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Unformatted text preview: 4.2 use the stopping criterion: .2; = {15): 152255 = 5.555
Tmevalue: 5111(3513}: U.3ﬁﬁ025___ zero order: sin[£] =5 = 1514:5193 3 3 E: : UBﬁﬁUEE —l_ﬂ4?193 x 103%: 2.332%
{1.355525 ﬁrst Dfdﬂl'.
sing] =1.54:r'195 — {K :53? = 5.555551 5, =1.15% an =‘ﬂ'355f‘ﬁﬁéﬂﬁwg x100%=22_36% Md 01115::
5
5' 5 =5_555551+ (”T ”r 3] =5_355255
3 125
5, =5.53155 a, = —5_355295—5_555351 x155%=1.211%
5.555295
11de urdar: T
s' 5 =5555295—W3] =5_555521
3 5515 ﬂBﬁﬁﬂll — 0.866295
{1.3661121 5} = ﬁ_ﬂﬂﬂ4?’?% Ed = x1ﬂﬂ% = ﬂ_ﬂ3lﬁ% Since the appmﬂmate mm is below 0.5%, the compmatiun can be terminated 4.5 T111»: value: 313} = 554. zero order:
f8)=f{l}=EE s,=‘554;;523x1oo=ya=111_19m
ﬁrst ordm'.
f{3)=ﬁl+f'(l){3—l)=—52+T0{2)=TB £,=85.9‘21%
second order:
f8)=?3+f2ﬂ) [3—1)2 =T3+¥4=354 £,=36_1ﬁ1%
ﬂaird mﬂar:
‘3} 150
f8)=354+f ﬁﬂ}(3_1}3 2354+?82554 53.20% Thus, 1h: ﬂ'ﬁrd—ordcr result is pﬂfect because the original flmctiun is a thiId—order polynomial. 4.6 True value:
f' {x} = ?5252 —12x+2
f' (2) = 25(2)2 —12{2)+ 2 = 233 ﬁmction values
xH = 1.8 fin.1) = 58.96
x! = 2 ﬁn) = 102
xm = 2.2 ﬁrm} 2 164.56
forward:
16456—102 283—3128
f'[2)=—=312.8 at: x100%=1l).53%
0.2 283
backward:
f'[2)=w=255_2 5!: ﬂ x1m=93239¢
0.2 283
centered:
16456—5036 283—234
f'(2)=—=284 a}: X100€6=0.353%
2(ﬂ.2} 283 Bod: the forward and backward have errors that can be approximated by (recall 1321.415), ‘E  g f"(x.) h
' 2
f"(2} = lﬁﬂx— 12 =150(2}—12 = 288 Et 22—3822 =2s.s Thisis very closetothe actual errorthat occurred inthe approximations forward: E, s5 233—312.3 =29.3
backward: E, a pas—2552‘ 222.3 The centered approximation has an error that can be approximated by (3:
E, as —f :2) 2:2 = —%022 = —1 which is exact: E = 283 — 284 = —1. This result occurs because the original ﬁmction is a cubic equation
which has zero fourth and higher derivatives. 4.? True value: f"(x) =15'Dx —l2
f"(2) 2150(2) —12= 283 ﬁ=02i
f"(2}=w=w=lgg
”25 0.25
h =o_125:
we) =W zw = 233 {11252 0.1252 Buﬂnresults ammm becausethe mars aIeaﬁmction of4ﬁ’ andhigher deﬁvaﬁves whicharezemﬁn
a Bed—order polynomial. 4.12 The condition number is computed as CNZEf'GE)
ft?)
J—‘ ]
1mm —
(a) CH: 2 1.00001—1 =1.nmm(153_1139)=m_6” «.Jl.00001—1+1 1.003162 The result is ill—conditioned because the derivative is large nearx = 1. —10 —5
a.) .93.;wa = —1o
3"“ 454x10—5 The result is ill—conditioned because A: is large. 31:] ﬂ_l
J3m2 +1 _ 300{—5.555556x10_6) {c} (EN = — = —0.99999444
430:)? +1_3m 0.001556? The result is well—conditioned x —xs_x — 3—: +1 2
([1) CN 2 x = 0.031(049'9657) = _0_ 000 5
g“ _1 — 0.9995
x The result is well—conditioned x (1+cos x)cosx+sinx(sinx} 2
[9) EN = {1 +_cos x) = 3.14190? [20,204,233 2 —10,001
smx — 63 65.2 l+cosx The result is ill—conditioned because, as in the following plug the ﬁmction has a singularity at x = Fr. 2000
1000 0 400013 3.1 3.12 3.1 3.16 3.13 3.2 2000 ...
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 Spring '10
 ROCKE

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