Linear systems
Solve Ax=b through direct and iterative methods while tracking sparsity, scale and structure.
Side 220Side Studies / Research
Subject
Numerical Linear Algebra
Purpose
Linear systems, eigenproblems and matrix factorizations studied through algorithms, conditioning, stability and finite-precision computation.
Structure
05 movesFormal systemV0
Definitions → structures → relations → proof → application
Separate exact linear algebra from computed linear algebra.
Numerical linear algebra asks not only whether a matrix problem has a solution, but whether that solution can be computed accurately and efficiently on real hardware.
Factorizations
Use LU, QR, Cholesky and related decompositions to expose stable computational pathways.
Conditioning
Measure how sensitive a problem's exact solution is to perturbations in data.
Stability
Analyze how an algorithm amplifies rounding and approximation error beyond the underlying problem sensitivity.
Eigen & singular-value problems
Compute dominant modes, ranks and low-dimensional structure using algorithms suited to dense or sparse matrices.
Keep the boundaries visible.
conditioning ≠ algorithm stability
exact arithmetic ≠ floating-point arithmetic
small residual ≠ small solution error
Questions that organize the Side.
How can a numerically stable algorithm still produce an inaccurate answer?
Why is QR often preferred to normal equations in least squares?
How do matrix structure and sparsity change algorithm choice?
What should carry weight here?
Report residuals, conditioning and convergence behavior; numerical agreement alone is weak evidence when the problem is ill-conditioned.