newtonian as we are for critical pf - university of...
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Newtonian As we are almostalways looking for criticalpoints i.e solution to Pf o it makes sense to look forgeneral tool of finding roots ie se Fca o F Un 71124
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a A Fca ta
If I is nonlinear just pretend it is usingDff as approxfor A So Newton methodjust iterates
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From pictures Newton method should be very fastGiang n I f a sis aNeuron FEI Et 1No LR 1617
zz 208426623244453492077799307 62 1.00601
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The If F u R is conkwously differentiable Ford 0and Da F is invertible then i
a There is an open ball Beta B for any startingHEBNES CB Rie k
dew Nn H o i.e gHae 2 11 3 AHae 2 11
b If F is 2Lipschitz then112 1 2 11 e K line 2 112
i.e He 2 11 0 e CI
Boot Coupute nice a me 2 42at FCskCDaEYCDECne a FCaDJ
CDfETEDxE re xD of DEF tea dthere Rt a C ne 2 5
we use standard funkFC2 j FEFofldqffkttcse HDdosbaf.cat Mdt
CBF I oaf Doc Coca x dt
a.ae lsH EYll.Hx x ltofIlDqf DxtFHdtC tAs Daf is coutureus in a ssumed small
for dat nut small i e for niceBpCx Gg if KDx.FIkMouBpCxwe can shrink p if needed so thatHDx F D FILE 44M on B and therefore we
M AXk 11 11 HXa x k o geometrically
This 42 is actually going done withq so that a holds
Now if DF is Lipschitz then G EM llxicX.tl Luxe 11ML HXa x H
This implies b
A useful and usually interting class of ineppiye is given byf i Cc a polynomial Julia sets emergingthere are examplesof fractalsWhat happens when 12 is not invertible Quadratic
convergence disappears Say tinka n o
Then Rice ace ZIE see12 bland geometricconvergence
Question about the regionfor starting pants to converge
to X is important one One typical resultvalid actually for s dinBanadspee
The Kautomicle I D R D openconvex
If for a ED llaoFTFCadlELillCDnoF5CDaF
DyFJllekkn yllfwa.y ED212 12
Bz a CD then Bada 3 K uniqueroot ICr 0 and HaeRHIS 92 t 9 22141
Nopoofhere.beof Newton method ie aphunzation Deprives finding
D F where Fiat Rfcal i.e Def Hf NM is therefore a
secondarymethod Coueephially one approximates f withits Taylor expansion to 2nd order and finds minimum
f Gfc a sett1YIHfExxd