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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2007 • Question 2

Question

Math 6610 • Analysis of Numerical Methods I Fall 2007 • Question 2

Least Squares

Suppose \(A\in \mathbb {R}^{m\times n}\) has linearly independent columns, and \(C\in \mathbb {R}^{\ell \times n}\) has linearly independent rows. Let \(d\in \mathbb {R}^\ell \) and \(b\in \mathbb {R}^m\). Show how two full-rank QR factorizations can be used to solve the linearly constrained least squares problem: \[ \text {Find } x^*\in \{x\in \mathbb {R}^n:Cx=d\} \text { which minimizes } \|b-Ax^*\|_2. \]

Raw LaTeX

{Least Squares}

Suppose \(A\in\R^{m\times n}\) has linearly independent columns, and
\(C\in\R^{\ell\times n}\) has linearly independent rows. Let \(d\in\R^\ell\) and
\(b\in\R^m\). Show how two full-rank QR factorizations can be used to solve the
linearly constrained least squares problem:
\[
  \text{Find } x^*\in\{x\in\R^n:Cx=d\} \text{ which minimizes }
  \|b-Ax^*\|_2.
\]

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Topic Tags

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