140:systèmes linéaires, systèmes échelonnés

x Ax =b A r
A x
A b ∈Im(A)
Di(λ)
Tij (λ) = Id+λ ij
GLn(k)
Ax =b A
xi=det(A1,...ai−1, b, . . .)
n!
r
r
Ax =b A
A r =rg(A)
r C
C bi
akj bkk>r
n−r

C
P det(C) = QkP(wk)w
det(C)≡c1+. . . +cn[n]
Cx =b c∗x=b c C
Fn(c∗x) = Fn(c)Fn(x) = Fn(b)
x=F−1
n(Fn(b)/Fn(c))
O(nlog(n)) O(n3)
Ax =b A
Ax =b A
nm
A
A
a11
gi1=ai1/a11
Aij , i >2, j >2
E0A0=L0
A
E A L =EA
Ax =b A LAx =Ex =Lb =b0E
n−r
r r
E
EX =b0E0x0=b00 x0r
O(n3)

b
A L
U P A =LU p
A=LtL
L L
lik = (aik −Pk−1
j=1 lij lkj )/(lkk)lkk =qakk −Pl2
kj
A
xk
xkA−1b
xk+1 =Bxk+c c = (I−B)A−1b
I−B
Bk
0B1
xk+1 =xk−θ(Axk−b)ρ(I−θA)<1
A θ
θ2/(λ++λ−)
(Ax, x)>α||x||2
0< θ < 2α/||A||2
(Ax, x)−2(b, x)
A M −N xk+1 =
M−1Nxk+M−1b ρ(Mn−1)<1
A
M∗+N A
A=D−E−F D
−E−F
M=D N =E+F D

M=D−E N =F
M= 1/w ∗(D−wE)N= (1 −w)/wD +F w < 1
w > 1
0< w < 2
Id
y0=A(t)y A
yn+1 =yn+hA(tn)yn+1 (I−A)yn+1 =yn
y0=y(1 + ah)n
(1 −ah)−n
yn+1 =yn+hf(tn+1, yn+1