411 REFERENCE · Wikibooks
Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Practice Problems and Solutions
Reference material by Wikibooks contributors. This retains its source identity and is not 411 news reporting.
Introduction
This is a compilation of problems and solutions from past numerical methods qualifying exams at the University of Maryland.
August 2008
Problem 1
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 1a
Show that minimizes .
Solution 1a
We wish to show that