AMS 527: Numerical Analysis II
(Fundamentals of Numerical Computations)
Spring 2024
Time: Monday & Wednesday 4:00–5:20 pm
Location: Psychology A 146

Lecture Schedule and Slides


Instructor: Prof. Xiangmin (Jim) Jiao
Email:
Office hours: Monday & Wednesday 2:30pm–3:30pm or by appointment
Office: Math Tower 1-117

TA: Haochun Wang
Email:
Office hours: Tuesday and Thursday 1:00pm–2:00pm
Office: Physics Building A-134 or via Zoom.


Course Description

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.

Required Textbook

Supplementary Materials

  • 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.

Prerequisite/Co-requisite

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

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

  • Approximations in scientific computing; nonlinear equations (2 weeks)
  • Optimization in 1-D; nonlinear least squares; constrained optimization (2 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 (2 week)
  • Trigonometric interpolation; Fourier transform (1.5 weeks)

You can find the lecture schedule and slides here.

Course Delivery Mode and Structure

This course will be in person. We will use Brightspace (https://it.stonybrook.edu/services/brightspace) for posting and submission of assignments, posting grades, and discussion forums. For personal/private issues, my preferred method of contact is email, as 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 email account and access to the Internet. Please plan on checking Brightspace regularly and your SBU email account for course-related messages or set up your SBU email account to forward to your preferred 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 Brightspace. These will be posted in the class and may or may not be sent by email. I will participate and post regularly on the discussion board in Brightspace and provide feedback on assignments within a week. When participating in class discussions, the expectation is that you will regularly respond to your peers and questions posed to your responses. Logging in regularly, checking the discussion board, and participating with your colleagues ensure that you can remain an active member of the class.

Technical Requirements

You are responsible for having a computer throughout the term. The following lists detail a minimum recommended computer set-up and the software packages you will need to access and use:

Hardware:

  • Intel Core i5 processor (or higher, including Apple silicon with Rosetta 2).
  • 250 GB hard drive and 8 GB RAM.
  • A reliable high-speed internet connection.

Software:

Course Policy and Assessment

Attendance & Lecture Information

Attendance is required for all lectures. It is imperative that before seeking help during office hours, students have adequately reviewed and familiarized themselves with the material covered in the lectures. This ensures a productive and efficient use of office hour time and facilitates a more effective learning process.

Assignment & Homework Details

Homework assignments are typically due two weeks after they are assigned and will be posted on Brightspace. Submissions should be made through Brightspace. It is encouraged to typeset homework solutions using LaTeX or LyX, or to submit a scanned handwritten copy. However, Microsoft Word is not recommended.

You are encouraged to discuss the course material and homework problems with peers, but these discussions should focus only on the general concepts and ideas, not on the solutions of specific homework problems. It is imperative that you produce and submit solutions on your own. Copying or sourcing solutions, be it from fellow students, previously distributed solutions, or other universities, is strictly prohibited and will be treated as a violation of academic integrity, resulting in disciplinary actions.

In the context of Generative AI platforms like ChatGPT, students are reminded of their limitations, especially for an advanced course such as AMS 527. While generative AI might prove useful for grammar refinement, sanity checks, and code debugging, they should not be trusted for content creation or problem solving. If you decide to incorporate substantial content or insights from these platforms into your work, ensure they are properly quoted and referenced. Misrepresenting AI-generated content as your own work will be flagged as an academic breach and could lead to severe consequences. However, for basic tasks like grammar corrections or sanity checks, there is no obligation to credit the AI.

Late Submission Policy

Understanding that unforeseen circumstances can arise, we aim to provide students with a policy that is both fair and encourages timely submission of assignments.

  • Grace Period: Homework assignments submitted within 24 hours after the deadline will incur a penalty of 10% off the earned points.
  • Extended Delays: Homework submitted between 24 to 48 hours after the deadline will incur a penalty of 20%. Assignments will not be accepted more than 48 hours past the original deadline, except for extensions granted by the instructor beforehand.
  • Documented Excuses: Exceptions will be made for documented medical reasons, family emergencies, or other unforeseen serious events. In these cases, students should notify the instructor as soon as possible, ideally before the assignment deadline. Documentation will be required.
  • Communication: If you anticipate not being able to submit an assignment on time, it is essential to communicate with the instructor or TA ahead of the deadline. While we cannot guarantee extensions, early communication may lead to more favorable considerations.
  • Pre-approved Extension: In certain unforeseen situations, the instructor may grant extensions to the entire class. In this situation, the grace period will start after the extended deadline.
  • Consistency: This policy is firm and will be consistently applied to all students to ensure fairness. Please plan your time accordingly and aim to submit assignments well before deadlines to account for any last-minute issues.

Examination Details

The exams, including two tests and the final exam, are closed-book. Students can bring a single-sided, one-page, letter-size cheat sheet prepared by themselves. The exams will be in person.

Midterm Dates:

  • First Midterm: February 26th in class
  • Second Midterm: April 8th in class

Final Exam:

  • May 13th between 5:30pm and 8pm Eastern Time

Sample tests will be available on Brightspace approximately one week before each exam.

Grading Criteria

  • Assignments: 30%
  • Two midterm exams: 40%
  • Final exam: 30%

Your cumulative score will be used to determine the final letter grade. There is no pre-established scale or curve for grading. However, for reference, historical average scores for letter grades in this course are as follows:

  • 90–100%: A
  • 85–90%: A-
  • 80–85%: B+/B
  • 70–80%: B-
  • 60–70%: C/C+
  • 0–60%: F

References and Other Resources

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

University Policies and Academic Integrity

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, Stony Brook Union Suite 107, (631) 632-6748, or 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. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures. Further information about most academic matters can be found in the Undergraduate Bulletin, the Undergraduate Class Schedule, and the Faculty-Employee Handbook.