Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Practice Problems and Solutions
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Introduction
[edit | edit source]This is a compilation of problems and solutions from past numerical methods qualifying exams at the University of Maryland.
August 2008
[edit | edit source]Problem 1
[edit | edit source]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
[edit | edit source]Show that minimizes .
Solution 1a
[edit | edit source]We wish to show that