The Charming Leading Eigenpair
Mu-Fa Chen
(Beijing Normal University)
FRONTIER PROBABILITY DAYS
Salt Lake City, Utah. May 8-11, 2016
2016 c 5
Mu-Fa Chen (BNU)
Leading eigenpair
10 F
2016 c 5
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1 / 35
Outline
Computing the maximal eigenpair
Theoretic results: unified speed
estimation of various stabilities
Original motivation: study on the
phase transitions
Mu-Fa Chen (BNU)
Leading eigenpair
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1)
The maximal eigenpair
Theorem (O. Perron, 1907; G.Frobenius 1912)
For A > 0, irreducible, ∃ρ(A) > 0-maximal
eigenvalue with left-eigenvector u > 0 and
right-eigenvector g > 0. Dimension: one.
uA = λu,
Ag = λg,
λ = ρ(A)
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Leading eigenpair
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1)
The maximal eigenpair
Theorem (O. Perron, 1907; G.Frobenius 1912)
For A > 0, irreducible, ∃ρ(A) > 0-maximal
eigenvalue with left-eigenvector u > 0 and
right-eigenvector g > 0. Dimension: one.
uA = λu,
Ag = λg,
λ = ρ(A)
Example (L.K. Hua, 1984)
!
1
25 14
A=
,
100 40 12
√
ρ(A) = 37 + 2409 /200.
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Leading eigenpair
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√
u = 5 13+ 2409 /7, 20 , ≈ 44.34397483
√
g = 13+ 2409 /4, 20
Let A > 0 be irreducible and invertible.
Input-output method: xn = x0A−n, n > 1.
(0)
(d)
xn = xn , · · · , xn : products in nth year.
Theorem (Hua’s Fundamental Theorem, 1984)
The optimal choice is x0 = u, it has the
fastest grow: xn = x0ρ(A)−n.
If A−16> 0 and x0 6= u, then collapse: ∃x(j)
n 6 0.
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Importance of u ≈(44.34397483, 20)
Table Input and collapse time
x0 [decimals]
Collapse time n
(44, 20)
3
(44.344, 20)
(44.34397483, 20)
Mu-Fa Chen (BNU)
Leading eigenpair
8
13
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Importance of u ≈(44.34397483, 20)
Table Input and collapse time
x0 [decimals]
Collapse time n
(44, 20)
3
(44.344, 20)
(44.34397483, 20)
8
13
(ρ(A), u)? High precision, large system
uA = λu ⇐⇒ A∗u∗ = λu∗
A: irreducible, off-diagonals > 0:
(A + mI)g = λg ⇐⇒ Ag = (λ − m)g
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Leading eigenpair
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Main Example







Q=





−1 1
0
0
0
0
0
0
1 −5 22
0
0
0
0
0
2
2
0 2 −13 3
0
0
0
0
0
0
32 −25 42
0
0
0
2
2
0
0
0
4
−41 5
0
0
2
2
0
0
0
0
5
−61 6
0
0
0
0
0
0
62 −85 72
0
0
0
0
0
0
72 −113
ρ(Q) ≈ −0.525268,













ρ(Infinite Q) = −1/4
g ≈ (55.878, 26.5271, 15.7059, 9.97983,
6.43129, 4.0251, 2.2954, 1)∗
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Leading eigenpair
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Power iteration, 1929
Given v0 ∈ RN +1, v0 6⊥ g with kv0k = 1, define
Avk−1
vk =
, zk = kAvk k,
k > 1,
kAvk−1k
Then vk → g and zk → ρ(A) as k → ∞.
vk =
Ak v0
“Power”
kAk v0k
ṽ0 = (1, 0.587624, 0.426178, 0.329975,
0.260701, 0.204394, 0.153593,
0.101142)∗,
P
v0 = ṽ0/kṽ0k, kvk = k |vk |, `1-norm
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Leading eigenpair
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0
1
2
3
4
5
6
7
8
9
10
2.11289
1.42407
1.37537
Computing
1.22712
180 times,
1.1711
103 iterations
1.10933
64 pages
1.06711
1.02949
0.998685
(k, −zk )
0.971749
0.948331
Mu-Fa Chen (BNU)
Leading eigenpair
11
12
13
14
15
16
17
18
19
20
30
0.927544
0.908975
0.892223
0.877012
0.86311
0.850338
0.838548
0.827619
0.817449
0.807953
0.738257
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40
50
60
70
80
90
100
120
140
160
0.694746
0.664453
0.641946
0.624473
0.610468
0.598963
0.589332
0.574136
0.56279
0.554157
Mu-Fa Chen (BNU)
180
200
300
400
500
600
700
800
900
> 990
Leading eigenpair
0.547529
0.542423
0.529909
0.526517
0.525603
0.525358
0.525292
0.525274
0.52527
0.525268
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2.0
1.5
The figure of − zk
for k = 0, 1, . . . , 1000.
1.0
200
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400
Leading eigenpair
600
800
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Inverse iteration, 1944
`2-norm
Rayleigh quotient iteration. Choose
(z0, v0) ≈ (ρ(A), g) with v0∗v0 = 1, where
v ∗ = transpose of v. Particular, z0 = v0∗Av0
for given v0. At the kth step (k > 1), define
vk =
(A−zk−1I)−1vk−1
k(A−zk−1I)−1vk−1k
,
zk = vk∗Avk
where I = identity. Then vk → g and zk → ρ(A)
provided (z0, v0) is closed enough to (ρ(A), g)
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Leading eigenpair
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What can we expect for the 2nd alg?
Search maximum on (0, 1)
with accuracy of 10−6.
Golden Section Search: 10−6 ≈ 0.61824.
Bisection Method: 10−6 ≈ 0.520.
Use the Rayleigh quotient iteration[RQI].
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Leading eigenpair
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What can we expect for the 2nd alg?
Search maximum on (0, 1)
with accuracy of 10−6.
Golden Section Search: 10−6 ≈ 0.61824.
Bisection Method: 10−6 ≈ 0.520.
Use the Rayleigh quotient iteration[RQI].
Example
The same matrix Q and ṽ0, need 2 steps only:
z1 ≈ −0.528215, z2 ≈ −0.525268.
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Leading eigenpair
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“Too Good” is dangerous. Pitfall
λj := λj (−Q). λ0 = −ρ(Q) > 0.
Example
Let Q be the same as above. Choose
√
v0 = {1, 1, 1, 1, 1, 1, 1, 1}/ 8.
Then
(z1, z2, z3, z4) ≈
ṽ0 efficient!
(4.78557, 5.67061, 5.91766, 5.91867).
(λ0, λ1, λ2)≈(0.525268, 2.00758, 5.91867).
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Example
Let Q and v0 be the same as in the last
example. Choose
z0 = 2.05768−1 ≈ 0.485985.
Then z1 ≈ 0.525313, z2 ≈ 0.525268.
z0 efficient!
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Comparison of different initials
Q
1
2
3
4
v0
z0
ṽ0
Power
ṽ0
Auto
Uniform Auto
Uniform δ1−1
# of Iterations
103
2
Collapse
2
RQI is much efficient than Power One
The initials (v0, z0) are very sensitive!
It is very hard to handle with the initials
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Leading eigenpair
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Google’s PageRank
Langville, A.N., Meyer, C. D. (2006).
Google’s PageRank and Beyond: The Science
of Search Engine Rankings.
Princeton University Press.
Power Iteration, included.
Inverse Iteration, not touched!
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Google’s PageRank
Langville, A.N., Meyer, C. D. (2006).
Google’s PageRank and Beyond: The Science
of Search Engine Rankings.
Princeton University Press.
Power Iteration, included.
Inverse Iteration, not touched!
For large N , guess: # of iterations ∼ N α.
81/3= 2. 104/3 ≈ 22.
Subvert
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Large N. λ0 = 1/4 if N=∞. 6 30 Sec
Use ṽ0 and δ1. Let z0 = 7/(8δ1)+v0∗(−Q)v0.
N+1
z0
z1
z2 = λ0 upper/lower
8 0.523309 0.525268 0.525268 1+10−11
100
500
1000
5000
7500
104
0.387333
0.349147
0.338027
0.319895
0.316529
0.31437
Mu-Fa Chen (BNU)
0.376393
0.338342
0.327254
0.30855
0.304942
0.302586
Leading eigenpair
0.376383
0.338329
0.32724
0.308529
0.304918
0.302561
2016 c 5
1+10−8
1+10−7
1+10−7
1+10−7
1+10−7
1+10−7
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Efficient initials. Tridiagonal case
E = {0, 1, . . . , N }.
P
Q = A − mI, m := maxi∈E j∈E aij .

−(b0 +c0 )
b0
0

a1
−(a1 +b1 +c1 )
b1


0
a2
−(a2 +b2 +c2 )
Q =

.
.
..
..
..

.
0
0
0
ai > 0,
µ0 = 1, µn = µn−1
Mu-Fa Chen (BNU)
0
0
b2
..
.
bn−1
an
aN
bi > 0,
···
···
···
..
.







−(aN +cN ))
ci > 0.
, n > 1. Speed measure
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Def {hn}, {ϕn}.
Explicit
h0 = 1, hn = hn−1rn−1, 1 6 n 6 N,
where r0 = 1+c0/b0,
an + cn
an
rn = 1 +
−
, 1 6 n < N,
bn
bnrn−1
hN +1 = cN hN + aN (hN −1 − hN ),
ci ≡ 0(i < N ) =⇒ hi ≡ 1. Q\the last rowh = 0
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Leading eigenpair
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Def {hn}, {ϕn}.
Explicit
h0 = 1, hn = hn−1rn−1, 1 6 n 6 N,
where r0 = 1+c0/b0,
an + cn
an
rn = 1 +
−
, 1 6 n < N,
bn
bnrn−1
hN +1 = cN hN + aN (hN −1 − hN ),
ci ≡ 0(i < N ) =⇒ hi ≡ 1. Q\the last rowh = 0
N
X
ϕn =
1
h h µ b
k=n k k+1 k k
Mu-Fa Chen (BNU)
, 0 6 n 6 N, bN := 1.
Leading eigenpair
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RQ Iteration: tridiagonal case
√
ṽ0(i) = hi ϕi, i 6 N ;
v0 = ṽ0/kṽ0kµ,2;
n
√ X
√
ϕn
µk h2k ϕk
δ1 = max
06n6N
k=0
3/2
+ϕn−1/2
µj h2j ϕj
n+16j6N
X
Solve wk : (−Q − zk−1I)wk = vk−1,
=: z0−1.
k > 1;
vk= wk /kwk kµ,2 → g, zk= (vk , −Qvk )µ,2→λ0
(ai, bi, ci) → (µi, ϕi, hi)
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Leading eigenpair
Generalization
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2) Unified speed estimation of various stabilities
Theorem (Informal ! 1988→ 2010–2014)
For tridiagonal matrix Q or one-dim elliptic
operator (order 2) with/without killing on a
finite/infinite interval, in each of 20 cases, there
exist explicit δ, δ1, δ10 (and then δn, δn0 ,
recursively) such that δn0 ↑, δn ↓ and
(4δ)−16 δn−1 6 λ0 6 δn0
−1
6 δ −1,
n > 1.
Besides, 1 6 δ10 −1/δ1−1 6 2.
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2) Unified speed estimation of various stabilities
Theorem (Informal ! 1988→ 2010–2014)
For tridiagonal matrix Q or one-dim elliptic
operator (order 2) with/without killing on a
finite/infinite interval, in each of 20 cases, there
exist explicit δ, δ1, δ10 (and then δn, δn0 ,
recursively) such that δn0 ↑, δn ↓ and
(4δ)−16 δn−1 6 λ0 6 δn0
−1
6 δ −1,
n > 1.
Besides, 1 6 δ10 −1/δ1−1 6 2.
One-dim elliptic operators, δ −→ κ.
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Leading eigenpair
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State space E = (−M, N ), M, N 6 ∞.
Eigenequation: Lg = −λg, g 6= 0.
Four boundaries. Use codes ‘D’ and ‘N’.
D: (Abs.) Dirichlet boundary g(−M )= 0 limM
N: (Ref.) Neumann boundary g 0(−M ) = 0.
•
•
•
•
λNN:
λDD:
λDN:
λND:
Neumann boundaries at −M and N .
Dirichlet boundaries at −M and N .
Dirichlet at −M and Neumann at N .
Neumann at −M and Dirichlet at N .
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Leading eigenpair
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Speed of L2-exp stability Pt = etL
L = a(x)
d2
d
Z
x
b
+ b(x) ,
C(x) =
dx2
dx
θ a
C
dµ e
dν̂
Speed
= , scale
= e−C, k·k = k·kL2(µ)
dx
a
dx
Z
kPtf k 6 kf k e−λ
NN t
,
µ(f ) :=
f dµ = 0,
E
−λ# t
kPtf k 6 kf k e
, t > 0, f ∈ L2(µ).
λNN = λ1: L2-exponentially ergodic rate.
Other λ#: L2-exponential decay rate.
∗
∗
kPt (x, ·)−πkVar6 c(x)e−α t , Pt (x, K)6 c(x, K)e−α t. α∗= λ#
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Theorem (C. 2010). For each # of 4 cases, the
−1
−1
unified estimates 4κ# 6 λ#6 κ#
,
Rβ
where
µ(α, β)= α dµ.
−1
= inf µ(−M, x)−1 + µ(y, N )−1 ν̂(x, y)−1
x<y
−1
κDD = inf ν̂(−M, x)−1 + ν̂(y, N )−1 µ(x, y)−1
κNN
x6y
κDN=
sup ν̂(−M, x) µ(x, N )
1920–1972
x∈(−M,N )
κND=
sup
µ(−M, x) ν̂(x, N ) µ and ν̂, factor 4
x∈(−M,N )
In particular, λ# > 0 iff κ# < ∞.
1988–2010. Three probabilistic tools + 5 steps.
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Intrinsic relation between 4 constants κ#
Remove ν̂(y,N )−1
DD
κ
−−−−−−−→
x

yRule
NN
κ
Remove µ(y,N )−1
DN
κ
x

yRule
ND
−−−−−−−→ κ
Mu-Fa Chen (BNU)
Leading eigenpair
Rule:
Codes D ↔ N
⇔ µ ↔ ν̂ in κ#
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Lc = a(x)d2/dx2+b(x)d/dx−c(x), a > 0, c > 0
Poincaré-type inequalities : (−M, N ), M, N6 ∞
λc# kf k2µ, 26kf 0k2ν,2 + kcf k2µ,2
= if f = g ←→ λc#
√
ν(dx) = eC(x)dx.
ν̂(dx) = e−C(x)dx
λ# kf kµ, 2 6 kf 0kν,2 (when c ≡ 0)
C(x) Hardy-type inequalities : ν̂(dx)= exp − p−1 dx
kf kµ, q 6 A#kf 0kν, p.
1/q
k|f |q kB
Mu-Fa Chen (BNU)
p, q∈(1, ∞). t−α
6 AB#kf 0kν, p, New analytic proof!
Leading eigenpair
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B, k · kB , µ)
Normed linear space (B
B subset of Borel meas funs on (X, X , µ).
Hypotheses (H). Norm on B is defined by
Z
|f | gdµ,
kf kB = sup
g∈G
R+.
where G ⊂ X /R
X
G = Lp(p > 1) =⇒ B = Lq .
kf kL1(π) 6 Ent(f ) 6 kf k1+ε
. Interpolation.
L1+ε (π)
R g
G = g > 0 : X e dπ 6 e2 +1 LogSobolev
DD: f (−M ) = f (N ) = 0 .
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κ#6 λ# −16 4κ#, B #6 A# 6 2B # DD
1/q
BB
sup x6y
k1(x, y) kB
ν̂(−M,x)1−p +ν̂(y,N )1−p
1/p
B =L1(µ)
µ(x,y)1/q
sup 1/p
B
x6y ν̂(−M,x)1−p +ν̂(y,N )1−p
q=p=2
sup ν̂(−M,x)µ(x,y)
−1
+ν̂(y,N )−1
κ
x6y
Killing c
c (x,y)
sup ν̂ (−M,x)µ−1
+ν̂c (y,N )−1
κc
x6y c
h2µ, h−2ν̂, Lch=0. In parallel for ridiagonal matrix
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3)
Exp stability in L2 or entropy
Semigroup Pt = etL. L2(π), k · k, (·, ·).
L self-adjoint: (f, Lg) = (Lf, g).
kPtf −π(f )k 6 kf ke−εt,
εmax= λ1 := λNN.
Exponential stability in entropy
Ent(Ptf ) 6 Ent(f )e−2σt,
t > 0,
Z
dµ
=f
Ent(f ) := H(µkπ) = f log f dπ if
dπ
E
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The ϕ4 Euclidean quantum field on the lattice
Zd
X
R 3 x : Z → R. H(x) = −2J xixj , J, β > 0.
d
hiji
X
0
∂ii −(u (xi)+∂iH)∂i , u(xi)= x4i−βx2i
L=
i∈Zd
Theorem (C. 2008)
inf
inf λβ,J
Λ,
ω
≈ inf
1
ΛbZd ω∈RZd
inf σ β,J Λ, ω
ΛbZd ω∈RZd
2
≈ exp −β /4−c log β −4dJ c ∈ [1, 2]
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Leading eigenpair
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Phase transition: the ϕ4 model
β,r
2
,
σ
≈
exp
−β
/4−c
log
β
− 2r
λβ,r
1
c = c(β) ∈ [1, 2]
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Leading eigenpair
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Phase transition: the ϕ4 model
β,r
2
,
σ
≈
exp
−β
/4−c
log
β
− 2r
λβ,r
1
c = c(β) ∈ [1, 2]
Looking for math tools to
study the phase transitions
Mu-Fa Chen (BNU)
Leading eigenpair
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For Further Reading I
C. (2008)
Spectral gap and logarithmic Sobolev constant
for continuous spin systems
Acta Math. Sin. New Ser. 24(5): 705–736
C. (2010)
Speed of stability for birth–death processes
Front. Math. China. 5:3, 379–515
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For Further Reading II
C. (2014)
Criteria for discrete spectrum of 1D operators
Commun. Math. Stat. 2014, 2(3): 279õ309
C. (2016) Survey article, free downloadable!
Unified speed estimation of various stabilities
Chin. J. Appl. Probab. Statis. 32(1), 1–22
C. (2016): Efficient initials for computing
the maximal eigenpair, preprint
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For Further Reading III
C. and Zhang, X. (2014).
Isospectral operators.
Commu. Math. Stat. 2, 17–32
C. and Zhang, Y.H. (2014).
Unified representation of formulas for single
birth processes.
Front. Math. China 9(4): 761–796
http://math0.bnu.edu.cn/˜chenmf 4 volumes
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The end!
Thank you, everybody!
Œ[œ
42
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