METHODS FOR THE. CONSTRUCTION OF. EXTRAPOLATION PROCESSES. Herbert H. H. Homeier, Regensburg. Plovdiv, 1996. ⢠General Approach.
METHODS FOR THE CONSTRUCTION OF EXTRAPOLATION PROCESSES Herbert H. H. Homeier, Regensburg Plovdiv, 1996
• General Approach • Iterative Approach • Variational Approach • Perturbative Approach
1
MANY THANKS TO Prof. Dr. Drumi D. Bainov Prof. Dr. Otto Steinborn Priv.-Doz. Dr. Joachim Weniger Dr. Holger Meiner Dr. Johannes Dotterweich Prof. Dr. Bernhard Dick Prof. Dr. Hartmut Yersin Dipl.-Chem. Jens Decker Prof. Dr. Josef Barthel Prof. Dr. Hartmut Krienke Dipl.-Chem. Sebastian Rast Prof. Dr. Peter Otto (Erlangen) Prof. Dr. Claude Daul (Fribourg) Dr. Miloslav Znojil (Rez) Deutsche Forschungsgemeinschaft Auswrtiges Amt 2
Development of Methods: Extrapolation Methods Problem: Slow Convergence Example sn = 1 +
x
+
x2
+
x3
+ ··· +
xn
1 2 3 n s = lim sn = 1 − ln(1 − x) n→∞ 4.4
x = 0.9,
4.2+ 4 3.8 3.6
+
3.4
+
+
+
+
+
+
3
3
3
+ 3
7
8
9
10
3.2 3 2.8 2.6 2.4 2.2
sn 3 s0n + s
3
3
3
3
3 2
3
4
5
6
n Sequence transformation {s0n} = T ({sn}): Accelerate convergence 3
Development of Methods: Extrapolation Methods Problem: No Convergence Example: sn = 1 +
x 1
+
x2 2
+
x3 3
+ ··· +
xn n
s = lim sn = 1 − ln(1 − x) n→∞ 1+ 3
x = −1.3,
0.8 0.6
3 +
0.4 0.2
+
3
3
+
+
+
+
+
sn 3 3 0 3 sn + s
+
+
+
0 -0.2
3
3
3
3
-0.4
3
-0.6 0
2
4
6
8
10
n Sequence transformation {s0n} = T ({sn}): Achieve convergence 4
PROBLEMS • Extrapolation • Convergence acceleration • Summation of divergent series • No universal method available • Linear methods often not efficient enough • Accuracy of single method unknown in practice AIMS • problem adapted • nonlinear • efficient APPROACH • Model sequences • Iteration of simple methods • Comparison of results of various methods 5
BASIC PRINCIPLE
EXAMPLE
Model sequences σn = σ + Mn(~ c, p ~(n)) σn = σ + c ωn Exact Calculation of σ σ = Tn({σn}, p ~(n)) σ = σn+1 − ωn+1
σn+1 − σn ωn+1 − ωn
Sequence transformation for problem {sn} s0n = Tn({sn}, p ~(n)) sn+1 − sn 0 sn = sn+1 − ωn+1 ωn+1 − ωn Acceleration for {sn} ≈ {σn} sn − s 6
ωn
= O(1)
ANNIHILATION OPERATORS k X
σn = σ + ωn
cj φj (n)
j=0
with linear operator On(φj (n)) = 0 ,
j = 0, . . . , k
then On
σn − σ
ωn
=0
or On(σn/ωn)
On(sn/ωn) 0 =⇒ sn = σ= On(1/ωn) On(1/ωn)
CONVERGENCE CLASSIFICATION lim
sn+1 − s
=ρ
sn − s linearly convergent: 0 < |ρ| < 1 logarithmically convergent: ρ = 1 n→∞
7
REMAINDER ESTIMATES sn − s ωn
= O(1) ,
n→∞
Series sn =
n X
aj → s ,
n→∞
j=0
Levin tω = a , n n uω = (n + β)a , n n v ω = anan+1 . n an − an+1
Smith and Ford t˜ω = a n n+1
8
β > 0,
Asymptotically related series a ˆ n ∼ an sˆn =
n X
a ˆ j → sˆ ,
n→∞
j=0
Linear remainder estimates ltω = a ˆn , n luω = (n + β)ˆ an , n ˆ na ˆ n+1 lv ω = a , n a ˆn − a ˆ n+1 lt˜ω = a ˆ n+1 n
β > 0,
Kummer-type remainder estimate kω = s ˆn − sˆ n
Tails Tn = an+1 + an+2 + · · · = s − sn t˜T n = sn+1 − sn = an+1 + 0 + · · · , lt˜T ˆn+1 − sˆn = a ˆ n+1 + 0 + · · · , n = s kT ˆ − sˆn = a ˆ n+1 + a ˆ n+2 + · · · . n = s 9
Example Fm(z) =
∞ X
(−z)j /j!(2m + 2j + 1) ,
j=0 kω = n
(1 − e−z )/z −
n X
! (−z)j /(j + 1)! .
j=0
uω n sn n 5 –13.3 0.3120747 6 14.7 0.3132882 7 –13.1 0.3132779 8 11.4 0.3133089 9 –8.0 0.3133083 z = 8 ,m = 0
2J-Transformation
tω n
kω n
0.3143352 0.3131147 0.3133356 0.3133054 0.3133090
0.3132981 0.3133070 0.3133087 0.3133087 0.3133087
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ITERATIVE SEQUENCE TRANSFORMATIONS Idea: • Simple sequence transformation {s0n} = T ({sn}) • Iteration: 0({s0 }) , {s000} = T 00({s00 }) , . . . {s00 } = T n n n n
• Example: Aitken ∆2 method (sn+1 − sn)2
(1)
sn = sn −
sn+2 − 2sn+1 + sn
.
Iteration (n) A0 = sn , (n)
(n)
Ak+1 = Ak
(n+1)
−
(Ak
(n)
− Ak )2
(n+1) (n) (n+2) Ak − 2Ak + Ak
11
.
Problem: • Iteration not unique, for example (k+1)
sn
(k)
= T (k)({sn })
with T (0) = T . • Example (n) A0 =sn , (n) (n) Ak+1 =Ak
(n+1) (n) 2 − Ak ) (Ak −Xk (n+2) (n+1) (n) Ak − 2Ak + Ak
with X0 = 1 “allowed”, i.e., Xk = (2k + 1)/(k + 1).
12
HIERARCHICAL CONSISTENCY Example: • Simple transformation: sn+1 − sn 0 sn = sn+1 − ωn+1 ωn+1 − ωn 0 ? • Problem: ωn
• For σn = σ + ωn(c0 + c1rn) we have 0 =σ−c σn 1
ωnωn+1 ωn+1 − ωn
(rn+1 − rn)
• Hence ωnωn+1 0 ωn = − (rn+1 − rn) ωn+1 − ωn
13
• Iteration =⇒ J transformation (0) sn = sn ,
(0) ωn = ωn , (k) (k) − s s n (k+1) (k) (k) n+1 sn = sn − ωn , (k) (k) ωn+1 − ωn (k) (k) ω n ωn+1 (k+1) (k) ωn =− δn , (k) (k) ωn+1 − ωn (k) (k) (k) Jn ({sn}, {ωn}, {rn }) = sn
• for power series, • very versatile and powerful. • related to E algorithm with model σn = σ +
k X j=0
14
cj gj (n)
• Analysed in H. H. H. Homeier, A hierarchically consistent, iterative sequence transformation, Numer. Algo. 8 (1994) 47-81. —, Analytical and numerical studies of the convergence behavior of the J transformation, J. Comput. Appl. Math. 69 (1996) 81-112. —, Determinantal representations for the J transformation, Numer. Math. 71 (1995) 275-288.
15
Numerical example: Gn =
Z
1
0
Z
Γ/π
∞
−∞ (x − ∆ω)2 + Γ2
"
Y q 2Γ/π exp − 2 x + Γ2
#
! −1
) dx
qn d q
Taylor series in Y =⇒ Power series Acceleration with J transformation
n 15 16 17 18 19 20 ∞
sn -3047434. 5412146. -9099655. 14525645. -22070655. 31994427. -0.16361729 (k)
(n) s0
-0.16361565 -0.16361782 -0.16361716 -0.16361732 -0.16361728 -0.16361729 -0.16361729
n = 2, ∆ω = 5, Γ = 1, Y = 100, r k = 1/(n + 1 + k), k variant 16
More general: • Hierarchy of model sequences: The higher, the more parameters • Iteration of simple transformation T is consistent, if – T is exact for lowest level sequences – Variant of T maps higher to lower level EXAMPLE: Hierarchy for J transformation σn = σ + ωn(c0 + c1
n−1 X
(0)
δ n1
n1=0 n −1 n−1 X (1) (0) 1X δ n2 δ n1 + c2 n1=0 n2=0
+ · · ·) at level k.
17
FORMAL DESCRIPTION • Simple model {σn(~ c, p ~)} → σ(~ p) • Simple transformation ∞ T (~ p ) : {σn(~ c, p ~ )}∞ −→ {σ(~ p )} n=0 n=0
• Hierarchy of model sequences (`) ∞ (`) (`) (`) (`) a }n=0}L {{σn (~ c ,p ~ )|~ c ∈C `=0 0) (`) (` with a > a for ` > `0
• Mapping between levels (`) (`) (`) T (~ p ) : {σn (~ c ,p ~ (`))}∞ n=0 (`−1) (`−1) −→ {σn (~ c ,p ~ (`−1))}∞ n=0
• Hierarchical consistent transformation T (`) = T (~ p (0)) ◦ T (~ p (1)) ◦ . . . ◦ T (~ p (`))
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METHODS FOR ORTHOGONAL SERIES • I transformation: – Simple model sequence: σn = σ+ωn(c exp(inν)+d exp(−inν)) – Simple sequence transformation sn+2 sn+1 sn − 2 cos(ν) + ωn+2 ωn+1 ωn 0 sn = 1 1 1 − 2 cos(ν) + ωn+2 ωn+1 ωn 0 ? – Compute ωn – More complicated model sequence:
σn = σ + ωn(einν (c0 + c1rn) +e−inν (d0 + d1rn)) yields 0 ≈ σ+ω 0 (c0 exp(inν)+d0 exp(−inν)) σn n 0 = ωn
−(rn+1 − rn) 1 1 1 − 2 cos(ν) + ωn+2 ωn+1 ωn 19
– Iteration =⇒ I transformation s(0) n = sn ,
ωn(0) = ωn (k)
sn+2 sn(k+1) =
(k) ωn+2
1 (k) ωn+2
ωn(k+1) =
1 (k) ωn+2
(k)
− 2 cos(ν) − 2 cos(ν)
sn+1 (k) ωn+1
1
(k) ωn+1 (k) −∆rn+1
− 2 cos(ν)
1 (k) ωn+1
+ +
s(k) n (k)
ωn 1
(k)
ωn
+
1 (k)
ωn
– for Fourier series – Notice three-term recurrence un+2 − 2 cos(ν)un+1 + un = 0 satisfied by exp(±inν) (or cos(nν), sin(nν)
20
• K transformation: – Simple model sequence : σn = σ + cωnPn(cos(ν)) – Three-term recurrence (0)
(1)
(2)
ζn Pn + ζn Pn+1 + ζn Pn+2 = 0 . (ν dependent) – Algorithm (analog to I transformation) s(0) n = sn ,
ωn(0) = ωn , (0)
sn(k)
(0)
ωn 1
ζn+k s(k+1) = n
ζn+k ωn(k+1) =
(0) ζn+k
(1)
(1)
ωn+1 1
(k)
+ ζn+k
(k)
+ ζn+k
ωn
1 (k) ωn
(k)
(k)
sn+1
(2)
ωn+1 1
(k)
+ ζn+k
(k)
+ ζn+k
ωn+1 δn(k) 1 (1)
+ ζn+k
(2)
sn+1
(k) ωn+1
+
(k)
(k)
ωn+1
(2) ζn+k
1 (k)
ωn+1
Kn(k)({δn(k)}, {ζn(j)}, {sn}, {ωn}) = s(k) n
– ν dependent, for orthogonal series. 21
MULTIPOLE EXPANSIONS r /r) m∗ 1 Y`m(~ UQ(~ r ) = 4π Q` `+1 2` + 1 `m r X
Qm ` =
Z
r 0`Y`m(~ r 0/r 0)ρ(~ r 0) d 3r 0 ,
Rotational symmetry UQ(~ r) =
∞ X
P`
`=0
Legendre Expansion
22
~ ~ r·R rR
!
q` r `+1
Example ~ ρ(~ r ) = exp(−αr) exp(−β|~ r − R|)
` − lg |1 − s`/s| − lg |1 − s0`/s| 2 2.6 5.1 4 4.3 9.7 6 5.9 11.5 8 7.6 16.0 10 9.2 16.0 K transformation, r = 12, θ = 60o
14 12 10 8 6 4 2
++ + s` 3 ++ s0` + + + 33 3 +++ 333 33 3 ++ 33 3 + ++ 33 3 3 3 3 ++ 33 3 + 3 3 3 ++33 3 3
0 5
10
15
20
25
`
Exact digits (r = 4, θ = 60o) 23
MANYFOLD FREQUENCIES • Important near singularities • Increases stability • Instead of s0, s1, s2, . . . , ω0, ω1, ω2, . . . take sτ ·0, sτ ·1, sτ ·2, . . . , ωτ ·0, ωτ ·1, ωτ ·2, . . . • for Fourier and orthogonal series put ν →τ ·ν i.e., for x = cos ν x → xτ = cos(τ · arccos x)
24
∞ X
1
`=0 ` + 1
s
P`(x) = ln 1 +
2 1−x
x = 0.9 K transformation τ =1 n − lg |1 − sn/s| 16 2.07 18 1.75 20 1.91 22 3.59 24 2.01 τ =3 m n 48 16 54 18 60 20 66 22 72 24
0 − lg 1 − sn/s 5.24 6.88 6.58 6.91 6.80
− lg |1 − sm/s| 2.51 2.45 2.48 2.59 2.80 25
0 − lg 1 − sn/s 9.31 10.40 11.63 13.18 14.47
ASSOCIATED POWER SERIES Example: ∞ X
1 + in
n=0
n2
∞ 1 + in 1 X inν cos(nν) = e 2 n=0 n2 ∞ 1 + in 1 X −inν + e 2 n=0 n2
Accelerate power series separately and add Adaptable to more complicated examples: X
cos((n + 1/2)ν)Pn(cos ν 0)
(singular at ν = ν 0) is sum of 4 power series ∞ 1 X 0) e±i(n+1/2)ν ρ± (ν n 4 n=0 2 ± 0 0 ρn (ν ) =Pn(cos ν ) ± i Qn(cos ν 0) π 0 exp(±inν ) ∼
√
× const.
n
26
τ = 10,
ν = 6π/10,
ν 0 = 2π/3
near singularity. n − lg |(sτ n − s)/s| 8 12 16 20 24 28
1.3 1.2 1.0 1.5 1.3 1.2
(τ ) − lg (Gn − s)/s 7.7 14.0 18.1 22.6 27.3 31.2
P4 (τ ) (0) Gn = j=1 Ln (1, [pj,τ n]|n=0, [(τ n+1)(pj,τ n−
pj,(τ n)−1)]|n=0)
27
VARIATIONAL METHODS Problem: Exact limit invariant under addition of null sequences Nonlinear sequence transformation usually not ! Idea: Restore invariance variationally for certain null sequences xn s0n = f ({sn + αxn}) ∂s0n ∂α
28
=0
Example: Aitken ∆2 method (1)
sn = sn −
(∆sn)2 ∆2s
.
n
Use sn → sn + αxn with lim xn = 0 n→∞ (1)
(1)
sn → sn (α) = sn+αxn−
(∆sn + α∆xn)2 ∆2s
n + α∆xn
(1) Choose α such that sn (α) is stationary:
(1) ∂sn ∂α
=0 α=α0
Result: For sn = 10 + 1/n2 new method accelerates (O(n−3) error), but Aitken does not(!) accelerate convergence.
29
.
Example s0n =
k X
cj sn+j ,
k X
cj = 1
j=0
j=0
Put k X
sn → sn +
(ν) αν xn
ν=1
Saddle point =⇒ Linear system ∂s0n ∂αµ
=
k X j=0
Result identical to E s n (1) xn .. (k) xn s0n = 1 (1) xn .. (k) xn
(µ) cj xn+j = 0
30
algorithm ! . . . sn+k (1) . . . xn+k .. ... (k) . . . xn+k . . . 1 (1) . . . xn+k .. ... (k) . . . xn+k
PERTURBATIONAL METHODS Rayleigh-Schrdinger Perturbation Theory H = H + βV yields E (n) = E0 + βE1 + . . . + β nEn Goldhammer-Feenberg H = (1 − α)H0 + [V + αH0] yields E (n)(α) = E0(α)+βE1(α)+. . .+β nEn(α) Choose α variationally (True E is α-independent) ∂E (n)(α) ∂α
=0
For n = 3 solution is α = E3/E2 −→ Feenberg series Fn = E (n)(E3/E2) 31
EFFECTIVE CHARACTERISTIC POLYNOMIALS ˇ ıˇ C´ zek
Pn(E)= det φj |H|φk − E δj,k =
n X j=0
Ej
n−j X
fn,j,k β k
k=0
Obtain f ’s from perturbation series Pn(E0 +βE1 +β 2E2 +. . .) = O(β n(n+3)/2) Zero of P2: Π2= E0 + E1 E22 E2 − E3 + 2 2 E E − E 3 q2 4 2 − 4 (E E − E 2) 2 (E − E ) E2 2 3 2 4 3 + 2 E2 E4 − E32 Invariant under Feenberg scaling Π2(E0, . . . , E4) = Π2(E0(α), . . . , E4(α)) . Scaling property Π2(c E0, . . . , c E4) = c Π2(E0, . . . , E4) . 32
MANY-BODY PERTURBATION THEORY Dissociation barrier (kJ/mol) for H2CO−→H2 + CO Method E0 + E1 E2 E3 E4 F4 [2/2] Π2 Best Estimate
Minimum -113.912879 -114.329202 -114.334186 -114.359894 -114.360838 -114.362267 -114.364840
Transition state -113.748693 -114.182435 -114.185375 -114.219892 -114.220603 -114.223409 -114.227767
Barr. 431.1 385.3 390.7 367.6 368.2 364.6 359.9 360
(TZ2P Basis at MP2 Geometries)
33
ITERATION SEQUENCES
Fixed-point equation x = Ψ(x)
Direct Iteration x0, x1 = Ψ(x0), . . . , xn+1 = Ψ(xn), . . .
Cycling s0 = xStart, s1 = Ψ(s0), . . . , sk = Ψ(sk−1) xStart = T (s0, . . . , sk )
Corresponds to new iteration function: yn+1 = T (yn, Ψ(yn), . . . , Ψ(Ψ(. . . Ψ(yn)))
34
ORNSTEIN-ZERNIKE-EQUATION • Classical many-particle systems (fluids) • Pair distribution function g(r) = 1 + h(r)
• Integral equation h = c + ρh ∗ c g(r) = exp(−βu(r)+h(r)−c(r)+E(r)) • Bridge diagrams E(r) =⇒ Closure relations
35
• Solution on lattice with FFT: Γi = (h(ri) − c(ri))ri, ri = i∆r ~ Γ = Ψ(~ Γ) via – direct iteration – direct iteration + vector extrapolation • Extrapolation reduces CPU time by up to 50 % • Extrapolation useful to achieve convergence
2
directit 0 -2
ln ζ
-4 -6 -8 -10 0
200
400
600
800
1000
N Figure 1: Unstable Fixed-point of Direct Iteration (Hard spheres, high density) 36
m2vj 0 -5
ln ζ
-10 -15 -20
200
300
400
500
600
700
N Figure 2: Convergence of the Cycling Method
37
800
900
