MATH 708


Foundations of Computational Mathematics I

Fall 2009


Note that this course was previously listed as MATH 726 Numerical Analysis.

This change is one of the steps in establishing a concentration in Applied and Computational Mathematics (ACM) as a part of the graduate program in Mathematics approved by the Department of Mathematics on February 19, 2009.



Meeting times: TTh  5:00 - 6:15 PM  at  LeConte (LC) 310.


Instructor: Dr. Peter G. Binev


phones:   576-6269 (at LC 425)   or   576-6304 (at SUM 206H)

Office hours:  TTH  1:00 - 2:30 PM  at LeConte 425/Sumwalt 206, or by appointment.



Text: An Introduction to Numerical Analysis by Endre Sόli and David F. Mayers, Cambridge University Press, 2003. The course will cover the material considered in chapters 1, 4, and 6-11. The students will receive handouts (lecture notes) for the topics not covered by the book.


Description: Computational Mathematics is build upon the understanding of computational processes provided by Numerical Analysis that studies the algorithms for the problems of continuous mathematics. The course will give an introduction to general ideas in Numerical Analysis and will discuss different aspects of the performance of the numerical procedures involved. In addition to the theoretical material, some numerical implementations in MATLAB will be considered on an elementary level. Topics include:

- number representations and loss of significance (handouts) – 0.5 weeks;

- nonlinear equations and systems of equations (chapters 1 and 4) – 2 weeks;

- polynomial interpolation, divided differences and numerical differentiation (chapter 6 and handouts) – 2 weeks;

- polynomial approximation in the infinity norm (chapter 8) – 1.5 weeks;

- polynomial approximation in the 2-norm, trigonometric polynomials and Fast Fourier Transform (chapter 9 and handouts) – 2 weeks;

- numerical integration (chapters 6 and 10, handouts) – 3.5 weeks;

- spline functions and computer aided geometric design (chapter 11 and handouts) – 2.5 weeks.


Prerequisites: Math 554 or equivalent upper level undergraduate course in Real Analysis.


Learning Outcomes: At the end of this course students will be able to read, interpret, use vocabulary, symbolism, and basic definitions and theorems from Numerical Analysis. The students will be able to use facts, formulas, and techniques learned in this course to apply algorithms and theorems to find numerical solutions and bounds on their errors to various types of problems including root finding, polynomial interpolation and approximation, fast Fourier transform, numerical differentiation and integration, and spline approximation.


Attendance: Regular class attendance is important. A grade penalty will be applied to any student missing three or more classes (10%) during the semester. The "10 percent rule" stated above applies to both excused and unexcused absences. Students who anticipate potential excessive absences due to participation in permissible events as described in the USC Academic Bulletins ( atten.) should receive prior approval from the instructor to potentially avoid such penalty.


Cell Phones: All cell phones must be turned off during the class.


Homework: A few homework problems will be assigned each class. Be sure to solve and write these problems before the next class. Some solutions will be collected (with or without preliminary notice).


Projects: Every student has to choose a project motivated by the computational or theoretical problems discussed in the course. Several possible themes for the projects will be suggested by the instructor in the length of the course. The project in a form of a poster, slides/presentation, or a short paper should be submitted on or before November 24, 2009.


Discussions: The homework and the projects will be discussed in class. The participation in the discussions will be taken into account as part of the homework grade.


Exams: There will be two exams both in a form of a test. The tentative date for the midterm exam is September 29, 2009. The tentative date for the second exam is November 10, 2009. The problems on the tests will be similar to the ones from the homework and the discussions in class.


Final Exam: The final exam in a form of a test will take place on Thursday, December 10 at 5:30 PM.


Grading: The final grade will be determined from the homework (25%), the exams (30%), the project (20%), and the final (25%).


Academic Dishonesty: Cheating and plagiarism will not be allowed. The University of South Carolina has clearly articulated its policy governing academic integrity and students are encouraged to carefully review the policy on the Honor Code in the Carolina Community (see


ADA: If you have special needs as addressed by the Americans with Disabilities Act and need any assistance, please notify the instructor immediately.


Important Dates:        

                                    August 26 – Last  day to drop without W

                                    September 29 – Midterm Exam

                                    October 1 – Last day to drop without WF

                                    November 10 – Second Exam

                                    November 24 – Deadline to submit the projects

                                    December 10 – Final Exam at 5:30 PM