Integraci´o num`erica
M`etodes simples
IS=f(a)+4f(xm)+f(b)
6(b−a)xm=(a+b)/2
IT=f(a)+f(b)
2(b−a)
Iinf =m(b−a)m= min
x∈[a,b](f(x))
Isup =M(b−a)M=max
x∈[a,b](f(x))
Idre =f(b)(b−a)
Iesq =f(a)(b−a)
M`etodes compostos
IS=
n−1
1
i=0
f(xi)+4f(xi,i+1)+f(xi+1)
6hixi,i+1 =(xi+xi+1)/2
IT=
n−1
1
i=0
f(xi)+f(xi+1)
2hi
Iinf =
n−1
1
i=0
mihimi=max
x∈Ii
(f(x))
Isup =
n−1
1
i=0
MihiMi=max
x∈Ii
(f(x))
Idre =
n−1
1
i=0
f(xi+1)hi
Iesq =
n−1
1
i=0
f(xi)hi
a=x0<x
1<...<x
n=b,Ii=[xi,x
i+1],hi=xi+1 −xi
nsubintervals
(intervals equiespaiats)
M`etodes compostos
IT=h
2(f(x0)+2f(x1)+... +2f(xn−1)+f(xn))
Idre =h(f(x1)+f(x2)+...+f(xn))
Iesq =h(f(x0)+f(x1)+...+f(xn−1))
nsubintervals , h=(b−a)/2,x
i=a+ih
ξ∈[a, b]
Error m`etodes simples
E2=−fiv)(ξ)
2880 h5=Ch5(simpson) Exact per grau 3
E1=−f′′(ξ)
12 h3=Ch3(trapezi) Exact per grau 1
E0=f′(ξ)
2h2=Ch2(aproximacions rectangulars) Exacte per constants
ξ∈[a, b]
Error m`etodes compostos
En particular: |E(2n)|≈|E(n)|
2q|E(10n)|≈|E(n)|
10q
Predicci´o de l’error en m`etode d’ordre q:|E(h/k)|=|E(kn)|≈|E(h)|
kq=|E(n)|
kq
M`etode d’ordre q:E(h)=E(n)=Chq=C/nq
ES=−fiv)(ξ)
2880 (b−a)h4=Ch4m`etode d’ordre 4 (q=4) Exact per grau 3
ET=−f′′(ξ)
12 (b−a)h2=Ch2m`etode quadr`atic (q=2) Exact per grau 1
Erect =f′(ξ)
2(b−a)h=Ch m`etode lineal (q=1) Exacte per constants
