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（4-如果對任意x(0)，

k時x(k)x*x(k)(x(k),x(k),,x(k))T

limx(k1)k

Blimx(k)k

x*x*Bx*三種基本的迭代一雅可比(Jacobi)迭代法。a11x1a12x2a1nxna21x1a22x2a2nxnan1x1an2x2annxnA(aij)nn非奇異，且aii0(i12,n)x1

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x(k1)Bx(k) 設

是(1)的解x(k

k若有l(wèi)imx(k kk注意.limx(k)x*limx(k)x*.k kx(k)

為任意向量范

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k

k

k ε(0) x(0)

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. . J

i

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x(0) .

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若||B||

||x(k)x*

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||x(k

x(k1)

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m

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x(k

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x(kp

p

)x(kp

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A 121 2 2x2x321 2 2 D1(LU)

10 0 detIBJ

3 (B)0 detI得

detI(DL)1Udet(DL)1D-LU

3424(2)2 進一步得(BG)2

A(aij

n|aii||aijnjj

(i

等式成立，稱矩陣A為嚴格對角占優(yōu)陣 A為非奇異矩陣，det(A) 2

0

1

1 6 6

04 04 定理4.3（充分性條件）Ax中的A為嚴格對角占優(yōu)陣則Jacobi證（1）Jacobi0 0a

a1na

nn a2n D1(LU)

a22

|a|

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所 1|a (i1,2,,|aii 即||

||

j

(D 的特征值 detI(DL)1Udet(DL)1det[(DL)U] det(DL)1 C(DL)U

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