Travaux Pratiques 10 : Calcul approché de la racine d`une équation

f(x)=0
f(x)=0 f
f:R→R
f(x) = 0.(E)
f f
2
x(E)
f
[a;b]
f[a;b]
f[a;b]
f(a)f(b)<0f(a)f(b)
c∈]a;b[f(c) = 0 (E)
c

c(E)ε
[a;b]ε
ab
a+εa+ 2εa+ 3ε...
f(a)f(a+ε)f(a)
c[a;a+ε]
f(a)f(a+ε)f(a)f(a+ε)
f(a)f(a+ε)
ε f(a+ε)f(a+ 2ε)
ε
a+kε a + (k+ 1)ε ε
f(x) = x2−2
[1 ; 3]
f(x) = 0 0.01
ε

[a;b]
m=a+b
2[a;m] [m;b]
f(a)f(m)<0c[a;m]c[m;b]
c
c[a;m]b←m
a←m
1.2.3.
f(x) = ln(x)−1
f(x)=0.(E2)
f[1 ; 4]
(E1)
2
50
N a b
c
0b−a
1b−a
2
2b−a
22
3b−a
23
Nb−a
2N
m c
b−a
2N
b−a
2N

c(E)ε
N
b−a
2Nc
b−a
2N≤ε⇐⇒ 2N≥b−a ε
⇐⇒ Nln 2 ≥ln b−a
ε
⇐⇒ N≥ln b−a
ε
ln 2
0,001
c ε
N ε
ε
c
ε