kr(n)k → 0 n → ∞
S S
kr(n)k = kG−1JnG r(0)k 6 kG−1k kJnk kGk kr(0)k → 0 n → ∞.
S
n → ∞
B = E
S = E − τA
S max|µk| τ max|µk| k k
τ A = A∗ > 0 A 0 < γ1 6 λk 6 γ2 k =
µk = 1 − τλk
0 < τ < 2/γ2 |µk| = |1−τλk| < 1
0 < τ < 2/γ2 τ = τ∗
|µ∗| = 0<τ<min2/γ2 1max6k6m |1 − τλk|
τ
γ1 < λ < γ2 gλ(τ) = 1−τλ
τ = τ∗ |gλ(τ∗)| 6 |gλ(τ)| γ1 < λ < γ2 0 < τ <
2/γ2 0 < τ < 1/γ2
|gγ2(τ)| 6 |gγ1(τ)| τ > 1/γ1 |gγ1(τ)| 6 |gγ2(τ)|
1/γ2 6 τ 6 1/γ1 τ0
1
kSk → 1 ζ → ∞
(n+1)
B = diag(a11,...,amm)
x(0)
n := 0
x(1)
Ax = b ε
n
N
n > N
A Ax = b aii =6 0
i
| + ... | | = b1 |
| + ... | + a2mxm(n) | = b2 |
...................................................
m = 2 (x1,x2)
I
II
x(0)
n := 0
i 1 m
n := n + 1
x∗
A = A∗ > 0
Φ(x) = (Ax − b,Ax − b) x ∈ Rm
F(x) = F(x1,x2,...,xm).
F(x)
x1
ϕ1(x1) = F(x1,x2(n),...,xm(n)),
x(1n+1)
x2
(n + 1)
C = 0
A1(x(1)1 ,x(1)2 )
C
Ax = b
Ax = b A = A∗ > 0
k
k k n + 1
x(kn+1)
Xk−1 aikx(in+1) + akkxk(n+1) + Xm aikxni = bk.
i=1 i=k+1
A = A∗ > 0
F(x) x
grad
x(n+1) = x(n) − αn gradF(x(n)), αn
gradF(xn) αn := αn/2
αn
αn N
x(n+1)
ε ε
“
“
αn |ϕ(αn)|
ϕ(αn) = F(x(n+1)) = F(x(n) − αn gradF(x(n))).
αn A = A∗ > 0
grad
αn
Ax = b A = A∗ > 0
A0 = (x01,x02)
gradF(x0) A0A1
A0A1
(x11,x12) A1
A0A1