Standard syllabus
Numerical linear algebra · Graduate · Math
Learning objectives from the Numerical linear algebra syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Numerical Linear Algebra — scope drawn from open NLA notes (e.g. Trefethen/Bau-style OER mirrors) and typical US graduate NLA syllabi.
Matrix factorizations and direct methods
- Review of LU and QR factorizations — Review of LU and QR factorizations
- Cholesky factorizations — Cholesky factorizations
- Pivoting strategies — Pivoting strategies
- Backward stability analysis — Backward stability analysis
- Sherman–Morrison–Woodbury — Sherman–Morrison–Woodbury
- Low-rank updates — Low-rank updates
- Sparse direct solvers — Sparse direct solvers
- Reordering (overview) — Reordering (overview)
- Block algorithms — Block algorithms
- Cache efficiency (introduction) — Cache efficiency (introduction)
Iterative methods for linear systems
- Krylov subspace methods: CG, MINRES, GMRES — Krylov subspace methods: CG, MINRES, GMRES
- Preconditioners: Jacobi, SSOR, incomplete factorizations — Preconditioners: Jacobi, SSOR, incomplete factorizations
- Convergence theory for SPD systems — Convergence theory for SPD systems
- Nonsymmetric systems and flexible Krylov variants — Nonsymmetric systems and flexible Krylov variants
- Restart strategies — Restart strategies
- Breakdown remedies — Breakdown remedies
Eigenvalue and SVD computations
- Power method and inverse iteration — Power method and inverse iteration
- Rayleigh quotient — Rayleigh quotient
- QR algorithm and Francis shifts — QR algorithm and Francis shifts
- Lanczos and Arnoldi processes — Lanczos and Arnoldi processes
- Singular value decomposition algorithms — Singular value decomposition algorithms
- Pseudospectra — Pseudospectra
- Sensitivity of eigenvalues (introduction) — Sensitivity of eigenvalues (introduction)
Large-scale and applied problems
- Sparse matrix formats — Sparse matrix formats
- Memory-aware implementations — Memory-aware implementations
- Parallel and distributed linear algebra (overview) — Parallel and distributed linear algebra (overview)
- Least squares — Least squares
- Ridge regression at scale — Ridge regression at scale
- Randomized numerical linear algebra (sketching, introduction) — Randomized numerical linear algebra (sketching, introduction)
- Applications in data science and imaging — Applications in data science and imaging
- PDE discretizations — PDE discretizations
Software and practice
- LAPACK/BLAS ecosystem and best practices — LAPACK/BLAS ecosystem and best practices
- Conditioning diagnostics in engineering workflows — Conditioning diagnostics in engineering workflows
- Mixed-precision algorithms (introduction) — Mixed-precision algorithms (introduction)
- Benchmarking — Benchmarking
- Reproducibility in HPC environments — Reproducibility in HPC environments
- Case studies from structural analysis — Case studies from structural analysis
- Machine learning — Machine learning
Learning objectives
Click an objective for study materials.
Matrix factorizations and direct methods
- Review of LU and QR factorizations — Review of LU and QR factorizations
- Cholesky factorizations — Cholesky factorizations
- Pivoting strategies — Pivoting strategies
- Backward stability analysis — Backward stability analysis
- Sherman–Morrison–Woodbury — Sherman–Morrison–Woodbury
- Low-rank updates — Low-rank updates
- Sparse direct solvers — Sparse direct solvers
- Reordering (overview) — Reordering (overview)
Iterative methods for linear systems
- Krylov subspace methods: CG, MINRES, GMRES — Krylov subspace methods: CG, MINRES, GMRES
- Preconditioners: Jacobi, SSOR, incomplete factorizations — Preconditioners: Jacobi, SSOR, incomplete factorizations
- Convergence theory for SPD systems — Convergence theory for SPD systems
- Nonsymmetric systems and flexible Krylov variants — Nonsymmetric systems and flexible Krylov variants
- Restart strategies — Restart strategies
- Breakdown remedies — Breakdown remedies
Eigenvalue and SVD computations
- Power method and inverse iteration — Power method and inverse iteration
- Rayleigh quotient — Rayleigh quotient
- QR algorithm and Francis shifts — QR algorithm and Francis shifts
- Lanczos and Arnoldi processes — Lanczos and Arnoldi processes
- Singular value decomposition algorithms — Singular value decomposition algorithms
- Pseudospectra — Pseudospectra
- Sensitivity of eigenvalues (introduction) — Sensitivity of eigenvalues (introduction)
Large-scale and applied problems
- Sparse matrix formats — Sparse matrix formats
- Memory-aware implementations — Memory-aware implementations
- Parallel and distributed linear algebra (overview) — Parallel and distributed linear algebra (overview)
- Least squares — Least squares
- Ridge regression at scale — Ridge regression at scale
- Randomized numerical linear algebra (sketching, introduction) — Randomized numerical linear algebra (sketching, introduction)
- Applications in data science and imaging — Applications in data science and imaging
- PDE discretizations — PDE discretizations
Software and practice
- LAPACK/BLAS ecosystem and best practices — LAPACK/BLAS ecosystem and best practices
- Conditioning diagnostics in engineering workflows — Conditioning diagnostics in engineering workflows
- Mixed-precision algorithms (introduction) — Mixed-precision algorithms (introduction)
- Benchmarking — Benchmarking
- Reproducibility in HPC environments — Reproducibility in HPC environments
- Case studies from structural analysis — Case studies from structural analysis
- Machine learning — Machine learning
Definitions and structure
- Review of LU and QR factorizations — Review of LU and QR factorizations
- Cholesky factorizations — Cholesky factorizations
- Pivoting strategies — Pivoting strategies
- Backward stability analysis — Backward stability analysis
- Sherman–Morrison–Woodbury — Sherman–Morrison–Woodbury
Proofs and reasoning
- Low-rank updates — Low-rank updates
- Sparse direct solvers — Sparse direct solvers
- Reordering (overview) — Reordering (overview)
- Block algorithms — Block algorithms
- Cache efficiency (introduction) — Cache efficiency (introduction)
Abstraction and generalization
- Krylov subspace methods: CG, MINRES, GMRES — Krylov subspace methods: CG, MINRES, GMRES
- Preconditioners: Jacobi, SSOR, incomplete factorizations — Preconditioners: Jacobi, SSOR, incomplete factorizations
- Convergence theory for SPD systems — Convergence theory for SPD systems
- Nonsymmetric systems and flexible Krylov variants — Nonsymmetric systems and flexible Krylov variants
- Restart strategies — Restart strategies
Modeling and computation
- Apply review of lu and qr factorizations in engineering contexts — Apply review of lu and qr factorizations in engineering contexts
- Apply cholesky factorizations in engineering contexts — Apply cholesky factorizations in engineering contexts
- Apply pivoting strategies in engineering contexts — Apply pivoting strategies in engineering contexts
- Apply backward stability analysis in engineering contexts — Apply backward stability analysis in engineering contexts
- Apply sherman–morrison–woodbury in engineering contexts — Apply sherman–morrison–woodbury in engineering contexts
Data and technology
- Use software to explore numerical linear algebra problems numerically — Use software to explore numerical linear algebra problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Low-rank updates — Low-rank updates
- Sparse direct solvers — Sparse direct solvers
- Reordering (overview) — Reordering (overview)
- Block algorithms — Block algorithms
- Cache efficiency (introduction) — Cache efficiency (introduction)
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Matrix factorizations and direct methods
Review of LU and QR factorizations
Coming soon - Iterative methods for linear systems
Krylov subspace methods: CG, MINRES, GMRES
Coming soon - Eigenvalue and SVD computations
Power method and inverse iteration
Coming soon - Large-scale and applied problems
Sparse matrix formats
Coming soon - Software and practice
LAPACK/BLAS ecosystem and best practices
Coming soon
Pricing calculator
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$1,162 · Numerical linear algebra · 18 tutoring hrs
Study guides, worksheets, reviews, practice tests, and answer keys for 1 class. 18 tutoring hours (1 hr / week · semester). Bundle discount applied vs buying separately. Pay in full via Zelle.