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Solving Ax=b

  • Robert M. Corless
  • Nicolas Fillion
Chapter

Abstract

This chapter first shows how to solve \(\mathbf{A}\mathbf{x} = \mathbf{b}\) in the simple cases in which \(\mathbf{A}\) is unitary or triangular, and then explains how the QR factoring can be used to reduce other problems to these simple cases. We show that these methods are backward stable; that is, they exactly solve a slightly perturbed problem. In order to understand how these small perturbations affect the solution, we then introduce the crucial notion of condition number in relation to the most important factoring, namely, the singular value decomposition (SVD). We also examine the LU factoring (equivalent to Gaussian elimination) and a number of applications of the main factorings. We end the chapter with a short discussion of nonlinear systems. ⊲

Keywords

Condition Number Singular Value Decomposition Gaussian Elimination Triangular System Partial Pivoting 
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.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Robert M. Corless
    • 1
  • Nicolas Fillion
    • 1
  1. 1.Applied MathematicsUniversity of Western OntarioLondonCanada

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