Numerical Linear Algebra

Module aims

Distinguish a mathematically unique solution from a numerically reliable computation, and quote backward-error or residual bounds when the synopsis does.

Course outline

1. 01 Norms and condition numbers 20h

   Learning result Compute or bound a condition number and interpret it as sensitivity, not as a residual.

   Formative check Four-option check

2. 02 LU and pivoting 25h

   Learning result State when LU exists, what pivoting changes, and a case where naive LU fails.

   Formative check Counterexample

3. 03 QR and least squares numerically 25h

   Learning result Relate QR to least squares and contrast it with forming AᵀA when A is ill-conditioned.

   Formative check Written derivation

4. 04 Residual versus forward error 20h

   Learning result Give an example with a small residual and a large forward error.

   Formative check Counterexample

5. 05 Iterative methods (taught grain) 25h

   Learning result State an iteration, a contraction or residual monitor, and a stopping caveat.

   Formative check Written derivation

6. 06 Eigenvalue computations (introduction) 15h

   Learning result Name a method at synopsis grain and a stability issue for non-normal matrices.

   Formative check Four-option check

7. 07 Synthesis 20h

   Learning result Choose a factorisation or iteration for a stated system and justify the choice by conditioning.

   Formative check Written derivation

Learning results

Describe LU/QR or an iterative method with an operation count and a stability or residual caveat.

Subject skills

Matrix factorisations, condition numbers, residual vs error, iterative methods at taught grain.

Transferable skills

Report a residual; refuse to treat floating-point output as exact algebra.