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Numerical Methods - Deep Dive

Mathematical Foundations​

Rigorous treatment of numerical linear algebra.

Norms Theory​

Vector Norms​

Definition: A norm on ℝⁿ satisfies:

  1. ||x|| ≥ 0 with equality iff x = 0
  2. ||cx|| = |c| ||x||
  3. ||x + y|| ≤ ||x|| + ||y||

Examples: (To be completed)

Matrix Norms​

Induced Norm: ∣∣A∣∣=max⁡∥x∥=1∣∣Ax∣∣||A|| = \max_{\|x\|=1} ||Ax||

Theorem: For induced 2-norm, ∣∣A∣∣2=σ1||A||₂ = σ₁ (largest singular value)

Proof: (To be completed)

Conditioning​

Condition Number Theory​

Theorem: The relative error in x satisfies: (1/κ(A)) (||δb||/||b||) ≤ ||δx||/||x|| ≤ κ(A) (||δb||/||b||)

Proof: (To be completed)

Implications​

(To be completed)

Eigenvalue Algorithms​

Power Method​

Theorem: The power method converges to dominant eigenvector if |λ₁| > |λ₂|

Proof: (To be completed)

QR Algorithm​

Algorithm: (To be completed)

Convergence: (To be completed)

Iterative Methods​

Convergence Theory​

Theorem: Jacobi iteration converges if A is strictly diagonally dominant

Proof: (To be completed)

Conjugate Gradient​

Theorem: CG finds exact solution in at most n steps

Proof: (To be completed)

Exercises​

(Advanced problems to be completed)