R3

Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 1 of 39 Go Back Full Screen Close Quit  HHHmmmŒŒŒÆÆÆêêêÆÆƉ‰‰ÆÆÆÆÆÆ��� ÚÚÚOOOOOOŽŽŽ†††ÚÚÚOOO^^^‡‡‡ Ìù: o§u ïЕ: ÚOL§›› e‡: zli@nankai.edu.cn ̐: http://202.113.29.3/˜zli/ þ‘:±�9-11(1��we);þÅ:±�7-8(1��¿) Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 2 of 39 Go Back Full Screen Close Quit §3 ```zzz†††¦¦¦)))ššš‚‚‚555§§§||| ÚOÆ¥éõ¯K�)ªÑ´`z¯K,X4Œq,�O§Bayesûü ¯K¥��ºx§š‚5��¦¯K�¦)¯K�. ù ¯K¥§· ‚Ï~ØU†���wª),I‡ÏLꊐ{��§�). þã¦)ÑáuXe�˜„¯Kµ argmax 휽∈퐷 푔(휽), (1) ∙ 푔´ëê•þ 휽�¼ê,¡ƒ8I¼ê (objective function) ∙ 휽·‚kž½¡ƒûüCþ (decision variable) ∙ ûüCþ��Š«퐷¡Œ18Ü (feasible set)½öÿÀ8Ü (can- didate set) ∙ du�z˜‡¼ê�duÙKŠ�Œz (−푔(휽)),·‚˜„òÄ ¦�ŒŠ�Ž{. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 3 of 39 Go Back Full Screen Close Quit ∙ «©k�åÚÃ�åü«`z¯K. � 퐷 Ò´ 푔(휽) �½Âž, ¯ K(1)Ò´˜Ã�å`z¯K (unconstrained optimization problem)ÄK, =´k�å`z¯K (constrained optimization problem) ∙ , ,3�Š«S 퐷�,‡A½fS,ŒUk,‡Û܁ŒŠ, 3, ˜‡A½fS3,˜‡Û܁ŒŠ. ·‚¡�ہ` (global optimum)=3�Š«S 퐷�ŒŠ; ¡Û܁` (local optimum)=3�Š«S퐷�,‡fS�Œ Š ∙ 3�Ù,·‚ò̇Ä 푔'u 휃1w…Œ‡�œ/. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 4 of 39 Go Back Full Screen Close Quit §3.1üüüCCCþþþ¯¯¯KKK ∙ éõ`z¯KÑ�du´˜‡§¦Š¯K.Ïd,·‚Äk5?ؘ �§¦Š¯K�êŠ){,=éu‰½�'u 푥�¼ê 푔 Ïé 푥¦ � 푔(푥) = 0. ∙ ¦)¼ê 푔(푥) = log 푥/(1 + 푥)�ŒŠ? Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 5 of 39 Go Back Full Screen Close Quit ∙ ¦)¼ê 푔(푥) = log 푥/(1 + 푥)�ŒŠ�du¦§ 푔′(푥) = 1 + 1/푥− log 푥 (1 + 푥)2 = 0⇔ 1 + 1/푥− log 푥 = 0 �). w,vkwª)" Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 6 of 39 Go Back Full Screen Close Quit �©{ (bisection method) ∙ �nµXJ 푔 3«m [푎0, 푏0]þëY,… 푔(푎0)푔(푏0) ≤ 0,KdŠ�3 ½n,–�3˜‡ 푥∗ ∈ [푎0, 푏0],¦� 푔(푥∗) = 0. ·‚¡ [푎0, 푏0]k Š«m; ∙ �«m [푎0, 푏0]¥: 푥0 = (푎0 + 푏0)/2ò§©üŒ,e 푔(푥0) = 0,K 푥0 Ò´ 푔(푥) = 0�˜‡Š; ∙ ÄKd 푥0 ©¤�ü‡«m¥7k˜‡´kŠ«m,P [푎1, 푏1]. dž =r«m [푎0, 푏0]  á– [푎1, 푏1], Ùݏ�«m�˜Œ. Xd‡E? 1,Œ��˜X�kŠ«m [푎0, 푏0] ⊃ [푎1, 푏2] ⊃ [푎2, 푏2] ⊃ ⋅ ⋅ ⋅ ,Ù¥z‡ «mÑ´c˜‡«m�˜Œ,Ïd [푎푘, 푏푘]�Ý 푏푘−푎푘 = (푏0−푎0)/2푘; ∙ � 푘 →∞žÙªu",Ò´`,XJ�©L§Ã/UYe�,ù « mòªÂ u˜:,T:w,¤¦�Š 푥∗; ∙ �,3¢SOŽž,ˆ�,«#NØ�Ò(å. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 7 of 39 Go Back Full Screen Close Quit �©{¦§Š (0) - 푘 = 0,…é�˜kŠ«m [푎0, 푏0]; (1) - 푘 = 푘 + 1… 푥푘 = (푎푘−1 + 푏푘−1)/2; (2) XJ sgn(푔(푥푘)) = sgn(푔(푎푘−1))K- 푎푘 = 푥푘;ÄK 푏푘 = 푥푘; (3) XJˆ�,˜ÂñOK,Kʎ¿ˆ£) 푥푘. ÄKˆ£1˜Ú. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 8 of 39 Go Back Full Screen Close Quit �©{¦§Š 푓(푥) = 푥3 − 4푥2 + 18푥− 115.(푥 ≈ 5) Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 9 of 39 Go Back Full Screen Close Quit �©{�`": ∙ `: – Ž{{ü, …Âñ5oU���y; – XJ 푔 3«m [푎0, 푏0]þëY,KØØ 푔′ ´Ä3½´ÄN´�Ñ, يь±é�,Ϗ§‚Ø7Ä 푔′ . ∙ ": – ÙÂñ„Ýéú; – XJ 푔 3Щ«mS�Šõu 1,KN´w��©{òé�Ù¥˜ ‡, éØ�Ù{�. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 10 of 39 Go Back Full Screen Close Quit ÂñOK ∙ ýéÂñ�ʎOÿµ ∣푥푡+1 − 푥푡∣ < 휖. (2) éuþã�©{,ØJwÑ ∣푥푡+1 − 푥푡∣ = (푏0 − 푎0)/2푡+1. ‡ 푡v Œ, o¤á; ∙ ƒéÂñOÿµ ∣푥푡+1 − 푥푡∣ ∣푥푡∣ < 휖 (3) žÊŽS“.dOKŒ±3Ø7Ä 푥�ü �œ¹eˆ�½�8 I°Ý,X1%. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 11 of 39 Go Back Full Screen Close Quit ÂñOK ∙ ·‚ATŠâ¢S¯K5ÀJA^ý鄴ƒéÂñOK.XJ 푥� Ýƒéu 휖錣½é�¤ž,ýéÂñOKkžòØX·‚¤�/ ʎS“½S“é¯Òʎ . ƒéÂñOKé 푥�Ý‰ ��,� � 푥(푡) �Š£½ýŠ¤† 0š~�Cž,§òC�Ø­½,dž,·‚Œ ±ÏL ∣푥푡+1−푥푡∣∣푥푡∣+휂 < 휖5?�ƒéÂñOK; ∙ 38�¦^ˆ«S“Ž{�L§¥,·‚Ï~„I‡‰Ñ˜‡IPØ Âñ�ʎOK.Ò´`,ØØÂñ†Ä,푁 ÚS“�ʎ$Ž; ∙ ¦^ÂñOK�žÿ, Œʇ½õ‡ÂñÝþ, 'X ∣푥푡+1 − 푥푡∣, ∣푥푡+1 − 푥푡∣ / ∣푥푡∣,½ ∣푔(푥푡+1)∣. XJz˜‡ÑØü~½eZgS“�Ñ y ±Ï,KS“ÊŽ; Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 12 of 39 Go Back Full Screen Close Quit m©kXʎ˜�­‡ ∙ ˜‡��ЊŒU�—Ž{uÑ!±Ï5!Ø\Üå�Û܁Œ½ �½Ù§¯K.ÑÑ(J6u¼ê 푔!Њڤ^�`zŽ{. ∙ �)ÜnЊ�{kã«{!Ð�O£XÝ�O¤!kŠâ�íÿ ڇEÁ†{�. ∙ XJOŽÅ$1„݁›\¤U«ú�å�S“gê§Kh²�‰{ ´Ø‡r¤k�$Ž] Ñ^�d`zŽ{�žm$1þ. A^õ ‡ÐŠ?1$Ž´˜‡Â¼Œ&$1(J�k�{…U;��Û Ü`Ú$ŽuÑ. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 13 of 39 Go Back Full Screen Close Quit ~3.1µ© ê�OŽ �ëY.‘ÅCþ 푋 �©Ù¼ê 퐹 (푥), é?‰� 훼 ∈ (0, 1), ·‚¦ � 퐹 (푥)� 훼e© :,=¦� 푔(푥) = 퐹 (푥)− 훼 = 0�Š. ∙ ·‚I‡é�˜‡kŠ«m5m©¦^Ž{ ∙ ˜‡{üB�‰{´|^ 푔(푥)�üN5?�˜Ð©: 푎. 'XÏ~ ·‚Œ� 푎þŠ; ∙ XJ 푔(푎) < 0,@o·‚Ïé1˜‡¦� 푔(푎 + 푘) > 0���ê 푘. 3 é�ù�� 푘ƒ�,w, [푎+ 푘 − 1, 푎+ 푘]Ò´kŠ«m; ∙ ‡ƒ, e 푔(푎) > 0, ·‚Ïé1˜‡¦� 푔(푎 + 푘) < 0 �K�ê 푘, [푎+ 푘, 푎+ 푘 + 1]kŠ«m; ∙ ·‚Œòù‡|¢�Ú 1UÙ§?ÛkŒU\¯|¢�kŠ«m ?§�Ú,'X,kžŒ�Tڏ 퐹 (푥)�IO�. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 14 of 39 Go Back Full Screen Close Quit §3.1.1ÚÚÚîîî{{{(Newton{{{) ���nnnµ ∙ b� 푔 ´ëYŒ‡�… 푔′(푥∗) ∕= 0. 31 푡 gS“,d{ÏL‚ 5 Taylor?êÐm 0 = 푔(푥∗) ≈ 푔(푥푡) + (푥∗ − 푥푡)푔′(푥푡) 5Cq 푔(푥∗). ∙ )þã'u 푥∗�§,·‚k 푥∗ = 푥푡 − 푔(푥푡) 푔′(푥푡) = 푥푡 + ℎ푡. d§wŠ·‚,é 푥∗�Cq6u�c��OŠ 푥푡ژ‡?� ℎ푡. ∙ ­EdL§,K Newton{�#§ 푥푡+1 = 푥푡 + ℎ푡, Ù¥ ℎ푡 = −푔(푥푡)/푔′(푥푡). Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 15 of 39 Go Back Full Screen Close Quit Úî{¦§Š: (0) - 푡 = 0,‰½˜‡ÐŠ 푥0. (1) - 푡 = 푡+ 1. XJ [푔′(푥푡−1)]−13, 푥푡 = 푥푡−1 − 푔(푥푡−1)푔′(푥푡−1) . (2) XJˆ�,˜ÂñOK,Kʎ¿ˆ£) 푥푡. (3) XJS“gêˆ�ýk½�gê 푁 , ½ö푔′(푥푡) = 0, K{”}¶ ÄKˆ£1˜ÚUYS“. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 16 of 39 Go Back Full Screen Close Quit Úî{�AÛ)º ∙ Úî{¢Ÿþ´˜«‚5z{,ÙÄ�gŽ´òš‚5§ 푔(푥) = 0ÅÚ8(,«‚5§5¦). Ò´`ƒ‚§ 푔(푥푡) + (푥− 푥푡)푔′(푥푡) = 0. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 17 of 39 Go Back Full Screen Close Quit Úî{�AÛ)º ∙ Úî{k²w�AÛ)º. § 푔(푥) = 0 �Š 푥∗ Œ)º­‚ 푦 = 푔(푥)† 푥¶��:�î‹I.� 푥푡 ´Š 푥∗ �,‡CqŠ,L­‚ 푦 = 푔(푥)þî‹I 푥푡 �: 푃푡 ڃ‚,¿òTƒ‚† 푥¶��:�î‹ I 푥푡+1Š 푥 ∗�#�CqŠ (�,·‚Ï"T:‚C 푔(푥)�":). duù«AÛ�µ, Newton{¡ƒ‚{. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 18 of 39 Go Back Full Screen Close Quit Âñ^‡ Newton{�Âñ56u 푔�/GÚЊ�À�. b� 푔äk��ëY�ê, 푔′(푥∗) ∕= 0,… 푥∗ 푔�˜‡üŠ,K3 푥∗� ˜‡�,�Њd�S?˜:ž, Newton{ÑÂñ� 푥∗. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 19 of 39 Go Back Full Screen Close Quit yyy²²²µÏ 푔′(푥∗) ∕= 0… 푔′3 푥∗?ëY,K73 푥∗�˜‡�,¦�3 d�S 푔′(푥) ∕= 0. ·‚=3d�SÄ,…½Â 휖푡 = 푥푡 − 푥∗. d TaylorÐmk 0 = 푔(푥∗) = 푔(푥푡) + (푥∗ − 푥푡)푔′(푥푡) + (푥∗ − 푥푡)2푔′′(휉)/2, Ù¥ 휉 ∈ (푥푡, 푥∗)ƒm. £‘�,·‚k 푥푡 + ℎ푡 − 푥∗ = (푥∗ − 푥푡)2 푔 ′′(휉) 2푔′(푥푡) , Ù¥ ℎ푡´ Newton#Oþ. duþª†>�u 푥푡+1 − 푥∗,� 휖푡+1 = (휖푡) 2 푔 ′′(휉) 2푔′(푥푡) . (4) Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 20 of 39 Go Back Full Screen Close Quit yé,‡ 훿 > 0,Ä 푥∗�� 풩훿(푥∗) = [푥∗ − 훿, 푥∗ + 훿]. P 푐(훿) = max 푥1,푥2∈풩훿(푥∗) ∣∣∣∣ 푔′′(푥1)2푔′(푥2) ∣∣∣∣ . (5) Ϗ� 훿 → 0 ž, 푐(훿) → ∣∣∣ 푔′′(푥∗)2푔′(푥∗)∣∣∣, ¤± 훿 → 0 ž, 훿푐(훿) → 0. ·‚�÷ v 훿푐(훿) < 1� 훿,Kd (4)ª� ∣푐(훿)휖푡+1∣ ≤ (푐(훿)휖푡)2 . (6) b�Њ÷v ∣휖0∣ = ∣푥0 − 푥∗∣ ≤ 훿,Kd (6)ª� ∣휖푡∣ ≤ (푐(훿)훿) 2푡 푐(훿) , §� 푡→∞žÂñ� 0,u´, 푥푡 → 푥∗. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 21 of 39 Go Back Full Screen Close Quit ∙ XJЊ u˜‡«m [푎, 푏],KI‡�ye�˜ ^‡. XJ (i) 3«m [푎, 푏]þ, 푔′(푥) ∕= 0, (ii) 3«m [푎, 푏]þ, 푔′′(푥)ØCÒ, (iii) 푔(푎)푔(푏) < 0, (iv) ∣푔(푎)/푔′(푎)∣ < 푏− 푎… ∣푔(푏)/푔′(푏)∣ < 푏− 푎, Kéud«mS�?˜‡ÐŠ 푥0, Newton{ÑòÂñ. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 22 of 39 Go Back Full Screen Close Quit ØÂñœ/: 푔′(푥푡)=0 Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 23 of 39 Go Back Full Screen Close Quit ØÂñœ/: ЊålýŠ�� Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 24 of 39 Go Back Full Screen Close Quit ÂÂÂñññ���êêê: ´^5ÝþX Newton{�¦Š{�Âñ„Ý�˜‡þ. ¡ ˜‡{�Âñ�ê 훽,XJ lim푡→∞ 휖푡 = 0… lim 푡→∞ ∣휖푡+1∣ ∣휖푡∣훽 = 푐, (7) Ù¥~ê 푐 ∕= 0… 훽 > 0. Ò´3ÂñL§¥S“Ø��eü„Ý. ∙ 3ŒU°(CqýŠ�¿Âe,p�Âñ`. ∙ , p�Âñ{´±GÑ­è�“d ���,, „Ý�ú� {¬'ÙéA�¯„Ž{S�. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 25 of 39 Go Back Full Screen Close Quit ∙ éu Newton{, (4)Ñ 휖푡+1 (휖푡)2 = 푔′′(휉) 2푔′(푥푡) . (8) XJ Newton {Âñ, KÙëY5wŠ·‚, d§�màÂñ � 푔 ′′(푥∗) 2푔′(푥∗) . u´, Newton{�gÂñ,= 훽 = 2… 푐 = 푔′′(푥∗) 2푔′(푥∗). ∙ éu�©{,XJ3ÙЩ«mk)�{,duÙzgS“«m�Ý þ~Œ… lim푡→∞ ∣휖푡∣ = 0,�§w«Ñäkaq‚5Âñ (훽 = 1)�A :. , ,ål 푥푡 − 푥∗ؘ½zgS“Ñ �,…§‚�'ŠŒU´Ã .�,u´,éu?Û 훽 > 0, lim푡→∞ ∣휖푡+1∣ ∣휖푡∣훽 ŒUØ3. ù�,�©{l /ªþÒØ÷vÂñ�ê�½Â. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 26 of 39 Go Back Full Screen Close Quit � 푔′Ã)Û/ª½Ø´w«Lў,·‚kžŒ¦^Xe�êŠCq5“ O§, 푔′(푥푡) = 푔(푥푡 + ℎ)− 푔(푥푡) ℎ . Ï~òù«{¡lÑÚî{,·‚òw��¡ò‡0���{ÓT lÑ{Ån´aq�. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 27 of 39 Go Back Full Screen Close Quit §3.1.2 Fisher���©©©{{{ Äc¡�~f, XJ 휃ˆ ´ MLE,K§ŒzÙéêq,, = 휃ˆ ´�© § 퐿′(휃) = 0�). dênÚO£·‚��*ÿ5guê©Ùxž, ·‚k Fisher&Eþ 퐼(휃) = 퐸[퐿′2(휃)] = −퐸(퐿′′(휃)). ÏdŒ^ −퐿′′(휃) 5Cq 퐼(휃) . u´, � 푔 éAXMLE�`z¯Kž, 3Newton#§¥,·‚Œ±^ 퐼(휃) 5O† −퐿′′(휃) ,džÙ#Oþ  ℎ푡 = 퐿 ′(휃푡)/퐼(휃푡),Ù¥ 퐼(휃푡)3 휃푡:�Ï" Fisher&Eþ. ù�,d #§ 휃푡+1 = 휃푡 + 퐿 ′(휃푡)/퐼(휃푡). (9) Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 28 of 39 Go Back Full Screen Close Quit §3.1.3���{{{(½½½uuu���{{{; Secant method) ���nnn: 3Newton{¥,ٍ#Oþ6٘��ê 푔′(푥푡). XJOŽd�ê '�(J,KŒ±^lÑ�© 푔(푥푡)−푔(푥푡−1)푥푡−푥푡−1 5Cqƒ. �{�#§ 푥푡+1 = 푥푡 − 푔(푥푡) 푥푡 − 푥푡−1 푔(푥푡)− 푔(푥푡−1), ∀ 푡 ≥ 1. (10) d{I‡ü‡ÐŠ 푥0, 푥1. ù��Ñ�S“úªƒ�uÚîS“ 푥푡+1 = 푥푡 − 푔(푥푡) 푔′(푥푡) ¥��ê 푔′(푥푡)^�û 푔(푥푡)−푔(푥푡−1) 푥푡−푥푡−1 �“�(J. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 29 of 39 Go Back Full Screen Close Quit ���{{{¦¦¦§§§ŠŠŠ: (0) - 푡 = 1,‰½ü‡ÐŠ 푥0Ú 푥1. (1) - 푡 = 푡+ 1. 푥푡 = 푥푡−1 − 푔(푥푡−1) 푥푡−1−푥푡−2푔(푥푡−1)−푔(푥푡−2) . (2) XJˆ�,˜ÂñOK,Kʎ¿ˆ£) 푥푡. (3) XJS“gêˆ�ýk½�gê 푁 ;ÄKˆ£1˜ÚUYS“. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 30 of 39 Go Back Full Screen Close Quit ���{{{���AAAÛÛÛ)))ººº ∙ ­‚ 푦 = 푔(푥)þî‹I 푥푡, 푥푡−1 �:©O 푃푡, 푃푡−1,Ku‚ 푃¯푡푃¯푡−1 ��Ç�u�ûŠ 푔(푥푡)−푔(푥푡−1)푥푡−푥푡−1 ,ِ§´ 푔(푥푡) + 푔(푥푡)−푔(푥푡−1) 푥푡 − 푥푡−1 (푥− 푥푡) = 0. Ïd,U#§¦Ñ� 푥푡+1¢Sþ´u‚ 푃¯푡푃¯푡−1† 푥¶�:�î‹ I.ÏdTŽ{½¡u�{. ∙ 5¿�T{†Úî{Ñ´‚5z{, �k�Ÿþ�«O. �ö3O Ž 푥푡+1ž^�c˜Ú� 푥푡Š, cö3¦ 푥푡+1ž^�c¡üÚ�( J 푥푡, 푥푡−1,Ïd¦^T{I‰Ñü‡Ð©Š. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 31 of 39 Go Back Full Screen Close Quit Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 32 of 39 Go Back Full Screen Close Quit Âñ�ê 3aqu Newton {�^‡e, �{òÂñ�Š 푥∗. ¦�Ù ñ�ê, ·‚=3,‡Ü·��«m [푎, 푏] SÄ, …b�d«m ¹ 푥0, 푥1 Ú 푥 ∗,…3d«mS 푔′(푥) ∕= 0, 푔′′(푥) ∕= 0. P 휖푡+1 = 푥푡+1 − 푥∗,K Œ†�y� 휖푡+1 = [ 푥푡 − 푥푡−1 푔(푥푡)− 푔(푥푡−1) ] [ 푔(푥푡)/휖푡 − 푔(푥푡−1)/휖푡−1 푥푡 − 푥푡−1 ] [휖푡휖푡−1] = 퐴푡퐵푡휖푡휖푡−1, Ù¥� 푥푡 → 푥∗… 푔′ëYž, 퐴푡 → 1/푔′(푥∗). Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 33 of 39 Go Back Full Screen Close Quit �� 퐵푡�4,é 푔3 푥∗??1TaylorÐmµ 푔(푥푡) ≈ 푔(푥∗) + (푥푡 − 푥∗)푔′(푥∗) + (푥푡 − 푥∗)2푔′′(푥∗)/2, u´, 푔(푥푡)/휖푡 ≈ 푔′(푥∗) + 휖푡푔′′(푥∗)/2. aq/, 푔(푥푡−1)/휖푡−1 ≈ 푔′(푥∗) + 휖푡−1푔′′(푥∗)/2. ù�, 퐵푡 ≈ 푔′′(푥∗) 휖푡 − 휖푡−1 2(푥푡 − 푥푡−1) = 푔 ′′(푥∗)/2, u´, 휖푡+1 ≈ 푑푡휖푡휖푡−1, (11) Ù¥� 푡→∞ž, 푑푡 → 푔 ′′(푥∗) 2푔′(푥∗) = 푑. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 34 of 39 Go Back Full Screen Close Quit ¦��{�Âñ�ê, ·‚7Lé� 훽 ÷vµlim푡→∞ ∣휖푡+1∣ ∣휖푡∣훽 = 푐, Ù ¥ 푐~ê. d,kb�þª¤á,¿^ 휖푡−1 † 휖푡+1 ƒ'5Lˆ(11)ª¥ � 휖푡,…“\þª,²�n�,k lim 푡→∞ ∣휖푡∣ 1−훽+1/훽 = 푐1+1/훽 푑 . (12) Ï(12)ªmà�ê,� 1 − 훽 + 1/훽 = 0,Ù) 훽 = (1 +√5)/2 ≈ 1.62. u´,�{�Âñ�ê$uNewton{. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 35 of 39 Go Back Full Screen Close Quit §3.1.4ØØØÄÄÄ:::SSS“““{{{(Fixed-Point Method) �nµ ∙ ˜‡¼ê�ØÄ:Ò´d:�¼êŠ�uÙg��:. ^ØÄ: {¦ŠÒ´‡(½˜‡¼ê 푓 ¦� 푔(푥) = 0 �…=� 푓(푥) = 푥. ù�Òr¦ 푔 �Š�¯KC†¤¦ 푓 �ØÄ:¯K, |^# § 푥푡+1 = 푓(푥푡)Ò´ÏéØÄ:�{ü{. ∙ éu§ 푔(푥) = 0, ‡A^ØÄ:S“{, 7Lkò§U�¤ 푥 = 푓(푥) �/ª,=I‡�餉�¼ê 푔(푥) �EÜ·�S“¼ê 푓(푥). ·‚Œ±}Á?ÛÜ·� 푓 ,�À� 푓(푥) = 푔(푥) + 푥´w,�,dž,Ù #§ 푥푡+1 = 푥푡 + 푔(푥푡). (13) Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 36 of 39 Go Back Full Screen Close Quit ∙ dŽ{�Âñ6u 푓 ´Ä´Â �(contractive). ‡¦ 푓 3« m [푎, 푏]þ´Â �§K§7L÷vµ 1. ‡ 푥 ∈ [푎, 푏]§푓(푥) ∈ [푎, 푏], 2. é,‡ 휆 ∈ [0, 1), ∣푓(푥1)− 푓(푥2)∣ ⩽ 휆∣푥1 − 푥2∣, ∀ 푥1, 푥2 ∈ [푎, 푏]. ∙ þã«m [푎, 푏] Œ±´Ã.�§1�‡^‡Ò´Lipschitz^‡§ ¡ 휆Lipschitz~ê. ∙ XJ 푓 3«m [푎, 푏] þ´Â �§K3d«mS3Ž˜�ØÄ : 푥∗§…éud«mS�?˜ÐŠ§dŽ{ÑòÂñ�dØÄ:. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 37 of 39 Go Back Full Screen Close Quit ÝN� ∙ ØÄ:S“XÂñ§KÙÂñ�ê6u 휆. �¿ØU(�ÙÂñ. ∙ Xéu¤k� 푥 ∈ [푎, 푏]§∣푓 ′(푥)∣ ⩽ 휆 < 1§KLipschitz^‡¤á. X J 푓(푥) = 푔(푥)+푥§Kþ˜^‡ƒ�u‡¦3«m [푎, 푏]þ ∣푔′(푥)+1∣ < 1. � 푔′3 [푎, 푏]þk.…ØCҞ§Ïé,‡ 훼 ∕= 0§훼푔(푥) = 0� …=� 푔(푥) = 0§�·‚Œ±ÏLÀ� 푓(푥) = 훼푔(푥) + 푥5­#N! ØÂñ¯K. ∙ �yÂñ§¤À�� 훼 7L÷vµ3¹ÐŠ�˜‡«m þ§∣훼푔′(푥) + 1∣ < 1. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 38 of 39 Go Back Full Screen Close Quit õŠ¯K éõ¯K¥·‚¡é�§äkõ‡Š,·‚�,F"ò¦‚�ÜéÑ5. Ï^�?n{´3cã�ù S“{¥}Áõ‡Ð©:. ¤±ù‡ž ÿ|^xã3éõœ¹e½k¤Ï.�´,˜„5`,XJ§Š�‡ê ™,·‚vk˜«˜„�{U �yé��Ü�Š. Home Page Title Page ◀◀ ▶▶ ◀ ▶ Page 39 of 39 Go Back Full Screen Close Quit þÅ ‡¦: rNÁÐ�§S“è()̇êŠ(J9gC©Û,^5ºI5)± ©�©‡(“.txt”)‚ªu�zli@nankai.edu.cn, Š’©‡Úe‡ÌK·¶‚ ª“R3 ÆÒ 6¶.txt” 1. ES�!‘¤ù§¦Š{��n,Ú½9ƒ'y² 2. ©O?�^�©{,Úî{,�{ÚØÄ:S“{¦š‚5§Š� ¼ê,¿¦§ 1 + 1/푥− log 푥 = 0 3«m[1, 5]S�Š,°Ý‡¦휖 = 10−4.