Mathematics 16:643:574 Numerical Analysis II
Schedule
The course is normally offered during the Spring semester.- Class meeting dates: Please visit the University's academic calendar.
- Schedule and Instructor: Please visit the University's schedule of classes for the instructor, time, and room.
- Instructor and Teaching Assistant Office Hours: Please visit the Mathematical Finance program's office hour schedule.
Course Abstract
This course is the second part, independent of 16:643:573 Numerical Analysis I, of a general survey of the basic topics in numerical analysis. We shall study and analyze a number of numerical algorithms for approximating the solution of a variety of generic problems which occur in applications. The course will begin with the description of the solution methods for the linear system of equations. Starting from the direct methods based on the Gaussian elimination, various classical iterative methods such as Gauss-Seidel, Jacobi and SOR will be discussed. Further, we study more advanced iterative methods, multigrid methods in this course. Large portion of the course will be devoted to numerical techniques for optimization, matrix eigenvalues and eigenvectors and numerical solutions to nonlinear equations. As a separate but important technique, finite difference and finite element discretization methods for simple partial differential equations such as Poisson's equations and Heat equations will be studied at the end of the course. Particular emphasis in this course is to interconnect the theorectical results and computer implementation. Students will study not only the solid theoretical backgrounds in developing and understanding the algorithms but also a hands-on experience to implement the methods.Pre-requisites and Co-requisites
Advanced Calculus, Linear Algebra, and familiarity with differential equations. Numerical Analysis I (16:643:573) is desirable but not required.Primary Textbooks
A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, 2nd ed., Springer, 2004.K. Atkinson, An Introduction to Numerical Analysis, 2nd ed., Wiley, 1989.
Grading
Please contact the instructor.Class Policies
Please see the MSMF common class policies.Assignments
Homework assignments in the course consist of both theoretical and computational work. For the computational component, the students should use a language/environment that possesses high level data types so that the students spend more time working with algorithms and not worrying about the details of writing computer code. MATLAB is a good choice. Fortran 77/90/95 and C++ with appropriate class libraries can also be used.Previous Instructor Course Websites
2010 Richard Falk2009 Young-Ju Lee
2008 Michael Vogelius
2007 Richard Falk
Weekly Lecturing Agenda and Readings
The lecture schedule below is a sample; actual content may vary depending on the instructor.
| Lecture | Topics |
|---|---|
| 1 | General course outline and Background for Programming projects. |
| Numerical Solution of Systems of Linear Equations. | |
| 2 | Gaussian Elimination |
| 3 | Choleski decomposition and pivoting |
| 4 | Perturbation theory for linear systems of equations |
| 5 | Matrix iterative method, Gauss-Seidel, Jacobi and SOR |
| 6 | Steepest descent and Conjugate Gradient Methods |
| Matrix Eigenvalues and Eigenvectors | |
| 7 | Calculation of Eigenvalues and Eigenvectors |
| 8 | Numerical Methods for Eigenvalues and Eigenvectors |
| 9 | QR algorithm I |
| 10 | QR algorithm II |
| Solution of Nonliear Equations | |
| 11 | Bisection and False Position |
| 12 | Secant, Newton's method and Fixed point iterations |
| 13 | Local Convergence Results and Order of Convergence |
| Solution of Nonliear Systems of Equations | |
| 14 | Newton and Broyden's method and their convergence |
| Minimization Problems | |
| 15 | Newton, quasi-Newton, Steepest descent and Levenberg-Marguardt method |
| 16 | Review |
| 17 | Midterm |
| Finite Difference Methods | |
| 18 | Shooting method and Finite Difference methods |
| 19 | Analysis of Finite Difference Methods |
| 20 | Finite Difference Methods for Elliptic Equations in two dimensions |
| 21 | Finite Difference Methods for Heat Equations |
| Finite Element Methods | |
| 22 | ntroduction to Finite Element Methods |
| 23 | Finite Element Method I |
| 24 | Finite Element Method II |
| 25 | Finite Element Method for Parabolic Equations |
| Multigrid Methods | |
| 26 | Brief Review on Iterative Methods, Gauss-Seidel and Jacobi |
| 27 | Introduction of Multigrid Methods for Eliptic Equations |
| 28 | Implementation of Multigrid Methods for Eliptic Equations |
| 29 | Convergence of Multigrid Methods |
| 30 | Review |
