Ax = b. 7x = 21. x = 21 7 = 3.
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1 3 s st 3 r 3 t r 3 3 t s st t 3t s 3 3 r 3 3 st t t r 3 s t t r r r t st t rr 3t r t 3 3 rt3 3 t 3 3 r st 3 t 3 tr 3 r t3 t 3 s st t Ax = b. s t 3 t 3 3 r r t n r A tr 3 rr t 3 t n ts b 3 t t r r t x t t r n ts Ax = b r s 3 x = A 1 b t3 t A 1 tr 3 A r r t3 3 t r t 3 r t r t 3 rr 3 3t rr r A 1 t3 r s rr 3 t t 3 3 t 3 rt 7x = 21. st r 3t r 3 t t x = 21 7 = 3. r t3 3 tr 3 r t3 t r r t t x = = =
2 s r t r t3 3 r t t r 3 t t t t r t t 3 t t t rr 3 rr t t3 r 3 3t s t1 t3 3 r t rt t3 s st t r 3 3 s st tr t r r t3 3 r rr t3 r 3 rr \ r AX = B s st 3t A r rr r t B r r t r r s st rr s 3 \ t rr X = A 1 B t 3 r 3 t t rr r 3 rr r XA = B s st 3t A r 3 t r t B r r t r r s st rr s 3 t rr X = BA 1 t s 3 t t t 3 r r t3 A rr t 3 r ts 3 r t 3 3 r s r 3 rr r 3 tr 3 rr t s st r t r x t st t 3t r t t r ts 3 Ax = b 3 s st x x 2 = x 3 6 r 3 r 3 s st r E 1 : 10x 1 7x 2 = 7, E 2 : 3x 1 +2x 2 +6x 3 = 4, E 3 : 5x 1 x 2 +5x 3 = 6. r t r rr ts 3 r t3 st 3 t x 1 3 t3 r rt3 E E 1 t E E 1 t E 1 3 x 1 3 t r t r t3 t r t3 r t3 st 3 t x 1 3 t t 3 3 t t 3 rt t t r rr ts t3 t x x 2 = x rt t t = ,
3 x t r t rr rr ts rr 3 r 3 r rr 3 x 2 3 t3 rr t rr 3 x 2 r 3 t 0.1 t r rr 3 t t1 r tr t3 r rr t t3 r t3 t s r 3 rr t r t3 rr r r r r r rr r 3 t t t r 3 r r rr t t r 3 r rr 3 x 2 3 t3 r rt r rr 3 r r rr t 3 ts r t3 rt 3 r x 1 x 2 x 3 x 1 x 2 x 3 = 6.2x 3 = 6.2 = t r 3 x 3 = 1 r rr 3 r x 2 +(5) (1) = 2.5 t x 2 s t 3 x 2 = r 3 t3 t x 2 r t x 3 r 10x 1 +( 7) ( 1) = 7 t x 1 s t 3 x 1 = 0 t3 x = t3 rr 3 3t t t rr 3 3 r = rt L 1 L 2 = L t t L 1 = , L 2 = , L = st A tr 3 rr t r rr rr tr t r t t t r P r t 3 tr 3 t 3 rr r t 3 rt PA = =
4 s r t r t s r tr t r 3 t t 3 tr 3 r 3 t r LU = PA, U = L tr 3 3 t3 r 3 s r t3 r t3 t t 3 U tr t tr 3 t P tr 3 t t3 r 3 s s r t3. P r t 3 t tr tr 3 P r t 3 tr 3 t t t t tr 3 t rr t 3 t tr t t r rr rr t 3 t t3 t 3 r r st P = A tr 3 3 rr t r t3 P r t 3 tr 3 t 3 ts PA A tr 3 r rr r t t3 r 3 s t r t3 ts AP r A tr 3 r 3 t r t t3 t r t 3 t r t tr 3 t rr 3 t rr r t3 P tr 3 r s r P t r r rt t tr 3 rr A r 3 rr A r rr A r t rr A r r P t r r A r 3 t r t 3 r s rt3 t Px = b 3 s st t r s 3 t3 rr r 3 r P 1 = P t t ts P rt x = P t b, tr 3 tr r t 3 r r s r 3 3t tr 3 tr r t 3 r r s r 3t 3 rr t L tr 3 tr rr t U tr 3 tr rr
5 P r t 3 t tr tr 3 r t st t tr 3 tr rr rr 3 3t 3 Ux = b s st tr r u 11 x 1 +u 12 x u 1,n 1 x n 1 +u 1n x n = b 1 u 22 x u 2,n 1 x n 1 +u 2n x n = b u n 1,n 1 x n 1 +u n 1,n x n = b n 1 u nn x n = b n. r 3 s 3 rt3 t3 r 3 r 3 t3 r 3 s rr t r ts t r x n = b n u nn t 3 r 3 3 r x n 1 = b n 1 u n 1,n x n u n 1,n 1. x i = b i u in x n u i,n 1 x n 1... u i,i+1 x i+1 u ii = i = n 1,n 2,...,3,2,1 r t r t rt 3 t3 b i n j=i+1 u ii u ij x j 1 3 r s r 1 1 st Lx = b s st tr r t L r t r x 1 = b 1 l 21 x 1 +x 2 = b l n1 x 1 +l n2 x l n,n 1 x n 1 +x n = b n 3 rt3 rr r 3 r 3 t3 r 3 s rr t r ts t r x 1 = b 1 x 2 = b 2 l 2,1 x 1.
6 s r t t 3 r 3 3 r x i = b i l i1 x 1 l i,2 x 2... l i,i 1 x i 1 = b i l ij x j i 1 j=1 i = 2,3,...,n 1,n r t s rt t r t r t P t t3 r rr U tr 3 r t r t3 t r t r t3 t t3 r 3 r 3 t 3 t t r r r 3 t t 3 r 3 r t r t t rr r r t t t 3 r r r st 1 t t r x 1 x 2 x 3 = tr 3 (2,2) t 3 t t 3 t r t r t3 r s t t3 r r 3 t (0, 1,1) t st 3 r s rr t r rt rr t t s 3 3 r rr ts t x 1 x 2 x 3 = r (2,2) t1 tr 3 st r t3 r tr r t 3 rr rr ts r rr 3 r r rr t.. (5+( ) 6)x 3 = 2.5+( ) r t3 r s = t t r 3 r 3 3 t3 t3 r rr r 3 s st t t r r r r r t3 r 3 r t t r rr r st s r t t t3 3 t3 t t3 r r t3 rr r t ( )x 3 =
7 P t t3 r rr r t r 3 r t t 3 3 rt x 3 = r 3 st t3 r 3 r 3 x 3 = = t3 3 t3 x 3 = 1 3 rr r 3 r s r r t1 rr 3 x 2 3 t t r t r 0.001x 2 +6 ( ) = 6.001, t rr t x 2 = = x t rt 3 10x 1 +( 7) ( 1.5) = 7, x 1 = (0, 1,1) t rt rr ( 0.35, 1.5, ) t rt rr r 3 r t3 rr r r t s rt t r t r t t 3 tr 3 3 s rr t s 3 rr rr ts t t1 t rt3 t t r r 3 r t t t 10 3 r t rr 3 r r 3 t r r t 3t rt 3 r rt t t3 t rr rr ts tr t t rr t r rr 3 r t3 3t s t t1 r t t s r t r t3 s t 3 3 t t t3 rt3 3 t t r 3 s r 3 P t t3 rt3 3 t3 r 3 s i rr rr i rr 3 t r t3 r rt ts a ji j = i,i+1,...,n t s t t r r p rr i rr t p rr rr tr t t max j i { a ji } = a pi, i rr t p rr rr tr t t r r t r r t3 r s b t r ts b p t b i r tr t3 r
8 s r t LU t r 3 3 r r 3 t3 ss rr t t rr r 3 3 t t3 r 3 3 rr r 3 3 n 1 rr ts t rr t s 3 i rr rr ts i rr t3 i rr 3 r t t3 3 r 3 x i r 3 t t1 rr r 3 s 3 tr t3 3 rr ts r t3 r s t r 3 3 tr t r t3 r t3 t r t s 3 r s 3 3 r rr 3 s s t r t3 r 3 3 r 3 r 3 rt 3 P i i = 1,...,n 1 3 r i rr rr ts r t r t 3 tr 3 3 M i i rr rr ts r t3 t r 3 s rt 3 rt t tr 3 tr r t t t rr tr 3 3 tr 3 r t3 3 U 3 r n 1 rr ts r rt t 3 tr 3 tr rr Pr 3 s s 3 t 3 s r t M n 1 P n 1...M 2 P 2 M 1 P 1 A = U. rr s A tr 3 3 s rr t P r t 3 tr 3 t r 3 PA =LU t3 3 t 3 A IR 3 3 tr 3 t r t 3 tr 3 P 1,P 2 t 3 tr 3 M 1,M 2 r t rt t3 M 2 P 2 M 1 P 1 A = U. P i rt 3 P t ip i = 1l s t r P i s tr r 3 P t i = P i r 3 P 2 P 2 = 1l r P 2 P 2 r 3 M 1 t P 1 rt s r 3 M 2 P 2 M 1 P 2 P 2 P 1 A = M 2 M M M 1 P 2 P 1 A = U, M M M 1 rr r t t 3 tr 3 t t M 1 s tr tr t t rr t 3 t M M M 1 = P 2 M 1 P r rt P 2 = 0 0 1, M 1 = m , m M M M 1 = P 2 M 1 P 2 = m m
9 LU t r 3 3 r t rr r 3 s r rr r 3 r ts M n 1 M M M n 2... M M M 1 P n 1...P 1 A = U, M M M k = P n 1...P k+1 M k P k+1...p n 1, k = 1,...,n 2. r r 3 t r 3t 3 L 1 L 2...L n 1 U = P n 1...P 2 P 1 A. 1 L k = M M M k t t r t3 M k t r 3 r t3 3 t 3 t r r t 3 t 3 r t3 r 3t 3 r 3 L = L 1 L 2...L n 1 P = P n 1...P 2 P 1, r LU = PA. r 3 L 3 r t r t3 3t r t3 t t P r t 3 tr 3 rr tr 3t r t3 t rr t r r A = r 3 t t tr 3 r 3 P 1 = P 2 = , M 1 =, M 2 = M M M 1 = P 2 M 1 P 2 = r r 3 L tr 3 L 1 = , L 2 = , , L =
10 s r t LU = PA r 3 r A r LU t r 3 3 s s 3 tr rr r t3 s LU t r t3 ss rr t 3 tr 3 r r 3 t t r 3 3 rr 3 s st r r t r Ax = b PAx = Pb t s st tr rr t rt3 { Ly = Pb Ux = y P t t3 3t r rr tr 3 t3 t r 3 rr 3 t t3 3 A IR n n tr 3 hertsiki diagonal menperatzailea s t t3 n a ii > a ij, i = 1,...,n. j=1,j i r t r r t3 t r A t r t3 A tr 3 LU t r 3 3 t t l ij 1 A t r t3 rr t LU = PA t r r P = 1l 3 t t r t t3 t3 t1 t r t t3 rr t r t t s 3 3 t3 r s t rt s r 3t r t t3 r r t d t3 t t 3 s(d) r r s 3 3 t3 d t r t r d t t t t1 s(d) t t1 t r r t t d t t r r t3 t s 3 d r t t1 s(d) t t r r t t d t t r r t1 rt t3 t
11 tr 3 r r t s d 1 t d 2 rt t 3 t t3 r Pr r t3 t3 3 3 r t 1 d 1 d 2 t1 s(d 1 ) s(d 2 ). d 1 d 2 r 3 r r t t3 r t t t t t s t r t3 rr r r rr r t t3 r r s t rr 3 t s rr t3 r (x 1) 4 = 0, 3 r rr r 3 3 t t1 t s t t r 10 8 r t3 r s r t3 r 3 (x 1) 4 = 10 8, rr 3 t r 3 3 r s t s(d 1 ) s(d 2 ) d 1 d 2 = ( ) = 10 2 = s 3 t t t1 t 10 8 s 3 r t t s rt r r 10 6 r t t t r t r s 3 r 3 r r t1 rt t3 t rt r t t rr 3 3 r s r r t t r tr 3 r tr 3 r t t r 3 t 3 3 r 3t t t r r 3 r r t t t t3 t A > 0, A 0 ca = c A c s r 3t t r A+B A + B tr 3 r t r r rr rr r t t r tr t s 3 t AB A B
12 s r t rr 3 t r r r rt r tr 3 r t t t A tr 3 r 3 r Ax t r r r t3 x t r r r r 3 3 t 3 A tr 3 t t t r r t A r tr 3 r t3 A = max x 0 Ax x. x r t r 3 3 t A r st 3 t A = max u =1 Au. A r r t r 3 x t r 3 t r Ax r 3 t r x r t 3 ts A Ax x. r t r t t t r r tr 3 r ❼ A 1 = max A :,j 1 j 3 t 3t r t 1 ❼ A 2 = σ 1 (A) s r ❼ A IR n n s tr A 2 = max 1 i n λ i λ i 1 i n A r t r ❼ A = max i A i,: 1 rr 3t r t 1 3 b t r r t t m tr 3 r t t r rr r s t A t x 3t t r t t3 Ax b A m x b. t r r t t r r tr 3 r t r t r rr s t r r t 3 r tr 3 r rr t3 ts t r s r r t3 A IR m n tr 3 t r m A F = 1/2 n a 2 ij i=1 j=1,
13 st t t3 3 3 r t t r t 3t t3 A 2 F = 3t r (A t A). r s r t r rr t r r rr r ts A t x 3t t r Ax 2 A F x 2. t r tr 3 t r r s r r r s r 3 rr 3 t3 t t3 t r t 3 3 t r r t r s t t r t c b t c g st t t r r t r rr x 3t t r t t3 t c b x x c g x. t3 t3 t 3t t3 tr 3 r t3 t r tr 3 r r t t t3 3 A IR m n tr 3 t r r t t t t3 r ❼ A 2 A F n A 2 ❼ 1 n A A 2 m A ❼ 1 m A 1 A 2 n A 1 Pr t t r r r t t t r s r r r 3 tr 3 r t 3 t3 st r t r 3 r t t r r 3 r 3 tr 3 t r st t3 r t t r r t3 3 r t r r t r X p r t3 r t3 r t X t r t tr 3 t 3 st t t A tr 3 3 s r t ts A 1 1 st t3 t rr s st Ax = b,
14 s r t r s 3 3 t3 t rr x = A 1 b r t3 r s b 3 t t b + δb 3 t r t3 ts b rt r t3 A r 3 3 x+δx b t t r r s 3 3 t3 A(x+δx b ) = b+δb. t t t r tr 3 t rr δ 3 t r ts r t t1 rt r 3 t s s t t r δb b r t t st δx b r b 3 3 s x r t b r t r Ax = b 3 r 3 Aδx b = δb t3 t r 3 δx b = A 1 δb. 3 r r r 3 δx b r r t rt 3 r s r t3 3 t δx b A 1 δb. P rt r 3 r t r t3 rt Ax = b r t3 t t t r 3 t t3 b A x 1 x A 1 b t r 3 r t 3 t r t 3 s r t3 r t3 rr t3 ts rt3 δx b x A 1 A δb b. rt r t3 r s t t t s 3 3 t3 rt r 3 r t r 1 s s t3 t 3 t r s 3 A b t δb r 3 t3 t r rr t t3 r 3 r t3 rr A tr 3 t 1 t ts rt r t t b ts r (A+δA)(x+δx A ) = b, δx A r A 3 3 s t x r t A r t r δa t1 A+δA 3 s rr r t3 r r t 3 Ax+A(δx A )+δa(x+δx A ) = b A(δx A ) = δa(x+δx A ) t rt A 3 s rr 3 3 r r 3t t3 δx A = A 1 δa(x+δx A ).
15 st t t3 3 3 r t r r 3 t r t3 t rt δx A A 1 δa(x+δx A ) A 1 δa x+δx A δx A x+δx A A 1 δa, t 3 t r t r r rt3 s 3 3 t3 δx A x+δx A A 1 A δa A, t r t3 rt t3 δa r t b r r 3 t3 t r s 3 3 A A 1 t t t rt3 t r t t rr r st s st t t t t r t 1 s s 3 3 t3 r 3 A tr 3 3 s r t t3 3 3 t3 rr t3 Ax = b s s t r 3 r κ(a) = A 1 A. 3 r r t r t t t tr 3 r r 3 r 1l = A 1 A t A 1 A A 1 A r κ(a) 1 r 3 tr 3 t3 t t t3 3 3 t t r r t tr 3 t1 rt t3 t t t3 3 3 t t s 3 3 t κ(a) t r r t r r r r r r rr r r t s t st s st r t rt r 3 t r s 3 3 t3 r t r t 3 t3 t3 3 3 r r t r 3 rr r t 3 t s r t3 t t rt 3 rt3 κ(a) δx b / x δb / b t κ(a) δx A /( x+δx A ) δa / A 3 s st 0.550x x 2 = x x 2 = st rr t r Ax = b A = [ ] t b = [ ] s s st rr A tr 3 t1 rt t3 t
16 s r t 3 3 b t r rt r t3 [ ] [ ] b = b+δb = + = [ Ax = b s st r s 3 3 t3 x = (1, 1) t A x = b s st r s 3 3 t3 x = (1.7, 1.91) t ts δx b = (0.7, 0.91) t t r r t3 b r t x r rt r 3 r t t3 3 r δb b = t ]. δx b x = r 3 δb rr t r s 3 t r t r 3 rr r st A r t3 3 3 t1 3 ts κ(a) 400 s A r t3 t1 rr s 3 a 21 1 t t 3 r t t t 3 [ ] A+δA = r (A + δa) x = b s st r s 3 3 t3 x = ( ,0.8900) t 3 δx A = x x = 1.89 t t s t s 3 3 t3 r t r t r A r t r t ts κ(a) 1890 s t A r r t3 3 3 t3 t t r r [ ] A 1 =, t t r r 3 A = t A 1 = t3 3 3 κ(a) = r 3 t t3 3 3 s 3 δb r t δa r t rt 3 r t t t rr 3 rr κ(a) r r A tr 3 t t1 rt t3 t t r t r s r t 3 r r r t r t s s r r x t r t r t r Ax t st t r t r t r t1 r A t1 rt t3 t Ax / x x 3t t r s t t r A = diag(10 8,10 8 ) s r A 3 t1 rt t3 t 3 t3 tr 3 t t r [ 10 4 ] [ [ 1 0 ÂA =, x = t x =, 0] 1] 10 4 x = x = 1. r [ 10 4] ÂAx = 0 [ t ÂA x = ],
17 st t t3 3 3 r t 3 r rt t 3 ÂAx = 10 4 t ÂA x = 10 4 t1 ÂA tr 3 t1 rt t3 t 3 r s rr 3 st t x t x t r r t r rt3 t t κ(a) = 10 8 A r t3 t r s rt s s t3 t r 3 t r r t1 rt t3 t t tr 3 t s rr 3 t tr 3 s r t t r t 3 r 3 t r t 3 r tr 3 t t3 3 t 3 t1 rr 3 r 3 n n ts A = diag(10 10 ) tr 3 r t r t 10 10n r t3 r t κ(a) = 1 st tr 3 s r t t1 3 t t r A s tr r t3 3 3 t t1 r 3 t r κ 2 (A) = max 1 i n λ i min 1 i n λ i. r 3 tr 3 s tr t1 rt t3 t t t t1 t t 3 r r r r 3 r A 2 = max 1 i n λ i r r r 3 r tr 3 s tr 3 P r t 3 tr 3 Px r s x r s rr t t r 3 Px = x t t3 x 3t t r t r 3 κ(p) = 1. A tr 3 c 0 s r t 3 r t3 ca = c A t (ca) 1 = 1/c A 1 r t r 3 κ(ca) = (ca) 1 ca = A 1 A = κ(a). D tr 3 r κ(d) = max d ii min d ii. 3 t3 t3 t3 3 3 t3 ❼ κ 2 (A) r r 3 t3 t3 3 s rr r t3 s t3 ❼ κ 1 (A) t r r 3 t3 t3 3 rr t3 r t3 r t t1 r t ❼ κ (A) t r r 3 t3 t3 3 rr t3 r t3 t r r
18 s r t s 2 r t r 3 3 t r 3 A IR n n tr 3 s tr t r A n t rr t λ 1,...,λ n t r rt t v 1,...,v n t t r t r IR n r rr rt r t s t3 t ts v t iv j = 1 i = j t v t iv j = 0 i j t r 3 A IR n n tr 3 s tr t r A t s t x t Ax > 0 x 0 t s t 3t s t r r t3 3 t 3 λ 1,...,λ n t t r 3 v 1,...,v n t t r rt r j t r λ j 0 r v t jav j = v t j(λ j v j ) = λ j v t jv j = λ j 0, rr r st A 3 t s t r 3 A t s t t 3t s t r r λ i > 0 i = 1,...,n r {v 1,...,v n } IR n r rr t s t3 t 3 t t3 3 v IR n t r 3 t r n n n n v = α i v i Av = A α i v i = α i Av i = α i λ i v i i=1 i=1 i=1 i=1 t α j t t r r {v 1,...,v n } rr rt r 3 t t n n n n n v t Av = (α i v i ) t (λ j α j v j ) == λ j α i α j v t iv j = λ j αj 2 > 0. i=1 j=1 r 3 A t s t i=1 j=1 j=1 t r rs r 3 A IR n n tr 3 s tr t λ 1,...,λ n t r min λ n i min 1 i n 1 i n a ii a ij max λ i max 1 i n 1 i n a ii + j=1,j i n j=1,j i a ij r r A tr 3 t s t i = 1,...,n 3t t r t t3 n a ii a ij > 0. j=1,j i
19 s 2 r t r 3 3 r t r t3 rt A rts r t3 rs r t r r s r t3 t t t1 3 r 3 A t s t A s tr r t t1 r r s rt n d = min a ii a ij. 1 i n j=1,j i A tr 3 s tr t t s t 3t s t r t tr 3 r t t r s 2r t r 3 3 A IR n n tr 3 s tr t t s t R IR n n tr 3 tr r r t 1 st t3 s t t A = R t R t t3 r t3 L t t tr 3 tr r r t t D = diag(d 1,...,d n ) tr 3 r t A = LDL t t t3 t s d k s t r 3 R t = Ldiag( d 1,..., d n ) tr 3 rr t tr rr s t r A = R t R t t3 rt s LDL t t r 3 3 r rt s t3 A s tr A = A t t t s t A = R t R t r 3 3 s s 3 3 R tr 3 tr r t t ts a 11 a a 1n r 11 r 11 r r 1n a 12 a 22 a 2n = r 12 r 22 r 22 r 2n a 1n a 2n... a nn r 1n r 2n... r nn r nn r t 3 r ❼ A r a 11 t rr r rr t 3 3 t r 2 11 = a 11 r 11 r 12 = a 12 r 11 r 1n = a 1n ❼ A r a 22 t rr r rr t 3 3 t r r 2 22 = a 22 r 12 r 13 +r 22 r 23 = a 23 r 12 r 1n +r 22 r 2n = a 2n
20 s r t ❼ rr rr t 3 i = 1,...,n 3t t r A r a ii t i rr r 3 t r 2 1i +r 2 2i +...+r 2 ii = a ii r 1i r 1,i+1 +r 2i r 2,i r ii r i,i+1 = a i,i+1 r 1i r 1n +r 2i r 2n +...+r ii r in = a in ❼ r 3 i = 1,...,n 3t t r R r 3 t t3 r ii = r ij = i 1 a ii rki 2, i = 1,...,n, k=1 i 1 a ij r ki r kj k=1 r ii, j = i+1,...,n, 3 3 tr 3 s tr t t s t s 2r t r A = rr r i = 1 r 3 3 r r 11 = a 11 = 6 = r r 3 3 t rt3 rr rr r i = 2 3 r r 22 = r 12 = a 12 r 11 = = r 13 = a 13 r 11 = = a 22 r12 2 = 55 ( ) 2 = r 23 = a 23 r 12 r 13 r 22 = = r rr rr r i = 3 rt r 33 = a 33 r13 2 r23 2 = 979 ( ) 2 ( ) 2 =
21 s 2 r t r 3 3 r t r 3 s 2r t r r t t r 3 3 rr t s t3 t R t R = A r t3 t t3 3t t 3 R t R = A t r 3 3 r rt r Ax = b s st t3 3 LU t r 3 3 r t 3 s st tr r t3 3 R t y = b Rx = y A tr 3 s tr t t s t 3 rr 3 rr tr t 3 LU t t 3 rt3 r 3 3 rr 3 t s 2r t 3 t At1 rt t3 t 3 3 t 3 r r 3 r i = 1,...,n 3t t r t t3 t r 2 1i +r 2 2i +...+r 2 ii = a ii t r 3 R 3t r t t r ji a ii, j = 1,...,i t s r t 3 3 r 3 3 rt s t t3 A tr 3 r s 2r t r t t r 3 A IR n n tr 3 s tr t t s t t A = R t R r s 2r s s 3 r κ 2 (A) max i r 2 ii min i r 2 ii r A = LDL t s 2r s s 3 r t3 L t t tr 3 tr rr t D = diag(d i ) tr 3 3 t t3 κ 2 (A) max i d i min i d i D r 1 r t r 3 t r A tr 3 r t3 r r t t t t 3 r t
22 s r t r t3 A 2 = max Au 2 max Ae i 2 = max a 2 1i +...+a 2 ni maxa ii maxr 2 u 2 =1 i i i i ii 3 s r t3 s r t3 r r 3 B = A 1 r B = (R t R) 1 = R 1 (R t ) 1 = R 1 (R 1 ) t r 3 1 B 2 maxb ii max i i rii 2 = 1 min i r 2 ii r 3 maxr 2 κ 2 (A) = A 2 B 2 i ii minr 2. i ii st A = LDL t = R t R 3 d i = r 2 ii t rr r t3 t3 t t r t LU t s 2r t r 3 3 t 3 3 r 3 r 3 r r t3 rr r s 3 rr ts r t t s 3 3 t3 r r t s st r tr 3 s s ts rt3 t t 3 r r ss t 3 r t3 t ts 3 3 r rt3 t r tr 3 r t r s s str t s t t3 s 3 t3 A tr 3 3 r 3 r r 3 t3 t t r t r tr 3 r t r r t1 t 3 r t t1 t t r r r t1 r t3 t r s t r s rr s st s r t rt3 t 3 s st t 3 r t r 3 t3 s st 4x y +z = 7 4x 8y +z = 21 2x+y +5z = 15.
23 t t r t r t 3 3 r t3 t3 7+y z x = 4 y = 21+4x+z 8 z = 15+2x y 5 t rt r 3 s t r t t r t3 x (k+1) = 7+y(k) z (k) 4 y (k+1) = 21+4x(k) +z (k) 8 z (k+1) = 15+2x(k) y (k). 5 k x (k) y (k) z (k) t r t r 3 r r t3 s r t (x (0),y (0),z (0) ) t = (1,2,2) t s t r 3 rr (2,4,3) t s 3 r t3 x (0) = 1 y (0) = 2 t z (0) = 2 r 3 t 3 s t t rt3 x (1) = = y (1) = z (1) = = = 3.00.
24 s r t Pr 3 s r rr t 3 t rt3 rr r 3t t r 3 (2,4,3) t s 3 r t3 Pr 3 s rr r t r 3 r t r t3 3 Ax = b s st t A tr 3 r s s 3 A = L+D+U, r t r a L = a 31 a a n1 a n2... a n,n 1 0 D = diag(a 11,...,a nn ) 0 a a 1n U = Ax = (L+D+U)x = b Dx = b (L+U)x. a n 2,n a n 1,n rt 3 t3 3 r k rr t r 3 Dx (k+1) = b (L+U)x (k). r 3 tr 3 3 r t3 t s s r r t t1 r t r t r 3 r t 3 x i t3 i = 1,...,n 3t t r 3 r a ii x (k+1) i = i 1 b i j=1 a ij x (k) j n j=i+1 a ij x (k) j 3 i = 1,...,n 3t t r r t r 3 x (k+1) i = i 1 b i j=1 a ij x (k) j n j=i+1 / a ij x (k) j a ii
25 ss t r 3 r t ss t r 3 r t r r t3 3 r t3 s 3 st t rr {x (k) } {y (k) } {z (k) } s r r t r t3 t rr 3 rr 3 r x (k+1) s r s x (k) r t r r 3 rr t3 t y (k+1) t3 x (k) r r 3 x (k+1) r t3 rt 3 3 t3 t z (k+1) y (k) r r 3 y (k+1) r t3 r r t rt 3 r t3 t r 3 x (k+1) = 7+y(k) z (k) 4 y (k+1) = 21+4x(k+1) +z (k) 8 z (k+1) = 15+2x(k+1) y (k+1). 5 t rr ss t r 3 r t r t3 t t r 3 r r 3 rt3 i = 1,...,n 3t t r x (k+1) i = i 1 b i j=1 a ij x (k+1) j n j=i+1 / a ij x (k) j a ii r tr 3 r 3 t Dx (k+1) = b Lx (k+1) Ux (k) (D+L)x (k+1) = b Ux (k) r r s st t3 ss t r 3 ts r y (0) = 2 t z (0) = 2 r 3 t 3 x (1) = = 1.75 t x (1) = 1.75 t z (0) = 2 r 3 t 3 3 r rt3 y (1) = = x (1) = 1.75 t y (1) = 3.75 r 3 t 3 rt3 z (1) = = t3 rr r t s 3 t (x,y,z) t = (2,4,3) t Pr 3 s rr rr t 3 t rt3
26 s r t k x (k) y (k) z (k) t ss t r 3 r r t3 r t3 r t3 3 Ax = b s st rr t t r t r t 3 t t3 rts r t3 t t3 s t t r t r t3 r t3 r 3 rr 3 r t3 r ts t 3 3 rr r t 3 t 3 QR t r 3 3 s r s s r s tr 3 tr s r 3 r t 3 3 r t rr t r rr t r t3 rr r s t 3 tr 3 rt s t r t3 A tr 3 t r 3 r rt3 r 3 A rr t s r tr s r 3 3 tr 3 tr r t r r u 0 t r t Householder en islapena Householder en transformazioa Householder en matrizea t1 r t tr 3 H = 1l ρuu t, ρ = 2, u 2 2 u t r r Householder en bektorea t r t3 H tr 3 s tr H = H t t rt H 1 = H t r 3t t rt H tr 3 u t r r rr
27 QR t r 3 3 r t r s r s Pr t H 3 3 r t3 3 r H r 3 x t r t t t r 3 t3 Hx Hx = (1l ρuu t )x = x ρu(u t x) t = ρu t x Hx = x tu. tr x t r u r t t3 t r x t r r 3 r t3 3 r ( u tu = ρ(u t t ) x u x)u = 2. u 2 u 2 r r st t u t r t t r 3 s 3 rt r rr 3 3 t u r x t y t r r Hx t Hy r st t H tr 3 3 t r r s rt3 u rr r 3 x t r t r t r t x r t Hx r rt r x (t/2)u t u rr 3 s 3 s 3 ts t 3 s 3 r u t r r r rr r rt 3
28 s r t r r st t r 3 r rt t3 u t r x t r t3 t s t t r t3 r Hx r t3 rr Hx r s 3t 3 r r t 3 3 r H rt 3 t r r 3 r r t3 3 r Hx 2 2 = (Hx) t (Hx) = x t H t Hx = x t x = x 2 2 Hx 2 = x 2. r 3 Hx t r r 3 r 3 s rr r ± x 2 r x t r t r Hx r k rr s 3 3 st 3t 3 r rt3 x t r r 3 r r 3 H s r tr 3 r r σ = ± x 2, u = x+σe k, ρ = 2/ u 2 2 = 1/(σu k ), H = 1l ρuu t. e k rr r k rr t r t r t3 rr r r rr t r t3 t rt 3 σ r 3 x k r r r rt3 σ = 3 (x k ) x 2 r t3 rt t t3 u 2 2 = u t u = (x+σe k ) t (x+σe k ) = x t x+σ 2 e t ke k +σx t e k +σe t kx = x 2 2 +σ 2 +2σx k = 2σ 2 +2σx k 3 = 2σ(x k +σ) = 2σu k r 3 u k 3 u t r r k rr s t 3 t r t3 t rt 3 rt3 Hx = x ρ(u t x)u = x 2 xt x+σx t e k u u 2 2 = x 2 σ2 +σx k 2σu k u = x u = σe k
29 QR t r 3 3 r t QR t r 3 3 A tr 3 rr t 3 s rr A IR n n tr 3 3 s rr r rr t r r r n 1 s r tr 3 r t3 t H n 1...H 2 H 1 A = R, R IR n n tr 3 tr rr t Pr 3 s rr ts H 1 tr 3 r t3 A r 3 t a 1 e 1 t rt3 e 1 rr r t r 3 t r r rr r 3 t n r 3 r rt 3 r t rt 3 3 r rt3 r 11 H 1 a 1 = (1l ρ 1 u 1 u t 1)a 1 = σ 1 e 1 = 0 0, r 11 = σ 1 t σ 1 = zeinu(a 11 ) a 1 2 r t3 rr r s st rr t t u 1 = a 1 +σ 1 e 1 rt r r 3 s r t r 3 a 11 +σ 1 a 21 u 1 = a n1 t r rr A tr 3 H 1 t 3 3 r r 11 a (2) a (2) 1n A (2) = H 1 A = 0 a (2) a (2) 2n 0 a (2) n2... a (2) nn A tr 3 r 3t t t r r 3 rt t3 3 t3 ss rr t rr t rr 3 t3 r tr 3 ÃA 2 rr t 3 t 3 r t3 (n 1) (n 1) tr 3 3 s r 3 t3 r rr ts tr 3 t r A =
30 s r t 3 t t 3 a 1 2 = 14 = 3.742, σ 1 = 3 (+1)3.742 = , r 11 = t r t rt 3 u 1 = = H 1 t A (2) t3 t r 3 rr 3 rr t3 t ρ 1 = 1/( ) = t H 1 = 1l ρ 1 u 1 u t 1 = , A (2) = H 1 A = , t 3 r s rr t r r t t rr s r tr s r 3 r r t3 r ÃA 2 r 3 t r r t3 ÃA 2 r rr t 3 t t r r rt3 u 2 r 3 r rt3 s r rr r t3 r rr H 2 r 3 t r r r t 3 t3 s t H 2 r 3 r 3t e 1 3 t r rr r t3 3 ÃA 2 = [ ] r ãa 2 3 t t ãa 2 = 3.139, σ 2 = 3 (+2.127)3.139 = , r 22 = 3.139, r r u 2 = = H 2 t A (3) t3 t r 3 rr 3 rr t3 t ρ 2 = 1/( ) =
31 QR t r 3 3 r t t H 2 = 1l ρ 2 u 2 u t 2 = , A (3) = H 2 A (2) = r 3 rt H 2 H 1 A = R R = A (3) IR 3 3 tr 3 tr rr t A IR n n 3 s rr s r 3 r n 1 rr ts t3 t ÃA i r tr 3 r ãa i 3 t 3 3 r 3 i rr rr ts t3 i = 1,...,n 1 ÃA 1 = A t 3 3 t 3 t 3 r r r H n 1...H 1 A = R, R IR n n tr 3 tr rr t 3 Q t IR n n tr 3 rt r 3 Q t = H n 1...H 1. Q = H 1...H n 1. r r 3 t3 r A r QR t r 3 3 r t3 Q t A = R A = QR. Pr t 3 Q t r s st t 3t 3 r Q t v tr 3 t r r t t rr rt3 t t r r t r 3 H n 1 H 1 s r tr s r 3 t 3 tr 3 r r 3 r s 3 t t r k rr s r tr 3 r 3t u k s r t r r n (k 1) s r r t t ρ k 3 t ρ k 3 u k r 3 r r r t 3 rr t3 r t3 t r 3 3 r t t t t t t A = PLU n 3 /3 n 3 /3 A = QR 2n 3 /3 2n 3 /3 A = R t R n 3 /6 n 3 /6 t tr 3 t r 3 3 st r t t t r r r QR t r 3 3 LU t r 3 3 r r t t3 r t3 t3 r tr 3 t t r 3 t3 r s t QR t r 3 3 r t3 r 3 3 rt s t Q rt 3 κ 2 (R) = κ 2 (A) QR t r rr t3 A r t3 r t r r t s
32 s r t st rr t t r t t 3 A r QR t r t t r AX = b t t rt 3 AX = QRx = b Rx = Q t b r 3 Q t b t3 t r s st 3t rr r rr t 3 s st x = r R = , Q t b = r s rr t r r t3 r s st x = (2, 1,1) t t3 rt3 R t Q t b r rts 3 t3 rr t s st t r t m t t 3 (t i,y i ) f(x,t) t3 t x 1,x 2,...,x n r tr s 3 (1,2) (2,3) t (3,5) r t f(x,t) = x 1 t+x 2 e t t3 t r t3 3 t3 t rt3 ts Ax = b s st t r 1 e A = 2 e 2, x = 3 e 3 f(x,t 1 ) = y 1 f(x,t 2 ) = y 2 f(x,t 3 ) = y 3, [ ] x1 x 2 2 t b = 3. 5 st t r t 3 m > n 3 t3 3 t3 3 t3 t t 3 3 r r t x t r Ax b r t r r rr r t 3 t3 3 r r t t rr r 3 r t3 r min x IR n Ax b 2,
33 rr t s st t r t r t r rr t A IR m n m n b IR m r m = 3 t n = 2 r s 3 t3 t (2,3,5) t t r t (1,2,3) t t (e,e 2,e 3 ) t t r 3 r r t3 r r r r rr r 3t r t r t b t A 3 t 3 r r tr rr s b t r t A r 3 t s rt t K(A) 3 s 3 n ts t r r r rr r t3 s r 3 t 3 a 1,...,a m IR n t r A tr 3 m 3 t r rr t s 3 b t K(A) 3 s 3 t r r Ax K(A) 3 3 r Ax b r rr K(A) 3 s 3 r r 3 x t r a t i(ax b) = 0, i = 1,...,n, A t (Ax b) = 0
34 s r t st Ax rr t A r 3 t s r x t r r rr r 3 x 3 s st s 3 s rr 3 (A t A)x = A t b, r 3 r rr t r rt3 t r 3 t 3 m n > 0 A IR m n b IR m r rr t r r s 3 {x A t (Ax b) = 0} t t3 A r 3 t s r x s 3 rr A t A 3 s rr t x = (A t A) 1 A t b r t3 3 f(x) t3 f(x) = Ax b 2 2 = (Ax b) t (Ax b) = (Ax) t (Ax) (Ax) t b b t (Ax)+b t b = x t A t Ax 2x t A t b+b t b. r min Ax b 2 = minf(x) t s 3 f(x ) = 0 t r f(x ) = 2A t Ax 2A t b = 0, r 3 x s 3 r t3 t r A t (Ax b) = 0. t A t A r t s t 3 f(x) t3 1 t r 3 x 3 t3 t r 3 t r r t r t t rr A r 3 t s r 3 A t A 3 s rr t r x 3 t3 rr rr r t 3 3 rt3 r (A t A)x = A t b, 3 r r t3 3 r tr 3 A t A 3 A t r tr 3 s tr t r t s t t t s t t s A tr 3 r 3 t s r ts A tr 3 3 t t st 3 r t r t r rr A t A s rr 3 rr 3 r rr t3 r t t 3 3 t t t t rr t s 3 t3 A t A t s t A t 3 r s 3 r t t s rr t rr t r t3 3 r t
35 rr t s st t r t r t r t t rr t r r 3 3 r t rt 3 rr ts rt A t b rr ts r t 3 r tr 3 A t A t A t b t r rr ts 3 3 s 2r t r 3 3 A t A = R t R R tr rr t rr ts t3 R t y = A t b rr r 3 r 3 3 r t3 Rx = y t3 r 3 r 3 3 rr ts t3 x t3 Ax = b s st t r t A = 1 0 t b = A tr 3 r 3 t t r s r 3 A t A t A = [ ] [ ] t A 1 2 t b = t (A 4 t A)x = A t b s st r s 3 x = (2, 3) t s 3 rr r t r 1 1 Ax b = [ ] 2 3 r 3 s st rr r t r r r rr min x Ax b 2 = Ax b 2 = 12 = s r r t t3 3 r 3 t s r f(x,t) = x 1 t+x 2 e t t3 r r tr t3 (1,2) (2,3) t (3,5) t t3 3 t A t 3 t r 3 3 r x s 3 r t 3 t r r t3 3 t rr t r 3t r A t A tr 3 r t3 κ 2 (A t A) = (κ 2 (A)) 2 t t3 t 3t t r t s 3 κ 2 (A) = 10 3 t3 3 s t1 rr κ 2 (A t A) = 10 6 s 3 t1 rr t 3 3 rt s r r 3 rr s r t3 r 3 t rr t = 2 2 2
36 s r t A tr 3 r rr 3 A r QR t r 3 3 r 3 Q IR m m rt R IR m n tr rr QR t r 3 3 r 3 A tr 3 rr t s 3 s r s r 3 t r s s 3 rr r A r 3 t rt r rr t r t r r st r t r 3 3 r 3t t r 3 t 3 m n > 0 b IR m t r t A IR m n 3 t t tr 3 r r A = QR s s 3 t 1 st t3 Q IR m n tr 3 rt t R IR m n tr rr t R u ts R r n rr tr 3 tr rr t 3 s rr r rr t r r s 3 rr x = R 1 u (Q t b) u, (Q t b) t u = ((Q t b) 1,...,(Q t b) n ), t rr r 3 r r rr t min x Ax b 2 2 = m i=n+1 (Q t b) 2 i. r t3 s r tr s r 3 t t r 3t t3 A r QR s s 3 r 1 st t3 t A r 3 t t 3 t t R u r 3 s r t t Q 3 t s rr r Q t rt 3 t r t 3 r t3 3 t3 r r rr Q t x 2 = x 2 t r 3 r Ax b 2 = QRx b 2 = Q t (QRx b) 2 = Rx Q t b 2, r t rr t3 3 r min b x IR n Rx Qt 2. r (Q t b) t l = ((Q t b) n+1,...,(q t b) m ) Rx Q t b 2 2 = R u x (Q t b) u (Q t b) l 2 2 t r 3 t3 x = Ru 1 (Q t b) u t 3 r t t r (Q t b) l 2 2 = m i=n+1 (Q t b) 2 i
37 rr t s st t r t r t r t t rr t r r 3 QR t r 3 3 r t rt 3 rr ts rt A t b rr ts 3 3 A r QR t r 3 3 s r tr s r 3 r 3 A = QR Q rt t t R tr rr rr ts r t b = Q t b t b u ts b r n s rr ts t3 R u x = b u t3 r 3 r 3 3 R u R r n rr s t t 3 tr 3 rr t tr rr t rr ts t3 x A IR m n tr 3 3 t t m > n t b IR m r t H n...h 1 A = R H n...h 1 b t r r Q t [A b] 3t t r t Q t = H n...h 1 3 r r t Hx t3 3 H r t r t QR t r t rt 3 t3 rr r Ax = b s st t r t 3 r A = 1 0 t b = t 3 a 1 = [1 1 0] t t a 2 = [1 0 1] t t s rr t r r 3 σ 1 = zeinu(a 11 ) a 1 2 = + 2 = 1.414, u 1 = t ρ 1 = 1 σ 1 [u 1 ] 1 = r 3 r 3 r 3 rt3 A tr 3 r 3 t t r r = = H 1 a 1 = a 1 ρ 1 (u t 1a 1 )u 1 = [ ] = , 0
38 s r t s r 3 s t r 3 ts r t r 3 t r r H 1 a 2 t H 1 a 2 = a 2 ρ 1 (u t 1a 2 )u 1 = [ ] 0 1 = r t A (2) = A (1) = A t s 3 3 H 1 t3 rr r t H 1 = 1l ρ 1 u 1 u t 1 = , r t s 3t t H 1 A = A (2) t t3 r Q t b t3 st b (2) = H 1 b b (1) = b t r t H 1 b = b ρ 1 (u t 1b)u 1 = [ ] 0 1 = r A (2) tr 3 r 3 t t rr 3 t 3 3 tr 3 r t3 [ ] ãa 2 =ÃA 2 =. 1 rr ãa 2 t r r 3 3 r rt3 σ 2 = zeinu( ) ãa 2 2 = 1.225, t r 3 A (2) r rr 3 t r r r u 2 s r t r 0 0 u 2 = = 1.932, ρ 2 = σ 2 [u 2 ] 2 = 1 ( 1.225)( 1.932) = , rt u 2 r 3 r A (2) r 3 t t rr 3 t3
39 rr t s st t r t r t r 3 r 3 r 3 rt3 H 2 a (2) 2 = a (2) 2 ρ 2 (u t 2a (2) 2 )u = [ ] = , 0 s r 3 ts r t r 3 t r r t A (3) = s 3 3 H 2 t3 rr r t H 2 = 1l ρ 2 u 2 u t 2 = , r t s 3t t H 2 A (2) = A (3) t t3 r Q t b r t3 b (3) = H 2 b (2) ts b (3) = Q t b = H 2 H 1 b r t H 2 b (2) = b (2) ρ 2 (u t 2b (2) )u = [ ] = R = A (3) t Q t b = b (3) r 3 R u x = (Q t b) u s st t3 [ ][ ] [ ] x R u x = = = (Q t b) u. st rr s 3 rr x 1 = 2 t x 2 = 3 r 3 st (Q t b) l = r t r r r rr s t s r t 3 s t r t t3 r rt t t3 x 2
40 QR t r 3 3 r r t t s r t rr r t r t3 QR t r st t3 3 K(A) A tr 3 3 t 3 t3 ts 3 t r s rt3 t 3 s 3 t r rr s 3 r r 3 Q r 3 t t K(A) t K(A) 3 s 3 rt t r A = QR s s 3 A IR m n r 3 t t QR t r 3 3 t A = [a 1,...,a n ] Q = [Q u,q l ] t Q u = [q 1,...,q n ] Q l = [q n+1,...,q m ] 3 t rt t r r t t3 K(A) = K(Q u ) K(A) = K(Q l ) t A = Q u R u R u IR n n R tr 3 r n rr t 3 tr 3 tr rr t r t3 r 3 Q = [Q u,q l ] rt t s rt t r 3 t t3 A = Q u R u +Q l 0 = Q u R u. k = 1,...,n 3t t r t t3 k a k = r ik q i K(Q u ), i=1 (r 1k,...,r kk,0...,0) t t r R u r k rr 3 t 3 r 3 K(A) K(Q u ) 3 A r t Q r r r 3 K(A) = K(Q u ) st A = QR Q t A = R [ Q t u Q t l ] A = [ ] Ru 0 Q t la = 0. r 3 K(Q l ) K(A) r K(A) r ts t Q l r r r m n r 3 K(Q l ) = K(A)
41 rr t s st t r t r t t r A IR m n tr 3 3 t t QR t r 3 3 r A = Q u R u rr Q u IR m n tr 3 3 t rt t t t R u tr rr t s t t r R u tr 3 A t A tr 3 r s 2r t r r t3 A t A = (Q u R u ) t (Q u R u ) = R t uq t uq u R u = R t ur u 3 r R u tr 3 A t A tr 3 r s 2r t r t r r rr t r t t Q u = AR 1 u 3 Q u r rr rr H 1 t H 2 t t r t s rr t r r Q = H 1 H 2 = r 3 3 r = Q u = t Q l = t t rr t r Q u R u = A t Q t la = 0 t t3 r t r 3 rr QR t r 3 3 A r rr QR t r t rr t K(A) 3 s 3 r Pr r A r rts r t t t QR t r 3 3 t 3 ts AP = QR P r t 3 tr 3 t t r r A r 3 t t tr r t tr 3 rr rr tr r t t t t3 t s r 3 s rr t 3 rt t 3 t st 3 t 3 r 3 t r t st st 3 t rt t t 3 r 3 3 t t s rr t r rr t 3 t r t 3 3 A tr 3 r 3 t t t r QR t r A = [a 1 a 2 a 3 ] = [q 1 q 2 q 3 ] 0 0 1, 0 0 1
42 s r t r hein(a) = 2 K(A) 3 K([q 1 q 2 ]) 3t K([q 1 q 3 ]) K([q 2 q 3 ]) r r 3 s r QR t r 3 3 rr 3 3 K(A) r rr rt r t rt3 P t t3 r str t r 3 t t s rt3 3 t k rr rr ts (H k 1...H 1 )A(P 1...P k 1 ) = R (k 1) = [ R (k 1) 11 R (k 1) ] 12 0 R (k 1), 22 R (k 1) 11 3 s rr t tr rr t R (k 1) 22 r 3 t r t t R (k 1) 22 = [ z (k 1) ],...,z (k 1) t p 3 3 k p n t t3 t1 z (k 1) p k 2 = max { z (k 1) } 2,..., z (k 1) n 2. k rt hein(a) = k 1 r 3 r 3 t t r 3 3 t st 3 P k r t 3 p t k 3 t tr t3 t t r t3 H k s r tr s r 3 R (k) = H k R (k 1) P k R (k) = R (k) (k +1 : m,k) = 0 ts (k,k) r 3 3 r r A IR m n r r r 3 t r t 3 r t r 3 r rr ts r r 3 (H r...h 1 )A(P 1...P r ) = R = n [ ] R11 R R 11 3 s rr t tr rr t R 11 3 t r A r t r Q t = H r 1...H 1 t P = P 1...P r t3 t r t rr r 3 Q t AP = R = [ ] R11 R r n r r m r AP = Q R r 3 rr t r 3t P t Q t rt r t rt 3 Ax b 2 2 = Q t (Ax b) 2 2 = (Q t AP)(P t x) Q t b t 3 P t x = [ ] y r z n r t Q t b = [ ] c r d m r,
43 rr t s st t r t r t r t 3 3 rt3 [ Ax b 2 2 = R11 R ][ ] y z [ ] c 2 = d 2 = R 11 y (c R 12 z) d 2 2. [ ] R11 y +R 12 z c 2 d 2 r t rr t s 3 R 11 y = (c R 12 z) t r ts 3 z r y = R 1 11(c R 12 z) s 3 3 t3 t 3 t r 3 [ ] [ y R 1 ] x = P = P 11(c R 12 z) z z z = 0 rt3 x B rr s 3 rt [ R 1 x B = P 3 r z = 0 rt3 t t r t A r t 3 tr 3 t r s 3 3 t t t r 11c 0 ]. r t rr rr t r r 3 QR t r 3 3 r t rt 3 rr ts rt A t b rr ts 3 3 A r QR t r 3 3 s r tr s r 3 r 3 AP = Q R Q rt t t R = [R 11 R 12 ] R 11 tr rr t t r = hein( R) rr ts t t t [ ] c d rr ts t3 R 11 y = (c R 12 z) rr ts r t x = P = Q t b c t r Q t b r r s s t3 [ ] y z z = 0 rt3 R 11 y B = c 3t [ yb z = 0 t y B t x B = P 0 rr ts t3 x z = 0 rt x B rr s 3 3 ] r r r 3 r x LM s 3 r r t3 r [ x LM 2 = min R 1 ] x B P z. z IR n r 2 11R 12 1l n r
44 s r t r t t t3 1 x B 2 1+ R 1 x LM 11R t A = t b = 8. 3 r t min Ax b 2 r r rr s 3 r t x LM s 3 r ts 3 r 3 r rr 3 t r r rr 3 t 3 t tr t t r t 3 tr 3 t rt P = ts AP = s r tr s r 3 r 3 AP = Q R s s 3 t Q = R = A r r = 2 t R 11 = [ ] [ ] , R =, Q t b = Q t b t r r r = 2 s c = [ ] t t r s t3 t t 3 m r = 4 2 = 2 s d = [ ] t t r r z = 0 rt 3 R 11 y B = c t s st r t3 3 rt3 y B = [ ] [ 1 yb, x 3 B = P 0 ] = = ,
45 rr t s st t r t r t r 3 min Ax b 2 = Ax B b 2 = d 2 = = r t r x LM t r 3 w = R 1 11R 12 r t ts R 11 w = R 12 t3 r [ r t r t [ R 1 v = x B P 11R 12 1l n r ] z = ][ ] w1 w 2 = t rt 3 1 [ ] z = [ ] 1 w = z = z 3+0.5z 1 z x LM 2 = min z v 2 min{z 2 +(3+0.5z) 2 +( 1 z) 2 } = min{2.25z 2 +5z+10} z = r 3 x LM 2 = t x LM = ( 1.111) = 1 ( 1.111) x B 2 = s s 3 rt s rr QR t r 3 3 R tr 3 rt s Q t AP = R = [ ] R11 R r n r r m r r 3 QR t r 3 3 r 3 t3 [ R Z 1...Z t ] 11 r R t = 12 [ T t ] 11 0 r n r Z i s r tr 3 t r t T t 11 tr rr r 3 Q t AZ = T = [ ] T r n r r m r Z = PZ r...z 1 s s 3 rr s s 3 rt s r t3
46 s r t r 3 rr t r t t3 Ax b 2 2 = (Q t AZ)Z t x Q t b 2 2 = T 11 w c d 2 2 Z t x = [ ] w r y n r t Q t b = [ ] c r d m r. st 3 x rr t 3 t r t r w = T 1 11c t r r x r r rr 3 t y 3 r 3 r t rr [ T 1 x LM = Z 11c 0 ].
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