ku1 (1108607), страница 8
Текст из файла (страница 8)
U =A : A = L^ DUD^ ;1L^ ;1 U D^ L^ ,U (L^ );1 = D^ ;1L^ ;1U D^ :(2)-*+)(*, 3)& L^ 2 RT(n), 7'(3+* %$-41-@ ,(-%&1-$: ;)&8 *-)'(0< /&/)&() (>+,(1(0. $+,&4-)+$:1& L^ 2 UT(n). 9&;)&*. 4 $+4&8 3-/)( '-4+1/)4- (2) /)&()*-)'(0- U (L^ );1 2 UT(n). 7'-4& A+ 3-/)( '-4+1/)4- (2) /)&() 7'&(>4+,+1(+ 1(A1(6 )'+.%&$:1<6 *-)'(0, 2&)&'&+ @4$@+)/@ &7@): 1(A1+8 )'+.%&$:1&8*-)'(0+8, ).+.
7'(1-,$+A() LT(n). 9&;)&*. (> (2) 4<)+2-+)U (L^ );1 2 UT(n) \ LT(n):,(1/)4+11&8 *-)'(0+8, 2&)&'-@ 7'(1-,$+A() &,1&4'+*+11& 7&,%'.77-* UT(n)( LT(n) @4$@+)/@ I { +,(1(31-@ *-)'(0-. $+,&4-)+$:1&,U (L^ );1 = D^ ;1L^ ;1U D^ = I:(3)^ . -$++, (> (3)-2(* &='->&*, U = L^ ( A = U DUI = D^ ;1L^ ;1 U D^ = D^ ;1 L^ ;1(L^ )D^ = D^ ;1L^ ;1L^ D^ = D^ ;1D^ .. !!x10.
# 1*(3 (/3 0)36).+. D^ = D^ . /($. (1) 7&$.3-+* lkk = lkk , ).+. lkk { 4+C+/)4+11<+ ,$@ 4/+6k = 1 : : : n . 9'+,/)-4(4 *-)'(0. D^ 4 4(,+ D^ = jD^ j1=2 D jD^ j1=2 , %,+qq1=2^D = diag (sign l11 : : : sign lnn) jDj = diag ( jl11 j : : : jlnnj)7&$.3-+* A = U jD^ j1=2 D jD^ j1=2 U . =&>1-3(* R = jD^ j1=2 U 2 RT (qn). &%,A = RDR , 7'(3+* ,(-%&1-$:1<+ ;$+*+1)< rii *-)'(0< R '-41< jliij > 0.$+,&4-)+$:1&, 7&$.3+11&+ '->$&A+1(+ @4$@+)/@ )'+=.+*<*.4 1. /$( *-)'(0- A { 4+C+/)4+11-@, )& 4/+ .3-/)4.LC(+ 4 )+&'+*+ 1 *-)'(0< 4+C+/)4+11<+.4 2. /$( 4 ./$&4(@6 )+&'+*< 1 *-)'(0- A 7&$&A()+$:1& &7'+,+$+1-, )& *-)'(0- D 4 )+&'+*+ 1 { +,(1(31-@, ).+.
'->$&A+1(+, ,-4-+*&+ ;)&8)+&'+*&8, (*++) 4(, A = RR .+8/)4()+$:1&, +/$( *-)'(0- A 7&$&A()+$:1& &7'+,+$+1- ( ,$@ 1++ /7'-4+,$(4& '->$&A+1(+ A = RD R , )& ,$@ 4/@2&%& x 2 Cn x 6= 0 4<'-A+1(+(Ax x) = (R D Rx x) = (D Rx Rx) 4+C+/)4+11& ( 7&$&A()+$:1&. $@ 4/@2&%&y 2 Cn 7&$&A(* x = R;1 y . &%,- 4<'-A+1(+ (Dy y) = (Ax x) 4+C+/)4+11& (7&$&A()+$:1& ,$@ 4/+6 y 2 Cn y 6= 0, ).+. *-)'(0- DP7&$&A()+$:1&P&7'+,+$+1-.9&/2&$:2. D = diag (d11 : : : dnn), )& (Dy y) = nj=1 djj yj yj = nj=1 djj jyj j2 .9& ,&2->-11&*., ;)- /.**- 4+C+/)4+11- ( 7&$&A()+$:1-.
<=('-@ >,+/: y =ek k = 1 : : : n , %,+ ek { &')< /)-1,-')1&%& =->(/-, 1-6&,(*, 3)& 4/+ dkk k =1 : : : n ,&$A1< =<): 4+C+/)4+11<*( ( 7&$&A()+$:1<*(. 9&/2&$:2. jdkk j = 1,)& ;)& &>1-3-+), 3)& dkk = 1 k = 1 : : : n .4 3. /$( *-)'(0- A 7&$&A()+$:1& &7'+,+$+1-, )& ,$@ 1++ 4<7&$1+1< ./$&4(@ )+&'+*< 1.
F)& 4<)+2-+) (> $+**< 9.2.x 10.2. "*% 5%A %6"7A 2"?*9&/)'&(* -$%&'()* 1-6&A,+1(@ '->$&A+1(@ (> )+&'+*< 1.9./): ,$@ /-*&/&7'@A+11&8 *-)'(0< A ( A = A ) )'+=.+)/@ 1-8)( 4+'61LL)'+.%&$:1.L *-)'(0. R = (rij ) c 4+C+/)4+11<*( 7&$&A()+$:1<*( ;$+*+1)-*(1- %$-41&8 ,(-%&1-$( ( rii > 0 ,$@ 4/+6 i = 1 : : : n ) ( ,(-%&1-$:1.L *-)'(0.D / '-41<*( 7& *&,.$L +,(1(0+ ,(-%&1-$:1<*( ;$+*+1)-*( ( jdiij = 1 ,$@ 4/+6i = 1 : : : n ) )-2.L, 3)& A = RDR .nF$+*+1) (k j ) *-)'(0< DR '-4+1 (DR)kj = P dkirij = dkk rkj , )-2 2-2 *-i=1)'(0- D { ,(-%&1-$:1-@S ;$+*+1) (i k) *-)'(0< R '-4+1 (R )ik = rki S ;$+*+1)nn(i j ) *-)'(0< RDR '-4+1 (R DR)ij = P (R)ik (DR)kj = P rkidkk rkj .
$+,&k=1k=14-)+$:1&, '-4+1/)4& A = R DR ,-+) 1-* .'-41+1(@nXk=1..rkidkk rkj = aij i j = 1 : : : n:(4) !!x10. # 1*(3 (/3 0)379&/2&$:2. *-)'(0< A ( R DR { /-*&/&7'@A+11<+, )& .'-41+1(+ / 1&*+'&*(j i) 7&$.3-+)/@ (> .'-41+1(@ / 1&*+'&* (i j ) 7.)+* 2&*7$+2/1&%& /&7'@A+1(@( 1+ ,-+) 1(3+%& 1&4&%&. 9&;)&*. /(/)+*- (4) ;24(4-$+1)1- /(/)+*+nXk=1i j i j = 1 : : : n:rkidkk rkj = aij (5)-2(* &='->&*, (5) 7'+,/)-4$@+) /&=&8 /(/)+*. (> n(n + 1)=2 .'-41+1(8 / n(n +1)=2 1+(>4+/)1<*( rij i j (1-7&*1(*, R 2 RT(n) ( rij = 0 7'( i > j ) ( n1+(>4+/)1<*( dkk k = 1 : : : n (7'( ;)&* rkk > 0 ( dkk 2 f;1 1g ).9&$.3(* 5&'*.$< ,$@ '+K+1(@ /(/)+*< (5), 2&)&'<+ ( /&/)-4$@L) -$%&'()**+)&,- &$+02&%&.9+'+7(K+* (5) 4 4(,+Xik=1rkidkk rkj +nXk=i+1rkidkk rkj = aij i j i j = 1 : : : n:(6)9&/2&$:2. *-)'(0- R { 4+'61@@ )'+.%&$:1-@, )& rki = 0 7'( k > i ( 4)&'-@ (>/.** 4 (6) '-41- 1.$L.
$+,&4-)+$:1&, /(/)+*- (5) ;24(4-$+1)1- /$+,.LC+8i;1Xk=1rkidkk rkj + riidiirij = aij i j i j = 1 : : : n(7)(>,+/: /3()-+)/@, 3)& /.**- 4(,- Pik;=11 '-41- 1.$L, +/$( 4+'61(8 7'+,+$ /.**('&4-1(@ *+1:K+ 1(A1+%&S ;)& 7&>4&$@+) 1+ '-//*-)'(4-): &),+$:1& /$.3-8i = 1S )-2A+ 4 (7) *< .3$(, 3)& rii { 4+C+/)4+11<8 ;$+*+1)). <,+$(* 4 /.**+ (7) &),+$:1& /$.3-8 i = j2iP;166 rii2 dii = aii ; k=1 jrkij2dkk 64i;1riidiirij = aij ; P rkidkk rkj k=1i = 1 : : : ni < j i j = 1 : : : n)/L,- 7&$.3-+* '-/3+)1<+ 5&'*.$<:2iP;166 dii = sign(aii ; k=1 jrkij2dkk )66s;166 rii = jaii ; iPjrkij2dkk j66k=164i;1rij = (aij ; P rki dkk rkj )=(riidii)k=1i = 1 : : : ni = 1 : : : n(8)i < j i j = 1 : : : n:9'&0+// 4<3(/$+1(8 7& ;)(* 5&'*.$-* /)'&()/@ /$+,.LC(* &='->&*: 41-3-$+4<3(/$@L)/@ 1+(>4+/)1<+ ;$+*+1)< 7+'4<6 /)'&2 *-)'(0 D ( R :qd11 = sign a11 r11 = ja11 j r1j = a1j =(r11d11) j = 2 : : : nS..
!!x10. # 1*(3 (/3 0)387&)&* 7& 5&'*.$-* (8) 7'( i = 2 4<3(/$@L)/@ 1+(>4+/)1<+ ;$+*+1)< 4)&'<6/)'&2 *-)'(0 D ( R :qd22 = sign(a22 ; jr12j2d11 ) r22 = ja22 ; jr12j2d11 jr2j = (a2j ; r12 d11r1j )=(r22d22 ) j = 3 : : : nS>-)+* 7& 5&'*.$-* (8) 7'( i = 3 4<3(/$@L)/@ 1+(>4+/)1<+ ;$+*+1)< )'+):(6/)'&2 *-)'(0 D ( R ( )-2 ,-$++.4 4. %*6?A %A %? A , R D & 5A.9&/2&$:2.
,$@ /-*&/&7'@A+11&8 *-)'(0< aji = aij , )& *&A1& 4*+/)& 4/+8*-)'(0< A 6'-1(): )&$:2& ++ 4+'61(8 )'+.%&$:1(2: aij i j i j = 1 : : : n .O&'*.$< (8) )-2&4<, 3)& 7'( 4<3(/$+1(( ;$+*+1)- rij (/7&$:>.L)/@ >1-3+1(@ ;$+*+1)- aij ( 4<3(/$+11<6 '-1++ ;$+*+1)&4 rkm k < i . F)& 7&>4&$@+)6'-1(): 4+'1LL )'+.%&$:1.L *-)'(0. R 1- *+/)+ 4+'61+%& )'+.%&$:1(2- *-)'(0< A : rij aij i j i j = 1 : : : n . -)'(0- D &=<31& 6'-1()/@ 4 4(,+&),+$:1&%& 4+2)&'- d = (d11 : : : dnn)t .4 5. "! 5"7"# %?' A .
/($. >-*+3-1(@ 2 4 ;)&* /$.3-+ '->$&A+1(+ *-)'(0< (*++) =&$++ 7'&/)&84(, A = RR . O&'*.$< (8) )&A+ .7'&C-L)/@:2si;166 rii = (aii ; P jrkij2)k=166i;4 r = (a ; P1 r r )=r ijijki kj iik=1i = 1 : : : ni < j i j = 1 : : : n:(9)/$( *-)'(0- A +C+ ( 4+C+/)4+11-@, )& 4 5&'*.$-6 (9) *&A1& .='-): >1-2*&,.$@ ( 2&*7$+2/1&%& /&7'@A+1(@:2si;166 rii = (aii ; P rki2 )k=166i;4 r = (a ; P1 r r )=r ijijki kj iik=1..i = 1 : : : ni < j i j = 1 : : : n: !!x11. # 3*+(39x 10.3.
? "& % 5%? & "*%5%A %6"7A 2"?*1. <3(/$+1(+ ;$+*+1)- dii i = 1 : : : n 7& 5&'*.$-* (8) )'+=.+) i ; 1 *.$:)(7$(2-)(41<6 ( /)&$:2& A+ -,,()(41<6&7+'-0(8. $+,&4-)+$:1&, 4<3(/$+1(+4/+6 ;$+*+1)&4 *-)'(0< D )'+=.+) Pni=1 (i ; 1) = n(n ; 1)=2 = O(n2) *.$:)(7$(2-)(41<6 ( /)&$:2& A+ -,,()(41<6 &7+'-0(8.2. <3(/$+1(+ ;$+*+1)- rii 7& 5&'*.$-* (8) )'+=.+) &,1&8 &7+'-0(( (>4$+3+1(@ 2&'1@ ( i ; 1 *.$:)(7$(2-)(41<6 ( /)&$:2& A+ -,,()(41<6 &7+'-0(8(.*1&A+1(+ 1- dii 2 f;1 1g *< >- &7+'-0(L 1+ /3()-+*). 9'( 5(2/('&4-11&* i = 1 : : : n 4<3(/$+1(+rij ,$@ 4/+6 j = i + 1 : : : n 7& 5&'Pn ;$+*+1)&4*.$-*(8))'+=.+)1+(i;1)=(n ; i)(i ; 1) + 1 *.$:)(7$(2-)(41<6j =i+1Pn( j=i+1(i ; 1) = (n ; i)(i ; 1) -,,()(41<6 &7+'-0(8. $+,&4-)+$:1&, 4<3(/$+1(+;$+*+1)&4 *-)'(0< R )'+=.+) n &7+'-0(8 (>4$+3+1(@ 2&'1@,Pn ((n ;4/+632i=1 P i)(i ; 1) + (i ; 1) + 1) = n =6 + O(n ) (n ! 1) *.$:)(7$(2-)(4n1<6 ( i=1 ((n ; i)(i ; 1)+(i ; 1)) = n3=6+ O(n2) (n ! 1) -,,()(41<6 &7+'-0(8(7&,'&=1&+ 4<3(/$+1(+ /*.
7'( 7&,/3+)+ 2&$(3+/)4- -'(5*+)(3+/2(6 &7+'-0(8,$@ -$%&'()*- LU -'->$&A+1(@).- 4<3(/$()+$:1<6 *-K(1-6, (*+LC(6 -77-'-)1.L 7&,,+'A2. ,$@ 4<3(/$+1(@ )'-1/0+1,+1)1<6 5.120(8 4+C+/)4+11&%& -'%.*+1)-, &7+'-0(@ (>4$+3+1(@ 2&'1@ 4<7&$1@+)/@ >- 4'+*@, 7& 7&'@,2. '-41&+ 4'+*+1( ,+$+1(@ ,4.6 4+C+/)4+11<6 3(/+$. 9&;)&*. 1- )-2(6 4<3(/$()+$:1<6 /(/)+*-6 &7+'-0(( (>4$+3+1(@ 2&'1@ 4 &7(/-11&* 4<K+ -$%&'()*+ >-8*.) 4'+*@, '-41&+ 4'+*+1(4<7&$1+1(@ O(n) *.$:)(7$(2-)(41<6 &7+'-0(8.- 4<3(/$()+$:1<6 *-K(1-6, 1+ (*+LC(6 -77-'-)1&8 7&,,+'A2( ,$@ 4<3(/$+1(@ )'-1/0+1,+1)1<6 5.120(8 4+C+/)4+11&%& -'%.*+1)-, &7+'-0(( (>4$+3+1(@ 2&'1@ 4 &7(/-11&* 4<K+ -$%&'()*+ >-8*.) 4'+*@, '-41&+ 4'+*+1( 4<7&$1+1(@ O(const n) = O(n) *.$:)(7$(2-)(41<6 &7+'-0(8.-2(* &='->&*, -$%&'()* 7&/)'&+1(@ '->$&A+1(@ &$+02&%& )'+=.+) ,$@/4&+%& 7'&4+,+1(@ 4<7&$1+1(@ n3 =6 + O(n2) (n ! 1) *.$:)(7$(2-)(41<6 (/)&$:2& A+ -,,()(41<6 &7+'-0(8, - 4 /.**+ { n3=3 + O(n2) (n ! 1) -'(5*+)(3+/2(6 &7+'-0(8, ).+.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
