-0174 able 15 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.761146326247330e-01 5.440971360286990e+001 3.016928648060250e-01 1.956911680269630e+002 1.167231718705550e-02 2.774260925258900e-023 -1.651276452041110e-03 -6.436890817427110e-034 2.666875570608340e-04 1.641299885337880e-035 -4.247191460497490e-05 -4.384480040071970e-046 4.247919767558590e-06 1.201938574075200e-047 7.240192379129270e-07 -3.345172414930230e-058 -5.468504974426770e-07 9.394648359883000e-069 1.621838761986780e-07 -2.653557174889410e-0610 -2.469682517941960e-08 7.531228230971810e-0711 -1.846812979021160e-09 -2.152447292666580e-0712 2.453530993347940e-09 6.237985768142160e-0813 -8.104795070564450e-10 -1.859198071101320e-0814 1.328212021134190e-10 5.827250350019090e-0915 9.746012118699450e-12 -1.970501685911350e-0916 -1.445305942484290e-11 7.299803884539640e-1017 5.164008080182300e-12 -2.939243341028580e-1018 -1.029938970952770e-12 1.250293085781470e-1019 -5.430549502597070e-1120 2.340535807434650e-1121 -9.802247791887960e-1222 3.924623772975130e-1223 -1.472095272217630e-1224 4.733000270115350e-132526272829 9.999145446729710e-01 k k k k -2.179745438075220e+00 -1.614273003898510e+0075 able 16 Chebyshev coefficients c n for h = 1 / . n left central middle0 1.064576523939530e-04 7.757528496943130e-04 6.583911279090610e-021 4.967483886634590e-05 3.963182148041290e-04 4.630386572794740e-022 -2.574907330657650e-06 1.452820177102820e-04 1.757544141274230e-023 -5.303715403034560e-07 4.855954814287430e-05 3.315284363718880e-034 2.663871929167150e-07 1.531863558980870e-05 1.968319378136490e-055 -8.401958707782960e-08 4.676444520070540e-06 -1.223428005218250e-046 2.541840517940180e-08 1.394003778702580e-06 -1.557379115932900e-057 -8.458536138015860e-09 4.088490384098540e-07 3.231990249141240e-068 3.265291063082850e-09 1.184092050832930e-07 9.273005743030430e-079 -1.446267628365560e-09 3.397158988245310e-08 -3.854154049131550e-0810 7.070186386866000e-10 9.672491318170320e-09 -3.568282952834910e-0811 -3.686990455737450e-10 2.737461712291000e-09 -8.276509465005390e-1012 2.008214150700500e-10 7.708833517653440e-10 1.206706537754180e-0913 -1.129090416703240e-10 2.161982864910220e-10 9.225936702119910e-1114 6.508194177274510e-11 6.042419184248020e-11 -3.617701898545360e-1115 -3.828935422940620e-11 1.683809589516920e-11 -4.590404543093130e-1216 2.291651606295020e-11 4.678693771234680e-12 9.605943640102260e-1317 -1.391339883329400e-11 1.290808850907550e-12 1.614097426563680e-1318 8.543046330402440e-12 3.320491288954080e-1319 -5.282416801082780e-1220 3.263505722460030e-1221 -1.979537858186990e-1222 1.125706972359250e-1223 -5.098805064619270e-1324252627282930 u k k k k k k -1.042176040321580e+00 1.000000000000000e+00 2.136086687920960e-0176 able 16 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 5.721713263001810e-01 5.012707101453440e+001 2.017820038440820e-01 2.062096651831810e+002 1.377375013917080e-02 3.999684930491530e-023 -1.806370775567440e-03 -1.094915283125730e-024 1.523593047177300e-04 3.280050342478840e-035 2.295752543436930e-05 -1.026359592164560e-036 -1.211645711839800e-05 3.290839108300630e-047 1.558915013492110e-06 -1.072510048733340e-048 3.379002506728430e-07 3.549425811224860e-059 -1.589627733030860e-07 -1.199878274669100e-0510 8.281053242094140e-09 4.199796533313170e-0611 9.754591213299930e-09 -1.552499344038120e-0612 -2.587101485419130e-09 6.174229919308370e-0713 -2.654922061326020e-10 -2.654025845860760e-0714 2.854986006586110e-10 1.213436337756580e-0715 -4.288462498279570e-11 -5.724458644126490e-0816 -1.614286040010390e-11 2.696385391883720e-0817 8.227664210952470e-12 -1.231399370142220e-0818 -8.469562416364850e-13 5.303258910110760e-0919 -2.080169945416400e-0920 6.955732723896000e-1021 -1.600540556356640e-1022 -1.256262769207190e-1123 4.730496708984270e-1124 -3.958827080866930e-1125 2.486066180938770e-1126 -1.342780201739020e-1127 6.510864817550600e-1228 -2.877910389935850e-1229 1.155822468250220e-1230 -3.885406981704210e-139.997327929473050e-01 k k k k -1.873568041871660e+00 -1.410192678438050e+0077 able 17 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.518991274756410e-05 8.392520820142320e-04 3.802327783487690e-021 1.590990829653510e-05 6.431136826076690e-04 2.293406644160830e-022 -1.491507257739220e-06 4.085596936397020e-04 6.950796412254640e-033 -9.066540515692110e-08 2.399296017984750e-04 1.080933113927390e-034 1.118185077651380e-07 1.344982201802300e-04 3.968276530040410e-055 -4.689629675507760e-08 7.316918708694770e-05 -1.510396050891140e-056 1.672578243190620e-08 3.898020499880180e-05 -2.164545910662930e-067 -5.962483120983330e-09 2.045075653316820e-05 1.031238628209580e-078 2.299928220409480e-09 1.060530105647130e-05 4.715822375760130e-089 -9.951311785026230e-10 5.449951689909780e-06 1.338195202962630e-0910 4.813973704272410e-10 2.780434074253660e-06 -7.162524074055580e-1011 -2.538921373245920e-10 1.410169685110300e-06 -5.838646258223070e-1112 1.421639472958470e-10 7.117308974598160e-07 9.035922879320330e-1213 -8.292852778673330e-11 3.577606972109210e-07 1.324669429761400e-1214 4.981742953446680e-11 1.792155094219710e-07 -9.136939392295850e-1415 -3.060881416945410e-11 8.951262809172500e-0816 1.915173995253600e-11 4.459614066323360e-0817 -1.216412493952420e-11 2.216977555006020e-0818 7.820244746397910e-12 1.100014354771840e-0819 -5.072097961571200e-12 5.448923967042200e-0920 3.302395224132890e-12 2.695162687572190e-0921 -2.138908948989580e-12 1.331357683924750e-0922 1.351588122026080e-12 6.569059660914260e-1023 -7.937121438141370e-13 3.237909926903940e-1024 3.670269446605750e-13 1.594506248721030e-1025 7.845610579728340e-1126 3.857391556723120e-1127 1.895058050737680e-1128 9.300136827782300e-1229 4.552865562504610e-1230 2.209843826831820e-1231 1.035453103819430e-1232 4.092881791896850e-13 u k k k k k k -1.029129856396840e+00 1.000000000000000e+00 4.634771910249610e-0178 able 17 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 4.996811103033990e-01 4.914258356400550e+001 2.283166265702920e-01 2.011738127244230e+002 2.514423513031150e-02 3.937356800757610e-023 -3.474198991631110e-03 -1.065266526129480e-024 5.077220214696990e-05 3.152291195579000e-035 1.613616422231280e-04 -9.738895553647930e-046 -3.880889586523480e-05 3.081786621477630e-047 -3.634159932107070e-06 -9.909792676876680e-058 3.510602114428830e-06 3.236113469325020e-059 -1.965557011348590e-07 -1.080367954217740e-0510 -2.893679425553000e-07 3.742488200776480e-0611 4.919338342933830e-08 -1.374128809584770e-0612 2.766224437511820e-08 5.450001386646280e-0713 -9.150633315570560e-09 -2.341737771120620e-0714 -2.629338358854130e-09 1.069346503420290e-0715 1.702910382225310e-09 -5.021967626980490e-0816 1.193636414090970e-10 2.344515094335100e-0817 -2.861147205442400e-10 -1.056108154875420e-0818 3.711031444447810e-11 4.460714134159530e-0919 3.822894605668730e-11 -1.700913808398100e-0920 -1.452954868328450e-11 5.418507670597160e-1021 -2.818928436440390e-12 -1.082511209350420e-1022 3.052195946657310e-12 -2.332308071729190e-1123 -4.763636711850320e-13 4.458678049756930e-1124 -3.438522185303330e-1125 2.066800196185070e-1126 -1.078966568122660e-1127 5.070919968888780e-1228 -2.172493412014030e-1229 8.448217334599220e-1330 -2.756966775927990e-133132 9.998681694019440e-01 k k k k -1.323148927258810e+00 -1.420354956151160e+0079 able 18 Chebyshev coefficients c n for h = 1 / . n left central middle0 5.483749876010110e-06 6.081052138201630e-05 5.473769059145160e-021 2.486444309865320e-06 3.802911084933040e-05 4.351538755149600e-022 -2.246254082499010e-07 1.823368632895480e-05 2.198780054496340e-023 -1.578587968151560e-08 7.946990081244110e-06 6.662984075703530e-034 1.732923293003170e-08 3.273356396625710e-06 7.907173467155090e-045 -7.081399642607440e-09 1.301100356483330e-06 -2.243067264619820e-046 2.482198403770580e-09 5.043448241124090e-07 -1.032261735179890e-047 -8.749831841180770e-10 1.919418161993120e-07 -5.350827374548080e-068 3.358446096487070e-10 7.202571800642060e-08 6.032870474106270e-069 -1.452715547964040e-10 2.672894811858390e-08 1.389775608451330e-0610 7.036053028305950e-11 9.830459878805920e-09 -1.667794341081260e-0711 -3.712078515936340e-11 3.588823452512350e-09 -1.098049445185060e-0712 2.075613750341240e-11 1.302082454421860e-09 -3.990118701505280e-0913 -1.206646596058050e-11 4.699379616814380e-10 6.210083081288950e-0914 7.203733700544880e-12 1.688418171221930e-10 9.633344895225680e-1015 -4.375271295344380e-12 6.042392709176170e-11 -2.609783367202760e-1016 2.674170121231750e-12 2.154618744858790e-11 -8.315421351488390e-1117 -1.612760826306620e-12 7.648543523947050e-12 6.884100040797630e-1218 9.148126116574220e-13 2.675596311035590e-12 5.334393632012140e-1219 -4.140244318492860e-13 8.418748326551260e-13 8.096094478676190e-14202122232425262728 u k k k k k k -1.030074288714920e+00 1.000000000000000e+00 4.184273889203900e-0180 able 18 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 5.722628305041320e-01 4.619812351486580e+001 2.017761232436850e-01 1.860589481022910e+002 1.372549559855640e-02 3.705233566867140e-023 -1.799486944230340e-03 -9.679912077752170e-034 1.548179312896890e-04 2.763008630922080e-035 2.188831960478410e-05 -8.227338295193510e-046 -1.205509962337280e-05 2.508114545966230e-047 1.626588892457300e-06 -7.771817988229730e-058 3.184920827863750e-07 2.450094311357910e-059 -1.604199428018640e-07 -7.933144007790290e-0610 1.026914924420960e-08 2.688223932510620e-0611 9.459364972251830e-09 -9.765416694004470e-0712 -2.712392941794810e-09 3.865875322747190e-0713 -2.064497813254320e-10 -1.657778704727940e-0714 2.839303432413100e-10 7.480889417663450e-0815 -4.871222002206250e-11 -3.419915104148750e-0816 -1.444200355589120e-11 1.528919983568170e-0817 8.372113940734360e-12 -6.479029251360280e-0918 -8.215034741114160e-13 2.513840140749050e-0919 -5.718274703852940e-13 -8.420176316921800e-1020 2.056434736961990e-1021 -1.906441397640450e-1222 -4.118010901086650e-1123 3.573866647613250e-1124 -2.210862584205230e-1125 1.157843167001790e-1126 -5.392753508375620e-1227 2.255758034817050e-1228 -7.825720526313930e-139.999476900342120e-01 k k k2 k2
Chebyshev coefficients c n for P = 10 . n left tail right tail0 -5.946145460337920e+00 5.964470382846980e+001 3.556004911984500e+00 3.570041770711440e+002 -6.801894131410160e-01 6.865586194977590e-013 1.092510150797910e-01 1.110985275994380e-014 -1.307725205702110e-02 1.349685679309470e-025 1.160328523663860e-03 1.237761453545620e-036 -8.240107479627320e-05 9.434859113712600e-057 5.926003936265940e-06 7.508635778142890e-068 -3.827985039853400e-07 5.673276390309780e-079 -4.939610258605130e-09 1.429761666233160e-0810 2.191431812357590e-09 -4.002728561697650e-1011 4.611932833470640e-10 6.107231293159660e-1012 -4.959733545874770e-11 6.167427914388110e-1113 -1.701165301087770e-11 -1.590293060974530e-1114 1.492500677577930e-12 -1.334659663440940e-1215 5.860252785293660e-13 6.358368604978550e-13 u k k -5.310898090226540e-01 5.311876095528130e-01 k k able 11 Chebyshev coefficients c n for h = 2 . n left left0 3.746732186396810e-01 1.246285546643250e+001 1.195997564921540e-01 3.358246308532060e-012 -8.070601474351940e-03 2.386732322681120e-023 1.339603001568510e-03 1.294898023275990e-024 -2.506080830464610e-04 2.637423242240120e-035 5.637743633696060e-05 1.091486949977880e-036 -1.413866007641720e-05 3.083088185622240e-047 3.869664142567620e-06 1.183343193111800e-048 -1.132969831139660e-06 3.838648490201040e-059 3.501617974093610e-07 1.440406811311990e-0510 -1.130976314574740e-07 5.002488355310480e-0611 3.788372056248530e-08 1.869987422785930e-0612 -1.308238855480900e-08 6.739900491698620e-0713 4.635781671551490e-09 2.526970560301390e-0714 -1.679323548784110e-09 9.311688370921460e-0815 6.200204087490220e-10 3.508754493020290e-0816 -2.327351020956820e-10 1.311638643142900e-0817 8.863586972126330e-11 4.968660909382380e-0918 -3.418842923801670e-11 1.875958145331950e-0919 1.333107561719920e-11 7.141781072249230e-1020 -5.234957827868200e-12 2.716166269988070e-1021 2.035050527498960e-12 1.038650818883500e-1022 -7.009107859412300e-13 3.972098592976600e-1123 1.524237223546780e-1124 5.838842289005870e-1225 2.206347821363880e-1226 7.400818875262400e-13 u k k k k -1.301445639726460e+00 -1.662175466937440e+0066 able 11 (cont.) Chebyshev coefficients c n for h = 2 . n middle right tail0 3.527966142386780e+00 1.003328309377590e+011 7.472142334253430e-01 2.496706503355630e+002 -1.333644417363060e-02 -1.589949448670880e-023 2.622018509548740e-03 3.113590178926820e-034 -6.027577883155430e-04 -6.963073964368400e-045 1.515242268297150e-04 1.678769292503920e-046 -4.036373098988620e-05 -4.250114645375870e-057 1.119644049107180e-05 1.113756440553480e-058 -3.199289309119440e-06 -2.994727316140500e-069 9.349240809409510e-07 8.214913272814210e-0710 -2.779917579878980e-07 -2.289772183834370e-0711 8.378973634959820e-08 6.466492239429990e-0812 -2.552869970129400e-08 -1.846256500319380e-0813 7.845488794437930e-09 5.320336235290850e-0914 -2.428118880696940e-09 -1.545402456085360e-0915 7.559005461053680e-10 4.520062644698680e-1016 -2.364994613022200e-10 -1.330078378584710e-1017 7.431800136657720e-11 3.934888781070520e-1118 -2.344374813365950e-11 -1.169587411803260e-1119 7.415598838082110e-12 3.489114825460500e-1220 -2.333558791113770e-12 -1.037689978383220e-1221 6.745346171083730e-13 2.859557405213800e-132223242526 9.997923134230980e-01 k k k k -6.514171570857550e-01 -1.450272673332570e+0067 able 12 Chebyshev coefficients c n for h = 1 / . n left central middle0 1.551440631887020e-02 5.965851817537560e-02 2.365922039769380e-011 7.154056831584140e-03 1.701537131814850e-02 7.509528018047980e-022 1.120793580526900e-04 2.597262497204220e-03 6.749060031298230e-033 -2.767589127786680e-05 4.709183006272860e-04 -9.756034732487190e-054 5.310776527089690e-06 7.145518311775060e-05 -4.546713797013440e-055 -1.242484981459680e-06 1.330260630541950e-05 2.265934249688400e-066 3.545760586969740e-07 1.967551836645490e-06 3.137406427278980e-077 -1.173714662110980e-07 3.949016650674680e-07 1.344476627067420e-088 4.313888580276150e-08 5.420695609123570e-08 -8.390259114557710e-099 -1.709037630074550e-08 1.243550067642890e-08 7.380356092377570e-1010 7.159028761307770e-09 1.439656776199120e-09 -2.253776759627590e-1011 -3.130641321239270e-09 4.213671249996360e-10 4.735083288679870e-1112 1.416650107438240e-09 3.320038533665110e-11 -6.226333936166760e-1213 -6.591639233672150e-10 1.566391114883240e-11 2.369177040131130e-1214 3.138939220131410e-10 4.052720676134970e-13 -5.354588204982440e-1315 -1.524291404102210e-1016 7.526974215608190e-1117 -3.770866706212000e-1118 1.912726184698180e-1119 -9.801629544246880e-1220 5.055418302750250e-1221 -2.598617692870460e-1222 1.287201224090770e-1223 -5.308318388467400e-1324 u k k k k k k -1.121868009800680e+00 1.000000000000000e+00 -1.516097941825860e+0068 able 12 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.766531067045780e-01 5.548556192934720e+001 3.018335097399800e-01 2.006312378503930e+002 1.137764493492600e-02 2.401553462574660e-023 -1.796551902064940e-03 -5.827238250950370e-034 2.961890615175630e-04 1.564108444981830e-035 -3.906140490786650e-05 -4.425250341488140e-046 3.228136981746220e-07 1.292806696330280e-047 1.990381470117290e-06 -3.860207865611520e-058 -7.180791153931000e-07 1.171917601240410e-059 1.237304075899040e-07 -3.609533592707940e-0610 4.797867870237380e-09 1.128193509794600e-0611 -1.022283804766720e-08 -3.587762176720570e-0712 3.242766128356910e-09 1.166622015545870e-0713 -3.934089138513020e-10 -3.905857435831940e-0814 -1.071903231878940e-10 1.356481292566670e-0815 7.087360088039800e-11 -4.912140911093550e-0916 -1.797818306057740e-11 1.854288283705740e-0917 1.002649821269050e-12 -7.243408051055730e-1018 1.130725736108390e-12 2.888927287597270e-1019 -5.251498162236730e-13 -1.156809097791070e-1020 4.568355593550080e-1121 -1.746849139303900e-1122 6.337718113837970e-1223 -2.123287366692640e-1224 6.058538275109010e-139.985870051144820e-01 k k k k -2.078549151656860e+00 -1.568565612274160e+0069 able 13 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.130510092436490e-03 2.226333345761950e-02 1.988982349173310e-011 1.417002017122940e-03 1.019151935447450e-02 8.748062377822970e-022 2.360674641499860e-05 2.673461828675060e-03 1.322178512127460e-023 -6.764554780577100e-06 7.541005867012190e-04 1.783172989082840e-054 1.073524111115770e-06 1.825724014495570e-04 -1.742061549930610e-045 -2.165240827636890e-07 5.070406265568330e-05 1.493474602053700e-066 5.714805445297300e-08 1.156695182629020e-05 3.359055560341550e-067 -1.802784817620440e-08 3.413930484845420e-06 5.333580684343860e-088 6.351907282247530e-09 6.943759577207300e-07 -5.521667146922240e-089 -2.410001034673690e-09 2.411908456774400e-07 -2.276235131191460e-0910 9.649353406509410e-10 3.679790048414060e-08 -2.893556379100810e-1011 -4.026554373758090e-10 1.888991234749750e-08 -6.436850207940650e-1112 1.736498040397040e-10 1.106003127127480e-09 2.350168269709410e-1113 -7.693121262340320e-11 1.752401085721140e-09 2.321358068727610e-1114 3.485269197769610e-11 -1.364762944049400e-10 2.636593526158320e-1215 -1.608506662519110e-11 2.000879783288790e-10 -1.139207052258690e-1216 7.530478144378150e-12 -4.354349576970780e-11 -7.314416583435220e-1317 -3.545863047530400e-12 2.738074123398660e-1118 1.631081082539700e-12 -8.687479490879170e-1219 -6.387767991337960e-13 4.196188738890250e-1220 -1.521565690726940e-1221 5.748616824737580e-13 u k k k k k k -1.150665798108190e+00 1.000000000000000e+00 -9.422665062905070e-0170 able 13 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 1.167088563196570e+00 5.557949689428240e+001 4.028821977072450e-01 1.804403214930680e+002 1.589726864878320e-02 2.001076313509200e-023 -2.776444180529080e-03 -4.157908839745580e-034 5.440877740378380e-04 9.533639401005070e-045 -1.067963570631740e-04 -2.298471294547880e-046 1.602535986540860e-05 5.706424934572420e-057 3.639164819106400e-07 -1.443342632426860e-058 -1.502345300041770e-06 3.696661627610680e-069 6.684757371238390e-07 -9.553281498410910e-0710 -1.716295249441850e-07 2.486726588153440e-0711 1.635394935133220e-08 -6.518766056569640e-0812 8.892938254607350e-09 1.724037249294330e-0813 -5.718412804481970e-09 -4.621243803810750e-0914 1.784898899321040e-09 1.266095041880910e-0915 -2.594370983148820e-10 -3.591505441133870e-1016 -6.261300169685750e-11 1.071702367595880e-1017 5.827210983507840e-11 -3.409479146600350e-1118 -2.189105354284160e-11 1.160098934787360e-1119 4.514597736601780e-12 -4.170286799495250e-1220 -6.083212355274480e-14 1.531989721713420e-1221 -5.089272134222670e-139.998315799958800e-01 k k k k -1.885294461971110e+00 -1.777456775656590e+0071 able 14 Chebyshev coefficients c n for h = 1 / . n left central middle0 8.320108386176220e-04 1.028289222803410e-02 1.796716788945920e-011 3.797922844843720e-04 6.051387649842750e-03 9.215513210293030e-022 6.119152516315670e-06 2.386515701734710e-03 1.798443115218630e-023 -2.216499535843730e-06 9.149383182711970e-04 3.536164396647120e-044 3.549796569970470e-07 3.260804312989750e-04 -3.290230257519100e-045 -6.909777679284210e-08 1.174302244391960e-04 -9.190331115977200e-066 1.800406773387920e-08 4.046558068563540e-05 8.941392960865100e-067 -5.727807832664590e-09 1.430511004864880e-05 4.569100427021660e-078 2.048306829365090e-09 4.845772394724280e-06 -1.898690252517320e-079 -7.892933375352240e-10 1.708561223033980e-06 -1.470693084569350e-0810 3.208272429161230e-10 5.709919634096540e-07 9.061517318686090e-1011 -1.358581479450610e-10 2.027725977295490e-07 -4.996134827280970e-1012 5.944136353935260e-11 6.667913363885630e-08 -3.085715272976990e-1113 -2.671169627505600e-11 2.408641753810920e-08 8.541875621658270e-1114 1.227327977365980e-11 7.729248497752890e-09 3.049756612877180e-1115 -5.743936308401000e-12 2.879366774439790e-09 7.727805117204190e-1316 2.726212185552570e-12 8.872469780543650e-10 -3.534457536346370e-1217 -1.300587961514960e-12 3.484429145625870e-10 -1.357036283678540e-1218 6.052241498351530e-13 1.001335286917030e-10 4.056653202382400e-1419 -2.390157663283510e-13 4.301663419431590e-11 2.166251245948460e-1320 1.093494057671210e-1121 5.462280485697060e-1222 1.109419945826560e-1223 6.264905101784660e-13 u k k k k k k -1.142970216093080e+00 9.999999999999990e-01 -6.895558541579670e-0172 able 14 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.761057883039990e-01 5.196073467821080e+001 3.017284864244540e-01 1.831141920349900e+002 1.166946937938660e-02 2.543065037748800e-023 -1.687052247930830e-03 -5.741796704905520e-034 2.745955374164150e-04 1.425454072455170e-035 -4.255962891388790e-05 -3.709730129757980e-046 3.654738784807000e-06 9.914921759807340e-057 9.678759239832630e-07 -2.692978838795350e-058 -5.962323527180660e-07 7.390951335687150e-069 1.619073566570090e-07 -2.044323535428510e-0610 -2.044848058779350e-08 5.700139242154520e-0711 -3.557663286168380e-09 -1.608555974873110e-0712 2.792106872070250e-09 4.637422326004230e-0813 -8.014791797406380e-10 -1.388223005657000e-0814 9.886383182987470e-11 4.408914855308610e-0915 2.338331909817720e-11 -1.513613329915950e-0916 -1.720362865683670e-11 5.636677579981080e-1017 5.096915629190340e-12 -2.238514411774360e-1018 -7.701281537440130e-13 9.204119152754320e-1119 -3.802089545459430e-1120 1.539622910176960e-1121 -5.993091125355200e-1222 2.193183001782910e-1223 -6.910989846330220e-139.997667653179200e-01 k k k k -2.168616210929900e+00 -1.650521806358000e+0073 able 15 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.342404855864130e-04 5.287504606339480e-03 1.679802218836370e-011 1.582119045977740e-04 3.609167042053250e-03 9.425159027828390e-022 -4.034459826869400e-06 1.936145076792520e-03 2.146641276528250e-023 -1.781674777546940e-06 9.633501098384940e-04 7.761771414171220e-044 6.111894067922420e-07 4.586062099584780e-04 -4.715700923865780e-045 -1.616413643230570e-07 2.128565367871240e-04 -2.911833910065400e-056 4.535507820139480e-08 9.711724060740940e-05 1.543681974168120e-057 -1.504566368490170e-08 4.380458926575550e-05 1.397441335389570e-068 5.916977370301490e-09 1.959490913685950e-05 -3.947158787904670e-079 -2.631104077272530e-09 8.713044455946110e-06 -4.582398590716490e-0810 1.265674285428750e-09 3.857017778878410e-06 4.537121001350180e-0911 -6.412984582939380e-10 1.701669482580570e-06 -8.587058106556170e-1012 3.373488093146030e-10 7.488328990975590e-07 -3.395277343066420e-1013 -1.827196103322590e-10 3.288850425725570e-07 1.497940638456500e-1014 1.013577890763340e-10 1.442278857066740e-07 9.047367570431180e-1115 -5.736165706600700e-11 6.317620038821080e-08 1.532712463574530e-1116 3.301900838213170e-11 2.764873413720030e-08 -5.899234983630290e-1217 -1.928175236669560e-11 1.209228320347630e-08 -4.759599480764640e-1218 1.139207390775470e-11 5.285996098982320e-09 -9.845099825540210e-1319 -6.784952996564550e-12 2.309885154214620e-09 3.353301735326350e-1320 4.046041780101790e-12 1.009129182529960e-0921 -2.377799356621700e-12 4.407929681530880e-1022 1.318077536426880e-12 1.925241011297400e-1023 -5.870490384521500e-13 8.408559765252580e-1124 3.672433365477270e-1125 1.603764366665520e-1126 6.998237196315440e-1227 3.040158479208750e-1228 1.288881492848470e-1229 4.729004482220720e-13 u k k k k k k -1.055332320056090e+00 1.000000000000000e+00 -5.434844889094470e-0174 able 15 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.761146326247330e-01 5.440971360286990e+001 3.016928648060250e-01 1.956911680269630e+002 1.167231718705550e-02 2.774260925258900e-023 -1.651276452041110e-03 -6.436890817427110e-034 2.666875570608340e-04 1.641299885337880e-035 -4.247191460497490e-05 -4.384480040071970e-046 4.247919767558590e-06 1.201938574075200e-047 7.240192379129270e-07 -3.345172414930230e-058 -5.468504974426770e-07 9.394648359883000e-069 1.621838761986780e-07 -2.653557174889410e-0610 -2.469682517941960e-08 7.531228230971810e-0711 -1.846812979021160e-09 -2.152447292666580e-0712 2.453530993347940e-09 6.237985768142160e-0813 -8.104795070564450e-10 -1.859198071101320e-0814 1.328212021134190e-10 5.827250350019090e-0915 9.746012118699450e-12 -1.970501685911350e-0916 -1.445305942484290e-11 7.299803884539640e-1017 5.164008080182300e-12 -2.939243341028580e-1018 -1.029938970952770e-12 1.250293085781470e-1019 -5.430549502597070e-1120 2.340535807434650e-1121 -9.802247791887960e-1222 3.924623772975130e-1223 -1.472095272217630e-1224 4.733000270115350e-132526272829 9.999145446729710e-01 k k k k -2.179745438075220e+00 -1.614273003898510e+0075 able 16 Chebyshev coefficients c n for h = 1 / . n left central middle0 1.064576523939530e-04 7.757528496943130e-04 6.583911279090610e-021 4.967483886634590e-05 3.963182148041290e-04 4.630386572794740e-022 -2.574907330657650e-06 1.452820177102820e-04 1.757544141274230e-023 -5.303715403034560e-07 4.855954814287430e-05 3.315284363718880e-034 2.663871929167150e-07 1.531863558980870e-05 1.968319378136490e-055 -8.401958707782960e-08 4.676444520070540e-06 -1.223428005218250e-046 2.541840517940180e-08 1.394003778702580e-06 -1.557379115932900e-057 -8.458536138015860e-09 4.088490384098540e-07 3.231990249141240e-068 3.265291063082850e-09 1.184092050832930e-07 9.273005743030430e-079 -1.446267628365560e-09 3.397158988245310e-08 -3.854154049131550e-0810 7.070186386866000e-10 9.672491318170320e-09 -3.568282952834910e-0811 -3.686990455737450e-10 2.737461712291000e-09 -8.276509465005390e-1012 2.008214150700500e-10 7.708833517653440e-10 1.206706537754180e-0913 -1.129090416703240e-10 2.161982864910220e-10 9.225936702119910e-1114 6.508194177274510e-11 6.042419184248020e-11 -3.617701898545360e-1115 -3.828935422940620e-11 1.683809589516920e-11 -4.590404543093130e-1216 2.291651606295020e-11 4.678693771234680e-12 9.605943640102260e-1317 -1.391339883329400e-11 1.290808850907550e-12 1.614097426563680e-1318 8.543046330402440e-12 3.320491288954080e-1319 -5.282416801082780e-1220 3.263505722460030e-1221 -1.979537858186990e-1222 1.125706972359250e-1223 -5.098805064619270e-1324252627282930 u k k k k k k -1.042176040321580e+00 1.000000000000000e+00 2.136086687920960e-0176 able 16 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 5.721713263001810e-01 5.012707101453440e+001 2.017820038440820e-01 2.062096651831810e+002 1.377375013917080e-02 3.999684930491530e-023 -1.806370775567440e-03 -1.094915283125730e-024 1.523593047177300e-04 3.280050342478840e-035 2.295752543436930e-05 -1.026359592164560e-036 -1.211645711839800e-05 3.290839108300630e-047 1.558915013492110e-06 -1.072510048733340e-048 3.379002506728430e-07 3.549425811224860e-059 -1.589627733030860e-07 -1.199878274669100e-0510 8.281053242094140e-09 4.199796533313170e-0611 9.754591213299930e-09 -1.552499344038120e-0612 -2.587101485419130e-09 6.174229919308370e-0713 -2.654922061326020e-10 -2.654025845860760e-0714 2.854986006586110e-10 1.213436337756580e-0715 -4.288462498279570e-11 -5.724458644126490e-0816 -1.614286040010390e-11 2.696385391883720e-0817 8.227664210952470e-12 -1.231399370142220e-0818 -8.469562416364850e-13 5.303258910110760e-0919 -2.080169945416400e-0920 6.955732723896000e-1021 -1.600540556356640e-1022 -1.256262769207190e-1123 4.730496708984270e-1124 -3.958827080866930e-1125 2.486066180938770e-1126 -1.342780201739020e-1127 6.510864817550600e-1228 -2.877910389935850e-1229 1.155822468250220e-1230 -3.885406981704210e-139.997327929473050e-01 k k k k -1.873568041871660e+00 -1.410192678438050e+0077 able 17 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.518991274756410e-05 8.392520820142320e-04 3.802327783487690e-021 1.590990829653510e-05 6.431136826076690e-04 2.293406644160830e-022 -1.491507257739220e-06 4.085596936397020e-04 6.950796412254640e-033 -9.066540515692110e-08 2.399296017984750e-04 1.080933113927390e-034 1.118185077651380e-07 1.344982201802300e-04 3.968276530040410e-055 -4.689629675507760e-08 7.316918708694770e-05 -1.510396050891140e-056 1.672578243190620e-08 3.898020499880180e-05 -2.164545910662930e-067 -5.962483120983330e-09 2.045075653316820e-05 1.031238628209580e-078 2.299928220409480e-09 1.060530105647130e-05 4.715822375760130e-089 -9.951311785026230e-10 5.449951689909780e-06 1.338195202962630e-0910 4.813973704272410e-10 2.780434074253660e-06 -7.162524074055580e-1011 -2.538921373245920e-10 1.410169685110300e-06 -5.838646258223070e-1112 1.421639472958470e-10 7.117308974598160e-07 9.035922879320330e-1213 -8.292852778673330e-11 3.577606972109210e-07 1.324669429761400e-1214 4.981742953446680e-11 1.792155094219710e-07 -9.136939392295850e-1415 -3.060881416945410e-11 8.951262809172500e-0816 1.915173995253600e-11 4.459614066323360e-0817 -1.216412493952420e-11 2.216977555006020e-0818 7.820244746397910e-12 1.100014354771840e-0819 -5.072097961571200e-12 5.448923967042200e-0920 3.302395224132890e-12 2.695162687572190e-0921 -2.138908948989580e-12 1.331357683924750e-0922 1.351588122026080e-12 6.569059660914260e-1023 -7.937121438141370e-13 3.237909926903940e-1024 3.670269446605750e-13 1.594506248721030e-1025 7.845610579728340e-1126 3.857391556723120e-1127 1.895058050737680e-1128 9.300136827782300e-1229 4.552865562504610e-1230 2.209843826831820e-1231 1.035453103819430e-1232 4.092881791896850e-13 u k k k k k k -1.029129856396840e+00 1.000000000000000e+00 4.634771910249610e-0178 able 17 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 4.996811103033990e-01 4.914258356400550e+001 2.283166265702920e-01 2.011738127244230e+002 2.514423513031150e-02 3.937356800757610e-023 -3.474198991631110e-03 -1.065266526129480e-024 5.077220214696990e-05 3.152291195579000e-035 1.613616422231280e-04 -9.738895553647930e-046 -3.880889586523480e-05 3.081786621477630e-047 -3.634159932107070e-06 -9.909792676876680e-058 3.510602114428830e-06 3.236113469325020e-059 -1.965557011348590e-07 -1.080367954217740e-0510 -2.893679425553000e-07 3.742488200776480e-0611 4.919338342933830e-08 -1.374128809584770e-0612 2.766224437511820e-08 5.450001386646280e-0713 -9.150633315570560e-09 -2.341737771120620e-0714 -2.629338358854130e-09 1.069346503420290e-0715 1.702910382225310e-09 -5.021967626980490e-0816 1.193636414090970e-10 2.344515094335100e-0817 -2.861147205442400e-10 -1.056108154875420e-0818 3.711031444447810e-11 4.460714134159530e-0919 3.822894605668730e-11 -1.700913808398100e-0920 -1.452954868328450e-11 5.418507670597160e-1021 -2.818928436440390e-12 -1.082511209350420e-1022 3.052195946657310e-12 -2.332308071729190e-1123 -4.763636711850320e-13 4.458678049756930e-1124 -3.438522185303330e-1125 2.066800196185070e-1126 -1.078966568122660e-1127 5.070919968888780e-1228 -2.172493412014030e-1229 8.448217334599220e-1330 -2.756966775927990e-133132 9.998681694019440e-01 k k k k -1.323148927258810e+00 -1.420354956151160e+0079 able 18 Chebyshev coefficients c n for h = 1 / . n left central middle0 5.483749876010110e-06 6.081052138201630e-05 5.473769059145160e-021 2.486444309865320e-06 3.802911084933040e-05 4.351538755149600e-022 -2.246254082499010e-07 1.823368632895480e-05 2.198780054496340e-023 -1.578587968151560e-08 7.946990081244110e-06 6.662984075703530e-034 1.732923293003170e-08 3.273356396625710e-06 7.907173467155090e-045 -7.081399642607440e-09 1.301100356483330e-06 -2.243067264619820e-046 2.482198403770580e-09 5.043448241124090e-07 -1.032261735179890e-047 -8.749831841180770e-10 1.919418161993120e-07 -5.350827374548080e-068 3.358446096487070e-10 7.202571800642060e-08 6.032870474106270e-069 -1.452715547964040e-10 2.672894811858390e-08 1.389775608451330e-0610 7.036053028305950e-11 9.830459878805920e-09 -1.667794341081260e-0711 -3.712078515936340e-11 3.588823452512350e-09 -1.098049445185060e-0712 2.075613750341240e-11 1.302082454421860e-09 -3.990118701505280e-0913 -1.206646596058050e-11 4.699379616814380e-10 6.210083081288950e-0914 7.203733700544880e-12 1.688418171221930e-10 9.633344895225680e-1015 -4.375271295344380e-12 6.042392709176170e-11 -2.609783367202760e-1016 2.674170121231750e-12 2.154618744858790e-11 -8.315421351488390e-1117 -1.612760826306620e-12 7.648543523947050e-12 6.884100040797630e-1218 9.148126116574220e-13 2.675596311035590e-12 5.334393632012140e-1219 -4.140244318492860e-13 8.418748326551260e-13 8.096094478676190e-14202122232425262728 u k k k k k k -1.030074288714920e+00 1.000000000000000e+00 4.184273889203900e-0180 able 18 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 5.722628305041320e-01 4.619812351486580e+001 2.017761232436850e-01 1.860589481022910e+002 1.372549559855640e-02 3.705233566867140e-023 -1.799486944230340e-03 -9.679912077752170e-034 1.548179312896890e-04 2.763008630922080e-035 2.188831960478410e-05 -8.227338295193510e-046 -1.205509962337280e-05 2.508114545966230e-047 1.626588892457300e-06 -7.771817988229730e-058 3.184920827863750e-07 2.450094311357910e-059 -1.604199428018640e-07 -7.933144007790290e-0610 1.026914924420960e-08 2.688223932510620e-0611 9.459364972251830e-09 -9.765416694004470e-0712 -2.712392941794810e-09 3.865875322747190e-0713 -2.064497813254320e-10 -1.657778704727940e-0714 2.839303432413100e-10 7.480889417663450e-0815 -4.871222002206250e-11 -3.419915104148750e-0816 -1.444200355589120e-11 1.528919983568170e-0817 8.372113940734360e-12 -6.479029251360280e-0918 -8.215034741114160e-13 2.513840140749050e-0919 -5.718274703852940e-13 -8.420176316921800e-1020 2.056434736961990e-1021 -1.906441397640450e-1222 -4.118010901086650e-1123 3.573866647613250e-1124 -2.210862584205230e-1125 1.157843167001790e-1126 -5.392753508375620e-1227 2.255758034817050e-1228 -7.825720526313930e-139.999476900342120e-01 k k k2 k2 -1.877416447914540e+00 -1.452884710450130e+0081 able 19 Chebyshev coefficients c n for h = 1 / . n left central middle0 1.702183541035500e-06 4.683227357260460e-05 5.536218616018370e-021 7.420902874786450e-07 3.703626675908400e-05 4.374721584420760e-022 -9.865066726948370e-08 2.459788457587220e-05 2.175988835715660e-023 2.116960181208090e-09 1.509706997971100e-05 6.418561137830710e-034 5.497757664443220e-09 8.833813587467830e-06 7.077802624591750e-045 -3.075679409634450e-09 5.007785871509330e-06 -2.234688941525440e-046 1.295461350315470e-09 2.775387397751700e-06 -9.419306294251860e-057 -5.101921537607570e-10 1.512369265686630e-06 -3.360150759204170e-068 2.046470586232400e-10 8.133846678811100e-07 5.606215973335830e-069 -8.795781406451580e-11 4.329046589636930e-07 1.138744283796750e-0610 4.150674524923850e-11 2.284469855279910e-07 -1.744975342137310e-0711 -2.146959372032940e-11 1.197027748370630e-07 -9.175604868426630e-0812 1.195357510557570e-11 6.234932494901730e-08 -9.405807379254040e-1013 -7.007921488895000e-12 3.231076406508420e-08 5.303111119723840e-0914 4.242800470479290e-12 1.667069009839600e-08 6.408776988724730e-1015 -2.605046952660340e-12 8.568338007765290e-09 -2.359584489072000e-1016 1.583112277512220e-12 4.389129098487210e-09 -5.912784735389670e-1117 -9.048703171667920e-13 2.241645822968350e-09 7.661640011719350e-1218 4.117767633668480e-13 1.141836261366420e-09 3.874828519769490e-1219 5.802414152021560e-10 -1.166357191529640e-1320 2.942280531036410e-1021 1.489078538692040e-1022 7.522864040175020e-1123 3.794343618365480e-1124 1.910697269597880e-1125 9.603319488497670e-1226 4.810076959555690e-1227 2.385153926155570e-1228 1.138578899937890e-1229 4.560886440536520e-13 u k k k k k k -1.022376759557920e+00 1.000000000000000e+00 4.064120883541780e-0182 able 19 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 5.722747148899260e-01 4.619756015894430e+001 2.017753450395620e-01 1.860594275953970e+002 1.371924357276870e-02 3.708005345183760e-023 -1.798581149904210e-03 -9.684906797069420e-034 1.551213775007570e-04 2.763643547482780e-035 2.175254385332560e-05 -8.226383894790440e-046 -1.204645343665130e-05 2.506774823695980e-047 1.634965165513770e-06 -7.763711144862420e-058 3.159981622016720e-07 2.446026791697170e-059 -1.605777694467310e-07 -7.914262660093390e-0610 1.051874919029200e-08 2.679801742363770e-0611 9.419608354447810e-09 -9.728790947989900e-0712 -2.727465609582260e-09 3.850313981959300e-0713 -1.988937692998890e-10 -1.651364788520010e-0714 2.836178919653520e-10 7.455668628814360e-0815 -4.943002831236940e-11 -3.410750194426710e-0816 -1.421606720565170e-11 1.526049674013000e-0817 8.387073685132130e-12 -6.472921009013150e-0918 -8.503982177047610e-13 2.514557909430010e-0919 -5.640963743958310e-13 -8.439824101491250e-1020 2.072428649649560e-1021 -2.907032521718900e-1222 -4.063479916620080e-1123 3.546906613254100e-1124 -2.198580726676460e-1125 1.152667195366090e-1126 -5.372666511779410e-1227 2.248692740626720e-1228 -7.804828886326530e-1329 9.999739151276640e-01 k k k k -1.877882960729700e+00 -1.452971340318820e+0083 able 20 Chebyshev coefficients c n for h = 1 / . n left central middle0 8.340782562108640e-08 2.468455017350810e-06 2.280294691861460e-021 3.790348335769390e-08 1.808380820210540e-06 1.944720833702610e-022 6.055403556177730e-10 9.977786192716350e-07 1.206597650481480e-023 -2.457590587255690e-10 5.095262839987460e-07 5.365044701332920e-034 3.852931864752020e-11 2.442450448287350e-07 1.591201684585550e-035 -7.060509039767990e-12 1.138166487478510e-07 2.189680324488170e-046 1.745854017090550e-12 5.143505829362970e-08 -5.028795904628400e-057 -4.817127130663810e-13 2.294108192095970e-08 -3.473771288547360e-058 1.003514899934200e-08 -6.612748781444730e-069 4.362945792279890e-09 8.987877800239300e-0710 1.869145494199760e-09 7.769588194187460e-0711 7.993315810705090e-10 1.334497286164700e-0712 3.374445612260970e-10 -2.638905677337210e-0813 1.426920238747580e-10 -1.641293598763500e-0814 5.956301256742110e-11 -1.798371164119150e-0915 2.498973580154360e-11 8.162891792784310e-1016 1.032764886165960e-11 3.081840920792170e-1017 4.294120630111010e-12 5.391475739175650e-1218 1.719237245981870e-12 -2.108451751530640e-1119 6.120456681919280e-13 -4.569166977955490e-1220 5.618348229993250e-132122232425262728 u k k k k k k -1.151540996208610e+00 1.000000000000000e+00 7.708355493813890e-0184 able 20 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 4.997785332154780e-01 4.423666819273100e+001 2.283167712864420e-01 1.759820399395020e+002 2.509022440364520e-02 3.541998526121500e-023 -3.473302792894470e-03 -9.020823900527130e-034 5.642398807896150e-05 2.508301403041810e-035 1.601477839497730e-04 -7.271867741631950e-046 -3.917135579658680e-05 2.157837314850230e-047 -3.440824241326100e-06 -6.511087849455140e-058 3.519318737665770e-06 2.002320021389290e-059 -2.194212259839990e-07 -6.350484440175690e-0610 -2.877017875529090e-07 2.122832316166750e-0611 5.198973063531290e-08 -7.670810351604760e-0712 2.710916358293470e-08 3.034734463075500e-0713 -9.510243857620720e-09 -1.296599648285690e-0714 -2.488075646152630e-09 5.774831521471910e-0815 1.740766101870470e-09 -2.574755859857430e-0816 8.803780581176810e-11 1.108702976174020e-0817 -2.866253352551910e-10 -4.459918192937890e-0918 4.280148433910300e-11 1.605691932192970e-0919 3.704110753342700e-11 -4.731139230730710e-1020 -1.527178121204120e-11 7.876459425075560e-1121 -2.422082175650230e-12 2.822685782970870e-1122 3.088328322933470e-12 -3.938365394966460e-1123 -5.645136949758680e-13 2.737131256274080e-1124 -1.510876686567670e-1125 7.257145569467040e-1226 -3.126553351859840e-1227 1.213046040716080e-1228 -3.939886674131800e-139.999869750663500e-01 k k k2
Chebyshev coefficients c n for P = 10 . n left tail right tail0 -5.946145460337920e+00 5.964470382846980e+001 3.556004911984500e+00 3.570041770711440e+002 -6.801894131410160e-01 6.865586194977590e-013 1.092510150797910e-01 1.110985275994380e-014 -1.307725205702110e-02 1.349685679309470e-025 1.160328523663860e-03 1.237761453545620e-036 -8.240107479627320e-05 9.434859113712600e-057 5.926003936265940e-06 7.508635778142890e-068 -3.827985039853400e-07 5.673276390309780e-079 -4.939610258605130e-09 1.429761666233160e-0810 2.191431812357590e-09 -4.002728561697650e-1011 4.611932833470640e-10 6.107231293159660e-1012 -4.959733545874770e-11 6.167427914388110e-1113 -1.701165301087770e-11 -1.590293060974530e-1114 1.492500677577930e-12 -1.334659663440940e-1215 5.860252785293660e-13 6.358368604978550e-13 u k k -5.310898090226540e-01 5.311876095528130e-01 k k able 11 Chebyshev coefficients c n for h = 2 . n left left0 3.746732186396810e-01 1.246285546643250e+001 1.195997564921540e-01 3.358246308532060e-012 -8.070601474351940e-03 2.386732322681120e-023 1.339603001568510e-03 1.294898023275990e-024 -2.506080830464610e-04 2.637423242240120e-035 5.637743633696060e-05 1.091486949977880e-036 -1.413866007641720e-05 3.083088185622240e-047 3.869664142567620e-06 1.183343193111800e-048 -1.132969831139660e-06 3.838648490201040e-059 3.501617974093610e-07 1.440406811311990e-0510 -1.130976314574740e-07 5.002488355310480e-0611 3.788372056248530e-08 1.869987422785930e-0612 -1.308238855480900e-08 6.739900491698620e-0713 4.635781671551490e-09 2.526970560301390e-0714 -1.679323548784110e-09 9.311688370921460e-0815 6.200204087490220e-10 3.508754493020290e-0816 -2.327351020956820e-10 1.311638643142900e-0817 8.863586972126330e-11 4.968660909382380e-0918 -3.418842923801670e-11 1.875958145331950e-0919 1.333107561719920e-11 7.141781072249230e-1020 -5.234957827868200e-12 2.716166269988070e-1021 2.035050527498960e-12 1.038650818883500e-1022 -7.009107859412300e-13 3.972098592976600e-1123 1.524237223546780e-1124 5.838842289005870e-1225 2.206347821363880e-1226 7.400818875262400e-13 u k k k k -1.301445639726460e+00 -1.662175466937440e+0066 able 11 (cont.) Chebyshev coefficients c n for h = 2 . n middle right tail0 3.527966142386780e+00 1.003328309377590e+011 7.472142334253430e-01 2.496706503355630e+002 -1.333644417363060e-02 -1.589949448670880e-023 2.622018509548740e-03 3.113590178926820e-034 -6.027577883155430e-04 -6.963073964368400e-045 1.515242268297150e-04 1.678769292503920e-046 -4.036373098988620e-05 -4.250114645375870e-057 1.119644049107180e-05 1.113756440553480e-058 -3.199289309119440e-06 -2.994727316140500e-069 9.349240809409510e-07 8.214913272814210e-0710 -2.779917579878980e-07 -2.289772183834370e-0711 8.378973634959820e-08 6.466492239429990e-0812 -2.552869970129400e-08 -1.846256500319380e-0813 7.845488794437930e-09 5.320336235290850e-0914 -2.428118880696940e-09 -1.545402456085360e-0915 7.559005461053680e-10 4.520062644698680e-1016 -2.364994613022200e-10 -1.330078378584710e-1017 7.431800136657720e-11 3.934888781070520e-1118 -2.344374813365950e-11 -1.169587411803260e-1119 7.415598838082110e-12 3.489114825460500e-1220 -2.333558791113770e-12 -1.037689978383220e-1221 6.745346171083730e-13 2.859557405213800e-132223242526 9.997923134230980e-01 k k k k -6.514171570857550e-01 -1.450272673332570e+0067 able 12 Chebyshev coefficients c n for h = 1 / . n left central middle0 1.551440631887020e-02 5.965851817537560e-02 2.365922039769380e-011 7.154056831584140e-03 1.701537131814850e-02 7.509528018047980e-022 1.120793580526900e-04 2.597262497204220e-03 6.749060031298230e-033 -2.767589127786680e-05 4.709183006272860e-04 -9.756034732487190e-054 5.310776527089690e-06 7.145518311775060e-05 -4.546713797013440e-055 -1.242484981459680e-06 1.330260630541950e-05 2.265934249688400e-066 3.545760586969740e-07 1.967551836645490e-06 3.137406427278980e-077 -1.173714662110980e-07 3.949016650674680e-07 1.344476627067420e-088 4.313888580276150e-08 5.420695609123570e-08 -8.390259114557710e-099 -1.709037630074550e-08 1.243550067642890e-08 7.380356092377570e-1010 7.159028761307770e-09 1.439656776199120e-09 -2.253776759627590e-1011 -3.130641321239270e-09 4.213671249996360e-10 4.735083288679870e-1112 1.416650107438240e-09 3.320038533665110e-11 -6.226333936166760e-1213 -6.591639233672150e-10 1.566391114883240e-11 2.369177040131130e-1214 3.138939220131410e-10 4.052720676134970e-13 -5.354588204982440e-1315 -1.524291404102210e-1016 7.526974215608190e-1117 -3.770866706212000e-1118 1.912726184698180e-1119 -9.801629544246880e-1220 5.055418302750250e-1221 -2.598617692870460e-1222 1.287201224090770e-1223 -5.308318388467400e-1324 u k k k k k k -1.121868009800680e+00 1.000000000000000e+00 -1.516097941825860e+0068 able 12 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.766531067045780e-01 5.548556192934720e+001 3.018335097399800e-01 2.006312378503930e+002 1.137764493492600e-02 2.401553462574660e-023 -1.796551902064940e-03 -5.827238250950370e-034 2.961890615175630e-04 1.564108444981830e-035 -3.906140490786650e-05 -4.425250341488140e-046 3.228136981746220e-07 1.292806696330280e-047 1.990381470117290e-06 -3.860207865611520e-058 -7.180791153931000e-07 1.171917601240410e-059 1.237304075899040e-07 -3.609533592707940e-0610 4.797867870237380e-09 1.128193509794600e-0611 -1.022283804766720e-08 -3.587762176720570e-0712 3.242766128356910e-09 1.166622015545870e-0713 -3.934089138513020e-10 -3.905857435831940e-0814 -1.071903231878940e-10 1.356481292566670e-0815 7.087360088039800e-11 -4.912140911093550e-0916 -1.797818306057740e-11 1.854288283705740e-0917 1.002649821269050e-12 -7.243408051055730e-1018 1.130725736108390e-12 2.888927287597270e-1019 -5.251498162236730e-13 -1.156809097791070e-1020 4.568355593550080e-1121 -1.746849139303900e-1122 6.337718113837970e-1223 -2.123287366692640e-1224 6.058538275109010e-139.985870051144820e-01 k k k k -2.078549151656860e+00 -1.568565612274160e+0069 able 13 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.130510092436490e-03 2.226333345761950e-02 1.988982349173310e-011 1.417002017122940e-03 1.019151935447450e-02 8.748062377822970e-022 2.360674641499860e-05 2.673461828675060e-03 1.322178512127460e-023 -6.764554780577100e-06 7.541005867012190e-04 1.783172989082840e-054 1.073524111115770e-06 1.825724014495570e-04 -1.742061549930610e-045 -2.165240827636890e-07 5.070406265568330e-05 1.493474602053700e-066 5.714805445297300e-08 1.156695182629020e-05 3.359055560341550e-067 -1.802784817620440e-08 3.413930484845420e-06 5.333580684343860e-088 6.351907282247530e-09 6.943759577207300e-07 -5.521667146922240e-089 -2.410001034673690e-09 2.411908456774400e-07 -2.276235131191460e-0910 9.649353406509410e-10 3.679790048414060e-08 -2.893556379100810e-1011 -4.026554373758090e-10 1.888991234749750e-08 -6.436850207940650e-1112 1.736498040397040e-10 1.106003127127480e-09 2.350168269709410e-1113 -7.693121262340320e-11 1.752401085721140e-09 2.321358068727610e-1114 3.485269197769610e-11 -1.364762944049400e-10 2.636593526158320e-1215 -1.608506662519110e-11 2.000879783288790e-10 -1.139207052258690e-1216 7.530478144378150e-12 -4.354349576970780e-11 -7.314416583435220e-1317 -3.545863047530400e-12 2.738074123398660e-1118 1.631081082539700e-12 -8.687479490879170e-1219 -6.387767991337960e-13 4.196188738890250e-1220 -1.521565690726940e-1221 5.748616824737580e-13 u k k k k k k -1.150665798108190e+00 1.000000000000000e+00 -9.422665062905070e-0170 able 13 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 1.167088563196570e+00 5.557949689428240e+001 4.028821977072450e-01 1.804403214930680e+002 1.589726864878320e-02 2.001076313509200e-023 -2.776444180529080e-03 -4.157908839745580e-034 5.440877740378380e-04 9.533639401005070e-045 -1.067963570631740e-04 -2.298471294547880e-046 1.602535986540860e-05 5.706424934572420e-057 3.639164819106400e-07 -1.443342632426860e-058 -1.502345300041770e-06 3.696661627610680e-069 6.684757371238390e-07 -9.553281498410910e-0710 -1.716295249441850e-07 2.486726588153440e-0711 1.635394935133220e-08 -6.518766056569640e-0812 8.892938254607350e-09 1.724037249294330e-0813 -5.718412804481970e-09 -4.621243803810750e-0914 1.784898899321040e-09 1.266095041880910e-0915 -2.594370983148820e-10 -3.591505441133870e-1016 -6.261300169685750e-11 1.071702367595880e-1017 5.827210983507840e-11 -3.409479146600350e-1118 -2.189105354284160e-11 1.160098934787360e-1119 4.514597736601780e-12 -4.170286799495250e-1220 -6.083212355274480e-14 1.531989721713420e-1221 -5.089272134222670e-139.998315799958800e-01 k k k k -1.885294461971110e+00 -1.777456775656590e+0071 able 14 Chebyshev coefficients c n for h = 1 / . n left central middle0 8.320108386176220e-04 1.028289222803410e-02 1.796716788945920e-011 3.797922844843720e-04 6.051387649842750e-03 9.215513210293030e-022 6.119152516315670e-06 2.386515701734710e-03 1.798443115218630e-023 -2.216499535843730e-06 9.149383182711970e-04 3.536164396647120e-044 3.549796569970470e-07 3.260804312989750e-04 -3.290230257519100e-045 -6.909777679284210e-08 1.174302244391960e-04 -9.190331115977200e-066 1.800406773387920e-08 4.046558068563540e-05 8.941392960865100e-067 -5.727807832664590e-09 1.430511004864880e-05 4.569100427021660e-078 2.048306829365090e-09 4.845772394724280e-06 -1.898690252517320e-079 -7.892933375352240e-10 1.708561223033980e-06 -1.470693084569350e-0810 3.208272429161230e-10 5.709919634096540e-07 9.061517318686090e-1011 -1.358581479450610e-10 2.027725977295490e-07 -4.996134827280970e-1012 5.944136353935260e-11 6.667913363885630e-08 -3.085715272976990e-1113 -2.671169627505600e-11 2.408641753810920e-08 8.541875621658270e-1114 1.227327977365980e-11 7.729248497752890e-09 3.049756612877180e-1115 -5.743936308401000e-12 2.879366774439790e-09 7.727805117204190e-1316 2.726212185552570e-12 8.872469780543650e-10 -3.534457536346370e-1217 -1.300587961514960e-12 3.484429145625870e-10 -1.357036283678540e-1218 6.052241498351530e-13 1.001335286917030e-10 4.056653202382400e-1419 -2.390157663283510e-13 4.301663419431590e-11 2.166251245948460e-1320 1.093494057671210e-1121 5.462280485697060e-1222 1.109419945826560e-1223 6.264905101784660e-13 u k k k k k k -1.142970216093080e+00 9.999999999999990e-01 -6.895558541579670e-0172 able 14 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.761057883039990e-01 5.196073467821080e+001 3.017284864244540e-01 1.831141920349900e+002 1.166946937938660e-02 2.543065037748800e-023 -1.687052247930830e-03 -5.741796704905520e-034 2.745955374164150e-04 1.425454072455170e-035 -4.255962891388790e-05 -3.709730129757980e-046 3.654738784807000e-06 9.914921759807340e-057 9.678759239832630e-07 -2.692978838795350e-058 -5.962323527180660e-07 7.390951335687150e-069 1.619073566570090e-07 -2.044323535428510e-0610 -2.044848058779350e-08 5.700139242154520e-0711 -3.557663286168380e-09 -1.608555974873110e-0712 2.792106872070250e-09 4.637422326004230e-0813 -8.014791797406380e-10 -1.388223005657000e-0814 9.886383182987470e-11 4.408914855308610e-0915 2.338331909817720e-11 -1.513613329915950e-0916 -1.720362865683670e-11 5.636677579981080e-1017 5.096915629190340e-12 -2.238514411774360e-1018 -7.701281537440130e-13 9.204119152754320e-1119 -3.802089545459430e-1120 1.539622910176960e-1121 -5.993091125355200e-1222 2.193183001782910e-1223 -6.910989846330220e-139.997667653179200e-01 k k k k -2.168616210929900e+00 -1.650521806358000e+0073 able 15 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.342404855864130e-04 5.287504606339480e-03 1.679802218836370e-011 1.582119045977740e-04 3.609167042053250e-03 9.425159027828390e-022 -4.034459826869400e-06 1.936145076792520e-03 2.146641276528250e-023 -1.781674777546940e-06 9.633501098384940e-04 7.761771414171220e-044 6.111894067922420e-07 4.586062099584780e-04 -4.715700923865780e-045 -1.616413643230570e-07 2.128565367871240e-04 -2.911833910065400e-056 4.535507820139480e-08 9.711724060740940e-05 1.543681974168120e-057 -1.504566368490170e-08 4.380458926575550e-05 1.397441335389570e-068 5.916977370301490e-09 1.959490913685950e-05 -3.947158787904670e-079 -2.631104077272530e-09 8.713044455946110e-06 -4.582398590716490e-0810 1.265674285428750e-09 3.857017778878410e-06 4.537121001350180e-0911 -6.412984582939380e-10 1.701669482580570e-06 -8.587058106556170e-1012 3.373488093146030e-10 7.488328990975590e-07 -3.395277343066420e-1013 -1.827196103322590e-10 3.288850425725570e-07 1.497940638456500e-1014 1.013577890763340e-10 1.442278857066740e-07 9.047367570431180e-1115 -5.736165706600700e-11 6.317620038821080e-08 1.532712463574530e-1116 3.301900838213170e-11 2.764873413720030e-08 -5.899234983630290e-1217 -1.928175236669560e-11 1.209228320347630e-08 -4.759599480764640e-1218 1.139207390775470e-11 5.285996098982320e-09 -9.845099825540210e-1319 -6.784952996564550e-12 2.309885154214620e-09 3.353301735326350e-1320 4.046041780101790e-12 1.009129182529960e-0921 -2.377799356621700e-12 4.407929681530880e-1022 1.318077536426880e-12 1.925241011297400e-1023 -5.870490384521500e-13 8.408559765252580e-1124 3.672433365477270e-1125 1.603764366665520e-1126 6.998237196315440e-1227 3.040158479208750e-1228 1.288881492848470e-1229 4.729004482220720e-13 u k k k k k k -1.055332320056090e+00 1.000000000000000e+00 -5.434844889094470e-0174 able 15 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 9.761146326247330e-01 5.440971360286990e+001 3.016928648060250e-01 1.956911680269630e+002 1.167231718705550e-02 2.774260925258900e-023 -1.651276452041110e-03 -6.436890817427110e-034 2.666875570608340e-04 1.641299885337880e-035 -4.247191460497490e-05 -4.384480040071970e-046 4.247919767558590e-06 1.201938574075200e-047 7.240192379129270e-07 -3.345172414930230e-058 -5.468504974426770e-07 9.394648359883000e-069 1.621838761986780e-07 -2.653557174889410e-0610 -2.469682517941960e-08 7.531228230971810e-0711 -1.846812979021160e-09 -2.152447292666580e-0712 2.453530993347940e-09 6.237985768142160e-0813 -8.104795070564450e-10 -1.859198071101320e-0814 1.328212021134190e-10 5.827250350019090e-0915 9.746012118699450e-12 -1.970501685911350e-0916 -1.445305942484290e-11 7.299803884539640e-1017 5.164008080182300e-12 -2.939243341028580e-1018 -1.029938970952770e-12 1.250293085781470e-1019 -5.430549502597070e-1120 2.340535807434650e-1121 -9.802247791887960e-1222 3.924623772975130e-1223 -1.472095272217630e-1224 4.733000270115350e-132526272829 9.999145446729710e-01 k k k k -2.179745438075220e+00 -1.614273003898510e+0075 able 16 Chebyshev coefficients c n for h = 1 / . n left central middle0 1.064576523939530e-04 7.757528496943130e-04 6.583911279090610e-021 4.967483886634590e-05 3.963182148041290e-04 4.630386572794740e-022 -2.574907330657650e-06 1.452820177102820e-04 1.757544141274230e-023 -5.303715403034560e-07 4.855954814287430e-05 3.315284363718880e-034 2.663871929167150e-07 1.531863558980870e-05 1.968319378136490e-055 -8.401958707782960e-08 4.676444520070540e-06 -1.223428005218250e-046 2.541840517940180e-08 1.394003778702580e-06 -1.557379115932900e-057 -8.458536138015860e-09 4.088490384098540e-07 3.231990249141240e-068 3.265291063082850e-09 1.184092050832930e-07 9.273005743030430e-079 -1.446267628365560e-09 3.397158988245310e-08 -3.854154049131550e-0810 7.070186386866000e-10 9.672491318170320e-09 -3.568282952834910e-0811 -3.686990455737450e-10 2.737461712291000e-09 -8.276509465005390e-1012 2.008214150700500e-10 7.708833517653440e-10 1.206706537754180e-0913 -1.129090416703240e-10 2.161982864910220e-10 9.225936702119910e-1114 6.508194177274510e-11 6.042419184248020e-11 -3.617701898545360e-1115 -3.828935422940620e-11 1.683809589516920e-11 -4.590404543093130e-1216 2.291651606295020e-11 4.678693771234680e-12 9.605943640102260e-1317 -1.391339883329400e-11 1.290808850907550e-12 1.614097426563680e-1318 8.543046330402440e-12 3.320491288954080e-1319 -5.282416801082780e-1220 3.263505722460030e-1221 -1.979537858186990e-1222 1.125706972359250e-1223 -5.098805064619270e-1324252627282930 u k k k k k k -1.042176040321580e+00 1.000000000000000e+00 2.136086687920960e-0176 able 16 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 5.721713263001810e-01 5.012707101453440e+001 2.017820038440820e-01 2.062096651831810e+002 1.377375013917080e-02 3.999684930491530e-023 -1.806370775567440e-03 -1.094915283125730e-024 1.523593047177300e-04 3.280050342478840e-035 2.295752543436930e-05 -1.026359592164560e-036 -1.211645711839800e-05 3.290839108300630e-047 1.558915013492110e-06 -1.072510048733340e-048 3.379002506728430e-07 3.549425811224860e-059 -1.589627733030860e-07 -1.199878274669100e-0510 8.281053242094140e-09 4.199796533313170e-0611 9.754591213299930e-09 -1.552499344038120e-0612 -2.587101485419130e-09 6.174229919308370e-0713 -2.654922061326020e-10 -2.654025845860760e-0714 2.854986006586110e-10 1.213436337756580e-0715 -4.288462498279570e-11 -5.724458644126490e-0816 -1.614286040010390e-11 2.696385391883720e-0817 8.227664210952470e-12 -1.231399370142220e-0818 -8.469562416364850e-13 5.303258910110760e-0919 -2.080169945416400e-0920 6.955732723896000e-1021 -1.600540556356640e-1022 -1.256262769207190e-1123 4.730496708984270e-1124 -3.958827080866930e-1125 2.486066180938770e-1126 -1.342780201739020e-1127 6.510864817550600e-1228 -2.877910389935850e-1229 1.155822468250220e-1230 -3.885406981704210e-139.997327929473050e-01 k k k k -1.873568041871660e+00 -1.410192678438050e+0077 able 17 Chebyshev coefficients c n for h = 1 / . n left central middle0 3.518991274756410e-05 8.392520820142320e-04 3.802327783487690e-021 1.590990829653510e-05 6.431136826076690e-04 2.293406644160830e-022 -1.491507257739220e-06 4.085596936397020e-04 6.950796412254640e-033 -9.066540515692110e-08 2.399296017984750e-04 1.080933113927390e-034 1.118185077651380e-07 1.344982201802300e-04 3.968276530040410e-055 -4.689629675507760e-08 7.316918708694770e-05 -1.510396050891140e-056 1.672578243190620e-08 3.898020499880180e-05 -2.164545910662930e-067 -5.962483120983330e-09 2.045075653316820e-05 1.031238628209580e-078 2.299928220409480e-09 1.060530105647130e-05 4.715822375760130e-089 -9.951311785026230e-10 5.449951689909780e-06 1.338195202962630e-0910 4.813973704272410e-10 2.780434074253660e-06 -7.162524074055580e-1011 -2.538921373245920e-10 1.410169685110300e-06 -5.838646258223070e-1112 1.421639472958470e-10 7.117308974598160e-07 9.035922879320330e-1213 -8.292852778673330e-11 3.577606972109210e-07 1.324669429761400e-1214 4.981742953446680e-11 1.792155094219710e-07 -9.136939392295850e-1415 -3.060881416945410e-11 8.951262809172500e-0816 1.915173995253600e-11 4.459614066323360e-0817 -1.216412493952420e-11 2.216977555006020e-0818 7.820244746397910e-12 1.100014354771840e-0819 -5.072097961571200e-12 5.448923967042200e-0920 3.302395224132890e-12 2.695162687572190e-0921 -2.138908948989580e-12 1.331357683924750e-0922 1.351588122026080e-12 6.569059660914260e-1023 -7.937121438141370e-13 3.237909926903940e-1024 3.670269446605750e-13 1.594506248721030e-1025 7.845610579728340e-1126 3.857391556723120e-1127 1.895058050737680e-1128 9.300136827782300e-1229 4.552865562504610e-1230 2.209843826831820e-1231 1.035453103819430e-1232 4.092881791896850e-13 u k k k k k k -1.029129856396840e+00 1.000000000000000e+00 4.634771910249610e-0178 able 17 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 4.996811103033990e-01 4.914258356400550e+001 2.283166265702920e-01 2.011738127244230e+002 2.514423513031150e-02 3.937356800757610e-023 -3.474198991631110e-03 -1.065266526129480e-024 5.077220214696990e-05 3.152291195579000e-035 1.613616422231280e-04 -9.738895553647930e-046 -3.880889586523480e-05 3.081786621477630e-047 -3.634159932107070e-06 -9.909792676876680e-058 3.510602114428830e-06 3.236113469325020e-059 -1.965557011348590e-07 -1.080367954217740e-0510 -2.893679425553000e-07 3.742488200776480e-0611 4.919338342933830e-08 -1.374128809584770e-0612 2.766224437511820e-08 5.450001386646280e-0713 -9.150633315570560e-09 -2.341737771120620e-0714 -2.629338358854130e-09 1.069346503420290e-0715 1.702910382225310e-09 -5.021967626980490e-0816 1.193636414090970e-10 2.344515094335100e-0817 -2.861147205442400e-10 -1.056108154875420e-0818 3.711031444447810e-11 4.460714134159530e-0919 3.822894605668730e-11 -1.700913808398100e-0920 -1.452954868328450e-11 5.418507670597160e-1021 -2.818928436440390e-12 -1.082511209350420e-1022 3.052195946657310e-12 -2.332308071729190e-1123 -4.763636711850320e-13 4.458678049756930e-1124 -3.438522185303330e-1125 2.066800196185070e-1126 -1.078966568122660e-1127 5.070919968888780e-1228 -2.172493412014030e-1229 8.448217334599220e-1330 -2.756966775927990e-133132 9.998681694019440e-01 k k k k -1.323148927258810e+00 -1.420354956151160e+0079 able 18 Chebyshev coefficients c n for h = 1 / . n left central middle0 5.483749876010110e-06 6.081052138201630e-05 5.473769059145160e-021 2.486444309865320e-06 3.802911084933040e-05 4.351538755149600e-022 -2.246254082499010e-07 1.823368632895480e-05 2.198780054496340e-023 -1.578587968151560e-08 7.946990081244110e-06 6.662984075703530e-034 1.732923293003170e-08 3.273356396625710e-06 7.907173467155090e-045 -7.081399642607440e-09 1.301100356483330e-06 -2.243067264619820e-046 2.482198403770580e-09 5.043448241124090e-07 -1.032261735179890e-047 -8.749831841180770e-10 1.919418161993120e-07 -5.350827374548080e-068 3.358446096487070e-10 7.202571800642060e-08 6.032870474106270e-069 -1.452715547964040e-10 2.672894811858390e-08 1.389775608451330e-0610 7.036053028305950e-11 9.830459878805920e-09 -1.667794341081260e-0711 -3.712078515936340e-11 3.588823452512350e-09 -1.098049445185060e-0712 2.075613750341240e-11 1.302082454421860e-09 -3.990118701505280e-0913 -1.206646596058050e-11 4.699379616814380e-10 6.210083081288950e-0914 7.203733700544880e-12 1.688418171221930e-10 9.633344895225680e-1015 -4.375271295344380e-12 6.042392709176170e-11 -2.609783367202760e-1016 2.674170121231750e-12 2.154618744858790e-11 -8.315421351488390e-1117 -1.612760826306620e-12 7.648543523947050e-12 6.884100040797630e-1218 9.148126116574220e-13 2.675596311035590e-12 5.334393632012140e-1219 -4.140244318492860e-13 8.418748326551260e-13 8.096094478676190e-14202122232425262728 u k k k k k k -1.030074288714920e+00 1.000000000000000e+00 4.184273889203900e-0180 able 18 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 5.722628305041320e-01 4.619812351486580e+001 2.017761232436850e-01 1.860589481022910e+002 1.372549559855640e-02 3.705233566867140e-023 -1.799486944230340e-03 -9.679912077752170e-034 1.548179312896890e-04 2.763008630922080e-035 2.188831960478410e-05 -8.227338295193510e-046 -1.205509962337280e-05 2.508114545966230e-047 1.626588892457300e-06 -7.771817988229730e-058 3.184920827863750e-07 2.450094311357910e-059 -1.604199428018640e-07 -7.933144007790290e-0610 1.026914924420960e-08 2.688223932510620e-0611 9.459364972251830e-09 -9.765416694004470e-0712 -2.712392941794810e-09 3.865875322747190e-0713 -2.064497813254320e-10 -1.657778704727940e-0714 2.839303432413100e-10 7.480889417663450e-0815 -4.871222002206250e-11 -3.419915104148750e-0816 -1.444200355589120e-11 1.528919983568170e-0817 8.372113940734360e-12 -6.479029251360280e-0918 -8.215034741114160e-13 2.513840140749050e-0919 -5.718274703852940e-13 -8.420176316921800e-1020 2.056434736961990e-1021 -1.906441397640450e-1222 -4.118010901086650e-1123 3.573866647613250e-1124 -2.210862584205230e-1125 1.157843167001790e-1126 -5.392753508375620e-1227 2.255758034817050e-1228 -7.825720526313930e-139.999476900342120e-01 k k k2 k2 -1.877416447914540e+00 -1.452884710450130e+0081 able 19 Chebyshev coefficients c n for h = 1 / . n left central middle0 1.702183541035500e-06 4.683227357260460e-05 5.536218616018370e-021 7.420902874786450e-07 3.703626675908400e-05 4.374721584420760e-022 -9.865066726948370e-08 2.459788457587220e-05 2.175988835715660e-023 2.116960181208090e-09 1.509706997971100e-05 6.418561137830710e-034 5.497757664443220e-09 8.833813587467830e-06 7.077802624591750e-045 -3.075679409634450e-09 5.007785871509330e-06 -2.234688941525440e-046 1.295461350315470e-09 2.775387397751700e-06 -9.419306294251860e-057 -5.101921537607570e-10 1.512369265686630e-06 -3.360150759204170e-068 2.046470586232400e-10 8.133846678811100e-07 5.606215973335830e-069 -8.795781406451580e-11 4.329046589636930e-07 1.138744283796750e-0610 4.150674524923850e-11 2.284469855279910e-07 -1.744975342137310e-0711 -2.146959372032940e-11 1.197027748370630e-07 -9.175604868426630e-0812 1.195357510557570e-11 6.234932494901730e-08 -9.405807379254040e-1013 -7.007921488895000e-12 3.231076406508420e-08 5.303111119723840e-0914 4.242800470479290e-12 1.667069009839600e-08 6.408776988724730e-1015 -2.605046952660340e-12 8.568338007765290e-09 -2.359584489072000e-1016 1.583112277512220e-12 4.389129098487210e-09 -5.912784735389670e-1117 -9.048703171667920e-13 2.241645822968350e-09 7.661640011719350e-1218 4.117767633668480e-13 1.141836261366420e-09 3.874828519769490e-1219 5.802414152021560e-10 -1.166357191529640e-1320 2.942280531036410e-1021 1.489078538692040e-1022 7.522864040175020e-1123 3.794343618365480e-1124 1.910697269597880e-1125 9.603319488497670e-1226 4.810076959555690e-1227 2.385153926155570e-1228 1.138578899937890e-1229 4.560886440536520e-13 u k k k k k k -1.022376759557920e+00 1.000000000000000e+00 4.064120883541780e-0182 able 19 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 5.722747148899260e-01 4.619756015894430e+001 2.017753450395620e-01 1.860594275953970e+002 1.371924357276870e-02 3.708005345183760e-023 -1.798581149904210e-03 -9.684906797069420e-034 1.551213775007570e-04 2.763643547482780e-035 2.175254385332560e-05 -8.226383894790440e-046 -1.204645343665130e-05 2.506774823695980e-047 1.634965165513770e-06 -7.763711144862420e-058 3.159981622016720e-07 2.446026791697170e-059 -1.605777694467310e-07 -7.914262660093390e-0610 1.051874919029200e-08 2.679801742363770e-0611 9.419608354447810e-09 -9.728790947989900e-0712 -2.727465609582260e-09 3.850313981959300e-0713 -1.988937692998890e-10 -1.651364788520010e-0714 2.836178919653520e-10 7.455668628814360e-0815 -4.943002831236940e-11 -3.410750194426710e-0816 -1.421606720565170e-11 1.526049674013000e-0817 8.387073685132130e-12 -6.472921009013150e-0918 -8.503982177047610e-13 2.514557909430010e-0919 -5.640963743958310e-13 -8.439824101491250e-1020 2.072428649649560e-1021 -2.907032521718900e-1222 -4.063479916620080e-1123 3.546906613254100e-1124 -2.198580726676460e-1125 1.152667195366090e-1126 -5.372666511779410e-1227 2.248692740626720e-1228 -7.804828886326530e-1329 9.999739151276640e-01 k k k k -1.877882960729700e+00 -1.452971340318820e+0083 able 20 Chebyshev coefficients c n for h = 1 / . n left central middle0 8.340782562108640e-08 2.468455017350810e-06 2.280294691861460e-021 3.790348335769390e-08 1.808380820210540e-06 1.944720833702610e-022 6.055403556177730e-10 9.977786192716350e-07 1.206597650481480e-023 -2.457590587255690e-10 5.095262839987460e-07 5.365044701332920e-034 3.852931864752020e-11 2.442450448287350e-07 1.591201684585550e-035 -7.060509039767990e-12 1.138166487478510e-07 2.189680324488170e-046 1.745854017090550e-12 5.143505829362970e-08 -5.028795904628400e-057 -4.817127130663810e-13 2.294108192095970e-08 -3.473771288547360e-058 1.003514899934200e-08 -6.612748781444730e-069 4.362945792279890e-09 8.987877800239300e-0710 1.869145494199760e-09 7.769588194187460e-0711 7.993315810705090e-10 1.334497286164700e-0712 3.374445612260970e-10 -2.638905677337210e-0813 1.426920238747580e-10 -1.641293598763500e-0814 5.956301256742110e-11 -1.798371164119150e-0915 2.498973580154360e-11 8.162891792784310e-1016 1.032764886165960e-11 3.081840920792170e-1017 4.294120630111010e-12 5.391475739175650e-1218 1.719237245981870e-12 -2.108451751530640e-1119 6.120456681919280e-13 -4.569166977955490e-1220 5.618348229993250e-132122232425262728 u k k k k k k -1.151540996208610e+00 1.000000000000000e+00 7.708355493813890e-0184 able 20 (cont.) Chebyshev coefficients c n for h = 1 / . n right tail right tail0 4.997785332154780e-01 4.423666819273100e+001 2.283167712864420e-01 1.759820399395020e+002 2.509022440364520e-02 3.541998526121500e-023 -3.473302792894470e-03 -9.020823900527130e-034 5.642398807896150e-05 2.508301403041810e-035 1.601477839497730e-04 -7.271867741631950e-046 -3.917135579658680e-05 2.157837314850230e-047 -3.440824241326100e-06 -6.511087849455140e-058 3.519318737665770e-06 2.002320021389290e-059 -2.194212259839990e-07 -6.350484440175690e-0610 -2.877017875529090e-07 2.122832316166750e-0611 5.198973063531290e-08 -7.670810351604760e-0712 2.710916358293470e-08 3.034734463075500e-0713 -9.510243857620720e-09 -1.296599648285690e-0714 -2.488075646152630e-09 5.774831521471910e-0815 1.740766101870470e-09 -2.574755859857430e-0816 8.803780581176810e-11 1.108702976174020e-0817 -2.866253352551910e-10 -4.459918192937890e-0918 4.280148433910300e-11 1.605691932192970e-0919 3.704110753342700e-11 -4.731139230730710e-1020 -1.527178121204120e-11 7.876459425075560e-1121 -2.422082175650230e-12 2.822685782970870e-1122 3.088328322933470e-12 -3.938365394966460e-1123 -5.645136949758680e-13 2.737131256274080e-1124 -1.510876686567670e-1125 7.257145569467040e-1226 -3.126553351859840e-1227 1.213046040716080e-1228 -3.939886674131800e-139.999869750663500e-01 k k k2 k2