Numerical Analysis I
This course introduces the basic numerical methods used in many applications areas in computational science and engineering.
Topics Covered:
- Introduction: Floating point arithmetics, sources of errors
- Systems of Linear Equations: Gaussian elimination, pivoting, norms, condition numbers
- Linear Least Squares: Normal equations method, orthogonalization methods for full rank problems
- Solution of Nonlinear Equations: Bisection and secant methods, fixed point iteration, Newton's method
- Interpolation: Lagrange interpolation, Newton interpolation, Chebyshev polynomials, Hermite interpolation, Splines, Fast Fourier Transformation
- Numerical Differentiation and Integration: Trapezoidal rule, Simpson's rule, Newton-Cotes quadrature, Gaussian quadrature, adaptive quadrature, finite difference, Richardson extrapolation
- Numerical Solutions of Ordinary Differential Equations: initial value problems, systems of equations, Euler method, Runge-Kutta method
- Optimization (if the schedule allows): Existence of solutions, Optimization in one dimension, unconstrained and constrained optimizations, optimality conditions, Newton's method, Steepest descent, Conjugate gradient method
Students will learn the high level concepts and rationale behind the methods (in contrast to just numerical recipes) and learn how to choose and apply them to solve complex problems using computers. The course strives to be reasonably broad and domain neutral, while achieving depth in some selected key topics including linear system solvers, systems of nonlinear equations, interpolation and approximation of functions, numerical integration and differentiation, optimization, and numerical handling of ordinary differential equations.
Scientific Computing: An Introductory Survey, Second Edition, Michael T. Heath, McGraw-Hill, 2002, ISBN 0-07-239910-4
Class attendance: 6%
4 Homeworks: 44 %. Written problems and programming in MATLAB or Python, 11% each
Midterm Exam 1: 15 %, Feb. 19, Th., in class, open books/notes, no electronic devices allowed
Midterm Exam 2: 15 %, Apr. 2, Th., in class, open books/notes, no electronic devices allowed
Final Exam: 20 %, May 5, 2:40pm - 5:30pm, in the lecture room (Clough UG Learning Room 423), open books/notes, no electronic devices allowed
Class attendance counts for 6% of the total score for grading.
Students are expected to maintain the highest standards of academic integrity. All work submitted must be original and properly cited. Plagiarism, cheating, or any form of academic dishonesty will result in immediate consequences as outlined in the university's academic integrity policy.