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Date: 11/24/98 at 16:29:56 From: Sam Subject: LaGrange Multiplier
¤º®e¦p¤U¡]ì¨÷¬O^¤å¡^ ¡½¡¼ §Ú¦³¤@Ó²³æªº°ÝÃD¡A¬O¦³Ãö©ó®Ô«a¦N«Y¼Æ ¡]LaGrange Multiplier¡^ªºÃD¥Ø¡A§Úªº®Ú¥» °ÝÃD¬O¦p¦ó¨M©w³o¨âÓ¤èµ{¦¡¤¤¡A¨º¤@Ó¬O ¨î±ø¥ó¦¡¡H¤j¦h¼ÆªºÃD¥Ø³£©ú½T«ü¥X¥Ø¼Ð¤èµ{ ¦¡¤Î©ú½T«ü¥X¨î¤èµ{¦¡¡A³oºØÃD¥Ø¡A§Ú¥un «ö·Ó¸ÑÃD¨BÆJ°µ¤U¥h¡C¦ý¬O¡A³o¤@¹DÃD¥Ø¤£¤Ó ¦n°µ¡AÃD¥Ø¤¤¨ºùØ©ú½T«ü¥X¨î±ø¥ó¦¡©O¡H¨Ò ¦p¡A³oùئ³¤@±ø§@·~ÃD¡C
¡e¡e ¦³¨â±ø¦±½u¡A¨C±ø½u¤W¦³¤@ÓÂI¥i¥H¦Û¥Ñªu½u ²¾°Ê¡A§ä¥X¨º¨âÓÂI¡]¤@½u¤@ÂI¡^ªº¶ZÂ÷³Ìªñ ¡C½Ð¥Î®Ô«a¦N«Y¼Æªk¸ÑÃD¡C y = 2x - 3 y = x^2 ¡f¡f
²{¦b§Úªº°ÝÃD¬O¡A³o¨âÓ¤èµ{¦¡¡A¨ì©³¨º¤@Ó ¬O¨î±ø¥ó¦¡¡H
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¤s©i¡A¹ê»Ú¤W©O¡]§A¦³¤ß²z·Ç³Æ¶Ü¡H¡^³o¨âÓ ¤½¦¡ y = 2x - 3 y = x^2 ³£¬O¨î¦¡¡I§A¥i¯àª¾¹DÃD¥Ø¥i¥H¥X²{¦hÓ ¨î¦¡¡C§A¥²¶·¹ï¨C¤@Ө¨Ï¥Î¤@Ó®Ô«a¦N «Y¼Æ¡C¨î¦¡ªº·N«ä¬O»¡¨¾¤î§AÂ÷¶}©w¸q°ì¡C ³o¤@¹DÃD¥Øªº«Ý·¥¤p¤Æ¥Ø¼Ð¨ç¼Æ¡]©Î«Ý³Ì¨Î¤Æ ¡^¬O¨âÂI¤§¶¡ªº¶ZÂ÷¡C (x[1],y[1]) ¤Î (x[2],y[2]) ¨âÂI¡C ÅãµM§ÚÌ¥i¥H¥O¨âÂIºÉ¥i¯à±µªñ¡Aª½¨ì¶ZÂ÷¬° ¹s¡]¨âÂI«¦X¡^¡C¦ý¬OÃD¥Ø«ü©w²Ä¤@ÂI¥²¶·¦b ²Ä¤@±ø¦±½u¤W¡B²Ä¤GÂI¥²¶·¦b²Ä¤G±ø¦±½u¤W¡C
©Ò¥H§A»Ýn¹ï¨âÂI¤§¶¡ªº¶ZÂ÷¬I¥H·¥¤j¤Æ¡C ¡]¦Û¥Ñ¤Hµù¡R¦¹³BÀ³¸Ó¬O¡u¬I¥H·¥¤p¤Æ¡v¡^ ¥Ø¼Ð¨ç¼Æ¬° sqrt[(x[1]-x[2])^2 + (y[1]-y[2])^2], ¦P®É¨ü¨ì¨î±ø¥ó y[1] = 2x[1] - 3, ¤Î y[2] = x[2]^2. ¨âÂI¥²¶·¦ì¦b¦U¦Û¦±½u¤W¡C
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G(x,y) = (x-1)*(x-1) + (y-2)*(y-2) R(x,y) = x + y - 3 = 0
G(x,y) ¥Nªí¥Ø¼Ð¨ç¼Æ (Goal) R(x,y) ¥Nªí¨î±ø¥ó (Restriction)
¥Ø¼Ð¨ç¼Æ¥i¥H·¥¤j¤Æ¡]³Ì¤j§Q¼í¡^¡A ¥Ø¼Ð¨ç¼Æ¥i¥H·¥¤p¤Æ¡]³Ì¤p¦¨¥»¡^¡C ·¥¤j¤Æªº¥Ø¼Ð¨ç¼Æ¼¥Ht¤@Åܬ° ·¥¤p¤Æªº¥Ø¼Ð¨ç¼Æ¡A©Ò¥H·¥¤j¤Æ©Î·¥¤p¤Æ³B²z ¤èªk¬Û¦P¡A³¦h³Ì¨Î³]p½Ò¥»²Î¤@¬°·¥¤p¤Æ°Ý ÃD¡A¨Ã¥BºÙ¥Ø¼Ð¨ç¼Æ¬°¦¨¥»¨ç¼Æ¡CÅ¥¨£¡u¦¨¥» ¡v¤j®a³£ª¾¹D¦¨¥»¶V§C¶V¦n¡C¤£½×¥Ø¼Ð¨ç¼Æ¬O §_¯uªº°Q½×°§C¦¨¥»¡A¡u¦¨¥»¨ç¼Æ¡vªº¡u¦¨¥» ¡v¥u¬O§Î®eµü¡C
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·¥¤p¤Æ¥Ø¼Ð¨ç¼Æ G(x,y) = (x-1)*(x-1) + (y-2)*(y-2)
¨ü¨ì¨î±ø¥ó R(x,y) = x + y - 3 = 0
¦b³Ì¨Î³]p½Ò¥»¤¤¡A©Ò¦³¨î±ø¥ó¦¡²Î¤@¬° ¨î±ø¥ó¤½¦¡¡@¡×¡@¢¯
¬°¤°»ò¨î±ø¥ó¥þ³¡½Õ¾ã¬°¡uµ¥©ó¹s¡v¡H
²Ä¤@ ¦]¬°¤£µ¥©ó¹sªº¨î¦¡¦b²¾¶µ¤§«áÅܬ°µ¥©ó¹s ¡A¨Ò¦p¨î¦¡ x+y=5 »P x+y-5=0 ¥þµ¥¡C
²Ä¤G «Ý·¥¤p¤Æ¥Ø¼Ð¨ç¼Æ G(x,y) n»P¡@¡@¨î±ø¥ó R(x,y) ¬Û¥[¡A¦¨¬°½Æ¦Xªº ®Ô«a¦N¨ç¼Æ L(x,y)=G(x,y)+lambda*R(x,y) ¨ä¤¤ lambda¬O®Ô«a¦N«Y¼Æ¡C¤]´N¬O ®Ô«a¦N¨ç¼Æ¡×¥Ø¼Ð¨ç¼Æ¡Ï®Ô«a¦N«Y¼Æ¡¯¨î±ø¥ó
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Y¨î±ø¥ó R(x,y) «íµ¥©ó¹s¡A¥Ø¼Ð¨ç¼Æ¥[¹s ¤£§ïÅܥؼШç¼ÆªºÈ¡C¤]´N¬O ¦pªG¨î±ø¥ó¥þ³¡½Õ¾ã¬°µ¥©ó¹s¡A¨º»ò¡A ®Ô«a¦N¨ç¼ÆªºÈ»P¥Ø¼Ð¨ç¼ÆªºÈ¬Ûµ¥¡C
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¦pªG¨î±ø¥ó¡@¡´¤£¡´¡@½Õ¾ã¬°¡uµ¥©ó¹s¡v ¡u·L¤Àªºµ²ªG¥O¬°¹s¡v¤£¯àÁÙì¨î±ø¥ó¦¡¡I
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¡´¡@¬JµM¨î±ø¥ó R(x,y) «íµ¥©ó¹s¡A ¡´¡@¬G¡u = 0¡v¸g±`²¤°£¡A
ÃD¥Ø²¤Æ¬°¡]²¤°£¡u = 0¡v¡^ ·¥¤p¤Æ G(x,y) = (x-1)*(x-1) + (y-2)*(y-2) ¨î¦¡ R(x,y) = x + y - 3
¬°¤°»òn¥Î G(x,y) ¤Î R(x,y) ¡H ¤£¥Î G(x,y) ¤Î¤£¥Î R(x,y) ¥i¥H¶Ü¡H
¥kºÝªºÁcÂø¤½¦¡¥H¥ªºÝ²²Å¥Nªí¡C
¸ÑÃD¨BÆJ»Ýn©w¸q®Ô«a¦N¨ç¼Æ¦p¤U L(x,y)=G(x,y)+lambda*R(x,y) ¤ñ¸ûºë²¡C¦pªG¤£¥Î²²Å¥Nªí¡Aª½±µ¼g¬° L(x,y)=(x-1)*(x-1) + (y-2)*(y-2)+lambda*(x + y - 3) ¤ñ¸ûÁcÂø¡A©Ò¥H¨Ï¥Î G(x,y) ¤Î R(x,y) ¡C
²Ö¿n¥H¤WªºI´º±ø¥ó¤§«á¡A ¤U±ªº¦C¦¡¡]²¤°£¡u = 0¡v¤Î¨Ï¥Î G, R¡^ G(x,y) = (x-1)*(x-1) + (y-2)*(y-2) R(x,y) = x + y - 3 ¨Ï·s¤â§x´b¡I
¬O ·¥¤p¤Æ G(x,y) = (x-1)*(x-1) + (y-2)*(y-2) ¨î¦¡ R(x,y) = x + y - 3 ¶Ü¡H
¬O ·¥¤p¤Æ R(x,y) = x + y - 3 ¨î¦¡ G(x,y) = (x-1)*(x-1) + (y-2)*(y-2) ¶Ü¡H
¬O»{©w G(x,y) ¬°¥Ø¼Ð¨ç¼Æ¶Ü¡H¦ý¬O G(x,y)¬OÀH·N¥N²Å¡A¦pªG§ï¥Î C(x,y) = x + y - 3 D(x,y) = (x-1)*(x-1) + (y-2)*(y-2) §Ú«ç»òª¾¹D½Ö¬O½Ö¡H¡I
¬°¦¹¡A¦Û¥Ñ¤H´¿¸gW«ä«Ü¤[¡C³Ì«áª`·N¨ä®t§O
·¥¤p¤Æ¤½¦¡ªº G(x,y) ¬OÅÜ°ÊÈ¡A¤£¯àÂꦺ¡C ¨î¦¡ªº¡@¡@ R(x,y) ¬O©T©wÈ¡A¤£¯àÅÜ°Ê¡C ¡´¡@½Ðª`·N¡A¤W±¨â¦æ¬O«ÂI¡I¡@¡´
·¥¤p¤Æ¤½¦¡ªº G(x,y) ¬OÅÜ°ÊÈ¡A¤£¯àÂꦺ¡C ¦]¬°¡u·¥¤p¤Æ¡vªº·N«ä¬O¦³¦h«¿ï¾Ü¡A ¦pªG«ü©w G(x,y) µ¥©ó¹s¡A©Î ¦pªG«ü©w G(x,y) µ¥©ó¤@¡A½Ð·Q¤@·Q¡A G(x,y) = (x-1)*(x-1) + (y-2)*(y-2) µ¥©ó±`¼Æ¡H ¤w¸gÂê©w¤F¡A¤£¥i¯à°õ¦æ·¥¤p¤Æ¡I¡I
¥t¤@¤è±¡A ¨î¦¡ªº¡@¡@ R(x,y) ¬O©T©wÈ¡A¤£¯àÅÜ°Ê¡C ¦pªG R(x,y) = x + y - 3 µ¥©óÅÜ°ÊÈ¡A³o´N ¤£¬O¨î±ø¥ó¤F¡I R(x,y) = x + y - 3 = 0 Âê©w x, y Ãö«Y R(x,y) = x + y - 3 = ¥ô·NÈ¡A x, y µLÃö
²{¦b¦^¨ì¤s©iªº°ÝÃD¡A¤wª¾¨âÓ¤½¦¡ y = 2x - 3 y = x^2 ¥i¥H§ï¼g¦p¤U y - 2x + 3 = 0 y - x^2 = 0 ³£¬Oµ¥©ó±`¼Æ¡]¹s¡^ªº¤½¦¡¡A©Ò¥H¤GªÌ³£¬O ¨î¦¡¡C¡]¦¹³B¦Û¥Ñ¤H¦^µª¤F¤s©iªº°ÝÃD¡^
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