In this section, we discuss the computation of the numerical value of G∗(2). First, we separate the integral into three parts
Z ∞
Now we consider the first part of the integral. Since g(z) is given by (3.12) and fe1(j)(0) = 0 for j ≤ b and ef1(b+1)(0) = 1, we have
for large s. Thus, from this and (2.21) we obtain for N tends to infinity,
This means that if we choose N large enough, then the first part can be safely neglected.
Next, we need a good way of computing g(z) for the second integral. We first consider fe1(z). Since ef1(z) = e−zP
and the error term introduced by this approximation is
e−z(log N )zN +1
which is very small if we choose T small enough, say N eT . So, in order to compute fe1(z), we just have to generate µb+1, . . . , µN (this can be done via the recurrence satisfied by µn) and then use (3.16). Since a similar approach also works for the derivatives of fe1(z), this can be used to compute g(z) as well.
Finally, we consider the last integral Z ∞ We use Watson’s lemma to get an asymptotic expansion of the Laplace transform of s/ϕ(2s):
Lemma 9. Consider a Laplace integral R∞
0 f (s)e−zsds and assume (i) f (s) has a power series expansion which converges for |s| < R.
(ii) There exists an α > 0 such that f (s) = O(eαs) as s → ∞.
We still need an asymptotic expansion of g(z) for z large. Again first consider ef1(z). From this and the expression above, we get
F (s) ≈ϕe0
Here, we have dropped all terms with si, i ≥ 0 (their contributions to ef1(z) is negligi-ble). Then, by inverse Laplace transform,
fe1(z) ∼ ϕe0
By differentiation we obtain similar expansions for the derivatives of ef1(z) and hence for g(z).
Plugging this expansion together with the expansion obtained from Watson’s lemma for the Laplace transform of s/ϕ(2s) into (3.17) and integration then yields an approxi-mation of (3.17) up to arbitrary large order. Choosing sufficiently many terms will then give a good approximation for very small T .
Finally, we explain how to compute ˆG∗(2 + χj) involved in (3.18). First notice that due to the very fast decay of ˆG∗(2+χj) only j = 0, 1, −1 are needed. For the computation we use the following result which is due to Flajolet and Richmond.
Lemma 10. The function J (s) defined in (2.26) admits the representation
J (s) = A0(2s) + (s − 1)A1(2s) + · · · + (s − 1)(s − 2) · · · (s − b + 1)Ab−1(2s), where Ak(x)’s are entire functions, and
Ak(x) = (−1)k k! · 1
Qb∞
∞
X
j=0
(−1)jb2−bj(j+1)/2
Qbj Yb−1−k(j)(x2b−1−k)j, with Qm =Qm
l=1(1 − 2−l), Q∞ =Q∞
l=1(1 − 2−l) and the Yβ(j) are defined by X
β≥0
Yβ(j)wβ = exp bX
α≥1
(−1)αwα
α (X
l6=j
1 (2l− 2j)α)
! .
Overall, by incorporating all the above ideas, we obtain the following program for computing the required values.
> b := 2; T := 30; N := 4*T;
> µ := vector(120);
for y from 1 to b do µ[y] := 0
end do;
> for y from 1 to 120-b do
µ[y+b] := y+2ˆ(1-y)*(sum(binomial(y, i)*µ[i], i=1..y)) end do;
> fnew := exp(-z)*(sum(µ[i]*zˆi/i!, i=1..120));
> f := vector(b);
for y from 1 to b do f[y] := diff(fnew, z$y+1) end do;
> evalf(evalf(Int(s*exp(-z*s)*((2*z+1)*f[1]ˆ2+z*f[2]ˆ2+4*z*f[1]*f[2])/
evalf(Product(1+s/2ˆi, i=0..60))ˆb, [z=0..T, s=0..N]))/ln(2));
0.1300797679
> with(PolynomialTools);
φ := sum(sum(b*(-1)ˆ(n-1)*sˆn/(n*(2ˆn-1)), n=1..12)ˆi/i!, i=0..12);
coeff phi := CoefficientVector(expand(φ), s);
> q := evalf(add((-1)ˆj*2ˆ(-binomial(j+1, 2)) /evalf(Product(1-2ˆ(-i), i=1..j)), j=0..60));
e := sum((-1)ˆi*wˆi*(sum(1/(2ˆd-2ˆj)ˆi, d=0..j-1)+sum(1/(2ˆd-2ˆj)ˆi, d=j+1..60 ))/i, i=1..b);
> coeff y := CoefficientVector(expand(sum((b*e)ˆi/i!, i=0..b-1)), w);
> a := vector(b);
for k from 0 to b-1 do
a[k+1] := evalf((-1)ˆk*add((-1)ˆ(b*j)*2ˆ(-(1/2)*b*j*(j+1))*coeff y[b-k]*2ˆ((b-1-k)
*j)/(Product(1-2ˆ(-i), i=1..j))ˆb, j=0..60)/(k!*qˆb)) end do;
> G := (s) → evalf(Pi*(sum((product(s-j, j=1..i-1))*a[i], i=1..b))/sin(Pi*s));
> χ := (2*Pi*I)/log(2);
func vec := vector(13);
func vec[1] := z*(-1+γ+ln(z))+G(χ)*zˆ(1+χ)/GAMMA(2+χ) +G(-χ)*zˆ(1-χ)/GAMMA(2-χ);
func vec[2] := γ+ln(z)+G(χ)*zˆχ/GAMMA(1+χ)+G(-χ)*zˆ(-χ) /GAMMA(1-χ);
for k from 3 to 13 do
func vec[k] := zˆ(-k+2)*(-1)ˆ(k-3)*(k-3)!+G(χ)*zˆ(-k+2+χ)/GAMMA(-k+3+χ) +G(-χ)*zˆ(-k+2-χ)/GAMMA(-k+3-χ)
end do;
> gnew := 0;
for k from 3 to 13 do
gnew := gnew+coeff phi[k]*func vec[k]
end do;
gnew := gnew/ln(2);
> g := vector(b);
for k from 1 to b do g[k] := diff(gnew, z$k+1)) end do;
> φ2 := s*(sum((sum(-b*(-1)ˆ(n-1)*2ˆn*sˆn/(n*(2ˆn-1)), n=1..12))ˆi/i!, i=0..12));
coeff phi 2 := CoefficientVector(expand(φ2), s);
> watson := 0;
for k from 1 to 13 do
watson := watson + coeff phi 2[k+1]*k!/zˆ(k+1) end do;
> evalf(Int(((2*z+1)*g[1]ˆ2+z*g[2]ˆ2+4*z*g[1]*g[2])*watson/log(2), z=T..∞, method = CCquad));
0.002522975851 + 1.220918538 · 10−19I
> 0.002522975851 + 0.1300797679;
0.1326027438
Chapter 4 Conclusion
To conclude this thesis, we briefly summarize the main contributions.
Our first goal was to give a self-contained survey of recent results in the analysis of DSTs. These results have been widely spread across the research literature before and this is up to our knowledge the first time that they appear in collected form. For clearance of presentation, we started by explaining the main techniques used in the analysis of DSTs in Chapter 1. Then, we showed applications of these techniques in Chapter 2. Therefore, we used the internal path length as guiding example and concentrated on mean value and variance. It should be stressed that all the results from Chapter 1 and Chapter 2 are not original. However, we improved and shortened several proofs, in particular, those concerned with the analysis of asymmetric DSTs.
Our second goal was to introduce a new approach of Fuchs, Hwang, and Zacherovas (which has not appeared yet) and to compare it with previous approaches from Chapter 2.
This new approach has two major improvements: one is the use of the Laplace transform which simplifies the overall analysis and the other is the consideration of eV (z) which makes the computation of the variance more easier. Then, we applied the new approach to the variance of the internal path length of b-DSTs and obtained more simpler expressions for the leading constant in the asymptotic expansion for b = 1, 2 (larger values of b can be treated as well). Finally, we explained how to obtain numerical values of the leading constant for small values of b via Maple. Our numerical computations suggest that previous computations contain several imprecisions.
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