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Unformatted text preview: Numerical Analysis II Dr Abigail Wacher Oct 27, 2010 1 Order of convergence (as in online notes) Definition Let p n converge to p and suppose p n s p c n The method converges with order α if lim n / N p n C 1 K p p n K p a = l If a = 1 and l s 0 then convergence is linear If a = 1 and l = 0 then p n converges superlinearly If a = 2 and l s 0 then p n converges quadratically. Problems For the equation x 3 K 13 x C 18 = 0 consider the rearrangements 1) x = x 3 C 18 13 2) x = 13 x K 18 1 3 3) x = 13 x K 18 x 2 4) x = x 3 K 12 x C 18 Root p=2. Which rearrangement can be used successfuly as the basis for an iterative method to find p ? If it's suitable find a neighbourhood of p in which you should choose your initial value p . 1) g x = x 3 C 18 13 , g ' x = 3 x 2 13 , g ' 2 = 12 13 d an interval surrounding 2 in which the confitions of the contraction mapping theorem are satisfied. Need to find an interval a , b for which g ' x % L ! 1 c x e a , b and g : a , b / a , b By solving g ' x = G...
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- Fall '11
- Numerical Analysis, Order theory, Monotonic function, Dr Abigail Wacher, Numerical Analysis II