PA=LU → •
[
L ,U , P
]
=lu
(
A
)
• y=L¿P∗b¿• x=U¿A=L L T→• R=chol
(
A
)
• y=R ' ¿b• x=R¿
A=QR → •
[
Q , R
]
=qr
(
A
)
• x=R
(
Q'∗b
)
A=US V T→•
[
U ,S , V
]
=svd
(
A
)
• y=S¿
(
U'∗b
)
• x=Vy
Gauss :for k=1 : n−1
for i=k+1 : n
A
(
i , k
)
=A
(
i , k
)
/A(k , k );
for j=k+1 : n
A
(
i, j
)
=A
(
i , j
)
−A
(
i , k
)
∗A
(
k , j
)
;
end
b
(
i
)
=b
(
i
)
−A
(
i , k
)
∗b
(
k
)
;
end
end
x
(
n
)
=b
(
n
)
/A(n , n);
for i=n−1 :−1 : 1
x
(
i
)
=
(
b
(
i
)
−A
(
i ,i +1: n
)
∗x
(
i+1 : n
)
)
/A
(
i ,i
)
;
end
tridiagonale →
Thomas :for k=1 : n−1
m
(
k
)
=c(k)/ d(k);
¿
function
[
x , k , ier
]
=Jacobi
(
A , b , x 0, tol , kmax
)
x=[ ] ;
k=[ ];
n=length
(
b
)
;
D=diag
(
diag
(
A
)
)
;tril
(
A
)
per GSmatrice
rho=max
(
|
(
eig
(
eye
(
n
)
−inv
(
D
)
∗A
)
)
|
)
;
if rho>1
ier=2;
return
end
C=A−D ;
for k =1 : kmax
¿;
if norm
(
x−x0,inf
)
≤ tol∗norm(x , inf )
ier=0;
break
end
x0=x;
end
ier=1;
¿
function
[
x , k , ier
]
=gaussseidel
(
A , b , x , tol , kmax
)
n=length
(
b
)
;
ier=1;
for k=1 : kmax
y=x
(
1
)
;
x
(
1
)
=
(
b
(
1
)
−A
(
1,2 : n
)
∗x
(
2 : n
)
)
/A(1,1);
xmax=
|
(
x
(
1
)
)
|
;
emax=
|
(
x
(
1
)
−y
)
|
;
for i=2 : n
y=x
(
i
)
;
x
(
i
)
=
(
b
(
i
)
−A
(
i , 1: i−1
)
∗x
(
1 : i−1
)
−A(i ,i +1 : n)∗x
(
i+1 : n
)
)
/A
(
i ,i
)
;
if
|
(
x
(
i
)
)
|
>xmax
xmax=
|
(
x
(
i
)
)
|
;
end
if
|
(
x
(
i
)
−y
)
|
>emax
emax=
|
(
x
(
i
)
−y
)
|
;
end
end
if emax<tol∗xmax
ier=0;
return
end
end
function
[
x , k , ier
]
=gauss¿
(
A , b , x , tol , kmax
)
n=length
(
b
)
;
d=diag
(
A
)
;
[
c , r , a
]
=find
(
A'−diag
(
diag
(
A
)
)
)
;
¿
