Assumptions: CS 357 Numerical methods and MATH 484 Nonlinear “Programming” (Optimization)

As a course

The “midterm” exams are taken as the best 3 out of 4 (this excludes the final) → skip exam4

Reminder

• Remember that condition number is not negative. Need to take absolute value.
• Distinguish “1-norm” and “1-norm condition number”
• Product of symmetric matrices is symmatric ↔ AB=BA

Review

I guess this is because condition number of A is about sensitivity to input, while pivoting affect the algorithm’s stability.

Math

Where H means Conjugate transpose. Note that orthogonal matrix is normal.

• Full rank matrix

  

• Thm. (Spectral theorem) Matrix A is symmetric ↔ $\exists$ orthogonal Q and diagonal $\Lambda \in \R^{n \times n}$ s.t. $A = Q \Lambda Q^T$, where $\Lambda = diag(\lambda_1 \dots \lambda_n)$ and columns of Q will be the eigenvectors of A. Note that this means there are $n$ real eigenvalues (but converse is not true).

  • In fact, Q can be chosen to be orthonormal (unitarily diagonalizable), s.t. $Q^T = Q^{-1}$. This happens ↔ A is normal ($A A^* = A^* A$)

Ch1 Error Analysis

Ch2 Linear System: Finding x for Ax=b

Ch3 Linear Least Squares (LLSQ): Finding x for Ax~b

Ch4 Eigenvalue Problems

Ch5 Nonlinear equations with $f:\R^n \to \R^n$

Ch6 Optimization with $f: \R^n \to \R$