MA780: Numerical
Analysis II
This is the second half of a one-year sequence courses (MA580/780) on
numerical analysis. The general goal of this sequence is to provide an encapsulation
of both the theory and the techniques that cast classical mathematics into
computational mathematics.
· General syllabus about this one-year sequence includes the following subjects (not necessarily in order).
· The emphasis in MA580 is on the linear problems which is the foundation of all scientific computation, with further applications to modern data science, neural science, and quantum computing. The emphasis in MA780 is on the nonlinear problems which cover approximation theory, numerical integration, numerical ODEs and others.
· The purpose of this course is to expose you to the full spectrum of basic computational techniques that will be needed in almost every discipline of science, whenever computation is needed. It is highly recommended that you take the full year course.
· This is also a Qualifying Exam course. The course is intended to carefully prepare you for the exam.
Reading materials:
· Many textbooks are available. I recommend the following list as the classical reference books, although some of which might be slightly more advanced than others.
o Introduction to Numerical Analysis, Third Edition, by Stoer and Bulirschs.
o Numerical Mathematics, Second Edition by Quarteroni, Sacco and Saleri,
o Matrix Computation, by Golub and Van Loan.
· For the MA580/780 combined, I recommend the newly released book, entitled Classical Numerical Analysis: A Comprehensive Course, by Salgodo and Wise, Cambridge University Press, 2022, as the comprehensive textbook.
· For
MA780 alone, the following books are freely available through NCSU libraries.
o
C. T. Kelley, Iterative Methods for
Linear and Nonlinear Equations, SIAM, 1995.
o
W. Gautschi, Numerical analysis,
2nd edition, 2011.
o
G. Dahlquist, A. Björk,
Numerical Methods in
Scientific Computing.
o
J. Trangenstein, Scientific Computing
Vol. III - Approximation and Integration.
· My own lecture notes for this course are available online.
· The slides prepared for undergraduate MA428 and MA427 (yes, in reversed order) highlight some must-known major points.
Some useful tools:
· Netlib: Numerical Analysis Library Network (Lots of established library routines).
· Some useful links to the world of mathematics:
o EqWorld: Offers a good collection of mathematical equations (Be warned, this is a Russian website, but is good.)
o Wolfram MathWorlds: Offers a good collection of glossary of terms.
o AMS: American Mathematical Society (Math Review online.)
· NCSU Library: Enter it with a solemn heart and a humble attitude; exit it with a cheerful spirit and a learned mind.
o MathSciNet: Allows you to download lots of papers through NCSU library subscription (NCSU login required)
o Encyclopedia and Dictionaries: Who says a mathematician doesn't need these?
· ODB Something good for your souls.
Grading Policy:
· final grade= (50% homework) + (35% computer project) + (15% final exam)
· Homework or projects may be worked on and turned in by study groups of up to 2 people per group. All group members receive the same grade.
o Homework is meant to fill in gaps of lectures. It usually involves some fact-checking mathematics.
- Projects are related to skills learned in class, but most topics require additional endeavors to investigate new territories.
· Final
exam will be a take-home exam with the expectation that you are able to manage time,
analyze problems, and develop thoughtful and clear responses to some general conceptual questions.
Old
Ph.D. Qualifying Exams: To prepare you for the Qualifying, many homework
assignments will be selected from for my collection
of previous qualifying exams on the subject of numerical analysis.