Latex Equation Compiler Code

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ÿÿÿÿïheader=pstricks.proïheader=pst-algparser.proïheader=pst-dots.pro ¹ðü ý;9‘ègiïcolor push gray 0’þñï	color popŽŽ ¦Æs‘ègi ýr9ïcolor push gray 0Ÿ<~ˆ’’ßNïcolor push gray 0ï	color popŽŽ’˜¿JóÆîê½q½qecrm2074»T‘þfKour–d—of“EQC“functionalit‘ÿwmyŽŸ(žæ’Ö±¸ó& Šffffecrm1440¼Jan‘‘Rheinl€änderŽŽŽŽŽŸ!\’ØP6Janžnuary–‘9,“2013ŽŸ:4ós©^dG®G®ecbx1728ÆCon–ÿ{kten“tsŽŸaeó¥!¢Necbx1200Ç1Ž‘žæNumeric–¸ev‘ÿ@alution“of“equations’
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sD"4ŽŽ¡4Ž‘žæAsking–¸for“v›ÿ@alues“of“v˜ariables’
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n%Ü7ŽŽ¤€‘žæóÓ·åecrm1200º5.1Ž‘,£žExample:–êlSolving“a“quadratic“equation‘™l‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘ëÍïcolor push gray 07Ž‘Ëoï	color popŽ¡‘žæ5.2Ž‘,£žExample:–êlFinding“the“extrema“of“a“function‘Ë̍‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘ëÍïcolor push gray 09Ž‘Ëoï	color popŽ¡‘žæ5.3Ž‘,£žLibrary–êlof“substititions‘+‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 010Ž‘Ëoï	color popŽ¤?DÇ6Ž‘žæMatrices’
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Ç'i14ŽŽ¤€‘žæº9.1Ž‘,£žUnexpSˆected‘êlnewlines‘j1‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 014Ž‘Ëoï	color popŽ¡‘žæ9.2Ž‘,£žMultiple–êlpSˆossible“v‘ÿXíalues“for“equations‘э‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 015Ž‘Ëoï	color popŽ¡‘žæ9.3Ž‘,£žMultiple‘êlsubstitutions‘¬~‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 016Ž‘Ëoï	color popŽ¡‘žæ9.4Ž‘,£žOutput–êlof“n•¬wum“bSˆers‘õA‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 016Ž‘Ëoï	color popŽ¡‘žæ9.5Ž‘,£žLibrary‘êlequations‘]Œ‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 016Ž‘Ëoï	color popŽ¡‘žæ9.6Ž‘,£žPstric•¬wks‘êlheadac“hes‘šo‘ÿý.ŽŽ–	CS‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ“‘ÿý.ŽŽ‘+ïcolor push gray 017Ž‘Ëoï	color popŽŽŸ‘ègiïcolor push gray 0’
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Ž‘UR¹10Ÿû¥2È5Ž‘Àº).–êlThis“options“givš¬we“the“smallest“n˜um˜bSˆer“that“is“still“prin˜ted“normally‘ÿe.ŽŸz‡‘â‡mïcolor push gray 0‘ßüÇhighsclimit–¸=“realï	color popŽŽ‘XPgºThis–êloptions“givš¬we“the“largest“n˜um˜bSˆer“that“is“still“prin˜ted“normally‘ÿe.Ž¦‘â‡mïcolor push gray 0‘ßüÇeqalign–¸=“none|eqnarra y|amsï	color popŽŽ’™‘¥ºThis–Õ8option“conš¬wtrols“equation‘Õ9formatting.‘ÖàÖnone“ºprin˜ts“no“ampSˆersands,Ž¡‘ÇUÖeqnarray–ž¨ºprin¬wts“ampšSˆersands“on“b˜oth“sides“of“the“equals“sign“(Ö&=&º),‘ž»and“Öams“ºprin¬wts“an“amp˜ersandŽ¡‘ÇUbšSˆefore–‹>the‘‹=equals“sign“(Ö&=º).‘Œ4By“default,‘‹gEQC‘‹automatically“adds“amp˜ersands“to‘‹=y¬wour“equationsŽ¡‘ÇUdepšSˆending–êlon“the“en•¬wvironmen“t–êlthey“app˜ear“in.ŽŸz‡‘â‡mïcolor push gray 0‘ßüÇeqc hain–¸=“true|falseï	color popŽŽ‘dpºOmits–Ó\the“left“hand“side›Ó[of“an“equation“if“it“is“iden¬wtical“to˜the“left“hand“side“ofŽ¡‘ÇUthe–êlequation“directly“preceding“it“in“an“Öeqnarray“ºor“AMS“en•¬wvironmen“t.Ž¦‘â‡mïcolor push gray 0‘ßüÇeqra w–¸=“true|falseï	color popŽŽ‘Z••ºPrinš¬wt–êlequations“as“y˜ou“t˜ypšSˆed“them“(if“p˜ossible),“or“with“EQC“formatting.ŽŸgb‘ègiThe›êlk•¬weyw“ord˜ÍnÇeqcoptions˜ºcan˜bSˆe˜used˜to˜set˜these˜options˜globally‘ÿe.ŽŸ-	,‘ègiÆ2Ž‘^…W‘þrDorking–6êwith“ph›ÿ{kysical“quan˜titiesŽŸY0‘ègiºEQC‘üÓadds›üÙthe–üØclass“Unit“to“the˜functionalitš¬wy“of“the“GiNaC‘üÓlibrary‘ÿe.‘üõThis“means“that“y˜ou‘üÙcan“use“unitsŽ¡‘ègiin–êla“natural“w•¬wa“y–êlinside“equations:ŽŸ a鍍‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\clearequationsŽ¡‘ègi\eqcoptions{units–,j=“{\mm}}Ž¡‘ègi$$\eq{x–,j=“\unit{3}{\mm}}$$Ž¡‘ègi$$\eq{y–,j=“\unit{4}{\mm}}$$Ž¡‘ègi$$\eq{z–,j=“x“+“y}“=“\quantity{z}$$ŽŸ aꡍ’ÖÃØÊx–UR¹=“3“mmŽŽ¤÷ˍ’×Êy‘˹=–UR4“mmŽŽ¡’ÀŒÍÊz‘Þ5¹=›URÊx–ª¨¹+“Êy‘˹=˜7‘UPmmŽŽŸ‘ègiºUpSˆon–³	startup,‘³}EQC›²“kno¬wws“only“the‘³base“SI˜units.‘µÆT‘ÿeo›³get“full“suppSˆort˜for“all“units“dened˜in“theŽ¤€‘ègiSIunits.st•¬wy›¼Cpac“k‘ÿXíage,‘¼ºinclude˜units.tex‘¼Dat˜the˜bSˆeginning˜of˜y“our˜doSˆcumen“t.‘¿New‘¼Dunits˜can˜easily˜bSˆeŽ¡‘ègidened–êlusing“the“ÍnÇdefunit“ºk•¬weyw“ord.–êlF‘ÿeor“example:ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹¶ ¹ðü ý;9‘ègiïcolor push gray 0Ø3‘¿DUSER–êlDEFINED“FUNCTIONS’
S׺4Ž’þñï	color popŽŽ ¦Æs ý~9‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\defunit['']{\inch}{2.54\mm}Ž¤€‘ègi\eqcoptions{units–,j=“{\inch}}Ž¡‘ègi$$z–,j=“x“+“y“=“\quantity{z}$$Ž© €¡’½WxÊz‘Þ5¹=›URÊx–ª¨¹+“Êy‘˹=˜2Ê:¹756‘UPŸû™ó#¾KÈcmsy8Î00ŽŽŽŸ‘ègiºThe–­option“Öunits›­ºtells“EQC‘­what“units“it“should“use˜in“the“output.‘­pIf“no˜units“are“giv¬wen,‘­"EQC‘­will“useŽ¡‘ègithe–DäSI‘DÍbase›Dåunits.‘EoThe“units“can˜also“bSˆe“giv¬wen“as˜an“optional“argumen¬wt˜to“the“ÍnÇv‘ÿ@al“ºk•¬weyw“ord˜(and‘DäwillŽ¡‘ègionly–êlapply“to“that“spSˆecic“v‘ÿXíalue):Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\defunit['']{\inch}{2.54\mm}Ž¡‘ègi\eqcoptions{units–,j=“{\inch}}Ž¡‘ègi$$z–,j=“x“+“y“=“\quantity{z}“=“\quantity[units“=“{\mm}]{z}$$Ž¦¡’¦Êz‘Þ5¹=›URÊx–ª¨¹+“Êy‘˹=˜2Ê:¹756‘UPŸû™Î00ŽŽ‘+ßâ¹=˜7‘UPmmŽŽŸ‘ègiºA–êlshort“form“also“exists,“with“the“syn¬wtax“Ö\quantity[\mm]{z}º.Ž©€‘ègiNote–êlthat“there“are“four“dierenš¬wt“k˜eyw˜ords“that“can“bSˆe“used“for“nding“the“v›ÿXíalue“of“a“v˜ariable:ŽŸ€‘â‡mïcolor push gray 0‘ßüÇn• um“v‘ÿ@alï	color popŽŽ‘tºThis–êlprinš¬wts“a“n˜umeric“v‘ÿXíalue,“or“prin˜ts“an“error“if“the“v‘ÿXíariable“doSˆes“not“ha˜v˜e“one.Ž¤€‘â‡mïcolor push gray 0‘ßüÇunitsï	color popŽŽ‘˜ÃºPrin¬wts–êlonly“the“units“of“the“v‘ÿXíariable.Ž¡‘â‡mïcolor push gray 0‘ßüÇquan• tit“yï	color popŽŽ‘”PºPrinš¬wts–êla“n˜umeric“v‘ÿXíalue“plus“units“(if“there“are“an˜y).Ž¡‘â‡mïcolor push gray 0‘ßüÇv‘ÿ@alï	color popŽŽ‘þöZºPrin•¬wts›êlan“y˜kind˜of˜sym“bSˆolic˜expressionŽŸ-4‘ègiÆ3Ž‘^…User–6êdened“functionsŽŸae‘ègiºEQC‘qnextends–qGiNaC‘qmb¬wy›q‘oering“user-dened“functions˜whic¬wh“can“bSˆe“created˜at“run¬wtime“in˜the“LatexŽ¤€‘ègile.›ÎA‘Vfunction–Œis“declared‘‹using“the“ÍnÇfunction“ºk•¬weyw“ord.˜A‘Vdenition–Œfor›‹the“function“ma¬wy“bSˆe˜giv¬wen“withŽ¡‘ègiÍnÇdeuncº,››Cbut–›/this“is“not“strictly“necessary‘ÿe.‘›©Of“course,˜only“functions“with“denitions“can“bSˆe“ev‘ÿXíaluated!Ž¦‘ègiF‘ÿeunctions–L*can“bSˆe‘L+givš¬wen“hin˜ts‘FÅwhen“they›L+are“dened“using“the˜synš¬wtax“Ö\function[hintlist]...º.‘MHin˜tsŽ¡‘ègienable––öEQC‘–áto“handle‘–÷and“prinš¬wt“functions“bSˆetter.‘—vF‘ÿeor“example,‘—the“hin˜t“Ötrig“ºwill“result‘–÷in“the“functionŽ¡‘ègibSˆeing–êlprin¬wted“as“¹sinŽ‘WŠŸû3ÎÈ2Ž‘ŒÊx“ºinstead“of“¹(sinŽ‘m(Êx¹))Ÿû¥2È2Ž‘Àº.Ž¦‘ègiConsider–êlthe“functionŽ©€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\function{f}{x}Ž¡‘ègi$$\deffunc{f}{ax^2–,j+“bx“+“c}$$Ž¦’ÅTÊf‘Q¹=‘URÊaxŸû™È2Ž‘j¬¹+–ª¨Êbx“¹+“ÊcŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹) ¹ðü ý;9‘ègiïcolor push gray 0Ø4‘¿DASKING–êlF¬wOR“V›þ±ÜALUES“OF“V˜ARIABLES’

÷º5Ž’þñï	color popŽŽ ¦Æs ý~9‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖWith–,j$\eq{a“=“4}$,“$\eq{b“=“-2}$“and“$\eq{c“=“7}$“we“can“compute“$f(x)$“for“$x“=“3$:Ž¤€‘ègi$$f(x)–,j=“\val{f(x)}$$Ž¡‘ègi$$f(3)–,j=“\val{f(3)}“=“\numval{\val{f(3)}}$$ŽŸ‘ègiºWith›êlÊa–UR¹=“4º,˜Êb“¹=“͹2˜ºand˜Êc“¹=“7˜ºw¬we˜can˜compute˜Êf‘
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ÿþÊb“¹+“9˜Êa–UR¹=“37ŽŸS¨‘ègiºNote–Ú©what›Ú¨eect“the“ÍnÇv‘ÿ@al˜ºstatemen¬wt“has“on˜the“dieren¬wt“parameters.‘ÚÁT‘ÿeo˜bSˆoth“expand“the˜function“andŽ¤€‘ègisubstitute–êlthe“v›ÿXíalues“of“kno¬wwn“v˜ariables,“a“double“ÍnÇv‘ÿ@al“ºstatemen¬wt“is“necessary‘ÿe.ŽŸ,,„‘ègiÆ4Ž‘^…Asking–6êfor“v›þöØalues“of“v˜ariablesŽŸµ
‘ègiºMost–a‚došSˆcumen¬wts‘awill“b˜e“of‘athe“t¬wyp˜e“that‘adene“a“n•¬wum“b˜er›aof–a‚equations“and˜then“try“to˜calculate“n¬wumericŽ¡‘ègiv›ÿXíalues–êlfor“the“v˜ariables“used“in“these“equations.“There“are“sev¬weral“methoSˆds“of“doing“this“ecienctly:ŽŸzù‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºThe›¹òmost‘¹ñstraigh•¬wt-forw“ard˜w“a“y–¹ñis˜to“dene˜an“equation˜and“then˜other“equations˜that“giv¬we˜theŽ¡‘ÇUkno¬wwn–&v›ÿXíalues‘'of“v˜ariables.‘jThe“ÍnÇv‘ÿ@al“ºstatemen¬wt›'can“then“bSˆe˜used“to˜calculated“the“v‘ÿXíalues˜of“unk•¬wo“wnŽ¡‘ÇUv‘ÿXíariables.Ž©íލçYïcolor push gray 0ï	color popŽŽ‘ÇUÖ\begin{eqnarray*}Ž¡‘xý\eq{x–,j=“3y“+“4}\\Ž¡‘xý\eq{y–,j=“5}\\Ž¡‘xýx–,j&=&“\numval{x}Ž¡‘ )\end{eqnarray*}ŽŸ&[¼’Üý“ÊxŽŽ’íªå¹=ŽŽ’
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Îm25ŽŽŽ¡¦‘ÇUºThe›{only‘zdra•¬wwbac“k˜is˜that˜y“ou–zcannot˜easily˜substitute“the˜denition˜Êx‘ƹ=‘Å3Êy‘6C¹+‘È
4“ºin¬wto˜otherŽ¤€‘ÇUequations.–êlY‘ÿeou“wš¬would“ha˜v˜e“to“do“something“lik˜e“Ö\eqsubst{z–,j=“3x}{x“=“\val{x}}º.ŽŸ€‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºThe–êllast“w•¬wa“y–êlis“to“use“the“Öwithº-form“of“the“ÍnÇv‘ÿ@al“ºstatemen¬wt.ŽŸ€çYïcolor push gray 0ï	color popŽŽ‘ÇUÖ\clearequations%Ž¡‘ÇU\begin{eqnarray*}Ž¡‘ )\eq{x–,j=“3y“+“4}\\Ž¡‘ )x–,j&=&“\numvalwith{x}{y“=“5}\\Ž¡‘ )x–,j&=&“\numvalwith{x}{y“=“7}\\Ž¡‘ÇU\end{eqnarray*}ŽŸ)€’Üý“ÊxŽŽ’íªå¹=ŽŽ’
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Îm25ŽŽŽ¡¦‘ÇUºAll–Èthe“assignmenš¬wts“giv˜en“as‘Éthe“second“argumen˜t“are“dened“as“equations,‘the“searc˜h“for“theŽ¤€‘ÇUv‘ÿXíalue–is“pšSˆerformed,‘$and“then“the“temp˜orary“equations‘are“deleted“again.‘eTherefore,‘$y¬wou“can“ndŽ¡‘ÇUev¬wen–êlv›ÿXíalues“of“v˜ariables“that“are“indirectly“dened:Ž©€çYïcolor push gray 0ï	color popŽŽ‘ÇUÖ\clearequationsŽ¡‘ÇU\begin{eqnarray*}Ž¡‘ )\eq{x–,j=“3y“+“4}\\Ž¡‘ )\eq{z–,j=“4x}\\Ž¡‘ )z–,j&=&“\numvalwith{z}{y“=“5}\\Ž¡‘ÇU\end{eqnarray*}Ž¦Ÿ€’Üý“ÊxŽŽ’íªå¹=ŽŽ’
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Ÿüfg’§ÜóX«Qffcmr12½arctanŽ’Ï(¾x½)ŽŽŽŽŽ’Äï ŸŽ0Fò"  tx@Dict begin STP newpath 0.8 SLW 0  setgray  /ArrowA { moveto } def /ArrowB { } def  [ 227.62204 78.75346 216.24023 77.27191 204.86017 75.66336 193.47838 73.91154 182.09831 71.9991 170.71652 69.90756 159.33473 67.61414 147.95467 65.09279 136.57288 62.31528 125.19281 59.25124 113.81102 55.867 102.42921 52.12675 91.04915 47.99577 79.66736 43.44258 68.2873 38.44112 56.9055 32.98055 45.52371 27.06627 34.14365 20.73196 22.76186 14.04164 11.38179 7.08974 0.0 0.0 -11.38179 -7.08974 -22.76186 -14.04164 -34.14365 -20.73196 -45.52371 -27.06627 -56.9055 -32.98055 -68.2873 -38.44112 -79.66736 -43.44258 -91.04915 -47.99577 -102.42921 -52.12675 -113.81102 -55.867   1. 0.1 0.  /c ED /b ED /a ED false OpenCurve  gsave 0.8 SLW 0  setgray  1. .setopacityalpha  0  setlinecap stroke  grestore end ò."  tx@Dict begin STP newpath 0.8 SLW 1 0 0  setrgbcolor  /ArrowA { moveto } def /ArrowB { } def  [ 227.62204 67.60002 216.24023 68.43361 204.86017 68.81783 193.47838 68.7538 182.09831 68.24149 170.71652 67.27983 159.33473 65.86992 147.95467 64.01173 136.57288 61.70529 125.19281 58.9495 113.81102 55.74544 102.42921 52.09311 91.04915 47.99144 79.66736 43.44258 68.2873 38.44438 56.9055 32.99683 45.52371 27.1021 34.14365 20.75801 22.76186 13.96565 11.38179 6.72397 0.0 -0.9649 -11.38179 -9.10313 -22.76186 -17.68962 -34.14365 -26.72546 -45.52371 -36.20956 -56.9055 -46.14194 -68.2873 -56.52257 -79.66736 -67.35255 -91.04915 -78.62973 -102.42921 -90.35623 -113.81102 -102.5321   1. 0.1 0.  /c ED /b ED /a ED false OpenCurve  gsave 0.8 SLW 1 0 0  setrgbcolor  1. .setopacityalpha  0  setlinecap stroke  grestore end ŽŽŽŸ%€‘_î]ïcolor push gray 0ºFigure–êl1:“Comparison“of“¹arctanŽ‘&âÊx“ºwith“its“T‘ÿea¬wylor“series“expansionï	color popŽŽŽï	color popŽŸŽm‘ègiThe–êlargumen¬wts“to“ÍnÇmmak epspicture“ºare:Ž¦‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºNegativš¬we–êlÊxº-axis“v‘ÿXíalue,“then“negativ˜e“Êyn9º-axis“v‘ÿXíalueŽ¤\v‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºP•¬wositiv“e–êlÊxº-axis“v›ÿXíalue,“then“pSˆositiv¬we“Êyn9º-axis“v˜alueŽ¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUÊxº-axis–êlstep“v›ÿXíalue,“then“Êyn9º-axis“step“v˜alue“(for“the“axis“labSˆels)ŽŸ\w‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUÊx–êlºand“Êy‘X¥ºaxis“namesŽ¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºAn–êlarra¬wy“of“¹(Êx¹;‘
ÿþÊyn9¹)“ºv‘ÿXíalues“to“plotŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹Šÿ ¹ðü ý;9‘ègiïcolor push gray 0Ø8‘¿DUTILITY‘êlFUNCTIONS’
u\åº14Ž’þñï	color popŽŽ ¦Æs ý~9‘ègiThe–9›last“is“generated‘9œbš¬wy“the“k˜eyw˜ord“ÍnÇprin• tv“ectorº,‘9Éwhic˜h–9›con˜v˜erts“the“output“of‘9œÍnÇeqev‘ÿ@al“ºin˜to“a“formatŽ¤€‘ègirecognized–!»bš¬wy“pstric˜ks.‘#˜ÍnÇeqev‘ÿ@al“ºev›ÿXíaluates“an“equation‘!ºfor“all“the“v˜alues“of‘!ºthe“indepSˆenden¬wt“v˜ariableŽ¡‘ègigiv¬wen–ªZas›ªYthe“second˜parameter.‘ª¼In˜the“example,‘ªj͹1–UR:“2“:“0Ê:¹1˜ºcreates–ªZa˜v¬wector“of˜n•¬wum“bSˆers–ªZfrom˜Í¹1“ºto˜¹2“ºinŽ¡‘ègiincremen¬wts›¥"of–¥!¹0Ê:¹1º.‘¥ŒThe“result˜is˜a“matrix˜with“the˜v‘ÿXíalues˜of“the˜indepSˆenden¬wt˜v‘ÿXíariable“in˜the“rst˜columnŽ¡‘ègiand–êlthe“ev‘ÿXíaluation“results“in“the“second“column:Ž¤<‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\eq{r–,j=“\eqeval{"eq:series"}{x“=“-1:2:0.5}}ŽŸE&•’­jÊr‘¨à¹=ŸËY‘URÐ0ŽŸǍ‘URBŽ¤38‘URBŽ¡‘URBŽ¡‘URBŽ¡‘URBŽ¡‘URBŽ¡‘URBŽ¡‘URBŽŸ®‘UR@ŽŽŸÔff‘g͹1Ž‘;+͹1Ê:¹441ŽŽ¤ÿ‘ÕT͹0Ê:¹5Ž‘8-͹0Ê:¹6487ŽŽ¡‘Ä0Ž‘5./͹0Ê:¹01357ŽŽ¡‘€0Ê:¹5Ž‘<ÈÙ0Ê:¹4639ŽŽ¡‘Ä1Ž‘<ÈÙ0Ê:¹7837ŽŽ¡‘€1Ê:¹5Ž‘<ÈÙ0Ê:¹9458ŽŽ¡‘Ä2Ž‘<ÈÙ0Ê:¹9504ŽŽŽŸËY‘júÐ1ŽŸǍ‘júCŽ¤38‘júCŽ¡‘júCŽ¡‘júCŽ¡‘júCŽ¡‘júCŽ¡‘júCŽ¡‘júCŽŸ®‘júAŽŽŽŽŸC·‘ègiºThere–êlis“also“a“simplied“vš¬wersion“ÍnÇmak epspicture“ºfor“dra˜wing“a“graph“in“the“rst“quadran˜t“only:Ž¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºLoSˆcation–êlof“origin“in“the“format“(x0,“y0)Ž¤Æ>‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºLoSˆcation–êlof“lo•¬ww“er–êlleft-hand“corner“in“the“format“(x1,“y1)Ž©Æ?‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºP•¬wositiv“e–êlÊxº-axis“v›ÿXíalue,“then“pSˆositiv¬we“Êyn9º-axis“v˜alueŽ¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUÊxº-axis–êlstep“v›ÿXíalue,“then“Êyn9º-axis“step“v˜alue“(for“the“axis“labSˆels)Ž¦‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUÊx–êlºand“Êy‘X¥ºaxis“namesŽ¡‘ègiïcolor push gray 0‘ïÍŽ‘ðï	color popŽŽ‘ÇUºAn–êlarra¬wy“of“¹(Êx¹;‘
ÿþÊyn9¹)“ºv‘ÿXíalues“to“plotŽŸ<‘ègiOf–0dcourse,‘0¶if“ÍnÇmakš epspicture‘0cºand“ÍnÇmmak˜epspicture›0cºdo“not“suit˜yš¬wour“purpSˆoses,‘0¶y˜ou“can‘0cuse“theŽ¤€‘ègistandard–êlpstricš¬wks“macros“instead“or“create“y˜our“o˜wn“shortcuts.ŽŸ+ž‘ègiÆ8Ž‘^…Utilit‘ÿ{ky‘6êfunctionsŽŸJ͑ègiºIf›KVy•¬wou‘KWw“an“t˜to˜mak“e–KWa˜fresh˜start˜in“the˜middle˜of“a˜doSˆcumen¬wt,‘Kuse˜ÍnÇclearequationsº.‘LJThis˜will“delete˜allŽ¡‘ègipreviously–‰Jdened“equations“except“library“equations,‘‰cbut‘‰Kkš¬weep“the“constan˜ts“and“an˜y“functions“denedŽ¡‘ègiwith–êlthe“hin¬wt“Ölibº.ŽŸ+Ÿ‘ègiÆ9Ž‘^…PitfallsŽŸ!ʍ‘ègiÙ9.1Ž‘y¯Unexppœected‘G\newlinesŽŸ(⍑ègiºIf–—€yš¬wou“use“EQC‘—Scommands“lik˜e›—Ö\eqmul*“ºthat“proSˆduce“no“Latex“output˜inside“an“equationarraš¬wy“en˜vi-Ž¡‘ègironmen•¬wt,‘‰/follo“w–‰them“bš¬wy“a“commen˜t‘‰to“a˜v˜oid“insertion“of“a‘‰newline“in“the“output“le.‘‰úEquationarra˜yŽ¡‘ègien•¬wvironmen“ts›êlma“y˜not˜con“tain˜newlines.˜F‘ÿeor˜example:ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹¬· ¹ðü ý;9‘ègiïcolor push gray 0Ø9‘¿DPITF‘þ±ÜALLS’
¹Šfº15Ž’þñï	color popŽŽ ¦Æs ý~9‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{eqnarray}Ž¤€‘ègi\eq{x–,j&“=“&“a“+“b}Ž¡‘ègi\eqsub*{"prev"}{b}Ž¡‘ègi\eqrev{"prev"}Ž¡‘ègi\end{eqnarray}ŽŸ±Ž‘ègiºwill–êlyield“the“follo¬wwing“output:Ž©±‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{eqnarray}Ž¡‘ègix&=&a+bŽ¡¡‘ègia&=&x-bŽ¡‘ègi\end{eqnarray}ŽŸ$ì荍‘ègiÙ9.2Ž‘y¯Multiple–G\ppœossible“v‘ÿÆalues“for“equationsŽŸ|‘ègiºConsider–êlthe“follo¬wwing“equation:ŽŸ^ø¡’Êp‚Êu–UR¹=“Êx–ª¨¹+“Êy‘n7¹sinŽ‘ÛSÊ'ŽŸ½ð‘ègiºNo•¬ww›êlw“e˜w“an“t˜to˜nd˜the˜v‘ÿXíalue˜ÊxŸ
ÌÌÈ1Ž‘ªpºwhic“h˜will˜mak“e˜Êu˜ºbSˆecome˜zero.Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{equation}Ž¡‘ègi\eqsubst[eq:usubst]{"prev"}{x–,j=“x_1}“=“0Ž¡‘ègi\end{equation}Ž©±Ž¡’¼9mÊu–UR¹=“ÊxŸ
ÌÌÈ1Ž‘j¬¹+‘ª¨Êy‘n5¹sinŽ‘ÛQÊ'“¹=“0Ž’
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ÌÌÈ1Ž‘ªpºto“bSˆe:Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\eqsubst*{"prev"}{u–,j=“0}Ž¡‘ègi\eqsub*{"prev"}{\sin\varphi‘,jy}Ž¡‘ègi$$\eqrev{"prev"}$$ŽŸ±¡’Ìú?ÊxŸ
ÌÌÈ1Ž‘V¹=‘URÍÊy‘n5¹sinŽ‘ÛQÊ'Ž©½ð‘ègiºBut–êlwhat“happSˆens“if“wš¬we“no˜w“ask“for“the“v‘ÿXíalue“of“Êuº?ŽŸ^ø‘ègiÖu–,j=“\val{u}Ž¡’ǁÊu–UR¹=“ÊxŸ
ÌÌÈ1Ž‘j¬¹+‘ª¨Êy‘n5¹sinŽ‘ÛQÊ'Ž¦‘ègiºEQC‘•prin•¬wts‘•Ca›•Bw“arning˜that˜there‘•Care˜m“ultiple˜pSˆossible˜v‘ÿXíalues‘•Cfor˜Êu˜º(i.e.,‘•nÊx–ÿ¹+‘Êy‘n7¹sinŽ‘ÛSÊ'˜ºand˜ÊxŸ
ÌÌÈ1Ž‘ß¹+“Êy‘n7¹sinŽ‘ÛSÊ'º)Ž¡‘ègiand–È~then›È}c¬whoSˆoses“the“last˜v‘ÿXíalue“(whic•¬wh˜migh“t–È~or“migh¬wt˜not“bSˆe“what˜wš¬we“w˜an˜ted).‘ȱThere“are“t˜w˜o‘È}w˜a˜ys“toŽ¡‘ègia•¬wv“oid‘êlthis:ŽŸ鍍‘ègiïcolor push gray 0‘\õ1.Ž‘ðï	color popŽŽ‘ÇUDo–êlbSˆoth“substitutions“at“the“same“time:“Ö\eqsubst{"prev"}{x–,j=“x_1;“u“=“0}º.ŽŸ¥‘ègiïcolor push gray 0‘\õ2.Ž‘ðï	color popŽŽ‘ÇUIn¬wtrošSˆduce–êla“temp˜orary“for“Êuº:“Ö\eqsubst{"prev"}{x–,j=“x_1;“u“=“u_{temp}}º.ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹¸÷ ¹ðü ý;9‘ègiïcolor push gray 0Ø9‘¿DPITF‘þ±ÜALLS’
¹Šfº16Ž’þñï	color popŽŽ ¦Æs ý~9‘ègiÙ9.3Ž‘y¯Multiple‘G\substitutionsŽŸ?„‘ègiºIf–Ûyš¬wou‘Üdo“m˜ultiple›Üsubstitutions“at˜the“same˜time,‘åthe“order˜in“whic¬wh˜the“substitutions˜are“done˜is“notŽ¤€‘ègidened.‘•By›”using–”the“k•¬weyw“ord–”ÍnÇeqsubstcº,‘”>substitutions“are“done“consecutiv¬wely“in“the˜order“they“areŽ¡‘ègilisted.–êlF‘ÿeor“example,“compare:Ž©€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ$$\eqsubst{x–,j=“3y}{y“=“4z;“z“=“3}$$Ž¡‘ègi$$\eqsubstc{x–,j=“3y}{y“=“4z;“z“=“3}$$Ž¦’ÛJ
Êx–UR¹=“12‘
ÿþÊzŽ¤’ßI'x–UR¹=“36Ž¡‘ègiºOnly–êluse“ÍnÇeqsubstc“ºwhen“order“is“impšSˆortan¬wt“b˜ecause“it“is“less“ecien¬wt“than“ÍnÇeqsubstº.Ž©'ɞ‘ègiÙ9.4Ž‘y¯Output–G\of“n•cum“bpœersŽŸ?„‘ègiºNot–Қall›ҙcom¬wbinations“of˜Ö\precisionº,‘ÓÖ\precision_type“ºand˜Ö\scientific_limits“ºmak¬we˜sense.‘ՇF‘ÿeorŽ¤€‘ègiexample,‘ú'y•¬wou›ùámigh“t˜set˜the‘ùâlo“w“er˜scien“tic˜limit–ùâto˜¹0Ê:¹001˜ºso˜that“n•¬wum“bSˆers˜lik“e˜¹0Ê:¹005˜ºwill‘ùâbSˆe˜prin“tedŽ¡‘ègiwithout–	ïexpSˆonenš¬wt.‘FIf“y˜ou‘	îno˜w“set“Öprecision_type“ºto“Öfixed_marker‘	îºand“precision“to“¹2º,‘
(n˜um˜bSˆers“smallerŽ¡‘ègithan–êl¹0Ê:¹01“ºwill“bSˆe“prin¬wted“as“zero!Ž¦‘ègiÙ9.5Ž‘y¯Library‘G\equationsŽŸ?„‘ègiºNote–§Öthat“library›§×equations“are“not“automatically˜used“for“nding“v‘ÿXíalues˜of“v‘ÿXíariables.‘©ÄThey“are“consideredŽ¡‘ègito–êlbšSˆe“purely“for“reference“purp˜oses.“An“example:Ž©€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\eq[lib:myeq]{x–,j=“7y}Ž¡‘ègi\eq{y–,j=“2}Ž¡‘ègix–,j=“\val{x}Ž¦‘ègiºThis–=
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to“use“a˜library“equation“for“nding˜v‘ÿXíalues,‘=y¬wouŽ¡‘ègineed–êlto“activ‘ÿXíate‘åit.“There“are“t•¬ww“o›êldieren“t˜pSˆossibilities:Ž¦‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\eq{"lib:myeq"}Ž¡‘ègi\eqsubst{"lib:myeq"}{x–,j=“x_1}Ž¦‘ègiºThe–
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9case“is“really˜what“the“library˜equations“are˜there“for:‘They“need˜to“bSˆe“adapted˜to“spSˆecialŽ¡‘ègipurpSˆoses–êlbš¬wy“substituting“custom“v‘ÿXíariables“in˜to“them.ŽŽŸ‘ègiïcolor push gray 0’þñï	color popŽŽŒ‹Àê ¹ðü ý;9‘ègiïcolor push gray 0Ø9‘¿DPITF‘þ±ÜALLS’
¹Šfº17Ž’þñï	color popŽŽ ¦Æs ý~9‘ègiÙ9.6Ž‘y¯Pstric•cks‘G\headac“hesŽŸ?„‘ègiºpstricš¬wks–Fìcan“giv˜e“y˜ou‘Fëheadac˜hes“when“the“function“that“is“to‘FëbSˆe“plotted“has“Êx“ºor“Êy‘µ$ºv‘ÿXíalues“of“less“thanŽ¤€‘ègi¹1Ê:¹0º,–êlfor“example,“¹sinŽ‘WˆÊxº:Ž©€‘â‡mïcolor push gray 0ï	color popŽŽ‘ègiÖ\begin{figure}[!htb]Ž¡‘ègi\begin{center}Ž¡‘ègi\pspicture(\val{-180\degree},-1)(\val{180\degree},1)Ž¡‘ôÀ=\psgrid(\val{-180\degree},-1)(\val{180\degree},1)Ž¡‘ôÀ=\psaxes[linewidth=2pt,‘,jlabels=none]{->}(0,0)(\val{-180\degree},-1)(\val{180\degree},1)Ž¡‘ôÀ=\pscurve{-}\printvector{\val{Ž¡‘
qå\eqeval{y–,j=“\sin“x}{x“=“-180\degree:180\degree:30\degree}}}Ž¡‘ègi\endpspictureŽ¡‘ègi\end{center}Ž¡‘ègi\caption{Function–,j$y“=“\sin“x$,“na€ïve“version}Ž¡‘ègi\end{figure}ŽŸ{g¢‘ègiŸŸBÞïcolor push gray 0Ÿ`½"ŸØ*¬’¨¹{ïcolor push gray 0ï	color popŽŽ’
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