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

CRU

emc c

it

a A Fca ta

If I is nonlinear just pretend it is usingDff as approxfor A So Newton methodjust iterates

2kt Nk af FleaFeatcuxu 1124

From pictures Newton method should be very fastGiang n I f a sis aNeuron FEI Et 1No LR 1617

zz 208426623244453492077799307 62 1.00601

i

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