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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