Lecture20101021_2

# Lecture20101021_2 - Numerical Analysis II Dr Abigail Wacher...

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Unformatted text preview: Numerical Analysis II Dr Abigail Wacher 1 O Proof (cont.) ii) Consider |p n-p| = |g(p n-1 )-g(p)| % L |p n-1-p| % L |p n-1-p n | + L |p n-p| => (1-L)|p n-p| ≤ L |p n-1-p n | = L|g(p n-2 )-g(p n-1 )| % L 2 |p n-2-p n-1 | % L n |p-p 1 | => |p n-p| ≤ L n 1 K |p-g(p )|-> 0 as n -> ∞ =>p n-> p as n -> ∞ QED Example f(x)= x K e K = 0 d : fx / C N as / C N / K N / K N !: f ' = 1 C K O Iteration: p n+1 = K pn =:g(p n ) Maple gives converges for p = 0, 10, -10 p:=-10.0: for n from 1 to 20 do p:=exp(-p); print(n,p,p-exp(-p)); end do: 1, 22026.46579, 22026.46579 2, 1.065250072 10-9566 , K 1. 3, 1., 0.6321205588 4, 0.3678794412, K 0.3243211863 5, 0.6922006275, 0.1917271269 6, 0.5004735006, K 0.1057700345 7, 0.6062435351, 0.0608477491 8, 0.5453957860, K 0.0342165495 9, 0.5796123355, 0.0194968741 10, 0.5601154614, K 0.0110276537 11, 0.5711431151, 0.0062637677 12, 0.5648793474, K 0.0035493776 13, 0.5684287250, 0.0020139918 14, 0.5664147332, K 0.0011419041 15, 0.5675566373, 0.0006477254 16, 0.5669089119, K 0.0003673203 17, 0.5672762322, 0.0002083338 Numerical Analysis II Dr Abigail Wacher 2 (2) O (1) O 18, 0.5670678984, K 0.0001181517 19, 0.5671860501, 0.0000670100 20, 0.5671190401, K 0.0000380039 Example f(x)=2x-cosx Try p n+1 = 1 2 cos(p n )=g(p n ) p:=1000.0: for n from 1 to 20 do p:=cos(p)/2; print(n,p,2*p-cos(p)); end do: 1, 0.2811895382, K 0.3983469464 2, 0.4803630114, 0.0738987896 3, 0.4434136166, K 0.0164651674 4, 0.4516462003, 0.0035625584 5, 0.4498649211, K 0.0007760066 6, 0.4502529244, 0.0001687886 7, 0.4501685301, K 0.0000367246 8, 0.4501868924, 0.0000079899 9, 0.4501828974, K 0.0000017384 10, 0.4501837666, 3.782 10-7 11, 0.4501835775, K 8.23 10-8 12, 0.4501836186, 1.78 10-8 13, 0.4501836097, K 3.9 10 3....
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## This note was uploaded on 02/09/2011 for the course MATH 2051 taught by Professor Dr.a.wacher during the Fall '11 term at Durham.

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Lecture20101021_2 - Numerical Analysis II Dr Abigail Wacher...

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