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Structured Linear Systems

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A Graduate Introduction to Numerical Methods

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Abstract

We define structured linear systems to include sparse systems or systems with correlated entries or both. We define the structured backward error and a structured condition number. We give examples of various classes of structured linear systems and examples of algorithms that take advantage of the special structure. ⊲

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Notes

  1. 1.

    See Davis and Hu (2011) for a beautiful collection of useful sparse matrices. See http://www.cise.ufl.edu/research/sparse/matrices/synopsis/ for an introduction to visualizing large sparse matrices by minimizing the “energy” in a graph related to the matrix.

  2. 2.

    Also, see Fig. 16.15 for spy pictures of sparse matrices arising in finite-difference solutions to partial differential equations.

  3. 3.

    Here, we only warn the reader. See Higham (2002 ​, chap. 13) for a detailed introduction to these issues.

  4. 4.

    See Davis (2006), whose techniques are used under the hood in Matlab and in many other problem-solving environments.

  5. 5.

    Available at http://www.cise.ufl.edu/research/sparse/matrices/Bai/bfwa398.html.

  6. 6.

    Using exact arithmetic, this is really practical only if k ≪ n (Chen et al. 2002).

  7. 7.

    This is a quote from David S. Watkins, in Hogben (2006 chapter 43).

  8. 8.

    To name two, Toh and Trefethen (1994) and Edelman and Murakami (1995).

  9. 9.

    See Corless et al. (1997) and, for example, Graillat and Trébuchet (2009).

  10. 10.

    See Manocha (1994) and Shakoori (2008).

  11. 11.

    For more on this, see Demmel and Koev (2005).

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Authors and Affiliations

  1. Applied Mathematics, University of Western Ontario, London, ON, Canada

    Robert M. Corless & Nicolas Fillion

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  1. Robert M. Corless
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  2. Nicolas Fillion
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Corless, R.M., Fillion, N. (2013). Structured Linear Systems. In: A Graduate Introduction to Numerical Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8453-0_6

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