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Dijkstra綆楁硶鏄敱鑽峰叞璁$畻鏈虹瀛﹀鑹懼吂鏍悸瘋開縐戞柉褰誨彂鐜扮殑銆傜畻娉曡В鍐崇殑鏄湁鍚戝浘鎴栨棤鍚戝浘涓渶鐭礬寰勯棶棰樸?br />綆楁硶鎻忚堪錛?br />聽聽聽聽聽聽聽聽聽榪欎釜綆楁硶鏄氳繃涓烘瘡涓《鐐?i>v淇濈暀鐩墠涓烘鎵鎵懼埌鐨勪粠s鍒皏鐨勬渶鐭礬寰勬潵宸ヤ綔鐨勩傚垵濮嬫椂錛屾簮鐐箂鐨勮礬寰勯暱搴﹀艱璧嬩負0錛?i>d[s]=0錛夛紝 鍚屾椂鎶婃墍鏈夊叾浠栭《鐐圭殑璺緞闀垮害璁句負鏃犵┓澶э紝鍗寵〃紺烘垜浠笉鐭ラ亾浠諱綍閫氬悜榪欎簺欏剁偣鐨勮礬寰勶紙瀵逛簬V涓墍鏈夐《鐐?i>v闄?i>s澶?i>d[v]= 鈭烇級銆傚綋綆楁硶緇撴潫鏃訛紝d[v]涓偍瀛樼殑渚挎槸浠?i>s鍒?i>v鐨勬渶鐭礬寰勶紝鎴栬呭鏋滆礬寰勪笉瀛樺湪鐨勮瘽鏄棤絀峰ぇ銆?Dijstra綆楁硶鐨勫熀紜鎿嶄綔鏄竟鐨勬嫇灞曪細濡傛灉瀛樺湪涓鏉′粠u鍒?i>v鐨勮竟錛岄偅涔堜粠s鍒?i>u鐨勬渶鐭礬寰勫彲浠ラ氳繃灝嗚竟(u,v)娣誨姞鍒板熬閮ㄦ潵鎷撳睍涓鏉′粠s鍒皏鐨勮礬寰勩傝繖鏉¤礬寰勭殑闀垮害鏄痙[u]+w(u,v)銆傚鏋滆繖涓兼瘮鐩墠宸茬煡鐨?i>d[v]鐨勫艱灝忥紝鎴戜滑鍙互鐢ㄦ柊鍊兼潵鏇夸唬褰撳墠d[v]涓殑鍊箋傛嫇灞曡竟鐨勬搷浣滀竴鐩存墽琛屽埌鎵鏈夌殑d[v]閮戒唬琛ㄤ粠s鍒皏鏈鐭礬寰勭殑鑺辮垂銆傝繖涓畻娉曠粡榪囩粍緇囧洜鑰屽綋d[u]杈懼埌瀹冩渶緇堢殑鍊肩殑鏃跺欐病鏉¤竟(u,v)閮藉彧琚嫇灞曚竴嬈°?br />
綆楁硶嫻佺▼錛?br />1. 鍦ㄧ畻娉曚腑姣忎釜緇撶偣涓兘榪涜鏍囨敞錛屾爣娉ㄥ垎涓?font color="#ff0000">涓存椂鎬ф爣娉?/font>鍜?font color="#ff0000">姘鎬箙鎬ф爣娉?/font>錛?
2.鍒濆鏃?/font>錛屾墍鏈夌粨鐐歸兘涓?font color="#ff0000">涓存椂鎬ф爣娉?/font>錛?font color="#ff0000">鏍囨敞鍊間負鏃犵┓澶?/font>錛?br />3.灝?font color="#ff0000">婧愮粨鐐?/font>鏍囨敞涓?錛屼笖涓烘案涔呮ф爣娉紝騫朵護鍏朵負宸ヤ綔緇撶偣錛?br />4.媯鏌ヤ笌宸ヤ綔緇撶偣鐩擱偦騫舵爣娉ㄤ負涓存椂鏍囨敞鐨?/font>緇撶偣錛?font color="#ff0000">鑻ヨ緇撶偣鍒板伐浣滅粨鐐圭殑璺濈涓庡伐浣滅粨鐐圭殑鏍囨敞涔嬪拰灝忎簬璇ョ粨鐐圭殑鏍囨敞錛屽垯鐢ㄦ柊璁$畻寰楀埌鐨勫拰閲嶆柊鏍囨敞璇ョ粨鐐癸紝騫跺皢宸ヤ綔緇撶偣璁板綍涓鴻緇撶偣鐨勪笂涓緇撶偣錛?br />5.鍦?/font>鏁翠釜鍥?/font>涓煡鎵懼叿鏈夋渶灝忓肩殑涓存椂鎬ф爣娉ㄧ粨鐐癸紝灝嗗叾鍙樹負姘鎬箙鎬х粨鐐癸紝騫舵垚涓轟笅涓杞鏌ョ殑宸ヤ綔緇撶偣錛?br />閲嶅絎洓銆佷簲姝ワ紝鐩村埌鐩殑緇撶偣鎴愪負宸ヤ綔緇撶偣銆傝繖鏃剁洰鏍囩粨鐐逛腑鐨勬爣娉ㄥ煎氨鏄洰鏍囩粨鐐瑰埌婧愮粨鐐圭殑鏈鐭礬寰勫箋傝屽洖婧?font color="#0000ff">鍚勭粨鐐圭殑涓婁竴緇撶偣鐩村埌婧愮粨鐐廣傚氨鏄渶鐭礬寰勩?br />
紺轟緥錛?br />
鏂囩珷璧勬簮鏉ユ簮浜庯細
1. http://www.roboticfan.com/college/knowledge/200608/195.shtml
2. http://johnfat.bokee.com/3435427.html
涓夐暅鍏堢敓
涓銆佺畻娉曚竴璇嶇殑鏉ユ簮
銆銆Algos鏄笇鑵婂瓧錛屾剰鎬濇槸鈥?/span>鐤?span lang="EN-US">鈥?/span>錛?span lang="EN-US">A1gor鏄媺涓佸瓧錛屾剰鎬濇槸鈥?/span>鍐峰嵈鈥?/span>銆傝繖涓や釜瀛楅兘涓嶆槸A1gorithm(綆楁硶)涓璇嶇殑璇嶆牴錛?span lang="EN-US">a1gorithm涓璇嶅嵈涓?span lang="EN-US">9涓栫邯鐨勯樋鎷変集瀛﹁?span lang="EN-US">al-Khwarizmi鏈夊叧錛屼粬鍐欑殑涔︺?span lang="EN-US">al-jabr w鈥檃l muqabalah銆?span lang="EN-US">(浠f暟瀛?span lang="EN-US">)婕斿彉鎴愪負鐜板湪涓鐨勪唬鏁版暀縐戜功銆?span lang="EN-US">Ad-Khwarizmi寮鴻皟姹傝В闂鐨勬湁鏉$悊鐨勬楠ゃ傚鏋滀粬鑳芥椿鍒頒粖澶╃殑璇濓紝浠栦竴瀹氫細琚互浠栫殑鍚嶅瓧鑰屽緱鍚嶇殑鏂規硶鐨勮繘灞曟墍鎰熷姩銆?/font>
浜屻?span lang="EN-US">20涓栫邯10鏈濂界殑綆楁硶
銆銆20涓栫邯鏈濂界殑綆楁硶錛岃綆楁満鏃朵唬鐨勬寫閫夋爣鍑嗘槸瀵圭瀛﹀拰宸ョ▼鐨勭爺絀跺拰瀹炶返褰卞搷鏈澶с備笅闈㈠氨鏄寜騫翠唬嬈″簭鎺掑垪鐨?span lang="EN-US">20涓栫邯鏈濂界殑10涓畻娉曘?span lang="EN-US">
1.聽 Monte Carlo鏂規硶
聽聽聽1946騫達紝鍦ㄦ礇鏂樋鎷夎帿鏂瀛﹀疄楠屽宸ヤ綔鐨?span lang="EN-US">John von Neumann錛?span lang="EN-US">Stan Ulam鍜?span lang="EN-US">Nick Metropolis緙栧埗浜?span lang="EN-US">Metropolis綆楁硶錛屼篃縐頒負Monte Carlo鏂規硶銆?span lang="EN-US">Metropolis綆楁硶鏃ㄥ湪閫氳繃妯′豢闅忔満榪囩▼錛屾潵寰楀埌鍏鋒湁闅句互鎺у埗鐨勫ぇ閲忕殑鑷敱搴︾殑鏁板奸棶棰樺拰鍏鋒湁闃朵箻瑙勬ā鐨勭粍鍚堥棶棰樼殑榪戜技瑙f硶銆傛暟瀛楄綆楁満鏄‘瀹氭ч棶棰樼殑璁$畻鐨勫己鏈夊姏
宸ュ叿錛屼絾鏄浜庨殢鏈烘э紙涓嶇‘瀹氭э級闂濡備綍褰撴椂騫朵笉鐭ユ檽錛?span lang="EN-US">Metropolis綆楁硶鍙互璇存槸鏈鏃╃殑鐢ㄦ潵鐢熸垚闅忔満鏁幫紝瑙e喅涓嶇‘瀹氭ч棶棰樼殑綆楁硶涔嬩竴銆?/font>
2.聽 綰挎ц鍒掔殑鍗曠函褰㈡柟娉?/font>
聽聽聽 1947騫達紝鍏板痙鍏徃鐨?span lang="EN-US">Grorge Dantzig鍒涢犱簡綰挎ц鍒掔殑鍗曠函褰㈡柟娉曘傚氨鍏跺箍娉涚殑搴旂敤鑰岃█錛?span lang="EN-US">Dantzig綆楁硶涓鐩存槸鏈鎴愬姛鐨勭畻娉曚箣涓銆傜嚎鎬ц鍒掑浜庨偅浜涜鎯沖湪緇忔祹涓婄珯浣忚剼錛屽悓鏃跺張鏈夎禆浜庢槸鍚﹀叿鏈夊湪棰勭畻鍜屽叾浠栫害鏉熸潯浠朵笅杈懼埌鏈浼樺寲鐨勮兘鍔涚殑宸ヤ笟鐣岋紝鏈夌潃鍐沖畾鎬х殑褰卞搷(褰撶劧錛屽伐涓氫腑鐨?span lang="EN-US">鈥?/span>瀹為檯鈥?/span>闂寰寰鏄潪綰挎х殑錛涗嬌鐢ㄧ嚎鎬ц鍒掓湁鏃跺欐槸鐢變簬浼拌鐨勯綆楋紝浠庤岀畝鍖栦簡妯″瀷鑰屼績鎴愮殑)銆傚崟綰艦娉曟槸涓縐嶈兘杈懼埌鏈浼樿В鐨勭簿緇嗙殑鏂規硶銆傚敖綆$悊璁轟笂璁插叾鏁堟灉鏄寚鏁拌“鍑忕殑錛屼絾鍦ㄥ疄璺典腑璇ョ畻娉曟槸楂樺害鏈夋晥鐨?span lang="EN-US">鈥斺?/span>瀹冩湰韜鏄庝簡鏈夊叧璁$畻鐨勬湰璐ㄧ殑涓浜涙湁瓚g殑浜嬫儏銆?/font>
3.聽 Krylov瀛愮┖闂村彔浠f硶
1950騫達紝鏉ヨ嚜緹庡浗鍥藉鏍囧噯灞鐨勬暟鍊煎垎鏋愮爺絀舵墍鐨?span lang="EN-US">Magnus Hestenes, Eduard Stiefel鍜?span lang="EN-US">Cornelius Lanczos寮鍒涗簡Krylov瀛愮┖闂村彔浠f硶鐨勭爺鍒躲傝繖浜涚畻娉曞鐞嗙湅浼肩畝鍗曠殑姹傝В褰負聽聽聽
Ax=b
鐨勬柟紼嬬殑闂銆傚綋鐒墮殣钘忕殑鍥伴毦鍦ㄤ簬A鏄竴涓法鍨嬬殑n*n 鐭╅樀錛岃嚧浣夸唬鏁拌В聽聽
x=b/A
鏄笉瀹規槗璁$畻鐨?span lang="EN-US">(紜疄錛岀煩闃電殑鈥?/span>鐩擱櫎鈥?/span>涓嶆槸涓涓疄闄呬笂鏈夌敤鐨勬蹇?span lang="EN-US">)銆傚彔浠f硶鈥斺?/span>璇稿姹傝В褰負
Kx(k+1)=Kx(k)+b-Ax(k)
鐨勬柟紼嬶紝鍏朵腑K 鏄竴涓悊鎯沖湴鈥?/span>鎺ヨ繎鈥滱 鐨勮緝涓虹畝鍗曠殑鐭╅樀鈥斺?/span>瀵艱嚧浜?span lang="EN-US">Krylov瀛愮┖闂寸殑鐮旂┒銆備互淇勭綏鏂暟瀛﹀Nikolai Krylov鍛藉悕鐨?span lang="EN-US">Krylov瀛愮┖闂寸敱浣滅敤鍦ㄥ垵濮?span lang="EN-US">鈥?/span>浣欓噺鈥?/span>鍚戦噺
聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽 r(0)=b-Ax(0)
涓婄殑鐭╅樀騫傚紶鎴愮殑銆傚綋 A鏄縐扮煩闃墊椂錛?span lang="EN-US">Lanczos鎵懼埌浜嗕竴縐嶇敓鎴愯繖縐嶅瓙絀洪棿鐨勬浜ゅ熀鐨勬瀬濂界殑鏂規硶銆傚浜庡縐版瀹氱殑鏂圭▼緇勶紝Hestenes 鍜?span lang="EN-US">Stiefel鎻愬嚭浜嗙О涓哄叡杞搴︽硶鐨勭敋鑷蟲洿濡欑殑鏂規硶銆傝繃鍘葷殑50騫翠腑錛岃澶氱爺絀朵漢鍛樻敼榪涘茍鎵╁睍浜嗚繖浜涚畻娉曘傚綋鍓嶇殑涓濂楁柟娉曞寘鎷潪瀵圭О鏂圭▼緇勭殑姹傝В鎶宸э紝鍍忓瓧棣栫緝鎷艱瘝涓?span lang="EN-US">GMRES鍜?span lang="EN-US">Bi-CGSTAB閭f牱鐨勭畻娉曘?span lang="EN-US">(GMRES鍜?span lang="EN-US">Bi-CGSTAB鍒嗗埆棣栨鍑虹幇浜?span lang="EN-US">1986鍜?span lang="EN-US">1992聽 SIAM journal on Scientific and Statistical computing(緹庡浗宸ヤ笟涓庡簲鐢ㄦ暟瀛﹀浼氱殑縐戝鍜岀粺璁¤綆楁潅蹇?span lang="EN-US">)銆?/font>
4.聽 鐭╅樀璁$畻鐨勫垎瑙f柟娉?/font>
聽聽聽1951騫達紝姍℃爲宀浗瀹跺疄楠屽鐨?span lang="EN-US">A1ston Householder緋葷粺闃愯堪浜嗙煩闃佃綆楃殑鍒嗚В鏂規硶銆傜爺絀惰瘉鏄庤兘鎶婄煩闃靛洜瀛愬垎瑙d負涓夎銆佸瑙掋佹浜ゅ拰鍏朵粬鐗規畩褰㈠紡鐨勭煩闃墊槸鏋佸叾鏈夌敤鐨勩傝繖縐嶅垎瑙f柟娉曚嬌杞歡鐮旂┒浜哄憳鑳界敓浜у嚭鐏墊椿鏈夋晥鐨勭煩闃佃蔣浠跺寘銆傝繖涔熶績榪涗簡鏁板肩嚎鎬т唬鏁頒腑鍙嶅鍑虹幇鐨勫ぇ闂涔嬩竴鐨勮垗鍏ヨ宸垎鏋愰棶棰樸?span lang="EN-US"> (1961騫翠雞鏁﹀浗瀹剁墿鐞嗗疄楠屽鐨?span lang="EN-US">James Wilkinson鍩轟簬鎶婄煩闃靛垎瑙d負涓嬪拰涓婁笁瑙掔煩闃靛洜瀛愮殑縐殑LU鍒嗚В錛屽湪緹庡浗璁$畻鏈哄崗浼?span lang="EN-US">(ACM)鐨勬潅蹇椾笂鍙戣〃浜嗕竴綃囬涓?span lang="EN-US">鈥?/span>鐭╅樀閫嗙殑鐩存帴鏂規硶鐨勮宸垎鏋?span lang="EN-US">鈥?/span>鐨勯噸瑕佹枃绔犮?span lang="EN-US">)
5.聽 Fortran鏈浼樼紪璇戠▼搴?/font>
聽聽聽 1957騫達紝John Backus鍦?span lang="EN-US">IBM棰嗗涓涓皬緇勭爺鍒?span lang="EN-US">Fortran鏈浼樼紪璇戠▼搴忋?/font>
Fortran
鐨勫垱閫犲彲鑳芥槸璁$畻鏈虹紪紼嬪巻鍙蹭笂鐙竴鏃犱簩鐨勬渶閲嶈鐨勪簨浠訛細縐戝瀹?span lang="EN-US">(鍜屽叾浠栦漢)緇堜簬鍙互鏃犻渶渚濋潬鍍忓湴鐙遍偅鏍峰彲鎬曠殑鏈哄櫒浠g爜錛屽氨鍙憡璇夎綆楁満浠栦滑鎯寵鍋氫粈涔堛傝櫧鐒剁幇浠g紪璇戠▼搴忕殑鏍囧噯騫朵笉榪囧垎鈥曗旻ortran I鍙寘鍚?span lang="EN-US">23錛?span lang="EN-US">500鏉℃眹緙栬璦鎸囦護鈥曗?/span>鏃╂湡鐨勭紪璇戠▼搴忎粛鐒惰兘瀹屾垚浠や漢鍚冩儕鐨勫鏉傝綆椼傚氨鍍?span lang="EN-US">Backus鏈漢鍦?span lang="EN-US">1998騫村湪IEEE annals of the History of computing 鍙戣〃鐨勬湁鍏?span lang="EN-US">Fortran I錛?span lang="EN-US">II, III鐨勮繎浠e巻鍙茬殑鏂囩珷涓洖蹇嗛亾錛氱紪璇戠▼搴?span lang="EN-US">鈥?/span>鎵浜х敓鐨勫姝ゆ湁鏁堢殑浠g爜錛屼嬌寰楀叾杈撳嚭浠ょ爺絀跺畠鐨勭紪紼嬩漢鍛橀兘鎰熷埌鍚撲簡涓璺熾?span lang="EN-US">鈥?/span>
6.聽 鐭╅樀鏈緛鍊艱綆楃殑QR綆楁硶
聽聽聽 1959鈥?1騫達紝浼︽暒Ferranti Ltd.鐨?span lang="EN-US">J.G. F. Francis鎵懼埌浜嗕竴縐嶇О涓?span lang="EN-US">QR綆楁硶鐨勮綆楁湰寰佸肩殑紼沖畾鐨勬柟娉曘傛湰寰佸煎ぇ姒傛槸鍜岀煩闃電浉榪炲湪鈥?/span>璧風殑鏈閲嶈鐨勬暟浜嗭紝鑰屼笖璁$畻瀹冧滑鍙兘鏄渶闇瑕佹妧宸х殑銆傛妸鈥?/span>涓柟闃靛彉鎹負涓涓?span lang="EN-US">鈥?/span>鍑犱箮鏄?span lang="EN-US">鈥?/span>涓婁笁瑙掔殑鐭╅樀鈥曗?/span>鎰忓嵆鍦ㄧ揣鎸ㄧ潃鐭╅樀涓誨瑙掔嚎涓嬮潰鐨勪竴鏂滃垪涓婂彲鑳芥湁闈為浂鍏冪礌鈥曗?/span>鏄浉瀵瑰鏄撶殑錛屼絾瑕佹兂涓嶄駭鐢熷ぇ閲忕殑璇樊灝辨妸榪欎簺闈為浂鍏冪礌娑堝幓錛屽氨涓嶆槸騫沖嚒鐨勪簨浜嗐?span lang="EN-US">QR 綆楁硶姝eソ鏄兘杈懼埌榪欎竴鐩殑鐨勬柟娉曪紝鍩轟簬QR 鍒嗚В錛?span lang="EN-US"> A鍙互鍐欐垚
姝d氦鐭╅樀Q 鍜屼竴涓笁瑙掔煩闃?span lang="EN-US">R 鐨勪箻縐紝榪欑鏂規硶鍙犱唬鍦版妸
聽A=Q(k)R(k)
鍙樻垚A(k+1)==Q(k)R(k) 灝卞姞閫熸敹鏁涘埌涓婁笁瑙掔煩闃佃岃█澶氬皯鏈夌偣涓嶈兘鎸囨湜銆?span lang="EN-US">20涓栫邯60騫翠唬涓湡QR 綆楁硶鎶婁竴搴﹂毦浠ュ浠樼殑鏈緛鍊奸棶棰樺彉鎴愪簡渚嬭紼嬪簭鐨勮綆椼?/font>
7.聽 蹇熷垎綾繪硶
聽聽聽1962錛氫雞鏁?span lang="EN-US">Elliott Brothers, Ltd.鐨?span lang="EN-US">Tony Hoare鎻愬嚭浜嗗揩閫?span lang="EN-US">(鎸夊ぇ灝?span lang="EN-US">)鍒嗙被娉?/font>
.鎶?span lang="EN-US">n涓簨鐗╂寜鏁版垨瀛楁瘝鐨勬搴忔帓鍒楄搗鏉ワ紝鍦ㄥ績鏅轟笂鏄笉浼氭湁浠涔堣Е鍔ㄧ殑鍗曡皟騫沖嚒鐨勪簨銆傛櫤鍔涚殑鎸戞垬鍦ㄤ簬鍙戞槑涓縐嶅揩閫熷畬鎴愭帓搴忕殑鏂規硶銆?span lang="EN-US">Hoare鐨勭畻娉曞埄鐢ㄤ簡鍙よ佺殑鍒嗗壊寮鍜屾帶鍒剁殑閫掑綊絳栫暐鏉ヨВ鍐抽棶棰橈細鎸戜竴涓厓绱犱綔涓?span lang="EN-US">鈥?/span>涓誨厓鈥?/span>銆佹妸鍏朵綑鐨勫厓绱犲垎鎴?span lang="EN-US">鈥?/span>澶х殑鈥?/span>鍜?span lang="EN-US">鈥?/span>灝忕殑鈥?/span>涓ゅ爢
(
褰撳拰涓誨厓姣旇緝鏃?span lang="EN-US">)銆佸啀鍦ㄦ瘡涓鍫嗕腑閲嶅榪欎竴榪囩▼銆傚敖綆″彲鑳借鍋氬彈鍒頒弗鍘夎矗澶囩殑鍋氬畬鍏ㄩ儴N(N-1)/2 嬈$殑姣旇緝(鐗瑰埆鏄紝濡傛灉浣犳妸涓誨厓浣滀負鏃╁凡鎸夊ぇ灝忓垎綾誨ソ鐨勮〃鍒楃殑絎竴涓厓绱犵殑
璇濓紒)錛屽揩閫熷垎綾繪硶榪愯鐨勫鉤鍧囨鏁板叿鏈?span lang="EN-US">O(Nlog(N)) 鐨勬湁鏁堟э紝鍏朵紭緹庣殑綆媧佹т嬌涔嬫垚涓鴻綆楀鏉傛х殑钁楀悕鐨勪緥瀛愩?/font>
8.聽 蹇?span lang="EN-US">Fourier鍙樻崲
聽聽聽 1965騫達紝IBM鐨?span lang="EN-US">T. J. Watson鐮旂┒涓績鐨?span lang="EN-US">James Cooley浠ュ強鏅灄鏂】澶у鍜?span lang="EN-US">AT錛?span lang="EN-US">T璐濆皵瀹為獙瀹ょ殑John Tukey鍚戝叕浼楅忛湶浜嗗揩閫?span lang="EN-US">Fourier鍙樻崲(鏂規硶)(FFT)銆傚簲鐢ㄦ暟瀛︿腑鎰忎箟鏈娣辮繙鐨勭畻娉曪紝鏃犵枒鏄嬌淇″彿澶勭悊瀹炵幇紿佺牬鎬ц繘灞曠殑FFT銆傚叾鍩烘湰鎬濇兂瑕佽拷
婧埌Gauss(浠栭渶瑕佽綆楀皬琛屾槦鐨勮建閬?span lang="EN-US">)錛屼絾鏄?span lang="EN-US">Cooley鈥擳ukey鐨勮鏂囧紕娓呮浜?span lang="EN-US">Fourier鍙樻崲璁$畻璧鋒潵鏈夊瀹規槗銆傚氨鍍忓揩閫熷垎綾繪硶涓鏍鳳紝FFT鏈夎禆浜庣敤鍒嗗壊寮鍜屾帶鍒剁殑絳栫暐錛屾妸琛ㄩ潰涓婁護浜鴻鍘岀殑O(N*N) 闄嶅埌浠や漢嬈箰鐨?span lang="EN-US">O(Nlog(N)) 銆備絾鏄笉鍍忓揩閫熷垎綾繪硶錛屽叾鎵ц(鍒濅竴鐪?span lang="EN-US">)鏄潪鐩磋鐨勮屼笖涓嶉偅涔堢洿鎺ャ傚叾鏈韓灝辯粰璁$畻鏈虹瀛︿竴縐嶆帹鍔ㄥ姏鍘葷爺絀惰綆楅棶棰樺拰綆楁硶鐨勫浐鏈夊鏉傛с?/font>
9.聽 鏁存暟鍏崇郴渚︽煡綆楁硶
聽聽聽 1977騫達紝BrighamYoung澶у鐨?span lang="EN-US">Helaman Ferguson 鍜?span lang="EN-US">Rodney Forcade鎻愬嚭浜嗘暣鏁板叧緋諱睛鏌ョ畻娉曘傝繖鏄竴涓彜鑰佺殑闂錛氱粰瀹?span lang="EN-US">鈥?/span>緇勫疄鏁幫紝渚嬪璇?span lang="EN-US">x(1),x(2),...,x(n) 錛屾槸鍚﹀瓨鍦ㄦ暣鏁?span lang="EN-US">a(1),a(2),..,a(n) 錛堜笉鍏ㄤ負闆訛級錛屼嬌寰?/font>
聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽聽 a(1)x(1)+a(2)x(2)+...+a(n)x(n)=0
瀵逛簬n=2 錛屽巻鍙叉偁涔呯殑嬈у嚑閲屽緱綆楁硶鑳藉仛榪欓」宸ヤ綔銆佽綆?span lang="EN-US">x(1)/x(2) 鐨勮繛鍒嗘暟灞曞紑涓殑鍚勯」銆傚鏋?span lang="EN-US">x(1)/x(2) 鏄湁鐞嗘暟錛屽睍寮浼氱粓姝紝鍦ㄩ傚綋灞曞紑鍚庡氨緇欏嚭浜?span lang="EN-US">鈥?/span>鏈灝忕殑鈥?/span>鏁存暟a(1)聽鍜?span lang="EN-US">a(2) 銆傛鍑犻噷寰楃畻娉曚笉緇堟鈥斺?/span>鎴栬呭鏋滀綘鍙槸綆鍗曞湴鐢變簬鍘屽﹁綆?span lang="EN-US">鈥斺?/span>閭d箞灞曞紑鐨勮繃紼嬭嚦灝戞彁渚涗簡鏈灝忔暣鏁板叧緋葷殑澶у皬鐨勪笅鐣屻?span lang="EN-US">Ferguson鍜?span lang="EN-US">Forcade鐨勬帹騫挎洿鏈夊▉鍔涳紝灝界榪欑鎺ㄥ箍鏇撮毦浜庢墽琛?span lang="EN-US">(鍜岀悊瑙?span lang="EN-US">)銆備緥濡傦紝浠栦滑鐨勪睛鏌ョ畻娉曡鐢ㄦ潵姹傚緱閫昏緫鏂皼(logistic)鏄犲皠鐨勭涓夊拰絎洓涓垎姝х偣錛?span lang="EN-US">b(3)=3.544090 鍜?span lang="EN-US"> b(4)=3.564407鎵婊¤凍鐨勫欏瑰紡鐨勭簿紜郴鏁般?span lang="EN-US">(鍚庤呮槸120 闃剁殑澶氶」寮忥紱瀹冪殑鏈澶х殑緋繪暟鏄?span lang="EN-US">257^30 銆?span lang="EN-US">)宸茶瘉鏄庤綆楁硶鍦ㄧ畝鍖栭噺瀛愬満璁轟腑鐨?span lang="EN-US">Feynman鍥劇殑璁$畻涓槸鏈夌敤鐨勩?/font>
10. 蹇熷鏋佺畻娉?/font>
聽聽聽 1987騫達紝鑰墮瞾澶у鐨?span lang="EN-US">Leslie Greengard 鍜?span lang="EN-US">Vladimir Rokhlin鍙戞槑浜嗗揩閫熷鏋佺畻娉曘?/font>
璇ョ畻娉曞厠鏈嶄簡N浣撴ā鎷熶腑鏈浠や漢澶寸柤鐨勫洶闅句箣涓錛氱粡鐢卞紩鍔涙垨闈欑數鍔涚浉浜掍綔鐢ㄧ殑N涓矑瀛愯繍鍔ㄧ殑綺劇‘璁$畻(鎯寵薄涓涓嬮摱娌崇郴涓殑鏄熶綋錛屾垨鑰呰泲鐧借川涓殑鍘熶簬)鐪嬫潵闇瑕?span lang="EN-US">O(N*N) 鐨勮綆楅噺
鈥斺?/font>
姣旇緝姣忎竴瀵硅川鐐歸渶瑕佷竴嬈¤綆椼傝綆楁硶鍒╃敤澶氭瀬灞曞紑(鍑鐢佃嵎鎴栬川閲忋佸伓鏋佺煩銆佸洓鐭╋紝絳夌瓑)鏉ヨ繎浼奸仴榪滅殑涓緇勮川鐐瑰褰撳湴涓緇勮川鐐圭殑褰卞搷銆傜┖闂寸殑灞傛鍒嗚В鐢ㄦ潵紜畾褰撹窛紱誨澶ф椂錛屾瘮浠ュ線浠諱綍鏃跺欓兘鏇村ぇ鐨勮川鐐圭粍銆傚揩閫熷鏋佺畻娉曠殑涓涓槑鏄句紭鐐規槸鍏鋒湁涓ユ牸鐨勮宸及璁★紝榪欐槸璁稿綆楁硶鎵緙哄皯鐨勬ц川銆?/font>
涓夈佺粨鏉熻
聽聽聽
銆銆