Система символьных вычислений Maxima для физиков-теоретиков

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ä㭪樨 (ª ª ¯à ¢¨«®) ¨§¡ëâ®ç­ë. ‘¨á⥬   ­ «¨â¨ç¥áª¨å ¢ëç¨á«¥­¨© áâ à ¥âáï "㬥âì ¢á¥" ¨ ­  ª ¤ãî § ¤ çã ¢ ­¥© ­ ©¤¥âáï ᢮ï äã­ªæ¨ï (⨯¨ç­ë¥ ¯à¨¬¥àë | MAXIMA ¨ Mathemati a). Šà®¬¥ ⮣®, á«¥¤ã¥â à §«¨ç âì à ¡®âã á á¨á⥬®©  ­ «¨â¨ç¥áª¨å ¢ëç¨á«¥­¨© ¢ ¨­â¥à ªâ¨¢­®¬ ¨ ¢ ¯ ª¥â­®¬ २¬¥. ‚ ॠ«ì­®© à ¡®â¥ ç é¥ ¢áâà¥ç ¥âáï ¯ ª¥â­ë© २¬. ‚ë ®âç¥â«¨¢® ¯à¥¤áâ ¢«ï¥â¥ ᥡ¥ ¯®á«¥¤®¢ â¥«ì­®áâì ¢ëª« ¤®ª, ª®â®àãî å®â¨â¥ ¯à®¢¥áâ¨, ¢ ⥪á⮢®¬ । ªâ®à¥ ­ ¡¨à ¥â¥ ᮮ⢥âáâ¢ãîéãî ¯à®£à ¬¬ã, § ¯ã᪠¥â¥ ¥¥ ­  áç¥â ¨ ᬮâà¨â¥ ­  १ã«ìâ âë ¥¥ à ¡®âë, ᮤ¥à é¨¥áï ¢ ¢ë室­®¬ ä ©«¥. ˆ­â¥à ªâ¨¢­ë© २¬ ¢áâà¥ç ¥âáï ⮣¤ , ª®£¤  ‚ë å®â¨â¥ çâ®-«¨¡® ¯à¨ª¨­ãâì. ‚ë ¢¢®¤¨â¥ ®¤­ã ª®¬ ­¤ã, ᬮâà¨â¥ çâ® ¯®«ã稫®áì, ¢ § ¢¨á¨¬®á⨠®â í⮣® ¢¢®¤¨â¥ á«¥¤ãîéãî ª®¬ ­¤ã ¨ â.¤. Ǒਠ¢ë¡®à¥ á¨áâ¥¬ë  ­ «¨â¨ç¥áª¨å ¢ëç¨á«¥­¨© á«¥¤ã¥â ãç¨â뢠âì: ­ áª®«ìª® 㤠祭 § ¬ëᥫ á¨á⥬ë: ¥áâ¥á⢥­­®áâì ᨭ⠪á¨á , ¯à®áâ®â  ॠ«¨§ æ¨¨ â¥å ¨«¨ ¨­ëå ¤¥©á⢨©,  ¢â®¬ â¨ç¥áª®¥ ¨«¨ ¯à¨­ã¤¨â¥«ì­®¥ ã¯à®é¥­¨¥, ¯®«­®â  ­ ¡®à  ä㭪権 ¨ â.¯. ‚®-¯¥à¢ëå,

­ áª®«ìª® 㤠筠 ¥¥ ª®­ªà¥â­ ï ॠ«¨§ æ¨ï: â ª, áâ à¥­ìª¨© REDUCE 3.0 ª®¬¯ ªâ¥­, ­® ªà ©­¥ ®£à ­¨ç¥­ ¨ ¯® ¯ ¬ï⨠¨ ¯® ᪮à®á⨠¯® áà ¢­¥­¨î á ¡®«¥¥ "­®¢ë¬" REDUCE 3.4,   ®ä®à¬«¥­¨¥ REDUCE 5.0 § ¬¥â­® «ãçè¥, 祬 ã ¢á¥å ¯à¥¤ë¤ãé¨å; Mathemati a 1.0 | íâ® ¯à®£à ¬¬ , ª®â®à ï ¯®ç⨠­¥ à ¡®â ¥â, Mathemati a 2.0 ¨ 2.2 à ¡®â îâ, ­® á "¯à¨ç㤠¬¨", Mathemati a 3.0 ¤«ï LINUX (®¤¨­ ¨§ ¢ à¨ ­â®¢ UNIX) à ¡®â ¥â ¢ 3 à §  ¡ëáâ॥, 祬 Mathemati a 3.0 ¤«ï Windows 95, ¥á«¨ ®­¨ ¨­áâ ««¨à®¢ ­ë ­  ¨¤¥­â¨ç­ëå ª®¬¯ìîâ¥à å, ®ä®à¬«¥­¨¥ Mathemati a 5.0 £®à §¤® «ãçè¥, 祬 㠯।ë¤ãé¨å ¢¥àᨩ ¨ ¡ëáâ७쪮 ¯à¨ª¨­ãâì çâ®-«¨¡® ¢ ¨­â¥à ªâ¨¢­®© ¬®¤¥ 㤮¡­¥¥ ¢á¥£® ¨¬¥­­® ­  ­¥© (Mathemati a 5.1 ¯®ç⨠­¥ ®â«¨ç ¥âáï ®â Mathemati '¨ 5.0), ¨ â.¤. ‚®-¢â®àëå,

«¥£ «ì­®áâì ¥¥ ¨á¯®«ì§®¢ ­¨ï: á⮨¬®áâì REDUCE 4.0 | (¡ë« ) 99$, á⮨¬®áâì Mathemati '¨ 3.0 ¯®¤ Solaris (®¤¨­ ¨§ ¢ à¨ ­â®¢ UNIX) | (¡ë« ) 500$, á⮨¬®áâì Mathemati '¨ 5.1 | ®â 2000$ ¤® 5000$, MAXIMA 5.13 ¯®¤ UNIX ¨ WinXXX | ¯à®£à ¬¬  ®¡é¥£® ¯®«ì§®¢ ­¨ï (¡®«¥¥ ⮣®, open sour e), ¥¥ ¨á室­¨ª¨ ¨ 㥠®âª®¬¯¨«¨à®¢ ­­ë¥ ¢¥àᨨ ¤«ï à §­ëå ®¯¥à æ¨®­­ëå á¨á⥬ «¥£ª® ­ ©â¨ á ¯®¬®éìî «î¡®£® ¨­â¥à­¥â®¢áª®£® ¯®¨áª®¢¨ª . …¥ â ª¥ «¥£ª® ­ ©â¨ ¢ ¯à®¥ªâ¥ GNU, ­® â ¬ ®­  áãé¥áâ¢ã¥â ¢ २¬¥ "not supported", çâ® ¤®¢®«ì­® ¯¥ç «ì­®.(2) ‚ ­ áâ®ï饥 ¢à¥¬ï ¥¥ ¬®­® ᪠ç âì á ‚-âà¥âì¨å,

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"http://sour eforge.net", ­®, ª ª ¯®ª §ë¢ ¥â ®¯ëâ, íâ® ã⢥थ­¨¥ ®áâ ­¥âáï ¢¥à­ë¬ ­¥ ᫨誮¬ ¤®«£®. ¯à¨¢ë窮© ¯®«ì§®¢ â¥«ï. â®â ä ªâ®à, ª ª ­¨ áâà ­­®, ­  ­ è ¢§£«ï¤, ï¥âáï ®á­®¢­ë¬. ‘ ®¤­®© áâ®à®­ë, ®á¢®¨¢ «î¡ãî ¨§ á¨á⥬, ‚ë á «¥£ª®áâìî ®¢« ¤¥¥â¥ ¨ ¢á¥¬¨ ®áâ «ì­ë¬¨, ­® ¢á¥ à ¢­® á ¬ ï ¯¥à¢ ï á¨á⥬  â ª ¨ ®áâ ­¥âáï ¤«ï ‚ á á ¬®© 㤮¡­®©, ¥¥ ᨭ⠪á¨á ¡ã¤¥â ª § âìáï á ¬ë¬ ¥áâ¥á⢥­­ë¬, ¨ ‚ë ­¥ ¡ã¤¥â¥ "§ ¬¥ç âì ¥£®" ¯à¨ ॠ«ì­®© ­ ãç­®© à ¡®â¥. €¢â®àë ¨¬¥îâ ®¯ëâ ॠ«ì­®© à ¡®âë á REDUCE, MAXIMA ¨ Mathemati a, ¨ ®¯ëâ ¨£àë á Maple, MathLab ¨ MathCad. ‘।¨ ¯¥à¥ç¨á«¥­­ëå ¯à®£à ¬¬ á ¬ë¬ ã¤®¡­ë¬ á¥à¢¨á®¬ (®ä®à¬«¥­¨¥¬) ®¡« ¤ ¥â, ­  ­ è ¢§£«ï¤, Mathemati a. …᫨ ‚ ¬ ­ã­® çâ®-«¨¡® ¡ëáâ७쪮 ¯à¨ª¨­ãâì (¯à®¢¥à¨âì ä®à¬ã«ã, ­ à¨á®¢ âì £à ä¨ª, ®æ¥­¨âì ¯®à冷ª ¢¥«¨ç¨­ë) ¢ ¨­â¥à ªâ¨¢­®¬ २¬¥, ¯® «ã©, á«¥¤ã¥â ¢®á¯®«ì§®¢ âìáï Mathemati '®©. (3) Ž¤­ ª® ‚ë ¤®«­ë ¯®¬­¨âì, çâ® á।¨ ¢á¥å ¯¥à¥ç¨á«¥­­ëå á¨á⥬ Mathemati a ᮤ¥à¨â ­ ¨¡®«ì襥 ª®«¨ç¥á⢮ ®è¨¡®ª. •®âï ®á­®¢­ë¥ ¥¥ ¨¤¥¨ ¨ ᨭ⠪á¨ç¥áª¨¥ ª®­áâàãªæ¨¨ ¡ë«¨ ¯®§ ¨¬á⢮¢ ­ë ã MAXIM'ë,(4) ­® ¨å ª®­ªà¥â­ ï ॠ«¨§ æ¨ï ®áâ ¢«ï¥â ¥« âì «ãç襣®. ˆáâ®à¨ï ¢§ ¨¬®®â­®è¥­¨© MAXIM'ë ¨ Mathemati '¨ ¯à ªâ¨ç¥áª¨ ¯®¢â®àï¥â ¨áâ®à¨î ¢§ ¨¬®®â­®è¥­¨© UNIX ¨ WinXXX. Š á® «¥­¨î, Mathemati a ¨§­ ç «ì­® à §¢¨¢ « áì ª ª ª®¬¬¥àç¥áª ï ¯à®£à ¬¬  ¨ ¡ë«  ®à¨¥­â¨à®¢ ­  ­  ¬ áᮢ®£® ¯®«ì§®¢ â¥«ï. ’ ª çâ® ª®­áâàã¨à®¢ « áì á¨á⥬ , ª®â®à ï "㬥¥â ¢á¥", ­® ¤¥« ¥â íâ® "¢á¥" ­¥ ᫨誮¬ å®à®è® ("ç⮡ë å®âì ª ª-­¨¡ã¤ì à ¡®â «®"). Œ «® ⮣®, ¥á«¨ ¢ ­¥ ᫨誮¬ á«®­ëå á«ãç ïå ®­  ®¡ëç­® ¢ë¤ ¥â ¯à ¢¨«ì­ë¥ ®â¢¥âë, â® ¢ ­¥âਢ¨ «ì­ëå á«ãç ïå ®­  ­¥à¥¤ª® (¨ ¡¥§ ª ª¨å-«¨¡® ¯à¥¤ã¯à¥¤¥­¨©) ¢ë¤ ¥â ‚-ç¥â¢¥àâëå,

Linux), ¨¬¥©â¥ ¢ ¢¨¤ã, çâ® ¯®á«¥¤­¨¥ ¢¥àᨨ MAXIMA ­¥ ¢á¥£¤  å®à®è® ¯®­¨¬ îâ ¯®á«¥¤­¨¥ ¢¥àᨨ GNU Common Lisp (GCL), â ª çâ® ¯à®é¥ ᮡ¨à âì ¥¥ ­  ¡ §¥ CLISP. ‚ ¯à¨­æ¨¯¥ ¢ ¡®«ì設á⢥ ¢¥àᨩ Linux ®¡¥ ¢¥àᨨ Common Lisp (¨ GCL, ¨ CLISP) ¢å®¤ïâ ¢ ãáâ ­ ¢«¨¢ ¥¬ë© ª®¬¯«¥ªâ ¯à®£à ¬¬, ­® (ª ª ¯à ¢¨«®) ­¥ 楫¨ª®¬,   ¢ ¢¨¤¥ ®£à맪®¢. â¨å ®£à맪®¢ ¢¯®«­¥ ¤®áâ â®ç­®, çâ®¡ë ¯®«ãç¨âì à ¡®â îéãî ¢ ⥪á⮢®¬ २¬¥ MAXIM'ã. Š ª ¯à ¢¨«®, ¤«ï ॠ«ì­®© ­ ãç­®© à ¡®âë ¡®«ì襣® ¨ ­¥ âॡã¥âáï, ­® ¥á«¨ ‚ë ᪫®­­ë ª ¯à®£à ¬¬¨áâ᪨¬ íªá¯¥à¨¬¥­â ¬ (âॢ®­ë© ¯à¨§­ ª | ¬®¥â ¡ëâì ‚ë ­¥ 䨧¨ª,   ¯à®£à ¬¬¨áâ?), â® ¯®áâ ¢ì⥠CLISP ¢ ¯®«­®¬ ®¡ê¥¬¥. (3) €¢â®àë ­¥ ­ áâ ¨¢ îâ ­  í⮬ ã⢥थ­¨¨. …᫨ Maple ¨«¨ MathLab ‚ ¬ ª ãâáï ¡®«¥¥ 㤮¡­ë¬¨, ¨á¯®«ì§ã©â¥ ¨å. (4) ‡­ ç¨â¥«ì­® ã«ãç襭 ⮫쪮  ¯¯ à â ¯®¤áâ ­®¢®ª ¯® è ¡«®­ã, ª®â®àë© ¢ MAXIM'¥ ªà ©­¥ ­¥ã¤®¡¥­. 3

­¥ç⮠ᮢ¥à襭­® ­¥¯à ¢¨«ì­®¥. Ž¯ïâì-â ª¨, ¯à¨ ãᮢ¥à襭á⢮¢ ­¨¨ á¨áâ¥¬ë ®á­®¢­®¥ ¢­¨¬ ­¨¥ 㤥«ï«®áì ®ä®à¬«¥­¨î,   ­¥ ®¯â¨¬¨§ æ¨¨ à ¡®âë. ¥ «¨§ æ¨ï ¨¬¥­­®  ­ «¨â¨ç¥áª¨å ¢ëç¨á«¥­¨© ¢ Mathemati '¥ ­¥ ®ç¥­ì 㤠筠 (¬¥¤«¥­­® ¨ ­¥ íª®­®¬­® ¯® ¯ ¬ïâ¨), ¨ ¯à¨ ¡®«ì讬 ®¡ê¥¬¥ ¢ëª« ¤®ª ®­¨ ­¥ ¬®£ãâ ¡ëâì ¢ë¯®«­¥­ë §  ª®­¥ç­®¥ ¢à¥¬ï. ¥á¬®âàï ­  ¢á¥ ᪠§ ­­®¥, ¤«ï à ¡®âë ¢ ¨­â¥à ªâ¨¢­®¬ २¬¥  ¢â®àë ¢á¥ ¥ ४®¬¥­¤ãîâ Mathemati 'ã.  ¯à®â¨¢, ¤«ï à ¡®âë ¢ ¯ ª¥â­®¬ २¬¥ ¬ë ४®¬¥­¤ã¥¬ MAXIM'ã.(5) Š ª ­ ¬ ª ¥âáï, ®­  ®¡« ¤ ¥â ¡®«ì襩 £¨¡ª®áâìî ¯à¨ à ¡®â¥ á ¢ëà ¥­¨ï¬¨, 祬 REDUCE, å®âï ¨­®£¤  ॠ«¨§ æ¨ï ­¥ª®â®à®£®  «£®à¨â¬  ¢ëç¨á«¥­¨© ­  MAXIM'¥ âॡã¥â ¡®«ìè¨å ãᨫ¨©, 祬 ­  REDUCE. ’¥¬ ­¥ ¬¥­¥¥, ¯à¨ à ¡®â¥ á MAXIM'®© ¬®­® ¤®áâ¨çì £®à §¤® ¡®«ì襣®, 祬 ­  REDUCE. „¥«® ­¥ á⮫쪮 ¢ ⮬, çâ® ¢ MAXIM'¥ ®¯à¥¤¥«¥­® ¬­®¥á⢮ ¯®«¥§­ëå ä㭪権, ª®â®àëå ­¥â ¢ REDUCE. ƒ« ¢­ë¬ ¯à¥¨¬ãé¥á⢮¬ MAXIM'ë ¯à¨ à ¡®â¥ á  ­ «¨â¨ç¥áª¨¬¨ ¢ëà ¥­¨ï¬¨ ï¥âáï â®, çâ® ®­  ­¥ áâ à ¥âáï "ã¯à®áâ¨âì" ¢ëà ¥­¨¥ ¤® ­¥ª®â®à®© ª ­®­¨ç¥áª®© ä®à¬ë, ¥á«¨ ¥¥ ®¡ í⮬ ­¥ ¯à®áïâ. „«ï ®ç¥­ì ¬­®£¨å ¢ëà ¥­¨© â ª®¥ "ã¯à®é¥­¨¥" ¯à¨¢®¤¨â ª ­¥¢¥à®ïâ­®¬ã ãá«®­¥­¨î (㤫¨­¥­¨î). REDUCE, ­ ¯à®â¨¢, ¢á¥£¤  ᢮¤¨â ¢ëà ¥­¨¥ ª ª ­®­¨ç¥áª®¬ã ¢¨¤ã, ¨ ¯à®¢®¤¨â ã¯à®é¥­¨ï ¤® â¥å ¯®à, ¯®ª  ¢ëà ¥­¨¥ ­¥ ¯¥à¥á⠥⠬¥­ïâìáï. …¤¨­á⢥­­®¥, ­  çâ® ¬®¥â ¯®¢«¨ïâì ¯®«ì§®¢ â¥«ì REDUCE | íâ® ¯®¬¥­ïâì á ¬ ª ­®­¨ç¥áª¨© ¢¨¤ á ¯®¬®éìî ä« £®¢. ƒ¨¡à¨¤­ ï ®¯¥à æ¨ï (­ ¯à¨¬¥à, ä ªâ®à¨§ æ¨ï ®â¤¥«ì­ëå á« £ ¥¬ëå ¢ á㬬¥) ¢ REDUCE âॡã¥â ®ç¥­ì ¡®«ìè¨å ¯à®£à ¬¬¨áâ᪨å ãᨫ¨©.  ¯à®â¨¢, MAXIMA ¤¥« ¥â ¢ â®ç­®á⨠â®, ® 祬 ¥¥ ¯à®á¨â ¯®«ì§®¢ â¥«ì á ¯®¬®éìî â¥å ¨«¨ ¨­ëå ä㭪権 ¨«¨ ä« £®¢ëå ¯¥à¥¬¥­­ëå. Ǒਠí⮬ ®­  ­¥ ¯à®¢®¤¨â ã¯à®é¥­¨ï ¨«¨ ¯à¥®¡à §®¢ ­¨ï "¤® ª®­æ " (â.¥. ¤® â¥å ¯®à, ¯®ª  ¢ëà ¥­¨¥ ­¥ ¯¥à¥á⠥⠬¥­ïâìáï), â ª çâ® ç áâ® ¯®¢â®à­ë© ¢ë§®¢ ⮩ ¥ á ¬®© ä㭪樨 ¬¥­ï¥â ¢ëà ¥­¨¥ (®¯ëâ ¯®ª §ë¢ ¥â, çâ® íâ® ¡®«ì讥 ¯à¥¨¬ãé¥á⢮,   ­¥ ­¥¤®áâ â®ª, ª ª ¬®¥â ¯®ª § âìáï ­  ¯¥à¢ë© ¢§£«ï¤). —â® ª á ¥âáï ¨­â¥à䥩ᮢ MAXIM'ë, â® ¨å ¢ ­ áâ®ï饥 ¢à¥¬ï ­ áç¨â뢠¥âáï ç¥âëà¥. à ¡®â  á ª®¬ ­¤­®© áâப¨ ("¯ ª¥â­ë© २¬"). ‚ «î¡®¬ ⥪á⮢®¬ । ªâ®à¥ ‚ë ­ ¡¨à ¥â¥ ¯à®£à ¬¬ã, § ¯ã᪠¥â¥ ¥¥ ­  áç¥â ¨ ¯®â®¬ (á

‚®-¯¥à¢ëå,

(5) €¢â®àë ­¥ ­ áâ ¨¢ îâ ­  í⮬ ã⢥थ­¨¨. …᫨ REDUCE ‚ ¬ ª ¥âáï ¡®«¥¥ 㤮¡­ë¬, ¨á¯®«ì§ã©â¥ ¥£®.

4

¨§ã¬«¥­¨¥¬) à áᬠâਢ ¥â¥ ¢ë室­®© ä ©«. „«ï í⮣® ¤®áâ â®ç­® 㬥âì § ¯ã᪠âì MAXIM'ã á ª®¬ ­¤­®© áâப¨. Ǒਠࠡ®â¥ ¯®¤ UNIX (¢¥à®ïâ­®, íâ® ¡ã¤¥â Linux) «¥£ª® ­ ©â¨ áªà¨¯â (®¡ëç­® ®­ ­ §ë¢ ¥âáï ¯à®áâ® "maxima"), § ¯ã᪠î騩 "£®«ãî" ⥪á⮢ãî ¬®¤ã MAXIM'ë. Ǒ® 㬮«ç ­¨î ®­  § ¯ãáâ¨âáï ¢ ¨­â¥à ªâ¨¢­®¬ २¬¥ (‚ë ¬®¥â¥ ¯®á«¥¤®¢ â¥«ì­® ¢¢®¤¨âì ª®¬ ­¤ë ¨ ­¥¬¥¤«¥­­® ¯®«ãç âì ­  ­¨å ®â¢¥â). ®, ¯®«ì§ãïáì ®¡ëç­ë¬ ¯¥à¥­ ¯à ¢«¥­¨¥¬ ¢¢®¤  á ¯®¬®éìî "<", ¬®­® áࠧ㠧 ¯ãáâ¨âì ­  áç¥â ‚ è ä ©«. Ǒ®¤ WinXXX (áâ à ©â¥áì ­¥ ¯®«ì§®¢ âìáï WinXXX!) ¢á¥ ᮢ¥à襭­®  ­ «®£¨ç­®, ⮫쪮 ­ ¤® ¨áª âì § ¯ã᪠î騩 ä ©« "maxima.bat". Š ª ¯®ª §ë¢ ¥â ®¯ëâ, ¯à¨ ॠ«ì­®© ­ ãç­®© à ¡®â¥ íâ®â २¬ á ¬ë© íä䥪⨢­ë©, ¨  ¢â®àë £®àïç® à¥ª®¬¥­¤ãîâ ¨¬¥­­® ¥£®. Šà®¬¥ ⮣®, íâ®â २¬ ¢ë£«ï¤¨â (¯®çâ¨) ®¤¨­ ª®¢® ­  à §­ëå ®¯¥à æ¨®­­ëå á¨á⥬ å, à ¡®â ¥â ¢ ⥪á⮢®© ¬®¤¥, ¨, á«¥¤®¢ â¥«ì­®, ¬®¥â ¡ëâì ¨á¯®«ì§®¢ ­ ¯à¨ à ¡®â¥ á 㤠«¥­­®£® â¥à¬¨­ « . à ¡®â  á ¯®¬®éìî । ªâ®à  ema s. „«ï íä䥪⨢­®© à ¡®âë ¢ í⮬ २¬¥ ­ ¤® ®á¢®¨âì ema s, çâ® ¯®«¥§­® ¨ á ¬® ¯® ᥡ¥, ¯®áª®«ìªã íâ® ¨áª«îç¨â¥«ì­ë© ¯® ¬®é­®á⨠। ªâ®à (íâ® ç áâì ¯à®¥ªâ  GNU, â ª çâ® ®­ áãé¥áâ¢ã¥â ¨ ¯®¤ UNIX, ¨ ¯®¤ WinXXX). ‡ â¥¬ á«¥¤ã¥â ¯®§­ ª®¬¨âì ema s á MAXIM'®©. „«ï í⮣® ¤®áâ â®ç­® ᪮¯¨à®¢ âì ᮤ¥à¨¬®¥ ¤¨à¥ªâ®à¨¨ "ema s", ª®â®à ï ¯à¨áãâáâ¢ã¥â ¢ ãáâ ­®¢«¥­­®© ‚ ¬¨ MAXIM'¥, ¢ «î¡®¥ ¬¥áâ®, ¨§ ª®â®à®£® ema s ᮣ« á¥­ ç¨â âì ä ©«ë "Ema s Lisp" (®­¨ ¨¬¥îâ ¨¬¥­  á à áè¨à¥­¨¥¬ ".el"). „«ï í⮣® ¯à®é¥ ¢á¥£® ­ ©â¨ ¢­ãâਠᠬ®£® ema s ¤¨à¥ªâ®à¨î, £¤¥ «¥ â ä ©«ë á à áè¨à¥­¨¥¬ ".el".  ª®­¥æ, á«¥¤ã¥â ᮧ¤ âì ª®­ä¨£ãà æ¨®­­ë© ä ©« ema s, ª®â®àë© ¤®«¥­ ¨¬¥âì ¨¬ï ".ema s" (­ ç¨­ ¥âáï á â®çª¨!) ¨ ᮤ¥à âì á«¥¤ãî騥 áâப¨ ‚®-¢â®àëå,

(autoload 'maxima "maxima" "Maxima intera tion" t) (autoload 'maxima-mode "maxima" "Maxima mode" t) (setq auto-mode-alist ( ons ' ("\\.mxm" . maxima-mode) auto-mode-alist))

(íâ® á«¥¤ã¥â ¢®á¯à¨­¨¬ âì  ªá¨®¬ â¨ç¥áª¨). Ǒ®¤ UNIX íâ®â ä ©« ¤®«¥­ «¥ âì ¢ ‚ è¥© home-¤¨à¥ªâ®à¨¨,   ¯®¤ WinXXX | ¢ ª®à­¥ ¤¨áª  "C:" Ǒ®á«¥ í⮣® ‚ë ¬®¥â¥ ­ ¡¨à âì ᢮¨ ¯à®£à ¬¬ë, ¯®«ì§ãïáì । ªâ®à®¬ ema s. …᫨ à áè¨à¥­¨¥ ä ©«  ¡ã¤¥â ".mxm", â® ema s  ¢â®¬ â¨ç¥áª¨ ¯¥à¥©¤¥â ¢ २¬ "maxima-mode". Ǒਠí⮬ ⥪á⠡㤥â á­ ¡¥­ "¡®¥¢®© à áªà áª®©" | ä㭪樨 MAXIM'ë ¡ã¤ãâ à¨á®¢ âìáï ®¤­¨¬ 梥⮬, ¯¥à¥¬¥­­ë¥ ¤à㣨¬, ª®¬¬¥­â à¨¨ | âà¥â쨬.(6) Šà®¬¥ ⮣®, ‚ë ᬮ¥â¥ ­¥¬¥¤«¥­­® ¯®áë« âì ­  ¨á¯®«­¥­¨¥ ª ª ®â¤¥«ì­ë¥ áâப¨ (Ctrl-C Ctrl-C) ¨«¨ ¬ àª¨à®¢ ­­ë¥ ¡«®ª¨ (Ctrl-C Ctrl-R, ¡ãª¢  "R" ¨§-§  ⮣®, çâ® ¢ ema s ¡«®ª¨ ­ §ë¢ îâáï

(6)   ¢§£«ï¤  ¢â®à®¢, í⮠᪮॥ ­¥¤®áâ â®ª, 祬 ¤®á⮨­á⢮. 5

"regions"), â ª ¨ ¢¥áì ­ ¡à ­­ë© ä ©« (Ctrl-C Ctrl-B, ¡ãª¢  "B" ¨§-§  ⮣®, çâ® ¢ ema s ä ©«ë ­ §ë¢ îâáï "buffers"). Žª­® । ªâ®à  ¡ã¤¥â¥ ¯®¤¥«¥­® ¯®¯®« ¬, ¨ ¢ ®¤­®© ¯®«®¢¨­¥ ‚ë 㢨¤¨â¥ ¨á室­ë© ⥪áâ,   ¢ ¤à㣮© | १ã«ìâ âë à ¡®âë MAXIM'ë. Šáâ â¨, १ã«ìâ âë à ¡®âë | í⮠⮥ "buffer", â ª çâ® ¥£® ⮥ ¬®­® á®åà ­¨âì ¢ ä ©«. ‚ ¯à¨­æ¨¯¥ ­  ¡ëáâன ¬ è¨­¥ íâ® ¤®¢®«ì­® 㤮¡­ë© ᯮᮡ à ¡®âë. ®«¥¥ ⮣®, ¯®¤ UNIX ema s áãé¥áâ¢ã¥â ª ª ¢ £à ä¨ç¥áª®©, â ª ¨ ¢ ⥪á⮢®© ¢¥àᨨ, â ª çâ® ¢®§¬®­  à ¡®â  á 㤠«¥­­®£® â¥à¬¨­ « . ® á«¥¤ã¥â ¯®¬­¨âì, çâ® ¢á¥  àå¨â¥ªâãà­ë¥ ãªà è¥­¨ï § ¬¥¤«ïîâ à ¡®âã. Šà®¬¥ ⮣®, ­¥®¡å®¤¨¬® ®á­®¢ â¥«ì­® ¨§ãç¨âì ema s. íâ® ¨­â¥à䥩á "xMaxima". â® áâண® £à ä¨ç¥áª¨© ¨­â¥à䥩á. Žª®èª® ¢ ­¥¬ ¯®¤¥«¥­® ¯®¯®« ¬, ¢ ¢¥àå­¥© ç á⨠¨¢¥â ¨­â¥à ªâ¨¢­ ï MAXIMA,   ¢ ­¨­¥© ¯à¨áãâáâ¢ã¥â ¯á¥¢¤®¡à ã§¥à. ‚ í⮬ ¯á¥¢¤®¡à ã§¥à¥ ¬®£ãâ ¡ëâì ®¯à¥¤¥«¥­ë áá뫪¨, ª®â®àë¥ à áᬠâਢ îâáï ª ª áªà¨¯âë. ˆå ⥪áâ ¯¥à¥áë« ¥âáï MAXIM'¥,   १ã«ìâ â à ¡®âë ¬®­® ®â¯à ¢¨âì ®¡à â­® ¢ ¯á¥¢¤®¡à ã§¥à. ‡ ¡ ¢­®, ç⮠⥪áâ ¢ ¯á¥¢¤®¡à ã§¥à¥ ¬®­® । ªâ¨à®¢ âì. â®â ¨­â¥à䥩á ॠ«¨§®¢ ­ ª ª ¯®¤ UNIX, â ª ¨ ¯®¤ WinXXX. Š ª ¨­â¥à ªâ¨¢­ë© help ¯® MAXIM'¥ íâ® ¢ë£«ï¤¨â ­¥¯«®å® (­¥¬¥¤«¥­­ ï ¨««îáâà æ¨ï à ¡®âë, ¯à¨ç¥¬ ¨á室­ãî ª®¬ ­¤ã ¬®­® ¯®¬¥­ïâì ¨ ¯®á¬®âà¥âì, çâ® ¯à¨ í⮬ ¢ë©¤¥â). Ž¤­ ª® (­  ­ è ¢§£«ï¤!) ¤«ï ᮡá⢥­­® ­ ãç­®© à ¡®âë íâ® ­¥ ®ç¥­ì 㤮¡­®. ‚-âà¥âì¨å,

íâ® ¨­â¥à䥩á "wxMaxima". â® áâண® £à ä¨ç¥áª¨© ¨­â¥à䥩á, ¢ë¯®«­¥­­ë© ¢ á⨫¥ WinXXX, ç⮠㥠­ áâ®à ¨¢ ¥â. ‚ ­¥¬ ॠ«¨§®¢ ­ë ¯®¤áª §ª¨, ã¯à®é¥­­ë© ¢¢®¤ ª®¬ ­¤ (á ¯®¬®éìî ª­®¯®ª) ¨ ¯à®ç¨© á¥à¢¨á, ª®â®àë© ®¡ëª­®¢¥­­® ᨫ쭮 § ¬¥¤«ï¥â à ¡®âã. (‹î¡®© 祫®¢¥ª, ­ ¡¨à ¢è¨© ­ áâ®ï騥 ä®à¬ã«ë ¢ TeX ¨ WinWord, §­ ¥â, ­ áª®«ìª® ¯à®áâ® ¨ ¡ëáâà® ­ ¡¨à âì ª®¬ ­¤ë TeX ¢ «î¡®¬ ⥪á⮢®¬ । ªâ®à¥ ¨ ­ áª®«ìª® ¬ãç¨â¥«ì­®¥ § ­ï⨥ ¯®«ì§®¢ âìáï "á¥à¢¨á®¬" á ª­®¯®çª ¬¨ ¢ WinWord. â® 㥠­¥ £®¢®àï ® ¯¥à¥­®á¨¬®á⨠䠩«®¢ å®âï ¡ë á ¬ è¨­ë ­  ¬ è¨­ã, ¤ ¥ ¢ à ¬ª å ®¤­®© ®¯¥à æ¨®­­®© á¨á⥬ë. ˆ ­¥ £®¢®àï ® ­¥¯à¥¤áª §ã¥¬®¬ ¯®¢¥¤¥­¨¨, ¥á«¨ £¤¥-â® § â¥á «áï ­¥¢¨¤¨¬ë© ᨬ¢®« ä®à¬ â¨à®¢ ­¨ï. ˆ ­¥ £®¢®àï ® ª ç¥á⢥ ¨â®£®¢ëå ä®à¬ã« | ­  ä®à¬ã«ë WinWord ®¡ëª­®¢¥­­® ¡¥§ á«¥§ ᬮâà¥âì ­¥¢®§¬®­®).   ­ è ¢§£«ï¤, íâ® (¯®ª ) ­¥ã¤ ç­®¥ ¯®¤à  ­¨¥ ®ä®à¬«¥­¨î Mathemati '¨. ‘¥à¢¨á ¯®«ã稫áï ­¥ ᫨誮¬ 㤮¡­ë©. ’ ª çâ® ¤«ï ᮡá⢥­­® ­ ãç­®© à ¡®âë íâ®â ¨­â¥à䥩á (¯®ª ) ­¥ ®ç¥­ì ¯®¤å®¤¨â. ‚-ç¥â¢¥àâëå,

6

‘«¥¤ã¥â ¯®¤ç¥àª­ãâì, çâ® ¢ í⮬ à㪮¢®¤á⢥ ®¯¨á ­ë ®â­î¤ì ­¥ ¢á¥ ¢®§¬®­®á⨠MAXIM'ë ¨ ¤ «¥ª® ­¥ ¢á¥ ®¯à¥¤¥«¥­­ë¥ ¢ ­¥© ä㭪樨.  §ã¬¥¥âáï, ªà¨â¥à¨¨ ®â¡®à  ¡ë«¨ ¢ §­ ç¨â¥«ì­®© ¬¥à¥ áã¡ê¥ªâ¨¢­ë¬¨.   ­ è ¢§£«ï¤, ¬ë ®¯¨á «¨ ¯à ªâ¨ç¥áª¨ ¢á¥ ᨭ⠪á¨ç¥áª¨¥ ª®­áâàãªæ¨¨, ¯®§¢®«ïî騥 ª®­â஫¨à®¢ âì ¨á¯®«­¥­¨¥ ¯à®£à ¬¬, ¯à ªâ¨ç¥áª¨ ¢á¥ ã­¨¢¥àá «ì­ë¥ ä㭪樨 ¤«ï à ¡®âë á  ­ «¨â¨ç¥áª¨¬¨ ¢ëà ¥­¨ï¬¨ ¨ ¡®«ì設á⢮ ®¡é¥ã¯®âॡ¨â¥«ì­ëå ¯à¨ª« ¤­ëå ä㭪権, ®à¨¥­â¨à®¢ ­­ëå ­  ç áâ­ë¥ á«ãç ¨. ˆ§ ¨§«®¥­¨ï ᮧ­ â¥«ì­® ¨áª«î祭® ®¯¨á ­¨¥ "­¨§ª®ã஢­¥¢ëå" ¢®§¬®­®á⥩ MAXIM'ë (¢§ ¨¬®¤¥©á⢨¥ á LISP'®¬, ¨ â.¯.). Ž¯ëâ ¯®ª §ë¢ ¥â, çâ® ­ ¯¨á ­¨¥ ¯®¤¯à®£à ¬¬ ­  LISP'¥, ª®â®àë¥ § â¥¬ § £àã îâáï ¢ MAXIM'ã, ¯®§¢®«ï¥â §­ ç¨â¥«ì­® ã᪮à¨âì ¯à®æ¥áá áç¥â  ¨ à áè¨à¨âì ¢®§¬®­®á⨠MAXIM'ë (ᮢ¥à襭­® â ª ¥, ª ª ¯à®£à ¬¬¨à®¢ ­¨¥ ­   áᥬ¡«¥à¥ à áè¨àï¥â ¢®§¬®­®á⨠ï§ëª  "‘"). Ž¤­ ª®, ­  ­ è ¢§£«ï¤, 䨧¨ª-⥮à¥â¨ª ­¥ ¤®«¥­ ¯à®£à ¬¬¨à®¢ âì ­¨ ­   áᥬ¡«¥à¥, ­¨ ­  LISP'¥. ã   ¥á«¨ ç¨â â¥«ì 㬥¥â ¯à®£à ¬¬¨à®¢ âì ­  LISP'¥, ®­ (­¥á®¬­¥­­®) ᬮ¥â ®á¢®¨âì "­¨§ª®ã஢­¥¢ë¥" ¢®§¬®­®á⨠MAXIM'ë á ¬®áâ®ï⥫쭮. Šà®¬¥ ⮣®, ¨§ ¨§«®¥­¨ï ¨áª«î祭® ®¯¨á ­¨¥ à¨á®¢ â¥«ì­ëå ä㭪権 MAXIM'ë. â¨ (¤®¢®«ì­® à §­®®¡à §­ë¥) ä㭪樨 ®à¨¥­â¨à®¢ ­ë ­  à ¡®âã ¢ ¨­â¥à ªâ¨¢­®¬ २¬¥,   (ª ª ­ ¬ ª ¥âáï) MAXIM'ã «ãçè¥ ¨á¯®«ì§®¢ âì ¢ ¯ ª¥â®¬ २¬¥. Šà®¬¥ ⮣®, ¨§ ¨§«®¥­¨ï ¨áª«î祭® ®¯¨á ­¨¥ ⥭§®à­ëå ¯ ª¥â®¢ MAXIM'ë. ’ ª¨å ¯ ª¥â®¢ áãé¥áâ¢ã¥â ¤¢  á ¯®«®¢¨­®î, ¨ (­  ¯¥à¢ë© ¢§£«ï¤) ®­¨ ¬®£«¨ ¡ë ¡ëâì ç१¢ëç ©­® ¯®«¥§­ë ¤«ï 䨧¨ª®¢-⥮à¥â¨ª®¢, ¨¬¥îé¨å ¤¥«® á £à ¢¨â æ¨¥©. Š á® «¥­¨î, ®¯à¥¤¥«¥­­ë¥ â ¬ ã­¨¢¥àá «ì­ë¥ ä㭪樨 ¢¥«¨ª®«¥¯­® à ¡®â îâ ⮫쪮 ¢ "¨£àãè¥ç­ëå" â¥á⮢ëå § ¤ ç å. Ǒਠ¯®¯ë⪥ ¨á¯®«ì§®¢ âì ¨å ¢ ॠ«ì­®© ­ ãç­®© à ¡®â¥ ®ª §ë¢ ¥âáï, ç⮠ᮢ¥à襭­® ­¥®¡å®¤¨¬® á ¬®áâ®ï⥫쭮, á ãç¥â®¬ ®á®¡¥­­®á⥩ ‚ è¥© ª®­ªà¥â­®© § ¤ ç¨, ¯¨á âì ¯à®æ¥¤ãàã ¢ëç¨á«¥­¨ï ¤ ¥ â ª¨å ­¥á«®­ëå ª®­áâàãªæ¨©, ª ª ᨬ¢®«ë Šà¨áâ®ä䥫ï. ‚ «î¡®¬ á«ãç ¥ ¯®¯®«­¨âì ­¥¤®áâ î騥 ᢥ¤¥­¨ï «¥£ª® á ¯®¬®éìî ®¯¨á ­¨ï MAXIM'ë (MAXIMA manual) ¨ á ¯®¬®éìî ä㭪樨 "des ribe", ª®â®à ï ¤ ¥â ªà âª®¥ (å®âï ¨ ­¥ ¢á¥£¤  ¤®áâ â®ç­® ¯®­ïâ­®¥) ®¯¨á ­¨¥ ª®¬ ­¤ ¨ ä㭪権 MAXIM'ë. Ǒਠࠡ®â¥ ¢ ¨­â¥à ªâ¨¢­®¬ २¬¥ ‚ë ¬®¥â¥ ¢¢¥á⨠¢ ª®¬ ­¤­®© áâப¥ ª®¬ ­¤ã "des ribe(des ribe);". Žâ¢¥â MAXIM'ë ¡ã¤¥â ¤®¢®«ì­® ¯®ãç¨â¥«ì­ë¬.

7

2. Ǒ¥à¢®­ ç «ì­ë¥ ᢥ¤¥­¨ï ® à ¡®â¥ á MAXIM'®© ˆ¤¥­â¨ä¨ª â®àë ¢ MAXIM'¥ á®áâ ¢«ïîâáï ¨§ 26  2 « â¨­áª¨å ¡ãª¢ (⥯¥àì ®­  à §«¨ç ¥â áâà®ç­ë¥ ¨ ¯à®¯¨á­ë¥ ¡ãª¢ë), 10 æ¨äà, ᨬ¢®«  ¯®¤ç¥àª¨¢ ­¨ï "_", ¯à®æ¥­â  "%". Š ª ¯à ¢¨«® á "%" ­ ç¨­ îâáï ᯥ樠«ì­ë¥ ¨¬¥­ , ­ ¯à¨¬¥à "%i" | íâ® ¬­¨¬ ï ¥¤¨­¨æ , "%pi" | íâ® ,   "%e" | ®á­®¢ ­¨¥ ­ âãà «ì­®£® «®£ à¨ä¬ . Ǒਠॠ«ì­®© à ¡®â¥ MAXIMA ¤ã¡«¨àã¥â ¢¢®¤ ¨ ¯¥ç â ¥â ¥£® ¢¯¥à¥¬¥ªã á ¢ë¢®¤®¬. „«ï 㤮¡á⢠ ç⥭¨ï ¬ë ¡ã¤¥¬ ¢ ¯à¨¬¥à å ¢ë¤¥«ïâì ¢ë¢®¤ ¤à㣨¬ èà¨ä⮬ ¨ ᤢ¨£ âì ¥£® ¢¯à ¢®. â® ¯®§¢®«ï¥â ᤥ« âì ­ è¨ ¯à¨¬¥àë ­¥ ᫨誮¬ ¯®å®¨¬¨ ­  â®, çâ® ‚ë 㢨¤¨â¥, ­¥¯®á।á⢥­­® à ¡®â ï á MAXIM'®©, ­® § â® ®­¨ ¡ã¤ãâ ¡®«¥¥ ç¨â ¡¥«ì­ë¬¨. ‚¢®¤ ¢ MAXIM'¥ § ¢¥àè ¥âáï ®¤­¨¬ ¨§ ¤¢ãå â¥à¬¨­ â®à®¢ | ";" ¨«¨ "$". ‚ ¯¥à¢®¬ á«ãç ¥ १ã«ìâ â ¢ëç¨á«¥­¨© ¯¥ç â ¥âáï, ¢® ¢â®à®¬ | ­¥â. Š ¤ë© ¢¢®¤ ­ã¬¥àã¥âáï á ¯®¬®éìî ¬¥â®ª "label" | "%i1", "%i2", "%i3", ¨ â.¤. Š ¤ ï ¨§ ­¨å ï¥âáï ¯¥à¥¬¥­­®©, ª®â®à®© ¯à¨á¢®¥­® §­ ç¥­¨¥, à ¢­®¥ ¢¢¥¤¥­­®© ª®¬ ­¤¥. ‘®®â¢¥âá⢥­­®, ª ¤ë© ¢ë¢®¤ â ª¥ ­ã¬¥àã¥âáï á ¯®¬®éìî ¬¥â®ª | "%o1", "%o2", "%o3", ¨ â.¤. Ž¯ïâì-â ª¨, ª ¤ ï ¨§ ­¨å ï¥âáï ¯¥à¥¬¥­­®©, ª®â®à®© ¯à¨á¢®¥­® §­ ç¥­¨¥, à ¢­®¥ १ã«ìâ â㠢몫 ¤®ª. ‘ãé¥áâ¢ãîâ ¬¥âª¨ âà¥â쥣® ⨯  "%t1", "%t2", ¨ â.¤., ® ª®â®àëå ¡ã¤¥â ᪠§ ­® ­¨¥. MAXIMA (¢ â®ç­®á⨠ª ª ï§ëª "‘") ¨£­®à¨àã¥â à §¡¨¥­¨¥ ⥪áâ  ­  áâப¨. Œ®­® ¢¢®¤¨âì ­¥áª®«ìª® ª®¬ ­¤ ¢ ®¤­®© áâப¥, ¬®­® à §¡¨¢ âì ®¤­ã ª®¬ ­¤ã ­  ­¥áª®«ìª® áâப. Š®¬¬¥­â à¨¨ ¢ MAXIM'¥ ⮥ ॠ«¨§®¢ ­ë ᮢ¥à襭­® â ª ¥, ª ª ¢ "‘" | ¤¢  ᨬ¢®«  "/*" ®âªà뢠îâ ª®¬¬¥­â à¨©,   ¤¢  ᨬ¢®«  "*/" § ªà뢠îâ ¥£®. Ǒ¥à¥¬¥­­®© "%" ¯® ®¯à¥¤¥«¥­¨î ¯à¨á¢ ¨¢ ¥âáï १ã«ìâ â ¯®á«¥¤­¥© ¢ëª« ¤ª¨. Žá­®¢­ë¥ ¬ â¥¬ â¨ç¥áª¨¥ ®¯¥à æ¨¨ ¢ MAXIM'¥ ¯¨èãâáï ®¡ëç­ë¬ ®¡à §®¬ | "+", "-", "*", "/"; ¢®§¢¥¤¥­¨¥ ¢ á⥯¥­ì | íâ® "^" (ªàëè¥çª ),   ¯à¨á¢®¥­¨¥ (¯® «ã©, íâ® ¤®¢®«ì­® ­¥ã¤ ç­ ï ¨¤¥ï) § ¯¨á뢠¥âáï ª ª ":" (¤¢®¥â®ç¨¥). Ǒ®¯ë⪠ § ¯¨á âì ¯à¨á¢®¥­¨¥ ¢ ¢¨¤¥ "=" | ¯®áâ®ï­­ ï ®è¨¡ª  ¯à¨ à ¡®â¥ á MAXIM'®©.

Ǒ¥à¥¬¥­­ë¥ ¬®£ã⠯ਭ¨¬ âì ç¨á«®¢ë¥ §­ ç¥­¨ï | 楫ë¥, à æ¨®­ «ì­ë¥, á ¯« ¢ î饩 â®çª®© 䨪á¨à®¢ ­­®© (¬ è¨­­®©) â®ç­®á⨠¨ á ¯« ¢ î饩 â®çª®© ­¥®£à ­¨ç¥­­®© â®ç­®áâ¨: x:-7; x:-13/5;

8

x:-0.012345; x:77.7777e-5; x:77.77777777777777777777777777777777b-5;

Šà®¬¥ ⮣®, ¯¥à¥¬¥­­ë¬ ¬®£ãâ ¡ëâì ¯à¨á¢®¥­ë §­ ç¥­¨ï ¢ ¢¨¤¥  ­ «¨â¨ç¥áª¨å ¢ëà ¥­¨© x:a^2+b;

‘®¡á⢥­­®, ¨¬¥­­® íâ® ¨ ¤¥« ¥â MAXIM'ã á¨á⥬®©  ­ «¨â¨ç¥áª¨å ¢ëç¨á«¥­¨©. ‡ ¬¥â¨¬, çâ® §¤¥áì ®¯à¥¤¥«¥­  äã­ªæ¨ï (¢ ¬ â¥¬ â¨ç¥áª®¬ á¬ëá«¥ í⮣® á«®¢ ) x(a; b). ‚ MAXIM'¥ ¥áâì ä㭪樨 ¨ ¢ ¯à®£à ¬¬¨áâ᪮¬ á¬ëá«¥, ­® ®­¨ ­ã­ë ᮢᥬ ¤«ï ¤à㣨å 楫¥©. …᫨ ‚ ¬ ­ã­  äã­ªæ¨ï f (x; y; z ), â® ­¥ ®¯à¥¤¥«ï©â¥ äã­ªæ¨î, ¯à®áâ® § ¢¥¤¨â¥ ¯¥à¥¬¥­­ãî f.

Šà®¬¥ ⮣®, ¯¥à¥¬¥­­ë¬ ¬®£ãâ ¡ëâì ¯à¨á¢®¥­ë §­ ç¥­¨ï ¢ ¢¨¤¥ áâப®¢ëå ¯¥à¥¬¥­­ëå x:"ab defgh";

‚áâ ¢¨âì ¢ áâப㠮¤¨­®ç­ãî ª ¢ëçªã "'", ¤¢®©­ãî ª ¢ëçªã """ ¨ ®¡à â­ë© á«íè "\" ¬®­® ᮢ¥à襭­® â ª¨¬ ¥ ®¡à §®¬, ª ª ¢ "C", â.¥. "\'", "\"" ¨ "\\" ᮮ⢥âá⢥­­®.  ª®­¥æ, ¯¥à¥¬¥­­ë¬ ¬®£ãâ ¡ëâì ¯à¨á¢®¥­ë «®£¨ç¥áª¨¥ §­ ç¥­¨ï | "true" ¨«¨ "false" ("¨á⨭ " ¨«¨ "«®ì"): x:true; y:false; ‡­ ª "::" (¤¢  ¤¢®¥â®ç¨ï ¯®¤àï¤) | íâ® ¯à¨á¢®¥­¨¥ á ¢ëç¨á«¥­¨¥¬ «¥¢®© ç -

áâ¨. ‡­ ç¥­¨¥¬ «¥¢®© ç á⨠¤®«¥­ ¡ëâì ®¡ê¥ªâ, ª®â®à®¬ã ¬®­® çâ®-«¨¡® ¯à¨á¢ ¨¢ âì. ’ ª çâ® ¢ १ã«ìâ â¥ ¨á¯®«­¥­¨ï s:2*a+b$ t:-b-a$ (s+t)::777$

¯¥à¥¬¥­­®© "a" ¡ã¤¥â ¯à¨á¢®¥­® §­ ç¥­¨¥ "777". Šàã£«ë¥ áª®¡ª¨ §¤¥áì ­¥®¡å®¤¨¬ë, â.ª. s+t::777$

¢ë§®¢¥â á®®¡é¥­¨¥ ®¡ ®è¨¡ª¥. Ž¯ëâ ¯®ª §ë¢ ¥â, çâ® ¢ ॠ«ì­®© ­ ãç­®© à ¡®â¥ ¯à®ªã ®â í⮩ ¢®§¬®­®á⨠­¥¬­®£®. ‡­ ª à ¢¥­á⢠ १¥à¢¨à®¢ ­ ¤«ï "ãà ¢­¥­¨©" ("equation"). “à ¢­¥­¨¥ | íâ® «¥¢ ï ¨ ¯à ¢ ï ç á⨠ࠢ¥­á⢠, ᮥ¤¨­¥­­ë¥ §­ ª®¬ "=". Š ãà ¢­¥­¨î ¬®­® çâ®-«¨¡® ¯à¨¡ ¢¨âì, ¨§ ­¥£® ¬®­® çâ®-«¨¡® ¢ëç¥áâì, ¥£® ¬®­® 㬭®¨âì ¨«¨ ¯®¤¥«¨âì ­  â® ¨«¨ ¨­®¥ ¢ëà ¥­¨¥. ’¥ ¥ ®¯¥à æ¨¨ ¬®­® ¯à®¤¥« âì á ¤¢ã¬ï ãà ¢­¥­¨ï¬¨: eq1:a=b$ eq1- ;

9

a- = b-

eq2:x=y$ eq1*eq2;

a x = b y

‡­ ª ":=" ¯à¨¬¥­ï¥âáï ¤«ï ®¯à¥¤¥«¥­¨ï ä㭪権, ¨¬¥­­®, § ¯¨á¨ f(x):=x^2+5$ g(a,b):=a^2+3/b$

®¯à¥¤¥«ïîâ ä㭪樨 ®¤­®£® ¨ ¤¢ãå  à£ã¬¥­â®¢ ᮮ⢥âá⢥­­®. ‡ ¬¥â¨¬ ¥é¥ à §, çâ® ®¡  ®¯à¥¤¥«¥­¨ï ä㭪権 ᮢ¥à襭­® ­¥«¥¯ë, ¥á«¨ ‚ë ᮡ¨à ¥â¥áì, Z  g (a; b) . „«ï íâ¨å ¬ ­¨¯ã«ï権 á ä㭪᪠¥¬, ¢ëç¨á«ïâì dx f (x) ¨«¨ a æ¨ï¬¨ (¢ ¬ â¥¬ â¨ç¥áª®¬ á¬ëá«¥) á«¥¤®¢ «® ¡ë ­ ¯¨á âì f:x^2+5$ integrate(f,x);

¨«¨

g:a^2+3/b$ diff(g,a);

Šà®¬¥ ä㭪権, ¢ MAXIM'¥ ¬®­® ®¯à¥¤¥«ïâì ¬ ªà®áë. „«ï í⮣® ¨á¯®«ì§ã¥âáï §­ ª "::=". f(x)::=x^2+5$ g(a,b)::=a^2+3/b$

Œ®­® áç¨â âì, çâ® ¬ ªà®áë à ¡®â îâ (¯®çâ¨) â ª ¥, ª ª ä㭪樨.  §­¨æã ¬¥¤ã ­¨¬¨ ¬ë ®¡á㤠âì ­¥ ¡ã¤¥¬. ’ ª çâ® ®á®¡®© ¯®«ì§ë ®â ­¨å ­¥â.  §ã¬¥¥âáï, ®¯à¥¤¥«¥­ë áâ ­¤ àâ­ë¥ ¬ â¥¬ â¨ç¥áª¨¥ ä㭪樨 exp(x) sin(x) sinh(x) asin(x) asinh(x) atan2(x,y)

log(x)

os(x)

osh(x) a os(x) a osh(x)

sqrt(x) tan(x) tanh(x) atan(x) atanh(x)

ot(x)

oth(x) a ot(x) a oth(x)

s (x)

s h(x) a s (x) a s h(x)

Ǒਠí⮬ MAXIMA §­ ¥â, çâ® sin( x) = sin(x), os(0) = 1, log(1) = 0, ¨ â.¤. Ž¯à¥¤¥«¥­ë ®¯¥à æ¨¨ ä ªâ®à¨ «  ¨ ¤¢®©­®£® ä ªâ®à¨ « : 5!;

6!!; 5!!;

120 48 15

10



max ¯¥à¥¡¨à ¥â ᢮¨  à£ã¬¥­âë ¨ ­ å®¤¨â ¬ ªá¨¬ «ì­®¥ ç¨á«® ”ã­ªæ¨ï

max(33,-22,11);



33

min ¯¥à¥¡¨à ¥â ᢮¨  à£ã¬¥­âë ¨ ­ å®¤¨â ¬¨­¨¬ «ì­®¥ ç¨á«® ”ã­ªæ¨ï

min(33,-22,11,44);

-22

11

3. ”㭪樨 ¢ë¢®¤  ­  íªà ­ Ǒਠ¨á¯®«ì§®¢ ­¨¨ â¥à¬¨­ â®à  ";" MAXIMA ¯¥ç â ¥â १ã«ìâ â ¢ëç¨á«¥­¨©, â.¥. §­ ç¥­¨¥ ᮮ⢥âáâ¢ãî饣® ¢ëà ¥­¨ï. Ž¤­ ª®, ¤«ï á«®­ëå ᨭ⠪á¨ç¥áª¨å ª®­áâàãªæ¨© ⨯  横«®¢ §­ ç¥­¨¥ ¡ã¤¥â "done", â.¥. ¢¯®«­¥ ¡¥á¯®«¥§­®¥. —â®¡ë ¬®­® ¡ë«® ॠ«¨§®¢ë¢ âì ®á¬ëá«¥­­ë© ¢ë¢®¤ ¨§ á«®­ëå ᨭ⠪á¨ç¥áª¨å ª®­áâàãªæ¨© ⨯  ¡«®ª®¢ ¨«¨ 横«®¢, ¯à¥¤ãᬮâ७ë ᯥ樠«ì­ë¥ ä㭪樨 ¢ë¢®¤ .



print ¯¥ç â ¥â §­ ç¥­¨ï ¢á¥å ᢮¨å  à£ã¬¥­â®¢ ¢ ®¤­ã áâப㠔㭪æ¨ï

(%i31) print("D=",a+A, ", x is equal to",77, "or",88," or ",99); D=a+A, x is equal to 77 or 88 (%o31)

or 99

99

â  äã­ªæ¨ï, ¯® «ã©, ï¥âáï ®á­®¢­®© ¨ á ¬®© 㤮¡­®© ¤«ï ¢ë¢®¤  ­  ¯¥ç âì.



disp ¯¥ç â ¥â §­ ç¥­¨ï ᢮¨å  à£ã¬¥­â®¢, ¯à¨ç¥¬ ª ¤®¥ §­ ç¥­¨¥ ¯¥ç â ¥âáï ¢ ®â¤¥«ì­®© áâப¥ ”ã­ªæ¨ï

(%i22) x:77$ y:44$ z:11$ (%i25) disp(x,y,z);

77 44 11 done

(%o25)



display ¯¥ç â ¥â §­ ç¥­¨ï ᢮¨å  à£ã¬¥­â®¢ ¢¬¥áâ¥ á ¨å ¨¬¥­¥¬, ª ¤®¥ ¢ ®â¤¥«ì­®© áâப¥ ”ã­ªæ¨ï

(%i12) display(x,y);

x=77 y=44 done

(%o12)

12



ldisp ¯¥ç â ¥â §­ ç¥­¨ï ᢮¨å  à£ã¬¥­â®¢ ¢¬¥áâ¥ á ¬¥âª ¬¨ "%t". â  äã­ªæ¨ï "ᡨ¢ ¥â" ­ã¬¥à æ¨î ¬¥â®ª "%i" ¨ "%o". ”ã­ªæ¨ï

(%i4) ldisp(x,y,z);

(%i7) x;



(%t4) (%t5) (%t6) (%o6)

77 44 11 [%t4,

(%o7)

77

%t5, %t6℄

ldisplay ¯¥ç â ¥â §­ ç¥­¨ï ᢮¨å  à£ã¬¥­â®¢ ¢¬¥áâ¥ á ¨å ¨¬¥­¥¬ ¨ ¬¥âª ¬¨ "%t". â  äã­ªæ¨ï â ª¥ "ᡨ¢ ¥â" ­ã¬¥à æ¨î ¬¥â®ª "%i" ¨ "%o". ”ã­ªæ¨ï

(%i18) ldisplay(x,y);

(%t18) (%t19) (%o19)

x=77 y=44 [%t18,

%t19℄

„«ï ¨««îáâà æ¨¨ ¯à¨¢¥¤¥¬ ª®à®â¥­ìª¨© ¯à¨¬¥à, ¨««îáâà¨àãî騩 ¢¢®¤ ¨ ¢ë¢®¤ ¢ MAXIM'¥. (%i1) 2+3; (%i2)

%;

(%i3) exp(x); (%i4) (a+b)^2/( +d);

(%o1)

5

(%o2)

5

(%o3)

%ex

(%o4)

(b+a) d+

(%o5)

d (f(x)) dx

(%o6)

%ex

2

(%i5) diff(f(x),x); (%i6)

%o3;

(%i7) 3+4$

13

(%i8)

%;

(%o8)

7

‚ ¤ «ì­¥©è¥¬ ¬ë (ª ª ¯à ¢¨«®) ¡ã¤¥¬ ®¯ã᪠âì ¬¥âª¨ ¢ ¯à¨¬¥à å.

Š ª ¢¨¤­® ¨§ ¯à¨¢¥¤¥­­®£® ¯à¨¬¥à , MAXIMA áâ à ¥âáï ­ à¨á®¢ âì á¢®î ¢ë¤ çã "ªà á¨¢®". ‘¯®á®¡ à¨á®¢ ­¨ï ®¯à¥¤¥«ï¥âáï ­¥áª®«ìª¨¬¨ ¯¥à¥¬¥­­ë¬¨, ¯¥à¥ç¨á«¨¬ ­¥ª®â®àë¥ ¨§ ­¨å.



linel ®¯à¥¤¥«ï¥â ¤«¨­ã áâப¨, ¢ ª®â®àãî ¤®«­  ¢¯¨á뢠âìáï ¢ë¤ ç . ˆ§­ ç «ì­® ãáâ ­®¢«¥­  ¢¥«¨ç¨­  "79". …᫨ ¥« â¥«ì­® ¯®«ãç¨âì ¡®«¥¥ 㧪ãî áâà ­¨æã (­ ¯à¨¬¥à, 60 ¯®§¨æ¨© ¤«ï ¤¢ã媮«®­®ç­®© ¯¥ç â¨), á«¥¤ã¥â ¯à¨á¢®¨âì ¯¥à¥¬¥­­®© "linel" §­ ç¥­¨¥ "60": Ǒ¥à¥¬¥­­ ï

linel:60;



display2d ¢ª«î砥⠨«¨ ¢ëª«îç ¥â "¤¢ã¬¥à­®¥" à¨á®¢ ­¨¥ ¤à®¡¥©, á⥯¥­¥©, ¨ â.¯. ˆ§­ ç «ì­® ãáâ ­®¢«¥­® §­ ç¥­¨¥ "true". Ǒਠí⮬ ¤à®¡¨ à¨áãîâáï ªà á¨¢®, ­® ¢ë¤ ç  ­¥ ¬®¥â ¡ëâì ¨á¯®«ì§®¢ ­  ¤«ï ¢¢®¤  ¢ MAXIM'ã. …᫨ ãáâ ­®¢¨âì §­ ç¥­¨¥ "false", â® ¢ë¢®¤ ¬®¥â ¡ëâì ¢¯®á«¥¤á⢨¨ ¨á¯®«ì§®¢ ­ ª ª ¢¢®¤: Ǒ¥à¥¬¥­­ ï

(x^2+a)/(y^2+b);

display2d:false$ (x^2+a)/(y^2+b);



60

2 x +a 2 y +b (x^2+a)/(y^2+b)

showtime ¢ª«î砥⠨«¨ ¢ëª«î砥⠯¥ç âì ¢à¥¬¥­¨, § âà ç¥­­®£® ­  ª ¤®¥ ¤¥©á⢨¥. ˆ§­ ç «ì­® ãáâ ­®¢«¥­® §­ ç¥­¨¥ "false". …᫨ ãáâ ­®¢¨âì §­ ç¥­¨¥ "true", â® ¡ã¤¥â ¯¥ç â âìáï, ¢®-¯¥à¢ëå, "¨¤¥ «ì­®¥" ¯à®æ¥áá®à­®¥ ¢à¥¬ï, ¢ â¥ç¥­¨¥ ª®â®à®£® ¢ë¯®«­ï« áì ¢ëª« ¤ª , ¨, ¢®-¢â®àëå, "䨧¨ç¥áª®¥" ¢à¥¬ï, § âà ç¥­­®¥ ­  ¢ëª« ¤ªã (¢ á¨á⥬ å á ¤¥«¥­¨¥¬ ¢à¥¬¥­¨ ¢â®à®¥ ¢á¥£¤  ¯à¥¢ë蠥⠯¥à¢®¥). Ǒ¥à¥¬¥­­ ï

showtime:true$ fun1(a,b, )$

Evaluation took 0.00 se onds (0.00 elapsed) Evaluation took 4.08 se onds (4.20 elapsed)

14



fortran ¯®§¢®«ï¥â ¯®«ãç¨âì ¢ë¤ çã, ª®â®àãî ¬®­® ¨á¯®«ì§®¢ âì ª ª äà £¬¥­â ä®àâà ­®¢áª®© ¯à®£à ¬¬ë ”ã­ªæ¨ï

fortran(sum(x^i,i,0,15));

x**15+x**14+x**13+x**12+x**11+ 1 x**10+x**9+x**8+x**7+x**6+x**5+ 2 x**4+x**3+x**2+x+1 ˆ¬¥©â¥ ¢ ¢¨¤ã, çâ® íâ  äã­ªæ¨ï ¨£­®à¨àã¥â ¯ à ¬¥âà linel | ¤«¨­  áâப

ᮮ⢥âáâ¢ã¥â áâ à®¬ã ä®àâà ­®¢áª®¬ã áâ ­¤ àâã | 72 ¯®§¨æ¨¨.

15

4.  ¡®â  á ä ©« ¬¨ 

bat h § ¯ã᪠¥â ä ©« á ¯à®£à ¬¬®©. Ž¯¥à â®àë ¢ë¯®«­ïîâáï ®¤¨­ §  ¤à㣨¬ «¨¡® ¤® ª®­æ  ä ©« , «¨¡® ¤® ᨭ⠪á¨ç¥áª®© ®è¨¡ª¨, «¨¡® ¤® ­¥ª®à४⭮© ®¯¥à æ¨¨. ”ã­ªæ¨ï

bat h("myfile.mxm");

ˆ¬¥©â¥ ¢ ¢¨¤ã, çâ® ¥á«¨ ‚ë à ¡®â ¥â¥ á ª®¬ ­¤­®© áâப¨, â® ¢«®¥­­ë© "bat h" ¯à ¢¨«ì­® à ¡®â âì ­¥ ¡ã¤¥â. â® §­ ç¨â, çâ® ¥á«¨ ‚ë § ¯ãá⨫¨ ­  ¨á¯®«­¥­¨¥ ­¥ª®â®àë© ä ©« (¯¥à¢ë©) á ¯®¬®éìî "bat h",   ¢­ãâਠí⮣® ¯¥à¢®£® ä ©«  ¥áâì ª®¬ ­¤  "bat h", § ¯ã᪠îé ï ­  ¨á¯®«­¥­¨¥ ¥é¥ ®¤¨­ ä ©« (¢â®à®©), â® íâ®â ¢â®à®© ä ©« ¡« £®¯®«ãç­® ®âà ¡®â ¥â, ­®, ª®£¤  ¥£® ¨á¯®«­¥­¨¥ ¯à¥ªà â¨âáï, § ®¤­® ¯à¥ªà â¨âáï ¨ ¨á¯®«­¥­¨¥ ¯¥à¢®£® ä ©« . ’ ª çâ® ­¨ ®¤¨­ ®¯¥à â®à, áâ®ï騩 ¯®á«¥ "bat h" ¢ ¯¥à¢®¬ ä ©«¥, ­¥ ¡ã¤¥â ¨á¯®«­¥­. Ǒਠ¨á¯®«ì§®¢ ­¨¨ ¤àã£¨å ¨­â¥à䥩ᮢ íâ® ¬®¥â ¡ëâì ­¥§ ¬¥â­®, ¯®â®¬ã çâ®, ­ ¯à¨¬¥à, ema s ¯à¨ § ¯ã᪥ ä ©«  ­  áç¥â ¯à®áâ® ¯¥à¥áë« ¥â MAXIM'¥ ®¤­ã áâப㠧  ¤à㣮©,   ­¥ ¨á¯®«ì§ã¥â ª®¬ ­¤ã "bat h".



load § £àã ¥â â®â ¨«¨ ¨­®© ä ©«. ”ã­ªæ¨ï

load(somefile);

’¨¯ § £à㧪¨ § ¢¨á¨â ®â ⨯  ä ©« . ˆ¬¥­­®, ¬®­® § £àã âì ä ©« á ¬ ªà®á ¬¨, â.¥. ä ªâ¨ç¥áª¨ ä ©« á ¯à®£à ¬¬®© ­  MAXIMA (⨯¨ç­ë¥ à áè¨à¥­¨ï ¨¬¥­ â ª¨å ä ©«®¢ ".max", ".mxm", ".m " ¨«¨ ".ma "), ¬®­® § £àã âì ä ©« á ¯à®£à ¬¬®© ­  LISP (⨯¨ç­ë¥ à áè¨à¥­¨ï ¨¬¥­ â ª¨å ä ©«®¢ ".lisp" ¨«¨ ".lsp"), ¨ ¬®­® § £àã âì ¤¢®¨ç­ë© ä ©« á 㥠®ââ࠭᫨஢ ­­ë¬¨ ª®¤ ¬¨ (⨯¨ç­®¥ à áè¨à¥­¨¥ ¨¬¥­ â ª¨å ä ©«®¢ ".o"). Š ª ¯à ¢¨«® íâ  äã­ªæ¨ï ­¥®¡å®¤¨¬  ¤«ï § £à㧪¨ ⮣® ¨«¨ ¨­®£® ¯ ª¥â , ª®â®àë© ­¥ § £àã ¥âáï  ¢â®¬ â¨ç¥áª¨. ˆ­â¥à¥á­®, çâ® à §­ë¥ ¯ ª¥âë åà ­ïâáï ¢ à §­ëå ä®à¬ â å, â ª çâ® ¬®¥â £à㧨âìáï ¨ ä ©« ¢ ä®à¬ â¥ MAXIMA, ¨ LISP-ä ©« ¨ ¤¢®¨ç­ë© ä ©«. ‘â ­¤ àâ­ë¥ ä㭪樨 ¢ MAXIMA «¨¡® ¢å®¤ïâ ¢ ï¤à® á¨áâ¥¬ë ¨ ¯®í⮬㠤®áâã¯­ë ¨§­ ç «ì­®, «¨¡® ïîâáï  ¢â®§ £à㧮ç­ë¬¨, â.¥. ¯à¨ ¨å ¯¥à¢®¬ ¢ë§®¢¥ ¯à®¨á室¨â  ¢â®¬ â¨ç¥áª ï § £à㧪  ­¥®¡å®¤¨¬®£® ä ©« , «¨¡® âॡãîâ ®© § £à㧪¨ ⮣® ¨«¨ ¨­®£® ä ©«  | ¢ í⮬ á«ãç ¥ ¤® ¨å ¢ë§®¢  ­¥®¡å®¤¨¬® ­ ¯¨á âì "load(pa ketname);". Ǒਠí⮬ ¤«ï § £à㧪¨, ­ ¯à¨¬¥à, ¯ ª¥â  ¨­â¥£à¨à®¢ ­¨ï ¯® ç áâï¬ ¬®­® ­ ¯¨á âì ª ª â ª ¨

load(bypart)$

16

load("bypart");

¨ ¢ ⮬ ¨ ¢ ¤à㣮¬ á«ãç ¥ § £à㧨âáï ä ©« "bypart.ma ".



writefile ­ ç¨­ ¥â ¯¨á âì ¢áî ¢ë¤ çã MAXIM'ë ¢ 㪠§ ­­ë© ä ©«. Ǒਠí⮬ (¢ ®â«¨ç¨¥ ®â REDUCE) ¢ë¤ ç  ­  â¥à¬¨­ « ­¥ ¯à¥ªà é ¥âáï: ”ã­ªæ¨ï

writefile("myoutput.mxm")$



losefile ¯à¥ªà é ¥â ¢ë¢®¤ ¢ ä ©«: ”ã­ªæ¨ï

losefile()$

‘ãé¥áâ¢ãîâ ¨ ¤à㣨¥ ä㭪樨, ¯®§¢®«ïî騥 ¯¨á âì ¢ ä ©«ë ¨ ç¨â âì ¨§ ­¨å, ­® (ª ª ­ ¬ ª ¥âáï) ¯à¨ ॠ«ì­®© ­ ãç­®© à ¡®â¥ ®­¨ ­¥ ®ç¥­ì ¯®«¥§­ë.

17

5. Ǒ८¡à §®¢ ­¨ï  ­ «¨â¨ç¥áª¨å ¢ëà ¥­¨© ®¡é¥£® ¢¨¤  

ev ï¥âáï ®á­®¢­®© ä㭪樥©, ®¡à ¡ â뢠î饩 ¢ëà ¥­¨ï. …¥ ᨭ⠪á¨á ¤®¢®«ì­® à §­®®¡à §¥­. ”ã­ªæ¨ï

ev(expr); ev(expr,flag1,flag2,...); ev(expr,x=a+b,y: /d,...); ev(expr,flag1,x=a,y:b,flag2,...); Œ®­® ¤ ¥ ®¯ã᪠âì ¨¬ï ä㭪樨 "ev" expr,flag1,flag2,...; expr,x=val1,y=val2,...; expr,flag1,x=val1,y=val2,flag2,...;

‘«¥¤ã¥â, ®¤­ ª®, ¨¬¥âì ¢ ¢¨¤ã, çâ® ¢ â® ¢à¥¬ï ª ª § ¯¨á¨ "ev(expr,flag);" ¨ "expr,flag;" ïîâáï ᨭ®­¨¬ ¬¨, § ¯¨á¨ "expr;" ¨ "ev(expr);" ­¥ ¨¤¥­â¨ç­ë,   ¨¬¥­­®: v:a+b$ a:7$ v;

b+a

ev(v);

b+7

v,expand;

b+7

  ¢ëà ¥­¨¥ "expr" ¯® 㬮«ç ­¨î ¤¥©áâ¢ã¥â äã­ªæ¨ï ã¯à®é¥­¨ï. …᫨ 㪠§ ­ë ä« £¨ (¨å ¨¬¥­  ª ª ¯à ¢¨«® ᮢ¯ ¤ îâ á ¨¬¥­ ¬¨ ¤à㣨å ä㭪権, ¯à¥®¡à §ãîé¨å ¢ëà ¥­¨ï), â® á ¢ëà ¥­¨¥¬ ¯à®¨§¢®¤ïâáï ¤¥©áâ¢¨ï ¢ ᮮ⢥âá⢨¨ á í⨬¨ ä« £ ¬¨. ‚®â ­¥ª®â®àë¥ ¨§ ä« £®¢: expand

fa tor

trigexpand

trigredu e

(­  ¢ëà ¥­¨¥ ¤¥©áâ¢ãîâ ®¤­®¨¬¥­­ë¥ ä㭪樨, ¨å ®¯¨á ­¨¥ á¬. ¤ «¥¥), pred

diff

simp

("pred" ¢ë§ë¢ ¥â ¢ëç¨á«¥­¨¥ §­ ç¥­¨ï «®£¨ç¥áª®£® ¢ëà ¥­¨ï, "diff" ¢ë§ë¢ ¥â ¢ë¯®«­¥­¨¥ "§ ¬®à®¥­­®£®" ¤¨ää¥à¥­æ¨à®¢ ­¨ï, "simp" ¢ë§ë¢ ¥â ¢ë¯®«­¥­¨¥ ä㭪樨 ã¯à®é¥­¨ï ¤ ¥ ¢ ⮬ á«ãç ¥, ª®£¤  ¯¥à¥¬¥­­ ï "simp" à ¢­  "false"). …᫨ 㪠§ ­ë ¯®¤áâ ­®¢ª¨ (¢ ¢¨¤¥ "x=val1" ¨«¨ "x:val2"), â® ®­¨ ¢ë¯®«­ïîâáï. 18

Ǒਠí⮬ ¯®¢â®à­ë© ¢ë§®¢ ä㭪樨 "ev" ¢¯®«­¥ ᯮᮡ¥­ ¥é¥ à § ¨§¬¥­¨âì ¢ëà ¥­¨¥, â.¥. ®¡à ¡®âª  ¢ëà ¥­¨ï ­¥ ¨¤¥â ¤® ª®­æ  ¯à¨ ®¤­®ªà â­®¬ ¢ë§®¢¥ ä㭪樨 "ev". ev((a+b)^2,expand);

2

ev((a+b)^2,a=x);

b

2

+ 2 a b + a

2 (x+b) ev((a+b)^2,a:x,expand,b=7); 2 x + 14 x + 49 (a+b)^2,a=x,expand,b=7; 2

x



+ 14 x + 49

simp à §à¥è ¥â «¨¡® § ¯à¥é ¥â ã¯à®é¥­¨¥ ¢ëà ¥­¨©. ˆ§­ ç «ì­® ®­  à ¢­  "true", ¥á«¨ ãáâ ­®¢¨âì ¥¥ à ¢­®© "false", â® ã¯à®é¥­¨ï ¯à®¨§¢®¤¨âìáï ­¥ ¡ã¤ãâ: Ǒ¥à¥¬¥­­ ï

simp:false$ x+y+x; simp:true$ x+y+x;

x + y + x

y + 2 x



fa tor ä ªâ®à¨§ã¥â ¢ëà ¥­¨¥. ”ã­ªæ¨ï

fa tor( a* +b* +a*d+b*d );

(b+a)(d+ ) fa tor( (x^3+2*x^2*y+y^3)/ (x^2+2*x*y+y^2)+ x^2*y/(x^2+2*x*y+y^2)+ 3*x*y^2/(x+y)^2 ); y+x

19



gfa tor ®â«¨ç ¥âáï ®â ä㭪樨 "fa tor" ⥬, ç⮠㬥¥â à ¡®â âì á ¬­¨¬®© ¥¤¨­¨æ¥©, â.¥. ¬®¥â ä ªâ®à¨§®¢ âì ¢ëà ¥­¨ï ⨯  x2 + a2 ¨ x2 + 2ixa a2 ”ã­ªæ¨ï

fa tor(x^2+a^2);

2 2 x +a fa tor(x^2+2*%i*x*a-a^2); 2 2 x + 2%i a x - a gfa tor(x^2+a^2); (x-%ia)(x+%ia) gfa tor(x^2+2*%i*x*a-a^2); 2 (x+%ia)



fa torsum ä ªâ®à¨§ã¥â ®â¤¥«ì­ë¥ á« £ ¥¬ë¥ ¢ ¢ëà ¥­¨¨. ”ã­ªæ¨ï

fa torsum( a^3+3*a^2*b+3*a*b^2+b^3 +x^2+2*x*y+y^2 ); 2 3 (y+x) +(b+a)

¥ á⮨⠮ᮡ¥­­® ­ ¤¥ïâìáï ­  íâã äã­ªæ¨î, ¯®áª®«ìªã ¬­®£®£® ®­  ­¥ § ¬¥ç ¥â: fa torsum(a+x^2+2*x*y+y^2); 2 (y+x) +a fa torsum(a+x^2-y^2); 2 2 -y +x +a



gfa torsum ®â«¨ç ¥âáï ®â "fa torsum" ⥬ ¥, 祬 "gfa tor" ®â«¨ç ¥âáï ®â "fa tor": ”ã­ªæ¨ï

gfa torsum( a^3+3*a^2*b+3*a*b^2+b^3 +x^2+2*%i*x*y-y^2 ); 3 (b + a) - (y -

%i

2

x)

  íâã äã­ªæ¨î ⮥ ­¥ á⮨⠭ ¤¥ïâìáï, ¯®áª®«ìªã ®­  ¬­®£®£® ­¥ § ¬¥ç ¥â: gfa torsum(a+x^2+2*%i*x*y-y^2);

a - (y - %i x) gfa torsum(a+x^2+y^2); 2 2 y + x + a

20

2



expand à áªà뢠¥â ᪮¡ª¨. ”ã­ªæ¨ï

expand( (a+b)*( +d) ); b d + a d + b + a expand( (x^3+2*x^2*y+y^3)/ (x^2+2*x*y+y^2)+ x^2*y/(x^2+2*x*y+y^2)+ 3*x*y^2/(x+y)^2 ); 2 3 3xy y + 2 2 2 2 y +2xy+x y +2xy+x 2 3 3x y x + + 2 2 2 2 y +2xy+x y +2xy+x



ombine ®¡ê¥¤¨­ï¥â á« £ ¥¬ë¥ á ¨¤¥­â¨ç­ë¬ §­ ¬¥­ â¥«¥¬ ”ã­ªæ¨ï

ombine( x^3/(x^2+2*x*y+y^2)+ 3*x^2*y/(x^2+2*x*y+y^2)+ 3*x*y^2/(x^2+2*x*y+y^2)+ y^3/(x^2+2*x*y+y^2)+ a/( +d)+b/( +d) ); 3 2 2 3 b+a y +3xy + 3x y+x + 2 2 d+ y +2xy+x



xthru ¯à¨¢®¤¨â ¢ëà ¥­¨¥ ª ®¡é¥¬ã §­ ¬¥­ â¥«î, ­¥ à áªà뢠ï ᪮¡®ª ¨ ­¥ ¯ëâ ïáì ä ªâ®à¨§®¢ âì á« £ ¥¬ë¥

”ã­ªæ¨ï

xthru( 1/(x+y)^10+1/(x+y)^12 ); 2 (y+x) + 1 12 (y+x) xthru( m/(x^2+2*x*y+y^2)+ n/(x+y)^4 ); 2 2 4 n (y + 2 x y + x ) + m (y + x) 4 2 2 (y + x) (y + 2 x y + x )

21

 §ã¬¥¥âáï, ¢ ¯®á«¥¤­¥¬ á«ãç ¥ ࠧ㬭¥¥ á­ ç «  ä ªâ®à¨§®¢ âì ª ¤®¥ á« £ ¥¬®¥,   ã ¯®â®¬ ¯à¨¬¥­ïâì "xthru". Ǒਬ¥­¨âì äã­ªæ¨î "fa tor" ª ®â¤¥«ì­ë¬ á« £ ¥¬ë¬ ¢ëà ¥­¨ï ¬®­® á ¯®¬®éìî ä㭪樨 "map": f:map(fa tor, m/(x^2+2*x*y+y^2)+ n/(x+y)^4 ); n m + 2 4 (y + x) (y + x) xthru(f); 2 m (y + x) + n 4 (y + x)



multthru 㬭® ¥â ª ¤®¥ á« £ ¥¬®¥ ¢ á㬬¥ ­  ¬­®¨â¥«ì, ¯à¨ç¥¬ ¯à¨ 㬭®¥­¨¨ ᪮¡ª¨ ¢ ¢ëà ¥­¨¨ ­¥ à áªà뢠îâáï. Ž­  ¤®¯ã᪠¥â ¤¢  ¢ à¨ ­â  ᨭ⠪á¨á  ”ã­ªæ¨ï

multthru(mult,sum); multthru(mult*sum);

(¯®à冷ª ᮬ­®¨â¥«¥© ¢ ¯®á«¥¤­¥¬ ¢ à¨ ­â¥ ­¥ áãé¥á⢥­¥­). multthru( (x+y)^2,(x+y)^5 +1/(x+y)^7+(x+y)^2 );

7

(y + x)

+ (y + x)

4

+

multthru( ( (x+y)^5+1/(x+y)^7 +(x+y)^2 ) * (x+y)^2 ); 7 4 (y + x) + (y + x) +

1 (y + x) 1 (y + x)

5

5

multthru( ( (x^3+2*x^2*y+y^3)/ (x^2+2*x*y+y^2)+ x^2*y/(x^2+2*x*y+y^2)+ 3*x*y^2/(x+y)^2 ) * (x^2+2*x*y+y^2)*(m+n)/(x+y) ); 3 2 3 (n + m) (y + 2 x y + x ) y + x 2 2 2 3 (n + m) x y (y + 2 x y + x ) + 3 (y + x)

22

2

(n + m) x + y + x

23

y

6. Ǒ८¡à §®¢ ­¨¥ à æ¨®­ «ì­ëå ¢ëà ¥­¨© •®âï ä㭪樨, ¯à¥®¡à §ãî騥 ¢ëà ¥­¨ï, ­¥ ¯à¨¢®¤ïâ ¨å ª ª ­®­¨ç¥áª®© ä®à¬¥ (¢ ®â«¨ç¨¥ ®â REDUCE), ª ­®­¨ç¥áª®¥ ¯à¥¤áâ ¢«¥­¨¥ (CRE) áãé¥áâ¢ã¥â | íâ® ª ­®­¨ç¥áª ï ä®à¬  ¤«ï ¤à®¡­®-à æ¨®­ «ì­ëå ¢ëà ¥­¨©. ‚ëà ¥­¨¥, ¯à¨¢¥¤¥­­®¥ ª CRE, á­ ¡ ¥âáï ¬¥âª®© /R/ áࠧ㠯®á«¥ ¬¥âª¨ "%o". „ «ì­¥©è ï à ¡®â  á ­¨¬ ¨¤¥â ¡ëáâ॥,   ¢¥à®ïâ­®áâì ¥£® ã¯à®é¥­¨ï ¢ëè¥, 祬 ¤«ï ¢ëà ¥­¨ï ®¡é¥£® ¢¨¤ .



rat ¯à¨¢®¤¨â ¢ëà ¥­¨¥ ª ª ­®­¨ç¥áª®¬ã ¯à¥¤áâ ¢«¥­¨î ¨ á­ ¡ ¥â ¥£® ¬¥âª®© "/R/". Ž­  ã¯à®é ¥â «î¡®¥ ¢ëà ¥­¨¥, à áᬠâਢ ï ¥£® ª ª ¤à®¡­®à æ¨®­ «ì­ãî äã­ªæ¨î, â.¥. à ¡®â ¥â á ®¯¥à æ¨ï¬¨ "+", "-", "*", "/" ¨ á ¢®§¢¥¤¥­¨¥¬ ¢ 楫ãî á⥯¥­ì. ¥æ¥«ë¥ á⥯¥­¨ ®­  ­¥ ã¯à®é ¥â, â.¥. ®­  ­¥ §­ ¥â, çâ® (xa=2 )2 = xa . Ǒਠí⮬ ¢¨¤ ®â¢¥â  § ¢¨á¨â ®â ⮣®, ª ª¨¥ ¯¥à¥¬¥­­ë¥ ®­  áç¨â ¥â ¡®«¥¥ £« ¢­ë¬¨,   ª ª¨¥ ¬¥­¥¥ £« ¢­ë¬¨. “¯®à冷祭¨¥ á­ ç «  ¨¤¥â ¯® á⥯¥­ï¬ á ¬®© £« ¢­®© ¯¥à¥¬¥­­®©, ¢­ãâਠª®íää¨æ¨¥­â®¢ ¯à¨ íâ¨å á⥯¥­ïå | ¯® á⥯¥­ï¬ ¬¥­¥¥ £« ¢­®© ¯¥à¥¬¥­­®©, ¨ â.¤. ˆ§­ ç «ì­® ¯¥à¥¬¥­­ë¥ 㯮àï¤®ç¥­ë ¢  «ä ¢¨â­®¬ ¯®à浪¥ ¨ ®â ­ ç «  ª ª®­æã "£« ¢­®áâì" ¢®§à áâ ¥â. â®â ¯®à冷ª ¬®­® ®âª®à४â¨à®¢ âì, ¤®¡ ¢¨¢ ¢  à£ã¬¥­âë ä㭪樨 ¨¬¥­  ¯¥à¥¬¥­­ëå ¢ ¯®à浪¥ ¢®§à áâ ­¨ï £« ¢­®áâ¨. ”ã­ªæ¨ï

(%i11) rat( (x^3+2*x^2*y+y^3)/ (x^2+2*x*y+y^2)+ x^2*y/(x^2+2*x*y+y^2)+ 3*x*y^2/(x+y)^2 ); (%o11)/R/ (%i12) v1:m/(a+b)+n/(x+y)$ (%i13) rat(v1); (%o13)/R/ (%i14) rat(v1,y,x,n,m,b,a);

(%o14)/R/ (%i15) rat(v1,m,n,a,b,x,y); (%o15)/R/ (%i16) rat(v1,m,n);

24

y + x

my+mx+(b+a)n (b+a)y+(b+a)x na+nb+(x+y)m (x+y)a+(x+y)b my+mx+nb+na (b+a)y+(b+a)x

(b+a)n+(y+x)m (b+a)y+(b+a)x

(%o16)/R/ (%i17) rat( (x^(a/2)-1)^2 * (x^(a/2)+1)^2 / (x^a-1) );

a/2 4 a/2 2 ) - 2(x ) + 1 a x - 1

(x

(%o17)/R/



ratvars ¯®§¢®«ï¥â ¨§¬¥­¨âì  «ä ¢¨â­ë© ¯®à冷ª "£« ¢­®áâ¨" ¯¥à¥¬¥­­ëå, ¯à¨­ïâë© ¯® 㬮«ç ­¨î. ”ã­ªæ¨ï

ratvars(z,y,x,w,v,u,t,s,r,q,p,o,n,m,l,k,j,i,h,g,f,e,d, ,b,a)$

¬¥­ï¥â ¯®à冷ª £« ¢­®á⨠¢ â®ç­®á⨠­  ®¡à â­ë©,   ratvars(m,n,a,b)$

㯮à冷稢 ¥â ¯¥à¥¬¥­­ë¥ "m, n, a, b" ¢ ¯®à浪¥ ¢®§à áâ ­¨ï "£« ¢­®áâ¨", ¨ ¤¥« ¥â ¨å ¡®«¥¥ £« ¢­ë¬¨, 祬 ¢á¥ ®áâ «ì­ë¥ ¯¥à¥¬¥­­ë¥. Ǒ®á«¥ í⮩ ª®¬ ­¤ë ¯®«ã稬 rat(v1);



nb+na+(y+x)m (y+x)b+(y+x)a

ratfa ¢ª«î砥⠨«¨ ¢ëª«îç ¥â ç áâ¨ç­ãî ä ªâ®à¨§ æ¨î ¢ëà ¥­¨© ¯à¨ ᢥ¤¥­¨¨ ¨å ª CRE. ˆ§­ ç «ì­® ãáâ ­®¢«¥­® §­ ç¥­¨¥ "false". …᫨ ãáâ ­®¢¨âì §­ ç¥­¨¥ "true", â® ¡ã¤¥â ¯à®¨§¢®¤¨âìáï ç áâ¨ç­ ï ä ªâ®à¨§ æ¨ï. Ǒ¥à¥¬¥­­ ï

(%i4) v2:m/(a+b)^2+n/(x+y)^2$

(%i5) rat(v2);

2 2 (my +2mxy+mx 2 2 2 2 2 +(b +2ab+a )n) / ((b +2ab+a )y 2 2 2 2 2 +(2b +4ab+2a )xy + (b +2ab+a )x )

(%o5)/R/

(%i6) ratfa :true$ (%i7) rat(v1);

(%o7)/R/ (%i8) rat(v2);

25

my+mx+(b+a)n (b+a)(y+x)

(%o8)/R/



2 2 2 2 my +2mxy+mx + (b +2ab+a )n 2 2 2 2 (y +2xy+x ) (b +2ab+a )

ratsimp ¯à¨¢®¤¨â ¢á¥ ªã᪨ (¢ ⮬ ç¨á«¥  à£ã¬¥­âë ä㭪権) ¢ëà ¥­¨ï, ª®â®à®¥ ­¥ ï¥âáï ¤à®¡­®-à æ¨®­ «ì­®© ä㭪樥©, ª ª ­®­¨ç¥áª®¬ã ¯à¥¤áâ ¢«¥­¨î, ¯à®¨§¢®¤ï ã¯à®é¥­¨ï, ª®â®àë¥ ­¥ ¤¥« ¥â äã­ªæ¨ï "rat". ¥ á­ ¡ ¥â ¢ëà ¥­¨¥ ¬¥âª®© "/R/". Ǒ®¢â®à­ë© ¢ë§®¢ ä㭪樨 ¬®¥â ¨§¬¥­¨âì १ã«ìâ â, â.¥. ã¯à®é¥­¨¥ ­¥ ¨¤¥â ¤® ª®­æ . ”ã­ªæ¨ï

(%i77) ratsimp( (x^(a/2)-1)^2 * (x^(a/2)+1)^2 / (x^a-1) ); 2a a x - 2x + 1 (%o77) a x - 1 (%i78) ratsimp(%); a (%o78) x - 1



fullratsimp ¢ë§ë¢ ¥â äã­ªæ¨î "ratsimp" ¤® â¥å ¯®à, ¯®ª  ¢ëà ¥­¨¥ ­¥ ¯¥à¥áâ ­¥â ¬¥­ïâìáï. ”ã­ªæ¨ï

fullratsimp( (x^(a/2)-1)^2 * (x^(a/2)+1)^2 / (x^a-1) ); a x - 1



ratsimpexpons ã¯à ¢«ï¥â ã¯à®é¥­¨¥¬ ¯®ª § â¥«¥© á⥯¥­¨ ¢ ¢ëà ¥­¨ïå. ˆ§­ ç «ì­® ãáâ ­®¢«¥­® §­ ç¥­¨¥ "false". ‡ ¡ ¢­®, çâ® ¯à¨ í⮬  à£ã¬¥­â «î¡®© ä㭪樨 ã¯à®é ¥âáï: Ǒ¥à¥¬¥­­ ï

fullratsimp( sin( (x^(a/2)-1)^2 * (x^(a/2)+1)^2 / (x^a-1) ) ); a sin(x - 1)

  ¯®ª § â¥«ì á⥯¥­¨ (¢ ⮬ ç¨á«¥ ¯®ª § â¥«ì íªá¯®­¥­âë) | ­¥â: fullratsimp( exp( (x^(a/2)-1)^2 * (x^(a/2)+1)^2 / (x^a-1) ) ); 2a a x 2x 1 + a a a x -1 x -1 x -1 %e

…᫨ ãáâ ­®¢¨âì §­ ç¥­¨¥ "true", â® ¯®ª § â¥«¨ á⥯¥­¨ ­ ç­ãâ ã¯à®é âìáï: 26

ratsimpexpons:true$ fullratsimp( exp( (x^(a/2)-1)^2 * (x^(a/2)+1)^2 / (x^a-1) ) ); a x - 1 %e



ratexpand à áªà뢠¥â ᪮¡ª¨ ¢ ¢ëà ¥­¨¨. ¥ á­ ¡ ¥â ¢ëà ¥­¨¥ ¬¥âª®© "/R/". Žâ«¨ç ¥âáï ®â ä㭪樨 "expand" ⥬, çâ® ¯à¨¢®¤¨â ¢ëà ¥­¨¥ ª ª ­®­¨ç¥áª®© ä®à¬¥, ¯®í⮬㠮⢥⠬®¥â ®ª § âìáï ª®à®ç¥, 祬 ¯à¨ ¯à¨¬¥­¥­¨¨ "expand": ”ã­ªæ¨ï

ratexpand( (x^3+2*x^2*y+y^3)/ (x^2+2*x*y+y^2)+ x^2*y/(x^2+2*x*y+y^2)+ 3*x*y^2/(x+y)^2 ); y + x (á¬. ¢ëè¥  ­ «®£¨ç­ë© ¯à¨¬¥à á "expand").

27

7. Ǒ८¡à §®¢ ­¨¥ âਣ®­®¬¥âà¨ç¥áª¨å ¢ëà ¥­¨© 

trigexpand à áª« ¤ë¢ ¥â ¢á¥ âਣ®­®¬¥âà¨ç¥áª¨¥ ä㭪樨 ®â á㬬 ¢ áã¬¬ë ¯à®¨§¢¥¤¥­¨© âਣ®­®¬¥âà¨ç¥áª¨å ä㭪権 ”ã­ªæ¨ï

trigexpand(sin(x+y));



os(x) sin(y) + sin(x) os(y)

trigexpand ã¯à ¢«ï¥â à ¡®â®© ä㭪樨 "trigexpand". ˆ§­ ç «ì­® ¯¥à¥¬¥­­ ï "trigexpand" à ¢­ï¥âáï "false", íâ® ¯à¨¢®¤¨â ª ⮬ã, çâ® äã­ªæ¨ï "trigexpand" ­¥ à ¡®â ¥â ¤® ª®­æ , â.¥. ¥¥ ¯®¢â®à­ë© ¢ë§®¢ ¬®¥â ¨§¬¥­¨âì ¢ëà ¥­¨¥. …᫨ ¯¥à¥¬¥­­ ï "trigexpand" ¡ã¤¥â à ¢­  "true", â® äã­ªæ¨ï "trigexpand" ¡ã¤¥â à ¡®â âì ¤® â¥å ¯®à, ¯®ª  ¢ëà ¥­¨¥ ­¥ ¯¥à¥áâ ­¥â ¬¥­ïâìáï. Ǒ¥à¥¬¥­­ ï

trigexpand(sin(2*x+y));

os(2 x) sin(y) + sin(2 x) os(y) trigexpand(%); 2 2 ( os (x) - sin (x)) sin(y) + 2 os(x) sin(x) os(y) trigexpand:true; true trigexpand(sin(2*x+y)); 2 2 ( os (x) - sin (x)) sin(y) + + 2 os(x) sin(x) os(y)



trigredu e ᢥàâ뢠¥â ¢á¥ ¯à®¨§¢¥¤¥­¨ï âਣ®­®¬¥âà¨ç¥áª¨å ä㭪権 ¢ âਣ®­®¬¥âà¨ç¥áª¨¥ ä㭪樨 ®â á㬬. ”ã­ªæ¨ï à ¡®â ¥â ­¥ ¤® ª®­æ , â ª çâ® ¯®¢â®à­ë© ¢ë§®¢ ¬®¥â ¨§¬¥­¨âì ¢ëà ¥­¨¥ ”ã­ªæ¨ï

trigredu e( ( os(x)^2sin(x)^2)*sin(y) + 2* os(x)*sin(x)* os(y) ); sin(y+2x) sin(y-2x) + os(2x)sin(y) 2 2 trigredu e(%); sin(y+2x)

28



”ã­ªæ¨ï

trigsimp

¢®¢á¥ ­¥ ã¯à®é ¥â ¢ëà ¥­¨¥,

os2 (x) = 1:

  ⮫쪮 ¯à¨¬¥­ï¥â ª ­¥¬ã ¯à ¢¨«® sin2 (x) +

trigsimp( ( os(x)^2sin(x)^2)*sin(y) + 2* os(x)*sin(x)* os(y) ); 2 (2 os (x) - 1) sin(y) + 2 os(x) sin(x) os(y)



trirat ¯ëâ ¥âáï ᢥá⨠¢ëà ¥­¨¥ á âਣ®­®¬¥âà¨ç¥áª¨¬¨ äã­ªæ¨ï¬¨ ª ­¥ª®¬ã ã­¨¢¥àá «ì­®¬ã ª ­®­¨ç¥áª®¬ã ¢¨¤ã (¢ ®¡é¥¬, ¯ëâ ¥âáï ã¯à®áâ¨âì ¢ëà ¥­¨¥). Š ª ¯à ¢¨«® áãé¥á⢥­­® ã¯à®é ¥â ¢ëà ¥­¨¥, ­® ¨­®£¤  à ¡®â ¥â ®ç¥­ì ¤®«£® (¨­®£¤  ¡¥áª®­¥ç­® ¤®«£®). ”ã­ªæ¨ï

trigrat( ( os(x)^2sin(x)^2)*sin(y) + 2* os(x)*sin(x)* os(y) ); sin(y+2x)

29

8. Ǒ८¡à §®¢ ­¨¥ ¢ëà ¥­¨© á® á⥯¥­ï¬¨ ¨ «®£ à¨ä¬ ¬¨ 

rad an ã¯à®é ¥â ¢ëà ¥­¨ï á® ¢«®¥­­ë¬¨ á⥯¥­ï¬¨ ¨ «®£ à¨ä¬ ¬¨: ”ã­ªæ¨ï

rad an( log( x^3*exp(4*y)* exp(5*log(w))/z^6 ) ); - 6 log(z) + 4 y + 3 log(x) + 5 log(w) rad an( (x^(a/2)-1)^2 * (x^(a/2)+1)^2 / (x^a-1) ); a x - 1



roots ontra t ª®¬¯ ªâ¨ä¨æ¨àã¥â ¢®§¢¥¤¥­¨ï ¢ á⥯¥­ì ¢ ¤ ­­®¬ ¢ëà ¥­¨¨ ”ã­ªæ¨ï

roots ontra t( x^(1/6)* y^(1/12)*z^(1/30) );

1/5 1/6 (x sqrt(y) z ) roots ontra t( x^(1/2)*y^(1/2)* z^(1/4) ); sqrt(x y sqrt(z))



log ontra t ª®¬¯ ªâ¨ä¨æ¨àã¥â «®£ à¨ä¬ë ¢ ¤ ­­®¬ ¢ëà ¥­¨¨ ”ã­ªæ¨ï

log ontra t( a*log(x)+ 3*log(y)-4*log(x) );

log(

y x

3 4

) + a log(x)

30

9. ‹®£¨ç¥áª¨¥ ¢ëà ¥­¨ï ¨ ¡ §  ¤ ­­ëå ‹®£¨ç¥áª¨¥ ¢ëà ¥­¨ï ®¡à §ãîâáï ¨§ ®¯¥à æ¨© áà ¢­¥­¨ï >

<

>=

<=

=

#

(ᨬ¢®« # ®§­ ç ¥â "­¥ à ¢­®",   § ¯¨áì "a=b" ¨¬¥¥â ᨭ®­¨¬ "equal(a,b)"). ‘âà ­­®¢ â®© ®á®¡¥­­®áâìî ®¯¥à æ¨© áà ¢­¥­¨ï ï¥âáï â®, çâ® ¥á«¨ ¨å ¯®áâ ¢¨âì ¢ ª ç¥á⢥ ãá«®¢¨© ¢ 横« å ¨ ãá«®¢­ëå ¢ëà ¥­¨ïå, â® ®­¨ ¡ã¤ãâ ¢ëç¨á«¥­ë, ­® ¢§ïâë¥ á ¬¨ ¯® ᥡ¥, ®­¨ ­¥ ¢ëç¨á«ïîâáï: 3>2;

equal(3,2); 3#2;

3>2 equal(3,2) 3#2

”« £ "pred" ¢ ä㭪樨 "ev" ¢ë§ë¢ ¥â ¢ëç¨á«¥­¨¥ «®£¨ç¥áª¨å ¢ëà ¥­¨©: ev(3#2,pred); 3#2,pred;

true true



is ¨­¨æ¨¨àã¥â ¢ëç¨á«¥­¨¥ «®£¨ç¥áª®£® ¢ëà ¥­¨ï ”ã­ªæ¨ï

is(3>2); is(3=2);

is(equal(3,2)); is(3#2);

true false false true

Šà®¬¥ ⮣®, ®¯à¥¤¥«¥­ë ¢áâ஥­­ë¥ «®£¨ç¥áª¨¥ ä㭪樨, ¯¥à¥ç¨á«¨¬ ­¥ª®â®àë¥ ¨§ ­¨å. 31



atom ¢®§¢à é ¥â "true", ¥á«¨  à£ã¬¥­â ­¥ ¨¬¥¥â áâàãªâãàë, â.¥. á®áâ ¢­ëå ç á⥩ (­ ¯à¨¬¥à, ç¨á«® ¨«¨ ¯¥à¥¬¥­­ ï ­¥ ¨¬¥îâ áâàãªâãàë). ”ã­ªæ¨ï

atom(x);

atom(f(x));

true false



zeroequiv ¯à®¢¥àï¥â, ï¥âáï «¨ ­¥ª®â®à ï äã­ªæ¨ï ®¤­®£®  à£ã¬¥­â  ­ã«¥¬. Ž­  ¢®§¢à é ¥â "true", ¥á«¨ äã­ªæ¨ï à ¢­  ­ã«î ¨ "false" ¢ ¯à®â¨¢­®¬ á«ãç ¥. ”ã­ªæ¨ï

zeroequiv(exp(2*x) - exp(x)^2, x) true



freeof ¢®§¢à é ¥â "true", ¥á«¨ ¢â®à®© ¥¥  à£ã¬¥­â ­¥ ᮤ¥à¨â ("᢮¡®¤¥­ ®â") ¯¥à¢®£® ”ã­ªæ¨ï

freeof(x,f(x+g(y))); freeof(g,f(x+g(y))); freeof(z,f(x+g(y)));

false false true



symbolp ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï ᨬ¢®«®¬: ”ã­ªæ¨ï

symbolp(f(x)); symbolp(3); symbolp(f);

false false true

32



s alarp ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï ª®­á⠭⮩: ”ã­ªæ¨ï

s alarp(f);

s alarp(sin(1/3)); s alarp(f(1/3)); s alarp(1/3);



false true

listp ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï ᯨ᪮¬. listp([x,y℄);

false true

matrixp ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï ¬ âà¨æ¥©. ”ã­ªæ¨ï

m:ident(2);

matrixp(x); matrixp(m);



true

”ã­ªæ¨ï

listp(x);



false



1 0 0 1



false true

numberp ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï ç¨á«®¬: ”ã­ªæ¨ï

numberp(1/3);

numberp(sin(1/3)); numberp(exp(1.0)); numberp(exp(1)); numberp(%pi); numberp(1.3b22);

true false true false false

33

true



integerp ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï æ¥«ë¬ ç¨á«®¬. ”ã­ªæ¨ï

integerp(-3);

integerp(1/5);



oddp ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï æ¥«ë¬ ­¥ç¥â­ë¬ ç¨á«®¬. oddp(4);

false

evenp ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï æ¥«ë¬ ç¥â­ë¬ ç¨á«®¬. evenp(-3);

true false

primep ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï æ¥«ë¬ ¯à®áâë¬ ç¨á«®¬. ”ã­ªæ¨ï

primep(11); primep(9);



true

”ã­ªæ¨ï

evenp(4);



false

”ã­ªæ¨ï

oddp(-3);



true

true false

floatnump ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï ¤¥©á⢨⥫ì­ë¬ ç¨á«®¬ ¬ è¨­­®© â®ç­®áâ¨. ”ã­ªæ¨ï

floatnump(1.0); floatnump(1);

floatnump(2.3e-4); floatnump(2.3b-4);

true false true false

34



bfloatp ¢®§¢à é ¥â "true", ¥á«¨ ¥¥  à£ã¬¥­â ï¥âáï ¤¥©á⢨⥫ì­ë¬ ç¨á«®¬ ­¥®£à ­¨ç¥­­®© â®ç­®áâ¨. ”ã­ªæ¨ï

bfloatp(1.0); bfloatp(1);

bfloatp(2.3b-4); bfloatp(2.3e-4);

false false true false

Šà®¬¥ ⮣®, «®£¨ç¥áª¨¬¨ ¢ëà ¥­¨ï¬¨ ïîâáï § ¯à®áë ¨§ ¡ §ë ¤ ­­ëå: is(a>3);

‘«¥¤ã¥â ¯®¤ç¥àª­ãâì, çâ® à¥çì ¨¤¥â ­¥ ® §­ ç¥­¨¨ ¯¥à¥¬¥­­®© "a" (ª®â®à®¥ ­¥ ¯à¨á¢®¥­® ¨, á«¥¤®¢ â¥«ì­®, ­¥¨§¢¥áâ­®),   ®¡ ¨­ä®à¬ æ¨¨, ¢ ª ª®© ®¡« á⨠íâ  ¯¥à¥¬¥­­ ï ¬®¥â ¬¥­ïâìáï.



assume ¢¢®¤¨â ¨­ä®à¬ æ¨î ® ¯¥à¥¬¥­­®© ¢ ¡ §ã ¤ ­­ëå. ”ã­ªæ¨ï

assume(n>4);

[n>4℄

Ǒ®á«¥ í⮣® ¬®­® ¢¢®¤¨âì § ¯à®á ⨯  is(n>1);

true

Ǒਠí⮬ § ¯à®áë ­  ¨­ä®à¬ æ¨î, ª®â®à®© ¢ ¡ §¥ ¤ ­­ëå ­¥â, ¢ë§®¢ãâ á®®¡é¥­¨¥ "unknown": is(n<7); is(n>7);

unknown

unknown ’®ç­® â ª ¥ ¢ë§®¢¥â á®®¡é¥­¨¥ "unknown" ¡®«¥¥ á«®­ë© § ¯à®á (ª®â®àë© ¢ ¯à¨­æ¨¯¥ ¤®«¥­ ¡ë« ¡ë ¤ âì §­ ç¥­¨¥ "true"): is(n^2+n>19); unknown

‡ ¡ ¢­®, ç⮠१ã«ìâ â ¤àã£¨å § ¯à®á®¢ ¤®¢®«ì­® § £ ¤®ç¥­: is(n^2+n>1);

unknown

35

is(n^2+n>0); true

Ǒ®¢â®à­®¥ ¯à¨¬¥­¥­¨¥ ä㭪樨 "assume" ¯à®¢¥àï¥âáï ­  ¯à®â¨¢®à¥ç¨¢®áâì ¨ ­  ¨§¡ëâ®ç­®áâì. ‚ á«ãç ¥, ¥á«¨ ­®¢ ï ¨­ä®à¬ æ¨ï ­¥ ¯à®â¨¢®à¥ç¨â ¯à¥¤ë¤ã騬 ¤ ­­ë¬ ¨ ­¥ ¢ë⥪ ¥â ¨§ ­¨å, ®­   ¤¤¨â¨¢­® ¤®¡ ¢«ï¥âáï ª ¡ §¥ ¤ ­­ëå. Š á® «¥­¨î, ¯à¥¤ë¤ã騥 ãá«®¢¨ï ­¥ ¯à®¢¥àïîâáï ­  ¨§¡ëâ®ç­®áâì ¯à¨ ¯®ï¢«¥­¨¨ ­®¢ëå ãá«®¢¨©: assume(n>3); assume(n<3); assume(n>10); assume(n<30); is(n<9); is(n>31);

[redundant℄ [in onsistent℄ [n>10℄ [n<30℄ false false



properties ¯¥ç â ¥â ᢮©á⢠ ¯¥à¥¬¥­­®© ¨, ⥬ á ¬ë¬, ¯®§¢®«ï¥â ¢ëïá­¨âì, ª ª ï ¨¬¥­­® ¨­ä®à¬ æ¨ï ᮤ¥à¨âáï ¢ ¡ §¥ ¤ ­­ëå ® ¤ ­­®© ¯¥à¥¬¥­­®© ”ã­ªæ¨ï

properties(n);

[database info, n > 4, n > 10, 30 > n℄

(­®¢®¥ ãá«®¢¨¥ n > 10 ­¥ ®â¬¥­¨«® ¨§¡ëâ®ç­®¥ ⥯¥àì ãá«®¢¨¥ n > 4). ˆ§ ¯à¨¢¥¤¥­­ëå ¯à¨¬¥à®¢ ¢¨¤­®, çâ® ¯®¬¥­ïâì ᢮©á⢮ ¯¥à¥¬¥­­®© ­  ¯à®â¨¢®¯®«®­®¥ á ¯®¬®éìî ä㭪樨 "assume" ­¥¢®§¬®­®: assume(x>0)$ assume(x<0); is(x>0);

[in onsistent℄ true

36



forget ®â¬¥­ï¥â ᢥ¤¥­¨¥, ¢¢¥¤¥­­®¥ ¢ ¡ §ã ¤ ­­ëå. â® ¯®§¢®«ï¥â ¯®¬¥­ïâì ᢮©á⢮ ¯¥à¥¬¥­­®© ­  ¯à®â¨¢®¯®«®­®¥. ”ã­ªæ¨ï

forget(n<30);

properties(n);

[n<30℄ [database info, n > 4, n > 10℄

‡ ¡ ¢­®, çâ® ¯®á«¥ ¢á¥å íâ¨å ¬ ­¨¯ã«ï権 ¬®­® ¯à¨á¢®¨âì ¯¥à¥¬¥­­®© "n" §­ ç¥­¨¥, ª®â®à®¥ ¡ã¤¥â ¯à®â¨¢®à¥ç¨âì ¨­ä®à¬ æ¨¨ ¨§ ¡ §ë ¤ ­­ëå: n:-77;

is(n>0); properties(n);



-77 false [value, database info, n > 4, n > 10℄

kill ã­¨çâ® ¥â ¢áî ¨­ä®à¬ æ¨î (ª ª ᢮©á⢠, â ª ¨ ¯à¨á¢®¥­­®¥ §­ ç¥­¨¥) ®¡ ®¡ê¥ªâ¥ ¨«¨ ­¥áª®«ìª¨å ®¡ê¥ªâ å: ”ã­ªæ¨ï

kill(x,y,z);

done

â  äã­ªæ¨ï ¯®§¢®«ï¥â §  ®¤¨­ à § «¨ª¢¨¤¨à®¢ âì ¢áî à ­¥¥ ¢¢¥¤¥­­ãî ¨­ä®à¬ æ¨î ® ¯¥à¥¬¥­­®© "n". kill(n);

properties(n);

done [℄

ˆ­â¥à¥á­®, çâ® íâã äã­ªæ¨î ¬®­® ¢ë§¢ âì á  à£ã¬¥­â®¬ "all". Ǒਠí⮬ ¡ã¤ãâ "㡨âë" ¢á¥ ®¯à¥¤¥«¥­­ë¥ ª ­ áâ®ï饬㠢६¥­¨ ¯¥à¥¬¥­­ë¥. Ž¤­ ª® ¯à¨ í⮬ MAXIMA ­¥ ¢®§¢à é ¥âáï ¢ áâ à⮢®¥ á®áâ®ï­¨¥, ¯®áª®«ìªã ¯ à ¬¥âà ¬ ¨ ä« £ ¬ ­¥ ¯à¨á¢ ¨¢ îâáï ¯¥à¢®­ ç «ì­ë¥ §­ ç¥­¨ï. …᫨ ¯¥à¥¬¥­­®© "linel" ¡ë«® ¯à¨á¢®¥­® §­ ç¥­¨¥ "40", â® ¯®á«¥ "kill(all);" ®­® â ª ¨ ®áâ ­¥âáï "40",   ­¥ ¢¥à­¥âáï ª ¨á室­®¬ã §­ ç¥­¨î "79". ‘®áâ ¢­ë¥ «®£¨ç¥áª¨¥ ¢ëà ¥­¨ï ä®à¬¨àãîâáï á ¯®¬®éìî «®£¨ç¥áª¨å ®¯¥à æ¨© "and", "or", "not". is( 3>1 and -1>=-3 and 2#1 and not(equal(2,1)) ); true

37

is( 3<1 or -1<=-3 or not 2#1 or 2=1 );

false

38

10. “á«®¢­ë¥ ¢ëà ¥­¨ï ¨ 横«ë ‘¨­â ªá¨á ãá«®¢­®£® ¢ëà ¥­¨ï ¬®¥â ¡ëâì ¯à®¨««îáâà¨à®¢ ­ ¯à¨¬¥à®¬ a:1$ if a>3 then x:1 else x:-1; -1 x; -1 if a<3 then x:x+1 else x:x-1; 0 x; 0 Š ª ®¡ëç­®, ç áâì á "else" ¬®­® ®¯ãáâ¨âì if a>3 then x:1; false if a<3 then x:1; 1

‘¨­â ªá¨á 横«  ¤®¯ã᪠¥â âਠ¢ à¨ ­â 

(%i5) for i:1 thru 3 step 2 do disp(i); 1 3 (%o5) done (%i6) for i:1 step 2 while i<6 do ldisplay(i); (%t6) i=1 (%t7) i=3 (%t8) i=5 (%o8) done (%i9) for i:1 step 2 unless i>4 do display(i); i=1 i=3 (%o9) done

(§ ®¤­® ¬ë ¥é¥ à § ¯à®¨««îáâà¨à®¢ «¨ à ¡®âã ä㭪権 "disp", "display" ¨ "ldisplay"). Š ª ®¡ëç­®, ¥á«¨ è £ à ¢¥­ ¥¤¨­¨æ¥, ¥£® ¬®­® ®¯ãáâ¨âì: x:0$ for i:1 thru 5 do x:x+1$ x; 5

Šà®¬¥ ⮣®, ¢®§¬®­ë 横«ë, ¢ ª®â®àëå ¯¥à¥¬¥­­ ï 横«  ¬¥­ï¥âáï ­¥ ­  䨪á¨à®¢ ­­ãî ¢¥«¨ç¨­ã,   ¯® ¯à®¨§¢®«ì­®¬ã § ª®­ã: 39

for i:1 next 2*i thru 5 do disp(i)$ 1 2 4

 §ã¬¥¥âáï, ¤®¯ãáâ¨¬ë ¢«®¥­­ë¥ 横«ë ¨ ¢«®¥­­ë¥ ãá«®¢­ë¥ ¢ëà ¥­¨ï. ‘ãé¥áâ¢ãîâ â ª¥ 横«ë á㬬¨à®¢ ­¨ï ¨ 㬭®¥­¨ï.



sum ॠ«¨§ã¥â 横« á㬬¨à®¢ ­¨ï ”ã­ªæ¨ï

sum(x^i,i,3,5);

sum(x^i,i,a+3,a+5);

5 4 3 x +x +x a+5

x



a+4

+x

a+3

+x

produ t ॠ«¨§ã¥â 横« 㬭®¥­¨ï ”ã­ªæ¨ï

produ t(x+i,i,3,5);

(x+3)(x+4)(x+5)

40

11. «®ª¨ Š ª ¢ ãá«®¢­ëå ¢ëà ¥­¨ïå, â ª ¨ ¢ 横« å ¢¬¥áâ® ¯à®áâëå ®¯¥à â®à®¢ ¬®­® ¯¨á âì á®áâ ¢­ë¥ ®¯¥à â®àë, â.¥. ¡«®ª¨. ‘â ­¤ àâ­ë© ¡«®ª ¨¬¥¥â ¢¨¤: blo k([r,s,t℄,r:1,s:r+1,t:s+1,x:t,t*t);

‘­ ç «  ¨¤¥â ᯨ᮪ «®ª «ì­ëå ¯¥à¥¬¥­­ëå ¡«®ª  (£«®¡ «ì­ë¥ ¯¥à¥¬¥­­ë¥ á ⥬¨ ¥ ¨¬¥­ ¬¨ ­¨ª ª ­¥ á¢ï§ ­ë á í⨬¨ «®ª «ì­ë¬¨ ¯¥à¥¬¥­­ë¬¨). ‘¯¨á®ª «®ª «ì­ëå ¯¥à¥¬¥­­ëå ¬®¥â ¡ëâì ¯ãáâë¬. „ «¥¥ ¨¤¥â ­ ¡®à ®¯¥à â®à®¢. “¯à®é¥­­ë© ¡«®ª ¨¬¥¥â ¢¨¤: (x:1,x:x+2,a:x);

Ž¡ëç­® ¢ 横« å ¨ ¢ ãá«®¢­ëå ¢ëà ¥­¨ïå ¯à¨¬¥­ïîâ ¨¬¥­­® íâã ä®à¬ã ¡«®ª . ‡­ ç¥­¨¥¬ ¡«®ª  ï¥âáï §­ ç¥­¨¥ ¯®á«¥¤­¥£® ¨§ ¥£® ®¯¥à â®à®¢. Ǒਢ¥¤¥¬ ­¥áª®«ìª® ¢¯®«­¥ ¡¥áá¬ëá«¥­­ëå ¯à¨¬¥à®¢ ¯à¨¬¥­¥­¨ï ¡«®ª®¢ ¢ 横« å ¨ ¢ ãá«®¢­ëå ¢ëà ¥­¨ïå a:1$ if a>3 then (r:y,r:(r+1)*2) else blo k([s℄,s:x,s:s+1,r:s^2);

(x + 1)

2

a:4$ if a>3 then (r:y,r:(r+1)*2) else blo k([s℄,s:x,s:s+1,r:s^2); 2 (y + 1) for i:1 thru 4 do (s:0,x:z^i,t:1, for j:i thru i+3 do blo k([℄,s:s+j*t,t:t*x), print("s(",i,")=",s) )$ 3 2 s( 1 )= 4 z + 3 z + 2 z + 1 6 4 2 s( 2 )= 5 z + 4 z + 3 z + 2 9 6 3 s( 3 )= 6 z + 5 z + 4 z + 3 12 8 4 s( 4 )= 7 z + 6 z + 5 z + 4

‚­ãâਠ¤ ­­®£® ¡«®ª  ¤®¯ã᪠îâáï ®¯¥à â®à ¯¥à¥å®¤  ­  ¬¥âªã ¨ ®¯¥à â®à "return". 41

Ž¯¥à â®à "return" ¯à¥ªà é ¥â ¢ë¯®«­¥­¨¥ ⥪ã饣® ¡«®ª  ¨ ¢®§¢à é ¥â ¢ ª ç¥á⢥ §­ ç¥­¨ï ¡«®ª  ᢮©  à£ã¬¥­â blo k([℄,x:2,x:x*x, return(x), x:x*x); 4 x; 4

‚ ®âáãâá⢨¥ ®¯¥à â®à  ¯¥à¥å®¤  ­  ¬¥âªã ®¯¥à â®àë ¢ ¡«®ª¥ ¢ë¯®«­ïîâáï ¯®á«¥¤®¢ â¥«ì­®. (‚ ¤ ­­®¬ á«ãç ¥ á«®¢® "¬¥âª " ®§­ ç ¥â ®â­î¤ì ­¥ ¬¥âªã ⨯  "%i5" ¨«¨ "%o7"). Ž¯¥à â®à "go" ¢ë¯®«­ï¥â ¯¥à¥å®¤ ­  ¬¥âªã, à á¯®«®¥­­ãî ¢ í⮬ ¥ ¡«®ª¥: blo k([a℄,a:1,metka, a:a+1, if a=1001 then return(-a), go(metka) ); -1001

‚ í⮬ ¡«®ª¥ ॠ«¨§®¢ ­ 横«, ª®â®àë© § ¢¥àè ¥âáï ¯® ¤®á⨥­¨¨ "¯¥à¥¬¥­­®© 横« " §­ ç¥­¨ï 1001. Œ¥âª®© ¬®¥â ¡ëâì ¯à®¨§¢®«ì­ë© ¨¤¥­â¨ä¨ª â®à. ‘«¥¤ã¥â ¨¬¥âì ¢ ¢¨¤ã, ç⮠横« á ¬ ¯® ᥡ¥ ï¥âáï ¡«®ª®¬, â ª çâ® (¢ ®â«¨ç¨¥ ®â ï§ëª  "C") ¯à¥à¢ âì ¢ë¯®«­¥­¨¥ 横«®¢ (®á®¡¥­­® ¢«®¥­­ëå 横«®¢) á ¯®¬®éìî ®¯¥à â®à  "go" ­¥¢®§¬®­® | ®¯¥à â®à "go" ¨ ¬¥âª  ®ª ãâáï ¢ à §­ëå ¡«®ª å. ’® ¥ á ¬®¥ ®â­®á¨âáï ª ®¯¥à â®àã "return". …᫨ 横«, à á¯®«®¥­­ë© ¢­ãâਠ¡«®ª , ᮤ¥à¨â ®¯¥à â®à "return", â® ¯à¨ ¨á¯®«­¥­¨¨ ®¯¥à â®à  "return" ¯à®¨§®©¤¥â ¢ë室 ¨§ 横« , ­® ­¥ ¢ë室 ¨§ ¡«®ª : blo k([℄,x:for i:1 thru 15 do if i=2 then return(555),display(x),777); x=555 777 blo k([℄,x:for i:1 thru 15 do if i=52 then return(555),display(x),777); x=done 777

…᫨ ­¥®¡å®¤¨¬® ¢ë©â¨ ¨§ ­¥áª®«ìª¨å ¢«®¥­­ëå ¡«®ª®¢ áà §ã (¨«¨ ­¥áª®«ìª¨å ¡«®ª®¢ ¨ 横«®¢ áà §ã) ¨ ¯à¨ í⮬ ¢®§¢à â¨âì ­¥ª®â®à®¥ §­ ç¥­¨¥, â® á«¥¤ã¥â ¯à¨¬¥­ïâì ¡«®ª " at h"

at h( blo k([℄,a:1,a:a+1, throw(a),a:a+7),a:a+9 ); a;

2

2

at h(blo k([℄,for i:1 thru 15 do

42

if i=2 then throw(555)),777); 555

at h(blo k([℄,for i:1 thru 15 do if i=52 then throw(555)),777); 777 Ž¯¥à â®à "throw" | íâ®  ­ «®£ ®¯¥à â®à  "return", ­® ®­ ®¡à뢠¥â ­¥ â¥-

ªã騩 ¡«®ª,   ¢á¥ ¢«®¥­­ë¥ ¡«®ª¨ ¢¯«®âì ¤® ¯¥à¢®£® ¢áâà¥â¨¢è¥£®áï ¡«®ª  " at h".  ª®­¥æ, ¡«®ª "err at h" ¯®§¢®«ï¥â ¯¥à¥å¢ â뢠âì ­¥ª®â®àë¥ (ª á® «¥­¨î, ­¥ ¢á¥!) ¨§ ®è¨¡®ª, ª®â®àë¥ ¢ ­®à¬ «ì­®© á¨âã æ¨¨ ¯à¨¢¥«¨ ¡ë ª § ¢¥à襭¨î áç¥â .  ¯à¨¬¥à, ¥á«¨ ¯¥à¥¬¥­­ë¬ "y" ¨ "z" ¯à¨á¢®¥­ë §­ ç¥­¨ï "b+ " ¨ "b- " ᮮ⢥âá⢥­­®, â® ®¯¥à â®à (y+z)::3;

¢ë§®¢¥â á®®¡é¥­¨¥ ®¡ ®è¨¡ª¥ ¨ ¯à¥à¢¥â ¨á¯®«­¥­¨¥ ä ©« . Ž¤­ ª®, ¥á«¨ ¯®¬¥áâ¨âì íâ®â § ¯à®á ¢ ¡«®ª "err at h", â® ¯à®¨§®©¤¥â ⮫쪮 ¢ë室 ¨§ í⮣® ¡«®ª . ‡­ ç¥­¨¥¬ ¡«®ª  ¢ í⮬ á«ãç ¥ ï¥âáï ¯ãá⮩ ᯨ᮪: (%i7) err at h(a:2,y:b+ ,z:b- ,(y+z)::3, a:-77); Improper value assignment: 2 b (%o7) [℄ (%i8) a; (%o8) 2

43

12. ‘¯¨áª¨ ‘¯¨á®ª ¢ MAXIMA | íâ® ®¡ê¥ªâ, ¢¯®«­¥  ­ «®£¨ç­ë© ᯨáªã ¢ LISP, â.¥. 㯮à冷祭­ ï ᮢ®ªã¯­®áâì ¯à®¨§¢®«ì­ëå (¢®§¬®­® à §­®à®¤­ëå) ®¡ê¥ªâ®¢ (çâ®-â® ¢à®¤¥ ®¤­®¬¥à­®£® ¬ áᨢ ). —â®¡ë § ¤ âì ᯨ᮪, ¤®áâ â®ç­® § ¯¨á âì ¥£® í«¥¬¥­âë ç¥à¥§ § ¯ïâãî ¨ ®£à ­¨ç¨âì § ¯¨áì ª¢ ¤à â­ë¬¨ ᪮¡ª ¬¨ li1:[a,b,11℄;

[a,b,11℄

«¥¬¥­â®¬ ᯨ᪠ ¬®¥â ¡ëâì «î¡®© ®¡ê¥ªâ, ¢ ⮬ ç¨á«¥ ¨ ¤à㣮© ᯨ᮪ li2:[a,b,[ ,d℄,e,f℄;

[a,b,[ ,d℄,e,f℄

‘¯¨á®ª ¬®¥â ¡ëâì ¯ãáâë¬ li3:[ ℄;

[℄

¨«¨ á®áâ®ïâì ¨§ ®¤­®£® í«¥¬¥­â  li4:[77℄;

[77℄

‘á뫪  ­  í«¥¬¥­â ᯨ᪠ ¯à®¨§¢®¤¨âáï â ª: li1[2℄;

li1[2℄: ; li1; li2[3℄; li2[3℄[2℄;

b

[a, ,11℄ [ ,d℄ d



length ¢®§¢à é ¥â ¤«¨­ã ᯨ᪠ ”ã­ªæ¨ï

length([a,b,[ ,d℄,e,f℄); 5

44



part ¯®§¢®«ï¥â ¢ë¤¥«¨âì â®â ¨«¨ ¨­®© í«¥¬¥­â ç áâì ᯨ᪠. ”ã­ªæ¨ï

part([a,b, ℄,2);

part([a,[b, ℄,d℄,2);

b [b, ℄

…᫨ ᯨ᮪ ¢«®¥­­ë©, â® ­¥®¡ï§ â¥«ì­® ¯¨á âì part(part([a,[b, ℄,d℄,2),2);

¬®­® ¯à®áâ® ­ ¯¨á âì

part([a,[b, ℄,d℄,2,2);

‘«¥¤ã¥â ¯®¤ç¥àª­ãâì, çâ® ¯à¨ ¯à¨á¢®¥­¨¨ ᯨ᪮¢ li1:[a,b, ℄$ li2:li1;

[a,b, ℄ ®â­î¤ì ­¥ ᮧ¤ ¥âáï ª®¯¨ï ᯨ᪠ "li1", ¯à®áâ® ¯¥à¥¬¥­­ ï "li2" áâ ­®¢¨âáï ¥é¥ ®¤­¨¬ 㪠§ â¥«¥¬ ­  â®â ¥ á ¬ë© á¯¨á®ª. Ǒ®í⮬ã li1[3℄:d$ li2; [a,b,d℄



opylist ¨§£®â®¢«ï¥â "­ áâ®ïéãî" ª®¯¨î ᯨ᪠ ”ã­ªæ¨ï

li1:[a,b, ℄$ li3: opylist(li1); li1[3℄:d$ li3;



[a,b, ℄ [a,b, ℄

makelist ¯®§¢®«ï¥â ᮧ¤ ¢ âì ᯨ᪨ ¨ ¤®¯ã᪠¥â 2 ¢ à¨ ­â  ᨭ⠪á¨á  ”ã­ªæ¨ï

makelist(i^3,i,1,3);

[1, 8, 27℄ (§¤¥áì ॠ«¨§ã¥âáï 横« ¯® ¯¥à¥¬¥­­®© "i" ¢ ¯à¥¤¥« å ®â 1 ¤® 3), «¨¡® makelist(x=i^2,i,[ ,d,e℄); [x= ^2, x=d^2, x=e^2℄ (§¤¥áì ¯¥à¥¬¥­­ ï "i" ¯à®¡¥£ ¥â ¢á¥ í«¥¬¥­âë § ¤ ­­®£® ᯨ᪠).

45



append ¯®§¢®«ï¥â ᪫¥¨¢ âì ¤¢  ᯨ᪠ ”ã­ªæ¨ï

append([a,b℄,[ ,d℄); li1:[a℄$ li2:[b, ℄$ append(li1,li2);



ons ¯®§¢®«ï¥â ¤®¡ ¢«ïâì í«¥¬¥­â ¢ ­ ç «® ᯨ᪠ [x, a, b℄

end ons ¯®§¢®«ï¥â ¤®¡ ¢«ïâì í«¥¬¥­â ¢ ª®­¥æ ᯨ᪠ ”ã­ªæ¨ï

end ons(x,[a,b℄);



[a, b, ℄

”ã­ªæ¨ï

ons(x,[a,b℄);



[a, b, , d℄

[a, b, x℄

reverse ¬¥­ï¥â ¯®à冷ª í«¥¬¥­â®¢ ¢ ᯨ᪥ ­  ®¡à â­ë© ”ã­ªæ¨ï

reverse([a,b,[ ,d℄,e,f℄); [f,e,[ ,d℄,b,a℄



sort 㯮à冷稢 ¥â í«¥¬¥­âë ᯨ᪠ ”ã­ªæ¨ï

sort([3,a,-1,x,b,0℄);

[-1,0,3,a,b,x℄

á­ ç «  ¨¤ãâ ç¨á«  (¢ ¯®à浪¥ ¢®§à áâ ­¨ï), § â¥¬ ¨¤¥­â¨ä¨ª â®àë (¢  «ä ¢¨â­®¬ ¯®à浪¥).



sublist á®áâ ¢«ï¥â ᯨ᮪ ¨§ â¥å í«¥¬¥­â®¢ ¨á室­®£® ᯨ᪠, ¤«ï ª®â®àëå § ¤ ­­ ï «®£¨ç¥áª ï äã­ªæ¨ï ¢®§¢à é ¥â §­ ç¥­¨¥ "true" ”ã­ªæ¨ï

sublist([3,a,-1,x,b,0℄,numberp); [3,-1,0℄ (äã­ªæ¨ï "numberp" ¢ë¤ ¥â "true" ¤«ï ç¨á¥« ¨ "false" ¢® ¢á¥å ®áâ «ì­ëå

á«ãç ïå). Ǒਠí⮬ ¬®¥â ¨á¯®«ì§®¢ âìáï «®£¨ç¥áª ï äã­ªæ¨ï, ®¯à¥¤¥«¥­­ ï ¯®«ì§®¢ â¥«¥¬ f(x):=is(x>5)$ sublist([7,3,6,4,5℄,f); [7,6℄

46



member ¢®§¢à é ¥â "true", ¥á«¨ ¥¥ ¯¥à¢ë©  à£ã¬¥­â ï¥âáï í«¥¬¥­â®¬ § ¤ ­­®£® ᯨ᪠, ¨ "false" ¢ ¯à®â¨¢­®¬ á«ãç ¥ ”ã­ªæ¨ï

li1:[a,b,[ ,d℄,e,f℄$ member(b,li1); member( ,li1); member([ ,d℄,li1);

true false true



first ¢ë¤¥«ï¥â ¯¥à¢ë© í«¥¬¥­â ᯨ᪠ ”ã­ªæ¨ï

first([a,b, ℄);

a



rest ¢ë¤¥«ï¥â ®áâ â®ª ¯®á«¥ 㤠«¥­¨ï ¯¥à¢®£® í«¥¬¥­â  ᯨ᪠ ”ã­ªæ¨ï

rest(a,b, );

[b, ℄



last ¢ë¤¥«ï¥â ¯®á«¥¤­¨© í«¥¬¥­â ᯨ᪠ ”ã­ªæ¨ï

last([a,b, ℄);



map ¯à¨¬¥­ï¥â § ¤ ­­ãî äã­ªæ¨î ª ª ¤®¬ã í«¥¬¥­âã ᯨ᪠ ”ã­ªæ¨ï

map(sin,[-1,0,1℄);

[-sin(1),0,sin(1)℄

¯à¨ í⮬ ¬®¥â ¨á¯®«ì§®¢ âìáï ª ª áâ ­¤ àâ­ ï äã­ªæ¨ï, â ª ¨ äã­ªæ¨ï, ®¯à¥¤¥«¥­­ ï ¯®«ì§®¢ â¥«¥¬ f(x):=2*x$ map(f,[1,2,3℄); [2,4,6℄

47



apply ¯à¨¬¥­ï¥â § ¤ ­­ãî äã­ªæ¨î ª® ¢á¥¬ã ᯨáªã (ᯨ᮪ áâ ­®¢¨âáï ᯨ᪮¬  à£ã¬¥­â®¢ ä㭪樨).  ¯à¨¬¥à, ¥á«¨ ‚ë á®®à㤨«¨ ᯨ᮪, á®áâ®ï騩 ¨§ ç¨á¥«: ”ã­ªæ¨ï

li1:[33,-22,11℄$

â®, çâ®¡ë ­ ©â¨ ¬ ªá¨¬ «ì­®¥ ¨«¨ ¬¨­¨¬ «ì­®¥ ç¨á«®, ­ ¤® ¢ë§¢ âì äã­ªæ¨î "max" ¨«¨ "min". Ž¤­ ª®, ®¡¥ ä㭪樨 ¢ ª ç¥á⢥  à£ã¬¥­â  ®¨¤ îâ ­¥áª®«ìª® ç¨á¥«,   ­¥ ᯨ᮪, á®áâ ¢«¥­­ë© ¨§ ç¨á¥«. Ǒਬ¥­ïâì ¯®¤®¡­ë¥ ä㭪樨 ª ᯨ᪠¬ ¨ ¯®§¢®«ï¥â äã­ªæ¨ï "apply": apply(max,li1); apply(min,li1);

33 -22

48

13. Œ áá¨¢ë Œ áᨢë | íâ® ®¡ê¥ªâë á ¨­¤¥ªá ¬¨. ‘á뫪  ­  í«¥¬¥­â ¬ áᨢ  ¯à®¨§¢®¤¨âáï ®¡ëç­ë¬ ®¡à §®¬: ª®¬ ­¤  ar[2,0,1℄;

¢ë¢¥¤¥â §­ ç¥­¨¥ í«¥¬¥­â  ¬ áᨢ ,   ª®¬ ­¤  ar[55℄:77+x$

¯à¨á¢®¨â í«¥¬¥­âã ¬ áᨢ  㪠§ ­­®¥ §­ ç¥­¨¥. ‚ MAXIM'¥ ®¯à¥¤¥«¥­ë ¬ áᨢë 3 ¢¨¤®¢. íâ® ¬ áá¨¢ë ­¥®¯à¥¤¥«¥­­®£® à §¬¥à  ("hashed" ¬ áᨢ). ‡­ ç¥­¨ï í«¥¬¥­â®¢ â ª®£® ¬ áᨢ  § ¤ ¥âáï «¨¡® "¨­¤¨¢¨¤ã «ì­®"

‚®-¯¥à¢ëå,

ar1[23℄:7777$ ar1[-4℄:5555$ ar2[2,3℄:777$ ar2[-22,-33℄:555;$

«¨¡® ª ª äã­ªæ¨ï ¨­¤¥ªá®¢ ar3[i,j℄:=i+j;

ar3i , j :=i+j

ˆ­¤¥ªáë í⮣® ¬ áᨢ  ¬®£ã⠯ਭ¨¬ âì «î¡ë¥ 楫®ç¨á«¥­­ë¥ (¢ ⮬ ç¨á«¥ ¨ ®âà¨æ â¥«ì­ë¥) §­ ç¥­¨ï. …᫨ í«¥¬¥­âë § ¤ ­ë ª ª äã­ªæ¨ï ¨­¤¥ªá®¢, â® ¨å §­ ç¥­¨ï ¡ã¤ãâ ¢ëç¨á«ïâìáï ¯® ¯à¨¢¥¤¥­­®© ä®à¬ã«¥. Ǒਠ¯®¢â®à­®© áá뫪¥ ­  íâ®â ¥ í«¥¬¥­â ¢ëª« ¤ª¨ ­¥ ¯à®¨§¢®¤ïâáï | ¨á¯®«ì§ã¥âáï ¢ëç¨á«¥­­®¥ ¯à¥¤ë¤ã騩 à § §­ ç¥­¨¥. Šà®¬¥ ⮣®, ¤ ¥ ¥á«¨ § ¤ ­  äã­ªæ¨ï ¨­¤¥ªá®¢, ®â¤¥«ì­ë¥ í«¥¬¥­âë ¬ áᨢ  ¬®­® ¯¥à¥®¯à¥¤¥«ïâì ¨­¤¨¢¨¤ã «ì­® ar3[1,2℄;

ar3[-2,-3℄; ar3[-2,-3℄:7; ar3[3,4℄:5;

3 -5 7 5

‚®-¢â®àëå,

íâ® ¬ áá¨¢ë § ¤ ­­®£® à §¬¥à  49



array ®¯à¥¤¥«ï¥â ¬ áᨢ á ¤ ­­ë¬ ¨¬¥­¥¬, ®¯à¥¤¥«¥­­ë¬ ª®«¨ç¥á⢮¬ ¨­¤¥ªá®¢ ¨ § ¤ ­­ë¬ à §¬¥à®¬. Œ®­® â ª¥ 㪠§ âì ¥£® ⨯ ”ã­ªæ¨ï

array(ar4,2,1); array(ar5,2);

array(ar6,float,2,1);

ar4 ar5

ar6 array(ar7,float,1,1,1); ar7

ˆ­¤¥ªáë íâ¨å ¬ áᨢ®¢ ¯à®¡¥£ îâ §­ ç¥­¨ï ®â 0 ¤® 㪠§ ­­®£® ç¨á« , â ª çâ® ä ªâ¨ç¥áª¨© à §¬¥à ¬ áᨢ®¢ "ar4" ¨ "ar6" | 3  2, ¬ áᨢ  "ar5" | 3,   ¬ áᨢ  "ar7" | 2  2  2. Ǒ¥à¢®­ ç «ì­ë¥ (áࠧ㠯®á«¥ ®¯à¥¤¥«¥­¨ï) §­ ç¥­¨ï í«¥¬¥­â®¢ ¤«ï ¬ áᨢ  á ®¯à¥¤¥«¥­­ë¬ ⨯®¬ | íâ® ­ã«¨,   ¤«ï ¬ áᨢ  á ­¥®¯à¥¤¥«¥­­ë¬ ⨯®¬ ¯¥à¢®­ ç «ì­ë¥ §­ ç¥­¨ï ­¥ ®¯à¥¤¥«¥­ë, ¨ ¯à¨ ¯®¯ë⪥ ¨å ¢ë¢®¤  ¯¥ç â îâáï 5 ¤¨¥§®¢ (#####).



íâ® "á ¬®¯¥ç â î騥áï" ¬ áá¨¢ë ”ã­ªæ¨ï make_array ᮧ¤ ¥â ¬ áᨢë âà¥â쥣® ¢¨¤ , ᮤ¥à¨¬®¥ ª®â®àëå ¯¥ç â ¥âáï  ¢â®¬ â¨ç¥áª¨. –¥«¥á®®¡à §­®áâì ¨á¯®«ì§®¢ ­¨ï ¬ áᨢ®¢ í⮣® ¢¨¤  ᮬ­¨â¥«ì­ . ‚-âà¥âì¨å,

ar8:make_array('any,3,2); {Array: #2A((NIL NIL)(NIL NIL)(NIL NIL)) } ar9:make_array('float,3); {Array: #(0.0 0.0 0.0) } ˆ­¤¥ªáë §¤¥áì ¯à®¡¥£ îâ §­ ç¥­¨ï ®â 0 ¤® 㪠§ ­­®£® ç¨á«  ¬¨­ãá 1 , â.¥. à §¬¥à­®áâì "ar8" | 3  2,   "ar9" | 3.

Ǒ¥à¢®­ ç «ì­ë¥ (áࠧ㠯®á«¥ ᮧ¤ ­¨ï) §­ ç¥­¨ï í«¥¬¥­â®¢ ¤«ï ¬ áᨢ  á ®¯à¥¤¥«¥­­ë¬ ⨯®¬ | íâ® ­ã«¨,   ¤«ï ¬ áᨢ  á ­¥®¯à¥¤¥«¥­­ë¬ ⨯®¬ (â.¥. á ⨯®¬ "any") | íâ® §­ ç¥­¨¥ "NIL". „«ï ¬ áᨢ®¢ ¯¥à¢®£® ¨ ¢â®à®£® ¢¨¤®¢ ¨¤¥­â¨ä¨ª â®à | íâ® ¨¬¥­­® ¨¬ï ¬ áᨢ  ¨ ¥£® ¨á¯®«ì§®¢ ­¨¥ ¡¥§ ª¢ ¤à â­ëå ᪮¡®ª ­¥ ¢ë§ë¢ ¥â ­¨ª ª¨å ¤¥©á⢨©: ar4

ar4

„«ï ¬ áᨢ®¢ âà¥â쥣® ¢¨¤  ¨¤¥­â¨ä¨ª â®à | íâ® ­¥ ¨¬ï ¬ áᨢ ,   ®¡ëç­ ï ¯¥à¥¬¥­­ ï, §­ ç¥­¨¥ ª®â®à®© ¥áâì ¬ áᨢ âà¥â쥣® ¢¨¤ . ® ᮤ¥à¨¬®¥ ¬ áᨢ  âà¥â쥣® ¢¨¤  ¯¥ç â ¥âáï  ¢â®¬ â¨ç¥áª¨, ¯®í⮬ã 50

ar9; {Array: #(0.0 0.0 0.0) }



arrays ᮤ¥à¨â ᯨ᮪ ¨¬¥­ ¬ áᨢ®¢ ¯¥à¢®£® ¨ ¢â®à®£® ¢¨¤®¢, ®¯à¥¤¥«¥­­ëå ­  ¤ ­­ë© ¬®¬¥­â Ǒ¥à¥¬¥­­ ï

arrays;

[ar1,ar2,ar3,ar4,ar5,ar6,ar7℄



arrayinfo ¯¥ç â ¥â ¨­ä®à¬ æ¨î ® ¬ áᨢ¥ | ¥£® ¢¨¤ (¤«ï ¬ áᨢ®¢ ¯¥à¢®£® ¢¨¤  | "hashed", ¤«ï ¬ áᨢ®¢ ¢â®à®£® ¢¨¤  á ­¥®¯à¥¤¥«¥­­ë¬ ⨯®¬ | "de lared", ¤«ï ¢â®à®£® ¢¨¤  á ®¯à¥¤¥«¥­­ë¬ ⨯®¬ | " omplete", ¤«ï ¬ áᨢ®¢ âà¥â쥣® ¢¨¤  | "de lared"). „ «¥¥ ¯¥ç â ¥âáï ç¨á«® ¨­¤¥ªá®¢ ¬ áᨢ . „ «¥¥ ¤«ï ¬ áᨢ®¢ ¯¥à¢®£® ¢¨¤  ¯¥ç â îâáï ¨­¤¥ªáë í«¥¬¥­â®¢, §­ ç¥­¨ï ª®â®àëå ¨§¢¥áâ­ë (â.¥. «¨¡® ¡ë«¨ ¯à¨á¢®¥­ë, «¨¡® ¡ë«¨ ¢ëç¨á«¥­ë ¯® ä®à¬ã«¥),   ¤«ï ¬ áᨢ®¢ ¢â®à®£® ¨ âà¥â쥣® ¢¨¤®¢ | à §¬¥à (¢ ¢¨¤¥ ᯨ᪠ ¬ ªá¨¬ «ì­ëå §­ ç¥­¨© ª ¤®£® ¨§ ¨­¤¥ªá®¢, ¯à¨ í⮬ ¤«ï ¬ áᨢ  âà¥â쥣® ¢¨¤  ¨§ § ¤ ­­®£® ¯à¨ ®¯à¥¤¥«¥­¨¨ à §¬¥à   ¢â®¬ â¨ç¥áª¨ ¢ëç¨â ¥âáï ¥¤¨­¨æ ). ”ã­ªæ¨ï

arrayinfo(ar1);

arrayinfo(ar2); arrayinfo(ar3); arrayinfo(ar4); arrayinfo(ar5); arrayinfo(ar6); arrayinfo(ar7); arrayinfo(ar8); arrayinfo(ar9);

[hashed, 1, [-4℄, [23℄℄ [hashed, 2, [-22, -33℄, [2, 3℄℄ [hashed, 2, [-2, -3℄, [1, 2℄, [3, 4℄℄ [de lared, 2, [2, 1℄℄ [de lared, 1, [2℄℄ [ omplete, 2, [2, 1℄℄ [ omplete, 3, [1, 1, 1℄℄ [de lared, 2, [2, 1℄℄ [de lared, 1, [2℄℄

51



listarray ¯¥ç â ¥â ᮤ¥à¨¬®¥ ¬ áᨢ®¢ ¯¥à¢®£® ¨ ¢â®à®£® ¢¨¤®¢. Ǒਠí⮬ ᮤ¥à¨¬®¥ ¯¥ç â ¥âáï ¢ â ª®¬ ¯®à浪¥: á­ ç «  ¢á¥ ¤®¯ãáâ¨¬ë¥ §­ ç¥­¨ï ¯à®¡¥£ ¥â ¯®á«¥¤­¨© ¨­¤¥ªá, ¯®â®¬ ¯à¥¤¯®á«¥¤­¨© ¨.â.¤.  ¯à¨¬¥à, ¤«ï ¬ áᨢ  á ¤¢ã¬ï ¨­¤¥ªá ¬¨ ¯®à冷ª ¨­¤¥ªá®¢ ¡ã¤¥â â ª®©: (0,0) (0,1) (0,2) (0,3) (1,0) (1,1) (1,2) (1,3) (2,0) (2,1) (2,2) (2,3). ‚맮¢ í⮩ ä㭪樨 ¤«ï ¬ áᨢ  âà¥â쥣® ¢¨¤  ¯à¨¢®¤¨â ª á®®¡é¥­¨î ®¡ ®è¨¡ª¥. Š ª 㥠¡ë«® ᪠§ ­®, ᮤ¥à¨¬ë¬ ¬ áᨢ  ¯¥à¢®£® ¢¨¤  áç¨â îâáï â¥ í«¥¬¥­âë, ª®â®àë¥ ¢ëç¨á«ï«¨áì «¨¡® ¯à¨á¢ ¨¢ «¨áì ®. Ǒ®í⮬㠔㭪æ¨ï

listarray(ar3); ar3[6,6℄;

listarray(ar3); listarray(ar4);

[7, 3, 5℄ 12 [7, 3, 5, 12℄

[#####, #####, #####, #####, #####, #####℄ ar4[2,0℄:3$ ar4[0,1℄:a+b$ listarray(ar4); [#####, a+b, #####, #####, 3, #####℄ listarray(ar6); [0.0, 0.0, 0.0, 0.0, 0.0, 0.0℄ ar8[1,0℄:77$ ar8; {Array: #2A((NIL NIL)(77 NIL)(NIL NIL)) }



fillarray ¯®§¢®«ï¥â § ¯®«­ïâì ®¤­®¨­¤¥ªá­ë¥ ¬ áᨢë âà¥â쥣® ¢¨¤  ¨§ ᯨ᪠ (¯à¨ í⮬ ¤«¨­  ᯨ᪠ ¬®¥â ­¥ ᮢ¯ ¤ âì á à §¬¥à­®áâìî ¬ áᨢ ) ”ã­ªæ¨ï

ar9:make_array('float,3); {Array: #(0.0 0.0 0.0) } fillarray(ar9,[4.0,5.0℄)$ ar9; {Array: #(4.0 5.0 0.0) } fillarray(ar9,[6.0,7.0,8.0,9.0℄)$ ar9; {Array: #(6.0 7.0 8.0) }

52



kill ã­¨çâ® ¥â 㪠§ ­­ë© ®¡ê¥ªâ ¨«¨ ®¡ê¥ªâë, ¢ ⮬ ç¨á«¥ ¨ ¬ áᨢ. ”ã­ªæ¨ï

kill(ar1,ar4); arrays;



done [ar2,ar3,ar5,ar6,ar7℄

remarray ã­¨çâ® ¥â ¬ áᨢ ¨«¨ ¬ áá¨¢ë ”ã­ªæ¨ï

remarray(ar2,ar3,ar5,ar7); [ar2,ar3,ar5,ar7℄ arrays; [ar6℄ remarray(ar6)$ arrays; [℄

53

14. Œ âà¨æë ‚ MAXIM'¥ ®¯à¥¤¥«¥­ë ¯àאַ㣮«ì­ë¥ ¬ âà¨æë.



matrix ¢®§¢à é ¥â ¬ âà¨æã, § ¤ ­­ãî ¯®í«¥¬¥­â­® ”ã­ªæ¨ï

ma1:matrix([a,b, ℄,[d,e,f℄);   a b d e f

‡­ ç¥­¨¥ í«¥¬¥­â  ¬ âà¨æë ¨§¢«¥ª ¥âáï â ª ma1[1,2℄;

b

’®ç­® â ª ¥ ¬®­® ¯®¬¥­ïâì ®â¤¥«ì­ë© í«¥¬¥­â ¬ âà¨æë ma1[2,3℄:77$ ma1;



a b d e

77



‘ãé¥áâ¢ãîâ ¬ âà¨æë, á®áâ®ï騥 ¨§ ®¤­®© áâப¨ ma2:matrix([a,b℄)

¨ ¨§ ®¤­®£® á⮫¡æ 

ma3:matrix([a℄,[b℄)



[a b℄   a b

genmatrix ¢®§¢à é ¥â ¬ âà¨æã § ¤ ­­®© à §¬¥à­®áâ¨, á®áâ ¢«¥­­ãî ¨§ í«¥¬¥­â®¢ ¤¢ã娭¤¥ªá­®£® ¬ áᨢ  ”ã­ªæ¨ï

ma4:genmatrix(ar1,2,2) "

ar11,1 ar12,1

ar11,2 ar12,2

#

¯à¨ í⮬ ¬®­® § ¤ âì í«¥¬¥­â ¬ áᨢ  ¢ ®¡é¥¬ ¢¨¤¥ ar2[i,j℄:=10*i+2*j$ ma5:genmatrix(ar2,2,2);   12 14 22 24

54



zeromatrix ¢®§¢à é ¥â ¬ âà¨æã § ¤ ­­®© à §¬¥à­®áâ¨, á®áâ ¢«¥­­ãî ¨§ ­ã«¥© ”ã­ªæ¨ï

zeromatrix(2,3);





0 0 0 0

0 0



ident ¢®§¢à é ¥â ¥¤¨­¨ç­ãî ¬ âà¨æã § ¤ ­­®© à §¬¥à­®á⨠”ã­ªæ¨ï

ident(2);



1 0 0 1



Ǒà¨á¢®¥­¨¥ ¬ âà¨æ ãáâ஥­® ¢ MAXIM'¥ â ª ¥, ª ª ¨ ¯à¨á¢®¥­¨¥ ᯨ᪮¢, ¨¬¥­­®, ­ ¡®à ®¯¥à â®à®¢ ma1:matrix([a,b℄,[ ,d℄)$ ma2:ma1$ ®â­î¤ì ­¥ ᮧ¤ ¥â ª®¯¨î ¬ âà¨æë "ma1" á ¨¬¥­¥¬ "ma2",   ¤¥« ¥â ¯¥à¥¬¥­­ãî "ma2" ¥é¥ ®¤­¨¬ 㪠§ â¥«¥¬ ­  âã ¥ á ¬ãî ¬ âà¨æã. ‚ १ã«ìâ â¥ ma1[2,2℄:77$

¯à¨¢¥¤¥â ª ma2;





a b

77



opymatrix ¨§£®â®¢«ï¥â "­ áâ®ïéãî" ª®¯¨î ¬ âà¨æë ”ã­ªæ¨ï

ma1:matrix([a,b℄,[ ,d℄)$ ma2: opymatrix(ma1)$ ma1[2,2℄:77$ ma2;   a b

d



row ¢ë¤¥«ï¥â § ¤ ­­ãî áâப㠬 âà¨æë ”ã­ªæ¨ï

ma1:matrix([a,b℄,[ ,d℄)$ row(ma1,2); [ d℄



ol ¢ë¤¥«ï¥â § ¤ ­­ë© á⮫¡¥æ ¬ âà¨æë ”ã­ªæ¨ï

ol(ma1,1);

  a

55



addrow ¤®¡ ¢«ï¥â áâப㠪 ¬ âà¨æ¥ ”ã­ªæ¨ï

addrow(ma1,[e,f℄);



3 a b 4 d5 e f

add ol ¤®¡ ¢«ï¥â á⮫¡¥æ ª ¬ âà¨æ¥ ”ã­ªæ¨ï

add ol(ma1,[e,f℄);



2



a b

d

e f



submatrix ¢ë¤¥«ï¥â ¨§ ¬ âà¨æë ¯®¤¬ âà¨æã. €à£ã¬¥­âë ä㭪樨 ¨¬¥îâ á«¥¤ãî騩 ¢¨¤: á­ ç «  ç¥à¥§ § ¯ïâãî ¨¤ãâ ­®¬¥à  ¢ëç¥àª¨¢ ¥¬ëå áâப, § â¥¬ á ¬  ¬ âà¨æ ,   § â¥¬ ­®¬¥à  ¢ëç¥àª¨¢ ¥¬ëå á⮫¡æ®¢ ”ã­ªæ¨ï

ar2[i,j℄:=10*i+j$ ma2:genmatrix(ar2,4,5); 2 11 12 6 21 22 4 31 32 41 42

13 23 33 43

14 24 34 44

15 3 25 7 35 5 45

submatrix(1,3,ma2,2,4,5);   21 23 41 43

  ¬ âà¨æ å ®¯à¥¤¥«¥­ë ®¡ëç­ë¥ ®¯¥à æ¨¨ 㬭®¥­¨ï ­  ç¨á«®, á«®¥­¨ï ¨ ¬ âà¨ç­®£® 㬭®¥­¨ï. Ǒ®á«¥¤­¥¥ ॠ«¨§ã¥âáï á ¯®¬®éìî ¡¨­ à­®© ®¯¥à æ¨¨ "." (â®çª ).  §ã¬¥¥âáï, à §¬¥à­®á⨠¬ âà¨æ ¤®«­ë ®¡¥á¯¥ç¨¢ âì ¬ â¥¬ â¨ç¥áªãî ª®à४⭮áâì ®¯¥à æ¨© ma1:matrix([a,b℄,[ ,d℄)$ ma1+3*ma1+ident(2);   4a+1 4b 4 4d+1

ma1.ma1;

2

2

6 b +a 4

d+a

56

3 bd+ab 7 5 2 d +b

Ǒ® ­¥ ®ç¥­ì ïá­®© ¯à¨ç¨­¥ ®¯à¥¤¥«¥­  § £ ¤®ç­ ï ®¯¥à æ¨ï ¢®§¢¥¤¥­¨ï ¬ âà¨æë ¢ "®¡ëç­ãî" á⥯¥­ì ma1^3;

2

3

3

6a 4 3

b

3

3 7 5

d

Ž¤­ ª®, ®¯à¥¤¥«¥­  ¨ ®¯¥à æ¨ï ¬ âà¨ç­®£® ¢®§¢¥¤¥­¨ï ¢ 楫ãî á⥯¥­ì ma1^^2;

ma1^^(-1);

2

3

2

6 b +a 4

d+a

bd+ab 7 5 2 d +b

2

3 -b ad-b 7 7 a 5 ad-b

d 6 ad-b 6 4 - ad-b

„¥â¥à¬¨­ ­â ¬®­® ¢ë­¥á⨠§  ¯à¥¤¥«ë ¬ âà¨æë, ¥á«¨ ¢ë§¢ âì äã­ªæ¨î "ev" á ä« £®¬ "detout": ev(ma1^^(-1),detout); 

d -b - a ad-b



Šà®¬¥ ⮣®, ®â¤¥«ì­® ®¯à¥¤¥«¥­  äã­ªæ¨ï ®¡à é¥­¨ï ¬ âà¨æë, ª®â®à ï ï¥âáï ᨭ®­¨¬®¬ ®¯¥à æ¨¨ ¢®§¢¥¤¥­¨ï ¬ âà¨æë ¢ á⥯¥­ì "-1". invert(ma1);

2

d 6 ad-b 6 4 - ad-b

3 -b ad-b 7 7 a 5 ad-b

‘«¥¤ã¥â ®â¤¥«ì­® ®¡á㤨âì ᢮©á⢠ 㬭®¥­¨ï ¬ âà¨æ ­  áâப¨ ¨ á⮫¡æë. Š ª íâ® ­¨ áâà ­­®, ¥á«¨ ¬ âà¨æ  á⮨â á«¥¢ , â® ¯à ¢ë¬ ᮬ­®¨â¥«¥¬ ¬®¥â ¡ëâì ­¥ ⮫쪮 á⮫¡¥æ, ­® ¨ áâப  ¨ ¤ ¥ ᯨ᮪. li1:[e,f℄$ str1:matrix([e,f℄)$

57

stlb1:matrix([e℄,[f℄)$

Ǒ®á«¥ í⮣® ®¯¥à â®àë ma1.li1; ma1.str1; ma1.stlb1;

¯à¨¢¥¤ãâ ª ®¤­®© ¨ ⮩ ¥ ¢ë¤ ç¥ 

bf+ae df+ e



…᫨ ¬ âà¨æ  á⮨â á¯à ¢ , â® ¢ ª ç¥á⢥ «¥¢®£® ᮬ­®¨â¥«ï ¤®¯ãá⨬ë ᯨ᮪ ¨ áâப , ­® ­¥ á⮫¡¥æ | ¢ á«ãç ¥ á⮫¡æ  ¯®ï¢¨âáï á®®¡é¥­¨¥ ®¡ ®è¨¡ª¥. ’ ª çâ® ®¯¥à â®àë li1.ma1; str1.ma1;

¯à¨¢¥¤ãâ ª ®¤­®© ¨ ⮩ ¥ ¢ë¤ ç¥ [ f+ae df+be ℄



transpose â࠭ᯮ­¨àã¥â ¬ âà¨æ㠔㭪æ¨ï

transpose(ma1);





a b d



determinant ¢ëç¨á«ï¥â ¤¥â¥à¬¨­ ­â ¬ âà¨æë ”ã­ªæ¨ï

determinant(ma1);

ad-b



mattra e ¢ëç¨á«ï¥â á«¥¤ ¬ âà¨æë (á㬬㠥¥ ¤¨ £®­ «ì­ëå í«¥¬¥­â®¢). Ǒ¥à¥¤ ⥬, ª ª ¢ë§ë¢ âì ¥¥ ¯¥à¢ë© à §, ­¥®¡å®¤¨¬® § £à㧨âì ¯ ª¥â "n hrpl". ”ã­ªæ¨ï

load(n hrpl)$ mattra e(ma1);

a+d

58



harpoly ï¥âáï ¤® ­¥ª®â®à®© á⥯¥­¨ ¨§¡ëâ®ç­®© | ®­  ¢ëç¨á«ï¥â å à ªâ¥à¨áâ¨ç¥áª¨© ¯®«¨­®¬ ¬ âà¨æë, â.¥. det(m ^ x) (ª®à­¨ í⮣® ¯®«¨­®¬  | ᮡá⢥­­ë¥ §­ ç¥­¨ï ¬ âà¨æë). Ǒ¥à¥¤ ⥬, ª ª ¢ë§ë¢ âì ¥¥ ¯¥à¢ë© à §, ­¥®¡å®¤¨¬® § £à㧨âì ¯ ª¥â "n hrpl". ”ã­ªæ¨ï

load(n hrpl)$

harpoly(ma1,t);



(a-t)(d-t)-b

n harpoly ¥áâì ã«ãç襭­ ï ¢¥àá¨ï ä㭪樨 " harpoly". Ǒ¥à¥¤ ⥬, ª ª ¢ë§ë¢ âì ¥¥ ¯¥à¢ë© à §, ­¥®¡å®¤¨¬® § £à㧨âì ¯ ª¥â "n hrpl". ”ã­ªæ¨ï

load(n hrpl)$ n harpoly(ma1,t);

2

t



+ (-a -d) t + a d - b

eigenvalues ¢ëç¨á«ï¥â ᮡá⢥­­ë¥ §­ ç¥­¨ï ¬ âà¨æë  ­ «¨â¨ç¥áª¨, ¥á«¨ íâ® ¢®§¬®­®. ‚ë¤ ç  í⮩ ä㭪樨 ¤®áâ â®ç­® ¯à¨å®â«¨¢  | ®­  ¢®§¢à é ¥â ᯨ᮪, á®áâ®ï騩 ¨§ ¤¢ãå ᯨ᪮¢. Ǒ¥à¢ë© ᮤ¥à¨â ᮡá⢥­­ë¥ §­ ç¥­¨ï,   ¢â®à®© | ¨å ªà â­®á⨠”ã­ªæ¨ï

ma2:matrix([0,1,0℄,[1,0,0℄, [0,0,1℄); 2

3 0 1 0 41 0 05 0 0 1 eigenvalues(ma2);



[[-1, 1℄, [1, 2℄℄

eigenve tors  ­ «¨â¨ç¥áª¨ ¢ëç¨á«ï¥â ᮡá⢥­­ë¥ §­ ç¥­¨ï ¨ ᮡá⢥­­ë¥ ¢¥ªâ®à  ¬ âà¨æë, ¥á«¨ íâ® ¢®§¬®­®. ‚ë¤ ç  í⮩ ä㭪樨 ç१¢ëç ©­® ¯à¨å®â«¨¢  | ®­  ¢®§¢à é ¥â ᯨ᮪, ¯¥à¢ë© í«¥¬¥­â ª®â®à®£® | íâ® ¢ â®ç­®á⨠¢ë¤ ç  ä㭪樨 "eigenvalues",   ¤ «¥¥ ¨¤ãâ ᮡá⢥­­ë¥ ¢¥ªâ®à , ª ¤ë© ¨§ ª®â®àëå ¯à¥¤áâ ¢«¥­ ª ª ᯨ᮪ ᢮¨å ª®¬¯®­¥­â (â.¥. ª ª áâப ) ”ã­ªæ¨ï

eigenve tors(ma2);

[[[-1, 1℄, [1, 2℄℄, [1, -1, 0℄, [1, 1, 0℄, [0, 0, 1℄℄

59



uniteigenve tors ®â«¨ç ¥âáï ®â ä㭪樨 "eigenve tors" ⥬, çâ® ¢®§¢à é ¥â ­®à¬¨à®¢ ­­ë¥ ­  ¥¤¨­¨æã ᮡá⢥­­ë¥ ¢¥ªâ®à  ”ã­ªæ¨ï

uniteigenve tors(ma2);

[[[-1,1℄,[1,2℄℄, [

1 1 ,, 0℄, sqrt(2) sqrt(2)

1 1 , , 0℄, [0,0,1℄℄ sqrt(2) sqrt(2)

[

60

15. ‚­ãâ७­¥¥ ¯à¥¤áâ ¢«¥­¨¥ ¢ MAXIM'¥ Š ª ¨ ¢ ¡®«ì設á⢥ ¤à㣨å á¨á⥬  ­ «¨â¨ç¥áª¨å ¢ëç¨á«¥­¨©,  ­ «¨â¨ç¥áª¨¥ ¢ëà ¥­¨ï ¢ MAXIM'¥ ¯à¥¤áâ ¢«ïîâáï ¢ ¢¨¤¥ ¢«®¥­­ëå ᯨ᪮¢. Ǒ®í⮬ã ⥠ä㭪樨, ª®â®àë¥ ¤¥©áâ¢ãîâ ­  ᯨ᪨, ¢¯®«­¥ ¬®­® ¯à¨¬¥­ïâì ¨ ª  ­ «¨â¨ç¥áª¨¬ ¢ëà ¥­¨ï¬. „«ï ®¯à¥¤¥«¥­¨ï ª®«¨ç¥á⢠ ç á⥩ ¢ ¢ëà ¥­¨¨ á«¥¤ã¥â ¯à¨¬¥­ïâì äã­ªæ¨î "length", ¤«ï ¢ë¤¥«¥­¨ï ®â¤¥«ì­®© ç á⨠| äã­ªæ¨î "part". Ǒਠí⮬ ­ ¤® ¯®¬­¨âì, çâ® ¯à¨¬¥­¥­¨¥ ä㭪樨 "part" ª ®¡ê¥ªâã ¡¥§ áâàãªâãàë ¢ë§®¢¥â ®è¨¡ªã. Ǒ®í⮬㠥᫨ ¯à¨à®¤  ¢ëà ¥­¨ï § à ­¥¥ ­¥¨§¢¥áâ­ , ¯à¥¤¥ 祬 ¯à¨¬¥­ïâì ª ¢ëà ¥­¨î äã­ªæ¨î "part", ­ ¤® ¯à¨¬¥­¨âì ª ­¥¬ã äã­ªæ¨î "atom". ’®«ìª® ¥á«¨ "atom" ¢ë¤ áâ "false", ¬®­® ¯à¨¬¥­ïâì "part". —â®¡ë ¯à®¨««îáâà¨à®¢ âì ®á®¡¥­­®á⨠¢­ãâ७­¥£® ¯à¥¤áâ ¢«¥­¨ï, ¯à¨¢¥¤¥¬ ­¥áª®«ìª® ¯à¨¬¥à®¢ length(a+b)

length(a+b+ ) length(-a) length(f(a)) part([a,b℄,0); part([a,b℄,1); part([a,b℄,2); part(a=b,0); part(a=b,1); part(a=b,2); part(a+b,0); part(a+b,1); part(a+b,2);

2 3 1 1 [ a b = a b + b a

61

part(a-b,0); part(a-b,1); part(a-b,2); part(-b,0); part(-b,1); part(a*b,0); part(a*b,1); part(a*b,2); part(a/b,0);

+ a -b b * a b //

(¨¬¥­­® â ª | ¤¢  §­ ª  ¤¥«¥­¨ï ¯®¤àï¤!) part(a/b,1); part(a/b,2); part(f(x),0); part(f(x),1); part(a+b+ ,0); part(a+b+ ,1); part(a+b+ ,2); part(a+b+ ,3); part(a*b* ,0); part(a*b* ,1); part(a*b* ,2);

a b f x +

b a * a

62

part(a*b* ,3); part((a+b)/( +d),0); part((a+b)/( +d),1); part((a+b)/( +d),2);

b

// b+a

d+ part((a+b)/( +d),2,2);

Žç¥­ì ¯®«¥§­ë ¯à¨ ¢ë¤¥«¥­¨¨ ç á⥩ ¢ëà ¥­¨ï ä㭪樨 "first", "rest" ¨ "last". first(x+y+z); rest(x+y+z); last(x+y+z);

z y+x x

‘ãé¥áâ¢ãîâ ¨ ᯥ樠«ì­ë¥ ä㭪樨 ¤«ï ¢ë¤¥«¥­¨ï ç á⥩ ¢ëà ¥­¨ï.



lhs ¢ë¤¥«ï¥â «¥¢ãî ç áâì ãà ¢­¥­¨ï ”ã­ªæ¨ï

lhs(a+b = +d);

b+a



rhs ¢ë¤¥«ï¥â ¯à ¢ãî ç áâì ãà ¢­¥­¨ï ”ã­ªæ¨ï

rhs(a+b = +d);

d+



num ¢ë¤¥«ï¥â ç¨á«¨â¥«ì ”ã­ªæ¨ï

num((a+b)/( +d)); num(a+b);

b+a b+a

63



denom ¢ë¤¥«ï¥â §­ ¬¥­ â¥«ì ”ã­ªæ¨ï

denom((a+b)/( +d));

denom(a+b);

d+ 1

„«ï ¬®¤¨ä¨ª æ¨¨ ¢ëà ¥­¨© ¬®­® ¯à¨¬¥­ïâì ä㭪樨 "map", "apply" ¨ "subst". ‘㬬  ï¥âáï ᯨ᪮¬ ᢮¨å á« £ ¥¬ëå, â ª çâ®

map(fa tor, m/(x^2+2*x*y+y^2)+ n/(x+y)^4); n m + 2 4 (y + x) (y + x) Ž¤­¨¬ ¨§ ¨¬¥­ ä㭪樨 á«®¥­¨ï ï¥âáï ᨬ¢®« "+". ’ ª çâ® apply("+",[a,b℄); b+a ”ã­ªæ¨ï "subst" ¢¨¤¨â ¢á¥ ç á⨠¢ëà ¥­¨ï, ¢ ⮬ ç¨á«¥ ¨ á ­®¬¥à®¬ "0".

Ǒ®í⮬ã

subst("+","[",[a,b℄);



b+a

pi kapart ¯®§¢®«ï¥â à §«®¨âì ¢ëà ¥­¨¥ ­  ç á⨠¢¯«®âì ¤® 㪠§ ­­®£® ã஢­ï. ”ã­ªæ¨ï

(%i23)

v1:pi kapart( (a+b)/2 + sin( )-d^2,1);

2 -d sin( ) b + a 2 %t23 +

(%t23) (%t24) (%t25) (%o25)

%t24

+

%t25

Ǒ®á«¥ í⮣® ¬®­® ã¯à®é âì ­¥ "v1",   à ¡®â âì á ¥£® ç áâﬨ | á ¯¥à¥¬¥­­ë¬¨ "%t23", "%t24" ¨ "%t25", ¯®áª®«ìªã ¯¥à¥¬¥­­ ï "v1" ¢á¥ à ¢­® ¢ëà  ¥âáï ç¥à¥§ ­¨å:

%t25:77$ ev(v1);

2

-d

+ sin( ) + 77

64



isolate ¯®§¢®«ï¥â ®â¤¥«¨âì ç áâì ¢ëà ¥­¨ï, ª®â®à ï ᮤ¥à¨â 㪠§ ­­ãî ¯¥à¥¬¥­­ãî, ®â ç áâ¨, ª®â®à ï ¥¥ ­¥ ᮤ¥à¨â ”ã­ªæ¨ï

(%i6) isolate(x*x+a*x+b*x+a+b+1,x); (%t6) b + a + 1 2 (%o6) x + b x + a x +

65

%t6

16. ‘¨­â ªá¨ç¥áª¨¥ ¯®¤áâ ­®¢ª¨ 

subst ॠ«¨§ã¥â "ç¨á⮠ᨭ⠪á¨ç¥áªãî" ¯®¤áâ ­®¢ªã. Ǒ®¤áâ ­®¢ª  | íâ® § ¬¥­  ­¥ª®â®à®© ¯¥à¥¬¥­­®© ¨«¨ ¡®«¥¥ á«®­®© ª®­áâàãªæ¨¨ ¢  ­ «¨â¨ç¥áª®¬ ¢ëà ¥­¨¨ ­  ­¥çâ® ¤à㣮¥.  ¯à¨¬¥à, ¢¬¥áâ® ¯¥à¥¬¥­­®© x ¬®­® "¯®¤áâ ¢¨âì" a + b. €à£ã¬¥­âë ä㭪樨 "subst" ¨¤ãâ ¢ â ª®¬ ¯®à浪¥: ­®¢®¥ (â®, çâ® ¬ë ¯®¤áâ ¢«ï¥¬ ¢¬¥áâ® áâ à®£®), áâ à®¥ (â®, ¢¬¥á⮠祣® ¬ë ¯®¤áâ ¢«ï¥¬), ¨, ­ ª®­¥æ, ¢ëà ¥­¨¥, ¢ ª®â®à®¬ ¯à®¨§¢®¤¨âáï ¯®¤áâ ­®¢ª . â  äã­ªæ¨ï ­¥ ¯®­¨¬ ¥â, çâ® x4 | íâ® x2  x2 , ¯®í⮬㠢 ¢ëà ¥­¨¨ ”ã­ªæ¨ï

subst(y,x,x^2+x^4+x^5); 5 4 2 y + y + y

¯®¤áâ ­®¢ª  ¢ë¯®«­ï¥âáï ¯®«­®áâìî, ¢ ¢ëà ¥­¨¨ subst(y^2,x^2,x^2+x^4+x^5); 2 5 4 y + x + x

¢ë¯®«­ï¥âáï ç áâ¨ç­®,   ¢ ¢ëà ¥­¨¨ subst(m,x+y,x+y+z);

z + y + x

­¥ ¢ë¯®«­ï¥âáï ¢®¢á¥. Ǒ®¤áâ ­®¢ª  ¢®§¬®­  ­¥ ⮫쪮 ¤«ï ¨¬¥­ ¯¥à¥¬¥­­ëå, ­® ¨ ¤«ï ¨¬¥­ ä㭪権 subst(gg,ff,ff(7+ff(5)); gg(gg(5) + 7)



exptsubst § ¯à¥é ¥â ¨«¨ à §à¥è ¥â "á⥯¥­­ë¥" ¯®¤áâ ­®¢ª¨. ˆ§­ ç «ì­® ãáâ ­®¢«¥­® §­ ç¥­¨¥ "false", ¢ १ã«ìâ â¥ Ǒ¥à¥¬¥­­ ï

subst(y,%e^x,%e^x+%e^(a*x)+ %e^(a*x+b*x) ); (b + a) x y + %e + …᫨ ãáâ ­®¢¨âì §­ ç¥­¨¥ "true", â® exptsubst:true$ subst(y^2,x^2,x^2+x^4+x^5); 4 2 5 y + y + x subst(y,%e^x,%e^x+%e^(a*x)+ %e^(a*x+b*x) ); b + a a y + y + y

66

%ea

x



ratsubst à ¡®â ¥â â ª ¥, ª ª "subst", ­® ¯®­¨¬ ¥â, çâ® x4 | íâ® x2  x2 , â ª çâ® ”ã­ªæ¨ï

ratsubst(y^2,x^2,x^2+x^4+x^5); 4 4 2 x y + y + y ratsubst(m,x+y,x+y+z); z + m ratsubst( m,3*x*y, expand((x+y)^3) ); 3 3 y + m y + x + m x

67

17. €«£¥¡à ¨ç¥áª¨¥ ¯®¤áâ ­®¢ª¨ ‚ à æ¨®­ «ì­ëå ¢ëà ¥­¨ïå áãé¥áâ¢ã¥â ®á®¡ë© ª« áá ¯®¤áâ ­®¢®ª, ¢ë¯®«­¥­¨¥ ª®â®àëå ¨­¨æ¨¨àã¥âáï ä« £®¬ "algebrai " ¢ ä㭪樨 "ev". ‘­ ç «  § ¤ ¥âáï ®¤­  ¨«¨ ­¥áª®«ìª® ¯®¤áâ ­®¢®ª.



tellrat  ¤¤¨â¨¢­® ¤®¡ ¢«ï¥â  «£¥¡à ¨ç¥áªãî ¯®¤áâ ­®¢ªã ª 㥠®¯à¥¤¥«¥­­ë¬ ¯®¤áâ ­®¢ª ¬ ¨ ¯¥ç â ¥â ¢¥áì ¨å ᯨ᮪. ”ã­ªæ¨ï

tellrat(z=y^3); tellrat(w^3=x); tellrat();

3 [z - y ℄ 3

3 - x, z - y ℄

3

3 - x, z - y ℄

[w [w

„«ï ॠ«¨§ æ¨¨ ¯®¤áâ ­®¢®ª ¢ëà ¥­¨¥ ¤®«­® ¡ëâì à æ¨®­ «ì­ë¬ (¤®«­® ¡ëâì á­ ¡¥­® ¬¥âª®© "/R/"), ¨ ª ­¥¬ã ­ ¤® ¯à¨¬¥­¨âì äã­ªæ¨î "ev" á ä« £®¬ "algebrai ": ev(z^2+z^3,algebrai );

rat(%); ev(%,algebrai );

3

z

/R/

2

+ z

3

z

9

/R/ y ev(rat(w^3+w^6),algebrai ); /R/

2

x

68

2

+ z

6

+ y + x

18. Ǒ®¤áâ ­®¢ª¨ ¯® è ¡«®­ã €¯¯ à â ¯®¤áâ ­®¢®ª ¯® è ¡«®­ã ¢ MAXIM'¥ § ¬¥â­® ãáâ㯠¥â ¯® 㤮¡áâ¢ã à ¡®âë ¨ ¯à®¤ã¬ ­­®á⨠ᨭ⠪á¨á   ­ «®£¨ç­®¬ã  ¯¯ à âã ¢ REDUCE. ®«¥¥ ⮣®, ¯® áã⨠¥¤¨­á⢥­­ë¬ ॠ«ì­ë¬ ãᮢ¥à襭á⢮¢ ­¨¥¬ Mathemati '¨ ¯® áà ¢­¥­¨î á MAXIM'®© ï¥âáï 㤮¡­ë©  ¯¯ à â ¯®¤áâ ­®¢®ª ¯® è ¡«®­ã. …᫨ ¢ REDUCE ¤®áâ â®ç­® ­ ¯¨á âì for all x,y let sin(x+y)=sin(x)* os(y)+ os(x)*sin(y); for all x,n su h that numberp(n) and n>1 let sin(n*x) = sin((n-1)*x)* os(x)+ os((n-1)*x)*sin(x);

çâ®¡ë § ¤ âì ¯à ¢¨«  ¯à¥®¡à §®¢ ­¨ï ¢ëà ¥­¨© ⨯  "sin(a+b+3*á)", â® ¢ MAXIM'¥ íâ® ¯®âॡã¥â §­ ç¨â¥«ì­® ¡®«ìè¨å ãᨫ¨©. ‡ â® ¢ à¨ ­â®¢ ä㭪権, ॠ«¨§ãîé¨å ¯®¤áâ ­®¢ª¨ ¯® è ¡«®­ã ¢ MAXIM'¥ £®à §¤® ¡®«ìè¥, ¨ ¢á¥ ®­¨ à ¡®â îâ ¯®-à §­®¬ã. Ǒ।¥ ¢á¥£®, á«¥¤ã¥â ®¯à¥¤¥«¨âì è ¡«®­.



mat hde lare ®¯à¥¤¥«ï¥â è ¡«®­, 㤮¢«¥â¢®àïî騩 ⮬㠨«¨ ¨­®¬ã ãá«®¢¨î. ”ã­ªæ¨ï

mat hde lare(a,true)$ mat hde lare(b,true)$ mat hde lare(n,numberp)$ mat hde lare(m,matrixp)$ ¯®á«¥ í⮣® è ¡«®­ë "a" ¨ "b" ®§­ ç îâ "ç⮠㣮¤­®", è ¡«®­ "n" ®§­ ç -

¥â "«î¡®¥ ç¨á«®" (â.¥. «î¡®© ®¡ê¥ªâ, ª®â®àë© ¯à¨ ¯®¤áâ ­®¢ª¥ ¢ äã­ªæ¨î "numberp" ¤ ¥â "true"), è ¡«®­ "m" ®§­ ç ¥â "«î¡ ï ¬ âà¨æ ". Š á® «¥­¨î, ¢ ®â«¨ç¨¥ ®â REDUCE ¨ Mathemati '¨, § ¯¨áì "a+b" ­¥ ®§­ ç ¥â ⥯¥àì "«î¡ ï á㬬 ". —â®¡ë ®¯à¥¤¥«¨âì è ¡«®­ "«î¡ ï á㬬 ", ¯à¨¤¥âáï ¯à®¤¥« âì á«¥¤ãî饥 summap(x):=blo k([℄,if atom(x) then return(false), if part(x,0)="+" then return(true) else return(false))$ mat hde lare(anys,summap)$

€­ «®£¨ç­®, çâ®¡ë ®¯à¥¤¥«¨âì è ¡«®­ "ç¨á«®, ¡®«ì襥 ¥¤¨­¨æë", á«¥¤ã¥â ­ ¯¨á âì num_g_1(x):=blo k([℄,if not numberp(x) then return(false), if x>1 then return(true) else return(false))$ mat hde lare(nnn,num_g_1)$

’¥¯¥àì áãé¥áâ¢ã¥â âਠ¢®§¬®­®áâ¨.

69

‚®-¯¥à¢ëå,



¬®­® ®¯à¥¤¥«ïâì ¯à ¢¨«  ¨ ¯à¨¬¥­ïâì ¨å.

defrule ®¯à¥¤¥«ï¥â "¯à ¢¨«®". …¥  à£ã¬¥­âë | ¨¬ï ¯à ¢¨« , áâ à®¥ ¢ëà ¥­¨¥, ­®¢®¥ ¢ëà ¥­¨¥. ”ã­ªæ¨ï

defrule( ru1, fff(m), m^^3 + m );

<3> ru1 : fff(m) -> m + m defrule( ru2, ff(n), n+x^n ); n ru2 : ff(n) -> x + n defrule( ru3, f(a), a+a^2 ); 2 ru3 : f(a) -> a + a defrule( ru4, sin(anys), sin(first(anys))* os(rest(anys)) +

os(first(anys))*sin(rest(anys)) )$ defrule( ru5, os(anys),

os(first(anys))* os(rest(anys)) sin(first(anys))*sin(rest(anys)) )$



disprule ¯¥ç â ¥â ¯à ¢¨« . ”ã­ªæ¨ï

(%i5) disprule(ru2,ru3);

n ru2 : ff(n) -> x + n 2 ru3 : f(a) -> a + a [%t5, %t6℄

(%t5)

(%t6) (%o6)



apply1 ¯à¨¬¥­ï¥â 㪠§ ­­ë¥ ¯à ¢¨«  ª ¢ëà ¥­¨î. Ǒà ¢¨«  ¯à¨¬¥­ïîâáï "ᢥàåã ¢­¨§" (®â ¡®«¥¥ ¢ë᮪®£® ã஢­ï ¢ ¢ëà ¥­¨¨ ª ¡®«¥¥ ¬¥«ª¨¬ ¥£® ç áâï¬), ¯à¨â®¬ ­  ª ¤®¬ ã஢­¥ ¢ëà ¥­¨ï á­ ç «  楫¨ª®¬ (¯®ª  ¢ëà ¥­¨¥ ­¥ ¯¥à¥áâ ­¥â ¬¥­ïâìáï) ®âà ¡ â뢠¥âáï ¯¥à¢®¥ ¯à ¢¨«®, ¯®â®¬ ¢â®à®¥, ¨ â.¤.



applyb1 ®â«¨ç ¥âáï ®â "apply1" ⥬, çâ® â®â ¥ á ¬ë©  «£®à¨â¬ ¯®¤áâ ­®¢®ª ¯à¨¬¥­ï¥âáï "á­¨§ã ¢¢¥àå".

”ã­ªæ¨ï

”ã­ªæ¨ï

70



apply2 ®â«¨ç ¥âáï ®â "apply1" ⥬, çâ® ­  ª ¤®¬ ã஢­¥ ¢ëà ¥­¨ï 楫¨ª®¬ ®âà ¡ â뢠¥âáï ¢¥áì 㪠§ ­­ë© ᯨ᮪ ¯à ¢¨«. ”ã­ªæ¨ï

ma1:matrix([2,0℄,[0,2℄)$ apply1(fff(ma1)*fff(7),ru1);   10 fff(7) 0 0 10 fff(7) apply1(ff(5)+ff(y),ru2); 5 ff(y) + x + 5 apply1(f(sin(z)^2),ru3); 4 2 sin (z) + sin (z) apply1( os(x+y+z),ru4,ru5); ( os(x) os(y) - sin(x) sin(y)) os(z) sin(y + x) sin(z) apply1(%,ru4,ru5); ( os(x) os(y) - sin(x) sin(y)) os(z) ( os(x) sin(y) + sin(x) os(y)) sin(z) apply1( os(x+y+z),ru5,ru4,ru5); ( os(x) os(y) - sin(x) sin(y)) os(z) ( os(x) sin(y) + sin(x) os(y)) sin(z) apply2( os(x+y+z),ru4,ru5); ( os(x) os(y) - sin(x) sin(y)) os(z) ( os(x) sin(y) + sin(x) os(y)) sin(z) applyb1( os(x+y+z),ru4,ru5);

os(y+x) os(z) - sin(y+x) sin(z)

Ǒਢ¥¤¥­­ë¥ §¤¥áì ¯à¨¬¥àë ¨««îáâà¨àãîâ à §­¨æã ¬¥¤ã âà¥¬ï ¢ à¨ ­â ¬¨ ä㭪樨 "apply". ˆå ¢ë§®¢ ¤«ï ®¤­®£® ¨ ⮣® ¥ ¢ëà ¥­¨ï " os(x+y+z)" ¤«ï ®¤­®£® ¨ ⮣® ¥ ᯨ᪠ ¯à ¢¨« "ru4, ru5" ¤ ¥â à §­ë¥ ®â¢¥âë. Ǒà¨ç¨­ë í⮣® ¤®¢®«ì­® ®ç¥¢¨¤­ë. ”ã­ªæ¨ï "applyb1" ¨¤¥â "á­¨§ã ¢¢¥àå", â ª çâ® ¯à ¢¨«® "ru5" áà ¡ â뢠¥â ⮫쪮 ­  á ¬®¬ ¯®á«¥¤­¥¬ ("¢¥àå­¥¬") íâ ¯¥, ®­® ¯à¨¬¥­ï¥âáï ª® ¢á¥¬ã ¢ëà ¥­¨î " os(x+y+z)" ¢ 楫®¬. Ǒ®á«¥ í⮣® ª ®â¤¥«ì­ë¬ á« £ ¥¬ë¬ ¯®«ã祭­®£® ¢ëà ¥­¨ï ­¨ª ª¨å ¯à ¢¨« 㥠­¥ ¯à¨¬¥­ï¥âáï, â.ª. í⨠᫠£ ¥¬ë¥ "«¥ â ­¨¥".  ¯à®â¨¢, äã­ªæ¨ï "apply2" ¯à¨¬¥­ï¥â ®¡  ¯à ¢¨«  á­ ç «  ª® ¢á¥¬ã ¢ëà ¥­¨î " os(x+y+z)" ¢ 楫®¬,   ¯®â®¬ ¯à¨¬¥­ï¥â ®¡  ¯à ¢¨«  ª ®â¤¥«ì­ë¬ á« £ ¥¬ë¬ ¯®«ã祭­®£® ¢ëà ¥­¨ï, ¢ १ã«ìâ â¥ ­¨ª ª¨å á㬬 ¢  à£ã¬¥­â å âਣ®­®¬¥âà¨ç¥áª¨å ä㭪権 ­¥ ®áâ ¥âáï. 71

 ª®­¥æ, äã­ªæ¨ï "apply1" á­ ç «  ®âà ¡ â뢠¥â ¯à ¢¨«® "ru4", ª®â®à®¥ ¢ ¢ëà ¥­¨¨ " os(x+y+z)" ¯à¨¬¥­ïâì ­¥ ª 祬ã. ’®«ìª® ¯®á«¥ í⮣® ®­  ­ ç¨­ ¥â ¯à¨¬¥­ïâì ¯à ¢¨«® "ru5". ‘­ ç «  ®­® ¯à¨¬¥­ï¥âáï ª® ¢á¥¬ã ¢ëà ¥­¨î " os(x+y+z)" ¢ 楫®¬,   ¯®â®¬ (¡®«¥¥ "­¨§ª¨©" ã஢¥­ì) ª ®â¤¥«ì­®¬ã á« £ ¥¬®¬ã " os(x+y)". Ǒà ¢¨«® "ru4" 㥠®âà ¡®â ­®, â ª çâ® ª á« £ ¥¬®¬ã "sin(x+y)") ®­® ­¥ ¯à¨¬¥­ï¥âáï (¢ ®â«¨ç¨¥ ®â ä㭪樨 "apply2"). Šà®¬¥ ⮣®, §¤¥áì ¯®ª § ­®, ª ª ¬®­® § áâ ¢¨âì "¤®à ¡®â âì ¤® ª®­æ " äã­ªæ¨î "apply1". â®£® ¬®­® ¤®¡¨âìáï ¨«¨ ¯®¢â®à­ë¬ ¢ë§®¢®¬ ä㭪樨 (¯à ¢¨«® "ru4" ¯à¨¬¥­¨âáï ª "­¥¤®à ¡®â ­­®¬ã" á« £ ¥¬®¬ã "sin(x+y)"), ¨«¨ 㤫¨­¥­¨¥¬ ᯨ᪠ ¯à ¢¨« §  áç¥â ¯®¢â®à¥­¨ï ®¤­¨å ¨ â¥å ¥ ¯à ¢¨« (¯à ¢¨«® "ru5" ¯à¨¬¥­¨âáï ª ¨á室­®¬ã ¢ëà ¥­¨î, ¯®á«¥ 祣® ¯à ¢¨«  "ru4" ¨ "ru5" ¯à¨¬¥­ïâáï ª ®â¤¥«ì­ë¬ á« £ ¥¬ë¬). Š ª 㥠¡ë«® ᪠§ ­®, ¯®¯ë⪠ à ¡®â âì á è ¡«®­®¬ "a+b" ("«î¡ ï á㬬 "), ¨«¨ á è ¡«®­®¬ "a+nnn" ("ç⮠㣮¤­® ¯«îá ç¨á«®, ¡®«ì襥 ¥¤¨­¨æë"), ¨«¨ á è ¡«®­®¬ "a*n" ("ç⮠㣮¤­® 㬭®¨âì ­  ç¨á«®") ­¥ª®à४â­ë ¨ ¢ë§ë¢ î⠯।ã¯à¥¤¥­¨ï ®¡ ®è¨¡ª¥: defrule(ru6,f(a+b), g(a)+g(b) ); b + a partitions `sum' ru6 : f(b + a) -> g(b) + g(a) defrule(ru7,f(a+nnn), g(a)+g(nnn) ); nnn + a partitions `sum' ru7 : f(nnn + a) -> g(nnn) + g(a) defrule(ru8,f(n*a), n*g(a) ); n a partitions `produ t' ru8 : f(a n) -> g(a) n ‚ áâண®¬ á¬ëá«¥ í⨠§ ¯¨á¨ ¤¥©á⢨⥫쭮 ­¥ª®à४â­ë: "a" | í⮠㥠"ç⮠㣮¤­®", â ª çâ® "a+b" íâ® ¯à®áâ® "a", â®ç­® â ª ¥ "a+nnn" ¨ "a*n" | íâ® "a". …᫨ ‚ë ¯à®¨£­®à¨àã¥â¥ í⨠¯à¥¤ã¯à¥¤¥­¨ï, â® ­ ç­ãâáï ç㤥á : apply2(f(5),ru7); g(5) + g(0)

â®â १ã«ìâ â ¢¯®«­¥ «®£¨ç¥­, å®âï ¨ ­¥¥« â¥«¥­. ‚¯à®ç¥¬, ¨­®£¤  ¯à ¢¨«® "ru7" à ¡®â ¥â â ª, ª ª § ¤ã¬ ­®: apply2(f(x+5),ru7);

apply2(f(x+1),ru7);

g(x) + g(5) f(x + 1)

„«ï ¯à ¢¨«  "ru6" á¨âã æ¨ï ¥é¥ åã¥:

72

apply2(f(x+y),ru6); apply2(f(x),ru6); apply2(f(0),ru6);

g(y + x) + g(0) g(x) + g(0) 2 g(0)

Š®à४â­ë© ᯮᮡ ®¯à¥¤¥«¨âì è ¡«®­ "ç⮠㣮¤­® 㬭®¨âì ­  ç¨á«®, ¡®«ì襥 ¥¤¨­¨æë", ¢ë£«ï¤¨â â ª: pronum(x):=blo k([aaa℄, if atom(x) then return(false), if not part(x,0)="*" then return(false), aaa:first(x), if not numberp(aaa) then return(false), if aaa>1 then return(true) else return(false) )$ mat hde lare(ppp,pronum)$

Ǒ®á«¥ í⮣® ¬®­® ®¯à¥¤¥«ïâì ¯à ¢¨«® ¯à¥®¡à §®¢ ­¨ï ¢ëà ¥­¨© ⨯  "sin(7x)": defrule(ru9a,sin(ppp), sin(rest(ppp))* os((first(ppp)-1)*rest(ppp))+

os(rest(ppp))*sin((first(ppp)-1)*rest(ppp)) )$ defrule(ru9b, os(ppp),

os(rest(ppp))* os((first(ppp)-1)*rest(ppp))sin(rest(ppp))*sin((first(ppp)-1)*rest(ppp)) )$ apply2(sin(4x),ru9a,ru9b); sin(x) ( os(x) ( os

2 2

- 2 os(x) sin

(x) - sin

2

(x))

(x))

2 + os(x) (sin(x) ( os (x) 2 2 - sin (x)) + 2 os (x) sin(x))

 §ã¬¥¥âáï, íâ® ªà ©­¥ ­¥ã¤®¡­® ¯® áà ¢­¥­¨î á REDUC'®¬ ¨«¨ á Mathmati '®©. ¬®­® ®¯à¥¤¥«¨âì ¡¥§ë¬ï­­ë¥ let-¯à ¢¨« , ª®â®àë¥ ­ ª ¯«¨¢ îâáï  ¤¤¨â¨¢­® ‚®-¢â®àëå,

73



let ®¯à¥¤¥«ï¥â let-¯à ¢¨«®. ”ã­ªæ¨ï

let( fff(m), m^^3 + m ); <3>

let( ff(n), n+x^n ); let( f(a), a+a^2 );

fff(m) --> m

n

ff(n) --> x 2

f(a) --> a



+ m

+ n + a

remlet ®â¬¥­ï¥â ®¯à¥¤¥«¥­­ë¥ à ­¥¥ ¯à ¢¨« . Ǒਠí⮬ ”ã­ªæ¨ï

remlet(ff(n))$

®â¬¥­¨â ⮫쪮 㪠§ ­­®¥ ¯à ¢¨«®,   remlet(all)$

®â¬¥­¨â ¢á¥ ®¯à¥¤¥«¥­­ë¥ ª ­ áâ®ï饬㠬®¬¥­â㠯ࠢ¨« .



letsimp ¯à¨¬¥­ï¥â ª ᢮¥¬ã  à£ã¬¥­â㠢ᥠ¨§¢¥áâ­ë¥ ­  ¤ ­­ë© ¬®¬¥­â let-¯à ¢¨« . Ǒà ¢¨«  ®âà ¡ â뢠îâáï 楫¨ª®¬, â.¥. ¤® â¥å ¯®à, ¯®ª  ¢ëà ¥­¨¥ ­¥ ¯¥à¥áâ ­¥â ¬¥­ïâìáï. ”ã­ªæ¨ï

letsimp(ff(5)+ff(y)+f(sin(z)^2); 4 2 5 sin (z) + sin (z) + ff(y) + x + 5 remlet(ff(n))$ letsimp(ff(5)+ff(y)+f(sin(z)^2); 4 2 sin (z) + sin (z) + ff(y) + ff(5) remlet(all)$ letsimp(ff(5)+ff(y)+f(sin(z)^2); 2 f(sin (z)) + ff(y) + ff(5)

¬®­® ¤®¢¥á⨠⮠¨«¨ ¨­®¥ ¯à ¢¨«® ¤® ᢥ¤¥­¨ï ®á­®¢­®© ä㭪樨, ã¯à®é î饩 ¢ëà ¥­¨ï. ‚ í⮬ á«ãç ¥ ¢á¥ § ¬¥­ë ¡ã¤ã⠢믮«­ïâìáï  ¢â®¬ â¨ç¥áª¨, ¡¥§ ª ª¨å «¨¡® ¤¥©á⢨© á® áâ®à®­ë ¯®«ì§®¢ â¥«ï. ‚-âà¥âì¨å,

74



tellsimp ¢¢®¤¨â ¯à ¢¨«® ¢ ¡ §ã ¤ ­­ëå ®á­®¢­®© ä㭪樨, ã¯à®é î饩 ¢ëà ¥­¨ï. ”ã­ªæ¨ï

tellsimp( sin(anys), sin(first(anys))* os(rest(anys)) +

os(first(anys))* sin(rest(anys)) ); [sinrule1, simp-%sin℄ tellsimp( os(anys),

os(first(anys))* os(rest(anys)) sin(first(anys))* sin(rest(anys)) ); [ osrule1, simp-% os℄

os(x+y+z); ( os(x) os(y) - sin(x) sin(y)) os(z) - ( os(x) sin(y) + sin(x) os(y)) sin(z)

75

19.  ¡®â  á oat-ç¨á« ¬¨ Ž¡ëª­®¢¥­­® MAXIMA áâ à ¥âáï à ¡®â âì á ¡¥áª®­¥ç­®© â®ç­®áâìî: exp(1);

%e

Ž¤­ ª® ¥¥ ¬®­® § áâ ¢¨âì à ¡®â âì ¨ á ¤¥©á⢨⥫ì­ë¬¨ ç¨á« ¬¨. „¥©á⢨⥫ì­ë¥ ç¨á«  ¬ è¨­­®© â®ç­®á⨠(ª ª ¯à ¢¨«® 16 §­ ª®¢) § ¯¨á뢠îâáï ®¡ëç­ë¬ ®¡à §®¬: 2.5

-1.0e20

5.768e-34

„¥©á⢨⥫ì­ë¥ ç¨á«  ­¥®£à ­¨ç¥­­®© â®ç­®á⨠®¡ï§ ­ë ¨¬¥âì ¯®ª § â¥«ì á⥯¥­¨ "b": 2.5b0



-1.0b20

fppre ®¯à¥¤¥«ï¥â ª®«¨ç¥á⢮ §­ ç é¨å æ¨äà ¤«ï ç¨á¥« ­¥®£à ­¨ç¥­­®© â®ç­®áâ¨. ˆ§­ ç «ì­® ®­  à ¢­  16. Ǒ¥à¥¬¥­­ ï

exp(1.0);

exp(1.0b0); fppre :21; exp(1.0b0);



5.768b-34

2.7182818284590451 2.718281828459045b0 21 2.71828182845904523536b0

float ª®­¢¥àâ¨àã¥â «î¡ë¥ ç¨á«  ¢ ¢ëà ¥­¨ïå ¢ ç¨á«  ¬ è¨­­®© â®ç­®áâ¨. ”ã­ªæ¨ï

float(1/3);

float(f(1/3)); float(%e); float(%pi); float(1.0b0); float(sin(%pi/6);

0.3333333333333333 f(0.3333333333333333)

%e %pi 1.0e0 0.5

76



bfloat ª®­¢¥àâ¨àã¥â «î¡ë¥ ç¨á«  ¢ ¢ëà ¥­¨ïå ¢ ç¨á«  ­¥®£à ­¨ç¥­­®© â®ç­®áâ¨. Ǒਠí⮬, ¥á«¨ ¢áâà¥ç ¥âáï ¤¥©á⢨⥫쭮¥ ç¨á«® ¬ è¨­­®© â®ç­®áâ¨, ®­  ¯¥ç â ¥â ¯à¥¤ã¯à¥¤¥­¨¥ ® ¨§¬¥­¥­¨¨ â®ç­®áâ¨: ”ã­ªæ¨ï

bfloat(1/3); bfloat(%e);

bfloat(%pi); bfloat(exp(1.0));

bfloat(sin(%pi/6);

0.3333333333333333b0 2.718281828459045b0 3.141592653589793b0 Warning:

float to bigfloat onversion of 2.7182818284590451 2.718281828459045b0

0.5b0

‘ãé¥áâ¢ã¥â ¨ ®¡à â­®¥ ¯à¥®¡à §®¢ ­¨¥. Š ­®­¨ç¥áª ï ä®à¬  à æ¨®­ «ì­®£® ¢ëà ¥­¨ï ­¥ ¤®«­  ᮤ¥à âì ¤¥©á⢨⥫ì­ëå ç¨á¥«. Ǒ®í⮬ã äã­ªæ¨ï "ratsimp" ¯à¥®¡à §ã¥â «î¡®¥ ¤¥©á⢨⥫쭮¥ ç¨á«® ¢ à æ¨®­ «ì­®¥ ¨ ¯¥ç â ¥â ¯à¨ í⮬ ¯à¥¤ã¯à¥¤¥­¨¥.



ratepsilon § ¤ ¥â â®ç­®áâì ¯à¥®¡à §®¢ ­¨ï ¤¥©á⢨⥫쭮£® ç¨á«  ¢ à æ¨®­ «ì­®¥. ˆ§­ ç «ì­® ãáâ ­®¢«¥­® §­ ç¥­¨¥ 2:0
10 8. Ǒ¥à¥¬¥­­ ï

ratsimp(x+1.23456789); `rat' repla ed 1.2345678900000001 by 100/81=1.2345679012345678 81 x + 100 81 ratepsilon:0.001$ ratsimp(x+1.23456789); `rat' repla ed 1.2345678900000001 by 21/17=1.2352941176470589 17 x + 21 17

‘ãé¥áâ¢ã¥â æ¥«ë© ­ ¡®à ä㭪権, ª®â®àë© à¥ «¨§ã¥â ­¥  ­ «¨â¨ç¥áª¨¥,   ç¨á«¥­­ë¥  «£®à¨â¬ë, ¨ ¢ë¤ ¥â ¢ ª ç¥á⢥ ®â¢¥â®¢ ¤¥©á⢨⥫ì­ë¥ ç¨á« . 77



allroots äã­ªæ¨ï, ª®â®à ï ­ å®¤¨â ¨ ¯¥ç â ¥â ¢á¥ (¢ ⮬ ç¨á«¥ ¨ ª®¬¯«¥ªá­ë¥) ª®à­¨ ¯®«¨­®¬¨ «ì­®£® ãà ¢­¥­¨ï á ¤¥©á⢨⥫ì­ë¬¨ «¨¡® ª®¬¯«¥ªá­ë¬¨ ª®íää¨æ¨¥­â ¬¨.



polyfa tor ®¯à¥¤¥«ï¥â ä®à¬ã ¢ë¤ ç¨ ä㭪樨 "allroots". ˆ§­ ç «ì­® ®­  à ¢­  "false", ¯à¨ í⮬ ª®à­¨ ¢ë¢®¤ïâáï ¢ ¢¨¤¥ ᯨ᪠. …᫨ ãáâ ­®¢¨âì ¥¥ à ¢­®© "true", â® ¢¬¥á⮠ᯨ᪠ ª®à­¥© ¡ã¤¥â ¢ ®¡é¥¬ á«ãç ¥ ¢ë¢®¤¨âìáï à §«®¥­¨¥ ¯®«¨­®¬  ­  «¨­¥©­ë¥ ᮬ­®¨â¥«¨. „«ï ¯®«¨­®¬  á ¤¥©á⢨⥫ì­ë¬¨ ª®íää¨æ¨¥­â ¬¨ ¡ã¤¥â ¢ë¢®¤¨âìáï à §«®¥­¨¥ ­  «¨­¥©­ë¥ ᮬ­®¨â¥«¨ ¤«ï ¤¥©á⢨⥫ì­ëå ª®à­¥© ¨ ­  ª¢ ¤à â¨ç­ë¥ ¤«ï ª ¤®© ¨§ ¯ à ᮯà省­ëå ¤à㣠¤àã£ã ª®¬¯«¥ªá­ëå ª®à­¥©.

”ã­ªæ¨ï

Ǒ¥à¥¬¥­­ ï

allroots(x^2-3*%i*x-2=0); [x = 1.0 %i + 9.4182784545287731E-21, x = 2.0*%i - 9.4182784545287731E-21℄ allroots(x^6+1=x^2+x^4); [x = 1.0 %i - 1.1323771713605928E-19, x = -1.0 %i - 1.1323771713605928E-19, x = 4.7121609153868797E-8 %i - 0.99999999999992, x = - 4.7121609153868797E-8 %i - 0.99999999999992, x = 0.99999944304722, x = 1.000000556952626℄ polyfa tor:true$ allroots(x^2-3*%i*x-2=0); 1.0 (x - 2.0 %i + 9.4182784545287731E-21) (x - 1.0 %i - 9.4182784545287731E-21) allroots(x^6+1=x^2+x^4); 1.0 (x - 1.000000556952626) (x - 0.99999944304722) 2 (x + 2.2647543427211855E-19 x + 1.0) 2 (x + 1.999999999999846 x + 0.99999999999985) ‡¤¥áì å®à®è® § ¬¥â­® ®¤­® ­¥¯à¨ïâ­®¥ ᢮©á⢮ ä㭪樨 "allroots": ¥á«¨

ª®à­¨ ­¥ ïîâáï ªà â­ë¬¨, â® â®ç­®áâì ®â¢¥â  ¯®à浪  ¬ è¨­­®£® ­ã«ï,   ¢®â ¯à¨ ¯®¨áª¥ ªà â­ëå ª®à­¥© â®ç­®áâì áãé¥á⢥­­® á­¨ ¥âáï (¢ ¤ ­­®¬ á«ãç ¥ â®ç­®áâì ¯®à浪  5
10 7 ) 78



find_root ­ å®¤¨â ª®à¥­ì ãà ¢­¥­¨ï ­  § ¤ ­­®¬ ¨­â¥à¢ «¥ ¬¥â®¤®¬ ¤¥«¥­¨ï ®â१ª  ¯®¯®« ¬. ”ã­ªæ¨ï

6*find_root(sin(x)=0.5,x,0,1);

3.141592653589795



newton ­ å®¤¨â ª®à¥­ì 㪠§ ­­®© ä㭪樨 ¬¥â®¤®¬ ìîâ®­ . ”ã­ªæ¨ï

Ǒ® ᮢ¥à襭­® ­¥®¡êïá­¨¬ë¬ ¯à¨ç¨­ ¬ ã í⮩ ä㭪樨 ¥áâì ¤¢¥ ¢¥àᨨ á à §­ë¬ ᨭ⠪á¨á®¬.

Ǒ¥à¢ ï ¨§ ¢¥àᨩ ᮤ¥à¨âáï ¢ ¯ ª¥â¥ "newton" ¨ ¨¬¥¥â ¤¢   à£ã¬¥­â  | äã­ªæ¨î, ª®à¥­ì ª®â®à®© ¬ë ¨é¥¬, ¨ ­ ç «ì­ãî â®çªã load(newton)$ 6*newton(sin(x)-1/2,0);

3.141592612362348b0

‚â®à ï ¨§ ¢¥àᨩ ᮤ¥à¨âáï ¢ ¯ ª¥â¥ "newton1" ¨ ¨¬¥¥â ç¥âëॠ à£ã¬¥­â  | äã­ªæ¨î, ª®à¥­ì ª®â®à®© ¬ë ¨é¥¬, ¯¥à¥¬¥­­ãî, ­ ç «ì­ãî â®çªã ¨ § ª § ­­ãî â®ç­®áâì ¯®¨áª  ª®à­ï load(newton1)$ 2*newton ( os (u), u, 1, 1/100);

3.141350554322501



mnewton ­ å®¤¨â ª®à¥­ì á¨á⥬ë ãà ¢­¥­¨© ¬­®£®¬¥à­ë¬ ¬¥â®¤®¬ ìîâ®­ . „«ï ¨á¯®«ì§®¢ ­¨ï ä㭪樨 ­¥®¡å®¤¨¬® á­ ç «  § £à㧨âì ¯ ª¥â "mnewton". ”ã­ªæ¨ï ¨¬¥¥â âਠ à£ã¬¥­â  | ᯨ᮪ ãà ¢­¥­¨© (¢ ¢¨¤¥ ᯨ᪠ ä㭪権, ­ã«¨ ª®â®àëå ¬ë ¨é¥¬), ᯨ᮪ ¯¥à¥¬¥­­ëå, ¨ ᯨ᮪ ­ ç «ì­ëå §­ ç¥­¨© ¯¥à¥¬¥­­ëå (­ ç «ì­ ï â®çª  ४ãàᨢ­®© ¯à®æ¥¤ãàë).



newtonepsilon § ¤ ¥â â®ç­®áâì ¯®¨áª  ª®à­ï ¤«ï ä㭪樨 "mnewton". Ǒ® 㬮«ç ­¨î ãáâ ­®¢«¥­® §­ ç¥­¨¥ 10^(-fppre /2), çâ® ¤ «¥ª® ­¥ ¢á¥£¤  ï¥âáï ®¯â¨¬ «ì­ë¬ ¢ë¡®à®¬.

”ã­ªæ¨ï

Ǒ¥à¥¬¥­­ ï

79



newtonmaxiter § ¤ ¥â ¬ ªá¨¬ «ì­®¥ ª®«¨ç¥á⢮ ¨â¥à æ¨© ¢ ¬­®£®¬¥à­®¬ ¬¥â®¤¥ ìîâ®­ . Ǒ® 㬮«ç ­¨î ãáâ ­®¢«¥­® §­ ç¥­¨¥ "50". Ž¡ëª­®¢¥­­® í⮣® ª®«¨ç¥á⢠ ¤¥©á⢨⥫쭮 ¡ë¢ ¥â ¤®áâ â®ç­® ¤«ï á室¨¬®áâ¨. Ǒ¥à¥¬¥­­ ï

load(mnewton)$ mnewton([x^2-y^2, x^2+y^2℄,[x,y℄,[2,1℄); [[x = 7.4505805969238281E-9, y = 3.7252902984619141E-9℄℄ newtonepsilon:1.0e-12$ mnewton([x^2-y^2, x^2+y^2℄,[x,y℄,[2,1℄); [[x = 9.0949470177292824E-13, y = 4.5474735088646412E-13℄℄



lsquares_estimates ä¨â¨àã¥â ¯ à ¬¥âàë ãà ¢­¥­¨ï (ãà ¢­¥­¨¥ ¯¨è¥âáï ¤«ï "¨§¬¥à塞ëå ¢¥«¨ç¨­") ¢ ᮮ⢥âá⢨¨ á "íªá¯¥à¨¬¥­â «ì­ë¬¨ ¤ ­­ë¬¨" ¬¥â®¤®¬ ­ ¨¬¥­ìè¨å ª¢ ¤à â®¢. „«ï ¨á¯®«ì§®¢ ­¨ï ä㭪樨 ­¥®¡å®¤¨¬® á­ ç «  § £à㧨âì ¯ ª¥â "lsquares". ”ã­ªæ¨ï ¨¬¥¥â ç¥âëॠ à£ã¬¥­â  | ᯨ᮪ "íªá¯¥à¨¬¥­â «ì­ëå ¤ ­­ëå" ¢ ¢¨¤¥ ¬ âà¨æë (ª ¤ ï áâப  ¬ âà¨æë | í⮠ᯨ᮪ §­ ç¥­¨© "¨§¬¥à塞ëå ¢¥«¨ç¨­", â.¥. "®¤­  íªá¯¥à¨¬¥­â «ì­ ï â®çª "); ᯨ᮪ ¨¬¥­ "¨§¬¥à塞ëå ¢¥«¨ç¨­"; £¨¯®â¥â¨ç¥áª®¥ ãà ¢­¥­¨¥ (¢ íâ® ãà ¢­¥­¨¥ ªà®¬¥ "¨§¬¥à塞ëå ¢¥«¨ç¨­" ¤®«­ë ¢å®¤¨âì ¨ ¨áª®¬ë¥ ¯ à ¬¥âàë); ¨ ᯨ᮪ ¨¬¥­ ¯ à ¬¥â஢. ”ã­ªæ¨ï

load(lsquares)$ lsquares_estimates(matrix([1,0.9℄,[2,2.1℄,[3,2.9℄), [x,y℄, y=a*x+b, [a,b℄); 1 ℄℄ [[a = 1, b = 30



romberg ç¨á«¥­­® ­ å®¤¨â ®¯à¥¤¥«¥­­ë© ¨­â¥£à « ä㭪樨 ­  § ¤ ­­®¬ ®â१ª¥. Ǒਠí⮬ ¨á¯®«ì§ã¥âáï  «£®à¨â¬ ®¬¡¥à£ , â.¥. íªáâà ¯®«ïæ¨ï ¨­â¥£à «ì­ëå á㬬, ¯®«ã祭­ëå ¯à¨ ã¡ë¢ î饬 è £¥ à¥è¥âª¨ h0 , h1 = h0 =2, h2 = h1 =2, h3 = h2 =2, : : : ¯® ¯¥à¥¬¥­­®© h(7) ¢ â®çªã h1 = 0, ᮮ⢥âáâ¢ãîéãî â®ç­®¬ã ®â¢¥âã. ”ã­ªæ¨ï

(7)   á ¬®¬ ¤¥«¥ ¯® ¯¥à¥¬¥­­®© h2 . 80



rombergtol § ¤ ¥â ®â­®á¨â¥«ì­ãî â®ç­®áâì, ª®â®à ï ¤®«­  ¡ëâì ¤®á⨣­ãâ  ¯à¨ ¨­â¥£à¨à®¢ ­¨¨. ˆ§­ ç «ì­® ãáâ ­®¢«¥­® §­ ç¥­¨¥ 1.0e-4. â® à §ã¬­ë© ¢ë¡®à, â.ª. ä ªâ¨ç¥áª ï â®ç­®áâì  «£®à¨â¬  ®¬¡¥à£  ®¡ëç­® § ¬¥â­® ¡®«ìè¥ § ª § ­­®© Ǒ¥à¥¬¥­­ ï

romberg( os(x)^2,x,0,2*%pi)4.0*atan(1.0); -4.2479656877873199E-9 rombergtol:1.0e-11$ romberg( os(x)^2,x,0,2*%pi)4.0*atan(1.0); -7.8179366199075434E-17



rombergit § ¤ ¥â ¬ ªá¨¬ «ì­®¥ ª®«¨ç¥á⢮ ¨â¥à æ¨©. ˆ§­ ç «ì­® ãáâ ­ ¢«¨¢ ¥âáï §­ ç¥­¨¥ 11. …᫨ âॡ㥬 ï â®ç­®áâì §  íâ® ª®«¨ç¥á⢮ ¨â¥à æ¨© ­¥ ¤®á⨣ ¥âáï, ¢ë¤ ¥âáï á®®¡é¥­¨¥ ®¡ ®è¨¡ª¥ ¨ ¯à®¨á室¨â ¢ë室 ¨§ á¨á⥬ë. Ǒ¥à¥¬¥­­ ï

rombergtol:1.0e-11$ err at h(romberg(sin(x)/ (x+0.01)^2,x,0,20)); `romberg' failed to onverge [℄ rombergit:20$ err at h(romberg(sin(x)/ (x+0.01)^2,x,0,20)); [4.042159703041129℄

…᫨ ‚ë áç¨â ¥â¥ ­ã­ë¬ ¢ëç¨á«ïâì ®¯à¥¤¥«¥­­ë© ¨­â¥£à «, ¯®«ì§ãïáì ¨¬¥­­® MAXIM'®© (¢ ¤¥©á⢨⥫쭮á⨠íâ® «ãçè¥ á¤¥« âì ­  ï§ëª¥ "C"), â® ¯®¤ë­â¥£à «ì­ãî äã­ªæ¨î á«¥¤ã¥â ®âª®¬¯¨«¨à®¢ âì, ¯à¨ç¥¬ ­¥¯à¥¬¥­­® á ¤¥ª« à æ¨¥© ⨯  ä㭪樨 ¨  à£ã¬¥­â  (á¬. à §¤¥« "’à ­á«ïâ®à ¨ ª®¬¯¨«ïâ®à ¢ MAXIM'¥").

81

20. Š®¬¯«¥ªá­ë¥ ç¨á«  ¨ ¢ëà ¥­¨ï Œ­¨¬ ï ¥¤¨­¨æ  ¢ MAXIM'¥ § ¯¨á뢠¥âáï ª ª "%i". ‘ ¥¥ ¯®¬®éìî ¬®­® ª®­áâàã¨à®¢ âì ª®¬¯«¥ªá­ë¥ ¢ëà ¥­¨ï: a+%i*b;



realpart ¢®§¢à é ¥â ¤¥©á⢨⥫ì­ãî ç áâì ¢ëà ¥­¨ï a

imagpart ¢®§¢à é ¥â ¤¥©á⢨⥫ì­ãî ç áâì ¢ëà ¥­¨ï ”ã­ªæ¨ï

imagpart(a+%i*b);



b + a

”ã­ªæ¨ï

realpart(a+%i*b);



%i

b

abs ¢®§¢à é ¥â ¬®¤ã«ì ª®¬¯«¥ªá­®£® ¢ëà ¥­¨ï ”ã­ªæ¨ï

abs(a+%i*b);

sqrt(b



2

2 + a )

arg ¢®§¢à é ¥â 䠧㠪®¬¯«¥ªá­®£® ¢ëà ¥­¨ï ”ã­ªæ¨ï

arg(a+%i*b);

atan2(b, a)

Š ª ¢¨¤­® ¨§ ¯à¨¢¥¤¥­­ëå ¯à¨¬¥à®¢, ¯® 㬮«ç ­¨î ¢á¥ ¯¥à¥¬¥­­ë¥ áç¨â îâáï ¤¥©á⢨⥫ì­ë¬¨. Œ®­® ¤¥ª« à¨à®¢ âì, çâ® â  ¨«¨ ¨­ ï ¯¥à¥¬¥­­ ï ï¥âáï ª®¬¯«¥ªá­®©: de lare(z, omplex);

done

â  ¨­ä®à¬ æ¨ï § ¯¨á뢠¥âáï ¢ ¡ §ã ¤ ­­ëå ¨ ï¥âáï ᢮©á⢮¬ ¯¥à¥¬¥­­®©.



properties ¯¥ç â ¥â ᢮©á⢠ ¯¥à¥¬¥­­®© ”ã­ªæ¨ï

properties(z);

[database info, kind(z, omplex)℄

82



remove 㤠«ï¥â ᢮©á⢮ ¯¥à¥¬¥­­®© ”ã­ªæ¨ï

remove(z, omplex); properties(z);

done [℄

…᫨ ¯¥à¥¬¥­­ ï "z" ¤¥ª« à¨à®¢ ­  ª ª ª®¬¯«¥ªá­ ï ¯¥à¥¬¥­­ ï, â® ä㭪樨 ⨯  "realpart" íâ® ãç¨â뢠îâ, ­ ¯à¨¬¥à: realpart(z^2);



2 2 realpart (z) - imagpart (z)

re tform ¯à¨¢®¤¨â ¢ëà ¥­¨¥ ª ¢¨¤ã Re(z ) + i  Im(z ) ”ã­ªæ¨ï

re tform(r*exp(%i*fi)); %i sin(fi) r + os(fi) r



polarform ¯à¨¢®¤¨â ¢ëà ¥­¨¥ ª ¢¨¤ã mod(z )  exp(i  arg(z )) ”ã­ªæ¨ï

polarform(x+%i*y);

2

sqrt(y



2 + x )

%e%i

exponentialize § ¬¥­ï¥â ¢á¥ âਣ®­®¬¥âà¨ç¥áª¨¥ ä㭪樨 ­  ᮮ⢥âáâ¢ãî騥 ª®¬¡¨­ æ¨¨ íªá¯®­¥­â ”ã­ªæ¨ï

exponentialize( os(a+%i*b)); %e%i (%i b + a) +



atan2(y,vx)

%e-%i (%i

b + a)

2

demoivre § ¬¥­ï¥â ¢á¥ íªá¯®­¥­âë á ¬­¨¬ë¬¨ ¯®ª § â¥«ï¬¨ ­  ᮮ⢥âáâ¢ãî騥 âਣ®­®¬¥âà¨ç¥áª¨¥ ä㭪樨

”ã­ªæ¨ï

demoivre(exp(a+%i*b)

%ea (%i

83

sin(b) + os(b))

21. „¨ää¥à¥­æ¨à®¢ ­¨¥ 

diff ¢ë¯®«­ï¥â ¤¨ää¥à¥­æ¨à®¢ ­¨¥. …¥ ᨭ⠪á¨á ¤®¢®«ì­® à §­®®¡à §¥­ ”ã­ªæ¨ï

diff(x^2,x)

diff(x^3,x,2)

2 x

6 x diff(y^3 * x^3,x,2,y,1) 18 x y

2

Ǒਠí⮬ ¯¥à¢®­ ç «ì­® ¢á¥ ¯¥à¥¬¥­­ë¥ áç¨â îâáï ­¥§ ¢¨á¨¬ë¬¨ diff(y,x)

0

 ¡®â âì á ¯à®¨§¢®¤­®© ®¤­®© ¯¥à¥¬¥­­®© ¯® ¤à㣮© ¬®­® â६ï ᯮᮡ ¬¨: "§ ¬®à®§¨âì" ®¯¥à æ¨î ¤¨ää¥à¥­æ¨à®¢ ­¨ï, 㪠§ âì ­  § ¢¨á¨¬®áâì ® ¨ ¤¥ª« à¨à®¢ âì § ¢¨á¨¬®áâì ­¥ï¢­®. ‡ ¯à¥â¨âì ¢ë¯®«­¥­¨¥ ä㭪樨 "diff" ¨ ¯®«ãç¨âì "§ ¬®à®¥­­ãî" ¯à®¨§¢®¤­ãî ¬®­®, ¥á«¨ ¯¥à¥¤ ¨¬¥­¥¬ ä㭪樨 "diff" ¯®áâ ¢¨âì ®¤¨­®ç­ãî ª ¢ëçªã: 'diff(y,x)

dy dx

Œ®­® ® 㪠§ âì ­  § ¢¨á¨¬®áâì, â.¥. à ¡®â âì á ä㭪樥©: diff(y(x),x)

diff(v(x,y),x,2,y,1)

d (y(x)) dx

3 d (v(x,y)) 2 dx dy (§¤¥áì ¯à¥¤¯®« £ ¥âáï, çâ® ä㭪樨 "y" ¨ "v" ­¥ ¡ë«¨ à ­¥¥ ®¯à¥¤¥«¥­ë á ¯®¬®éìî ®¯¥à â®à  ®¯à¥¤¥«¥­¨ï ä㭪樨 ":=").



depends ¯®§¢®«ï¥â ¤¥ª« à¨à®¢ âì, çâ® ¯¥à¥¬¥­­ ï § ¢¨á¨â ®â ®¤­®© ¨«¨ ­¥áª®«ìª¨å ¤àã£¨å ¯¥à¥¬¥­­ëå ”ã­ªæ¨ï

depends(y,x);

depends(u,[x,y℄);

[y(x)℄ [u(x, y)℄

84



Ǒ¥à¥¬¥­­ ï

dependen ies

ᮤ¥à¨â ᯨ᮪ "§ ¢¨á¨¬®á⥩", ®¯à¥¤¥«¥­­ëå ­  ¤ ­­ë© ¬®¬¥­â depende ies;

[y(x), u(x, y)℄

‡ ¢¨á¨¬®áâì "u" ®â "x" ¨ "y" ï¥âáï "᢮©á⢮¬" ¯¥à¥¬¥­­®© "u".



”ã­ªæ¨ï

properties

¯¥ç â ¥â ᯨ᮪ ᢮©á⢠§ ¤ ­­®© ¯¥à¥¬¥­­®© properties(u);

[dependen y℄



”ã­ªæ¨ï

remove

㤠«ï¥âáï 㪠§ ­­®¥ ᢮©á⢮ ¤ ­­®© ¯¥à¥¬¥­­®© remove(u,dependen y); dependen ies;

done [y(x)℄

ˆ­â¥à¥á­®, çâ® MAXIMA ¯à ¢¨«ì­® ãç¨â뢠¥â § ¢¨á¨¬®á⨠¯¥à¥¬¥­­ëå ¤«ï á«ãç ï ¢«®¥­­ëå ä㭪権: diff(v(x,y),x,1,y,1)

diff(u,x,1,y,1)

2 d (v(x,y)) dx dy 2

2 u dy d u + dx dy 2 dx

d

dy

(ª ª ­¥âà㤭® ¯®­ïâì, ®¡¥ § ¯¨á¨  ¡á®«îâ­® ª®à४â­ë ¬ â¥¬ â¨ç¥áª¨). 85



gradef ®¯à¥¤¥«ï¥â १ã«ìâ â ¤¨ää¥à¥­æ¨à®¢ ­¨ï ä㭪樨 ¯® ᢮¨¬  à£ã¬¥­â ¬ ”ã­ªæ¨ï

gradef(f(a,b, ),g(a,b, ), 77, *f(a,b, ));

⥬ á ¬ë¬ ®¯à¥¤¥«¥­®, çâ®

f(a,b, )

 f (a; b; ) = g(a; b; ); a

 f (a; b; ) = 77; b

 f (a; b; ) =  f (a; b; ) 

(§¤¥áì ¯à¥¤¯®« £ ¥âáï, çâ® äã­ªæ¨ï "f" ­¥ ¡ë«  à ­¥¥ ®¯à¥¤¥«¥­  á ¯®¬®éìî ":="). ƒà ¤¨¥­â ä㭪樨, ª ª ¨ § ¢¨á¨¬®áâì ¯¥à¥¬¥­­ëå, ¥áâì ᢮©á⢮: properties(f);



[gradef℄

prinprops ¯¥ç â ¥â 㪠§ ­­®¥ ᢮©á⢮ ¤ ­­®© ¯¥à¥¬¥­­®© ”ã­ªæ¨ï

printprops(f,gradef);

d (f(a,b, )) = g(a,b, ) da d (f(a,b, )) = 77 db d (f(a,b, )) = f(a,b, ) d

ˆ¬¥©â¥ ¢ ¢¨¤ã, çâ® ¨­ä®à¬ æ¨ï ® § ¢¨á¨¬®á⨠¯¥à¥¬¥­­ëå ¬®¥â ¡ëâì à á¯¥ç â ­  ⮫쪮 á ¯®¬®éìî ¯¥à¥¬¥­­®© "dependen ies". Ǒ® ­¥ ®ç¥­ì ¯®­ïâ­ë¬ ¯à¨ç¨­ ¬ äã­ªæ¨ï "prinprops" ®âª §ë¢ ¥âáï ¯¥ç â âì íâã ¨­ä®à¬ æ¨î, å®âï "dependen y" | í⮠᢮©á⢮ ¯¥à¥¬¥­­®©. ’ ª çâ® ª®¬ ­¤  prinprops(y,dependen y);

¢ë§®¢¥â á®®¡é¥­¨¥ ®¡ ®è¨¡ª¥.

86

22. Ǒ।¥«ë 

limit ¢ëç¨á«ï¥â ¯à¥¤¥« § ¤ ­­®£® ¢ëà ¥­¨ï ¯à¨ áâ६«¥­¨¨ ¯¥à¥¬¥­­®© ª 㪠§ ­­®¬ã §­ ç¥­¨î. ‚ â¥å á«ãç ïå, ª®£¤  «¥¢ë© ¨ ¯à ¢ë© ¯à¥¤¥« ­¥ ᮢ¯ ¤ îâ, ¬®­® ãâ®ç­¨âì, á ª ª®© áâ®à®­ë ¡¥à¥âáï ¯à¥¤¥«. ‘ãé¥áâ¢ã¥â ç¥âëॠᯥ樠«ì­ë¥ §­ ç¥­¨ï | "inf" (+1), "minf" ( 1), "und" (1), "ind" (­¥®¯à¥¤¥«¥­­®áâì): ”ã­ªæ¨ï

limit(sin(x)/x,x,0); limit(1/x,x,0); limit(1/x,x,0,plus);

limit(1/x,x,0,minus); limit(sin(1/x),x,0);

1 und inf minf

ind limit((x+1)/(x+2),x,inf); 1

”ã­ªæ¨ï ¯à¨¬¥­ï¥â ¯à ¢¨«® ‹®¯¨â «ï.



lhospitallim ®£à ­¨ç¨¢ ¥â ç¨á«® ¤¨ää¥à¥­æ¨à®¢ ­¨© ¯à¨ ¢ëç¨á«¥­¨¨ ¯à¥¤¥« , ¯à¨ç¥¬ ¥á«¨ í⮣® ª®«¨ç¥á⢠ ¤¨ää¥à¥­æ¨à®¢ ­¨© ­¥ 墠⠥â, â® äã­ªæ¨ï ¢®§¢à é ¥â á ¬ã ᥡï. ˆ§­ ç «ì­® ãáâ ­®¢«¥­® §­ ç¥­¨¥ "4", ¯®í⮬ã Ǒ¥à¥¬¥­­ ï

limit((sin(x)-x)^2/ (x^4*( os(x)-1)),x,0);

limit x->0 lhospitallim:16$ limit((sin(x)-x)^2/ (x^4*( os(x)-1)),x,0); 1 18

2 (sin(x)-x) 4 4 x os(x)-x

‚®®¡é¥ äã­ªæ¨ï ॠ«¨§®¢ ­  ­¥ ®ç¥­ì  ªªãà â­®, ¯à¨¬¥à®¬ 祬㠬®¥â á«ã¨âì â ª ï ¢ëª« ¤ª  lhospitallim:16$

87

limit((sin(x)-x)^2/x^6,x,0); 1 36 limit(( os(x)-1)^3/x^6,x,0); 1 8 limit(( os(x)-1)^3/ (sin(x)-x)^2,x,0); 9 2 limit((sin(x)-x)^2/ ( os(x)-1)^3,x,0); inf

Ǒ®á«¥¤­¨© ®â¢¥â, ®ç¥¢¨¤­®, ­¥¢¥à¥­.



tlimit ®â«¨ç ¥âáï ®â ä㭪樨 "limit" ⮫쪮  «£®à¨â¬®¬ | ®­  à áª« ¤ë¢ ¥â ¢ëà ¥­¨¥ ¢ àï¤ ’¥©«®à . « £®¤ àï í⮬㠮­  (¢ ®â«¨ç¨¥ ®â äã­ªæ¨ï "limit") ¢ë¤ ¥â ¢¥à­ë© ®â¢¥â ¨ ¤«ï á«ãç ï ”ã­ªæ¨ï

limit((sin(x)-x)^2/ ( os(x)-1)^3,x,0);

-

2 9

88

23. ˆ­â¥£à¨à®¢ ­¨¥ 

integrate ¢ë¯®«­ï¥â ¨­â¥£à¨à®¢ ­¨¥ § ¤ ­­®£® ¢ëà ¥­¨ï ¯® 㪠§ ­­®© ¯¥à¥¬¥­­®© (­¥®¯à¥¤¥«¥­­ ï ª®­áâ ­â  ­¥ ¤®¡ ¢«ï¥âáï). Œ®­® â ª¥ 㪠§ âì ¯à¥¤¥«ë ¨­â¥£à¨à®¢ ­¨ï | ¢ í⮬ á«ãç ¥ ¢ëç¨á«ï¥âáï ®¯à¥¤¥«¥­­ë© ¨­â¥£à «. ”ã­ªæ¨ï

integrate(1/(x+a),x); integrate(x^3,x,a,b);

log(x+a) 4 4 a b 4 4

Ž¯à¥¤¥«¥­­ë© ¨­â¥£à «, § ¢¨áï騩 ®â ¯ à ¬¥âà , ¬®¥â ¡ëâì ¯® ­¥¬ã ¯à®¤¨ää¥à¥­æ¨à®¢ ­

w:integrate(f(x,y),x,a(y),b(y)); b(y) R f(x,y) dx a(y) diff(w,y); d d f(b(y),y) (b(y)) - f(a(y),y) (a(y)) dy dy b(y) R d (f(x,y)) dx + dy a(y)



hangevar ॠ«¨§ã¥â § ¬¥­ã ¯¥à¥¬¥­­ëå ¢ ¨­â¥£à «¥. …¥  à£ã¬¥­âë ¤®«­ë ¨¬¥âì ¢¨¤: á ¬ ¨­â¥£à «, á¢ï§ì áâ à®© ¨ ­®¢®© ¯¥à¥¬¥­­®©, ­®¢ ï ¯¥à¥¬¥­­ ï, áâ à ï ¯¥à¥¬¥­­ ï. ‘¢ï§ì ¯¥à¥¬¥­­ëå § ¤ ¥âáï «¨¡® ¢ ®© ä®à¬¥ ("x=g(t)"), «¨¡® ¢ ¢¨¤¥ ¢ëà ¥­¨ï, ®¤¨­ ¨§ ª®à­¥© ª®â®à®£® ¨ ¤ ¥â á¢ï§ì ¯¥à¥¬¥­­ëå ("x-g(t)"). ”ã­ªæ¨ï

w:integrate(f(x),x)$

hangevar(w,x=g(t),t,x); R d f(g(t)) ( (g(t))) dt dt

hangevar(w,x^2-g(t),t,x); R f(-sqrt(g(t))) d ( (g(t))) dt sqrt(g(t)) dt 2

89

„«ï ®¯à¥¤¥«¥­­ëå ¨­â¥£à «®¢ ¢ë¯®«­ï¥âáï ¯®¤áâ ­®¢ª  ¢ ¯à¥¤¥« å ¨­â¥£à¨à®¢ ­¨ï, â ª çâ® äã­ªæ¨ï, á¢ï§ë¢ îé ï áâ àãî ¨ ­®¢ãî ¯¥à¥¬¥­­ãî, ¤®«­  ¡ëâì ®¡à â¨¬ . Ǒ®í⮬㠯®¯ë⪠ ­ ¯¨á âì w:integrate(f(x),x,a,b)$

hangevar(w,x=g(t),t,x); Unable to solve for t [℄

¢ë§®¢¥â á®®¡é¥­¨¥ ®¡ ®è¨¡ª¥ ("g 1 (a)" ¨ "g 1 (b)") ­¥ ¬®£ãâ ¡ëâì ¢ëç¨á«¥­ë). Ž¤­ ª®, ¢¯®«­¥ ¤®¯ãá⨬®

hangevar(w,x=t+ ,t,x); b- R

¨

a-

f(t+ ) dt

hangevar(w,x^2-t, t, x);

2 bR 2

-



a

f(- sqrt(t)) dt sqrt(t) 2

byparts ¢ë¯®«­ï¥â ¨­â¥£à¨à®¢ ­¨¥ ¯® ç áâï¬. Ǒ¥à¥¤ ¯¥à¢ë¬ ¢ë§®¢®¬ ä㭪樨 ­¥®¡å®¤¨¬® § £à㧨âì ¯ ª¥â "bypart", ¢ ª®â®à®¬ ®­  ®¯à¥¤¥«¥­ : ”ã­ªæ¨ï

load(bypart)$

€à£ã¬¥­âë ¤®«­ë ¨¬¥âì ¢¨¤: ¨­â¥£à «, ¯¥à¥¬¥­­ ï ¨­â¥£à¨à®¢ ­¨ï, "u " ¨ R R 0 0 0 " v " (¨á¯®«ì§ã¥âáï ä®à¬ã«  " uv = uv u v "). ”ã­ªæ¨ï ­¥ ᫨誮¬ 㤠筮 ॠ«¨§®¢ ­  | ®­  ­¥ ®â«¨ç ¥â ®¯à¥¤¥«¥­­ë© ¨­â¥£à « ®â ­¥®¯à¥¤¥«¥­­®£® ¨ ¢á¥£¤  ¢®§¢à é ¥â ­¥®¯à¥¤¥«¥­­ë©, ¯®í⮬㠭 ¡®àë ®¯¥à â®à®¢ ¨

w:integrate(f(x)* os(x),x)$ byparts(w,x,f(x), os(x));

w:integrate(f(x)* os(x),x,a,b)$ byparts(w,x,f(x), os(x));

¯à¨¢¥¤ãâ ª ®¤­®© ¨ ⮩ ¥ ¢ë¤ ç¥

f(x) sin(x) -

R

sin(x) (

d (f(x))) dx dx

”ã­ªæ¨ï "integrate" ¬®¥â á®®¡à §¨âì, çâ® ¨­â¥£à « ®â ¯à®¨§¢®¤­®© ¯à®¨§¢®«ì­®© ä㭪樨 ¥áâì á ¬  äã­ªæ¨ï 90

integrate(diff(u(x),x,2),x); d (u(x)) dx

® ¢ ¡®«¥¥ á«®­ëå á«ãç ïå ®­ ­¥ § ¬¥ç ¥â ¯®«­®© ¯à®¨§¢®¤­®©

integrate(diff(u(x),x,2)*sin(u(x)) +diff(u(x),x)^2* os(u(x)),x); 2 R d (sin(u(x)) (u(x)) 2 dx d 2 (u(x))) ) dx + ( os(u(x)) ( dx



antidiff ¢ë¯®«­ï¥â ¨­â¥£à¨à®¢ ­¨¥ ¢ëà ¥­¨© á ¯à®¨§¢®«ì­ë¬¨ äã­ªæ¨ï¬¨, ¯¥à¥¤ ¥¥ ¯¥à¢ë¬ ¢ë§®¢®¬ á«¥¤ã¥â § £à㧨âì ¯ ª¥â "antid" ”ã­ªæ¨ï

load(antid)$

 §ã¬¥¥âáï, ¨­â¥£à¨à®¢ ­¨¥ ¢ë¯®«­ï¥âáï ⮫쪮 ¤«ï á«ãç ï ¯®«­®© ¯à®¨§¢®¤­®© antidiff(diff(u(x),x,2)*sin(u(x)) +diff(u(x),x)^2* os(u(x)),x); sin(u(x)) (

91

d (u(x))) dx

24. ï¤ë, ¯ ¤¥- ¯¯à®ªá¨¬ æ¨ï ¨ 楯­ë¥ ¤à®¡¨ 

taylor à áª« ¤ë¢ ¥â äã­ªæ¨î ¢ àï¤ ’¥©«®à . ¥§ã«ìâ â ¢ë§®¢  ä㭪樨 ï¥âáï ®á®¡ë¬ ¢ëà ¥­¨¥¬ | "à冷¬", íâ® ¢ëà ¥­¨¥ á­ ¡ ¥âáï ¬¥âª®© "/T/" áࠧ㠯®á«¥ ¬¥âª¨ "%o". €à£ã¬¥­âë ä㭪樨 â ª®¢ë: ¢ëà ¥­¨¥, ª®â®à®¥ ¡ã¤¥â à §«®¥­®; ¯¥à¥¬¥­­ ï, ¯® ª®â®à®© ¨¤¥â à §«®¥­¨¥; â®çª , ¢ ª®â®à®© ¬ë à áª« ¤ë¢ ¥¬; ¨ ¯®à冷ª, ¤® ª®â®à®£® ¨¤¥â à §«®¥­¨¥: ”ã­ªæ¨ï

(%i7) ta1:taylor(sin(x),x,0,3)

3 x + . . (%o7) /T/ x 6 (%i8) ta2:taylor( os(x),x,0,6)

.

2 4 6 x x x (%o8) /T/ 1 + + . . . 2 24 720

ï¤ë ¬®­® ᪫ ¤ë¢ âì, ¢ëç¨â âì, 㬭® âì ¨ ¤¥«¨âì ¤à㣠­  ¤à㣠, ¯à¨ í⮬ â®ç­®áâì à §«®¥­¨ï ãç¨â뢠¥âáï  ¢â®¬ â¨ç¥áª¨, ­ ¯à¨¬¥à (%i9) ta1/ta2;

3 x (%o9) /T/ x + + . . 3

.

â.¥. à §«®¥­¨¥ ¨¤¥â ¤® âà¥â쥣® ¯®à浪  (â®ç­®áâì ¯¥à¢®£® à鸞). Ǒਠí⮬ ä ªâ¨ç¥áª¨ à¥çì ¨¤¥â ® à拉 ‹®à ­ , â.¥. ¤®¯ã᪠¥âáï taylor(sin(x)/x^3,x,0,5);

2 4 1 1 x x + + . . . 6 120 5040 2 x

®«¥¥ ⮣®, áãé¥áâ¢ã¥â íª§®â¨ç¥áª ï ¢®§¬®­®áâì taylor(exp(1/x),[x,0,3,'asymp℄); 1 +

1 + x

92

1 2 x

2

+

1 6 x

3

+ . .

.



pade  ¯¯à®ªá¨¬¨àã¥â ®â१®ª à鸞 ’¥©«®à , ᮤ¥à é¨© á« £ ¥¬ë¥ ¤® N -£® ¯®à浪  ¢ª«îç¨â¥«ì­®, ¤à®¡­®-à æ¨®­ «ì­®© ä㭪樥©. …¥  à£ã¬¥­âë | àï¤ ’¥©«®à , ¯®à冷ª ç¨á«¨â¥«ï n, ¯®à冷ª §­ ¬¥­ â¥«ï m.  §ã¬¥¥âáï, ª®«¨ç¥á⢮ ¨§¢¥áâ­ëå ª®íää¨æ¨¥­â®¢ à鸞 ’¥©«®à  ¤®«­® ᮢ¯ ¤ âì á ®¡é¨¬ ª®«¨ç¥á⢮¬ ª®íää¨æ¨¥­â®¢ ¢ ¤à®¡­®-à æ¨®­ «ì­®© ä㭪樨 ¬¨­ãá ®¤¨­ (¯®áª®«ìªã ç¨á«¨â¥«ì ¨ §­ ¬¥­ â¥«ì ®¯à¥¤¥«¥­ë á â®ç­®áâìî ¤® ®¡é¥£® ¬­®¨â¥«ï). ˆ­ë¬¨ á«®¢ ¬¨, N + 1 = (n + 1) + (m + 1) 1. ”ã­ªæ¨ï

ta1:taylor(log(1+x),x,0,6); 2 3 4 5 6 x x x x x x + + + . . . 2 3 4 5 6 pade(ta1,3,3) 3 2 11 x + 60 x + 60 x [ ℄ 3 2 3 x + 36 x + 90 x + 60 pade(ta1,4,2) 4 3 2 x - 12 x - 150 x - 180 x ℄ [2 72 x + 240 x + 180



”ã­ªæ¨ï



Ǒ¥à¥¬¥­­ ï



”ã­ªæ¨ï

f ‘®§¤ ¥â 楯­ãî ¤à®¡ì,  ¯¯à®ªá¨¬¨àãîéãî ¤ ­­®¥ ¢ëà ¥­¨¥. ‚ëà ¥­¨¥ ¤®«­® á®áâ®ïâì ¨§ 楫ëå ç¨á¥«, ª¢ ¤à â­ëå ª®à­¥© 楫ëå ç¨á¥« ¨ §­ ª®¢  à¨ä¬¥â¨ç¥áª¨å ®¯¥à æ¨©. ‚ë¤ ç  ¨¬¥¥â ¢¨¤ ᯨ᪠.

flength ®¯à¥¤¥«ï¥â ª®«¨ç¥á⢮ ¯¥à¨®¤®¢ 楯­®© ¤à®¡¨. ˆ§­ ç «ì­® ãáâ ­®¢«¥­® §­ ç¥­¨¥ 1.

fdisrep ¯à¥®¡à §ã¥â ᯨ᮪ (ª ª ¯à ¢¨«® ¢ë¤ çã ä㭪樨 " f") ¢ ᮡá⢥­­® 楯­ãî ¤à®¡ì.

f(sqrt(3));

fdisrep(%);

[1,1,2℄ 1+

float(%-sqrt(3.0));

1 1 1+ 2

-0.065384140902211

93

flength:2$

f(sqrt(3));

fdisrep(%);

[1,1,2,1,2℄ 1+

flength:5$

f(sqrt(3));

fdisrep(%)$ float(%-sqrt(3.0));

1 1+

2+

1 1

1+

1 2

[1,1,2,1,2,1,2,1,2,1,2℄ -1.7707912940423398E-6

94

25. Ǒ८¡à §®¢ ­¨¥ ‹ ¯« á  ¨ ¢ëç¥âë 

lapla e ॠ«¨§ã¥â ¯àאַ¥ ¯à¥®¡à §®¢ ­¨¥ ‹ ¯« á  ”ã­ªæ¨ï

lapla e(exp(t),t,s);

lapla e(sin(t),t,s);

1 s - 1 1

2 s + 1 lapla e(diff(x(t),t),t,s); s lapla e(x(t), t, s) - x(0)

(¯®á«¥¤­¥¥ ᮮ⭮襭¨¥ ¯®§¢®«ï¥â à¥è âì «¨­¥©­ë¥ ¤¨ää¥à¥­æ¨ «ì­ë¥ ãà ¢­¥­¨ï).



ilt ॠ«¨§ã¥â ®¡à â­®¥ ¯à¥®¡à §®¢ ­¨¥ ‹ ¯« á  ”ã­ªæ¨ï

ilt(1/(s-1),s,t);

%et

ilt(lapla e(x(t),t,s),s,t); x(t)

(¯®á«¥¤­¥¥ ᢮©á⢮ ¯®§¢®«ï¥â ¯®«ãç âì ï¢­ë¥ ®â¢¥âë ¯à¨ à¥è¥­¨¨ «¨­¥©­ëå ¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥­¨©).



residue ¯®§¢®«ï¥â ¢ëç¨á«ïâì ¢ëç¥âë ”ã­ªæ¨ï

residue(1/(s^2+a^2),s,%i*a); %i 2 a

95

26. “à ¢­¥­¨ï — áâì ä㭪権, ª®â®àë¥ ¯®§¢®«ïîâ à¥è âì ãà ¢­¥­¨ï, ¯¥à¥ç¨á«¥­  ¢ à §¤¥«¥ " ¡®â  á oat-ç¨á« ¬¨". â® ä㭪樨, ª®â®àë¥ ¢ë¤ î⠮⢥⠢ â¥à¬¨­ å float-ç¨á¥« (allroots, find_root, newton, mnewton).



realroots äã­ªæ¨ï, ª®â®à ï ¢ë¤ ¥â ¤¥©á⢨⥫ì­ë¥ ª®à­¨ ¯®«¨­®¬¨ «ì­®£® ãà ¢­¥­¨ï á ¤¥©á⢨⥫ì­ë¬¨ ª®íää¨æ¨¥­â ¬¨. Ǒਠí⮬ äã­ªæ¨ï à ¡®â ¥â ¢®¢á¥ ­¥ á float-ç¨á« ¬¨, ®­  ¢ë¤ ¥â ®â¢¥â ¢ â¥à¬¨­ å à æ¨®­ «ì­ëå ç¨á¥«. …᫨ ª®íää¨æ¨¥­âë ¢ ¨á室­®¬ ãà ¢­¥­¨¨ ïîâáï float-ç¨á« ¬¨, â® ®­¨ ª®­¢¥àâ¨àãîâáï ¢ à æ¨®­ «ì­ë¥. ’®ç­®áâì í⮩ ®¯¥à æ¨¨ § ¤ ¥âáï ¯ à ¬¥â஬ "ratepsilon" (á¬. à §¤¥« " ¡®â  á oat-ç¨á« ¬¨"). ”ã­ªæ¨ï



Ǒ¥à¥¬¥­­ ï



Ǒ¥à¥¬¥­­ ï

rootsepsilon § ¤ ¥â § ª § ­­ãî â®ç­®áâì ¯®¨áª  ª®à­¥© ¤«ï ä㭪樨 "realroots". Ǒ® 㬮«ç ­¨î § ¤ ­® ¤®¢®«ì­® áâà ­­®¥ §­ ç¥­¨¥ 1.0e-7. ˆ¬¥©â¥ ¢ ¢¨¤ã, çâ® íâ® â®ç­®áâì ¯®¨áª  ª®à­¥© ­¥ ¨á室­®£® ãà ¢­¥­¨ï,   ⮣® ãà ¢­¥­¨ï, ª®â®à®¥ ¯®«ãç ¥âáï ¯®á«¥ ¯¥à¥å®¤  ª à æ¨®­ «ì­ë¬ ª®íää¨æ¨¥­â ¬. ’ ª çâ® â®ç­®áâì à¥è¥­¨ï ¨á室­®£® ãà ¢­¥­¨ï á float-ª®íää¨æ¨¥­â ¬¨ ¡ã¤¥â § ¢¨á¥âì ¥é¥ ¨ ®â ¯ à ¬¥âà  "ratepsilon". multipli ities ¯®á«¥ ¢ë¯®«­¥­¨ï ä㭪樨 "realroots" ᮤ¥à¨â ᯨ᮪ ªà â­®á⥩ ª®à­¥©. realroots(x^6+1=x^2+x^4); [x = -1, x = 1℄ multipli ities; [2, 2℄



nroots äã­ªæ¨ï, ª®â®à ï ¢ë¤ ¥â ª®«¨ç¥á⢮ ¤¥©á⢨⥫ì­ëå ª®à­¥© ¯®«¨­®¬¨ «ì­®£® ãà ¢­¥­¨ï á ¤¥©á⢨⥫ì­ë¬¨ ª®íää¨æ¨¥­â ¬¨, ª®â®àë¥ «®ª «¨§®¢ ­ë ¢ 㪠§ ­­®¬ ¨­â¥à¢ «¥ ”ã­ªæ¨ï

nroots(x^6+1=x^2+x^4,minf,inf); 4 nroots(x^6+1=x^2+x^4,0,inf); 2

96





”ã­ªæ¨ï

algsys

à¥è ¥â ¯®«¨­®¬¨ «ì­ë¥ á¨á⥬ë ãà ¢­¥­¨©. „®¯ã᪠îâáï á¨áâ¥¬ë ¨§ ®¤­®£® ãà ¢­¥­¨ï á ®¤­®© ­¥¨§¢¥áâ­®©. Šà®¬¥ ⮣®, ¤®¯ã᪠îâáï ­¥¤®®¯à¥¤¥«¥­­ë¥ á¨á⥬ë. €à£ã¬¥­âë ä㭪樨 "algsys" | í⮠ᯨ᮪ ãà ¢­¥­¨© ¨ ᯨ᮪ ¯¥à¥¬¥­­ëå,   ¥¥ ¢ë¤ ç  | í⮠ᯨ᮪ à¥è¥­¨©. Ǒ®áª®«ìªã ª ¤®¥ à¥è¥­¨¥ ¥áâì ᯨ᮪ §­ ç¥­¨© ª ¤®© ¨§ ¯¥à¥¬¥­­ëå, ⮠ᯨ᮪ à¥è¥­¨© | íâ® ¤¢®©­®© ¢«®¥­­ë© ᯨ᮪. „«ï ãà ¢­¥­¨© á ­ã«¥¢®© ¯à ¢®© ç áâìî íâã ­ã«¥¢ãî ¯à ¢ãî ç áâì ¬®­® ®¯ã᪠âì. ‹®£¨ª  ä㭪樨 "algsys" ¤®¢®«ì­® à §¢¥â¢«¥­­ ï, ¢ § ¢¨á¨¬®á⨠®â ¢¨¤  ª®­ªà¥â­®© á¨á⥬ë ãà ¢­¥­¨© ®­  ¬®¥â ¢ë§ë¢ âì ä㭪樨 "allroots", "realroots", "solve". (Šáâ â¨, ¢ ­¥ª®â®àëå á¨âã æ¨ïå äã­ªæ¨ï "solve", «®£¨ª  ª®â®à®© â ª¥ ¢¥á쬠 à §¢¥â¢«¥­­ ï, ¬®¥â, ¢ á¢®î ®ç¥à¥¤ì, ¢ë§ë¢ âì äã­ªæ¨î "algsys"). ”ã­ªæ¨ï "algsys" ­¥ ª®­¢¥àâ¨àã¥â oat-ç¨á« , ¢å®¤ï騥 ¢ á¨á⥬ã, ¢ à æ¨®­ «ì­ë¥. algepsilon „®«­  § ¤ ¢ âì â®ç­®áâì à¥è¥­¨ï á¨áâ¥¬ë ¤«ï ä㭪樨 "algsys". Ǒ® 㬮«ç ­¨î ãáâ ­®¢«¥­® §­ ç¥­¨¥ 108 , çâ® ®§­ ç ¥â 8 ¢¥à­ëå §­ ª®¢. Ǒ®¢ë襭¨¥ â®ç­®á⨠ᮮ⢥âáâ¢ã¥â 㢥«¨ç¥­¨î ¯®ª § â¥«ï á⥯¥­¨. Ǒ¥à¥¬¥­­ ï

Š á® «¥­¨î, íâ®â ¯ à ¬¥âà ­¥ à ¡®â ¥â. ”ã­ªæ¨ï " algsys" ¨£­®à¨àã¥â ¯¥à¥¬¥­­ãî " algepsilon".



Ǒ¥à¥¬¥­­ ï

%rnum_list

¯®á«¥ ¢ë§®¢  ä㭪樨 "algsys" ᮤ¥à¨â ᯨ᮪ ­¥®¯à¥¤¥«¥­­ëå ¯ à ¬¥â஢, ¢å®¤ïé¨å ¢ à¥è¥­¨¥ ¤«ï ­¥¤®®¯à¥¤¥«¥­­®© á¨á⥬ë. ˆ¬¥­  íâ¨å ¯ à ¬¥â஢ ª®­áâàã¨àãîâáï ¨§ ¯à¥ä¨ªá  "%r" ¨ 楫®£® ç¨á« , ­ ¯à¨¬¥à "%r7". algsys([x^2-3*x+2=0℄,[x℄); [[x = 1℄, [x = 2℄℄ algsys([x^2-3*y+2,x=y℄,[x,y℄); [[x = 2, y = 2℄, [x = 1, y = 1℄℄ algsys([x^2-y℄,[x,y℄); 2 [[x = %r13, y = %r13 ℄℄ %rnum_list; [%r13℄

97



solve à¥è ¥â ãà ¢­¥­¨ï ¨ á¨á⥬ë ãà ¢­¥­¨©. …¥  à£ã¬¥­âë | ᯨ᮪ ãà ¢­¥­¨© ¨ ᯨ᮪ ¯¥à¥¬¥­­ëå,   ¢ë¤ ç  | ᯨ᮪ à¥è¥­¨©. Ǒਠí⮬ ¤«ï ãà ¢­¥­¨© á ­ã«¥¢®© ¯à ¢®© ç áâìî íâ㠯ࠢãî ç áâì ¬®­® ®¯ã᪠âì. ‚ ®â«¨ç¨¥ ®â ä㭪樨 "algsys", äã­ªæ¨ï "solve" ª®­¢¥àâ¨àã¥â oat-ç¨á« , ¢å®¤ï騥 ¢ á¨á⥬ã, ¢ à æ¨®­ «ì­ë¥. Ǒ®í⮬㠥¥ ¯®¢¥¤¥­¨¥ ¬®¥â § ¢¨á¥âì ®â ¯ à ¬¥âà  "ratepsilon" (á¬. à §¤¥« " ¡®â  á oat-ç¨á« ¬¨"). ”ã­ªæ¨ï

solve([x^2-3*x+2=0℄,[x℄); [x = 1, x = 2℄ solve([sin(x)-1/2℄,[x℄); `solve' is using ar -trig fun tions to get a solution. Some solutions will be lost. %pi ℄ [x = 6



multipli ities ᮤ¥à¨â ᯨ᮪ ªà â­®á⥩ ª®à­¥©, ­ ©¤¥­­ëå ä㭪樥© "solve" Ǒ¥à¥¬¥­­ ï

solve([x^4-3*x^3+2*x^2=0℄,[x℄); [x = 1, x = 2, x = 0℄ multipli ities; [1, 1, 2℄

„«ï á¨á⥬ ãà ¢­¥­¨© à¥è¥­¨¥ | íâ® ¤¢®©­®© ¢«®¥­­ë© ᯨ᮪ (á¬. äã­ªæ¨î "algsys"): solve([x+y=4,x-y=2℄,[x,y℄); [[x = 3, y = 1℄℄

(¢ ¤ ­­®¬ á«ãç ¥ à¥è¥­¨¥ ®¤­®). „«ï ­¥¤®®¯à¥¤¥«¥­­ëå á¨á⥬ ¢ à¥è¥­¨¥ ¢å®¤ïâ ­¥®¯à¥¤¥«¥­­ë¥ ¯ à ¬¥âàë ¢¨¤  "%r4",   ¯¥à¥¬¥­­ ï "%rnum_list" ¯®á«¥ ¢ë§®¢  ä㭪樨 "solve" ᮤ¥à¨â ¨å ᯨ᮪ (á¬. äã­ªæ¨î "algsys"): solve([x^2-y℄,[x,y℄);

%rnum_list;

[[x =

%r5,

y =

%r52

℄℄

[%r5℄

”ã­ªæ¨ï "solve" ¨¬¥¥â ¤®¢®«ì­® à §¢¥â¢«¥­­ãî «®£¨ªã. ‚ § ¢¨á¨¬®á⨠®â ª®­ªà¥â­®£® ¢¨¤  ãà ¢­¥­¨ï ¨«¨ á¨áâ¥¬ë ®­  ¢¥¤¥â á¥¡ï ®ç¥­ì ¯®-à §­®¬ã ¨ 98

¬®¥â ¢ë§ë¢ âì ¤à㣨¥ ä㭪樨 ("linsolve", "algsys", ¨.â.¯.), ª®â®àë¥ ¯à¥¤­ §­ ç¥­ë ¤«ï ¯®¨áª  à¥è¥­¨© ¢ â¥å ¨«¨ ¨­ëå ç áâ­ëå á«ãç ïå. ‘«¥¤ã¥â â ª¥ ¨¬¥âì ¢ ¢¨¤ã, çâ® äã­ªæ¨ï "solve" ã¯à ¢«ï¥âáï ¤®¢®«ì­® ¡®«ì訬 ª®«¨ç¥á⢮¬ ¯¥à¥¬¥­­ëå (ä« £®¢), ª®â®àë¥ ¬¥­ïîâ ¥¥ ¯®¢¥¤¥­¨¥. Š á® «¥­¨î, ¯à¨ ॠ«ì­®© à ¡®â¥ ®­¨ ¯®ç⨠¡¥á¯®«¥§­ë. „¥«® ¢ ⮬, çâ® äã­ªæ¨ï "solve" ­ ¤¥­® à ¡®â ¥â £« ¢­ë¬ ®¡à §®¬ ¤«ï â¥å ãà ¢­¥­¨© ¨«¨ á¨á⥬, ª®â®àë¥ á ®ç¥¢¨¤­®áâìî ¨¬¥îâ à¥è¥­¨¥. ‚ í⮬ á«ãç ¥ ä« £¨ ­¥ ­ã­ë. ‚ ¡®«¥¥ ­¥âਢ¨ «ì­ëå á«ãç ïå à¥è¥­¨¥ ®¡ëç­® ¯®«ãç¨âì ­¥¢®§¬®­®, ¢­¥ § ¢¨á¨¬®á⨠®â §­ ç¥­¨ï ä« £®¢ëå ¯¥à¥¬¥­­ëå. —â®¡ë ¯à®¤¥¬®­áâà¨à®¢ âì, ­ áª®«ìª® ¯à¨ç㤫¨¢ë¬ ¬®¥â ¡ëâì ¯®¢¥¤¥­¨¥ ä㭪樨 "solve", ¯à¨¢¥¤¥¬ ­¥áª®«ìª® ¯à¨¬¥à®¢:

solve([x^5+y=7,x=y℄,[x,y℄); [[x = 1.36861648832 %i + 0.5084694089, y = 1.36861648832 %i + 0.5084694089℄, [x = 0.5084694089 - 1.36861648832 %i, y = 0.5084694089 - 1.36861648832 %i℄, [x = 0.9241881109 %i - 1.21387633450, y = 0.9241881109 %i - 1.21387633450℄, [x = - 0.9241881109 %i - 1.21387633450, y = - 0.9241881109 %i - 1.21387633450℄, [x = 1.41081382385, y = 1.41081382385℄℄ solve(x^5+x-7); 5 [0 = x + x - 7℄ solve(x^6+x-7.0000000123456789); `rat' repla ed -7.00000001234568 by -7/1 = -7.0 6 [0 = x + x - 7℄ solve([sin(x)-y,sin(x)+y=1℄,[x,y℄); [℄



eliminate ¨áª«î砥⠨§ á¨á⥬ë ãà ¢­¥­¨© 㪠§ ­­ë¥ ¯¥à¥¬¥­­ë¥. Žá⠢訥áï ãà ¢­¥­¨ï ¯à¨¢®¤ïâáï ª ¢¨¤ã á ­ã«¥¢®© ¯à ¢®© ç áâìî, ª®â®à ï ®¯ã᪠¥âáï. ”ã­ªæ¨ï "eliminate" ª®­¢¥àâ¨àã¥â oat-ç¨á« , ¢å®¤ï騥 ¢ á¨á⥬ã, ¢ à æ¨®­ «ì­ë¥. ”ã­ªæ¨ï

eliminate([x+y+z=1, x-y+z=2,x+y-z=3℄,[z℄); [2 (y + x - 2), - 2 y - 1℄ eliminate([x+y+z=1, x-y+z=2,x+y-z=3℄,[x,y℄);

99

[- 2 (z + 1)℄ eliminate([sin(x)-y=0,sin(x)+y=0.5℄,[y℄); `rat' repla ed -0.5 by -1/2 = -0.5 [1 - 4 sin(x)℄ Ž¤­ ª® ­¥ á«¥¤ã¥â ¤ âì ®â ä㭪樨 "eliminate" ᫨誮¬ ¬­®£®£®: eliminate([sin(x)-sin(y)=0,sin(x)+sin(y)=1/2℄,[y℄); [1℄

100

27. „¨ää¥à¥­æ¨ «ì­ë¥ ãà ¢­¥­¨ï 



ode2 à¥è ¥â ¤¨ää¥à¥­æ¨ «ì­ë¥ ãà ¢­¥­¨ï ¯¥à¢®£® ¨ ¢â®à®£® ¯®à浪®¢. …¥  à£ã¬¥­âë | á ¬® ¤¨ää¥à¥­æ¨ «ì­®¥ ãà ¢­¥­¨¥ ¢ ä®à¬¥ á "§ ¬®à®¥­­®©" ¯à®¨§¢®¤­®© (â.¥. á ¯à®¨§¢®¤­®©, ¢ëç¨á«¥­¨¥ ª®â®à®© § ¯à¥é¥­® á ¯®¬®éìî ®¤¨­®ç­®© ª ¢ë窨: " 'diff(y,x) "), äã­ªæ¨ï ¨ ¯¥à¥¬¥­­ ï. ¥®¯à¥¤¥«¥­­ë¥ ª®­áâ ­âë ¤«ï ãà ¢­¥­¨© ¯¥à¢®£® ¯®à浪  ¯¨èãâáï ª ª "% ",   ¤«ï ãà ¢­¥­¨© ¢â®à®£® ¯®à浪  | ª ª "%k1", "%k2". ”ã­ªæ¨ï

method ¯®á«¥ à ¡®âë ä㭪樨 "ode2" ᮤ¥à¨â 㪠§ ­¨¥ ­  ⨯ ãà ¢­¥­¨ï. ”ã­ªæ¨ï à á¯®§­ ¥â «¨­¥©­ë¥ ãà ¢­¥­¨ï ¯¥à¢®£® ¯®à浪  Ǒ¥à¥¬¥­­ ï

ode2('diff(y,x)=2*y+exp(x),y,x); method;

%e-x) %e2

y = (% -

x

linear

”ã­ªæ¨ï à á¯®§­ ¥â ãà ¢­¥­¨ï ¯¥à¢®£® ¯®à浪  à §¤¥«ïî騬¨áï ¯¥à¥¬¥­­ë¬¨ ode2('diff(y,x)=(x^2+1)*y^4,y,x); method;

1

x

=

3

3 y

3

+ 3 x + 3

%

separable

”ã­ªæ¨ï à á¯®§­ ¥â â®ç­® ¨­â¥£à¨àã¥¬ë¥ ãà ¢­¥­¨ï ¯¥à¢®£® ¯®à浪  ode2('diff(y,x)=(x^2+3*y^2)/(2*x*y),y,x); y method;

2

+ x 3 x

2

=

exa t

„«ï ãà ¢­¥­¨© í⮣® ⨯  ¢¢®¤¨âáï ¥é¥ ¨ 101

%



intfa tor ¯®á«¥ à ¡®âë ä㭪樨 "ode2" ¤«ï ãà ¢­¥­¨© ⨯  "exa t" ᮤ¥à¨â ¨­â¥£à¨àãî騩 ¬­®¨â¥«ì Ǒ¥à¥¬¥­­ ï

intfa tor;

1 4 x

”ã­ªæ¨ï à á¯®§­ ¥â «¨­¥©­ë¥ ­¥®¤­®à®¤­ë¥ ãà ¢­¥­¨ï ¢â®à®£® ¯®à浪  ode2('diff(y,x,2)-3*'diff(y,x)+ 2*y=4*exp(3*x),y,x); 3 x 2x y = 2%e + %k1 %e + method; variationofparameters

%k2 %ex

„«ï ãà ¢­¥­¨© í⮣® ⨯  ¢¢®¤¨âáï ¥é¥ ¨



yp ¯®á«¥ à ¡®âë ä㭪樨 "ode2" ¤«ï ãà ¢­¥­¨© ⨯  "variationofparameters" ᮤ¥à¨â ç áâ­®¥ à¥è¥­¨¥ Ǒ¥à¥¬¥­­ ï

yp;

2



%e3

x

i 1 ¯®§¢®«ï¥â ãç¥áâì ­ ç «ì­®¥ ãá«®¢¨¥ ¢ à¥è¥­¨ïå ¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥­¨© ¯¥à¢®£® ¯®à浪 , ¥¥  à£ã¬¥­âë | à¥è¥­¨¥, §­ ç¥­¨¥ "x" ¢ ¢¨¤¥ ãà ¢­¥­¨ï ¨ ᮮ⢥âáâ¢ãî饥 §­ ç¥­¨¥ "y" ⮥ ¢ ¢¨¤¥ ãà ¢­¥­¨ï. ”ã­ªæ¨ï

ode2('diff(y,x)=2*y+exp(x),y,x); i 1(%,x=0,y=1);

y = (% y = 2



%e2

%e-x) %e2 x

-

x

%ex

i 2 ¯®§¢®«ï¥â ãç¥áâì ­ ç «ì­ë¥ ãá«®¢¨ï ¢ à¥è¥­¨ïå ¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥­¨© ¢â®à®£® ¯®à浪 , ¥¥  à£ã¬¥­âë | à¥è¥­¨¥, §­ ç¥­¨¥ "x" ¢ ¢¨¤¥ ãà ¢­¥­¨ï ¨ ᮮ⢥âáâ¢ãî騥 §­ ç¥­¨ï "y" ¨ "§ ¬®à®¥­­®©" ¯à®¨§¢®¤­®© "dy=dx" (" 'diff(y,x) ") ⮥ ¢ ¢¨¤¥ ãà ¢­¥­¨ï. ”ã­ªæ¨ï

102



b 2 ¯®§¢®«ï¥â ãç¥áâì ªà ¥¢ë¥ ãá«®¢¨ï ¢ à¥è¥­¨ïå ¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥­¨© ¢â®à®£® ¯®à浪 , ¥¥  à£ã¬¥­âë | à¥è¥­¨¥, §­ ç¥­¨¥ "x" ¢ ¯¥à¢®© â®çª¥ ¢ ¢¨¤¥ ãà ¢­¥­¨ï ¨ ᮮ⢥âáâ¢ãî饥 §­ ç¥­¨¥ "y", §­ ç¥­¨¥ "x" ¢® ¢â®à®© â®çª¥ ¨ ᮮ⢥âáâ¢ãî饥 §­ ç¥­¨¥ "y" ⮥ ¢ ¢¨¤¥ ãà ¢­¥­¨©. ”ã­ªæ¨ï

re1:ode2('diff(y,x,2)-3*'diff(y,x)+ 2*y=4*exp(3*x),y,x); 3 x 2 x y = 2 %e + %k1 %e + i 2(re1,x=0,y=4,'diff(y,x)=9); 3 x 2 x x y = 2 %e + %e + %e b 2(re1,x=0,y=4,x=1, y=exp(1)+exp(2)+2*exp(3) ); 3 x 2 x x y = 2 %e + %e + %e

%k2 %ex

‘®¢¥à襭­® ¯®-¤à㣮¬ã ®à£ ­¨§®¢ ­   «ìâ¥à­ â¨¢­ ï äã­ªæ¨ï, ª®â®à ï â ª¥ 㬥¥â à¥è âì ¤¨ää¥à¥­æ¨ «ì­ë¥ ãà ¢­¥­¨ï, ¨, ªà®¬¥ ⮣®, á¨áâ¥¬ë ¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥­¨©. Ž­  áãé¥á⢥­­ë¬ ®¡à §®¬ ¨á¯®«ì§ã¥â ᢮©á⢮ ä㭪権, ª®â®à®¥ ­ §ë¢ ¥âáï "atvalue".



atvalue ¯®§¢®«ï¥â § ¤ âì §­ ç¥­¨¥ ä㭪樨 ¨ ¥¥ ¯à®¨§¢®¤­ëå ¯à¨ ­¥ª®â®àëå §­ ç¥­¨ïå  à£ã¬¥­â®¢. ”ã­ªæ¨ï

atvalue(x(t),t=0,5);

5 atvalue(diff(x(t),t),t=0,55); 55 atvalue(diff(x(t),t),t=1,77); 77 atvalue(f(a,b),[a=0,b=1℄,555); 555 atvalue(diff(f(a,b),b), [a=1,b=0℄,777); 777

â  ¨­ä®à¬ æ¨ï ï¥âáï ᢮©á⢮¬ ä㭪権 "x(t)" ¨ "f(a,b)". 103



properties ¯¥ç â ¥â ᢮©á⢠ ¯¥à¥¬¥­­®© ”ã­ªæ¨ï

properties(x);

[atvalue℄



printprops ¯¥ç â ¥â ¨­ä®à¬ æ¨î ® § ¤ ­­®¬ ᢮©á⢥ ¯¥à¥¬¥­­®© ”ã­ªæ¨ï

printprops(x,atvalue);

d (x(1)) d1 d (x(1)) d1

1=1 1=0

= 77 = 55

x(0)=5 printprops(f,atvalue);

d (f(1,2)) d2

1=1, 2=0

= 777

f(0,1)=555



remove ®â¬¥­ï¥â 㪠§ ­­®¥ ᢮©á⢮ ¯¥à¥¬¥­­®© ”ã­ªæ¨ï

remove(x,atvalue); properties(x);

done [℄



at ¢ëç¨á«ï¥â §­ ç¥­¨¥ ¢ëà ¥­¨ï ¢ § ¤ ­­®© â®çª¥ á ãç¥â®¬ ᢮©á⢠ "atvalue". ”ã­ªæ¨ï

at(x(t)+10*diff(x(t),t), t=0); 555 at( f(a,b) + diff(f(a,b),b) , [a=1,b=0℄); f(1, 0) + 777

104



desolve à¥è ¥â ¤¨ää¥à¥­æ¨ «ì­ë¥ ãà ¢­¥­¨ï ¨ á¨áâ¥¬ë ¤¨ää¥à¥­æ¨ «ì­ëå ãà ¢­¥­¨©. …¥  à£ã¬¥­âë | ᯨ᮪ ãà ¢­¥­¨©, ¢ ª®â®àëå ä㭪樨 § ¯¨á ­ë ® ("y(t)") ¨ ᯨ᮪ ­¥¨§¢¥áâ­ëå ä㭪権 (â ª¥ ¢ ¢¨¤¥ "y(t)"). ”ã­ªæ¨ï

re2:desolve( [diff(y(t),t)= 2*y(t)+%e^t℄, [y(t)℄); atvalue(y(t),t=0,1)$ ev(re2,at);

y(t) = (y(0) + 1)

%e2

t

-

%et

2 t t y(t) = 2 %e - %e desolve( [diff(y(t),t)= 2*y(t)+%e^t℄, [y(t)℄ ); 2 t t y(t) = 2 %e - %e atvalue(x(t),t=0,2)$ atvalue(y(t),t=0,0)$ desolve( [diff(x(t),t)=y(t), diff(y(t),t)=x(t)℄, [x(t),y(t)℄); t -t t -t [x(t) = %e + %e , y(t) = %e - %e ℄ desolve( [diff(x(t),t,2) -3*diff(x(t),t)+2*x(t)℄,[x(t)℄); 2 t d (x(t)) - x(0)) x(t) = %e ( dt t = 0 +

%et

105

(2 x(0) -

d (x(t)) ) dt t = 0

28. ‘¯¥æ¨ «ì­ë¥ ä㭪樨 ‚ MAXIM'¥ ®¯à¥¤¥«¥­ë á«¥¤ãî騥 ᯥ樠«ì­ë¥ ä㭪樨: ”㭪樨 ©à¨

airy_ai(x)

airy_bi(x)

airy_dai(x)

airy_dbi(x)

¨ ¨å ¯à®¨§¢®¤­ë¥

–¨«¨­¤à¨ç¥áª¨¥ ä㭪樨 ¥áᥫï, ¥©¬ ­ , ˆ­ä¥«ì¤  ¨ Œ ª¤®­ «ì¤  ¨­¤¥ªá  m bessel_j(m,x) bessel_i(m,x)



bessel_y(m,x) bessel_k(m,x)

besselexpand ®¯à¥¤¥«ï¥â, ¡ã¤ãâ «¨ 樫¨­¤à¨ç¥áª¨¥ ä㭪樨 ¯®«ã楫®£® ¨­¤¥ªá  § ¬¥­ïâìáï ­  ᮮ⢥âáâ¢ãî騥 ¢ëà ¥­¨ï, á®áâ ¢«¥­­ë¥ ¨§ "í«¥¬¥­â à­ëå" ä㭪権. Ǒ® 㬮«ç ­¨î ãáâ ­®¢«¥­® §­ ç¥­¨¥ "false": Ǒ¥à¥¬¥­­ ï

bessel_j(1/2,x);

besselexpand:true$ bessel_j(1/2,x);

1 bessel_j( , x) 2 sqrt(2) sin(x) sqrt(%pi) sqrt(x)

ƒ ¬¬ -äã­ªæ¨ï, ¡¥â -äã­ªæ¨ï ¨ ¯á¨-äã­ªæ¨ï («®£ à¨ä¬¨ç¥áª ï ¯à®¨§¢®¤­ ï £ ¬¬ -ä㭪樨)

gamma(x) beta(x,y) psi[m℄(x) ‡¤¥áì "psi[m℄(x)" | íâ® m- ï ¯à®¨§¢®¤­ ï ä㭪樨 í⮬ á ¬  äã­ªæ¨ï (x) | íâ® "psi[0℄(x)".

(x) = 0 (x)= (x). Ǒà¨

®«ì設á⢮ áâ ­¤ àâ­ëå ᥬ¥©á⢠®à⮣®­ «ì­ëå ¯®«¨­®¬®¢ ®¯à¥¤¥«¥­ë ¢ ¯ ª¥â¥ "orthopoly". ’ ª çâ® ¯®á«¥ ¢ë§®¢  load(orthopoly)$

áâ ­ãâ ¨§¢¥áâ­ë¬¨ á«¥¤ãî騥 ä㭪樨: ä㭪樨 ‹¥ ­¤à  1-£® ¨ 2-£® த  ¨­¤¥ªá  "n" legendre_p(n,x)

legendre_q (n, x)

¯à¨á®¥¤¨­¥­­ë¥ ä㭪樨 ‹¥ ­¤à  1-£® ¨ 2-£® த  ¨­¤¥ªá®¢ "n,m" asso _legendre_p(n,m,x)

asso _legendre_q(n,m,x)

106

¯®«¨­®¬ë —¥¡ë襢  1-£® ¨ 2-£® த  ¨­¤¥ªá  "n"

hebyshev_t(n,x)

hebyshev_u(n,x)

¯®«¨­®¬ë ‹ï£¥àà  ¨­¤¥ªá  "n" ¨ ®¡®¡é¥­­ë¥ ¯®«¨­®¬ë ‹ï£¥àà  ¨­¤¥ªá®¢ "n,a" laguerre(n,x)

gen_laguerre(n,a,x) ¯®«¨­®¬ë à¬¨â  ¨­¤¥ªá  "n" hermite(n,x) ¯®«¨­®¬ë Ÿª®¡¨ ¨­¤¥ªá®¢ "n,a,b" ja obi_p(n,a,b,x) ¯®«¨­®¬ë ƒ¥£¥­¡ ãíà  ¨­¤¥ªá®¢ "n,a" ultraspheri al(n,a,x);

Šà®¬¥ ⮣®, ®¯à¥¤¥«¥­ë ¤¢¥ "á¯à ¢®ç­ë¥" ä㭪樨.





orthopoly_re ur ¯¥ç â ¥â ४ãàᨢ­ãî ä®à¬ã«ã ¤«ï ¯®«¨­®¬®¢. …¥ ¯¥à¢ë©  à£ã¬¥­â | ¨¬ï ä㭪樨, ¢ëç¨á«ïî饩 ¯®«¨­®¬ë, ¢â®à®© | ᯨ᮪, á®áâ ¢«¥­­ë© ¨§ ¨¬¥­ ¨­¤¥ªá®¢ ¨ ¨¬¥­¨ ¯¥à¥¬¥­­®©. Ǒ®à冷ª ¨¬¥­ â ª®© ¥, ª ª ¯à¨ ¢ë§®¢¥ ä㭪樨. ”ã­ªæ¨ï

orthopoly_re ur( hebyshev_t,[k,z℄); Tk+1 (z) = 2 Tk (z) z - Tk-1 (z)

orthopoly_weight ¯¥ç â ¥â ᯨ᮪, á®áâ ¢«¥­­ë© ¨§ ¢¥á , ­¨­¥© £à ­¨æë ¨­â¥à¢ «  ¨ ¢¥àå­¥© £à ­¨æë ¨­â¥à¢ « , ­  ª®â®à®¬ ®¯à¥¤¥«¥­ë ¯®«¨­®¬ë. …¥ ¯¥à¢ë©  à£ã¬¥­â | ¨¬ï ä㭪樨, ¢ëç¨á«ïî饩 ¯®«¨­®¬ë, ¢â®à®© | ᯨ᮪, á®áâ ¢«¥­­ë© ¨§ ¨¬¥­ ¨­¤¥ªá®¢ ¨ ¨¬¥­¨ ¯¥à¥¬¥­­®©. Ǒ®à冷ª ¨¬¥­ â ª®© ¥, ª ª ¯à¨ ¢ë§®¢¥ ä㭪樨. ”ã­ªæ¨ï

orthopoly_weight( hebyshev_t,[n,x℄); 1 , - 1, 1℄ [ 2 sqrt(1 - x )

‘¥¬¥©á⢮ í««¨¯â¨ç¥áª¨å ä㭪権. Ž¯à¥¤¥«¥­ë ®¡ëç­ë¥ í««¨¯â¨ç¥áª¨¥ ä㭪樨 Ÿª®¡¨: ja obi_sn(x,m)

ja obi_ n(x,m)

ja obi_dn(x,m)

ja obi_ns(x,m)

ja obi_n (x,m)

ja obi_nd(x,m)

ja obi_s (x,m)

ja obi_sd(x,m)

Šà®¬¥ ⮣®, ®¯à¥¤¥«¥­ë ® ¨§¡ëâ®ç­ë¥ ä㭪樨:

ns(x; m) = 1=sn(x; m), n (x; m) = 1= n(x; m), nd(x; m) = 1=dn(x; m)

107

s (x; m) = sn(x; m)= n(x; m), sd(x; m) = sn(x; m)=dn(x; m) ja obi_ s(x,m)

ja obi_ d(x,m)

ja obi_ds(x,m)

ja obi_d (x,m)

s(x; m) = n(x; m)=sn(x; m), d(x; m) = n(x; m)=dn(x; m) ds(x; m) = dn(x; m)=sn(x; m), d (x; m) = dn(x; m)= n(x; m)

Šà®¬¥ ⮣®, ®¯à¥¤¥«¥­ë ä㭪樨, ®¡à â­ë¥ ª® ¢á¥¬ ¯¥à¥ç¨á«¥­­ë¬ inverse_ja obi_sn(x,m) inverse_ja obi_dn(x,m) inverse_ja obi_ns(x,m) inverse_ja obi_nd(x,m) inverse_ja obi_s (x,m) inverse_ja obi_ s(x,m) inverse_ja obi_ds(x,m)

inverse_ja obi_ n(x,m) inverse_ja obi_n (x,m) inverse_ja obi_sd(x,m) inverse_ja obi_ d(x,m) inverse_ja obi_d (x,m)

Šà®¬¥ ⮣®, ®¯à¥¤¥«¥­ë í««¨¯â¨ç¥áª¨¥ ¨­â¥£à «ë ellipti _k (m)

k (m) =

Z=2

0

ellipti _e (m)

1

p

e (m) =

1 m sin2 x

dx

Z=2p

1 m sin2 x dx

0

 ª®­¥æ, ®¯à¥¤¥«¥­ë ­¥¯®«­ë¥ í««¨¯â¨ç¥áª¨¥ ä㭪樨 ellipti _f(x,m)

f ('; m) =

Z'

0

p

ellipti _e(x,m)

1

1 m sin2 x

ellipti _eu(x,m)

e('; m) =

dx

Z' p

0

1 m sin2 x dx

ellipti _pi(n,x,m)

eu(x; m) =

snZ(x;m)

p i(n; '; m) =

Z'

0

0

p p1

mt2 dt 1 t2

1

p

p

1 n sin2 x 1 m sin2 x

108

dx

29. ’à ­á«ïâ®à ¨ ª®¬¯¨«ïâ®à ¢ MAXIM'¥ Ž¯à¥¤¥«¨¢ âã ¨«¨ ¨­ãî äã­ªæ¨î, ¬®­® § ¬¥â­® ã᪮à¨âì ¥¥ ¢ë¯®«­¥­¨¥, ¥á«¨ ¥¥ ®ââ࠭᫨஢ âì ¨«¨ ®âª®¬¯¨«¨à®¢ âì. â® ¯à®¨á室¨â ¯®â®¬ã, çâ® ¥á«¨ ‚ë ­¥ ®ââ࠭᫨஢ «¨ ¨ ­¥ ®âª®¬¯¨«¨à®¢ «¨ ®¯à¥¤¥«¥­­ãî ‚ ¬¨ äã­ªæ¨î, â® ¯à¨ ª ¤®¬ ®ç¥à¥¤­®¬ ¥¥ ¢ë§®¢¥ MAXIMA ª ¤ë© à § § ­®¢® ¢ë¯®«­ï¥â ⥠¤¥©á⢨ï, ª®â®àë¥ ¢å®¤ïâ ¢ ®¯à¥¤¥«¥­¨¥ ä㭪樨, â.¥. ä ªâ¨ç¥áª¨ à §¡¨à ¥â ᮮ⢥âáâ¢ãî饥 ¢ëà ¥­¨¥ ­  ã஢­¥ ᨭ⠪á¨á  MAXIM'ë.



translate â࠭᫨àã¥â äã­ªæ¨î MAXIM'ë ­  ï§ëª LISP. ”ã­ªæ¨ï

f(x):=1+x+x^2+x^3+x^4+x^5+x^6+x^7$ translate(f); [f℄

Ǒ®á«¥ í⮣® äã­ªæ¨ï (ª ª ¯à ¢¨«®) ­ ç¨­ ¥â áç¨â âìáï ¡ëáâ॥.



ompile á­ ç «  â࠭᫨àã¥â äã­ªæ¨î MAXIM'ë ­  ï§ëª LISP,   § â¥¬ ª®¬¯¨«¨àã¥â íâã äã­ªæ¨î LISP'  ¤® ¤¢®¨ç­ëå ª®¤®¢ ¨ § £àã ¥â ¨å ¢ ¯ ¬ïâì. ”ã­ªæ¨ï

(%i9)

ompile(f); Compiling /tmp/gazonk_1636_0.lsp. End of Pass 1. End of Pass 2. OPTIMIZE levels: Safety=2, Spa e=3, Speed=3 Finished ompiling /tmp/gazonk_1636_0.lsp. (%o92) [f℄

Ǒ®á«¥ í⮣® äã­ªæ¨ï (ª ª ¯à ¢¨«®) ­ ç¨­ ¥â áç¨â âìáï ¥é¥ ¡ëáâ॥, 祬 ¯®á«¥ âà ­á«ï樨. ‘«¥¤ã¥â ¨¬¥âì ¢ ¢¨¤ã, çâ® ª ª ¯à¨ âà ­á«ï樨, â ª ¨ ¯à¨ ª®¬¯¨«ï樨 MAXIMA áâ à ¥âáï ®¯â¨¬¨§¨à®¢ âì äã­ªæ¨î ¯® ᪮à®áâ¨, ­¥ § ¡®âïáì ®¡  ªªãà â­®áâ¨. Ǒ®í⮬㠯ਠࠡ®â¥ á ¡®«ì訬¨ ¯® ®¡ê¥¬ã äã­ªæ¨ï¬¨ ¬®£ãâ ¢®§­¨ª­ãâì ç㤥á . ‚ í⮬ á«ãç ¥ á«¥¤ã¥â ®âª § âìáï ®â âà ­á«ï樨 ¨«¨ ª®¬¯¨«ï樨, «¨¡® ¯¥à¥¯¨á âì äã­ªæ¨î. ‚먣àëè ¢® ¢à¥¬¥­¨ áãé¥á⢥­­ë¬ ®¡à §®¬ § ¢¨á¨â ®â ⨯  ¬ è¨­ë, ®â ¢¨¤  ä㭪樨, ®â ⮣®, ¤¥ª« à¨à®¢ ­ «¨ ⨯ ä㭪樨 ¨ ¥¥  à£ã¬¥­â  ¯à¨ ®¯à¥¤¥«¥­¨¨ ä㭪樨, ®â ⨯   à£ã¬¥­â , á ª®â®àë¬ ¢ë§ë¢ ¥âáï äã­ªæ¨ï. …᫨ ¯à¥¤¯®« £ ¥âáï ¨á¯®«ì§®¢ âì äã­ªæ¨î ⮫쪮 ¤«ï à ¡®âë á ¤¥©á⢨⥫ì­ë¬¨ ç¨á« ¬¨ (­ ¯à¨¬¥à ¤«ï ¢ëç¨á«¥­¨ï ®¯à¥¤¥«¥­­®£® ¨­â¥£à «  á ¯®¬®éìî 109

ä㭪樨 "romberg" ¨«¨ ¯®¨áª  ª®à­ï á ¯®¬®éìî ä㭪権 "find_root" ¨ "newton"), â® ®¡ï§ â¥«ì­® á«¥¤ã¥â ¤¥ª« à¨à®¢ âì ⨯  à£ã¬¥­â  ¨ á ¬®© ä㭪樨 ª ª "float". â® ¢® ¬­®£® à § ãᨫ¨â íä䥪⠮â âà ­á«ï樨 ¨«¨ ª®¬¯¨«ï樨. „«ï ⮣®, çâ®¡ë ¤ âì ®¡é¥¥ ¯à¥¤áâ ¢«¥­¨¥ ® ¢«¨ï­¨¨ âà ­á«ï樨 ¨ ª®¬¯¨«ï樨 ­  ᪮à®áâì áç¥â  à §­ëå ⨯®¢ ä㭪権 ¨ à §­ëå ⨯®¢  à£ã¬¥­â , ¯à¨¢¥¤¥¬ â ¡«¨çªã á ¢à¥¬¥­ ¬¨ ¨á¯®«­¥­¨ï ä㭪権 ­  ®¤­®© ª®­ªà¥â­®© ¬ è¨­¥. ë«¨ ®¯à¥¤¥«¥­ë ç¥âëॠࠧ­ë¥ ä㭪樨, ¢ëç¨á«ïî騥 ®¤­® ¨ â® ¥ ¢ëà ¥­¨¥ f1(x):=1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9$ f2(x):=blo k([s℄,s:1,for i:1 thru 9 do s:s+x^i,s)$ f3(x):=blo k([℄,mode_de lare([fun tion(f),x℄,float), 1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)$ f4(x):=blo k([s℄,mode_de lare([fun tion(f),x,s℄,float), s:1, for i:1 thru 9 do s:s+x^i,s)$ ”®à¬ã § ¯¨á¨ ¤¢ãå ¯®á«¥¤­¨å ä㭪権 ("f3" ¨ "f4") á«¥¤ã¥â ¢®á¯à¨­¨¬ âì

 ªá¨®¬ â¨ç¥áª¨. „ «¥¥ ª ¤ ï ¨§ íâ¨å ä㭪権 ¢ë§ë¢ « áì á  ­ «¨â¨ç¥áª¨¬  à£ã¬¥­â®¬ "s" (ªà®¬¥ "f3" ¨ "f4", ¤«ï ª®â®àëå  ­ «¨â¨ç¥áª¨©  à£ã¬¥­â ­¥¢®§¬®¥­), 楫®ç¨á«¥­­ë¬  à£ã¬¥­â®¬ "7" ¨ ¢¥é¥á⢥­­ë¬  à£ã¬¥­â®¬ "0.3". Ǒ®á«¥ í⮣® ¢á¥ ç¥âëॠä㭪樨 ¡ë«¨ ®ââ࠭᫨஢ ­ë ([t℄) ¨ íªá¯¥à¨¬¥­â ¡ë« ¯®¢â®à¥­.  ª®­¥æ, ¢á¥ ç¥âëॠä㭪樨 ¡ë«¨ ®âª®¬¯¨«¨à®¢ ­ë ([ ℄) ¨ íªá¯¥à¨¬¥­â á­®¢  ¡ë« ¯®¢â®à¥­. ¨¥ ¯à¨¢¥¤¥­ë ᮮ⢥âáâ¢ãî騥 (ãá«®¢­ë¥) ¢à¥¬¥­  ¨á¯®«­¥­¨ï ä㭪権. f1 f2 f3 f4

s 8.75 17.73

7 4.75 12.44 5.39 13.30

0.3 4.22 12.03 4.87 12.75

s 7.32 9.11

[t℄

7 3.52 4.80 2.72 3.45

[t℄

0.3 3.04 4.09 2.17 2.91

[t℄

s 6.98 7.83

[ ℄

7 3.25 3.20 1.67 2.47

[ ℄

0.3 2.75 2.67 1.23 1.94

[ ℄

Ǒਢ¥¤¥­­ë¥ æ¨äàë ¤ îâ ¢¥á쬠 ¡®£ âë© ¬ â¥à¨ « ¤«ï  ­ «¨§ . ‚®-¯¥à¢ëå, å®à®è® § ¬¥â­®, ­ áª®«ìª® âà㤮¥¬ª®© ®ª §ë¢ ¥âáï ¯à®æ¥¤ãà  ã¯à®é¥­¨ï. ”ã­ªæ¨ï "f1" § ¤ ­  ¢ ¢¨¤¥ ®© ä®à¬ã«ë, ¨ ¥¥ ¢ëç¨á«¥­¨¥ ᢮¤¨âáï ª ®¤­®ªà â­®© ¯®¤áâ ­®¢ª¥, ¢ â® ¢à¥¬ï ª ª äã­ªæ¨ï "f2" âॡã¥â ã¯à®é¥­¨ï ­  ª ¤®¬ ®¡®à®â¥ 横« , çâ® ä â «ì­® ᪠§ë¢ ¥âáï ­  ᪮à®á⨠110

¥¥ ¢ëç¨á«¥­¨ï. ‡ â® ¨ íä䥪⠮â âà ­á«ï樨 ¨«¨ ª®¬¯¨«ï樨 ¤«ï ­¥ï¢­ëå ä㭪権 ⨯  "f2" ®ª §ë¢ ¥âáï £®à §¤® § ¬¥â­¥¥. ‚®-¢â®àëå, § ¬¥â­®, çâ® íª®­®¬¨ï ¢à¥¬¥­¨ ¤«ï "¡®«¥¥ ¯à®áâëå" ç¨á«®¢ëå  à£ã¬¥­â®¢ (¯® áà ¢­¥­¨î á ᨬ¢®«ì­ë¬¨  à£ã¬¥­â ¬¨) ®ª §ë¢ ¥âáï ­¥ â ª®© ã à ¤¨ª «ì­®©, å®âï ¨ ¤®¢®«ì­® áãé¥á⢥­­®©. â® á¢ï§ ­® á ⥬, çâ® ç¨á«  ¢á¥ à ¢­® à áᬠâਢ îâáï ª ª í«¥¬¥­â  ­ «¨â¨ç¥áª®£® ¢ëà ¥­¨ï, å®âï ã¯à®é¥­¨¥  ­ «¨â¨ç¥áª®£® ¢ëà ¥­¨ï, á®áâ ¢«¥­­®£® ¨áª«îç¨â¥«ì­® ¨§ ç¨á¥«, ¨¤¥â ¡ëáâ॥. ‚-âà¥âì¨å, § ¬¥â­®, çâ® ¤«ï ä㭪権, ª®â®àë¥ ¤¥ª« à¨à®¢ ­ë ª ª ¢¥é¥á⢥­­ë¥ ä㭪樨 ®â ¢¥é¥á⢥­­ëå  à£ã¬¥­â®¢, ¢ëç¨á«¥­¨¥ ®â 楫®ç¨á«¥­­®£®  à£ã¬¥­â  ¨¤¥â ¬¥¤«¥­­¥¥, 祬 ®â ¢¥é¥á⢥­­®£®  à£ã¬¥­â . ‚-ç¥â¢¥àâëå, ®ç¥­ì å®à®è® ¢¨¤­®, çâ® ¥á«¨ ‚ë ­¥ ¢ë¯®«­ï¥â¥ ª®¬¯¨«ïæ¨î ¤«ï ¢¥é¥á⢥­­®© ä㭪樨, â® ‚ë à¨áªã¥â¥ § ¬¥¤«¨âì ¥¥ ¢ëç¨á«¥­¨¥ ¡®«¥¥ 祬 ¢ 6 à §.

111

‘®¤¥à ­¨¥ 1. Ž¡é¨¥ ᢥ¤¥­¨ï ® MAXIM'¥ .......................................................................... 1 2. Ǒ¥à¢®­ ç «ì­ë¥ ᢥ¤¥­¨ï ® à ¡®â¥ á MAXIM'®© ......................................... 8 3. ”㭪樨 ¢ë¢®¤  ­  íªà ­ ............................................................................. 12 4.  ¡®â  á ä ©« ¬¨ ......................................................................................... 16 5. Ǒ८¡à §®¢ ­¨ï  ­ «¨â¨ç¥áª¨å ¢ëà ¥­¨© ®¡é¥£® ¢¨¤  ......................... 18 6. Ǒ८¡à §®¢ ­¨¥ à æ¨®­ «ì­ëå ¢ëà ¥­¨© ............................................... 24 7. Ǒ८¡à §®¢ ­¨¥ âਣ®­®¬¥âà¨ç¥áª¨å ¢ëà ¥­¨© ..................................... 28 8. Ǒ८¡à §®¢ ­¨¥ ¢ëà ¥­¨© á® á⥯¥­ï¬¨ ¨ «®£ à¨ä¬ ¬¨ ....................... 30 9. ‹®£¨ç¥áª¨¥ ¢ëà ¥­¨ï ¨ ¡ §  ¤ ­­ëå ........................................................ 31 10. “á«®¢­ë¥ ¢ëà ¥­¨ï ¨ 横«ë ................................................................... 39 11. «®ª¨ ........................................................................................................... 41 12. ‘¯¨áª¨ .......................................................................................................... 44 13. Œ áᨢë........................................................................................................ 49 14. Œ âà¨æë....................................................................................................... 54 15. ‚­ãâ७­¥¥ ¯à¥¤áâ ¢«¥­¨¥ ¢ MAXIM'¥ ...................................................... 61 16. ‘¨­â ªá¨ç¥áª¨¥ ¯®¤áâ ­®¢ª¨ ..................................................................... 66 17. €«£¥¡à ¨ç¥áª¨¥ ¯®¤áâ ­®¢ª¨....................................................................... 68 18. Ǒ®¤áâ ­®¢ª¨ ¯® è ¡«®­ã ............................................................................. 69 19.  ¡®â  á oat-ç¨á« ¬¨ ............................................................................... 76 20. Š®¬¯«¥ªá­ë¥ ç¨á«  ¨ ¢ëà ¥­¨ï ............................................................... 82 21. „¨ää¥à¥­æ¨à®¢ ­¨¥ .................................................................................... 84 22. Ǒ।¥«ë ....................................................................................................... 87 23. ˆ­â¥£à¨à®¢ ­¨¥ ........................................................................................... 89 24. ï¤ë, ¯ ¤¥- ¯¯à®ªá¨¬ æ¨ï ¨ 楯­ë¥ ¤à®¡¨ ............................................ 92 25. Ǒ८¡à §®¢ ­¨¥ ‹ ¯« á  ¨ ¢ëç¥âë ............................................................ 95 26. “à ¢­¥­¨ï..................................................................................................... 96 27. „¨ää¥à¥­æ¨ «ì­ë¥ ãà ¢­¥­¨ï...................................................................101 28. ‘¯¥æ¨ «ì­ë¥ ä㭪樨 ...............................................................................106 29. ’à ­á«ïâ®à ¨ ª®¬¯¨«ïâ®à ¢ MAXIM'¥ .....................................................109

112