Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 1995 • Question 2

Question

Math 6610 • Analysis of Numerical Methods I Fall 1995 • Question 2

Linear Least Squares Let \(A\) be a real \(m\times n\) matrix of full rank and let \(b\in \mathbb {R}^m\) and consider the Least Squares problem: \[ \text {Find }x\in \mathbb {R}^n\text { such that }\lVert Ax-b\rVert _2\text { is minimized.} \] What are the ‘normal equations’ for the Least Squares problem? How can they be used to solve the Least Squares problem? What is the QR factorization of \(A\) and how can it be used to solve the Least Squares problem? What is the Singular Value Decomposition (SVD) of \(A\) and how can it be used to solve the Least Squares problem? Compare and contrast these approaches for numerically solving the Least Square problem.

{Linear Least Squares}
Let \(A\) be a real \(m\times n\) matrix of full rank and let \(b\in\R^m\) and consider
the Least Squares problem:
\[
  \text{Find }x\in\R^n\text{ such that }\lVert Ax-b\rVert_2\text{ is minimized.}
\]
What are the `normal equations' for the Least Squares problem? How can they be used
to solve the Least Squares problem? What is the QR factorization of \(A\) and how can
it be used to solve the Least Squares problem? What is the Singular Value
Decomposition (SVD) of \(A\) and how can it be used to solve the Least Squares
problem? Compare and contrast these approaches for numerically solving the Least
Square problem.

Topic Tags

Least Squares and Normal Equations Matrix Rank and Fundamental Linear Algebra QR Factorization Singular Value Decomposition