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.
