ࡱ> _a^9 4bjbj.LXl , ,#9jX X X X X X X X f8h8h8h8h8h8h8$: <z8QX X X X X 8X X 8X .X X f8X f8: .AZ8X L dZ :7ZZ8 80#97'=4'=Z801>@0B>@=0O @01>B0 3 5H5=85 =5;8=59=KE 0;351@08G5A:8E C@02=5=89 1. >AB0=>2:0 CG51=>-?@0:B8G5A:>9 7040G8 0?8A0BL 8 >D>@<8BL 2 2845 <>4C;O (Unit) =01>@ DC=:F89, A ?><>ILN :>B>@KE <>6=> =0E>48BL @5H5=8O =5;8=59=KE 0;351@08G5A:8E C@02=5=89 2840: F(x) = 0. $C=:F88 4>;6=K >15A?5G8BL 2>7<>6=>ABL: 7040=8O 2840 C@02=5=8O F(x) = 0; 7040=8O 8=B5@20;0 ?>8A:0 @5H5=8O 8 =0G0;L=>3> ?@81;865=8O; 7040=8O B@51C5<>9 B>G=>AB8 @5H5=8O; 2K1>@ <5B>40 @5H5=8O. 2. #G51=>-<5B>48G5A:85 F5;8 @01>BK K?>;=5=85 ;01>@0B>@=>9 @01>BK =0?@02;5=> =0: ?@0:B8G5A:>5 >A2>5=85 ?@>AB59H8E ?@85<>2 ?@>3@0<<8@>20=8O G8A;5==KE <5B>4>2; >A2>5=85 ?@85<>2 ?@8<5=5=8O @07@01>B0==>3> 8 181;8>B5G=KE <>4C;59 4;O @5H5=8O 8AA;54>20B5;LA:8E 7040G;. 3. 5:><5=40F88 ?> 2K?>;=5=8N @01>BK @8 @07@01>B:5 <>4C;O @5H5=8O =5;8=59=KE C@02=5=89 A;54C5B >1@0B8BL 2=8<0=85 =0 >1ICN AE5<C G8A;5==>3> <5B>40 @5H5=8O =5;8=59=KE 0;351@08G5A:8E C@02=5=89 8 >A>15==>AB8 >B45;L=KE <5B>4>2. 3.1. 1I0O AE5<0 G8A;5==>3> <5B>40 @5H5=8O =5;8=59=KE C@02=5=89 '8A;5==>5 @5H5=85 =5;8=59=KE 0;351@08G5A:8E C@02=5=89 >A=>20=> =0 ?>AB@>5=88 ?>A;54>20B5;L=>AB8 7=0G5=89 2840: { Xi }, i = 0, 1, 2, ..., AE>4OI59AO : @5H5=8N C@02=5=8O. -B0 ?>A;54>20B5;L=>ABL AB@>8BAO A ?><>ILN 8B5@0F8>==>3> ?@>F5AA0 2840: Xn+1 = Pm(Xn, Fn), 345 Fn = F(Xn), 0 8B5@0F8>==0O D>@<C;0 Pm >?@545;O5BAO ?@8<5=O5<K< <5B>4>< @5H5=8O C@02=5=8O. ;O =5:>B>@KE <5B>4>2 2 D>@<C;C Pm <>3CB 2E>48BL 7=0G5=8O ?@81;865=89, ?>;CG5==KE =0 1>;55 @0==8E H030E 8B5@0F8>==>3> ?@>F5AA0: Xn-1, Xn-2, ..., Fn-1, Fn-2,... 8 7=0G5=8O ?@>872>4=KE DC=:F88 F(X) 2 MB8E ?@81;865=8OE. #A;>285< 7025@H5=8O 8B5@0F8>==>3> ?@>F5AA0 ?>8A:0 @5H5=8O O2;O5BAO: Abs( F(Xn) ) ( (. =0G5=85 (>0 70405BAO 70@0=55 :0: B@51C5<0O B>G=>ABL @5H5=8O. >A:>;L:C =0 ?@0:B8:5 CA;>28O AE>48<>AB8 <5B>4>2 @5H5=8O 2K?>;=ONBAO =5 2A5340, 8 8B5@0F8>==K9 ?@>F5AA <>65B 84B8 15A:>=5G=>, B> =0@O4C A B@51C5<>9 B>G=>ABLN @5H5=8O 70405BAO nmax - <0:A8<0;L=>5 :>;8G5AB2> 8B5@0F89. ;O 70?CA:0 8B5@0F8>==>3> ?@>F5AA0 =5>1E>48<> 7040BL =0G0;L=>5 ?@81;865=85 X0. @><5 B>3>, 70405BAO 8=B5@20; [a, b], =0 :>B>@>< =04> 8A:0BL @5H5=85 C@02=5=8O. 06=K< B0:65 O2;O5BAO 2>?@>A > ACI5AB2>20=88 @5H5=8O =0 8=B5@20;5 [a, b] 8 53> 548=AB25==>AB8 (=5:>B>@K5 <5B>4K 30@0=B8@CNB AE>48<>ABL B>;L:> ?@8 548=AB25==>AB8 @5H5=8O =0 8=B5@20;5). @>AB59H8< 4>AB0B>G=K< CA;>285< ACI5AB2>20=8O @5H5=8O O2;O5BAO: DC=:F8O F(X) =5?@5@K2=0 =0 8=B5@20;5 [a, b]; F(a)*F(b) <= 0. "0:8< >1@07><, 7040G0 =0 G8A;5==>5 @5H5=85 =5;8=59=>3> C@02=5=8O AB028BAO B0:: =0 8=B5@20;5 [a, b] A B>G=>ABLN (, => =5 1>;55 G5< 70 nmax H03>2, =09B8 @5H5=85 C@02=5=8O F(X) = 0, 8A?>;L7CO X0 2 :0G5AB25 =0G0;L=>3> ?@81;865=8O. 3.2. 5B>4 48E>B><88 $>@<C;0 <5B>40 48E>B><88 (45;5=8O ?>?>;0<) 8<55B 284: Xn = P4 (Xn-2, Xn-1, Fn-2, Fn-1), 345: Fn-2* Fn-1 ( 0, B.5. >B@57>: [Xn-2, Xn-1] A>45@68B :>@5=L C@02=5=8O. 0 n->< H035 <5B>40 >B@57>: [Xn-2, Xn-1] 45;8BAO ?>?>;0<, 8 2 :0G5AB25 A;54CNI53> >B@57:0 15@5BAO B0 ?>;>28=0 >B@57:0, :>B>@0O A>45@68B :>@5=L C@02=5=8O: X( = (Xn-2 + Xn-1) / 2; 5A;8 F(X()*Fn-1 ( 0, B>: X n= Xn-1,Xn-1 = X( 8=0G5: Xn-1 = Xn-2,Xn= X(  :0G5AB25 =0G0;L=>3> >B@57:0 15@5BAO >B@57>: [a, b].  8A. 5.1. 5B>4 A5:CI8E 3.3. 5B>4 A5:CI8E 5B>4 A5:CI8E (<5B>4 E>@4) >B;8G05BAO >B <5B>40 48E>B><88 B5<, GB> >B@57>: [Xn-2, Xn-1] 45;8BAO =5 ?>?>;0<, 0 ?@>?>@F8>=0;L=> 7=0G5=8O< DC=:F88 =0 :>=F0E >B@57:0, B.5.: X( = Xn-2 - (Xn-1 - Xn-2) / (Fn-1 - Fn-2) * Fn-2. 5><5B@8G5A:8 (A<. @8A. 5.1) B>G:0 X( 87>1@0605BAO :0: B>G:0 ?5@5A5G5=8O ?@O<>9, A>548=ONI59 B>G:8 (Xn-1, Fn-1), (Xn-2, Fn-2) A >ALN X.  8A. 5.2. 5B>4 :0A0B5;L=KE 3.4. 5B>4 :0A0B5;L=KE (<5B>4 LNB>=0)  <5B>45 LNB>=0 :064>5 A;54CNI55 ?@81;865=85 8I5BAO :0: @5H5=85 ;8=59=>3> C@02=5=8O 2840: Fn-1 + F(n-1 * (X - Xn-1) = 0, :>B>@>5 O2;O5BAO >B@57:>< AB5?5==>3> @O40 DC=:F88 F(X) 2 >:@5AB=>AB8 B>G:8 Xn-1. $>@<C;0 <5B>40 LNB>=0 8<55B 284: Xn = Xn-1 - Fn-1/F(n-1 5><5B@8G5A:8 (A<. @8A. 5.2) B>G:0 Xn 87>1@0605BAO :0: B>G:0 ?5@5A5G5=8O :0A0B5;L=>9 : 3@0D8:C DC=:F88 F(X) 2 B>G:5 Xn-1 A >ALN X. 4. 5:><5=40F88 ?> >@30=870F88 ?@>3@0<<=>3> 8=B5@D59A0 $C=:F88 @5H5=8O =5;8=59=KE 0;351@08G5A:8E C@02=5=89 A;54C5B >D>@<8BL 2 2845 >B45;L=>3> <>4C;O Solver. 5:><5=4C5<K9 A>AB02 703>;>2>G=>3> D09;0 MB>3> <>4C;O A>AB>8B 2 A;54CNI5<. // >4C;L @5H5=8O =5;8=59=KE C@02=5=89 typedef double (TFun)(double); // 09B8 @5H5=85 =5;8=59=>3> C@02=5=8O long Root(double x0, double a, double b, double eps, long kStp, TFun Fun, double& root); // E>4 : // x0 - =0G0;L=>5 ?@81;865=85 @5H5=8O // a, b - 8=B5@20; ?>8A:0 @5H5=8O // eps - B@51C5<0O B>G=>ABL @5H5=8O // kStp - <0:A8<. :-2> 8B5@0F89 // Fun - DC=:F8O ;52>9 G0AB8 C@-O // KE>4 : // root - ?>;CG5==>5 @5H5=85 // 2>72@0I05<>5 7=0G5=85  :>4 7025@H5=8O: // = -1 - =0@CH5=85 CA;>28O Fun(a)*Fun(b)<= 0 // > 0 - :-2> 8B5@0F89, 70 :>B>@>5 @5H5=85 ?>;CG5=> A 7040==>9 B>G=>ABLN // = 0 - 2K?>;=5=> kStp 8B5@0F89, 7040==0O B>G=>ABL =5 4>AB83=CB0 5. $>@<C;8@>2:0 7040=89 A=>2=>5 7040=85 0?8A0BL ?@>3@0<<C, 2:;NG0NICN: >A=>2=>9 <>4C;L 2 A>AB025: B5AB>2>9 DC=:F88 2KG8A;5=8O 7=0G5=89 ?@02>9 G0AB8 @5H05<>3> C@02=5=8O 8 3>;>2=>9 DC=:F88 main, >15A?5G820NI59 22>4 8AE>4=KE 40==KE 7040G8 8 2K2>4 @57C;LB0B0; <>4C;L Solver, >15A?5G820NI89 @5H5=85 =5;8=59=>3> C@02=5=8O <5B>4>< 48E>B><88. @>25AB8 A@02=8B5;L=K9 0=0;87 AE>48<>AB8 <5B>40 48E>B><88 4;O @07;8G=KE =0G0;L=KE 7=0G5=89. >?>;=8B5;L=>5 7040=85  <>4C;L Solver 4>1028BL DC=:F88 @5H5=8O =5;8=59=>3> C@02=5=8O <5B>4>< A5:CI8E 8 <5B>4>< :0A0B5;L=KE. ;O MB>3> 2 703>;>2>G=CN G0ABL <>4C;O 4>1028BL >?8A0=85 B8?0 <5B>40 (enum Method) 8 DC=:F8N CAB0=>2:8 <5B>40 @5H5=8O (SetMethod): enum Method {m_dih, m_sek, m_tan}; // m_dih  <5B>4 48E>B><88 // m_sek  <5B>4 A5:CI8E // m_tan  <5B>4 :0A0B5;L=KE // #AB0=>28BL <5B>4 p5H5=8O C@02=5=8O void SetMethod(Method meth);  >A=>2=>9 <>4C;L 4>1028BL =5A:>;L:> B5AB>2KE DC=:F89 2KG8A;5=8O ?@02KE G0AB59 C@02=5=8O. 15A?5G8BL 2>7<>6=>ABL 2K1>@0 @5H05<>3> C@02=5=8O A ?5@540G59 A>>B25BAB2CNI59 DC=:F88 G5@57 C:070B5;L. @>25AB8 A@02=8B5;L=K9 0=0;87 AE>48<>AB8 <5B>4>2 4;O @07;8G=KE C@02=5=89 8 @07;8G=KE =0G0;L=KE 7=0G5=89. & .024ft " jemH sH  jmH sH H*6H* H*mH sH  H*mH sH mH sH V. 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A!"R#n$n% gDdz  <  C Ab ``2+DEn ``2+PNG  IHDRz42=gAMAPLTEٟ pHYs*IDATxڱn0Cʘ.(X x r%72!^*WuXHJH.BjNl۹&_/Xjr|?A`/@uq`_~>RO;/Pp d6|/rHKpn|̼7SߠdA `D { .d'B||#{k|~vc ?Pzw0 *b S ,D% ~ bjQPǾD@DPP"rHЇ;H;8G@9CTptg82QpdQ4 2 b`P !AЌ=A*Xr% f Bk)4 @;yogh5t  cH@et'cteJZt P ]2t#d1 ܣ LMAWDzSE4d`оp}&bE,Pо vf P@ hM L@=>+'#MW 'Ɲ9AL1=5g@)OYmuE݇5J3uQ/>ߟ@SNG^r{TQihfӹ2pf @i oH 5b ĄRXk0RK`C5s[JQ҇Wܵ0_m@ӰbdӭNY0 T[nL{Aմiw-h u]/c@ۅ8.[AԴ"c5 [T_P#2aphS ;>do&E=PUNhQbg$Fywb2@:t9҆'(ԁ ${ʺnC ԧ fO@P>9ɁC K89( ?(t0`V z 385 ~A`'"Oī H`0 bQG >5X0~D  A`j3j`c .7J BA3Q݃G3$w0$a BAPH PI(p@ ` [F " lU%@k[/Z,: @f{\b rcfh=,>7͙iQ}C1i;F0y}Mq8;"DdRc K^mہeB XbdY: WUj: <ZV)Bc~ڰlȗʠbT*A.w{`jpg NUgSrh*8 @-yϦ*~Fj3Q&L@5Aʁ;4Qۮʹh'L/?!V+"d5ڈ׬\2ٔ`p%+I` NuNM͊ojd F:f•@+˰ K7˨/M,A#5FYM:Ʋ% ,jvsh1 $6t 4YųX7<}b nG%+CA7#1{Jco.Y-\ɔvIENDB` i:@: 1KG=K9CJ_HaJmHsHtHP@P 03>;>2>: 1$<@&5CJKHOJQJaJJ@J 03>;>2>: 2$<@&56OJQJaJD@D 03>;>2>: 3$<@& OJQJaJ:A@: A=>2=>9 H@8DB 0170F0X LCl,Ma)no0 ] m Q g  I X Y 6]LMfg!E#Jo%x tst+-WZ0000C0C0C0C0C 0C 0C 0C 0C00 0 000(000000000000 0 00(00 0 0 0 0 0 0 0 (00 0 0 0 0 0 0 0 (00n0n0n0n0n00"0"0"0"0"0"0"0"0"0"0"0"0"0"0"0"0"0"0"0"0(06 07 07 06 0(08 000@0@0@0@0008 08 00hJ)4 "$ %/4!#%4Z\`avxpt   & ( , . 0 2 7 9 @ B   _ a g i n p "$MOux),PTaeux8<6:gptx$(Z\^ru] _ ; B  EI_`ab8<txZ3333333333333tZ 0@?5=:> wD:\karpenko\-!-! 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