Fixed-point iteration
General · Mathematics
Study notes
Q: Solve x = cos x by fixed-point iteration from x₀ = 0.5. Does it converge? g(x) = cos x: |g'(x)| = |sin x| < 1 near 0.7 ✓: converges! x₁ = cos(0.5) ≈ 0.8776. x₂ = cos(0.8776) ≈ 0.6390. x₃ = cos(0.6390) ≈ 0.8027. → 0.739085... (Dottie number!)! (Contraction: |g'|<1!)