Mathematics / advanced
Numerical linear algebra
Separate conditioning from algorithms while building stable factorizations, iterative methods, and low-rank approximations.
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Inside this module
Module 01 advanced
Numerical linear algebra
Separate conditioning from algorithms while building stable factorizations, iterative methods, and low-rank approximations.
- 01
Matrix norms and condition numbers
Measure amplification and sensitivity with a declared norm, perturbation model, and scale.
28 min - 02
Backward stability and perturbation bounds
Interpret residuals as data perturbations and turn backward accuracy into qualified forward-error estimates.
29 min - 03
Pivoted LU and growth factors
Track permutations, elimination multipliers, and intermediate growth before judging a solve's stability.
30 min - 04
Householder QR and least squares
Use orthogonal transformations to solve full-rank least squares without squaring the matrix condition number.
31 min - 05
Singular values and low-rank approximation
Use the SVD to quantify discarded directions, select a numerical rank, and understand truncation bias.
30 min - 06
Krylov subspaces and Arnoldi
Build an orthonormal basis from repeated matrix actions and retain the small projected problem.
31 min - 07
Conjugate gradients and preconditioning
Exploit positive-definite geometry, energy minimization, and a compatible preconditioner.
33 min - 08
GMRES and nonnormality
Minimize residuals over Krylov spaces and see why eigenvalues alone do not determine convergence.
32 min - 09
Symmetric eigensolvers and Ritz values
Preserve symmetry, project onto useful subspaces, and interpret eigenpair residuals with spectral gaps.
32 min - 10
Randomized range finding and sketching
Build a compressed basis with random probes while separating expectation bounds from verified approximation error.
34 min