Gauss-Siedel 
Theorima:An o pinakas A exei diagonia
iperoxi kata grammes,tote o pinakas C
tis method ikanopoiei ti sxesi:
||C||oo<=||Cj||<1
opou Cj o jacobi kai i gauuss siglinei
fia kathe xo.Telos,ektimisi sfalmatos:
||xk-x||oo<=||Cj||oo * ||xk-xk-1||
/(1-||Cj||oo
Apod:Apo ti diag.iperoxi tou A:
||Cj||oo=max sum(j<>i)(|aij/aii|)<1
          i
Thetontas gia u anikei Rn me ||u||oo
=1, v=Q**-1Pu=Cu exoume:
vi=-sum(j<i)((aij/aii)vj)-
-sum(j>i)((aij/aii)uj)
Tha deiksoume prota me epagogh oti:
|vi|<=||Cj||oo gia i=1..n
Gia i=1 isxiei:
|v1|<=sum(j<i)(|aij/a11||uj|)
<=||u||oo sum(j>i)(|aij/a11|)<=||Cj||oo
An ipothesoume oti gia i stath 
kai j<i isxiei |vj|<=||Cj||oo<=1 tote
|vi|<=sum(j<i)(|aij/aii||vj|)+
+sum(j>i)(|aij/aii||uj|)<=(max|vj|)*
                           j<i
*sum(j<i)(|aij/aii|) + ||u||oo*
*sum(j>i)(|aij/aii|)<=sum(j<>i)(|aij/aii|)
<=||Cj||oo
Epomenos ||v||oo=||Cu||oo<=||Cj||oo 
gia kathe u me ||u||oo=1 kai ara
||C||oo=max||Cu||oo<=||Cj||oo<1 kai meth. 
siglin. ||u||=1
Epipleon exoume:||xk-x||oo<=
||C||oo||xk-xk-1||oo/(1-||C||oo)<=
||Cj||oo||xk-xk-1||/(1-||Cj||oo).

Idiot kai idiodian.
Theorima1:Sigklisi method-isxiei: 
lim ek.xk/||xk||2=u1 ,lim rik=lim xik/xi,k-1=l1
k->oo     (1)         k->oo      (2)
opou l1 i megaliteri iditimi kai
u1 to antist.idiodian. me ||u||2=1
Apod:Epidi ta idiodian.uj aneksart.
to xo grafetai:xo=sum(j=1..n)(cjuj)
Apo xk=A**k xo pairnoume:
xk=sum(j=1..n)(lj**kcjuj)=l1**k[
[c1u1+sum(j=2..n)((lj/l1)**k cjuj)]
gia k=1,2,...Alla |lj/l1|<1 opote
gia j>1 h kateuth. tou dian. xk
siglinei sthn kateuth. tou u1 otan k->oo
(ipoth.c1<>0) opote i (1)isxiei.Meta 
thetoume: rik=xik/xi,k-1=
l1[c1ui1+sum(j=2..n)((lj/l1)**k cjuij)]/
[c1ui1+sum(j=2..n)((lj/l1)**k-1 cjuij)]
opou xk=(xik,...,xnk)
An c1<>0 kai dialeksoume to i oste ui1<>0
tote lim rk=l1
     k->oo
An c1=0 allazoume arxiko dianisma.

Theorima2:O pinakas A1(simmetrikos)exei 
idiotimes:0,l2,...,ln.kai antist.idiodian 
u1,u2,..un (A1=A-l1u1u1T) T
Apod:Exoume: ||u1||2 **2 =u1u1=1,ara:
A1u1=Au1-l1u1(u1Tu1)=l1u1-l1u1=0
Gia j>=2 exoume: A1uj=Auj-u1(u1Tuj)=
=Auj=ljuj opou ta uj orthokanonika.

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Paremboli Lagrange
Theorima1:Esto f kai xo,x1,..,xn 
tou [a,b] simeia opou f(x) gnosta.
a)Yparxei pn anikei Pn :pn(xi)=f(xi)(1) 
b)pn monadiko
c)pn(x)=sum(i=0..n)(li(x)f(xi)) me:
li(x)=(x-xo)..(x-xi-1)(x-xi+1)..(x-xn)
(2)  /(xi-xo)..(xi-xi-1)(x-xi+1)..(xi-xn)
Apod:Einai fanero oti to polionimo 
pou orizetai apo ti (2) paremb. tin f
sta xo,x1,..,xn,diladi isxeiuoun oi
prot (a) kai (c).Gia na apodeiksoume ti
monadik. tou (b) esto q anikei Pn ena 
allo tixon polionimo pou paremb. tin f
sta idia simia.Tote to polion. r=pn-q
anikei sto Pn kai exei profanos toul.
n+1 rizes.Ara r=0 kai to pn einai monadiko.

Theorima2:(Sfalma Lagrange)An f 
       n+1
anikei C[a,b] tote iparxei ksi(x) sto 
(a,b):     (n+1)
f(x)-pn(x)=f(ksi(x))GIN(i=0..n)(x-xi)
           /(n+1)!
Apod:Gia x sto [a,b] stathero kai 
diaforo tou kathe xi,orixoume tin
F(t)=f(t)-pn(t)-[f(x)-pn(x)]*
*GIN(i=0..n)(t-xi)/GIN(i=0..n)(x-xi)
Parat. oti ta xo,x1,..xn,x einai n+2 
rizes tis F.Ara apo to theor. Rolle 
se kathena apo ta n+1 diastim. pou 
orizoun autes oi rizes simperain. oti
i F' exei n+1 rizes sto [a,b].Efarm.
n fores to Rolle stin F',n-1 stin F''
k.o.k. katalig. stin iparksi enos 
ksi(x) sto (a,b) tetoio oste 
(n+1)
F(ksi)=0.To theorima epetai amesos afou
      (n+1)                      (n+1)
[pn(t)] =0 kai [GIN(i=0..n)(t-xi)] =(n+1)!

Mi gramm. algebr. eksisoseis
Method dixotomisis
Theorima1:An f sinex. sto [a,b] kai
f(a)f(b)<=0 i {xk} tis dixot. siglinei
se mia riza tis f(x)=0.
Apod:Apo idiot. sinex. sinart. iparxei 
mia riza tis f sto [a,b], afou f(a)f(b)<=0.
i sigl.tis {xk} apotel. mia katask.
apod. tis iparksis mias rizas.
i akol. {ak} auksousa,ano fragmeni apo 
to bo, eno i {bk} fthinousa kai kato
fragm. apo to ao.Ara oi akolouth. autes 
sigkl.: ak->a- ,bk->b-, a-,b- sto [a,b].
Epidi isxiei:
bk-ak=(bo-ao)/2**k ->0 kai xk=(bk+ak)/2
anikei sto [ak,bk] blep. oti anagkastika:
a-=b-=x-=lim xk
         k->oo
Tora apo tin katask. ton ak,bk, exoume:
f(ak)f(bk)<=0 kai gia k->oo brisk. 
sto orio: 0<=f(x-)**2<=0 afou f sinex.
Sfalma:|xk-x-|=|(ak+bk)/2 - x-|<=
(bk-ak)/2=(bo-ao)/2**(k+1) =ek.

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)

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---------------
