Numerical Methods for Computer Science

Numerical methods convert mathematical models into reliable finite-precision computation.

Why This Section Matters

Many mathematically correct formulas become computationally fragile on real hardware. Numerical methods teach how to compute answers that are not only fast, but trustworthy.

Core Themes

  • Floating-point representation and rounding error
  • Algorithmic stability vs instability
  • Conditioning of problems vs quality of methods
  • Iterative vs direct methods
  • Convergence guarantees and stopping criteria

Dependency Map

flowchart LR
  A[Floating Point and Error] --> B[Root Finding]
  A --> C[Linear Systems]
  C --> D[Conditioning]
  B --> E[Optimization Methods]
  C --> E

Practical Outcomes

After this section, you should be able to: 1. predict when computations are numerically unsafe, 2. choose suitable solvers, 3. justify convergence/stability decisions, 4. validate outputs with diagnostics.

Exercises

  1. Give one example where mathematically equivalent formulas differ numerically.
  2. Explain difference between model error and numerical error.
  3. Define conditioning in one sentence.