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Ax=f.
A=LU
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L U
(11)
L U, A=LU
LU
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MathLab L U
[L,U,P]=lu(A),
L U .
(1) :
(12)
, i=1,2,...,n; , aii¹0.
(12) :
(13)
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(12) i=1
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(14) :
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. n´n , n2 . . QR - A Ax = b
,
U = E − 2 wwT.
E ; w n - ,
(w, w) = 1, wwT :
.
, U , U = UT. , U , UT = U −1, :
UUT = (E − 2 wwT)(E − 2 wwT) T = E − 2 wwT − 2 wwT + 4 wwT wwT = E,
wTw = (w, w) = 1.
U , U 2 = UUT = E, U , , λ = −1 w:
Uw = λ w = Ew − 2 wwT w = w − 2 w = − w = (−1) w → λ = −1.
λ = 1 , w. v w, (v, w) = 0,
Uv = Ev − 2 wwT v = v − 2 w (w, v) = v.
U y, : y = z + v. z w y w:
z = dw, d = (y, w), v : (v, w) = 0, v = y −(y, w) w.
Uy = U (z + v) = E (z + v) − 2 wwT (z + v) = z + v − 2 wwT z − 2 wwTv =
= z + v − 2 wwTdw = z + v − 2 z = − z + v.
, Uy y , w. , w , y ≠ 0 Uy e.
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Uy =α e Uy = −α e, α = (y, y), (19)
(U ). , , (y −αe) (y +αe).
,
, (20)
, , . y e , ( ).
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A n.
. y A:
.
(i= 2,3,,n), , A (1) = A; U = E 1 . A U1=En−2w1w1T, w1 , U1y1 e1= [1, 0, 0,, 0] T, (19), (20). En n, e 1, y1 n . A (1) , , .
A(1)
.
A (1)
,
Sn-1=En-1-2wn-1wn-1T (n −1) - , (19), (20). , A(2) =U2 A(1).
A . k - A ( k ), , k - , : i>j, j=1,2,,k.
(k +1) -
.
( i =(k+ 2), (k +3),, n), A (k +1) = A (k); Uk+1 = En .
Sk+1=En-k-2wk+1wk+1T
( wk+1 Sk+1 (n − k)) yk+1 , ek+1= [1, 0, 0,, 0] T ( (n − k)), A (k) A (k +1):
A(k+1) = Uk+1A(k),
, (n −1) -
,
. U , A (n −1)= UA, A = QR, Q = UT , R = A (n −1) .
Ax = b .
,
UAx = Ub Þ Rx = Ub, A (n −1) x = Ub.
A(n−1) , xi i = n, (n −1),,1 , , .
, A (n −1) x = Ub , .
-
, . QR - A n (4 / 3) n 3 .
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LU-. , .ID_1
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(22)
(2)
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, i=j
(24)
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(24) (25) S.
.
A=STS m .