CSc/MATH 4610– Learning Outcomes

Title: Numerical Analysis I

  • 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 iterative methods, including Bisection, Newton, Secant and Fixed Point, to compute solutions of nonlinear equations to within a specified tolerance
  • Students will be able to construct polynomial and piecewise polynomial interpolants of functions of one or two variables in a variety of ways including Lagrange Interpolants, Divided Differences, Chebychev polynomials, and Splines.
  • Students will be able to derive approximation formulae for derivatives using Taylor’s Theorem and use these formulae and their error bounds.
  • Students will be able to o btain approximate values of definite integrals in one or two dimensions, as well as bound their error terms, via Newton-Cotes methods, Gaussian Quadrature methods, and Adaptive methods.

Assessment of Learning Outcomes

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.