lizfcm/notes/Nov-6.org

• Power Method
  • Properties
  • Shifting Eigenvalues
• Lanczos Algorithm

Power Method

v_k+1 = A v_k, k = 0,1,2

Properties

1. \frac{A v_k}{||v_k||} → v_1
2. \frac{v_k^T A v_k}{v_k^T v_k} → λ_1
3. If λ is a n eigenvalue of A, then \frac{1}{λ} is an eigenvalue of A^-1
4. Av = λ v Av - μ v = (λ-μ)v = (A - μ I)v
5. If λ is an eigenvalue of A, then λ - μ is an eigenvalue of A ⋅ μ I

Shifting Eigenvalues

1. Partition [λ_n, λ_1]

Lanczos Algorithm

  for (int i = 0; i < n; i++) {
    sum = a0;
    v_dot_v(a[i], x);

    b[i] = sum;
  }