This is an archived course. A more recent version may be available at ocw.mit.edu.
Lectures: 2 sessions / week, 1.5 hours / session
This course offers an advanced introduction to numerical linear algebra. Topics include direct and iterative methods for linear systems, eigenvalue decompositions and QR/SVD factorizations, stability and accuracy of numerical algorithms, the IEEE floating point standard, sparse and structured matrices, preconditioning, linear algebra software. The problem sets require some knowledge of MATLAB®.
Differential Equations (18.03) and Linear Algebra (18.06).
The required textbook for this class is:
Bau III, David, and Lloyd N. Trefethen. Numerical Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1997. ISBN: 0898713617.
Other readings include:
Bai, et al. Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000. ISBN: 0898714710.
Barrett, et al. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1993. ISBN: 0898713285.
Shewchuk, Jonathan R. "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain." Carnegie Mellon University (August 1994). (
PDF)
Goldberg, David. "What Every Computer Scientist Should Know About Floating Point Arithmetic." ACM Computing Surveys 23, no. 1 (March 1991): pp. 5-48.
| ACTIVITIES | PERCENTAGES |
|---|---|
| Homework Assignments | 60% |
| Midterm Exam | 40% |
Collaboration on the homeworks is encouraged, but each student must write his/her own solutions, understand all the details of them, and be prepared to answer questions about them.
No books, notes, or calculators are allowed on the Midterm exam.
| LEC # | TOPICS | KEY DATES |
|---|---|---|
| 1 | Introduction, Basic Linear Algebra | |
| 2 | Orthogonal Vectors and Matrices, Norms | |
| 3 | The Singular Value Decomposition | |
| 4 | The QR Factorization | |
| 5 | Gram-Schmidt Orthogonalization | Homework 1 due |
| 6 | Householder Reflectors and Givens Rotations | |
| 7 | Least Squares Problems | |
| 8 | Floating Point Arithmetic, The IEEE Standard | |
| 9 | Conditioning and Stability I | Homework 2 due |
| 10 | Conditioning and Stability II | |
| 11 | Gaussian Elimination, The LU Factorization | |
| 12 | Stability of LU, Cholesky Factorization | Homework 3 due |
| 13 | Eigenvalue Problems | |
| 14 | Hessenberg / Tridiagonal Reduction | |
| 15 | The QR Algorithm I | |
| 16 | The QR Algorithm II | Homework 4 due |
| 17 | Other Eigenvalue Algorithms | |
| Midterm Exam | ||
| 18 | The Classical Iterative Methods | |
| 19 | The Conjugate Gradients Algorithm I | |
| 20 | The Conjugate Gradients Algorithm II | |
| 21 | Sparse Matrix Algorithms | Homework 5 due |
| 22 | Preconditioning, Incomplete Factorizations | |
| 23 | Arnoldi / Lanczos Iterations | |
| 24 | GMRES, Other Krylov Subspace Methods | |
| 25 | Linear Algebra Software | Homework 6 due |
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