Вычислительные методы линейной алгебры (стр. 1 из 5)

A x = b,

A m {Ab}

{Ab} A m

{Ab} A

m = 2

a11x1 + a12x2 = b1 a21x1 + a22x2 = b2

5x₁ + 7x₂ = 12,

7x₁ + 10x₂ = 17,

x₁ = 1 x₂ = 1 F

t = 2 β = 10 t

F β F

x₁ = 2.4 x₂ = 0 12 16.8

0 0.2 1.4 −1

F x₁ = 2.4 x₂ = 0

F

x ∈ Rm A m × m

,

k^Ak

kAk > 0 A 6= 0 kAk = 0 ⇔ A = 0

m × m

k^Ak

kAkα

kxkα kAkβ kxkα = kxkβ

E

E

Ax = b

∆A

b

A A + ∆A

x∗

.

, .

(A + ∆A)−¹ − A−¹ = A−¹ A (A + ∆A)−¹ − A−¹ (A + ∆A) (A + ∆A)−¹ = = A−¹ (A − (A + ∆A)) (A + ∆A)−¹ = −A−¹ ∆A (A + ∆A)−¹.

δ(x) 6 cond(A)k∆Ak/kAk δ(x) 6 cond(

,

cond(A) = kA−¹k kAk

k∆Ak → 0

cond(A) = kA−¹k kAk

t t

O(2−^t)

O(2t/2) O(2−t/2)

cond(A) = kA−¹k kAk

cond(A) ≥ 1 A A−¹ = E ⇒ 1 = kEk = kA A−¹k > kAk kA−¹k = cond(A) cond(c A) = cond(A) c cond(A B) 6 cond(A) cond(B) cond(A−¹) = cond(A)

max d_ii

cond(

D D = diag(d_ii)

16i6m

cond(A) = kAk2 kA−¹k2

cond(A)

A = A∗ > 0

i = 1,...,m

Rm

,

.

b

.

ε_i

λ_l

A−1

A−1

ε “

δ

A x = b,

x

aij

a_ij = 0 i > j (i < j)

U

UT U−1

U^TU = UU^T = E

|det(U)| = 1 1 = det(E) = det(UU^T) =

det(U) det(U^T) = det²(U)

1

P_ij

i j

i j P₂₄ 5 × 5

0 0 0 1 0

 0 A	0		A
i	j	A

24  1 0 0 0 0 

P =0 0 1 0 0

0 1 0 0 0

P_ij

Q_ij(ϕ)

i j

Q₂₄(ϕ) 5 × 5

₂₄1 0 0 0 0 

 0 cosϕ 0 −sinϕ 0

Q (ϕ) =0 0 1 0 0

0 sinϕ 0 cosϕ 0

0 0 0 0 1

Q_ij

P

m

v₁ > 0,

e = (1,0,...,0)^T

v₁ < 0.

,

u = v−σkvke P

.

u₁ u

P

y = αu + βs

Aij a_ij = 0 i > j + 1(i < j − 1)

“

“

,

α = 1.2.3