CSc/MATH 4620– Learning Outcomes

Title: Numerical Analysis II

Students will understand the basic theory of and use O notation appropriately. Students will understand floating-point arithmetic including issues of overflow and underflow.
Students will understand the different issues surrounding errors in using numerical methods including machine epsilon, error analysis, convergence, rounding error, truncation error, and norms.
Students will understand the difficulties involved in finding reliable solutions as well as be able to apply various methods for estimating errors in solutions in order to judge how reliable those solutions are.
Students will understand conditioning of problems and stability of algorithms and the difference between the two.
Students will be able to compute a Taylor polynomial and bound its error term.
Students will be able to apply methods to compute solutions of systems of linear equations via Gaussian Elimination (including pivoting), Iterative techniques (Gauss-Seidel, Jacobi, SOR), and Least Squares.
Students will be able to obtain approximate solutions to Initial Value Problems via Euler’s method, Runge-Kutta methods, and Multi-step methods.
Students will be able to obtain approximate solutions to Boundary Value Problems via Difference methods and Shooting methods.
Students will be able to o btain approximate solutions of Partial Differential Equations using finite difference methods including Explicit methods, Implicit Methods, and Iterative methods.

Problems based on the learning objectives will be assigned on a regular basis and may appear in a variety of contexts:

Classroom discussion provides an indication of the students' understanding of newly presented topics, of old material, and of their ability to relate new topics to old ones. Class meetings may involve a combination of lecture, questions and discussion, and small group work.
Homework problem sets serve as both learning and assessment tools for understanding how numerical methods work and for their theoretical underpinnings. Homework is an important part of the course (numerical work is done to illustrate concepts and justification for the ideas behind the methods is investigated)
Programming assignments are designed to assess the students' ability to synthesize the ideas discussed in class by writing programs that solve problems for which numerical solution has clear advantages over analytical techniques. The analysis of the output of these programs will give students the opportunity to compare the accuracy of their implementation of numerical schemes to theoretical bounds.
Exams may be in-class, take-home or both. In-class exams give students the opportunity to demonstrate their ability to work simple problems and to demonstrate their understanding of fundamental concepts. Take-home exams provide the opportunity for students to delve more deeply into the subject, to work more complicated problems, and thus arrive at a deeper understanding of the material. The exams are used to determine whether the student understands the concepts behind the methods.