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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2014 • Question 4

Question

Math 6610 • Analysis of Numerical Methods I Fall 2014 • Question 4

Suppose \(A\) is an \(m\times n\) real matrix with \(m>n\), and \(b\in \mathbb {R}^m\). For any vector \(v\in \mathbb {R}^m\), let \[ \|v\|_2=\sqrt {\sum _{i=1}^m v_i^2} \] as usual. Describe how to solve the Least Squares Problem \[ \|Ax-b\|_2=\min \] using the \(QR\) factorization of \(A\). Briefly state how you would compute the \(QR\) factorization.

Raw LaTeX

Suppose \(A\) is an \(m\times n\) real matrix with \(m>n\), and
\(b\in\R^m\). For any vector \(v\in\R^m\), let
\[
  \|v\|_2=\sqrt{\sum_{i=1}^m v_i^2}
\]
as usual. Describe how to solve the Least Squares Problem
\[
  \|Ax-b\|_2=\min
\]
using the \(QR\) factorization of \(A\). Briefly state how you would compute the
\(QR\) factorization.

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Topic Tags

Cholesky Factorization and SPD Matrices Least Squares and Normal Equations Linear Programming and Optimization QR Factorization