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% $ "*%-//*&)'(* /7&/&=<, 7'(*+1@+*<+ ,$@ ./2&'+1(@ /6&,(*&/)( 7&/$+,&4-)+$:1&/)( *-)'(0 fAk g 2 )'+.%&$:1&8 *-)'(0+. F)( /7&/&=< &,(1-2&4< 2-2 ,$@LR --$%&'()*-, )-2 ( ,$@ '-//*-)'(4-+*<6 1(A+ -$%&'()*- &$+02&%& ( QR -$%&'()*- 1-6&A,+1(@ /&=/)4+11<6 >1-3+1(8. 9&/2&$:2. 4/+ ;)( -$%&'()*< *$ 1+ 7'(*+1@L)/@ ,$@ *-)'(0 7'&(>4&$:1&%& 4(,-, 4/L,. 1(A+ *< =.,+*/3()-):, 3)& (/6&,1-@ *-)'(0- .A+ 7'(4+,+1- .1()-'1<* 7&,&=(+* 2 7&3)( )'+.%&$:1&*.
($( )'+6,(-%&1-$:1&*. 4(,.. -2(* &='->&*, 1-3-$:1-@ *-)'(0- A1| 7&3)( )'+.%&$:1-@ (($( )'+6,(-%&1-$:1-@). 9& ,&2->-11&*. 4<K+ ;)& &>1-3-+), 3)& 4/+ *-)'(0< Ak | 7&3)( )'+.%&$:1<+ ()'+6,(-%&1-$:1<+).4 3. 7(/-11<+ 1(A+ 7'(+*< 1+ )&$:2& ./2&'@L) /6&,(*&/): -$-%&'()*&4 1-6&A,+1(@ /&=/)4+11<6 >1-3+1(8, 1& ( '-/K('@L) *1&A+/)4& *-)'(0, ,$@ 2&)&'<6 &1( /6&,@)/@. '.%(*( /$&4-*(, 7'( (/7&$:>&4-1(( ;)(67'(+*&4 -$%&'()*< 1-6&A,+1(@ /&=/)4+11<6 >1-3+1(8 3-/)& '-=&)-L) ,$@ *-)'(0, ,$@ 2&)&'<6 1+ 4<7&$1+1< ./$&4(@ 7'(4+,+11<6 )+&'+* & /6&,(*&/)(-$%&'()*&4...
x7. LR "4&1054 4. +> (/7&$:>&4-1(@ &7(/-11<6 1(A+ 7'(+*&4 /2&'&/): /6&,(*&/)( -$%&'()*&4 1-6&A,+1(@ /&=/)4+11<6 >1-3+1(8 &2-><4-+)/@ 4+/:*- 1(>2&8, - *1&A+/)4& *-)'(0, ,$@ 2&)&'<6 &1( 7'(*+1(*<, ,&/)-)&31& .>2(*. 9&;)&*. ;)( 7'(+*< &*$ 7'(*+1@L)/@ 7'( 4<3(/$+1(@6 7& ;)(* -$%&'()*-*.x 7.3.1. %5'& %6'- k -&* K-%+ *-)'(0- Ak (*++) 4(,0 (k) (k)BB a(11k) a(12k)BB a21 a22BBa(32k)BBBAk = BBBBBBBBB@: : : a(1ki): : : a(2ki). . . ....
. . a(k)ii(k)ai+1i::::::::::::......a(1kn) 2a(2kn) 2...a(ink) 2...a(nk) 2n 2a(nk) 1n 2a(1kn) 1a(2kn) 1...a(ink) 1...a(nk) 2n 1a(nk) 1n 1k)a(nn1;;;;;;;;a(1kn)a(2kn)...a(ink)...;;;;;;a(nk) 2na(nk) 1na(nnk);;;1CCCCCCCCCCCCCCCCCCAk) j < "kA k ,/$( ,$@ 1+2&)&'&%& i , i = 1 2 : : : n ; 1 4<7&$1+1& ./$&4(+ ja(i+1ki%,+ " | )&31&/):, / 2&)&'&8 )'+=.+)/@ 1-8)( /&=/)4+11<+ >1-3+1(@, )& 7&$-%-+*k) = 0. 9&/2&$:2.
/&=/)4+11<+ >1-3+1(@ @4$@L)/@ 1+7'+'<41<*( 5.120(a(i+1i@*( ;$+*+1)&4 *-)'(0<, )& /&=/)4+11<+ >1-3+1(@ *-)'(0< Ak (>*+1@)/@ 7'(;)&8 >-*+1+ 1- 4+$(3(1. 7&'@,2- " . &4-@ *-)'(0- Ak (*++) =$&31.L /)'.2).'.:10 (k) (k)BB aa11(k) aa(12k)BB 21 22BBa(32k)BBBBAk = BBBBBBBBBB@: : : a(1ki): : : a(2ki). . . .... . . a(k)ii0a(1ki)+1a(2ki)+1...a(iik)+1k)a(i+1i+1(k)ai+2i+1:::::::::::::::......a(1kn) 2a(2kn) 2...a(ink) 2k)a(i+1n 2...a(nk) 2n 2a(nk) 1n 2;;;;;;;;a(1kn) 1a(2kn) 1...a(ink) 1k)a(i+1n 1...a(nk) 2n 1a(nk) 1n 1k)a(nn1;;;;;;;;;a(1kn)a(2kn)...a(ink)k)a(i+1n...a(nk) 2na(nk) 1na(nnk);;1CCCCCCCCCCCCCCCCCCCCA( ++ *1&A+/)4& /&=/)4+11<6 >1-3+1(8 '-41& &=S+,(1+1(L *1&A+/)4 /&=/)4+11<6 >1-3+1(8 7&,*-)'(0 (a(jlk) )jl=12:::i 2 Mi ( (a(jlk) )jl=i+1i+2:::n 2 Mn i .
$+,&4-)+$:1&, *&A1& 7'(*+1(): -$%&'()* 2 2-A,&8 (> ;)(6 7&,*-)'(0 7& &),+$:1&/)(.;.. x7. LR "4&106k) j <-/)& ,$@ .*+1:K+1(@ 4<3(/$()+$:1<6 >-)'-) 4*+/)& ./$&4(@ ja(i+1i(k)"kAk k (/7&$:>.L) ./$&4(+ jai+1ij < "kA1k .11x 7.3.2. $&*$@ *-)'(0< A;I /&=/)4+11<+ >1-3+1(@ '-41< i ; , i { /&=/)4+11<+ >1-(k)3+1(@ *-)'(0< A . 2&'&/): /6&,(*&/)( 7&,,(-%&1-$:1<6 ;$+*+1)&40 ai+1i 2k1.1$L ,$@ ;)&8 *-)'(0< 7& )+&'+*+ & /6&,(*&/)( -$%&'()*- +/): O @ i+1;; A .i(k )/$( =$(>2& 2 i+1 , )& /2&'&/): /6&,(*&/)( ;$+*+1)- ai+1i 2 1.$L =.,+) &3+1:4</&2&8.&,(5(0('&4-11<8 LR --$%&'()*, &/1&4-11<8 1- ;)&8 (,++, 4<%$@,() /$+,.LC(* &='->&*..,+* /)'&(): ,$@ *-)'(0< A 2 Mn 7&/$+,&4-)+$:1&/): fAk g *-)'(0 Ak 2Mn 7& /$+,.LC(* 7'-4($-*:1) A1 = A T2) ,$@ 4/+6 k = 1 2 : : : *-)'(0- Ak+1 7&$.3-+)/@ (> *-)'(0< Ak /$+,.LC(*&='->&*:-) &7'+,+$@+* )'+=.+*<8 /,4(% sk (+%& &7)(*-$:1<8 4<=&' { &),+$:1-@>-,-3-),=) /)'&(* LR -'->$&A+1(+ *-)'(0< Ak ; sk I : Ak ; sk I = Lk Rk ,4) 4<3(/$@+* *-)'(0.
Ak+1 2-2 7'&(>4+,+1(+ *-)'(0 Rk ( Lk 7$L/ sk I :Ak+1 = Rk Lk + sk I .,+/: *< 7'+,7&$-%-+*, 3)& ,$@ 2-A,&%& k = 1 2 : : : LR -'->$&A+1(+ *-)'(0<Ak ; sk I /.C+/)4.+), ).+. ,$@ 1++ 4<7&$1+1< ./$&4(@ )+&'+*< 1. /$( ;)& 1+ )-2,)& 1-,& (>*+1(): >1-3+1(+ /,4(%- sk . - 7'-2)(2+ ./$&4(@ )+&'+*< 1 1+ 7'&4+'@L), - 4<7&$1@L) -$%&'()* 7&/)'&+1(@ LR -'->$&A+1(@. /$( 4 -$%&'()*+)'+=.+)/@ &/.C+/)4(): ,+$+1(+ 1- 0, )& 1+*1&%& (>*+1@L) >1-3+1(+ /,4(%- sk (>-1&4& 4<7&$1@L) -$%&'()* 7&/)'&+1(@ LR -'->$&A+1(@.+)'.,1& 7'&4+'():, 3)& *-)'(0- Ak+1 7&,&=1- Ak : Ak+1 = Rk Lk + sk I =(Lk 1 Lk )(Rk Lk + sk I ) = Lk 1 (Lk Rk )Lk + sk Lk I = Lk 1(Lk Rk + sk I )Lk = Lk 1 Ak Lk (,/$+,&4-)+$:1&, 4/+ *-)'(0< Ak , k = 1 2 : : : (*+L) )+ A+ /&=/)4+11<+ >1-3+1(@,3)& ( *-)'(0- A .;;;;x 7.3.3.
.%@ %*:6@ &'" & LR "*%9./): )'+=.+)/@ &7'+,+$(): 4/+ /&=/)4+11<+ >1-3+1(@ *-)'(0< A 2 Mn /)&31&/):L " .1-3-$+ 7'(4&,(* *-)'(0. 2 7&3)( )'+.%&$:1&*. 4(,. A1 .1()-'1<* 7&,&=(+* &,1(* (> -$%&'()*&4, &7(/-11<6 4 x I.14 ( x I.15...
x8. & )"$4107-)+* 2 *-)'(0+ A1 7'(*+1@+* LR --$%&'()* /& /,4(%-*(. - K-%+ k 42-3+/)4+ /,4(%- sk 4&>:*+* a(nnk) , ).+. sk = a(nnk) . 9&/2&$:2. a(nnk) ! n , )& skk) =.@4$@+)/@ 7'(=$(A+1(+* 2 n ( /2&'&/): /6&,(*&/)( 2 1.$L ;$+*+1)- a(nn1,+) &3+1: 4</&2&8. -2 )&$:2& 1- 1+2&)&'&* K-%+ k =.,+) 4<7&$1+1& ./$&4(+ja(nnk) 1j < "kAk , 4 2-3+/)4+ n =+'+* a(nnk) ( 7'(*+1@+* -$%&'()* 2 7&,*-)'(0+(aij )ij=12:::n 1 2 Mn 1 1- 1 *+1:K+8 '->*+'1&/)(. -2 7&/).7-+* ,& )+6 7&',7&2- '->*+'1&/): *-)'(0< 1+ /)-1+) '-41&8 2. $@ ;)&8 *-)'(0< /&=/)4+11<+>1-3+1(@ &7'+,+$@L)/@ 2-2 '+K+1(@ /&&)4+)/)4.LC+%& 24-,'-)1&%& .'-41+1(@.;1;;;x8. +)&, &$+02&%& 7&>4&$@+) 1-6&,(): 4/+ /&=/)4+11<+ >1-3+1(@ /-*&/&7'@A+11&8 7&$&A()+$:1& &7'+,+$+11&8 *-)'(0< A 2 Mn >- 4,4&+ *+1:K++ 2&$(3+/)4& -'(5*+)(3+/2(6 &7+'-0(8, 3+* LR --$%&'()*.x 8.1.
:"F 2"6*, 5"#:! & $ 2"6*% 1. ( '->$&A+1(( &$+02&%&) 2 A = A > 0 | , A = LL , L | c & .:"#&. &%$-/1& )+&'+*+ I.10.1 ( >-*+3-1(@* I.10.2, I.10.3 *-)'(0-A ,&7./2-+) 7'+,/)-4$+1(+ 4 4(,+ A = R R , %,+ R | 4+'61@@ )'+.%&$:1-@*-)'(0- c 4+C+/)4+11<*( 7&$&A()+$:1<*( ;$+*+1)-*( 1- %$-41&8 ,(-%&1-$(.9&$&A(* L = R 2 LT(n).
&%,- A = LL ( *-)'(0- L | 1(A1@@ )'+.%&$:1-@c 4+C+/)4+11<*( 7&$&A()+$:1<*( ;$+*+1)-*( 1- %$-41&8 ,(-%&1-$(. +&'+*,&2->-1-.x 8.1.1. "*% 5%@ %:"F@ 2"6* $"@5%:&"# 5%@F %6'$%&'()* 7&/)'&+1(@ '->$&A+1(@ &$+02&%& /-*&/&7'@A+11&8 *-)'(0< A 2Mn &3+1: 7&6&A 1- -$%&'()* 7&/)'&+1(@ '->$&A+1(@ &$+02&%&, (/7&$:>.+*&%&4 *+)&,+ &$+02&%& '+K+1(@ $(1+81<6 /(/)+* / /-*&/&7'@A+11&8 *-)'(0+8 (/*./)'. 36).9./): ,$@ /-*&/&7'@A+11&8 *-)'(0< A = (aij ) ( A = A , ).+. aij = aji ))'+=.+)/@ 1-8)( 1(A1LL )'+.%&$:1.L *-)'(0. L = (lij ) c 4+C+/)4+11<*( 7&$&A()+$:1<*( ;$+*+1)-*( 1- %$-41&8 ,(-%&1-$( ( lii > 0 ,$@ 4/+6 i = 1 : : : n ),)-2.L, 3)& A = LL , ).+.nXk=1..lik ljk = aij i j = 1 : : : n:(1) x8. & )"$41089&/2&$:2. *-)'(0< A ( LL { /-*&/&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&;)&*. /(/)+*- (1) ;24(4-$+1)1- /(/)+*+nXk=1i j i j = 1 : : : n:lik ljk = aij (2)-2(* &='->&*, (2) 7'+,/)-4$@+) /&=&8 /(/)+*. (> n(n + 1)=2 .'-41+1(8 / n(n +1)=2 1+(>4+/)1<*( lij , i j (1-7&*1(*, L 2 LT(n) ( lij = 0 7'( i < j , 7'( ;)&*lkk > 0). 9&$.3(* 5&'*.$< ,$@ '+K+1(@ /(/)+*< (2), 2&)&'<+ ( /&/)-4$@L)-$%&'()* 1-6&A,+1(@ '->$&A+1(@ &$+02&%&.9+'+7(K+* (2) 4 4(,+Xik=1lik ljk +nXlik ljk = aij k=i+1i j i j = 1 : : : n:(3)9&/2&$:2.
*-)'(0- L { 1(A1@@ )'+.%&$:1-@, )& lik = 0 7'( i < k ( 4)&'-@ (>/.** 4 (3) '-41- 1.$L. $+,&4-)+$:1&, /(/)+*- (2) ;24(4-$+1)1- /$+,.LC+8i 1X;k=1lik ljk + lii lji = aij i j i j = 1 : : : n(>,+/: /3()-+)/@, 3)& /.**- 4(,- Pik=11 '-41- 1.$L, +/$( 4+'61(8 7'+,+$ /.**('&4-1(@ *+1:K+ 1(A1+%&T ;)& 7&>4&$@+) 1+ '-//*-)'(4-): &),+$:1& /$.3-8i = 1). 9'(*+1(* 2 ;)(* .'-41+1(@* 2&*7$+2/1&+ /&7'@A+1(+ ( .3)+*, 3)&aij = aji ( lii { 4+C+/)4+11<8 ;$+*+1):;i 1X;k=1lik ljk + lii lji = ajii j i j = 1 : : : n<,+$(* 4 >,+/: &),+$:1& /$.3-8 i = j :2iP166 lii2 = aii ; k=1 jlik j264i 1liilji = aji ; P lik ljk;i = 1 : : : n;i < j i j = 1 : : : nk=1)/L,- 7&$.3-+* '-/3+)1<+ 5&'*.$<:s2i 166 lii = aii ; kP=1 jlik j264i 1lji = (aji ; P lik ljk )=lii;;k=1i = 1 : : : ni < j i j = 1 : : : n:(4)9'&0+// 4<3(/$+1(8 7& ;)(* 5&'*.$-* /)'&()/@ /$+,.LC(* &='->&*: 41-3-$+4<3(/$@L)/@ 1+(>4+/)1<+ ;$+*+1)< 7+'4&%& /)&$=0- *-)'(0< L :l11 = pa11 lj1 = aj1=l11 j = 2 : : : nT..
x8. & )"$41097&)&* 7& 5&'*.$-* (4) 7'( i = 2 4<3(/$@L)/@ 1+(>4+/)1<+ ;$+*+1)< 4)&'&%&/)&$=0- *-)'(0< L :ql22 = a22 ; jl21j2lj2 = (aj2 ; l21lj1)=l22 j = 3 : : : nT>-)+* 7& 5&'*.$-* (4) 7'( i = 3 4<3(/$@L)/@ 1+(>4+/)1<+ ;$+*+1)< )'+):+%&/)&$=0- *-)'(0< L ( )-2 ,-$++.4 1. %*:6@ %@ %6 A L & 5@. 9&-/2&$:2. ,$@ /-*&/&7'@A+11&8 *-)'(0< aji = aij , )& *&A1& 4*+/)& 4/+8 *-)'(0< A 6'-1(): )&$:2& ++ 1(A1(8 )'+.%&$:1(2: aij i j i j = 1 : : : n .^&'*.$< (4) )-2&4<, 3)& 7'( 4<3(/$+1(( ;$+*+1)- lji (/7&$:>.L)/@ >1-3+1(@;$+*+1)- aji ( 4<3(/$+11<6 '-1++ ;$+*+1)&4 lmk , k < i . F)& 7&>4&$@+) 6'-1():1(A1LL )'+.%&$:1.L *-)'(0. L 1- *+/)+ 1(A1+%& )'+.%&$:1(2- *-)'(0< A :lji aji i j i j = 1 : : : n .6 "& % 5%6 & "*%5%@ %:"F@ 2"6*<3(/$+1(+ ;$+*+1)- lii 7& 5&'*.$-* (4) )'+=.+) &,1&8 &7+'-0(( (>4$+3+1(@2&'1@ ( i ; 1 *.$:)(7$(2-)(41<6 ( /)&$:2& A+ -,,()(41<6 &7+'-0(8.
9'(5(2/('&4-11&* i = 1 : : : n P4<3(/$+1(+ ;$+*+1)&4 lji ,$@ 4/+6 j = i + 1 : : : n7&(4) )'+=.+) 1+ nj=i+1(i;1) = (n;i)(i;1)+1 *.$:)(7$(2-)(41<6 (Pn5&'*.$-*j =i+1(i ; 1) = (n ; i)(i ; 1) -,,()(41<6 &7+'-0(8. $+,&4-)+$:1&, 4<3(/$+1(+n ((n ;4/+6 ;$+*+1)&4 *-)'(0< L )'+=.+) n &7+'-0(8 (>4$+3+1(@ 2&'1@, Pi=1Pi)(i ; 1) + (i ; 1) + 1) = n3=6 + O(n2) (n ! 1) *.$:)(7$(2-)(41<6 ( ni=1((n ;i)(i ; 1) + (i ; 1)) = n3 =6 + O(n2) (n ! 1) -,,()(41<6 &7+'-0(8 (7&,'&=1&+4<3(/$+1(+ /*.
7'( 7&,/3+)+ 2&$(3+/)4- -'(5*+)(3+/2(6 &7+'-0(8 ,$@ *+)&,&$+02&%& '+K+1(@ $(1+81<6 /(/)+*).x 8.1.2. "*% 5%@ %:"F@ 2"6* $"@%$*"# %6'-//*&)'(* /$.3-8, 2&%,- /-*&/&7'@A+11-@ *-)'(0- A 2 Mn 4 7'(4+,+11&*4<K+ -$%&'()*+ )'+6,(-%&1-$:1-@. > &7'+,+$+1(@ 7'&(>4+,+1(@ *-)'(0 4<)+2-+), 3)& *-)'(0- L 4 '->$&A+1(( &$+02&%& A = LL =.,+) ,4.6,(-%&1-$:1&8:0a a11 12BaB21 a22Ba32A=BBBB@..a23a33.........ann;an 1nann;110CCBB ll1121 l22CCBl32 l33CC L = BBB...CAB@...lnn1;lnn1CCCCCC : (5)CA x8. & )"$4110^&'*.$< (4), /$+,&4-)+$:1&, 7'(*.) 4(,qlii = aii ; jlii 1j2 i = 1 : : : n(6)li+1i = ai+1i=liii = 1 : : : n ; 1:$%&'()* 7&/)'&+1(@ '->$&A+1(@ &$+02&%& 7& ;)(* 5&'*.$-* )'+=.+) ,$@/4&+%& 7'&4+,+1(@ 4<7&$1+1(@ n &7+'-0(8 (>4$+3+1(@ 2&'1@, 2n ; 1 *.$:)(7$(2-)(41<6 ( n -,,()(41<6 &7+'-0(8.";x 8.2.