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$ 2"6* F$@ >&' :.,+* /)'&(): ,$@ /-*&/&7'@A+11&8 7&$&A()+$:1& &7'+,+$+11&8 *-)'(0<A 2 Mn 7&/$+,&4-)+$:1&/): fAk g *-)'(0 Ak 2 Mn 7& /$+,.LC(* 7'-4($-*:1) A1 = A T2) ,$@ 4/+6 k = 1 2 : : : *-)'(0- Ak+1 7&$.3-+)/@ (> *-)'(0< Ak /$+,.LC(*&='->&*:-) /)'&(* '->$&A+1(+ &$+02&%& *-)'(0< Ak : Ak = Lk Lk ,=) 4<3(/$@+* *-)'(0. Ak+1 2-2 7'&(>4+,+1(+ *-)'(0 Lk ( Lk : Ak+1 =Lk Lk .
1. k = 1 2 : : : Ak A .:"#&. *++*: Ak+1 = Lk Lk = (Lk 1Lk )Lk Lk = Lk 1 (Lk Lk )Lk =1;;Lk Ak Lk . $+,&4-)+$:1&, *-)'(0- Ak+1 7&,&=1- Ak . 9&/2&$:2. A1 = A , )& 7&(1,.20(( 7&$.3-+*, 3)& Ak 7&,&=1- A ,$@ 4/+6 k = 1 2 : : : , 7'(3+* Ak+1 =L1 1 : : : Lk 1 A1 Lk : : : L1 = (Lk : : : L1 ) 1A(Lk : : : L1 ).;;;;"$& 1.
3 Ak , k = 1 2 : : : ' , A . 2. k = 1 2 : : : Ak { .:"#&. +8/)4()+$:1&, A1 = A { /-*&/&7'@A+11-@ 7&$&A()+$:-1& &7'+,+$+11-@ *-)'(0-. 9./): /,+$-1& k = 1 2 : : : K-%&4 &7(/-11&%& 4<K+7'&0+//-. -)'(0- Ak = (Lk 1 : : : L1 ) 1A(Lk 1 : : : L1 ) 7&,&=1- *-)'(0+ A ( (*++) )+ A+ /&=/)4+11<+ >1-3+1(@. 9&/2&$:2. Ak = Lk 1Lk 1 ( Ak = Lk 1(Lk 1 ) =Ak , )& Ak { /-*&/&7'@A+11-@ *-)'(0-. 9& $+**+ I.9.3 4/+ /&=/)4+11<+ >1-3+1(@*-)'(0< A 4+C+/)4+11< ( 7&$&A()+$:1<.
$+,&4-)+$:1&, 4/+ /&=/)4+11<+ >1-3+1(@ *-)'(0< Ak 4+C+/)4+11< ( 7&$&A()+$:1<. 9& $+**+ I.9.3 ;)& &>1-3-+),3)& Ak { /-*&/&7'@A+11-@ 7&$&A()+$:1& &7'+,+$+11-@ *-)'(0-.;;;;;;; 3. 5 6 k = 1 2 : : : ... x8. & )"$4111:"#&. +8/)4()+$:1&, 7& $+**+ 2 Ak { /(**+)'(31-@ 7&$&A()+$:1& &7'+,+$+11-@ *-)'(0-, ( ,$@ 1++ '->$&A+1(+ &$+02&%& /.C+/)4.+) 7&)+&'+*+ 1.% 2. (+> ,&2->-)+$:/)4-.) ( A 2 Mn | , fig A , 1 > 2 > : : : > n:* Ak ! diagU1 : : : n] k ! 1 ( ). , & Ak = (a(ijk) ) A , :a(iik) ! i k ! 1 i = 1 2 : : : n:4 Ak "0 k 1(k)(k)aij = aji = O @ i A k ! 1 i > j:j9'(*+1+1(+ -$%&'()*- 2 7'&(>4&$:1&8 /-*&/&7'@A+11&8 *-)'(0+ A 2 Mn)'+=.+) /$(K2&* =&$:K&%& 3(/$- -'(5*+)(3+/2(6 &7+'-0(8: n3=3 + O(n2) (n !1) 1- 7&/)'&+1(+ '->$&A+1(@ &$+02&%& *-)'(0< Ak ( 1+ =&$++ n3 +O(n2) (n !1) 1- 4<3(/$+1(+ *-)'(0< Ak+1 2-2 7'&(>4+,+1(@ ,4.6 )'+.%&$:1<6 *-)'(0.9&;)&*.
*+)&, &$+02&%& *$ 1+ 7'(*+1@+)/@ 2 *-)'(0-* 7'&(>4&$:1&%&4(,-.x 8.2.1. $ 2"6* F$@ >&' : $"@%$*"# %6' 4. 2 A | , Ak , k =1 2 : : : 6 | .:"#&. -)'(0- A1 = A | )'+6,(-%&1-$:1-@. 9'+,7&$&A(*, 3)&*-)'(0- Ak | )'+6,(-%&1-$:1-@. &%,- 4 '->$&A+1(( &$+02&%& Ak = Lk Lk *-)'(0- Lk (*++) 4(, (5) ().+. @4$@+)/@ ,4.6,(-%&1-$:1&8). > &7'+,+$+1(@ 7'&(>4+,+1(@ *-)'(0 4<)+2-+), 3)& *-)'(0- Ak+1 = Lk Lk @4$@+)/@ )'+6,(-%&1-$:1&8*-)'(0+8.
+**- ,&2->-1-.F)- $+**- 7&>4&$@+) >1-3()+$:1& ./2&'(): '-=&). *+)&,- &$+02&%&. 9+'+,+%& 7'(*+1+1(+* (/6&,1-@ *-)'(0- A 7'(4&,()/@ 2 )'+6,(-%&1-$:1&*. 4(,. A.1()-'1<* 7&,&=(+* &,1(* (> -$%&'()*&4, &7(/-11<6 4 x I.14 ( x I.15. -)+* 2*-)'(0+ A 7'(*+1@+)/@ *+)&, &$+02&%&.00.. x8. & )"$4112"*% &'"@ 5%:&$@ %6 L L9'&(>4+,+1(+ *-)'(0 A = L L , %,+ L (*++) 4(, (5), *&A+) =<): 4<3(/$+1&>1-3()+$:1& =</)'++, 3+* 7'&(>4+,+1(+ 7'&(>4&$:1<6 )'+.%&$:1<6 *-)'(0. 9&&7'+,+$+1(L 7'&(>4+,+1(@ *-)'(02 a =l l 66 aiiii =1 jliij2ii +ii jl1i+1ij264 a = l l ii+1ii ii+1ann = jlnnj2;i = 2 3 : : : ni = 1 2 : : : n ; 1i = 1 2 : : : n ; 1;(7)<3(/$+1(+ 7'&(>4+,+1(@ A = L L 7& ;)(* 5&'*.$-* )'+=.+) 4n +O(1) (n ! 1) *.$:)(7$(2-)(41<6 ( n + O(1) (n ! 1) -,,()(41<6 &7+'-0(8.6 "& % 5%6 $ 8* $2"6* $"@ %$*"# %6'1) 9&/)'&+1(+ '->$&A+1(@ &$+02&%& *-)'(0< Ak = Lk Lk 7& 5&'*.$-* (6))'+=.+) n &7+'-0(8 (>4$+3+1(@ 2&'1@, 2n + O(1) (n ! 1) *.$:)(7$(2-)(41<6( n + O(1) (n ! 1) -,,()(41<6 &7+'-0(8.2) <3(/$+1(+ 7'&(>4+,+1(@ Ak+1 = Lk Lk 7& 5&'*.$-* (7) )'+=.+) 4n +O(1) (n ! 1) *.$:)(7$(2-)(41<6 ( n + O(1) (n ! 1) -,,()(41<6 &7+'-0(8.$+,&4-)+$:1&, &,(1 K-% -$%&'()*- ,$@ )'+6,(-%&1-$:1&8 *-)'(0< )'+=.+)n &7+'-0(8 (>4$+3+1(@ 2&'1@, 6n + O(1) (n ! 1) *.$:)(7$(2-)(41<6 ( 2n +O(1) (n ! 1) -,,()(41<6 &7+'-0(8.x 8.3.
% $ "*%-//*&)'(* /7&/&=<, 7'(*+1@+*<+ ,$@ ./2&'+1(@ /6&,(*&/)( 7&/$+,&4-)+$:1&/)( *-)'(0 fAk g 2 ,(-%&1-$:1&8 *-)'(0+. -2 &)*+3-$&/: 4<K+ (/*. /)'. 104),;)( /7&/&=< 4& *1&%&* /6&A( /& /7&/&=-*( ./2&'+1(@ /6&,(*&/)( LR - ( QR -$%&'()*&4. -2A+ /7'-4+,$(4< >-*+3-1(@ 7.3 ( 7.4.9&/2&$:2. *+)&, &$+02&%& *$ 1+ 7'(*+1@+)/@ ,$@ *-)'(0 7'&(>4&$:1&%& 4(,-, 4/L,. 1(A+ *< =.,+* /3()-):, 3)& (/6&,1-@ *-)'(0- .A+ 7'(4+,+1- .1()-'1<* 7&,&=(+* 2 )'+6,(-%&1-$:1&*.
4(,.. -2(* &='->&*, 1-3-$:1-@*-)'(0- A1 | )'+6,(-%&1-$:1-@. 9& ,&2->-11&*. 4<K+ ;)& &>1-3-+), 3)& 4/+*-)'(0< Ak | )'+6,(-%&1-$:1<+.x 8.3.1. %5'& %6',+@ (/3+'7<4-1(@ *-)'(0< ,$@ *+)&,- &$+02&%& )- A+, 3)& ( ,$@ LR -$%&'()*- (/*. /)'. 105)... x8. & )"$4113x 8.3.2. $&*,+@ (/7&$:>&4-1(@ /,4(%&4 ,$@ *+)&,- &$+02&%& )- A+, 3)& ( ,$@ LR -$%&'()*- (/*. /)'. 106). &,(5(0('&4-11<8 *+)&, &$+02&%&, &/1&4-11<8 1;)&8 (,++, 4<%$@,() /$+,.LC(* &='->&*..,+* /)'&(): ,$@ *-)'(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-),=) /)'&(* '->$&A+1(+ &$+02&%& *-)'(0< Ak ; sk I : Ak ; sk I = Lk Lk ,4) 4<3(/$@+* *-)'(0.
Ak+1 2-2 7'&(>4+,+1(+ *-)'(0 Lk ( Lk 7$L/ sk I :Ak+1 = Lk Lk + sk I .,+/: *< 7'+,7&$-%-+*, 3)& ,$@ 2-A,&%& k = 1 2 : : : '->$&A+1(+ &$+02&%& *-)'(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(@ )+&'+*< 17'&4+'(): 1+4&>*&A1&, 7&;)&*.
4<7&$1@L) -$%&'()* 7&/)'&+1(@ '->$&A+1(@&$+02&%&. 9'( ;)&*, +/$( 4 -$%&'()*+ )'+=.+)/@ (>4$+3: 2&'+1: (> &)'(0-)+$:1&%& 3(/$- ($( &/.C+/)4(): ,+$+1(+ 1- 0, )& 1+*1&%& (>*+1@L) >1-3+1(+/,4(%- sk ( >-1&4& 4<7&$1@L) -$%&'()* 7&/)'&+1(@ '->$&A+1(@ &$+02&%&.+)'.,1& 7'&4+'():, 3)& *-)'(0- Ak+1 7&,&=1- Ak : Ak+1 = Lk Lk + sk I =(Lk 1 Lk )(Lk Lk + sk I ) = Lk 1(Lk Lk )Lk + sk Lk I = Lk 1(Lk Lk + 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 8.3.3. .%@ %*:6@ &'" & $ 2"6*9./): )'+=.+)/@ &7'+,+$(): 4/+ /&=/)4+11<+ >1-3+1(@ /-*&/&7'@A+11&8 7&$&A()+$:1& &7'+,+$+11&8 *-)'(0< A 2 Mn / )&31&/):L " .1-3-$+ 7'(4&,(* *-)'(0. 2 )'+6,(-%&1-$:1&*. 4(,. A1 .1()-'1<* 7&,&=(+* &,1(* (> -$%&'()*&4, &7(/-11<6 4 x I.14 ( x I.15.-)+* 2 *-)'(0+ A1 7'(*+1@+* *+)&, &$+02&%& /& /,4(%-*(.
- K-%+ k4 2-3+/)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;;..; x9. QR "4&114x9. QR QR -$%&'()* 7&>4&$@+) 1-6&,(): 4/+ /&=/)4+11<+ >1-3+1(@ *-)'(0< AMn .x 9.1.2QR -%:"F, 5"#:! & QR "*% QR --$%&'()*+ (/7&$:>.+)/@ )& A+ QR -'->$&A+1(+, 3)& /)'&($&/: 4 *+)&,+4'-C+1(8 (/*.
x I.12, )+&'+*- I.12.1) ( 4 *+)&,+ &)'-A+1(8 '+K+1(@ $(1+81<6/(/)+* (/*. x I.13, )+&'+*- I.13.1).x 9.1.1. "*% 5%@ QR -%:"F@ $"@ 5%:&"#%6'$%&'()* 7&/)'&+1(@ QR -'->$&A+1(@ ,$@ 7'&(>4&$:1&8 *-)'(0< =<$ &7(/-1 '-1++, /*. x I.12.4, \9&/)'&+1(+ QR -'->$&A+1(@ *+)&,&* 4'-C+1(8", /. 50,($( x I.13.4, \9&/)'&+1(+ QR -'->$&A+1(@ *+)&,&* &)'-A+1(8", /. 59.
-* A+7&,/3()-1- 4<3(/$()+$:1-@ /$&A1&/): ;)(6 -$%&'()*&4.x 9.1.2. "*% 5%@ QR -%:"F@ $"@ 5%!*"# %6'-//*&)'(* /$.3-8, 2&%,- *-)'(0- A 2 Mn 4 -$%&'()*+ 7&/)'&+1(@ QR -'->$&A+1(@ 7&3)( )'+.%&$:1-@:0a aBB a2111 a2212BBBBa32BBA=BBBBBBBB@: : : a1i: : : a2i.
. . ...... aiiai+1i::::::::::::......a1n 2a2n 2...ain 2...an 2n 2an 1n 2;;;;;;;a1n 1a2n 1...ain 1...an 2n 1an 1n 1ann 1;;;;;;;;a1na2n...ain...an 2nan 1nann;;1CCCCCCCCCCCCCCCCA"*% 5%@ QR -%:"F@ $"@ 5 %!*"#%6' $ &%?=&>1-3(* a1 = (a11 a21 0 : : : 0)t 2 Rn { 7+'4<8 /)&$=+0 *-)'(0< A . &-%$-/1& $+**+ I.12.3 /.C+/)4.+) *-)'(0- T12 = T12 ('12), )-2-@, 3)& T12 a1 = ka1k e1..
x9. QR "4&115(7'(3+* >1-3+1(+ .%$- '12 &7'+,+$@+)/@ $+**-*( I.12.2, I.12.3). "*1&A(* *-)'(0. A 1- T12 /$+4-, 7&$.3(* *-)'(0. A(1) = T12A ,012BB ka1 k r(1)a22BB(1)(1)BaA =B32BB@: : : r1n: : : a(1)2n: : : a(1)3n. . . ...a(1)nn;111;;1;r1 na(1)2na(1)3n...a(1)nn1CCCCCC :CCA(1)-$++ 7'&0+// 7'(*+1@+)/@ 2 7&,*-)'(0+ (a(1)ij )ij =2:::n .9./): /,+$-1< k ; 1 k = 1 : : : n ; 1 K-%&4 ;)&%& 7'&0+//-, ).+. *-)'(07'+&='->&4-1- 2 4(,.Y1A(k 1) =Tii+1A(2);i=k 1;%,+0 ka k rr1312BB 1 ka(1)1 k r23BBka(2)1kBBBBBA(k 1) = BBBBBBBBBB@: : : r1k 1 r1k: : : r2k 1 r2k: : : r3k 1 r3k.........ka(1k 2)k rk(k 11)kakka(kk+11)k;;;;;;;;:::::::::...:::::::::...r1nr2nr3n...111;;;rk 1n 1k 1)a(kn1a(kk+11)n 1...k 1)a(nn1;;;;;;;;r1nr2nr3n...rk 1na(knk 1)a(kk+11)n...a(nnk 1);;;;(>,+/: Q &>1-3-+), 3)& /&*1&A()+$( =+'.)/@ 4 7&'@,2+ k ; 1 : : : 1).i=k 1=&>1-3(*a(1k 1) = (a(kkk 1) a(kk+11)k 0 : : : 0)t 2 Rn k+11CCCCCCCCCCCC (3)CCCCCCCA1;;;;;(4){ 7+'4<8 /)&$=+0 7&,*-)'(0< (a(ijk 1) )ij=k:::n .
&%$-/1& $+**+ I.12.3 /.C+/)4.+)*-)'(0- Tkk+1 = Tkk+1('kk+1), )-2-@, 3)&;Tkk+1a(1k;= ka(1k1)1);k e(1n k+1);(5)(>1-3+1(@ .%$- 'kk+1 &7'+,+$@L)/@ $+**-*( I.12.2, I.12.3), >,+/: e(1m) =(1 0 : : : 0)t 2 Rm . "*1&A(* *-)'(0. (3) 1- Tkk+1 /$+4-, 7&$.3(*A(k)..= Tkk+1A(k 1);=Y1i=kTii+1A(6) x9. QR "4&116%,+ A(k) =0ka1k r(1)12 r13BBka1 k r(2)23BBBka1 kBBBBBBBBBBBBBBBBB@: : : r1k 1r1 k: : : r2k 1r2k: : : r3k 1r3 k.........ka(1k 2)k rk(k 11)kka1 kr1k+1 : : :r2k+1 : : :r3k+1 : : :......rk 1k+1 : : :rkk+1 : : :a(kk+1) k+1 : : :a(kk+2) k+1 : : :...;;;;;;;r1nr2nr3n...111;;;rk 1n 1rkn 1a(kk+1) n 1a(kk+2) n 1...k)a(nn1;;;;;;r1nr2nr3n...rk 1nrkn(k)ak+1na(kk+2) n...a(nnk);1CCCCCCCCCCCCCC :CCCCCCCCA(7))*+)(*, 3)& 4 (6) *-)'(0- ;$+*+1)-'1&%& 4'-C+1(@ Tkk+1 .*1&A-+)/@ )&$:2&1- 7&,*-)'(0.
(a(ijk 1) )ij=k:::n *-)'(0< A(k 1) '->*+'- n ; k + 1 (&/)-$:1-@3-/): A(k 1) 4 7'+&='->&4-1(( (6) 1+ .3-/)4.+)). $+,&4-)+$:1&, *-)'(0- A(k)7&$.3-+)/@ (> A(k 1) (>*+1+1(+* ,4.6 /)'&2 ( k -&8 ( (k +1)-&8) ,$(1< n ; k +1.9&/$+ n ; 1 K-%&4 ;)&%& 7'&0+//- ().+. 7+'+6&,- &) *-)'(0 (3) 2 (7)), *-)'(07'(*+) 4(,;;;;R = A(n1);=Y1i=n 1Tii+1A(8);%,+0r1312BB ka1 k kar(1)1 k r23BBBBka(2)1 kBR=BBBBBB@: : : r1n 2: : : r2n 2: : : r3n 2......ka(1n 3)k;;;;r1nr2nr3n...111;;;r1 nr2 nr3 n...rn 2n 1 rn 2nka(1n 2)k rn(n 11)nann;;;;;;1CCCCCCCCCCCCCA(9)(1-7&*1(*, &7'+,+$+1(@ 4+2)&'&4 a(1k 1) k = 1 : : : n ; 1 ,-L)/@ 4 (4), %,+ /3()-+*, 3)& a(0)1 = a1 ).1 = Tt-2 2-2 *-)'(0< 4'-C+1(@ &')&%&1-$:1<+, )& Tii+1ii+1 ( (> (8) 7&$.3-+*nY1A = Tiit +1R QR(10);;;i=1| (/2&*&+ QR -'->$&A+1(+.2% %6 Q R & 5@.