Newton-Ralphson
Theorima1:Esto x- anikei A riza tis f(x)=0
a)An f orism. kai parag. gia |x-x-|<g,
f' sin sto x- kai f'(x-)<>0, tote 
g'(x-)=0 kai ara i akol. {xk} tis 
Newton sigl. ipergram. sto x- gia 
|xo-x-|<=d, gia kapio d>0/
b)An epipleon |f'(x)-f'(x-)|<=M|x-x-|
gia |x-x-|<g' tote sigl. tetrag.
Apod:Apo ipoth. prokiptei oti oi 
f kai g orism. kai oi antist. eksis.
isodin se anoikti perioxi tou x-.
a)Apo ton orismo tis parag kai tis
ipoth exoume afou f(x-)=0 kai f'(x-)<>0
g'(x-)=1-lim f(x)/((x-x-)f'(x)) =
         x->x-
=1-lim (1/f'(x))[(f(x)-f(x-))/(x-x-)]
   x->x-
=1-f'(x-)/f'(x-)=0
Sinep. i {xk} sigl. ipergram. gia
|xo-x-|<=d gia kapoio d>0.
b)Esto N tetoio oste |xk-x-|<g kai
|xk-x-|<g' gia k>N.Apo theor.mesis.
timis kai tis ipoth exoume gia k>N
|f(xk)-f(x-)-f'(x-)(xk-x-)|=
=|f'(ksi k)-f'(x-)||xk-x-|
<=M|xk-x-|**2 .Apo auti tin anisot.
kai tis ipoth exoume gia k>N:
|xk+1-x-|=|xk-(f(xk)/f'(xk))-x-|
<=|-[f(xk)-f(x-)-f'(x-)(xk-x-)]/f'(xk)|
+|[(f'(xk)-f'(x-))(xk-x-)]/f'(xk)|
<=c|xk-x-|**2 ->tetr.sigklisi.

Theorima2:Esto x- anikei A rixa f(x)=0
An se mia anoikti perioxi tou x-,i f
grafetai f(x)=(x-x-)**m h(x) me m>1,
h'(x-)<>0,h orism,parag,h' sinex 
sto x-,tote: g'(x-)=1-1/m<1
kai i {xk} tis Newton sigl. gramm.
sto x-,gia |xo-x-|<d
Apod:Thetontas z=f/f', konta sto x:
z(x)=(x-x-)h(x)/(mh(x)+(x-x-)h'(x)
ara oi f,g orisk kai oi eksis f(x)=0
kai g(x)=x-f(x)/f'(x) isodin se anoikt
perioxi tou x-.Epipleon,z(x-)=0 kai
g'(x-)=1-lim z(x)/(x-x-)=1-1/m<1
         x->x-
Sthn peript tou Theor1 apo theor
mesis timis gia x konta sto x-:
f(x)=(x-x-)h(x) ,h(x)=f'(ksi(x)),
ksi(x) anikei (x,x-) gia x<>x-,h(x-)=
=f'(x-)<>0 kai l h sinexis sto x- 
kai ara riza x- apli.Stin peript tou
theor2 x- poll/tas m.
-----------------TELOS---------------
