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A8AB5<5 0@001 >15A?5G8205BAO ?>445@6:0 A;54CNI8E B8?>2KE B>?>;>389: ?>;=K9 3@0D (completely-connected graph 8;8or clique)  A8AB5<0, 2 :>B>@>9 <564C ;N1>9 ?0@>9 ?@>F5AA>@>2 ACI5AB2C5B ?@O<0O ;8=8O A2O78; :0: @57C;LB0B, 40==0O B>?>;>38O >15A?5G8205B <8=8<0;L=K5 70B@0BK ?@8 ?5@540G5 40==KE, >4=0:> O2;O5BAO A;>6=> @50;87C5<>9 ?@8 1>;LH>< :>;8G5AB25 ?@>F5AA>@>2; ;8=59:0 (linear array 8;8or farm)  A8AB5<0, 2 :>B>@>9 :064K9 ?@>F5AA>@ 8<55B ;8=88 A2O78 B>;L:> A 42C<O A>A54=8<8 (A ?@54K4CI8< 8 ?>A;54CNI8<) ?@>F5AA>@0<8; B0:0O AE5<0 O2;O5BAO, A >4=>9 AB>@>=K, ?@>AB> @50;87C5<>9, 0 A 4@C3>9 AB>@>=K, A>>B25BAB2C5B AB@C:BC@5 ?5@540G8 40==KE ?@8 @5H5=88 <=>38E 2KG8A;8B5;L=KE 7040G (=0?@8<5@, ?@8 >@30=870F88 :>=2595@=KE 2KG8A;5=89); :>;LF> (ring)  40==0O B>?>;>38O ?>;CG05BAO 87 ;8=59:8 ?@>F5AA>@>2 A>548=5=85< ?5@2>3> 8 ?>A;54=53> ?@>F5AA>@>2 ;8=59:8; 725740 (star)  A8AB5<0, 2 :>B>@>9 2K45;5==K9 F5=B@0;L=K9 ?@>F5AA>@, C :>B>@>3> A ;N1K< ?@>F5AA>@>< ACI5AB2C5B ?@O<0O ;8=8O A2O78; 2A5 >AB0;L=K5 ?@>F5AAK 8<5NB A2O7L B>;L:> A F5=B@0;L=K<; :0: @57C;LB0B, 40==0O B>?>;>38O >15A?5G8205B <0;K5 70B@0BK ?@8 @50;870F88, => ?@8 1>;LH>< :>;8G5AB25 ?@>F5AA>@>2 2K7K205B >G5=L 1>;LHCN =03@C7:C =0 F5=B@0;L=K9 ?@>F5AA>@; @5H5B:0 (mesh)  A8AB5<0, 2 :>B>@>9 3@0D ;8=89 A2O78 >1@07C5B ?@O<>C3>;L=CN 42C<5@=CN A5B:C; ?>4>1=0O B>?>;>38O <>65B 1KBL 4>AB0B>G=> ?@>AB> @50;87>20=0 8, :@><5 B>3>, <>65B 1KBL MDD5:B82=> 8A?>;L7C5<0 ?@8 ?0@0;;5;L=>< 2K?>;=5=88 <=>38E G8A;5==KE 0;3>@8B<>2 (=0?@8<5@, ?@8 @50;870F88 <5B>4>2 1;>G=>3> C<=>65=8O <0B@8F); 38?5@:C1 (hypercube)  40==0O B>?>;>38O ?@54AB02;O5B G0AB=K9 A;CG09 AB@C:BC@K N-<5@=>9 @5H5B:8, :>340 ?> :064>9 @07<5@=>AB8 A5B:8 8<55BAO B>;L:> 420 ?@>F5AA>@0 (B.5. 38?5@:C1 A>45@68B 2N ?@>F5AA>@>2 ?@8 @07<5@=>AB8 N); 40==K9 20@80=B >@30=870F88 A5B8 ?5@540G8 40==KE 4>AB0B>G=> H8@>:> @0A?@>AB@0=5= 2 ?@0:B8:5 8 E0@0:B5@87C5BAO A;54CNI8< @O4>< >B;8G8B5;L=KE ?@87=0:>2: 420 ?@>F5AA>@0 8<5NB A>548=5=85, 5A;8 42>8G=>5 ?@54AB02;5=85 8E =><5@>2 8<55B B>;L:> >4=C @07;8G0NICNAO ?>78F8N; 2 N -<5@=>< 38?5@:C15 :064K9 ?@>F5AA>@ A2O70= @>2=> A N A>A54O<8; N -<5@=K9 38?5@:C1 <>65B 1KBL @0745;5= =0 420 (N 1)-<5@=KE 38?5@:C10 (2A53> 2>7<>6=> N @07;8G=KE B0:8E @07185=89); :@0BG09H89 ?CBL <564C 42C<O ;N1K<8 ?@>F5AA>@0<8 8<55B 4;8=C, A>2?040NICN A :>;8G5AB2>< @07;8G0NI8EAO 18B>2KE 7=0G5=89 2 =><5@0E ?@>F5AA>@>2 (40==0O 25;8G8=0 8725AB=0 :0: @0AAB>O=85 %M<<8=30). @028;0 8A?>;L7>20=8O A8AB5<K 0@001 1. 0?CA: A8AB5<K. ;O 70?CA:0 A8AB5<K 0@001 2K45;8B5 ?8:B>3@0<<C A8AB5<K 8 2K?>;=8B5 42>9=>9 I5;G>: ;52>9 :=>?:>9 <KH8 (8;8 =06<8B5 :;028HC Enter). 0;55 2K?>;=8B5 :><0=4C K?>;=8BL =>2K9 M:A?5@8<5=B (?C=:B <5=N 0G0;>) 8 =06<8B5 2 480;>3>2>< >:=5 0720=85 M:A?5@8<5=B0 :=>?:C  (?@8 65;0=88 4> =060B8O :=>?:8  <>65B 1KBL 87<5=5=> =0720=85 A>740205<>3> >:=0 4;O ?@>2545=8O M:A?5@8<5=B>2). 2. K1>@ B>?>;>388 2KG8A;8B5;L=>9 A8AB5<K. ;O 2K1>@0 B>?>;>388 2KG8A;8B5;L=>9 A8AB5<K A;54C5B 2K?>;=8BL :><0=4C ">?>;>38O ?C=:B0 <5=N !8AB5<0.  ?>O282H5<AO 480;>3>2>< >:=5 (@8A. 13.2) I5;:=8B5 ;52>9 :;028H59 <KH8 =0 ?8:B>3@0<<5 =C6=>9 B>?>;>388 8;8 2=87C 2 >1;0AB8 A>>B25BAB2CNI59 :@C3;>9 :=>?:8 2K1>@0 (@048>:=>?:8). @8 =060B88 :=>?:8 !?@02:0 <>6=> ?>;CG8BL A?@02>G=CN 8=D>@<0F8N > @50;87>20==KE B>?>;>38OE. 06<8B5 :=>?:C  4;O ?>4B25@645=8O 2K1>@0 8 :=>?:C B<5=0 4;O 2>72@0B0 2 >A=>2=>5 <5=N A8AB5<K 0@001.  80;>3>2>5 >:=> 4;O 2K1>@0 B>?>;>388 040=85 :>;8G5AB20 ?@>F5AA>@>2 8 O45@ ;O 2K1@0==>9 B>?>;>388 A8AB5<0 0@001 ?>72>;O5B CAB0=>28BL =5>1E>48<>5 :>;8G5AB2> ?@>F5AA>@>2 8 O45@. K?>;=O5<K9 ?@8 MB>< 2K1>@ :>=D83C@0F88 A8AB5<K >ACI5AB2;O5BAO 2 A>>B25BAB288 A B8?>< 8A?>;L7C5<>9 B>?>;>388. "0:, G8A;> ?@>F5AA>@>2 2 42CE<5@=>9 @5H5B:5 4>;6=> O2;OBLAO ?>;=K< :204@0B>< (@07<5@K @5H5B:8 ?> 3>@87>=B0;8 8 25@B8:0;8 A>2?040NB), 0 G8A;> ?@>F5AA>@>2 2 38?5@:C15  AB5?5=LN G8A;0 2.  40==>9 @50;870F88 A8AB5<K O45@ =0 ?@>F5AA>@5 <>65B 1KBL >4=>, 420 8;8 G5BK@5. >4 ?@>872>48B5;L=>ABLN ?@>F5AA>@0 (:064>3> O4@0) 2 A8AB5<5 0@001 ?>=8<05BAO :>;8G5AB2> >?5@0F89 A ?;020NI59 70?OB>9, :>B>@>5 ?@>F5AA>@ <>65B 2K?>;=8BL 70 A5:C=4C (floating point operations per second  flops). 06=> >B<5B8BL, GB> ?@8 ?>AB@>5=88 >F5=>: 2@5<5=8 2K?>;=5=8O M:A?5@8<5=B0 ?@54?>;0305BAO, GB> 2A5 <0H8==K5 :><0=4K O2;ONBAO >48=0:>2K<8 8 A>>B25BAB2CNB >4=>9 8 B>9 65 >?5@0F88 A ?;020NI59 B>G:>9. @028;0 8A?>;L7>20=8O A8AB5<K 0@001 1. 040=85 :>;8G5AB20 ?@>F5AA>@>2 8 O45@. ;O 2K1>@0 G8A;0 ?@>F5AA>@>2 8 O45@ =0 ?@>F5AA>@5 =5>1E>48<> 2K?>;=8BL :><0=4C >;8G5AB2> @>F5AA>@>2 ?C=:B0 <5=N !8AB5<0.  ?>O282H5<AO 480;>3>2>< >:=5 (@8A. 13.3) 0< ?@54>AB02;O5BAO =5A:>;L:> ?8:B>3@0<< A> AE5<0B8G5A:8< 87>1@065=85< G8A;0 C7;>2 2KG8A;8B5;L=>9 A8AB5<K. ;O :064>3> C7;0 5ABL 2>7<>6=>ABL 7040BL :>;8G5AB2> ?@>F5AA>@>2 8 O45@ =0 ?@>F5AA>@5. ;O 2K1>@0 I5;:=8B5 ;52>9 :;028H59 <KH8 =0 =C6=>9 ?8:B>3@0<<5. 8:B>3@0<<0, A>>B25BAB2CNI0O B5:CI5<C G8A;C C7;>2, 2K45;5=0 O@:>-A8=8< F25B><. 06<8B5 :=>?:C  4;O ?>4B25@645=8O 2K1>@0 8;8 :=>?:C B<5=0 4;O 2>72@0B0 2 >A=>2=>5 <5=N A8AB5<K 0@001 157 87<5=5=8O G8A;0 ?@>F5AA>@>2. 2. ?@545;5=85 ?@>872>48B5;L=>AB8 O45@ ?@>F5AA>@0. ;O 7040=8O ?@>872>48B5;L=>AB8 O45@ ?@>F5AA>@>2, A>AB02;ONI8E <=>3>?@>F5AA>@=CN 2KG8A;8B5;L=CN A8AB5<C, A;54C5B 2K?>;=8BL :><0=4C @>872>48B5;L=>ABL @>F5AA>@>2 ?C=:B0 <5=N !8AB5<0. 0;55 2 ?>O282H5<AO 480;>3>2>< >:=5 (@8A. 13.4) ?@8 ?><>I8 153C=:0 8;8 ?>;O 22>40 7040BL 25;8G8=C ?@>872>48B5;L=>AB8. ;O ?>4B25@645=8O 2K1>@0 =06<8B5 :=>?:C  (8;8 :;028HC Enter). ;O 2>72@0B0 2 >A=>2=>5 <5=N A8AB5<K 0@001 157 87<5=5=89 =06<8B5 :=>?:C B<5=0 (8;8 :;028HC Escape).  80;>3>2>5 >:=> 4;O 7040=8O G8A;0 ?@>F5AA>@>2  80;>3>2>5 >:=> 4;O 7040=8O ?@>872>48B5;L=>AB8 ?@>F5AA>@0 040=85 E0@0:B5@8AB8: A5B8 @5<O ?5@540G8 40==KE <564C ?@>F5AA>@0<8 >?@545;O5B :><<C=8:0F8>==CN A>AB02;ONICN (communication overhead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store-and-forward routing 8;8or SFR). @8 B0:>< ?>4E>45 ?@>F5AA>@, A>45@60I89 8AE>4=>5 A>>1I5=85, 3>B>28B 25AL >1J5< 40==KE 4;O ?5@540G8, >?@545;O5B B@0=78B=K9 ?@>F5AA>@, G5@57 :>B>@K9 40==K5 <>3CB 1KBL 4>AB02;5=K F5;52><C ?@>F5AA>@C, 8 70?CA:05B >?5@0F8N ?5@5AK;:8 40==KE. @>F5AA>@, :>B>@><C =0?@02;5=> A>>1I5=85, 2 ?5@2CN >G5@54L >ACI5AB2;O5B ?@85< ?>;=>ABLN 2A5E ?5@5AK;05<KE 40==KE 8 B>;L:> 70B5< ?@8ABC?05B : ?5@5AK;:5 ?@8=OB>3> A>>1I5=8O 40;55 ?> <0@H@CBC. @5<O ?5@5AK;:8 40==KE T 4;O <5B>40 ?5@540G8 A>>1I5=8O @07<5@>< m ?> <0@H@CBC 4;8=>9 l >?@545;O5BAO 2K@065=85<: EMBED Equation.3. B>@>9 A?>A>1 :><<C=8:0F88 >A=>2K205BAO =0 ?@54AB02;5=88 ?5@5AK;05<KE A>>1I5=89 2 2845 1;>:>2 8=D>@<0F88 <5=LH53> @07<5@0 (?0:5B>2), 2 @57C;LB0B5 G53> ?5@540G0 40==KE <>65B 1KBL A2545=0 : ?5@540G5 ?0:5B>2 (). @8 B0:>< <5B>45 :><<C=8:0F88 (cut-through routing or CTR) B@0=78B=K9 ?@>F5AA>@ <>65B >ACI5AB2;OBL ?5@5AK;:C 40==KE ?> 40;L=59H5<C <0@H@CBC =5?>A@54AB25==> A@07C ?>A;5 ?@85<0 >G5@54=>3> ?0:5B0, =5 4>6840OAL 7025@H5=8O ?@85<0 40==KE 2A53> A>>1I5=8O. >;8G5AB2> ?5@540205<KE ?@8 MB>< ?0:5B>2 @02=>  EMBED Equation.3 , 345 V 5ABL @07<5@ ?0:5B0, 0 25;8G8=0 V0 >?@545;O5B >1J5< A;C651=KE 40==KE, ?5@540205<KE 2 :064>< ?0:5B5 (703>;>2>: ?0:5B0). 0: @57C;LB0B, 2@5<O ?5@540G8 A>>1I5=8O 2 MB>< A;CG05 A>AB028B  EMBED Equation.3  (A:>1:8  EMBED Equation.3  >1>7=0G0NB >?5@0F8N ?@82545=8O : F5;><C A 871KB:><). !@02=820O ?>;CG5==K5 2K@065=8O, <>6=> 70<5B8BL, GB> 2 A;CG05, :>340 4;8=0 <0@H@CB0 1>;LH5 548=8FK, <5B>4 ?5@540G8 ?0:5B>2 ?@82>48B : 1>;55 1KAB@>9 ?5@5AK;:5 40==KE; :@><5 B>3>, 40==K9 ?>4E>4 A=8605B ?>B@51=>ABL 2 ?0<OB8 4;O E@0=5=8O ?5@5AK;05<KE 40==KE 4;O >@30=870F88 ?@85<0-?5@540G8 A>>1I5=89, 0 4;O ?5@540G8 ?0:5B>2 <>3CB 8A?>;L7>20BLAO >4=>2@5<5==> @07=K5 :><<C=8:0F8>==K5 :0=0;K. @028;0 8A?>;L7>20=8O A8AB5<K 0@001 1. ?@545;5=85 E0@0:B5@8AB8: :><<C=8:0F8>==>9 A@54K. ;O >?@545;5=8O E0@0:B5@8AB8: A5B8 2K?>;=8B5 :><0=4C %0@0:B5@8AB8:8 A5B8 ?C=:B0 <5=N !8AB5<0.  >B:@K2H5<AO 480;>3>2>< >:=5 (@8A. 13.5) ?@8 ?><>I8 153C=:>2 <>6=> 7040BL 2@5<O =0G0;L=>9 ?>43>B>2:8 40==KE (;0B5=B=>ABL) 2 <8:@>A5:C=40E 8 ?@>?CA:=CN A?>A>1=>ABL :0=0;>2 A5B8 (109B/A). ;O ?>4B25@645=8O 2K1>@0 =06<8B5 :=>?:C . ;O 2>72@0B0 2 >A=>2=>5 <5=N A8AB5<K 0@001 157 87<5=5=8O MB8E ?0@0<5B@>2 =06<8B5 :=>?:C B<5=0.  80;>3>2>5 >:=> 4;O 7040=8O E0@0:B5@8AB8: A5B8 2. ?@545;5=85 <5B>40 ?5@540G8 40==KE. ;O >?@545;5=8O <5B>40 ?5@540G8 40==KE, :>B>@K9 1C45B 8A?>;L7>20BLAO ?@8 ?@>2545=88 2KG8A;8B5;L=>3> M:A?5@8<5=B0 8 ?@8 ?>AB@>5=88 2@5<5==KE E0@0:B5@8AB8:, =5>1E>48<> 2K?>;=8BL :><0=4C 5B>4 5@540G8 0==KE ?C=:B0 <5=N !8AB5<0.  >B:@K2H5<AO 480;>3>2>< >:=5 (@8A. 13.6) A;54C5B I5;:=CBL ;52>9 :;028H59 <KH8 2 >1;0AB8 @048>:=>?:8, :>B>@0O A>>B25BAB2C5B 65;05<><C <5B>4C ?5@540G8 40==KE. A;8 2K1@0= <5B>4 ?5@540G8 ?0:5B>2, ?@8 ?><>I8 153C=:>2 2>7<>6=> 7040BL 4;8=C ?0:5B0 8 4;8=C 703>;>2:0 ?0:5B0 2 109B0E. ;O ?>4B25@645=8O 2K1>@0 <5B>40 ?5@540G8 40==KE 8 53> ?0@0<5B@>2 =06<8B5 :=>?:C .  80;>3>2>5 >:=> 4;O 7040=8O <5B>40 ?5@540G8 40==KE 3. 025@H5=85 @01>BK A8AB5<K. ;O 7025@H5=8O @01>BK A8AB5<K 0@001 A;54C5B 2K?>;=8BL :><0=4C 025@H8BL (?C=:B <5=N @E82). >AB0=>2:0 2KG8A;8B5;L=>9 7040G8 8 2K1>@ ?0@0;;5;L=>3> <5B>40 @5H5=8O ;O ?0@0;;5;L=>3> @5H5=8O B5E 8;8 8=KE 2KG8A;8B5;L=KE 7040G ?@>F5AA 2KG8A;5=89 4>;65= 1KBL ?@54AB02;5= 2 2845 =01>@0 =57028A8<KE 2KG8A;8B5;L=KE ?@>F54C@, 4>?CA:0NI8E 2K?>;=5=85 =0 =57028A8<KE ?@>F5AA>@0E. 1I0O AE5<0 >@30=870F88 B0:8E 2KG8A;5=89 <>65B 1KBL ?@54AB02;5=0 A;54CNI8< >1@07><: @0745;5=85 ?@>F5AA0 2KG8A;5=89 =0 G0AB8, :>B>@K5 <>3CB 1KBL 2K?>;=5=K >4=>2@5<5==>, @0A?@545;5=85 2KG8A;5=89 ?> ?@>F5AA>@0<, >15A?5G5=85 2708<>459AB28O ?0@0;;5;L=> 2K?>;=O5<KE 2KG8A;5=89. >7<>6=K5 A?>A>1K ?>;CG5=8O <5B>4>2 ?0@0;;5;L=KE 2KG8A;5=89: @07@01>B:0 =>2KE ?0@0;;5;L=KE 0;3>@8B<>2, @0A?0@0;;5;820=85 ?>A;54>20B5;L=KE 0;3>@8B<>2. #A;>28O MDD5:B82=>AB8 ?0@0;;5;L=KE 0;3>@8B<>2: @02=><5@=0O 703@C7:0 ?@>F5AA>@>2 (>BACBAB285 ?@>AB>52), =87:0O 8=B5=A82=>ABL 2708<>459AB28O ?@>F5AA>@>2 (=57028A8<>ABL).  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K?>;=8B5 B@8 ?>A;54>20B5;L=KE M:A?5@8<5=B0 A 8A?>;L7>20=85< B@5E @07;8G=KE 0;3>@8B<>2 C<=>65=8O <0B@8FK =0 25:B>@  0;3>@8B<>2, >A=>20==KE =0 3>@87>=B0;L=><, 25@B8:0;L=>< 8 1;>G=>< @07185=88 <0B@8FK. !@02=8B5 2@5<5==K5 E0@0:B5@8AB8:8 0;3>@8B<>2, :>B>@K5 >B>1@060NBAO 2 ?@02>9 =86=59 G0AB8 >:=0. 7<5=8B5 >1J5< 8AE>4=KE 40==KE (2K?>;=8B5 :><0=4C 0@0<5B@K 7040G8 ?C=:B0 <5=N 040G0). !=>20 ?@>2548B5 M:A?5@8<5=BK. !@02=8B5 2@5<5==K5 E0@0:B5@8AB8:8 0;3>@8B<>2. 7<5=8B5 :>;8G5AB2> ?@>F5AA>@>2, CAB0=>28B5 :>;8G5AB2> C7;>2, @02=>5 16 (2K?>;=8B5 :><0=4C >;8G5AB2> ?@>F5AA>@>2 ?C=:B0 <5=N !8AB5<0). @>2548B5 2KG8A;8B5;L=K5 M:A?5@8<5=BK 8 A@02=8B5 2@5<5==K5 E0@0:B5@8AB8:8. !45;09B5 2K2>4K > A@02=8B5;L=>9 MDD5:B82=>9 <5B>4>2 <0B@8G=>-25:B>@=>3> C<=>65=8O. 0B@8G=>5 C<=>65=85 040G0 C<=>65=8O <0B@8FK =0 <0B@8FC >?@545;O5BAO A>>B=>H5=8O<8:  EMBED Equation.3  (4;O ?@>AB>BK 87;>65=8O <0B5@80;0 1C45< ?@54?>;030BL, GB> ?5@5<=>605<K5 <0B@8FK A 8 B O2;ONBAO :204@0B=K<8 8 8<5NB ?>@O4>:  EMBED Equation.3 ). 0: A;54C5B 87 ?@82545==KE A>>B=>H5=89, 2KG8A;8B5;L=0O A;>6=>ABL 7040G8 O2;O5BAO 4>AB0B>G=> 2KA>:>9 (>F5=:0 :>;8G5AB20 2K?>;=O5<KE >?5@0F89 8<55B ?>@O4>:  EMBED Equation.3 ). A=>2C 2>7<>6=>AB8 ?0@0;;5;L=KE 2KG8A;5=89 4;O <0B@8G=>3> C<=>65=8O A>AB02;O5B =57028A8<>ABL @0AG5B>2 4;O ?>;CG5=8O M;5<5=B>2 Aij @57C;LB8@CNI59 <0B@8FK C. "5< A0<K<, 2A5 M;5<5=BK <0B@8FK C <>3CB 1KBL 2KG8A;5=K ?0@0;;5;L=> ?@8 =0;8G88 n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mm). ">340 >?5@0F8N <0B@8G=>3> C<=>65=8O <0B@8F A 8 B 2 1;>G=>< 2845 <>6=> ?@54AB028BL A;54CNI8< >1@07><:  EMBED Equation.3 , 345 :064K9 1;>: Cij <0B@8FK C >?@545;O5BAO 2 A>>B25BAB288 A 2K@065=85<:  EMBED Equation.3 . >;CG5==K5 1;>:8 Cij B0:65 O2;ONBAO =57028A8<K<8 8, :0: @57C;LB0B, 2>7<>6=K9 ?>4E>4 4;O ?0@0;;5;L=>3> 2K?>;=5=8O 2KG8A;5=89 <>65B A>AB>OBL 2 2K?>;=5=88 @0AG5B>2, A2O70==KE A ?>;CG5=85< >B45;L=KE 1;>:>2 Cij, =0 @07=KE ?@>F5AA>@0E. @8<5=5=85 ?>4>1=>3> ?>4E>40 ?>72>;O5B ?>;CG8BL <=>385 MDD5:B82=K5 ?0@0;;5;L=K5 <5B>4K C<=>65=8O 1;>G=>-?@54AB02;5==KE <0B@8F.  A8AB5<5 0@001 @50;87>20=K ?0@0;;5;L=K9 0;3>@8B< C<=>65=8O <0B@8F ?@8 ;5=B>G=>9 AE5<5 @0745;5=8O 40==KE 8 420 <5B>40 (0;3>@8B<K $>:A0 8 M==>=0) 4;O 1;>G=>-?@54AB02;5==KE <0B@8F. >;55 ?>;=0O 8=D>@<0F8O >1 0;3>@8B<0E C<=>65=8O <0B@8F, @50;87>20==KE 2 A8AB5<5 0@001, A>45@68BAO 2 3;025 7. 1. 5=B>G=K9 0;3>@8B<. @8 ;5=B>G=>9 AE5<5 @0745;5=8O 40==KE 8AE>4=K5 <0B@8FK @071820NBAO =0 3>@87>=B0;L=K5 (4;O <0B@8FK A) 8 25@B8:0;L=K5 (4;O <0B@8FK B) ?>;>AK. >;CG05<K5 ?>;>AK @0A?@545;ONBAO ?> ?@>F5AA>@0<, ?@8 MB>< =0 :064>< 87 8<5NI53>AO =01>@0 ?@>F5AA>@>2 @0A?>;0305BAO B>;L:> ?> >4=>9 ?>;>A5 <0B@8F A 8 B. 5@5<=>65=85 ?>;>A (0 2K?>;=5=85 ?@>F5AA>@0<8 MB>9 >?5@0F88 <>65B 1KBL 2K?>;=5=> ?0@0;;5;L=>) ?@82>48B : ?>;CG5=8N G0AB8 1;>:>2 @57C;LB8@CNI59 <0B@8FK C. ;O 2KG8A;5=8O >AB02H8EAO 1;>:>2 <0B@8FK C A>G5B0=8O ?>;>A <0B@8F A 8 B =0 ?@>F5AA>@0E 4>;6=K 1KBL 87<5=5=K.  =081>;55 ?@>AB>< 2845 MB> <>65B 1KBL >15A?5G5=>, =0?@8<5@, ?@8 :>;LF52>9 B>?>;>388 2KG8A;8B5;L=>9 A5B8 (?@8 G8A;5 ?@>F5AA>@>2, @02=>< :>;8G5AB2C ?>;>A)  2 MB>< A;CG05 =5>1E>48<>5 4;O <0B@8G=>3> C<=>65=8O 87<5=5=85 ?>;>65=8O 40==KE <>65B 1KBL >15A?5G5=> F8:;8G5A:8< A4283>< ?>;>A <0B@8FK B ?> :>;LFC. >A;5 <=>3>:@0B=>3> 2K?>;=5=8O >?8A0==KE 459AB289 (:>;8G5AB2> =5>1E>48<KE ?>2B>@5=89 O2;O5BAO @02=K< G8A;C ?@>F5AA>@>2) =0 :064>< ?@>F5AA>@5 ?>;CG05BAO =01>@ 1;>:>2, >1@07CNI89 3>@87>=B0;L=CN ?>;>AC <0B@8FK C. 0AA<>B@5==0O AE5<0 2KG8A;5=89 ?>72>;O5B >?@545;8BL ?0@0;;5;L=K9 0;3>@8B< <0B@8G=>3> C<=>65=8O ?@8 ;5=B>G=>9 AE5<5 @0745;5=8O 40==KE :0: 8B5@0F8>==CN ?@>F54C@C, =0 :064>< H035 :>B>@>9 ?@>8AE>48B ?0@0;;5;L=>5 2K?>;=5=85 >?5@0F88 ?5@5<=>65=8O ?>;>A 8 ?>A;54CNI53> F8:;8G5A:>3> A42830 ?>;>A >4=>9 87 <0B@8F ?> :>;LFC. >4@>1=>5 >?8A0=85 ;5=B>G=>3> 0;3>@8B<0 ?@82>48BAO 2 3;025 7. 040=8O 8 C?@06=5=8O !>7409B5 2 A8AB5<5 0@001 =>2>5 >:=> 2KG8A;8B5;L=>3> M:A?5@8<5=B0. ;O MB>3> >:=0 2K15@8B5 7040GC <0B@8G=>3> C<=>65=8O (I5;:=8B5 ;52>9 :=>?:>9 <KH8 =0 AB@>:5 0B@8G=>5 C<=>65=85 ?C=:B0 <5=N 040G0). B:@>9B5 480;>3>2>5 >:=> 2K1>@0 B>?>;>388 8 2K15@5B5 B>?>;>38N :>;LF>. B:@>9B5 480;>3>2>5 >:=> 2K1>@0 <5B>40 8 C1548B5AL 2 B><, GB> 2K1@0= <5B>4 ;5=B>G=>3> C<=>65=8O <0B@8F. @>2548B5 =5A:>;L:> 2KG8A;8B5;L=KE M:A?5@8<5=B>2. 7CG8B5 7028A8<>ABL 2@5<5=8 2K?>;=5=8O 0;3>@8B<0 >B >1J5<0 8AE>4=KE 40==KE 8 >B :>;8G5AB20 C7;>2, ?@>F5AA>@>2 8 O45@. 2. ;>G=K5 0;3>@8B<K $>:A0 8 M==>=0. @8 1;>G=>< ?@54AB02;5=88 40==KE ?0@0;;5;L=0O 2KG8A;8B5;L=0O AE5<0 <0B@8G=>3> C<=>65=8O 2 =081>;55 ?@>AB>< 2845 <>65B 1KBL ?@54AB02;5=0, 5A;8 B>?>;>38O 2KG8A;8B5;L=>9 A5B8 8<55B 284 ?@O<>C3>;L=>9 @5H5B:8 (5A;8 @50;L=0O B>?>;>38O A5B8 8<55B 8=>9 284, ?@54AB02;5=85 A5B8 2 2845 @5H5B:8 <>6=> >15A?5G8BL =0 ;>38G5A:>< C@>2=5). A=>2=K5 ?>;>65=8O ?0@0;;5;L=KE <5B>4>2 4;O 1;>G=> ?@54AB02;5==KE <0B@8F A>AB>OB 2 A;54CNI5<: 064K9 87 ?@>F5AA>@>2 @5H5B:8 >B25G05B 70 2KG8A;5=85 >4=>3> 1;>:0 <0B@8FK C;  E>45 2KG8A;5=89 =0 :064>< 87 ?@>F5AA>@>2 @0A?>;0305BAO ?> >4=><C 1;>:C 8AE>4=KE <0B@8F A 8 ; @8 2K?>;=5=88 8B5@0F89 0;3>@8B<>2 1;>:8 <0B@8FK  ?>A;54>20B5;L=> A42830NBAO 24>;L AB@>: ?@>F5AA>@=>9 @5H5B:8, 0 1;>:8 <0B@8FK B ( 24>;L AB>;1F>2 @5H5B:8;  @57C;LB0B5 2KG8A;5=89 =0 :064>< 87 ?@>F5AA>@>2 2KG8A;O5BAO 1;>: <0B@8FK !, ?@8 MB>< >1I55 :>;8G5AB2> 8B5@0F89 0;3>@8B<0 @02=>  EMBED Equation.3  (345 @  G8A;> ?@>F5AA>@>2),  3;025 7 ?@82>48BAO ?>;=>5 >?8A0=85 ?0@0;;5;L=KE <5B>4>2 $>:A0 8 M==>=0 4;O C<=>65=8O 1;>G=>-?@54AB02;5==KE <0B@8F. 040=8O 8 C?@06=5=8O  0:B82=>< >:=5 2KG8A;8B5;L=>3> M:A?5@8<5=B0 A8AB5<K 0@001 CAB0=>28B5 B>?>;>38N 5H5B:0. K15@8B5 G8A;> C7;>2, @02=>5 452OB8. !45;09B5 B5:CI59 7040G59 MB>3> >:=0 7040GC <0B@8G=>3> C<=>65=8O. K15@8B5 <5B>4 $>:A0 C<=>65=8O <0B@8F 8 ?@>2548B5 2KG8A;8B5;L=K9 M:A?5@8<5=B. K15@8B5 0;3>@8B< M==>=0 <0B@8G=>3> C<=>65=8O 8 2K?>;=8B5 2KG8A;8B5;L=K9 M:A?5@8<5=B. @>=01;N409B5 @07;8G=K5 <0@H@CBK ?5@540G8 40==KE ?@8 2K?>;=5=88 0;3>@8B<>2. !@02=8B5 2@5<5==K5 E0@0:B5@8AB8:8 0;3>@8B<>2. 7<5=8B5 G8A;> C7;>2 2 B>?>;>388 5H5B:0 =0 H5AB=04F0BL. >A;54>20B5;L=> 2K?>;=8B5 2KG8A;8B5;L=K5 M:A?5@8<5=BK A 8A?>;L7>20=85< <5B>40 $>:A0 8 <5B>40 M==>=0. !@02=8B5 2@5<5==K5 E0@0:B5@8AB8:8 MB8E M:A?5@8<5=B>2. 5H5=85 A8AB5< ;8=59=KE C@02=5=89 !8AB5<K ;8=59=KE C@02=5=89 2>7=8:0NB ?@8 @5H5=88 @O40 ?@8:;04=KE 7040G, >?8AK205<KE A8AB5<0<8 =5;8=59=KE (B@0=AF5=45=B=KE), 48DD5@5=F80;L=KE 8;8 8=B53@0;L=KE C@02=5=89. =8 <>3CB ?>O2;OBLAO B0:65 2 7040G0E <0B5<0B8G5A:>3> ?@>3@0<<8@>20=8O, AB0B8AB8G5A:>9 >1@01>B:8 40==KE, 0??@>:A8<0F88 DC=:F89, ?@8 48A:@5B870F88 :@052KE 48DD5@5=F80;L=KE 7040G <5B>4>< :>=5G=KE @07=>AB59 8;8 <5B>4>< :>=5G=KE M;5<5=B>2 8 4@. 8=59=>5 C@02=5=85 A n =58725AB=K<8 x0, x1, & , xn 1 <>65B 1KBL >?@545;5=> ?@8 ?><>I8 2K@065=8O  EMBED Equation.3  345 25;8G8=K a0, a1, & , a -1 8 b ?@54AB02;ONB A>1>9 ?>AB>O==K5 7=0G5=8O. =>65AB2> n ;8=59=KE C@02=5=89  EMBED Equation.3  =07K205BAO A8AB5<>9 ;8=59=KE C@02=5=89 8;8 ;8=59=>9 A8AB5<>9.  1>;55 :@0B:>< (<0B@8G=><) 2845 A8AB5<0 <>65B 1KBL ?@54AB02;5=0 :0:  EMBED Equation.3 , 345 A=(aij) 5ABL 25I5AB25==0O <0B@8F0 @07<5@0 n(n, 0 25:B>@K b 8 x A>AB>OB 87 n M;5<5=B>2. >4 7040G59 @5H5=8O A8AB5<K ;8=59=KE C@02=5=89 4;O 7040==KE <0B@8FK  8 25:B>@0 b >1KG=> ?>=8<05BAO =0E>645=85 7=0G5=8O 25:B>@0 =58725AB=KE x, ?@8 :>B>@>< 2K?>;=ONBAO 2A5 C@02=5=8O A8AB5<K.  A8AB5<5 0@001 @50;87>20=K 420 0;3>@8B<0 @5H5=8O A8AB5< ;8=59=KE C@02=5=89: H8@>:> 8725AB=K9 <5B>4 0CAA0 (?@O<>9 <5B>4 @5H5=8O A8AB5< ;8=59=KE C@02=5=89, =0E>48B B>G=>5 @5H5=85 A8AB5<K A =52K@>645==>9 <0B@8F59 70 :>=5G=>5 G8A;> H03>2) 8 <5B>4 A>?@O65==KE 3@0485=B>2  >48= 87 H8@>:>3> :;0AA0 8B5@0F8>==KE <5B>4>2 @5H5=8O A8AB5< ;8=59=KE C@02=5=89 A <0B@8F59 A?5F80;L=>3> 2840. >;55 ?>;=0O 8=D>@<0F8O >1 0;3>@8B<0E @5H5=8O A8AB5< ;8=59=KE C@02=5=89, @50;87>20==KE 2 A8AB5<5 0@001, A>45@68BAO 2 3;025 8. 1. ;3>@8B< 0CAA0. 5B>4 0CAA0 2:;NG05B ?>A;54>20B5;L=>5 2K?>;=5=85 42CE MB0?>2. 0 ?5@2>< MB0?5  ?@O<>9 E>4 <5B>40 0CAA0  8AE>4=0O A8AB5<0 ;8=59=K9 C@02=5=89 ?@8 ?><>I8 ?>A;54>20B5;L=>3> 8A:;NG5=8O =58725AB=KE (?@8 ?><>I8 M:2820;5=B=KE ?@5>1@07>20=89) ?@82>48BAO : 25@E=5<C B@5C3>;L=><C 284C. 0 >1@0B=>< E>45 <5B>40 0CAA0 (2B>@>9 MB0? 0;3>@8B<0) >ACI5AB2;O5BAO >?@545;5=85 7=0G5=89 =58725AB=KE. 7 ?>A;54=53> C@02=5=8O ?@5>1@07>20==>9 A8AB5<K <>65B 1KBL 2KG8A;5=> 7=0G5=85 ?5@5<5==>9 xn-1, ?>A;5 MB>3> 87 ?@54?>A;54=53> C@02=5=8O AB0=>28BAO 2>7<>6=K< >?@545;5=85 ?5@5<5==>9 xn-2 8 B.4. @8 2=8<0B5;L=>< @0AA<>B@5=88 <5B>40 0CAA0 <>6=> 70<5B8BL, GB> 2A5 2KG8A;5=8O A2>4OBAO : >4=>B8?=K< 2KG8A;8B5;L=K< >?5@0F8O< =04 AB@>:0<8 <0B@8FK :>MDD8F85=B>2 A8AB5<K ;8=59=KE C@02=5=89. 0: @57C;LB0B, 2 >A=>2C ?0@0;;5;L=>9 @50;870F88 0;3>@8B<0 0CAA0 <>65B 1KBL ?>;>65= ?@8=F8? @0A?0@0;;5;820=8O ?> 40==K<.  MB>< A;CG05 <0B@8FC  <>6=> @0745;8BL =0 3>@87>=B0;L=K5 ?>;>AK, 0 25:B>@ ?@02KE G0AB59 b  =0 1;>:8. >;>AK <0B@8FK 8 1;>:8 25:B>@0 ?@02KE G0AB59 @0A?@545;ONBAO <564C ?@>F5AA>@0<8 2KG8A;8B5;L=>9 A8AB5<K. ;O 2K?>;=5=8O ?@O<>3> E>40 <5B>40 0CAA0 =5>1E>48<> >ACI5AB28BL (n 1) 8B5@0F8N ?> 8A:;NG5=8N =58725AB=KE 4;O ?@5>1@07>20=8O <0B@8FK :>MDD8F85=B>2 A : 25@E=5<C B@5C3>;L=><C 284C. K?>;=5=85 8B5@0F88 i, 0(i<n 1, ?@O<>3> E>40 <5B>40 0CAA0 2:;NG05B @O4 ?>A;54>20B5;L=KE 459AB289. @>F5AA>@, =0 :>B>@>< @0A?>;>65=0 AB@>:0 i <0B@8FK, 4>;65= @07>A;0BL 55 8 A>>B25BAB2CNI89 M;5<5=B 25:B>@0 b 2A5< ?@>F5AA>@0<, :>B>@K5 A>45@60B AB@>:8 A =><5@0<8 k, k>i. >;CG82 254CICN AB@>:C, ?@>F5AA>@K 2K?>;=ONB 2KG8B0=85 AB@>:, >15A?5G820O B5< A0<K< 8A:;NG5=85 A>>B25BAB2CNI59 =58725AB=>9 xi. @8 2K?>;=5=88 >1@0B=>3> E>40 <5B>40 0CAA0 ?@>F5AA>@K 2K?>;=ONB =5>1E>48<K5 2KG8A;5=8O 4;O =0E>645=8O 7=0G5=8O =58725AB=KE. 0: B>;L:> :0:>9-;81> ?@>F5AA>@ >?@545;O5B 7=0G5=85 A2>59 ?5@5<5==>9 xi, MB> 7=0G5=85 4>;6=> 1KBL @07>A;0=> 2A5< ?@>F5AA>@0<, :>B>@K5 A>45@60B AB@>:8 A =><5@0<8 k, k<i. 0;55 ?@>F5AA>@K ?>4AB02;ONB ?>;CG5==>5 7=0G5=85 =>2>9 =58725AB=>9 8 2K?>;=ONB :>@@5:B8@>2:C 7=0G5=89 4;O M;5<5=B>2 25:B>@0 b. 040=8O 8 C?@06=5=8O  0:B82=>< >:=5 2KG8A;8B5;L=>3> M:A?5@8<5=B0 A8AB5<K 0@001 CAB0=>28B5 B>?>;>38N >;=K9 @0D. K15@8B5 G8A;> C7;>2, @02=>5 45AOB8. !45;09B5 B5:CI59 7040G59 MB>3> >:=0 7040GC @5H5=8O A8AB5<K ;8=59=KE C@02=5=89. K15@8B5 <5B>4 0CAA0 @5H5=8O 7040G8 8 2K?>;=8B5 2KG8A;8B5;L=K9 M:A?5@8<5=B. @>=01;N409B5 <0@H@CBK ?5@540G8 40==KE ?@8 2K?>;=5=88 0;3>@8B<0. 2. 5B>4 A>?@O65==KE 3@0485=B>2. 5B>4 A>?@O65==KE 3@0485=B>2  >48= 87 H8@>:>3> :;0AA0 8B5@0F8>==KE <5B>4>2 @5H5=8O A8AB5< ;8=59=KE C@02=5=89, :>B>@K9 <>65B 1KBL ?@8<5=5= 4;O @5H5=8O A8<<5B@8G=>9 ?>;>68B5;L=> >?@545;5==>9 A8AB5<K ;8=59=KE C@02=5=89 1>;LH>9 @07<5@=>AB8. @8 2K?>;=5=88 8B5@0F89 <5B>40 A>?@O65==KE 3@0485=B>2 : 8A:><><C B>G=><C @5H5=8N x* A8AB5<K Ax=b AB@>8BAO ?>A;54>20B5;L=>ABL ?@81;865==KE @5H5=89 x0, x1,& , xk,& @8 MB>< ?@>F5AA 2KG8A;5=89 >@30=87C5BAO B0:8< A?>A>1><, GB> :064>5 >G5@54=>5 ?@81;865=85 405B >F5=:C B>G=>3> @5H5=8O A> 2A5 C<5=LH0NI59AO ?>3@5H=>ABLN, 8 ?@8 ?@>4>;65=88 @0AG5B>2 >F5=:0 B>G=>3> @5H5=8O <>65B 1KBL ?>;CG5=0 A ;N1>9 B@51C5<>9 B>G=>ABLN. B5@0F8O <5B>40 A>?@O65==KE 3@0485=B>2 A>AB>8B 2 2KG8A;5=88 >G5@54=>3> ?@81;865=8O : B>G=><C @5H5=8N 2 A>>B25BAB288 A ?@028;><:  EMBED Equation.3 , 2 A>>B25BAB288 A :>B>@K< =>2>5 7=0G5=85 ?@81;865=8O xk 2KG8A;O5BAO A CG5B>< ?@81;865=8O, ?>AB@>5==>3> =0 ?@54K4CI5< H035 xk 1, A:0;O@=>3> H030 sk 8 25:B>@0 =0?@02;5=8O dk. 5@54 2K?>;=5=85< ?5@2>9 8B5@0F88 25:B>@0 x0 8 d0 ?>;030NBAO @02=K<8 =C;N, 0 4;O 25:B>@0 g0 CAB0=02;8205BAO 7=0G5=85 @02=>5  b. 0;55 :0640O 8B5@0F8O 4;O 2KG8A;5=8O >G5@54=>3> 7=0G5=8O ?@81;865=8O xk 2:;NG05B 2K?>;=5=85 G5BK@5E H03>2: (03 1: KG8A;5=85 3@0485=B0:  EMBED Equation.3 ; (03 2: KG8A;5=85 25:B>@0 =0?@02;5=8O:  EMBED Equation.3 , 345 (gT,g) 5ABL A:0;O@=>5 ?@>872545=85 25:B>@>2; (03 3: KG8A;5=85 25;8G8=K A<5I5=8O ?> 2K1@0==><C =0?@02;5=8N:  EMBED Equation.3 ; (03 4: KG8A;5=85 =>2>3> ?@81;865=8O:  EMBED Equation.3 .  F5;><, A;>6=>ABL 0;3>@8B<0 8<55B ?>@O4>: O(n3). @8 @07@01>B:5 ?0@0;;5;L=>3> 20@80=B0 <5B>40 A>?@O65==KE 3@0485=B>2 4;O @5H5=8O A8AB5< ;8=59=KE C@02=5=89 2 ?5@2CN >G5@54L A;54C5B CG5ABL, GB> 2K?>;=5=85 8B5@0F89 <5B>40 >ACI5AB2;O5BAO ?>A;54>20B5;L=> 8, B5< A0<K<, =081>;55 F5;5A>>1@07=K9 ?>4E>4 A>AB>8B 2 @0A?0@0;;5;820=88 2KG8A;5=89, @50;87C5<KE 2 E>45 2K?>;=5=8O >B45;L=KE 8B5@0F89. A=>2=K5 2KG8A;5=8O, 2K?>;=O5<K5 2 A>>B25BAB288 A <5B>4><, ?@54AB02;ONB A>1>9 >?5@0F88 <0B@8G=>-25:B>@=>3> C<=>65=8OA>AB>OB 2 C<=>65=88 <0B@8FK A =0 25:B>@K x 8 d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n 7=0G5=89; 0;3>@8B<K A <5=LH59 B@C4>5<:>ABLN <>3CB 1KBL ?>;CG5=K B>;L:> 4;O G0AB=KE 20@80=B>2 7040G8. #A:>@5=85 A>@B8@>2:8 <>65B 1KBL >15A?5G5=> ?@8 8A?>;L7>20=88 =5A:>;L:8E (p, p>1) ?@>F5AA>@>2. AE>4=K9 C?>@O4>G8205<K9 =01>@ 2 MB>< A;CG05 @0745;O5BAO <564C ?@>F5AA>@0<8; 2 E>45 A>@B8@>2:8 40==K5 ?5@5AK;0NBAO <564C ?@>F5AA>@0<8 8 A@02=820NBAO <564C A>1>9. 57C;LB8@CNI89 (C?>@O4>G5==K9) =01>@, :0: ?@028;>, B0:65 @0745;5= <564C ?@>F5AA>@0<8, ?@8 MB>< 4;O A8AB5<0B870F88 B0:>3> @0745;5=8O 4;O ?@>F5AA>@>2 22>48BAO B0 8;8 8=0O A8AB5<0 ?>A;54>20B5;L=>9 =C<5@0F88 8 >1KG=> B@51C5BAO, GB>1K ?@8 7025@H5=88 A>@B8@>2:8 7=0G5=8O, @0A?>;0305<K5 =0 ?@>F5AA>@0E A <5=LH8<8 =><5@0<8, =5 ?@52KH0;8 7=0G5=89 ?@>F5AA>@>2 A 1>;LH8<8 =><5@0<8.  A8AB5<5 0@001 2 :0G5AB25 <5B>4>2 C?>@O4>G5=8O 40==KE ?@54AB02;5=K ?C7K@L:>20O A>@B8@>2:0, A>@B8@>2:0 (5;;0, 1KAB@0O A>@B8@>2:0. AE>4=K5 (?>A;54>20B5;L=K5) 20@80=BK MB8E <5B>4>2 87;>65=K 2> <=>38E 8740=8OE (A<., =0?@8<5@, [71]), A?>A>1K 8E ?0@0;;5;L=>3> 2K?>;=5=8O 87;030NBAO 2 3;025 9. 1. C7K@L:>20O A>@B8@>2:0. ;3>@8B< ?C7K@L:>2>9 A>@B8@>2:8 2 ?@O<>< 2845 4>AB0B>G=> A;>65= 4;O @0A?0@0;;5;820=8O: A@02=5=85 ?0@ A>A54=8E M;5<5=B>2 ?@>8AE>48B AB@>3> ?>A;54>20B5;L=>. 0@0;;5;L=K9 20@80=B 0;3>@8B<0 >A=>2K205BAO =0 <5B>45 G5B  =5G5B=>9 ?5@5AB0=>2:8, 4;O :>B>@>3> =0 =5G5B=>9 8B5@0F88 2K?>;=5=8O A@02=820NBAO M;5<5=BK ?0@  EMBED Equation.DSMT4  A;8 ?0@0 =5 C?>@O4>G5=0, B> 55 M;5<5=BK ?5@5AB02;ONBAO. 0 G5B=>9 8B5@0F88 C?>@O4>G820NBAO ?0@K  EMBED Equation.DSMT4  >A;5 n-:@0B=>3> ?>2B>@5=8O ?>4>1=KE 8B5@0F89 <0AA82 >:07K205BAO >BA>@B8@>20==K<. >;55 ?>4@>1=0O 8=D>@<0F8O > ?0@0;;5;L=>< 20@80=B5 0;3>@8B<0 ?@82>48BAO 2 3;025 9. 040=8O 8 C?@06=5=8O 1. 0?CAB8B5 A8AB5<C 0@001 8 A>7409B5 =>2K9 M:A?5@8<5=B. K15@8B5 ?C=:B <5=N 040G0 8 C1548B5AL, GB> 2 0:B82=>< >:=5 B5:CI59 7040G59 O2;O5BAO 7040G0 A>@B8@>2:8. B:@>9B5 480;>3>2>5 >:=> 2K1>@0 <5B>40 8 ?>A<>B@8B5, :0:85 0;3>@8B<K A>@B8@>2:8 <>3CB 1KBL 2K?>;=5=K =0 B5:CI59 B>?>;>388. 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">G:8 87 >1;0AB8 ?>8A:0, C4>2;5B2>@ONI85 2A5< >3@0=8G5=8O<, =07K20NBAO 4>?CAB8<K<8 B>G:0<8 8;8 4>?CAB8<K<8 @5H5=8O<8. =>65AB2> Q((y( y(D, gj(y)(0, 1(j(m( 2A5E B0:8E B>G5: =07K20NB 4>?CAB8<>9 >1;0ABLN. 06=K9 2 ?@8:;04=>< >B=>H5=88 ?>4:;0AA 7040G 2840 E0@0:B5@87C5BAO B5<, GB> 2A5 DC=:F8>=0;K, 2E>4OI85 2 >?@545;5=85 7040G8, 7040=K =5:>B>@K<8 (?@>3@0<<=> @50;87C5<K<8) 0;3>@8B<0<8 2KG8A;5=8O 7=0G5=89 ((y), gj(y), 1(j(m, 2 B>G:0E >1;0AB8 ?>8A:0 D. @8 MB>< @5H5=85 7040G8 A2>48BAO : ?>AB@>5=8N >F5=:8  EMBED Equation.3 , >B25G0NI59 =5:>B>@><C ?>=OB8N 1;87>AB8 : B>G:5 y* (=0?@8<5@, GB>1K  EMBED Equation.3  8;8  EMBED Equation.3 , 345 (>0 5ABL 7040==0O B>G=>ABL) =0 >A=>25 =5:>B>@>3> G8A;0 k 7=0G5=89 DC=:F8>=0;>2 7040G8, 2KG8A;5==KE 2 B>G:0E >1;0AB8 D.  7040G0E <=>3>M:AB@5<0;L=>9 >?B8<870F88 2>7<>6=>ABL 4>AB>25@=>9 >F5=:8 3;>10;L=>3> >?B8<C<0 ?@8=F8?80;L=> >A=>20=0 =0 =0;8G88 0?@8>@=>9 8=D>@<0F88 > DC=:F88, ?>72>;ONI59 A2O70BL 2>7<>6=K5 7=0G5=8O <8=8<878@C5<>9 DC=:F88 A 8725AB=K<8 7=0G5=8O<8 2 B>G:0E >ACI5AB2;5==KE ?>8A:>2KE 8B5@0F89. 5AL<0 G0AB> B0:0O 0?@8>@=0O 8=D>@<0F8O > 7040G5 ?@54AB02;O5BAO 2 2845 ?@54?>;>65=8O, GB> F5;520O DC=:F8O ( (2 40;L=59H5< >1>7=0G05<0O B0:65 gm(() 8 ;52K5 G0AB8 >3@0=8G5=89 gj,1(j(m, C4>2;5B2>@ONB CA;>28N 8?H8F0 A A>>B25BAB2CNI8<8 :>=AB0=B0<8 Lj,1(j(m((, 0 8<5==> (gj(y1)(gj(y2)((Lj||y1(y2||, 1(j(m((, y1,y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A;CG05, :>340 B>G:0 yk 4>?CAB8<0, B. 5. :>340 ys(xk)(Qm+1, 8A?KB0=85 2:;NG05B 2KG8A;5=85 7=0G5=89 2A5E DC=:F8>=0;>2 7040G8. @8 MB>< 7=0G5=85 8=45:A0 ?@8=8<05BAO @02=K< 25;8G8=5 ((m((, 0 B@>9:0 ys(xk), ((((xk), zk(g((ys(xk)), O2;O5BAO @57C;LB0B>< 8A?KB0=8O 2 B>G:5 xk. 4. A;8 ((xk)>0, B> 5ABL yk(D, B> >?@545;8BL ?@>>1@07K xkl([0,1],0(l(L, B>G:8 yk, 8 8=B5@?@5B8@>20BL 8A?KB0=85, ?@>2545==>5 2 B>G:5 yk(D, :0: ?@>2545=85 8A?KB0=89 2 L+1 B>G:5 xk0, xk1,& , xkL, A >48=0:>2K<8 @57C;LB0B0<8 ((xk0)(((xk1)(& (((xkL)(((xk), g((y0(xk0))(g((y1(xk1))(& (g((yL(xkL))(zk. @>8=D>@<8@>20BL >AB0;L=K5 C7;K > @57C;LB0B0E 8A?KB0=8O 2 B>G:5 yk.  A;CG05 5A;8 ((xk)(0, B> 5ABL yk(D, @57C;LB0B 8A?KB0=8O >B=>A8BAO B>;L:> : s-9 7040G5. 064K9 C75; 8<55B A2>N :>?8N ?@>3@0<<=KE A@54AB2, @50;87CNI8E 2KG8A;5=85 DC=:F8>=0;>2 7040G8, 8 @5H0NI55 ?@028;> 0;3>@8B<0. ;O >@30=870F88 2708<>459AB28O =0 :064>< C7;5 A>7405BAO L+1 >G5@54L, 2 :>B>@K5 ?@>F5AA>@K ?><5I0NB 8=D>@<0F8N > 2K?>;=5==KE 8B5@0F8OE 2 2845 B@>5:: B>G:0 >G5@54=>9 8B5@0F88, 8=45:A 8 7=0G5=85, ?@8G5< 8=45:A 701;>:8@>20==>9 B>G:8 ?>;0305BAO @02=K<  1, 0 7=0G5=85 DC=:F88 2 =59 =5 >?@545;5=>. @54;0305<0O AE5<0 =5 A>45@68B :0:>3>-;81> 548=>3> C?@02;ONI53> C7;0, GB> C25;8G8205B =0456=>ABL 2K?>;=O5<KE 2KG8A;5=89. ;O @5H5=8O 7040G8 >?B8<870F88 =5>1E>48<> 7040BL ?0@0<5B@K, C:070==K5 2 480;>3>2>< >:=5 0@0<5B@K 7040G8 (A<. @8A. 13.14).  040=85 ?0@0<5B@>2 7040G8 <=>3>M:AB@5<0;L=>9 >?B8<870F88 040=8O 8 C?@06=5=8O 0?CAB8B5 A8AB5<C 0@001.  0:B82=>< >:=5 2KG8A;8B5;L=>3> M:A?5@8<5=B0 CAB0=>28B5 B>?>;>38N >;=K9 3@0D. 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A;8 87CG0NBAO ?0@0;;5;L=K5 0;3>@8B<K <0B@8G=>3> C<=>65=8O, B> =0 :>=25@B5 87>1@0605BAO =><5@ 1;>:0, :>B>@K9 ?5@5AK;05BAO. @8 2K?>;=5=88 0;3>@8B<>2 =0 3@0D0E 2A5 8B5@0F88 ?0@0;;5;L=>3> 0;3>@8B<0 >4=>B8?=K 8 G8A;> 8E 25;8:> (@02=> G8A;C 25@H8= 2 3@0D5). ;O >B>1@065=8O 2A5E 8B5@0F89 ?>=04>18BAO 4>AB0B>G=> <=>3> 2@5<5=8. >MB><C 2 A8AB5<5 0@001 @50;87>20=0 2>7<>6=>ABL >B>1@060BL =5 2A5 8B5@0F88, 0 ;8HL =5:>B>@K5 87 =8E. @028;0 8A?>;L7>20=8O A8AB5<K 0@001 1. 7<5=5=85 A?>A>10 >B>1@065=8O ?5@5AK;:8 40==KE. ;O 7040=8O A?>A>10 >B>1@065=8O :><<C=8:0F88 ?@>F5AA>@>2 2K?>;=8B5 :><0=4C 5@5AK;:0 40==KE ?C=:B0 <5=N 87C0;870F8O.  ?>O282H5<AO A?8A:5 2K45;8B5 =0720=85 65;05<>3> A?>A>10 >B>1@065=8O.  K1>@ A?>A>10 >B>1@065=8O ?5@5AK;:8 40==KE 2. K1>@ B5<?0 45<>=AB@0F88. ;O 2K1>@0 B5<?0 45<>=AB@0F88 =5>1E>48<> 2K?>;=8BL :><0=4C "5<? 5<>=AB@0F88 ?C=:B0 <5=N 87C0;870F8O.  ?>O282H5<AO 480;>3>2>< >:=5 (@8A. 13.17) ?@54>AB02;O5BAO 2>7<>6=>ABL 2K1>@0 25;8G8=K 7045@6:8 <564C 8B5@0F8O<8 0;3>@8B<0 8 A:>@>AB8 42865=8O ?0:5B>2 (2@5<5=8 F25B>2>3> 2K45;5=8O :0=0;0) ?@8 >B>1@065=88 :><<C=8:0F88 ?@>F5AA>@>2.  80;>3>2>5 >:=> 2K1>@0 B5<?0 45<>=AB@0F88 06<8B5  (Enter) 4;O ?>4B25@645=8O 2K1>@0 B5<?0 45<>=AB@0F88. ;O 2>72@0B0 2 >A=>2=>5 <5=N A8AB5<K 0@001 157 A>E@0=5=8O 87<5=5=89 =06<8B5 B<5=0 (Escape). 3. 7<5=5=85 H030 287C0;870F88. ;O B>3>, GB>1K 2 @01>G59 >1;0AB8 A8AB5<K 0@001 >B>1@060;0AL =5 :0640O 8B5@0F8O, 0 ;8HL =5:>B>@K5 87 =8E, 2K?>;=8B5 :><0=4C >;8G5AB2> 8B5@0F89 =0 >4=C 287C0;870F8N ?C=:B0 <5=N 87C0;870F8O.  ?>O282H5<AO 480;>3>2>< >:=5 CAB0=>28B5 ?@8 ?><>I8 ?>;7C=:0 153C=:0 65;05<CN G0AB>BC >B>1@065=8O 8B5@0F89. ;O 2K1>@0 H030 287C0;870F88 =06<8B5 OK (Enter), 4;O 2>72@0B0 2 >A=>2=>5 <5=N A8AB5<K 0@001 =06<8B5 B<5=0 (Escape) 4. 0AB@>9:0 F25B>2>9 ?0;8B@K. ;O 87<5=5=8O F25B>2, :>B>@K5 8A?>;L7CNBAO 2 A8AB5<5 0@001 4;O 287C0;870F88 ?@>F5AA0 @5H5=8O 7040G, 2K?>;=8B5 :><0=4C 0@0<5B@K 287C0;870F88 ?C=:B0 <5=N 87C0;870F8O.  ?>O282H5<AO 480;>3>2>< >:=5 (@8A. 13.18) ;O :064>3> 0;3>@8B<0 <>6=> 7040BL F25B0 >B>1@065=8O @07;8G=KE A>AB>O=89 40==KE, 0 B0:65 <5B>4 >B>1@065=8O @57C;LB0B>2 (42CE- 8;8 B@5E- <5@=K9). 0 ?@82545==>< @8AC=:5 ?@54AB02;5= 480;>3 4;O 1;>G=>3> ?5@5<=>65=8O <0B@8F. !>>B25BAB25==>, <0B@8FC <>6=> >B>1@078BL B>;L:> =0 42CE<5@=>9 >1;0AB8 8 5ABL 2>7<>6=>ABL 7040BL F25B 4;O 2KG8A;5==KE 1;>:>2 8 1;>:>2, 5I5 ?>4;560I8E 2KG8A;5=8N. 'B>1K 87<5=8BL F25B0, I5;:=8B5 ;52>9 :;028H59 <KH8 =0 F25B=>9 :=>?:5 A>>B25BAB2CNI53> ?0@0<5B@0.  @57C;LB0B5 ?>O28BAO AB0=40@B=>5 480;>3>2>5 >:=> 2K1>@0 F25B0 >?5@0F8>==>9 A8AB5<K Windows.  MB>< >:=5 2K15@8B5 F25B 8 =06<8B5 :=>?:C OK. &25B 1C45B 87<5=5=. ;O B>3>, GB>1K 87<5=8BL F25B 2K45;5=8O, I5;:=8B5 ;52>9 :;028H59 <KH8 =0 >B>1@060NI5< MB>B F25B :204@0B5. @8 ?><>I8 AB0=40@B=>3> 480;>3>2>3> >:=0 70409B5 =5>1E>48<K9 F25B.  80;>3>2>5 >:=> =0AB@>9:8 287C0;870F88 ;O 87<5=5=8O F25B>2>9 ?0;8B@K =06<8B5 :=>?:C  (Enter) 2 480;>3>2>< >:=5 0@0<5B@K 287C0;870F88. ;O 2>72@0B0 2 >A=>2=>9 @568< @01>BK A8AB5<K 0@001 =06<8B5 B<5=0 (Escape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b. 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"0:8< >1@07><, 2A5 1;>:8 25:B>@0 AB0=>2OBAO B5<=>-A8=8<8. @8 2K?>;=5=88 0;3>@8B<0, >A=>20==>3> =0 ;5=B>G=>< 25@B8:0;L=>< @07185=88 <0B@8FK, :064>5 O4@> 2KG8A;O5B 25:B>@ G0AB8G=KE @57C;LB0B>2 ?CB5< C<=>65=8O ?>;>AK <0B@8FK =0 1;>: 25:B>@0-0@3C<5=B0 b; 2A5 1;>:8 @57C;LB8@CNI53> 25:B>@0 2 >1;0AB8 57C;LB0B C<=>65=8O <0B@8FK =0 25:B>@ ?>4A25G820NBAO A25B;>-A8=8< F25B><. >A;5 2K?>;=5=8O :><<C=8:0F8>==>3> H030 =0 :064>< O4@5 @0A?>;0305BAO 1;>: @57C;LB8@CNI53> 25:B>@0, 1;>: 0:B82=>3> ?@>F5AA>@0 >B>1@0605BAO 2 >1;0AB8 57C;LB0B C<=>65=8O <0B@8FK =0 25:B>@ B5<=>-A8=8< F25B><. @8 2K?>;=5=88 0;3>@8B<0, >A=>20==>3> =0 1;>G=>< @07185=88 <0B@8FK, 25:B>@ b @0A?@545;5= <564C C7;0<8, A>AB02;ONI8<8 AB>;1FK 2KG8A;8B5;L=>9 A8AB5<K. >A;5 C<=>65=8O 1;>:0 <0B@8FK  =0 1;>: 25:B>@0 b O4@0 ?@>F5AA>@>2 2KG8A;O5B 1;>: 25:B>@0 G0AB8G=KE @57C;LB0B>2  >= ?>4A25G8205BAO A25B;>-A8=8< F25B><. >A;5 >1<5=0 1;>:0<8 2 @0<:0E >4=>9 AB@>:8 C7;>2 2KG8A;8B5;L=>9 A8AB5<K, :064K9 C75; MB>9 AB@>:8 A>45@68B 1;>: @57C;LB8@CNI53> 25:B>@0, 1;>: 0:B82=>3> O4@0 >B>1@0605BAO 2 >1;0AB8 57C;LB0B C<=>65=8O <0B@8FK =0 25:B>@ B5<=>-A8=8< F25B><. 1;0ABL 57C;LB0B C<=>65=8O <0B@8F -B0 >1;0ABL =0E>48BAO 2 ?@02>9 25@E=59 G0AB8 >:=0 8 >B>1@0605B A>AB>O=85 <0B@8FK  @57C;LB0B0 2 O4@5 2K?>;=5=8O ?0@0;;5;L=>3> 0;3>@8B<0 <0B@8G=>3> C<=>65=8O. 0B@8F0 C ?@54AB02;O5BAO @0718B>9 =0 :204@0B=K5 1;>:8. 064>5 O4@> <=>3>?@>F5AA>@=>9 2KG8A;8B5;L=>9 A8AB5<K >B25G05B 70 2KG8A;5=85 >4=>3> (0;3>@8B<K $>:A0 8 M==>=0) 8;8 =5A:>;L:8E (;5=B>G=K9 0;3>@8B<) 1;>:>2 @57C;LB8@CNI59 <0B@8FK !. @8 2K?>;=5=88 ;5=B>G=>3> 0;3>@8B<0 C<=>65=8O B5<=>-A8=8< F25B>< 70:@0H820NBAO B5 1;>:8, :>B>@K5 C65 2KG8A;5=K : 40==><C <><5=BC. 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