lanczos
DESCRIPTION
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Lanczos Method for Eigensystems
Aiichiro Nakano Collaboratory for Advanced Computing & Simulations
Department of Computer Science Department of Physics & Astronomy
Department of Chemical Engineering & Materials Science University of Southern California
Email: [email protected]
B. N. Parlett The Symmetric Eigenvalue Problem
(Prentice-Hall, ʼ80) Secs. 11-13
Rayleigh Quotient Theorem Let A be an n×n real symmetric matrix, λ1[A] ≤ … ≤ λn[A] its eigenvalues in ascending order, x ∈ Rn, & the Rayleigh quotient
then
Proof Let q(k) be the k-th orthonormalized eigenvector of A, , & orthogonal transformation matrix, , then €
ρ(x;A) =xTAxxTx
€
λ1[A] =x∈ℜnmin ρ(x;A)
λn[A] =x∈ℜnmax ρ(x;A)
€
Aqk = λkqk
€
Q = q1q2qn[ ]
Let x = Qz (note QTQ = I), then
which is a weighted average of λ1, …, λn, & the minimum is when zT = (1,0,…,0) = e1 & x = Qe1 = q1.
€
QTAQ =
λ1
λn
€
ρ(x;A) =zTQTAQzzTQTQz
=z12λ1 +zn
2λnz12 +zn
2
Rayleigh-Ritz Procedure Theorem Let {q1,…,qm} be an orthonormal set that spans Rm (m < n) ⊂ Rn, so that any vector x ∈ Rm is expressed as a linear combination of q1,…,qm:
or
then the best approximations for λ1[A] & λn[A] are obtained by diagonalizing
as λ1[H] & λn[H].
Proof Note
then
the minimum of which is λ1[H].
€
x = z1q1 ++ zmqm
€
1
nx1
xn
=
m
n q1 qm
1z1
zm
m =Qz
€
m ×mH =
m × nQT
n × nA
n ×mQ
€
ρ(x;A) =zTQTAQzzTQTQz
=zTHzzTz
=z12λ1(H )++ zm
2 λm (H )z12 ++ zm
2
€
QTQ( ) ij = QkiQkjk=1
n∑ = qi • q j = δ ij 1≤ i, j ≤ m
Orthogonalization by QR Decomposition • Gram-Schmidt orthonormalization: The orthonormal set Q required for
the Rayleigh-Ritz procedure is obtained starting from an arbitrary set of m vectors, S = [s1…sm] (sj ∈ Rn) as:
€
q1 = s1 / | s1 |for i = 2 to m
′ q i = si − q j q j • si( )j=1
i−1∑
qi = ′ q i / | ′ q i |endfor
€
si =| ′ q i | qi + q j q j • si( )j=1
i−1∑
• The Gram-Schmidt amounts to QR decomposition, S = QR, where R is an m×m right-triangle matrix:
€
m
n s1 s2 s3 s4
=
m
n q1 q2 q3 q4
m| ′ q 1 | q1 • s2 q1 • s3 q1 • s40 | ′ q 2 | q2 • s3 q2 • s40 0 | ′ q 3 | q3 • s40 0 0 | ′ q 4 |
m
Projection!
Rayleigh-Ritz Algorithm
1. Start from S = [s1…sm] (sj ∈ Rn) & do Gram-Schmidt orthonormalization, S = QR, to obtain an orthonormal set Q = [q1…qm]
2. Form H = QTAQ
3. Diagonalize H to get λ1[H],…,λm[H]:
4. The approximations of λ1[A] & λn[A] are given by λ1[H] & λm[H] with the corresponding eigenvectors, yk = Qgk (k = 1 & m).
€
QTAQH
gk = λk[H]gk
↓ Q ×
∴AQgky k
= λk[H ]Qgky k
€
Hgk = λk[H]gk (k = 1,…,m)
Krylov Subspace • Krylov subspace Sm is spanned by a Krylov matrix, Km(f) = [f Af … Am-1f]
(f ∈ Rn) Theorem Let Qm be the orthonormal basis obtained by QR factorization, Km(f) = QmR, then Tm = Qm
TAQm is a tridiagonal matrix Proof For i > j+1, qi
T(Aqj) = 0, since Aqj ⊂ Sj+1 by construction & qi ⊥ Sj+1 by Gram-Schmidt orthonormalization for i > j+1. By the symmetry of A, qi
T(Aqj) = qj
T(ATqi) = qjT(Aqi) = 0 for j > i+1 or i < j-1.
€
Tm =
α1 β1β1 α2 β2
βm−2 αm−1 βm−1βm−1 αm
€
α j = q jTAq j j = 1,…,m
β j = q j+1T Aq j j = 1,…,m −1
Recursion Formula
• Due to the tridiagonality, Aqi is a linear combination of qi-1, qi & qi+1:
If we define q0 = 0, the above equation is valid for i = 1 as well. Let ri ≡ βiqi+1 (ri is a component of Aqi orthogonal to qj for j ≤ i), then
• Lanczos algorithm:
€
Aqi = β i−1qi−1 +α iqi + β iqi+1 (2 ≤ i ≤ m −1)
€
ri = Aqi − β i−1qi−1 −α iqi (1≤ i ≤ m −1)
€
Given r0,β0 = r0 q0 = 0( )for i = 1,,mqi ← ri−1 /β i−1ri ← Aqi − β i−1qi−1αi ← qi
Triri ← ri −αiqiβ i = ri only when i ≤ m −1( )
endfor
€
qiT (Aqi − β i−1qi−1) = qi
TAqi =α i (orthogonality)
Keep increasing m until λ1[Tm] converges
An Application of Rayleigh-Ritz/Lanczos • Search for transition states (with a negative eigenvalue of the Hessian matrix, ∂2E/∂ri∂rj, by following the eigenvector with the smallest eigenvalue —Rayleigh-Ritz: Kumeda, Wales & Munro, Chem. Phys. Lett. 341, 185 (ʼ01) —Lanczos: Mousseau et al., J. Mol. Graph. Model. 19, 78 (ʼ01)
Figure from Prof. H. B. Schlegel; http://chem.wayne.edu/schlegel
Lanczos Algorithm for Hessian Calculation
Sample Run of Lanczos Program
Electronic Energy Bands of GaAs
C. Pryor, Phys. Rev. B 57, 7190 (’98)
• 8-band k•p model
Band-edge Wave Functions • Band-edge states in an array of GaN quantum dots in AlN matrix
S. Sburlan, Ph.D. dissertation, USC (‘13)
Valence-band states
Conduction-band states