AMS 527: Numerical Analysis II

Spring 2012
Time: Monday & Wednesday  3:50pm - 5:10pm
Location: Light Engineering 152
Lecture schedule (tentative)


Instructor: Prof. Xiangmin (Jim) Jiao
Email: xiangmin.jiao@stonybrook.edu; Phone: 631-632-4408
Office hours: Mon. 11:00am-noon, Fri. 2:00-3:00pm
Office: 1-115 Math Tower 		TA: Mr. Vladimir Dyedov
Email: vladimir@ams.sunysb.edu
Office hours: Tue. & Thur. 16:00-17:00
Office: S-250 Math Tower


[ Course Description | Course Outline | Course Policy | Homework and Sample Tests | References ]

Course Description (back to top)

The objective of this course is to introduce students to the fundamentals of numerical computations. The course focuses on numerical methods for nonlinear equations, optimization, interpolation, differentiation and integration, ordinary differential equations, boundary-value problems, and Fourier transform.

Prerequisite: Prior knowledge of linear algebra and calculus (at the level of AMS 510). Basic skills of UNIX systems and programming in C.

Required Textbook

Supplementary Materials (Optional)
  • Gilbert Strang, Computational Science and Engineering, Wellesley Cambridge Press, 2007. Chapters 3 & 4.
  • A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Texts in Applied Math, Vol 37, Springer, 2007.
  • Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, 2007.
Course Outline (back to top)


Outline

  • Approximations in scientific computing; nonlinear equations (2 weeks)
  • Optimization in 1-D; nonlinear least squares; constrained optimization (1.5 weeks)
  • Polynomial interpolation; piecewise polynomial interpolation (1 week)
  • Numerical integration and differentiation; Richardson extrapolation (1.5 weeks)
  • Initial value problems; single-step methods; multi-step methods; Runge-Kutta methods (2.5 weeks)
  • Boundary-value problems; shooting method; finite-difference methods; finite element methods (3 week)
  • Trigonometric interpolation; Fourier transform (1.5 weeks)
  • Solve large systems; direct sparse linear solvers; multigrid methods (1 week)
Course Policy (back to top)


Assignments

You are allowed to discuss course materials and homework problems in small groups, but limited to discussion of general ideas only. You must write your solutions completely independently. Under no circumstances may you copy solutions from any source, including but not limited to other students solutions, official solutions distributed in past terms, and solutions from courses taught at other universities. Violation of these rules may result in disciplinary actions.

Exams

The exams (including two tests and the final exam) are closed-book. A one-page cheat sheet is allowed, but you must prepare the page yourself.

Attendance

All students are expected to attend all the lectures and exams.

Grading

  • Assignments: 30%
  • Two tests: 40%
  • Final exam: 30%
Homework and Sample Tests (back to top)

For the computing assignments, you are encouraged to use the Mathlab SINC Site at Math Tower S-235. You can remotely log onto the Linux computer compute.mathlab.sunysb.edu using ssh. Before you can login, you may need to go to Math Tower S-235 to activate your account. You may use your own computer if it runs a UNIX system (such as Linux or Mac OS X), has a C compiler (such as gcc) and debugger (such as gdb and ddd), and has Octave,  gnuplot, and gv (for plotting).

Sample tests
References (back to top)
 

Less Advanced References 

These references are appropriate for students who have not been exposed to the material in this course before.
  • K. Atkinson, Elementary Numerical Analysis, 3rd edition, Wiley, 2003.
  • W. Cheney and D. Kincaid, Numerical Mathematics and Computing, 6th ed., Brooks Cole, 2007.
  • M. Grasselli and D. Pelinovsky, Numerical Mathematics , Jones & Bartlett, 2008.
  • C. Moler, Numerical Computing with MATLAB, SIAM, 2004. Available online for free.

More Advanced References

The following references are appropriate for students who have taken numerical analysis courses at an undergraduate level before.
  • G. Dahlquist and A. Bjorck, Numerical Methods in Scientific Computing, Vol. 1 , SIAM, 2008.
  • D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd ed., Brooks Cole, 2002.
  • A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Texts in Applied Math, Vol 37, Springer, 2007.
  • J.F. Epperson, An Introduction to Numerical Methods and Analysis, Wiley, 2007.

References on Selected Advanced Topics

The following references are for students who want to specialize in numerical analysis. See Numerical Analysis I for references on numerical linear algebra, and Numerical Analysis III for references on numerical methods for partial differential equations.
  • K. Atkinson and W. Han. Theoretical Numerical Analysis: A Functional Analysis Framework, 2nd ed., Springer, 2005.
  • J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed., Wiley, 2008.
  • N. Higham. Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996.
  • J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, 2006. (First Edition is available on Google Books. Suggested readings: Sections 3.x, 5.x, and 6.1.)
  • G. M. Phillips, Interpolation and Approximation by Polynomials, Springer, 2003.

References on C Programming

For students who are not familiar with C, you are encouraged to read some books on C programming. There are some free online books linked at this "C Programming Language" page. Among these, the following book might be most appropriate.
  • M. Banahan, D. Brady and M. Doran, The C Book, second edition, Addison Wesley, 1991.
Another good starting point is the community-written C book is
If you want to purchase a C book, a classical one is
  • B.W. Kernighan, D.M. Ritchie, C Programming Language (2nd Edition). Prentice Hall, 1988.

Other Links