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(1)

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......

an1 an2 ... ann

x =

x1

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b =

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b2...

bn

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a(k)kk

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(k)kk , k = n − 1, ..., 2, 1

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A = L−11 (−l1)L

−12 (−l2) · · ·L−1

n−1(−ln−1)A(n) = LU (11)

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=

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ln1 ln2 . . . 1

I

U = A(n) =

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(1)1n

a(2)22 ... a

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u22 ... u2n

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unn

I á=��

u1j = a1j , j = 1, 2, ..., n;

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likukj , j = i , i + 1, ..., n

¦U�i1§

uij = aij −i−1∑k=1

likukj , j = i , i + 1, ..., n (14)

I ¦L�i�§Kk

lij =1

ujj(aij −

j−1∑k=1

likukj), i = i , j + 1, ..., n (15)

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. . .

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an bn

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l3. . .. . . 1

ln 1

u1 c1

u2 c2

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un−1 cn−1

un

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I O�úª

u1 = b1, li =ai

ui−1, ui = bi − lici−1, i = 2, 3, ..., n

(17)

I ¦)Ly = fÚUx = y

y1 = f1, yi = fi − liyi−1, i = 2, 3, ..., n (18)

xn =yn

un, xi =

yi − cixi+1

ui, i = n − 1, ..., 2, 1 (19)

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lnn

I

ljj = (ajj −j−1∑k=1

l2jk)1/2, j = 1, 2, ..., n (20)

lij = (aij −j−1∑k=1

lik ljk)/ljj , i = j + 1, ..., n (21)

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I n«~��þ�ê

(1)1-�êµ||x ||1 =∑n

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i )1/2

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i )1/p

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I ½Â3.3µ�x ∈ Rn§A ∈ Rn×n§�½,«�þ�

ê|| · ||ν§�±½Â

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I ½n3.7

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n∑j=1

|aij |

||A||1 = max1≤j≤n

n∑i=1

|aij |

||A||2 =√

λmax(ATA)

I Frobenius�ê

||A||F = (n∑

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I ½n3.9 �B ∈ Rn×n�||B|| < 1§KI − B�ÛÉ§�

||(I − B)−1|| ≤ 1

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cond(A) ≥ 1 cond(A) = cond(A−1)

cond(αA) = cond(A)

cond(A)2 =λ1

λn

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