Mathematics / advanced
Reliable numerical computation
Turn continuous mathematics into controlled approximations, from conditioning and stable algebra to integration and time stepping.
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Module 01 advanced
Reliable numerical computation
Turn continuous mathematics into controlled approximations, from conditioning and stable algebra to integration and time stepping.
- 01
Conditioning and backward error
Separate sensitivity of a mathematical problem from the stability of the algorithm that solves it.
26 min - 02
Floating-point arithmetic and stable formulas
Track representation, rounding, cancellation, and reformulation instead of assuming algebraic equality implies numerical equality.
26 min - 03
Root finding and safeguarded Newton iteration
Combine local speed with brackets, residual interpretation, and stopping rules that survive difficult starting points.
28 min - 04
Polynomial interpolation and node geometry
Build exact fits at samples while separating interpolation error, data sensitivity, and stable evaluation.
27 min - 05
Numerical differentiation and competing errors
Derive difference formulas, balance truncation against uncertainty, and recognize when smaller steps become worse.
27 min - 06
Quadrature and adaptive integration
Approximate integrals with explicit rules, refinement estimates, and a clear distinction between an estimated and guaranteed error.
28 min - 07
Linear systems and trustworthy factorization
Solve by structured factorizations, choose pivots sensibly, and verify the result with residuals rather than explicit inverses.
29 min - 08
Iterative linear solvers and preconditioning
Connect residual corrections, spectral convergence, Krylov spaces, and the practical role of a preconditioner.
30 min - 09
Numerical eigenvalue problems and residual certificates
Find spectral information with power iteration and structured algorithms while respecting sensitivity and invariant subspaces.
29 min - 10
Time stepping, stiffness, and stability regions
Separate an equation's dynamics from its discrete update and balance accuracy with stability for fast and slow modes.
30 min