MTH 451 Course Description and Syllabus
What is Numerical Analysis?
There are 2 things that you learn as you progress in your science/engineering:
(1) most every problem you solved in your basic linear algebra, ordinary differential equations, partial differential equations, is extremely special in the sense that it is solvable by analytical means and with few resources. These are the exception, rather than the rule, in practice. (2) Finite speed, storage and
precision computing (e.g. using a computer) brings in new challenges in the computation: there are finite resources and everything is approximated. Numerical analysis, more than anything, focuses on recognizing how the issues in (2) play out in (1) and on developing skills to do something about these. The impact of finite resources is obvious, what is less obvious is how using approximate numbers over a finite bandwidth of numbers affect computations.
The Basic Numerical Analysis Sequence
MTH451 is part of a 3 course sequence in introductory numerical analysis:
MTH 451, 452, 453 (and its graduate counterparts).
Typically 451 is offered summer and fall, 52 and 53 are offered in winder and spring. 451 covers linear algebra. Along the way you learn about numerical stability, computational complexity, and how to use theoretical
estimates to determine whether a code is producing the expected outcomes. The audience for this course include science and engineering students, including CS.
452 covers differentiation/integration, interpolation, and the approximation of solutions to ordinary differential equations, with an emphasis on initial value problems. The audience is engineers and science students.
453 covers approximate methods for partial differential equations, and extends further the approximation of boundary value problems and the application of
iterative linear algebraic problems. The audience fo this class typically has had a course like MTH482 (minimally) and an interest or need to know how to
find appproximations to PDEs.
The fields of numerical analysis and scientific computing are rich and are
specializations with high employment demand. People make full scientific careers in these two fields. There are a variety of 500 and 600 level courses you can take that follow where this sequence left off. You are encouraged to consider these.
MTH451 Topics:
All of the topics covered in Trefethen and Bau's book will be denoted by the specific lecture number in the book. We will have to sacrifice coverage of eigenvalues in order to cover linear iterative solvers (Notes).
Study Schedule (will be built as the class proceeds).
| Week |
Lecture |
Topic |
| Week 1 |
Lectures 1, 2, 3 |
Review
|
| Week 2 |
4, 5, 6 |
SVD, QR, Least Squares |
| Week 3 |
12, 13, 14 |
Numerical Stability and Condition Number. |
| Week 4 |
15, 16, 17 |
Stability in Least Squares and Householder Triangularization. |
| Week 5 |
18, 19 |
Stability and Computational Complexity |
| Week 6 |
20, 21, 22 |
Systems of Equations |
| Week 7 |
24, 25, 27 |
Eigenvalues |
| Week 8 |
31, Notes |
SVD, Iterative Methods |
| Week 9 |
Notes, 32 |
Iterative Methods |
| Week 10 |
Notes, 38 |
Conjugate Gradient |
| Week 11 |
Notes, 35, 40 |
GMRES, Preconditioning |