Numerical linear algebra
Graduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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$1,162 · Numerical linear algebra · 18 tutoring hrs
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Topics typically covered
Standard syllabus
Matrix factorizations and direct methods
- Review of LU, QR, and Cholesky factorizations
- Pivoting strategies and backward stability analysis
- Sherman–Morrison–Woodbury and low-rank updates
- Sparse direct solvers: fill-in and reordering (overview)
- Block algorithms and cache efficiency (introduction)
Iterative methods for linear systems
- Krylov subspace methods: CG, MINRES, GMRES
- Preconditioners: Jacobi, SSOR, incomplete factorizations
- Convergence theory for SPD systems
- Nonsymmetric systems and flexible Krylov variants
- Restart strategies and breakdown remedies
Eigenvalue and SVD computations
- Power method, inverse iteration, and Rayleigh quotient
- QR algorithm and Francis shifts
- Lanczos and Arnoldi processes
- Singular value decomposition algorithms
- Pseudospectra and sensitivity of eigenvalues (introduction)
STEM / applied
Large-scale and applied problems
- Sparse matrix formats and memory-aware implementations
- Parallel and distributed linear algebra (overview)
- Least squares and ridge regression at scale
- Randomized numerical linear algebra (sketching, introduction)
- Applications in data science, imaging, and PDE discretizations
Software and practice
- LAPACK/BLAS ecosystem and best practices
- Conditioning diagnostics in engineering workflows
- Mixed-precision algorithms (introduction)
- Benchmarking and reproducibility in HPC environments
- Case studies from structural analysis and machine learning
Notes
Topics reflect common graduate numerical linear algebra syllabi at US universities. This course is distinct from a first numerical analysis course in its depth on matrix algorithms and large-scale computation.