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Solving Ax=b

In: A Graduate Introduction to Numerical Methods

Author

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  • Robert M. Corless

    (University of Western Ontario, Applied Mathematics)

  • Nicolas Fillion

    (University of Western Ontario, Applied Mathematics)

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. ⊲

Suggested Citation

  • Robert M. Corless & Nicolas Fillion, 2013. "Solving Ax=b," Springer Books, in: A Graduate Introduction to Numerical Methods, edition 127, chapter 0, pages 167-237, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-8453-0_4
    DOI: 10.1007/978-1-4614-8453-0_4
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