1.Olokliromata
                   x1
Trapez:(a=xo,b=x1) S f(x)dx=
                   x0
       =(h/2)(fo+f1)-(h**3/12)
       f''(m) m anikei (a,b)
                   x2
Simpso:(a=xo,b=x2) S f(x)dx=
                   xo
       =(h/3)(fo+4f1+f2)-
       -(h**5/90)f(4)(m)
                   x3
3/8   :(a=xo,b=x3) S f(x)dx=
                   xo
       =(3h/8)(fo+3f1+3f2+f3)-
       -(3h**5/80)f(4)(m)
Sinthetoi Tipoi
Trapez: h(fo/2+f1+f2+..+fN/2)-
        -(h**3/12)sum(i=1..N)(f''(m))
Simpso: (h/3)(fo+4f1+2f2+4f3+..+fN)
        -(h**5/90)sum(k=1..M)(f(4)(m))

Genikos tipos:
 b
 S f(x)dx=sum(i=1..n)(cif(xi))+En(f)
 a
        b                   b
 me: ci=S li(x)dx kai En(f)=Sen(x)dx
        a                   a
     gia en(x), li(x) bl. paremboli

Paremboli

pn(x)=sum(i=0..n)(li(x)f(xi))
li(x)=(x-xo)..(x-xi-1)(x-xi+1)..(x-xn)
     /(xi-xo)..(xi-xi-1)...(xi-xn)
Sfalma:           (n+1)
      en(f)=f(x)-pn(x)=f(ksi(x))
      GIN(i=0..n)*(x-xi)  /(n+1)!

Newton: pn(x)=f[xo]+(x-xo)f[xo,x1]+
              +f[xo,x1,x2](x-xo)(x-x1)
              (x-x2)...

Diaforikes eksisoseis.

Taylor 1 order:(euler)yk+1=yk+hy'k=
                      =yk+hf(xk,yk)
  ''   2 ''   :yk+1=yK+hy'k+h**2/2y''k
               =yk+hf(xk,yk)+h**2/2
               [df/dx +(df/dy)f]
               meriki  meriki

RK 1 order:Idia me taylor(Euler) 
'' 2  ''  :m=p=2.Diakr. periptoseis
i)(a1=1/2) Belt.Euler:
   yk+1=yk+hf(xk+h/2,yk+z)
   z=(h/2)f(xk,yk)
ii)(a1=1) RK 2 order :
   yk+1=yk+(1/2)(z1+z2) me:
   z1=hf(xk,yk), z2=hf(xk+h,yk+z1)
iii)RK 4 order:
   yk+1=yk+(1/6)(z1+2z2+2z3+z4) me:
   z1=hf(xk,yk),z2=hf(xk+h/2,yk+z1/2)
   z3=hf(xk+h/2,yk+z2/2)
   z4=hf(xk+h,yk+z3)

Mi grammikes alg.eksis.

Dixotom:Sfalma:|xk-x-|<=(bo-ao)/2**(k+1)
                       =ek
Temnousa:xk+1=xk-f(xk)(xk-xk-1)/
              (f(xk)-f(xk-1))
Gen.Epanal: xk+1=g(xk)
Sfalma:
|xk-x-|<=(a**k/(1-a))|x1-xo|=a**k|xo-x-|
opou a=|g'(x)| to fragma
Newton: xk+1=xk-f(xk)/f'(xk)


Grammika Sistimata

Normes
||A+B||<=||A||+||B||
||A*B||<=||A||||B||
||A||g=max sum(j=1..n)(|aij)|
       1<=i<=n
||A||s=max sum(i=1..n)(|aij)|
       1<=j<=n
||A||e=(sum(i,j)(ai,j*ai,j))**1/2
||A||oo=||A||g
||A||1=||A||s
||A||2=[r(ATA)]**1/2
      =r(ATA) A simmetr
Eustatheia
Deiktis katast, m=||A||||A**-1||
Epanal.Method
||x-xk||=(||C||**k/(1-||C||))||x1-x0||
Fasm Aktina: r(C)=max|li|
                  1<=i<=n
------------------TELOS-------------------
