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

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

August 2008[edit]

Problem 1[edit]

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]

Show that minimizes .

Solution 1a[edit]

We wish to show that