The course covers the standard techniques of numerical computation from a theoretical as well as a practical perspective with a particular emphasis on large-scale high-performance computation, and provides the foundation for more specialized third year courses in computation and modeling.
It is assumed that participants have the mathematical background equivalent to two semesters of Engineering and Science Mathematics - attendance of the first year B track (Multivariable Calculus, ODE; Linear Algebra, Fourier Methods, Probability) is highly recommended but not formally required - or Analysis I/II and Linear Algebra I.
The course is appropriate as a home school elective for students of all majors with a particular interest in computation. It is recommended that students commit to this course for the full year. Students interested in a more compact introduction to methods of numerical computation are advised to take the one-semester Engineering and Science Mathematics 4A (Numerical Methods) instead.
Topics covered throughout the year are: computer arithmetic, condition of algorithms, systems of linear equations including iterative methods, computation of eigenvalues, interpolation and least square methods, numerical quadrature, numerical solution of ordinary differential equations (including stiff equations and boundary value problems); introduction to linear and nonlinear optimization. The last part of this course consists of special topics which may change from year to year.
|Office hours:||WF after class in Research I, 107|
|Lectures:||WF 9:45-11:00 in East Hall 4|
Collaborative project work in groups of two or three is permissible provided:
You may consult books and internet resources provided you always quote the source.
|07/09/2005:||Computer arithmetic, condition of algorithms|
|09/09/2005:||Solving scalar nonlinear algebraic equations: bisection, Newton's method|
|14/09/2005:||Solving scalar nonlinear algebraic equations: analysis of Newton's method, secant method|
|16/09/2005:||Matrix norms and condition numbers|
|21/09/2005:||Linear systems: Gauss elimination|
|23/09/2005:||LU decomposition without pivoting|
|28/09/2005:||LU decomposition with pivoting; error analysis|
|30/09/2005:||QR decomposition; least square solutions to linear systems|
|05/10/2005:||Iterative methods: Jacobi and Gauss-Seidel method|
|12/10/2005:||Conjugate Gradient method|
|14/10/2005:||Review for midterm exam|
|26/10/2005:||Lagrange interpolation; estimates of the interpolation error; numerical differentiation|
|28/10/2005:||Lagrange interpolation (continued)|
|02/11/2005:||Splines: minimality properties and approximation error estimates|
|04/11/2005:||Splines: Derivation of the coefficients for the natural spline; introduction to B-splines|
|09/11/2005:||Trigonometric interpolation: basic setup, orthogonality properties of DFT|
|11/11/2005:||Trigonometric interpolation: approximation error estimates; introduction to the FFT|
|16/11/2005:||Numerical integration: Newton Cotes formulae|
|18/11/2005:||Numerical integration: Gauss quadrature|
|23/11/2005:||Numerical integration: extrapolation|
|30/11/2005:||Review of ordinary differential equations|
|02/12/2005:||One step methods for ordinary differential equations|
|07/12/2005:||Review for final exam|
|19/12/2005:||Final Exam, 9:00-11:00 in East Hall 5|