Syllabus

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Description

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®.

Prerequisites

Differential Equations (18.03) and Linear Algebra (18.06).

Texts

The required textbook for this class is:

Amazon logo Bau III, David, and Lloyd N. Trefethen. Numerical Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1997. ISBN: 0898713617.

Other readings include:

Amazon logo 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.

Amazon logo 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.

Grading


ACTIVITIES PERCENTAGES
Homework Assignments 60%
Midterm Exam 40%

Policies

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.

Calendar


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