Course Description: This is first part of a two semester sequence in numerical analysis, giving an introduction to various numerical methods widely used in scientific computing. The content of the course is divided into three primary parts.
The study of several numerical methods for solving nonlinear equations, such as the Newton, Bisection, Secant Methods. In general, all these methods employ some iteration until certain convergence criteria is reached. Theoretical considerations explaining the advantages and limitations associated with these methods shall be briefly described. Some applications of these methods shall be explored through the computer laboratory exercises. The suggested time frame for this part is around five weeks.
The study of numerical methods to solve system of linear equations, written in matrix form Ax=b. Two distinct categories of methods for solving systems of this type are direct and iterative solvers. We will study LU,Cholesky decompositions and Gaussian elimination for the direct solvers. As for iterative solvers, we shall study Jacobi, Gauss-Seidel, SOR and CG solvers. We shall try to complete this part in five weeks.
We finish the first part with an introduction to approximation theory and its applications. In practical computations, one has to deal with sets of scattered paired data of certain variables, and often times there is a need for information about these variables in the region where they are not initially defined. We shall study various ways of representing these data through polynomial interpolation and provide algorithms for computing the "best approximation" of a given function. Application of polynomial interpolation to perform numerical integration of a function will be described. We will also apply polynomial interpolation to the numerical differentiation of functions.
Recommended Text: David Kincaid and Ward Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd ed., The Brooks/Cole Series in Advanced Mathematics, 2004.
Grading System, Homeworks, & Tests: Your grade will be based on two in-class tests, homeworks and projects, and a take-home final exam. The exact dates of tests will be announced later during the semester but can be expected in Week 5 and Week 10. If you are going to miss a test, please notify me at least one week prior to the exam. There will be weekly computer assignments containing some mixture of theoretical and computational work. Homework is typically due on Wednesdays after class. The assignment grade is divided between completion(did you do all the work) and accuracy. I grade all problems for completion, but not all problems for accuracy. The relative weights of these components are given below.
Test I Test II Final Exam Homeworks and Projects 20% 20% 30% 30%Your course grade is your numerical score, subject to curving(details later)
A: 100%-90% B: 89%-80% C: 79%-70% D: 69%-60% F: 59% or below