Math 270B: Numerical Analysis (Part B) - Bo Li

Math 270B: Numerical Analysis (Part B)
Winter quarter 2024

Description This is the second part of a series of three parts of an introductory numerical analysis course for graduate students. The entire course series serves as a graduate qualifying exam course. The second part focuses on nonlinear equations, approximation, interpolation, and numerical integration. (More detailed topics are listed below on this web page.)
Prerequisites  Math 270A or consent of instructor.
Lectures 1:00 - 1:50, Mondays, Wednesdays, and Fridays, AP&M 5402.
Instructor Professor Bo Li
Office: AP&M 5723. Office phone: (858) 534-6932. E-mail: bli@math.ucsd.edu
Office hours: 2:00 - 3:00, Mondays and Fridays.
Teaching Assistant  Silvia Liu
Office: HSS 4056. E-mail: mil094@ucsd.edu
Office hours: 9:30 - 10:30 Tuesdays and Thursdays
Lecture Notes and Reference Books We will mainly use instructor's lecture notes. These lecture notes will be distributed in the class. The following two textbooks are good reference books, and their UCSD electronic veersions are available to UCSD students:
  • A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, 2nd ed., Springer, 2007.
  • J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd ed., Springer, 2004. <\ul> Some additional references are listed below on this web page.
Homework Assigned, collected, and graded regularly. Note: No homework assignments completed by using ChatGPT or other AI tools is are acceptable.
Exams There will be one midterm exam and one final, close-book and close-note exams. The final exam will be cumulative.
        Midterm exam: 1:00 pm - 1:50 pm, Friday, February 9, AP&M 5402.
        Final exam: 11:30 am - 2:30 pm, Friday, March 22, AP&M 5402.
Note: Neither make-up nor rescheduled exams will be allowed unless a proved written excuse (such as hospitalization, a family emergency, and a major religious activity) is provided sufficiently ahead of time.
Grading The final course grade will be determined based on the homework and exams with the weight: homework - 30%, midterm exam 20%, and final exam - 50%.
Disability Accommodations Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter (paper or electronic) issued by the Office for Students with Disabilities (OSD) Students are required to discuss accommodation arrangements with instructors and OSD liaisons in the department in advance of any exams or assignments.
Academic Integrity All students are expected to conduct themselves with academic integrity. Violations of academic integrity will be treated seriously. See UCSD Academic Integrity page

Topics to be Covered
  1. Nonlinear Equations (~ 3 lectures)
    1. Examples and notations
    2. Newton iteration
    3. Fixed-point iteration
  2. Optimization (~ 3 lectures)
    1. Necessary and sufficient conditions
    2. The gradient descent method
    3. The method of Lagrange multipliers
    4. A penalty method
  3. Polynomial Approximation (~ 8 lectures)
    1. The Weierstrass Theorem
    2. Best uniform approximations
    3. Least-squares approximations
    4. Orthogonal polynomials
  4. Polynomial Interpolation (~ 8 lectures)
    1. Lagrange interpolation
    2. Newton's formula and divided differences
    3. Hermite interpolation
    4. Piecewise polynomial interpolation
  5. Numerical Integration (~ 5 lectures)
    1. Interpolatory quadrature
    2. Euler-Maclaurin formula
    3. Gaussian quadrature
References
  1. E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, 1966.
  2. E. K. P. Chong and S. H. Zak, An Introduction to Optimization, 2nd ed., Wiley, 2010.
  3. P. J. Davis, Interpolation and Approximation, Dover, 1975.
  4. P. J. Davis and P. H. Rabinowitz, Methods of Numerical Integration, Academic Press, 1975.
  5. C. de Boor, A Practical Guide to Splines, Springer-Verlag, 2001.
  6. H. Engles, Numerical Quadrature and Cubature, Academic Press, 1980.
  7. R. Fletcher, Practical Methods of Optimization, 2nd ed., Wiley, 1987.
  8. O. Guler, Foundations of Optimization, Springer, 2010. (UCSD e-version available.)
  9. F. B. Hildebrand, Introduction to Numerical Analysis, 2nd ed., Dover, 1987.
  10. E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover, 1994.
  11. G. G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, 1966.
  12. J. M. Ortega and W. C. Rheinboldt, Iterarive Solution of Nonlinear Equations in Several Variables, Academic Press, 1970.
  13. R. Plato, Concise Numerical Mathematics, Amer. Math. Soc., 2003.
  14. P. M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981.
  15. T. J. Rivlin, Introduction to the Approximation of Functions, Dover, 1987.
  16. A. Ruszczynski, Nonlinear Optimization, Princeton University Press, 2006.
  17. I. Schoenberg, Cardinal Spline Functions, SIAM, 1973.
  18. L. L. Schumaker, Spline Functions: Basic Theory, 3rd ed., Cambridge University Press, 2007.
  19. A. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall, 1971.
  20. E. Suli and D. F. Mayer, An Introduction to Numerical Analysis, Cambridge University Press, 2003.
  21. G. Szgo, Orthogonal Polynomials, 3rd ed., Amer. Math. Soc., 1967.

Last updated by Bo Li on February 27, 2024.