Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Practice Problems and Solutions

IntroductionEdit

This is a compilation of problems and solutions from past numerical methods qualifying exams at the University of Maryland.

August 2008Edit

Problem 1Edit

Consider the system  . The GMRES method starts with a point   and normalizes the residual   so that   has 2-norm one. It then constructs orthonormal Krylov bases   satisfying

 


where   is a   upper Hessenberg matrix. One then looks for an approximation to   of the form

 

choosing   so that   is minimized, where   is the usual Euclidean norm.

Part 1aEdit

Show that   minimizes  .

Solution 1aEdit

We wish to show that