Courses:

Introduction to Numerical Methods >> Content Detail



Syllabus



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



ACTIVITIESPERCENTAGES
Homework Assignments60%
Midterm Exam40%



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 #TOPICSKEY DATES
1Introduction, Basic Linear Algebra
2Orthogonal Vectors and Matrices, Norms
3The Singular Value Decomposition
4The QR Factorization
5Gram-Schmidt OrthogonalizationHomework 1 due
6Householder Reflectors and Givens Rotations
7Least Squares Problems
8Floating Point Arithmetic, The IEEE Standard
9Conditioning and Stability IHomework 2 due
10Conditioning and Stability II
11Gaussian Elimination, The LU Factorization
12Stability of LU, Cholesky FactorizationHomework 3 due
13Eigenvalue Problems
14Hessenberg / Tridiagonal Reduction
15The QR Algorithm I
16The QR Algorithm IIHomework 4 due
17Other Eigenvalue Algorithms
Midterm Exam
18The Classical Iterative Methods
19The Conjugate Gradients Algorithm I
20The Conjugate Gradients Algorithm II
21Sparse Matrix AlgorithmsHomework 5 due
22Preconditioning, Incomplete Factorizations
23Arnoldi / Lanczos Iterations
24GMRES, Other Krylov Subspace Methods
25Linear Algebra SoftwareHomework 6 due

 








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