Вычислительные методы линейной алгебры (стр. 4 из 5)

kr(n)k → 0 n → ∞

S S

n → ∞ |µk| < 1 ,

.

kr(n)k = kG−1JnG r(0)k 6 kG−1k kJnk kGk kr(0)k → 0 n → ∞.

S

ε

ⁿ → ∞

B = E

S = E − τA

S max|µ_k| τ max|µ_k| k k

τ A = A∗ > 0 A 0 < γ₁ 6 λ_k 6 γ₂ k =

λ_k S

µ_k = 1 − τλ_k

0 < τ < 2/γ₂ |µ_k| = |1−τλ_k| < 1

0 < τ < 2/γ₂ τ = τ∗

|µ∗| = 0<τ<min2/γ2 1max6k6m |1 − τλk|

τ

γ₁ < λ < γ₂ g_λ(τ) = 1−τλ

τ = τ∗ |g_λ(τ∗)| 6 |g_λ(τ)| γ₁ < λ < γ₂ 0 < τ <

2/γ₂ 0 < τ < 1/γ₂

|g_γ2(τ)| 6 |g_γ1(τ)| τ > 1/γ₁ |g_γ1(τ)| 6 |g_γ2(τ)|

1/γ₂ 6 τ 6 1/γ₁ τ₀

|g_γ2(τ₀)| = |g_γ1(τ₀)|, τ₀

cond(A)

1

kSk → 1 ζ → ∞

a_ii =6 0 i = 1,...,m

(n+1)

B = diag(a₁₁,...,a_mm)

= b ⇒ x(n+1) = (E − B−1A)x(n) + B−1b, A

,

.

x(0)

n := 0

x(1)

Ax = b ε

n

N

n > N

A Ax = b aii =6 0

(n + 1)

i

	+ ...		= b1
	+ ...	+ a2mxm(n)	= b2

...................................................

.

m = 2 (x₁,x₂)

,

I

,

II

x(0)

n := 0

i 1 m

n := n + 1

Ax = b

x∗

A = A∗ > 0

.

Φ(x) = (Ax − b,Ax − b) x ∈ R^m

x∗

F(x) = F(x₁,x₂,...,x_m).

F(x)

x₁

ϕ1(x1) = F(x1,x2(n),...,xm(n)),

_x(₁n+1)

.

x₂

.

(n + 1)

A = A∗ > 0

C = 0

a₁

A₁(x(1)₁ ,x(1)₂ )

C

Ax = b

Ax = b A = A^∗ > 0

k

k k n + 1

_x(_kn+1)

.

.

.

X^k−¹ aikx(in+1) + akkxk(n+1) + X^m aikxni = bk.

i=1 i=k+1

A = A∗ > 0

F(x) x

grad

x(n+1)

x(n+1) = x(n) − αn gradF(x(n)), α_n

x(n+1)

gradF(xⁿ) α_n := α_n/2

x(n+1)

α_n

α_n N

x(n+1)

ε ε

“

,

“

,

α_n |ϕ(α_n)|

ϕ(αn) = F(x(n+1)) = F(x(n) − αn gradF(x(n))).

α_n A = A∗ > 0

grad

.

α_n

Ax = b A = A^∗ > 0

A0 = (x01,x02)

gradF(x⁰) A0A1

A0A1

(x11,x12) A₁

A0A1

A = A∗ > 0