Gen.Epanal.Methodos x=g(x) (1)

Therima1:(periliptika)An x- mia riza
tis f i tis (1)kai a=|g(x-)|<1 tote
i {xk} periex. sto I=[x- -d,x- +d] kai
siglinei sto x- gia kathe xo.
Apod:Epidi i g(x) parag. sto x- exoume:
e(x)=|(g(x)-g(x-))/(x-x-) -g'(x-)|->0
otan x->x-.Epomenos:
|g(x)-x-|=|g(x)-g(x-)|<=|g(x)-g(x-)-
-g'(x-)(x-x-)|+|g'(x-)||x-x-|=[e(x)+a]|x-x-|
Tora gia b sto (a-,1) esto d>0 arketa 
mikro oste:I=[x- -d,x- +d] iposin tou A
kai a(x)=a- +e(x)<=b gia x sto I.
Gia xo anikei sto I i {xk} tis epanal
methodou ikanop. tote ris anisot:
|xk+1-x-|=|g(xk)-x-|<=a(xk)|xk-x-|
<=..<=b**(k+1) |xo-x-|
pou dixn. oti i {xk} periex. sto I
kai oti sigl. sto x-,afou b<1.

Protasi2:An g sistol. sto S iposinol
tou R kai isx. mia apo tis 3:
(i)g(x) anikei S gia kathe x sto S
(ii)S=[z-d,z+d] kai |g(z)-z|<=(1-a)d 
(iii)S=[x- -d,x- +d] kai f(x-)=0
Apod:
(i)Exoume xo anikei S kai an xk anikei 
S, tote xk+1=g(xk) anikei S
(ii) An x anikei S tote |g(x)-z|
<=|g(x)-g(z)|+|g(z)+z|<=a|x-z|+(1-a)d
<=ad+(1-a)d=d Ara isx. i sinthiki (i)
(iii)An x sto S tote:
|g(x)-x-|=|g(x)-g(x-)|<=a|x-x-|<=ad<=d
opote isx. i (i).

Theorima2:(perilipt)An g sistol.kai i 
{xk} periex. sto S :
a)i {xk} sigl. se mia riza x- tis f(x)=0
b)i x- monad. riza sto S
c)ektimiseis sfalmatos:
i)|xk-x-|<=a**k|xo-x-|
ii)|xk-x-|<=(a/(1-a))|xk-1-xk|
iii)|xk-x-|<=(a**k/(1-a))|x1-xo|
Apod:Kat arxin exoume gia kathe k:
|xk+1-xk|=|g(xk)-g(xk-1)|<=a|xk-xk-1|
<=...<=a**k|x1-xo|   (1)
eksalou i sinart. g einai sinexis 
sto S, dil. gia x' anikei sto S
lim g(x)=g(x') (x anikei S) afou g sist. sto S
x->x'
a)As deiks. oti i akol. {xk} tipou 
Cauchy dil:lim |xk-xk+p| =0
           k->oo,p->oo
Exoume apo tin anisot. (1) :
|xk-xk+p|<=|xk-xk+1|+|xk+1-xk+2|+..
+|xk+p-1-xk+p|<=(a**k + ...+a**(k+p-1))
                |x1-xo|=(1-a**p)/(1-a)
                        *a**k |x1-xo|
I teleut. ekfr. teinei sto 0
otan k->oo,p->oo.Ara h {xk} teinei se 
kapoio x- anikei S.epidi g sinex. einai:
x-=lim xk=lim g(xk-1)=g(x-)
   k->oo  k->oo
dil. to x- riza tis g(x)=x.
b)Esto z- mia alli riza tis eksis.
g(x)=x.Tote:|x- -z-|=|g(x-)-g(z-)| 
<=a|x- - z-|<|x- - z-| ara x-=z-
c)Gia tin ektim (i) exoume:
|xk-x-|=|g(xk-1)-g(x-)|<=a|xk-1 -x-|
<=...<=a**k |xo-x|
Gia tin (ii) exoume:
|x- -xk-1|<=|x- -xk|+|xk-xk-1|<=
a|x- -xk-1|+|xk-xk-1|->|x- -xk-1|
<=1/(1-a) |xk-xk-1| opote->(ii)
Telos apo tin (ii) kai tin(1)->(iii)

