Abstract
This chapter looks at iterative methods to solve linear systems and at some alternative methods to solve eigenvalue problems. That is, we now look at iteration instead of using a finite number of noniterative steps. Iterative methods for solving eigenvalue problems are, of course, completely natural. We looked at power iteration and at the QR iteration in Chap. 5; here we look at some methods that take advantage of sparsity or structure. We also use one pass of iterative refinement to improve structured backward error. ⊲
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- 1.
Yes, even for n = 2, because while square roots are “legal,” they are not finite—extracting them is iterative, too.
- 2.
See Bostan et al. (2003). For a history of the transposition principle, see http://cr.yp.to/transposition.html.
- 3.
- 4.
For more on this, see the discussion in Higham (2002).
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Corless, R.M., Fillion, N. (2013). Iterative Methods. In: A Graduate Introduction to Numerical Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8453-0_7
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