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Lecture 15 Flashcards

(8 cards)

1
Q

Inverse power iteration

A
A-1 x = 1/λ x
|1/λn| > ... > |1/λ1|
xk+1 = A-1 xk
xk → (1/λn)^k αn un, converges to un
λn = (unT A un) / (unT un)
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2
Q

Inverse power iteration steps

A
1. factorize PLU=A
Ax{k+1}=xk so LUx{k+1}=P.T xk
2. Solve Ly=P.T xk
3. Solve U.x{k+1}=y
4. Normalize x{k+1}=x{k+1}/||x{k+1}||
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3
Q

Cost of computing largest eigenvalue? Smallest?

A

O(n2) vs O(n3) – indeed factorization n3

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4
Q

eigenvalue of (A+B/2)^-1?

A

2/(2λ1+λ2)

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5
Q

Inverse power method shifted matrix

A

(A-σI)^-1 x = ƛx
xk+1=(A-σI)^-1 xk
converges to largest ƛ, smallest (λ-σ), that is λ closest to σ (not |λ|!!!)

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6
Q

Inverse power method shifted matrix steps

A
  1. Factorize B=(A-σI)=PLU
  2. Solve Ly=P.T xk
  3. Solve Uxk+1 = y
    O(n^3))
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7
Q

Convergence inverse power method shifted matrix

A

||ek+1|| = |(λclosest - σ)/(λ2ndclosest - σ)| ||ek||

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8
Q

Rayleigh quotient iteration

A

inverse power method shifted matrix xk+1=(A-σkI)^-1 xk

(A-σkI) with σk updated at each iteration s.t. σk=xkT.A.xk/xkT.xk, convergence close to cubic, but cost per it n^3

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