Chap 1 : Résolution d`équations non-linéaires

f(x) = 0
f
−→
•f
x0
•x0x1, x2, x3, ...
lim
n→∞
xn= ¯x¯x f(¯x)=0.
f I R R
f(x)=0.
−→ x
<5
f:I−→ Rf¯x∈I
f(¯x) = 0.
¯x f

f¯x
f(¯x) = ¯x.
f(x) = x2−5x+ 6
g(x) = x2−4x+ 6
¯x f ¯x g :x7→ f(x) + x
f
I= [a, b]f f(a)f(b)
∀d∈[f(a), f(b)] c∈I f(c) = d
f:I= [a, b]→R
f(a)f(b)<0
f(a)f(b) ¯x∈]a, b[
f(¯x)=0
f¯x
•R
•x(1 + 2x) = ex]0,1[
g: [a, b]→[a, b] [a, b]g
¯x[a, b]
g[a, b]
(g(a)> a
g(b)< b
f(x) = g(x)−x f g
(f(a)>0
f(b)<0.
f[a, b] ¯x
f g

xn+1 =F(xn), n = 0,1,2, ...
x0x1x1x2
xn+1
xnn
p p
C¯x= lim
nxn
|¯x−xn+1| ≤ C|¯x−xn|p
lim
n→∞ |¯x−xn+1|
|¯x−xn|p=C.
p= 1 p= 2
p= 1 C < 1 (xn)
¯x
f[a, b]Rf(a)f(b)<0
f I0= ]a, b[ ¯x¯x
In= [an, bn]In−1¯x
In
f(an)<0f(bn)>0xn=
an+bn
2f(xn)
•f(xn) = 0 ¯x=xn
•f(xn)<0f]xn, bn[
an+1 =xnbn+1 =bn
•f(xn)>0f]an, xn[
an+1 =anbn+1 =xn
(an) (bn)
¯x¯x

a, b ∈Ra < b f : [a, b]7→ R
¯x∈]a, b[f(a)f(b)<0 (an)
(bn) ¯x
∀n≥0,0≤¯x−an≤b−a
2n,0≤bn−¯x≤b−a 2n.
(an) (bn)
n
∀n≥1, an−1≤an, bn≤bn−1, bn−an=b−a
2nan=bn= ¯x .
(an) (bn)|an−
bn| → 0n→ ∞ (an) (bn)
l∈R∀n an≤l≤bn
•n= 1 f(x0)f(a)a1=x0=a0+b0
2b1=b0=b
a0≤a+b
2=a1≤b1=b0
b1−a1=b−a+b
2=b−a 2.
f(x0)f(a)<0
•n f(xn) =
fan+bn
2f(xn)f(a)>0
an+1 =xn
an≤an+bn
2=an+1
bn+1 −an+1 =bn−an+bn
2=bn−an 2=1 2
b−a
2n=b−a 2n+1 ,
(an) (bn)l
l= ¯x
f(l)=0 n
(f(an)f(a)≥0
f(bn)f(b)≥0
f
(f(l)f(a)≥0
f(l)f(b)≥0

a b
(f(l)≤0
f(l)≥0=⇒f(l)=0.
0≤¯x−an≤bn−an≤b−a
2n
0≤bn−xn≤bn−an≤b−a
2n.
m
|¯x−xm| ≤ |Im|< ε
ε|Im|=b−a
2mε
m
m≥log b−a
ε
log(2) = log2b−a ε.
|¯x−xn+1| ≤ Cn|¯x−xn|n≥0
Cn<1
x7→ x2
In=
[an, bn] [an, xn] [xn, bn]xn(an, f(an))
(bn, f(bn)) xn
f(bn)−f(an)
bn−an
(xn−bn) + f(bn) = 0,
xn=anf(bn)−bnf(an)
f(bn)−f(an).
f(xn)