AMS 527: Numerical Analysis II

Spring 2022
Time: Monday & Wednesday  4:25pm - 5:45pm
Location: Physics P128 or online via Zoom

Lecture Schedule and Slides



Instructor: Prof. Xiangmin (Jim) Jiao
Email: ; Phone: 631-632-4408
Office hours: Mondays 1:15--2:45pm, Wednesdays 2:30–4:00pm or by appointment
Office: In-person in Math Tower 1-117 or online via Zoom
TA: Mr. Hongji Gao
Email:
Office hours: Tue. 4:00pm--5:00pm ET &
                      Thur. 6:00pm--7:00pm ET
Office: Online via Zoom



Description | Outline | Delivery Mode | Tech Requirements | Grading PolicyHomework and Exams | References | University Policy ]

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 and approximation, 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.

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.

Learning Objectives

The objective of this course is to introduce the fundamentals of numerical computations. The course focuses on numerical methods for nonlinear equations, optimization, interpolation and approximation, differentiation and integration, ordinary differential equations, and boundary-value problems. Key leaning outcomes include the following:

  1. Build understanding of fundamentals of numerical approximations
    • classification of sources of errors
    • effect of floating-point arithmetic
    • accuracy and stability
  2. Master concepts and numerical methods for solving nonlinear equations
    • methods for nonlinear equations in 1-D: interval bisection method, fixed-point iteration, Newton's method, secant method
    • methods for nonlinear equations in n-D: Newton's method, Newton-like method
    • sensitivity, convergence rates, and stopping criteria 
  3. Build fundamental understanding of concepts and numerical methods for optimization
    • unconstrained vs. constrained optimization, global vs. local minimum, convexity, optimality conditions
    •  algorithms for unconstrained optimization in 1-D and n-D: golden section search, Newton's method, Quasi-Newton methods, steepest descent, and conjugate gradient
    • algorithms for constrained optimization: Lagrange multiplier
  4. Build fundamental understanding of interpolation and approximation
    • interpolation versus approximation, basis functions, convergence, Taylor polynomial
    • polynomial interpolation, piecewise polynomial interpolation, orthogonal polynomial interpolation, lease squares approximations
    • trigonometric interpolation
  5. Master concepts and numerical methods for numerical integration and differentiation
    • Newton-Cotes rules, Gaussian quadrature rules, change of interval
    • derivation with method of undetermined coefficients and orthogonal polynomials
    • finite difference approximation, forward difference, backward difference, and centered difference
  6. Master basic numerical methods for initial-value and boundary-value problems
    • stability of solutions of ODEs; global error vs. local error; stiffness; explicit vs. implicit methods; analysis of stability
    • basic algorithms/schemes and their derivations: Euler's methods (forward and backward); trapezoid method; Heun's method; fourth-order Runge-Kutta method
    • finite-difference methods and finite element methods
  7. Demonstrate programming skills for numerical methods
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)

Course Delivery Mode and Structure (back to top)
This course will be in person with simultaneous delivery via Zoom (https://it.stonybrook.edu/services/zoom) for lectures and office hours. We will use Microsoft Teams (https://it.stonybrook.edu/services/office-365) to facilitate communications between faculty and students, submission of assignments, and posting grades. Teams will also be used as a backup for office hours if there are issues with Zoom. All links are available through the Blackboard learning management system. See the “Technical Requirements” section for more information. In Teams, you will access online lessons, course materials, and resources.

Course-related questions should be posted in Teams. For personal/private issues, my preferred method of contact is private chat messages on Teams or email listed at the top of this syllabus. I strive to respond to your emails as soon as possible, but please allow between 24-48 hours for a reply. All email communication will be sent to your Stony Brook University email account. You must have an active Stony Brook University e-mail account and access to the Internet. Please plan on checking Teams regularly and your SBU email account for course related messages or set up your SBU email account to forward to your personal email account. To log in to Stony Brook Google Mail, go to http://www.stonybrook.edu/mycloud and sign in with your NetID and password.

Important announcements will be sent on Blackboard and Teams. These will be posted in the class and may or may not be sent by email I will participate and post regularly in the discussion board in Teams and provide feedback on assignments within a week. Regular communication is essential in online classes. When we are participating in class discussions the expectation is that you will respond regularly to your peers and questions posed to your responses. Logging in regularly, checking the discussion board and participating with your colleagues ensures that you are able to remain an active member of the class.
Technical Requirements (back to top)

You are responsible for having a computer and a reliable Internet connection throughout the term. The following lists detail a minimum recommended computer set-up and the software packages you will need to access and use:
All the software with a stonybrook.edu link above is supported by the University and is available to Stony Brook students at no additional charge. Please make sure that you must sign up for the software using your Stony Brook account, instead of your personal account. Your login password for Teams and Zoom is the same as your NetID password for logging into Blackboard. If you are unsure of your NetID password, visit https://it.stonybrook.edu/help/kb/finding-your-netid-and-password for more information.
 
If you need any technical assistance for the University-licensed and support software,
For other questions on computer hardware, software, or Internet connection, post them on Teams to get help from the instructor, the TA, or your peer students.
Grading Policy (back to top)


Assignments

Homework assignments are due typically two weeks after they are assigned. 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 student's 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, but you are allowed to bring a single-sided, one-page, letter-size cheat sheet, which you must prepare by yourself. The exams will be in person unless the student is remote due to travel restrictions; in the latter case, the exam can be proctored via Zoom.

Attendance

All students are expected to attend all the lectures (either synchronously or asynchronously) and exams (synchronously).

Grading

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

Homework

All the assignments will be posted on Microsoft Teams and should be submitted through Teams, with links available on Blackboard. You are encouraged to typeset your homework solutions using LaTeX or LyX (a very easy-to-use document processor, using LaTeX in the backend), or write your answers by hand and submit a scanned copy. However, Microsoft Word is not recommended.

Exams

The two midterms are tentatively scheduled to be held on
  • February 28th 4:25--5:45pm Eastern Time
  • April 6th 4:25--5:45pm Eastern Time
Specific times of the midterms are subject to change.

The final exam will be on
  • May 17th 2:15--05:00pm Eastern Time

Sample tests will be posted on Teams about one week before the exam.

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 MATLAB Programming

References on NumPy Programming

References on C/C++ Programming

There are many reference on C and C++. For example,

  • M. Banahan, D. Brady and M. Doran, The C Book, 2nd Edition, Addison Wesley, 1991.
  • B. Stroustrup, A Tour of C++, 2nd Edition, 2018.

References on Technical Writing

Other Links

Policies and Academic Integrity (back to top)


Student Accessibility Support Center Statement

If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact the Student Accessibility Support Center, 128 ECC Building, (631) 632-6748, or via e-mail at:  sasc@stonybrook.edu. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.

Academic Integrity Statement


Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty is required to report any suspected instances of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology & Management, Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty please refer to the academic judiciary website at http://www.stonybrook.edu/commcms/academic_integrity/index.html

Critical Incident Management

Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Student Conduct and Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Until/unless the latest COVID guidance is explicitly amended by SBU, during Spring 2022 "disruptive behavior” will include refusal to wear a mask during classes.