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Condition pp 3-19 | Cite as

Normwise Condition of Linear Equation Solving

  • Peter Bürgisser
  • Felipe Cucker
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 349)

Abstract

Every invertible matrix \(A\in \mathbb {R}^{n\times n}\) can be uniquely factored as A=QR, where Q is an orthogonal matrix and R is upper triangular with positive diagonal entries. This is called the QR factorization of A, and in numerical linear algebra, different ways for computing it are studied. From the QR factorization one obtains the solution of the system Ax=b by y=Q T b and x=R −1 y, where the latter is easily computed by back substitution.

The Householder QR factorization method is an algorithm for computing the QR-decomposition of a given matrix. It is one of the main engines in numerical linear algebra. The following result states a backward analysis for this algorithm.

Theorem 1.1Let \(A\in \mathbb {R}^{n\times n}\) be invertible and \(b\in \mathbb {R}^{n}\). If the system Ax=b is solved using the Householder QR factorization method, then the computed solution \(\tilde{x}\) satisfies
$$\tilde{A}\tilde{x}=\tilde{b}, $$
where \(\tilde{A}\) and \(\tilde{b}\) satisfy the relative error bounds
$$\|\tilde{A}-A\|_F\leq n\gamma_{cn}\|A\|_F \quad\mathit{and}\quad \|\tilde{b}-b\|\leq n\gamma_{cn}\|b\| $$
where \(\gamma_{cn}:=\frac{cn\epsilon _{\mathsf {mach}}}{1- cn\epsilon _{\mathsf {mach}}}\) with a small constant c.  □
This yields \(\|\tilde{A} - A \| \le n^{3/2} \gamma_{cn}\, \| A\|\) when the Frobenius norm is replaced by the spectral norm. It follows from this backward stability result that the relative error for the computed solution \(\tilde{x}\) satisfies
$$ \frac{\|\tilde{x}-x\|}{\|x\|} \leq cn^{5/2}\epsilon _{\mathsf {mach}}\mathsf {cond}(A,b) +o(\epsilon _{\mathsf {mach}}), $$
(*)
and the loss of precision is bounded by
$$ \mathsf {LoP}\bigl(A^{-1}b\bigr) \leq \frac{5}{2}\log n+\log \mathsf {cond}(A,b) + \log c+o(1) . $$
Here cond(A,b) is the normwise condition number for linear equation solving,
$$\mathsf {cond}(A,b)=\lim_{\delta\to0} \sup_{\max\{\mathsf {RelError}(A),\mathsf {RelError}(b)\}\leq\delta} \frac{\mathsf {RelError}(A^{-1}b)}{\max\{\mathsf {RelError}(A),\mathsf {RelError}(b)\} } , $$
where RelError(A) is defined with respect to the spectral norm and RelError(b) with respect to the Euclidean norm. Inequality (*) calls for a deeper understanding of what cond(A,b) is than the equality above. The pursuit of this understanding is the goal of this chapter.

Keywords

Condition Number Matrix Norm Frobenius Norm Spectral Norm Numerical Linear Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. 221.
    A.M. Turing. Rounding-off errors in matrix processes. Quarterly Journal of Mechanics and Applied Mathematics, 1:287–308, 1948. MathSciNetzbMATHCrossRefGoogle Scholar
  2. 226.
    J. von Neumann and H.H. Goldstine. Numerical inverting matrices of high order. Bulletin of the American Mathematical Society, 53:1021–1099, 1947. MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Peter Bürgisser
    • 1
  • Felipe Cucker
    • 2
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsCity University of Hong KongHong KongHong Kong SAR

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