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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

1996 • Question 8

Question

Math 6610 • Analysis of Numerical Methods I 1996 • Question 8
**Discrete Approximation.** Let \(A\) be an \(m\times n\) matrix and \(b\in\mathbb{R}^m\) a given vector. Consider the problem of finding \(x\in\mathbb{R}^n\) such that \[ \lVert Ax-b\rVert_p=\min \] Depending on the value of \(p\) you obtain different kinds of problems. Describe and derive these problems for \(p=1\), \(p=2\), and \(p=3\). ****Math 663: Solution of Differential Equations.****

Raw LaTeX

\textbf{Discrete Approximation.} Let \(A\) be an \(m\times n\) matrix and
\(b\in\R^m\) a given vector. Consider the problem of finding \(x\in\R^n\) such that
\[
  \lVert Ax-b\rVert_p=\min
\]
Depending on the value of \(p\) you obtain different kinds of problems. Describe and
derive these problems for \(p=1\), \(p=2\), and \(p=3\).

\topic{Math 663: Solution of Differential Equations.}

Topic Tags

Discrete Maximum Principles Linear Programming and Optimization Low-Rank Approximation