Abstract
Iterative methods formally yield the solution x of a linear system after an infinite number of steps. At each step they require the computation of the residual of the system. In the case of a full matrix, their computational cost is therefore of the order of n2 operations for each iteration, to be compared with an overall cost of the order of EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIYaaabaGaaG4maaaaaaa!377D!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\frac{2}{3}$$n3 operations needed by direct methods. Iterative methods can therefore become competitive with direct methods provided the number of iterations that are required to converge (within a prescribed tolerance) is either independent of n or scales sublinearly with respect to n.
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© 2007 Springer Science+Business Media New York
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Quarteroni, A., Sacco, R., Saleri, F. (2007). Iterative Methods for Solving Linear Systems. In: Numerical Mathematics. Texts in Applied Mathematics, vol 37. Springer, New York, NY. https://doi.org/10.1007/978-0-387-22750-4_4
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DOI: https://doi.org/10.1007/978-0-387-22750-4_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-7394-1
Online ISBN: 978-0-387-22750-4
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