This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Differential Equations with MATLAB® (Second Edition)
Brian R. Hunt Ronald L. Lipsman John E. Osborn Jonathan M. Rosenberg
Differential Equations with MATLAB® (Second Edition)
Updated for MATLAS
*' 7 and Simulink -
6
Brian R. Hunt Ronald L. Lipsman John E. Osborn Jonathan M. Rosenberg All 01the UnWefsity 01 Maryland. College Park
Kevin R. Coombes Garrett J. Stuck
WIllY
JOl iN WIU;¥ & SONS, INC.
MATI.A8. Simulink, ond Handle Graph ics are rcgi~ l crcd Imdcrnarks of the MatbWorb, Inc.
C : ,4! t 0
zoo.s Jolla Wiley &i; SoIu. Inc. All nghlJ ,......td ,
tn".. mmtd
,. . . . IIoob .. lOr ............ s
'f I
'I'"be UIIiwd Slain of Am
109""4)21 )'tt II had bound by M olloy I.illlop phing. In<
,
Preface A\ !he subjec' mailer of d,ffe ... n';al equations ronunu.. '0 grow, a> "",.. I,""h, nologie.,; IIlrr.... ntlal v,q .... Uo ... , E.ghth Ed",on Tr.sdnional ,ntrOOUClory cOlI",". in ord,n:ory diff.... mial ~U.I'onS (ODE) hal" OOfICen' ItIlI.d on leaching .... perloi ... of,,",,hn;qu.' for find,ng formul ..... llIlion. of nnous d.,!<:! of diffe.enlial C<jua"oRS_ T)'p,cally, th ....,ull W3, rOle applica,ion of formula I«:MiqU" ,..i!hou, a ~riOUI quaIo13''''. un&"tanrung of such fundam.D on three popular rompu~r pla,forms' ll-'indo"-J. /itKin/usn, aOO UNIX worUlauOIII .• nclud,n, Lrnu~, This Ilool is not a comprrhe:nllve ,"uoducli"" Or ref",once manual (0 e,(her MATI.AII or any of the compuler plalfonm, In.~ad, ,I forus., on lhe 'flC<'ific f.a' ...... of MATI.AII'h.1( .... u~ful for anaJYLinl d,ff~ ... n1J.1 ~Ud,ng of d,ff.... nllal ~ua"onl by g,.ing 1hc:m
,,,,,,hod., n....
,wo
'yslem. for lJl~lylmll d, ffe,en!,al ~'1u.!ions. S..:ond. 10. lewl or n,,"n; ~ '0 lho nulho rnalic.l ... ~y"em in 0Ib0. malhornahcl. f o,;"""noll. or science COII ....s. We 'oo, ~"" ,,~,. W. hope llus . uppk",.nl w,ll in.ll'Uel'o achlCnnl lhe 101",. goal • .
f,......,
•• ",_, m
W. at< role" led 10 Kknowlcdgc support of OUr .e· , oooln buled ow. nuny y~:tr! 10 1M wnl ~I~' Our WOI'k 011 1M Stcond cd,'ion w;>, r~n,a Uy 5uppurl.d by NSF' GmnlS ::::~:: , ~. DMS· 034 1982. Any opin ions. find,n,!. and co,\C 11! • • presSt
;
8rIIa R. HIIIII ' p,1dL l lf ,... Jolla B. Oabuob
-
Contents ,
I'rduce
,
,,
Introd ucllon
"
" "
Guid ing PllI lo'ophy S!"
In,(IUC!O"s GUide 1.3.1
132
MATLAB and Simulm~ ODE Cha",<",
I .D
,"
("ompu." Problem Set< A Word About Sofl,, ~ V."ion,
f:Wlnll. StartiPd ..-jl h MATIoAl:I
"2.2 " "" 2.' ,"
PI.lforml and V."ion, In' lallation SIMing MATLAB Typing in lhe Command Window
""
""
"
.'22
Aboning Calculltlon ' S}mbolic Coml"'lahoo 1.3. I SUb
"
JA I
3,~
I'"..c!iom
,.
V.,..'10r.l
35 1
•, ,, , 9 9
Online Help MAT LA li Wi ndows linding a S." ion
1)(>loR Mathernali", " ith ,\1,\1"1.,\ K AnthOltl,. 32 R«O'~riRg from Problems 32.1 bTOn in Input
HIP- Ordtr Rqll.llloll! Mnd S).I..... or 1" ".. O rd~r Etjun lloM 13.1 HIJhtI Ordt, l inear Equ"ion. IU, Syllems of hl>l Q;dn Equation> 1121 lint ... hl'1 Order Sy'''ms 1311 MAll-AB '0 Find b,enpai.. 13,) " - I'ortrai" 13.31 PloIII", & Single Trajectory Ill} Plonin, S.",",! Tnjrc\offlS 13 3) NII""'",al So\u';on. or fIrs, Q;do r Syllem,
u,,'"
.. Q .,e"" 11Itw)' to.r S),I,mJ of mlf'""llal t:qu.llol\ll
..'"'"'
'" '" 20' 20' '" '" m
'" "" 2S9
'"
Chapter 1 Introduction We ""Kin by "'<eribing the philowphy behInd OUr Ipproach 10 the study of ordinary dtfftr enll.] equatIons. ThIs phlio'lOphy h.. It.. fOOl. in the ....1 "'c undtnW>d MId ~rply dIfferfnll~1 equation.; It h... inftll(oced our (cach,ng and gu,lkd m. de"ol"""",m of 111 .. book. Thi. chlpier al'>O contaJns ''''0 UStr's i"'dos. Oflt fOl"OOrn,\ and """ fo< inillUrton.
1.1
Guiding Philosophy
In Kicntific inqulIY. whon ...., are 101...... u.d In undrrIlMd'"g. dtsOmlc. Or other ~YII.m hy one Or more functions of One or ""eml '"anable., r.." example:
• The po"I,O" 8(1) of a panlde is" funclion oflh. (ilnt'. o The tempera,u", T(l', ~, " I) in a body;, a [unclion of the I',,,i,ion (r, 1/. :) and the time l.
• 111<= populallons .r(t) and I/(t) of ,"0 compchn& 'peCI.S an: '"f>Clions of lime. • ~ ","vlLlllOOal
Of
cl«IronLIIIPCIi< force on &II objoct IS • f"f>Cllon of lIS po.ilion.
• ~ _ y .upply is a fUf>Clion ofume Nul. "'0 a!lempllo formulale, in marMrnalicallcm" , !he fuMamtntllll.,,· SO"emins!he r/Knomenon. Typicall)" IlIls formulallon "'sulls 'n....,., Of rtoOft d,ffe,enllol equ.tions;, '., equatIons involvins de';''lIh'"es of tt.. (.f>Chon. do \.Cnl>lng the .tlto of !he sy,trm With ""1'«t 10 the varilbles lhey Clioo l
,,,,I,,,,,,),
,
Ch"l" rr /. IUlroductitm
:.:: ~I&i la..... "'( u,ually durnbol 1M ,nlll.1 S!:I'O of the SY""m """holll&lKally by . p«ifyinE ,",,;,,1 ",/un .r{O) • .r'(0). tlC_ 10 th" : ... _abc II III UIir/GI.""", pmbltm------an onImllJ}' d,Il'" ",o.i" "lIIn agarn, E"en if we . . 22 1 m's' With !be model. MW lechnology. new ",quirrmCn' l, or I\twly d,scm,.",d fca_oflbl l)'Jlem may rendtr oUlol1I model obsolore, Aiain . " '0 "'spond hy rt"u minins
... ......, n.
1IJ\da:1 til difk",oti" ~Ullionl
t .•
, I
if
111 :
I ' 101 :
pru1 of buildIng. solving. and
* /M'"'' I'I'KJOkb, Manyfan ""ults and """ hod. h."" beon dc".loped for this w,th,n one or foll"""nE thtmr,: 27
r
II w"lIId 11IC!hod<
!lIOn'
01 the:
anclnp"I''''MU oholutioos.
2. Dept: .. _ of solutions on initilh'aluts. ,. DeEh aa;(I! of fonnulll for solutions. 4. N!mMdce' ca/aLlabon of $O]u.ion., , Ol4 l11uilllalYli' of $Olulioo •.
I. QlllUIIIive analYI;' of diff.reo,i.1 oqua1ion, and tho'. lolullon.
'l1li bale ruulll on uillcncc and uniquen." {the"", ]) and del"'ndc:lICe on ioi, i31 val _ IiIII CIIIea 2) form IlIc fo undation of tht subj(ct of diff.,."" al cqUl" ons. 1be ~mlllCn'
PJ -, m.........""ian J"'IIIC I Ilad.lmard r.rm~d to. probkm I.
wtll-pomJ if good re_ .... nlftillblr COIICmun. tht'" .... 0 the:"", •• ;,r .. if one hilS '~IS'.ncC and uniq ... ncJS
"!"7"'.IIII! Ihmall cllangn In initial ' ",",!irS make only I small change m ,Ite solution.
al 'hi
"dOl'! oIformull.oluhons (thtmr 3) i~. rKh and 'mportant pu! of 1M subjeCt; a for fil>dlllj fonnul. t.Oluhonslo !p«lal dantl of can be .olved describe the: originllphroomollOfl (which led 10 tbe ri .. lOW diff.","ti"1 .i " Oll" , 'iool (""u~ on rllC'hods for liei , . _. included lome simple numerical and . ill compiling nUIlJ<.ical or graphical ,mpl,m( ollog more ad''anccd quall.ative 01' qua"""'i'-. han reduced 1M", olmoele,. Sop!riSliclled ",fr...a,. and
"
1.2, Sr!l
3
mainf"'ItIc oompu~rs cnho""cd Iht u'>e of qu,n"'all~."d qu~I'lIu"c ,",,111<.:.:.11 in rhe lhe. ory "nd .pplkalronj of d,rr .. enli.1 ~qualron" Wilh Ihe arrival of comprehensive malh<mOl. ieal wfLw.rc .y~lem. un per"'n.1 compule..,. thi, fIl()(km appruath hal become lIIX~n,bk Iu .ru
[n th" houl, we U.l4' the mathemalLcll .ofI""" .yMem MATIJ\BlU 'mplc .... nt th,s a~h. w. u~ MATI-AB', symbolic. nu .... neal. and gnph'cal c~l'"bih"~. to analya differential equal,on, and lhei. wlution,_ I'on.lly, enginee" and scienli,,, ha,·c 10 d.,-dop 1101 only .k,lI. ,n Mary,inS problcml and intcrpret"'g 'OIUlion', but alent cohc"'nI condYJionl ,n a logic31 and c<mY'lI
1.2
Student's Guide
1l>c ch.pI'''' of thi. book em he d"'idcd ,nlo th_ das.n: I"""ral dlsC\""OfI of MAT. LAB, wppkn-..nl8Jy rn31enal on ord,nuy d,rr.",nl'31 eqU.bOflS (ODE). and c01llpu1.r p.oblem \CI< 11.", is a brief dcICriphon of lhe Contents. Chapt.. 2 c~pla;n> how 1" ',an and ",n MATLAI:I on )·ou. compute" Cbapl •• 3 introduces ba.ic "IATLAB comm.nd,. Unlen rOIl h~'·. pre"ioul e'poriell<e ... ,th MATLAB, you should wor~ through ("h,plt. 3 "hile 'm,nl ar you. comput~. l1Icn you .hould ",ad Clurple. 4, "hoch cool~in' rk"',l~d ins1rUClion, for using MATLo\B M.fil~l..,d prin1.Jng or "p"bh\h,ni" you. "(I,l_ Af, •• !hat, "Ofl the probleml on Problem Set A 10 ~hce lho Il,lIs you·,-. leamed ill ("h'rlers 3 and 4 Tllne lIeps ""II bring )'0" ro. balic le"el of comretence on lhe u>e of MATlA' 1:1, s"ffieien! for Ihe 1hn:. of 1he d,rr.ren".! equations chapler<, Charlers 5-7. and for Problem S.t B. So,"" mort ad, """cd "'peets of MATI.AB. needed for SOme of the problem. H.rt,ng on Problem Srt C ..... d'~S$td in Chlrple. 8. Chapt(f 9 ,nlroduc •• the MATI.AB eomp.onlOO program lnoo." .. S,muhnl<. IIlucll pro"irk. a "\Crul graphiul tool for sol' 'ng ,n,lilll1ll"" problem' lhe primuy purpo<e (>( lhi. book .. to .tudy diffc",,,ull ~quati""'. "e h,,'e not .ltemp1..J 1,. dCl<:riix "II tho major ",,,",," of "",TLAB or S,onuhnk. ~'ou can •.' plore MATLA II in mOfe depth uI,ng ,I; dcltl
fi,,,,
s,"""
• Amo. Gilal, M A1 1.A N: An IntroduC110n wit h AppllnrUOfl~, 2nd cdltiOO. Jolin W,. ley III Son,. III< .. 200S_ • B,;an It. ilunt, Roo.ld L L,p.man .• nd Jon,than M i(oS(nbrrg. A Guld. to ~ IAT· LAH: for H.W"",,,, ~",I t: xporlon«d U",,,,, Cambndg<' Uni,.. rslty ..... n. 200L 4Th,. «111;00 .;.,.. wnllcn for MATLAR 6 •• 2nd c,htlOO fOf MATLAB 7 I. in fKt"pa. ",lion)
• Wilham J. Palm III , I nlrodurt4ol1 tl> MATLAB 7 for ~:nWII'.rs. McG",w· II,II. 200.~
Ihal found In I U'Bdl1ion.1 ODE l<:xt TI>e mam ..anh for ~'lI<:l formula w lulion •• and ,,,,aler "",pIIa' of lh~ ~quUllon' and Ihri' solUlionl. II appear In I
•
f~ ~:n•• :~',~::"':"::'rlnu....,rical of the book Solvin, 1hc.. prot> foalu"" of Ihr ,"bj~~l
the prot>lom .. a ;f)'ou plan.o do liItm in 1""(1 "lSion~. and thinlin, obou. lilt i"u.e. in'"h"d. n..n go 10 1hc
. If)QU 1<' 1'\1<1, 1-1" your "'Ofk ond 10on'o 1hc IR ....lIj' you'd lilo 10 d,«u ,' ",th your Inllnl< a.,ignmen!;s and OOIuined anl"ell'O your q\l
II
the
firs. few
,ugi~,;;on'
.oove,
I u.ing MATI.A13 WI bookeonllin. Iwo useful ><'Clio"s, I GI'ma.y "lid a cullcclioo of .... lolutllJlll. 'Thr Glo.ury con lain, a brief ,unllll>!)' uf rdevan. MATLA13 com· ~~7t~, bull. in fllllC'bon" and proBr.tmming
1.3 Instruclor's Guide '!lie philcnophy thOI WUlded Ihr "riling of Ihis bool;~ .~pl'lII.d al tho iJcglnmng of Ihis
dMptv. He", i,. ~"r\Ulo ,umma.)' oflhal philosorhy, w.
~I.'
• 10 auldc- .rudenl' inlo a 010", intorpre li,'. "'od. or Ih,nl.lnR, • 'Jb uw .....lhrnlalKal ,ofl..-= rad_gt 10
to co"'pule
,
/.3. Inmucr...,·, GUJ(k
symboht and numeriC1l1 wlutionl. and to pt"rforrn quahlati"e and grarh,cal analy.i. of d,rre~ntia.l equalions, • To develop toll"'" matt".llhat ~ne.ts lhc Current .!ate of ODE and emphuil.C. tile m.lhcm.t,c.1 modeling of phy.ital rrobkm~ . • TO minimiu!he hme lajuircd to learn to UK the ~ft ... ..., path",.
As ment,oned on S«lion 1.2, OUr m.terial con'ists of MATtAH dis.cu"ion. DOE lUI'. plements. and tompute. problem KIs. lIe~ an our rccommen,ul,<>n. for inttgralin, thll m~teri.l in", I Iypiral1Jn.1 COU"", In dirrerential ~U.tiOflI for scionh;" Of engmeell.
1.3.1 MATLAB and Simulink Our ~!Udeot< r.3d ChOrle .. 2. 3, and 4. and worl /'mbl.,. ScI A w,min m. first "'«~ of the ",mester. Alibough Ch~ptet 8 is 001 e'KOhal fOf Protokm ScI B. "lldcnl. oflen find il uK/ul, An.nl;Ofllo!he", ehJpttr> qu,dJy ludllludcnl\ 10 I 1....1 of proficiency, Charter 9 leach.s enough aboul Simuhok 10 ITI3ke il JIO .. ible for ,tuoknl' to u>c il fOf many pmblefTl'l in Problem Set.< C·p
bu,.
1.3.2 ODE Chapters l1rc", .,ght eh'pler. (~-7 and 10--14) ,upplemenl the ITI3lcrial in I t:ad'liooal te~1. We usc MATLAB 10 sludy dirrcrtnli.1 equalion, "'ing symoolic. nun ... ric,l. ""phi.al. Wld qu.hlah~ meillods. We .m~\I"" u.. followinglopic" d,~ction fIelds. ".hilll)'. numer· ical merhods. comparr"'" mrrhod •. and phase porU1Uts. "These IOp'C~ an "<.>1 emph .. lled to lhe ~ame deg~e in lradilional "XIS. We incorpornlc mi, new cmph,"is iOl0 our d>1$ di§cuSliOR'. lIc>llIin& wmc clUllime to each chapter. Spttifi~ &UldelrllCl m dIfficult 10 prtscrihc. and !he requ,rc.j lime >·aries ",im each chapler. bul On a>'enge we Ipt"nd up 10 an hour pcr chapl.. in cl.I' di§cuI,ion. "The S!nXlure of !hi. bool ~quire. Iblot numerical mtrhods Ix diseu'sed caly io Ihc COUr\e. immed,ately after Ihc dl>cussioo of firl.1 Ofder ~u.honl. "The d'scu~ .. un of numer· ical .nclho1£
soh''''
1.3.3
Computer I'robl r rn S~ts
"The", are
.i~
romputer probltm .. I., "The tOJ'lCI addrcu ed In the problom KI< arc:
{AI l'I"aclicr" l!h MATLAB. (H) Firsl Order EqU'lIon,. (C) Nume ...:,,1 Solulloo, .
(0) s<:cond Order Equalion s.
., ' ,. of DilJrmllialllqu.ociool
, ." ".d co acqu a,nl llUde ol l w,th the !>UK: symboI,c .:utd to .ucclimate wm to calculus We assign all and 1IIo".;t turned in nllhor qu,ckly, W. ge"""'Uy .»iEn )-5
,
lhat Sludrnts present lheir i I app.opriately On the" prinlOlFlS, tbry mu\l nwCer M,fi ic •• and W "p"!>li sh feature of MA11.AH (the ~;:: M 'boob. 01 w pllting of MATLAH rode and uutput ,nlo word I nw... tooll (ac,IiI,le pre,lfntal;om ,"'ich '"~~l.d inpul. OIllpul. M
formaIled Ie\l. Chaplel 4 contains detailed in.truCI,ons on !be "publish"
,,
,
,
.~
to the homewurk
'-c
'"~"', .i&hch E Boyce.l Ri1M, 1.., for a fi ..t . """" in diff., r"l,al equalioo. fur ~ orn..-n- Some 'O"",hOOI on bow tu 1«0000plim chi, may be (""nd on OIlr
h.".
~;:'~dJ Iw.,.w
http. llwww,ma t h , umd , edu /u nderqraduata/achol /ode
1.4 A Word Aboul Software Versions Hftt wnlOllI of ",n,,&I't apfFeat frequenlly, Who:n 3 cOII1ple~ prog .... m 10k. MATLAH *'1I\1?l, IMDJI rommand< " o rl ben.. than they d,d befort, some wOlk d,ff'renlly. and. I'tw II1II)' no loniC'l ""'lot all AI tlt i' book goo, 10 press, the ClIffe", '''l3ion is MATt .A" Ytnion lIR.I •• .., 14); thi. ;51h. ". .. ion wc d.",,,I>e.
Chapter 2 Getting Started with MATLAB I" thi, chapter... e will introduce 101110 the 1001\ ~ou need In order 10 beg,n using MATLA" effocli,-dy. These indude: SOme ",levlnt iofonnalion on comput~' pl~t(onru and ..,rt ... "",; Inllallation pro/ocols; how 10 Ilun(llor 4, and .... e lUll m .~'1 on d,ff.",nti.1 cquIIJo<\l In Chapter S.
,,>IUd
2.1
I'latforms and Versions
It i! likely th., you will ",n MATlAB on" I'C (",nning Window!. Or Lin"x), or on ,ome fom! of UNIX operalio!: syslem. Some .,n:v;ou\ >'minos of MAlLAB (Rele_, II .nd 12) did not IUppo" ~hdn(osh. \)u( (he mOIl Cum:nl I'mion. (Rclca"" Il ""d 14) do, {On • Mucin(olh. MATLAB 7 ",qui",. M"" OS X 1032 (Panm..),) If you runnin, • MlICinlosh plalform, you should find tlul OIU ,n,tnJ(I'()fI\ for Windows plalf"",,, "ill $U(fICe for your nted, 1.,1. MATLAD 6 (Rdeal<:l 12 and 13). and unille eallie. "c .. i()fl'. MATW'.B 7 lools ~i"ually identical on Wind,,.H and UNIX pIOlfo""" For definimentll, "c ~ll.n"m< (he rude. i, " " n,. PC ,n . W,ntJoo,I,.en",ron"",nl In those ,'cry few ,n\(lOCe. ",he.e OUr mUJ\lction. mu.t be la,lom! d,ffc",nlly for t..n"~. UN IX or Ml
=
R~m~rk
1.1 We "'"' the "'Ofd lVi"do"" 10 "'fer 10.11 8 . ....... of m. Window, optl'1lhna 'Yilem MATI.AD 7 ( R 14) " 'ill run lin W,ntJoo,I, . 2000. W,nd""," NT (40 and higher). and On Window. XP, MAnAB 7 w,lI nQr run lin Window. 95. Wi""",,', 98. 0. on Wmdows MI!, How~'·
7
•• .., r-n
of 1M MATl-"B 1nconK. (~-J. M · fi~' and M hooi:l] are .-a1id for ,Of_ . . 6.5 (R1lJ.1IId 0.,1... ",,,;001 in some <~ •. W. 1]..0 1\01< that lho: d,ff.Il'nces be· "' HE !lie MATi-'B Pruf•• "onal Vc ... ion and .he: MATLAO StUiknt VrrslOfl are ,,!.lilt. lei"", md VJrtu,olly uMObcuhlr 10 • btginncr. or ••'Cn I nud-Ic....,]II§e' Agam. In the few Ip 7 wi1m' ~ """,nht. MA11.AB realll,," WIll only .nil.I'M In '''' ProfeSSIOnal 22
Ya_
WI!
hlJhliahl thll r"", dearly.
2.2 Installation U you m\cnd 10 run MATLAB nn n pc, .'pe<:ially Ih~ Studt"l Vm ion:;1 i ~ quite f>O,,;ble dill Y"" WIll hi...., 10 in,,,,11 il yountlf. You can eas,ly O«Olnphlh Ih " lIitol: .the product PoD"", 1M in.wllMn inmuctions aI yOIl ""ould wnh nny new 10ft ....... ,"ltaHalloo. At _ point in lht In,II.11,';on yOll may be as~ "'hlCh ,oolbo
co.
2.3 Starting MATLAB You oWl MATlA 9 .. )'OU w"uld any othn $Ofl ....... arrhcltiOll. On. PC you acccU " vii 11K' Sfllrt mon u. in thr l"roaranu folder under MATI.A II 7.0 0/ Student MAT LA II Al~l1\.IlI.rly. )'nu moy h",~ a de.ktor ieon th.1 .nable~ you 10 lInn MATLAlI wilh a simrlc dOllblr.t h<~ On B l TN IX mach ine. generally you need only Iype . . tlab in 8lenmn,1 window. 111",,1" you ",oy fi ...., hal'. l() find the n\atlab7. O/bin dlf«IOIY and add "10 )'OW'r-III. Or )nu may ha,·~ an kon or > speci,l !N1I0Il on your ooktop lh>t IChiev~.!he
...
you
bri.n,
H_f,,,r )0" ,wt MATLA9. "'ill >ee • WIDOOw 11111 dlJ.plays tile MAl1...AlI Ioaou "fill' ,omo MATLAB prod""l informalion. and tlltn aMAnA H IHSklop wir>dovo WIn .... n.:h Ttu.t wondo"" w>11 cont:un a htl. bar. I monu bu.• tool Nr and four .mbeddr-d w\ndIJIo. ........ "hl.k.fl"'rt 8 .....·ur. !he {""...",mld 11m".., lI'indoo.·. and (he C"N"..'" ( h"" ."", 8",w,,,, For now ". COftCenlnlie onlhe Comlll.1nd \\ ,"jndow In ordr. 10 ... t you Mar1.d I"uong MA11JI1I ,,,,mm:lnd, a, qukkly.s posI,hle Atllle top of lhe Command Windo\\', )ou.my ~ w nle gonual",fO/rnllion aboul MA:n.AK. perhaps"""", Iped&lm'lruClIon, rur grlling >1,,"od Or <>cocssing hdp. bul rn(»t ""l'Ollant of all. you will I« • Cmpt (> > or IlDU »), If the Command Window t< "a.ch"c". il' IIllc bar will be d ... ~. alld th~ l'ro"'pl will he follo..-ed by a cursu. (I hlinking ,'crllell h~). lb.11$ Ibrpllcc .... he .... )'00 will ~nt.r your MATLAB commandl (>ee Chapter 3) Iflhe Comm>nd WIDdow I. not ""II' •. ju,1 dlc~ In it any"hetc. I'igu", 2.1 conlaln, an eurnple o( a .... \\'Iy laWII:htd MAil.AB lJe'lt<>p
"r
,
24 lYI'In/,' on 11K: Comrn;lnd ll'indo ..
•
_......
.
Fig"", 2.1: A MATLAR [)•• llO9
Rc ..... rk 1.1 MATU\R 6 has a OClklOp. but.n oklcl ,o""nn. of MA11..AB. for v.amplo ~,.l. I~'" wa"1() Int(c,rlUro Do,klop. Only tho C"omllW>d Window appeared "bra you laul",hed t~ awllealion (On UNIX Iyllom\. the lcnm.w from "'mob you in. '"(Ikod MArLAR 5 ile<-IlII"It the Command W"..I...... ,) Command, ~hal ",. ;n;ll\I<1 you 10 en~ .. on "'" COnlnl:tJId Window in,;de the Do,llor fo, , ... ion 7 can be cntr ...d dil«lly In10 11M: Command W;ndo,,· in Iny older "o ..,On.
'''ndow
2.4
Typing in the Command Window
Lhe: ClIfIUJlJnd Window tQ ntal.c It att,,· •. W~n a "".dow becom.. aclive. m LIlI.bar d:ulcn. and. bhnklng
Click
In
.,,,,U
hI;(- hgu,. 22
2.5
O nline Help
MATI.AJI h., o~tcnl,,"e onhne help In flrcomo qu,le profic1enl w,th MATI.AII YOII can ac«\< the onhne hell' In one of ... 'cr~1 "·oys. 'I"\"p,ng balp .tthe command prompl will "",oal a long li,l of 101" . ' fot "Inch help" .,ailablt, lust 10 IlIu",",ue. try typo jllg bdp vana,,_l Now)'"" !.Co . long h l l of "g. ,..,,,,lpII""""" MATI-All cOrrlrn)nd!, h(311)', tf~ balp aolv a LO I.arn _bool Lhe: _olva romm,nd In c"cf)' in§lancc abo,·c, .""'" informallOn lh.n your S<:rccn can hold ",II ",roll by, Y"" ...... "§c tho s<:roll bat Qn
--
-
•
!he riJhI of the win.dow 10 Kron bad up "llem'li<'oly. you can for<:. MATLA B to dl~rl.y IIIformItIOII oat ""lttoful al a lim< by I)'plol.on ot\ You press thr ,pac. bar 10 d"play IlIr Dallf;1ftofu]; l)'p( bdp .or. for d $(arc""~ the fiBI line of .~ry MATLAH Ilelp m. for a spec ilod ,1riJI1 (U!.( lookfor -aU 10 ..arth alllin.I). For unmple, if you wanted to _ Iii •• of all MATI.AII romm>nd, lha! conloin the word "foe lor" n parI of It.. <noonlnd ....., ... brief dcKnption. tlltn you ,,'ould Iyl'" loolofor f.c~or If (he command y(>lO 1ft IooktnII for Ippt..... In tbt Ii; !, tb
i. a much """.. u,,". friendly way to acreS! lhe online Ilclp, namely ~;a [he MA:rLAB H.lp B..""It,_ You can 100i-.1< tl on "'.....1 ways. for example , typmg h e lpwln. "1~.1r. u&b Ibc menu tw undc. Htlp I·Iu lly.!he qUcsl,oo m•• k bullOO 00 (he tool bu w,lI .JIO Unw tho Uolp B..... ""., Upon no launch you "'Ill ~ I"" .. ,IIIkno.'*. TIle Ii~ " wIN tho HttpNI/nNI/'''" il i, " ",d IQ find IlQrUmtnl>IKJII TIle iCCOrId. c,lkd ,110 duplfl)' ".,..,. i~ uw:! fm ,.,.... !loxUmtn,. lIQ'l 1be d,splay pane "'Va)' 10 in",~o!he Uelp Brow" 1 I. 10 Iype 40c . o h . Th" 1M Help Brow"". and d"play. lh< I.re.eroce pajt rQmcllll ..U ha.
""""'"'t.
,"g
'-tin
more information. You C&II abo u ... the 11011' Na\'igator to local. lbe docu .... nl.>l'On !hal y"" ",II .~ plOfe 10
16_ MATI.i\6 l\'indo .. ,
"
the di~pby paM, The l lelp Nnisalor tw four Llbs th..1 allow you 10 Ifnnge you. >earch for documcnlllion in d,fferenl wayl, 'The 6,,1 ,~ the Llh Wt drI,sate 10 11M: "m.l ~lph>he1ic reaction mo.k The thi,d lab pro. HieS tho Starrh ,nccha""m Vou type: In "hal yOIl .~ek. o"i>t. a fUl'I(:lioo 0, .0 .... othe. er. if the dre,
Con,,",.
"",pood,
R~mark 1,3
If lou type: helpwi n 10 launch the 'kIp Hrov...". tilt dilpl.y ]WI' .... ,11 coota,,, lilt SlIme I'O>t"' that yOY 0« I I lhc rel"lt oftYPlnll h.lp ~Ithe command prom pl. bullhe enl".\ "'ill be links, To ,umrrwiu. lhe IIdp Drow"" 'II robu<1 hype. by clicking. t(l brow~ through a I>o,t of C
,ob,,,,
If you .re pOli'''I. and not O"crly an~iO\u to gel '0 Ch.:tplcr 3. lOU can typ< dltJDCl 10 try §Onle of MATLAl:I'l oo1>ne dtnlOrll""tioo"
2.6
MA'TLAU Windows
We ha"" . Iready dt:ICnbcd the MATLAB command Window and the lIelp Brov.'se,. and hl"e mcnl'OMd In pamng tilt Command HmO<)' Window. 1M Cum'nl thre<:tory Dro.... se,. nnd tile Wo,kip'ce DrowJ.c', These. and J.c",,,,1 othe, window. yO" w;llcnco"nt ..... you work w,.h MATLAI:!. will .llow yo.. 10' (oolrol file ••1Id foldtl'lth.:r.1 yo.. Ind MATI.A B will -.110 ""'""..; "''I1te and ~dlllho small MATLAH prDlRI1l' (M, fil.,lWt you will usc to run MATLAB effecll'-.:Iy; ~c,p If1ICk of lilt ,,,n.bl .. and funotionl Ih.1 you define ...
Ch' prcr 2 OcrrJlI$ Sr;1md willI MATl..AB
FIJIIfI1J: The MATLAB Help Brow"'
ooh .. probkms and ,imuili. procesoes. ,
You can
I i Ill. !}t.kIO(> Illrough \be l>ttktop . l)ock monu d",,"nwanl cu,wd arrow in !he .oolbM. You can separate !}tlUOP 001 (0 yoor compo ... du krop by clicking on \be com.' of window', roolbnr. 4 ~nd 8, For now, we wan. to WIll . ncounlor, ",moly grnphics I ' These will Ii windows. nlballibo ull them graphic. wiooow., In Chapl.. 8, teach you ID_ .. Uld.1IIIII1",,1 ... MATlAB p phic! Wl ndovos d f«,,,,..,ly_Sec: Figu", 31 in ElF' for IIUllpIc. ...mplo of a BJ*opll~J window
.h.
.
'J.I In MATLAB b Of .arli., ,..,,,ion., you cannot dock figu'" WIndows. /II", can ia t'ftIioa 7 ,f)1'u 4ft t>ptnrring On a M""inu..h plalfOl'fl1.
2.7 Ending a Session 1b" brllllesl w.~ to c"nduJ. a MATI,AB "~Iion is (0 Iypc: q\lit al the prompt. You can IlIo click 0/1 1M 'proia] ,)'ml>ollh.1 doses yOUT windows (lIIu. lly un x in (he uPI'I" ugll l
iliad comer).
!I'.
Snll Molber way.o u il i, to th. ~:xll M ATI.,\ II opliOfl from.he: f ile - II Ihe Dc.ktop f/,j",y you oxil MAT I.AD. yoo .hould be SOI'O 10 UVe rOOT "'OIk, p i lii)' pphi" or mher lile s you need, ""d in 1 . I.. n up afr.. yourself. Some . ' I , . rordoin, so ..... ddn: .Kd In Chapto, 4
or
a."...
Chapter 3 DOing Mathematics with MATLAB Thl, <'hap'" dt.cn bc. so"", of !he b,o,ic MATl-AB comml/ld, (hal )'011 will u," in this book. We rocornrnorld thaI you ,WI MATI..AB 111<1 cnlCf the oornmand$ di~pla)N In this chaple. IU )'00 road it.
3.1
Arithmetic
MATLAR ~an do ari ll1melic just hl t I ulculllQr. You ,an odd wilh .... ubtrKt "nil ~ . multiply w, llI ... divide with I . and c. poncnlia'c ""b· _Th . ,'alu le an anthm
I'orualnpl. : u 2" 3 - ('
.. 5) 16 .. 7 *8
'"62.5000 MATLA R pnnll (he alii"'.' and a"'gn, the ,.Iuc 10 a "ltllbl. <:tiled ..... If )'OU wan l 10 pcrfonn furIM. calculation. willi the an,we •. you can u'" the \~~bl . ..... ",lbo, than "'typo 1M an,wer. FOf uampk. you can lale the squan: root of the pre' ;.... , IlS fall"",";
all,,,,,
~>
·lIrtl ..... ]
•••7.9051 NOIICC Ihal MATLAIJ lSligm a JlCYI' ' ...1".., 10 ...... ""th carll uleul.lion. To do lOOn: romplu colculations. you can "";1\11 COIl11"',ed '-011><"1 10 \-anablc$ of you, (boo.in,. For cumrl~: ~~
•
. . . co . III
0.5403
"
,"00
,Roo'''' ",pointonlylIithm(lic. "hidl '~ KtPlIIle 10 applo15 digl1 ' by dofau\\. To di splay rno'"
;~,;";;il numerical oulput will have )5 digits di ll'loy.
_ferIDl from Problems
Sl.1 Knan lD lraput .:raa1lDlb 1/1 ~1TOf In III 1111"'1 hnc. MATLAB will print"" error meuage. r'01" r umple,
'*" ....
IllbllIIappIDl wbtn you tty \0 . ,.,.I ul le 2a'
..
I M1.a1nv MATLAB
operat o r.
'Ib. .nor Is II'DI""'I muluph.alion opcralOf · The COIT«I input would be 2 · a Note
.... MAnAB placn. mutt. II ,~nical1in...g.... nl) I llhe pllCe where il thInk« lhe ..TOr ...... be; bowewI'. 1M -"luI .nor may l1li...,
oc:c:urml u .hcr or lale, in lhe upreuion.
" 7 Ij IDlllliplielllOII <>f'(l1IlOrS, bracUtl. and paR'Olhe.u are the most rommon errors. You oaa edil llllllfllll hilt by IISinS lht up-........ key LO , edlspllY \be ,,","IOUS rommand. edidD& III!! eommand lI,mg. !he left·arrow key• .00 \be n p'e."ng E.NTu or R 6TUJN. Thc up d UIW key a1I(l'Oo' )'0010 $Croll back lhroIIgh an the command. yoo"~ \yp«!.n I MAT· L4B _ _ .aDd i, "CIY u;cfu l .,hen you wanl lOcorm:l. mod,fy. and ",.n~' I previous 1
c·"........ Abortlnx Calculations 1tMA11.AB 1"1> huog up in a (.loul.bon, or K en!' 10 be 3.2.)
1aklnil loo lon, 10 perform .. fIIIN'IIoII. yoo ~an u,uJlly abo., il by Iyping C TML+C. Wh ile I>Ol foolp roof, holding ..... 1IIt by I.belled tnt. Of CONTROL , and pR'ss,nl C i. the method o f choice wlltn
IL\1L\I .. DOI R..pood'n~,
3.3 Symbolic Computation
"
U\lnll MATlAB'5 Symbolic TooIbo •• lOll CI1l cany 0111 ~ymbohc calculation•• lKh as fklOn", poIynomll]. or ..,].ing a1~llfaic tquaMllI, 1),," b . lJ:l aymbolic to make 'UI. Ihll!he: Symbolic Tooilloi 11 in.wlled Od.-
.. .."
H. ~
»
x
y
Ix . ylO(x + y) " l
"
Ix.y)'J » .Iq)aruil.".)
..
x"3']·x-2·y.3 · x · y-2'Y"] ~.
,
f . ctor ' .... )
(x'yl" ] NOle IhOl ,ymllolic: 0II1p111 illefl-ju,lified, ... h,l. nument OII' pIIl illl>donl~ TIti. fe.""" "ollen u... ful In d"URluisItiJIl.ymbolit 001('111 from Rumcneal OUlpul MA:rt.AB oflen mmor .implifH:.""", 10 !he: "'fft"ioo.)ou I)'pe bu, Il00:. IlOl male any b,g change, u"len )'011 ,ell ;110. You "111 "lot MAl1.AB's d.ll;pl ify eomllllJld to try to <>1'"''' an • • fft";on as "mply as pu
nul.,
....., ... ~ ~
»
. Y'U a dlOl)li f y(.xp(x + log(y· . xp(a)l))
y · explxu.)
3,3.1 Substit ut ing in Symbolic Elpressions When)'011 ",-orl with .ymbolic C.' fftS"Oshl.. Ie I nume .......l '-allK. or .,'cn another .ymbolit exprU,iOblc. in !he: npr."ion Thi~ i, dooot by uSIng !he: . ubl cormnand For .. ampl'.l""s.. rru"g the symbol uprc,,;oo w i. dofined and i, in,"OI,'" Ihc: .ymbollc "Imble II. !he:n !he: comllllJld l ub. (w, II, 2) wIll ,ub.mul. lhe , ... IIK 2 for the '<mabie U In the "pres,ion w More uarnple,:
,
•• "
•
1,
IY'U U v
"
in
i
lJI
,'-3Iuohon, we would
!oCt
twO OU lput lines_
III... if you .srpar>lle them by commas .
Precision Arithmetic I for II I .akul~lIo.", U,ing''''' ~.o do U kl arithmetic with symbolic ulltusions. Conskkr II
lloatin£ point formal (.. ~,.nl;lic JlOIauon") and meaM 1.2246 It thol oin(".) " ",ally cqll
,
Iyping . :.--I' p i') So,lOlUclhesme
1M apk\N &/I"H'. •
quotr. aroou>d pi in .yat ' pi ') crtau: . J1rtnl cOOSISunK of 1"" chll'x.e .. pi
....... wl MATLAB ffOOlC,,,lualm, pi allbl,ng point nurl'lbct The .ya comnund t) . . . . Ibe 1trinJ: 10. 'ymbolic "1""1"00 '!be ~ .,.. ""d .~ "'" d OiCly ""l l tcd. In fact lyaa It .. cqu""knt \0 • • ~C 'a') While _ h), II 11\ 11"11 effect 00 us "'lIom,,"1 (01 deda,.., n 10 he IlTtoIW from now on1. , , . has only a lcmpor.trY d fecl unlen Y()\ll"ig" the oulpU' 10. "",. bIt, U in _ • 1,.1'''').
YIJ\I can I]", do ,'Uiab]~ pruisiQlllnlhmelic ""Ih lhe 1"101 SO d'll,l. nf.;2. 1)1"':
If ~IJ\I dnn'l specIfy lhe nllmb<:, of dii'ls. 1i"H: deflul1 !-tillna is 32. YIJ\I un change !he defl"11 "'"h 1M comllUlld dill! t.
3.4
Vectors
1\ '-':01<" i. an ordered 11" of numbers. You CWI
.""'pi(:
,
.. r1,:I;,ll
n
Y .. r. 6 5 :I; 81
~~X
, , , • roo ... , 6
,
•
Suppose ·..",IOC... 31. a ,·",torof'·a]""" running from ] wllhoultYl'rng .a.:h numbe., »X .. l,9
,
, ,
,
•
,
,
109. H .... 's
how 10 do II
• ,
6 The lIOlalinn 1 " ;. ",ed 10 represonl a .ecto, of numbo .. running from I 109 in incre..,.n" of I. The incre..,.nl can be Sp«lfie
»
,
X .. 1,:1;,9
, ,
, , ,
You can II", U~ fllCli1ln.)1 or .... gah'.., Inc ... menll. as In a , a • 1 ,lor lao , -1 ,0 10 .~tllCl a romponrnl of a ,"eCtor.... y 1he founh componrnl In X. 1)"1'"
I[
I~nally.
U ),
To (I.ange X from a'.....· "octor to • column ,ttt"'. put • ,,"me (') oft.r Jt. ~~
X'
".",
,, , ,
You c~n perform mad~"",'ical opo.alloo, on \ OXIOflo. 1'0< •.• ampl •. men" of tho \'ColO< x . Ilpe:
In
.
,
2~
49
8l
1 .... ill dDt uprruioD il wry ,mporlanl; " uyllhallh( n~n>btn. ,n It .hould be m iDdiYI~y. or T)-p,n, It " 2 ,, 001<11.11 MATLAB 10 u§c ma· IIIIIdpIiCMIOIIIO lPulliply Jt by itself and ""00 1.1 prod~ In .rror n... '~ case (WI wID diJCUu matricn in ('haP''' S.) Sm"l;trly. you mU I\ lyf'C' . ' Of • I ,f)ou "an\ 10 IlllLllipIy or divide \'KWf> .ltlntnl- hy .. lrmcn!. hi, . ... mple. 10 mull'ply I~ clement-> of
,'rIft<",-by.",rNm
loa,' th,.
l1li 'iL.M :I by !he .00000.ponding .1.n",nI5 of lhe ,'ecl0' Y. Iyl"': ... :I.'"
•
"
Mall MAtlJ\B ..".rall(>fll ~ aUlomalicall y prrfO)l"Tll(d clemonl by-d." .. n!. Fox u _ • 11'.)'011 do DOt !)pi I prriod fOf addilion and jub\.:oo;I,on.• nd you canl)'pt' - xP ~~ 10 . . till uponcl'ltl.li of <:OO;h numht-r in X (!be nllltn~ .,ponoenllal funcl10n ,5 _ l ...... "" of III! III of MAn.AB " ", abilily 10 .ffic ienlly pt'rfOfm ""'''',11)0\ 1)11 ,-eeION.
,.lIt
3.4.1 SUppreWnK Output typiq • HlIIIOOluo lIllie rnd of • MATLAB command . upprt'I'U p<1nll", of !be OUlp'" '1bc lemKOlOll """"Id ,.""",lIy be u",d wbe n oI1$ Of malri.es (.ueh os
:I • _110.112, 1_ 11 .-an .110 be u",d in any o1h. , l il" ~I'on ",here lhe MATI-AS OUlpul Deed DOl be d.. pl.ycd You <~n also use a ",micolon, lil<e " C""'''''', 10 .. parOle MATtA B COCIIIIIII\dI oR 1M .
3.5 Funttlons MAn.AB .011""'1 )'OU 10 u« bolh bu,Il·,n f"nclion. and f"ncuonl you define )'0'-' . ..11.
3.5.1 Bulll·ln "' uortions "';:'n.AB h• • an of the ul ul l "d oJtl".nll,) function of K . and 1011 (,,) (nol In (,,) ),, Ihe nalural)oganlhm oI~ :.0:.0
In d"llotC!i'''' ". w,lI UamiM IWO meth..xh 10 ~f",. you. ""'0 function. ,n MA1l.AD Pl mg the cOlllmand h.11 n •. and u,;nlthe 0JI<11I10I _ 10 cre~le .... hal ;s c~11ed an ~lUIOIIy· mOIl' fUnellQll," ".., lalte. method ;s ""'" ;n MATI.AII 7 1M 14). and ;> the ~f.m-d mtlhod fot u~n of MATLAB 1. W••Iull mcnllQll iD11n. periodJcally for the ...... of u)ttl of . arlle. ,..,..;ons. bul " ••tronlly recommrlld thaI u<ers of MATIAB 7. and U~fl of olde. "...,on, wlltn lhey u~ 10 R14. adopt _.1 ,he u,ual method for ~fin.ng lhe;, f."". lion " l' ulICllOns can 11'0 be defined ,n "paral. f,I •• cilled M f,lel-<et Cb~rlc. 4, lie .. ', 00 .... 10 deEne Ih. function J(II x' .\lng lhe,. commandl, H t · _ l x ) x ·2
Inli na function: f1 {:o' l ,, ' 2 Onc. lhe fUIICI,on ,. drfined-----by .,the. method loo cln ."al.~I. II' tl t l
u 0 00
"
A< we "b ........ d u,h••. moll MATtAII fUIICI,On\ cln upcmlc on "tCIOfl as ",.11 3, ",ala.. , To ,mu,," Ihot yoo. use.,defined f.nclion con .cl 0<1 ,.ctnN. in .. n dots befon: the I'. lllathelll"" ..1 operat""' · . I and" Thus 10 obta,n. ,e.;loo,.d .'.... iOll of J(II dlhe. type ~~
Of
f
__ (xl
x."2
.11.e
~~
t l . inUn.('x.-2', 'x'i Now ..... can cul"a! ••;ther fUlIChOl'l 01'11 ''telot. fot .umple H
f(1,5)
"""1
4 9 16 2~ You can 11'0 "I.e !hel't mrthod s 10 define I fUIIC"un of "",eral "ariables, For """'pic. c,lher of ,he foll"",mg ~> g __ (x, y) ,, "2 • y "2, g,l, 2) » g l . in11n."" " 2 + y "2', 'x', 'y')' gl(l, 2) ",.ulIS in lhe ,,,,,"".r S. If ,nlt.ad you
.~aJuate
01'1 ,..,<:ton; thus
IS 'J) ~,,,,,, ,.....,,'" 11.3) 3Il
U
MlDaglue Variables M b
.~• •~ •• -
d" ,re nl dllS'.' of MA"1 LAII dala : 1I"' "0 K[IO;nl MmbofJ. lid ymboIic: uI" •• i",,' In a long MATL,\ II SO" i"" ,I may be hard 10 re,ncmbof IICllnnl O'ln ,_u u..
• • lIIIIcI_ ' of III tho ,-.nabl., you h.,·r dof,""" You can 1)'pe whoa 10 SU a - oldie II1/II<"_ pd ' ),,"' of )'0'" cu""ntly
••
: 51
.... •
ByteS
Size
,., ,., '"' ,.., '"' '"' ,",..
,,
.
•
,"
"• • ."" ..", '" '" no •• '"
", '"
"
,"
'"' '"' '" '", '"'
• ,• •
Class
d oublc array doub le array d oublo array dOUble array do uble array double array [uno::tion_handle array inline objeo::t funo::tlon_handle array inHne objOlCt sym. object sym object sym object double "rray char array sym object
Grand tOl.li io UB eleme nts using 238B bytoS w. "'. \IIJ! Il>n. a.... cUI",mly si~l.e n "'s'gned ",,,able. in OUf M A"n.A n ses.ion "'p. racnlUll din..... d>la fouf.re of da" ··'Y'" oil,""" i t_. tbey af. symbolic ob,l«" 111< ,-ariabl. , u and v are symbolic bcc.u", ". lk<;1.... d them 1<1 be so u"ng .,.7 . " ..., ddi",," in l. mI ' of u ""d v . and _ '''IS lkf,,,,,d us,ng 8ya ~""" are of
Ii,.
"1
d.,,,,,,
21 clas. "dOlI hie .rrllY··, i.~,. lhcy an: ana)1 of doobl. pm:i.ion "umbers. In lhll c,"", nlO'1 of 11>0: arra)s are of \,u 1 ~ L j.~,. ",.Iars. TIle ,-anable. X. Y••. b . '" and .. an: of III" bul ....... lhe , •• uh of lhc wi eotnrnand ,n Ihc J'rC"OII' !ottl,OO. i •• I >< 2 anay, The ".nah~ y" of cln. "cluJ arnoy", j.~,. a .lnn••• pre .. i"". The I.. IIW(I dau d ..ou .... rtpro lenled by £ 1 and 9 1. wruell .... of ew~ "mlnle 00.1«1." and by f and 9 . ",hleh an: of "fu""lIon handl. arnoy" 1l>e " Bylo," column ,bow. bow m",,11 computer momory " I IiOC. I"
"u.
cI."
Wh,le who •• Ilows inform.li"" aboul 011 do:fi~d .anlhl.,." d"", nOl,1low lhe , .. Iuo, of llle ,an"hlo •. To .. e Ihe , .. Iu. of. ,'a,;.bl •. 'imply Iype lIIc: namo of lhe vanabl. and Pfc" H~lI k nr RHUI1N, MATl.AII cummand, •• peel panicolar d ..... of dalO I I ,"PUI. and ,I i. imponanllO know "hal of dl10 is • • peclrd by a &,,'.n conu"'nd. Ih. help 1"1 for a ronun.1Jld u,ually ,ndic,", Ihe ell.' 0< cI ..... of inpul II npec" Tn. wrona d .... of inpul u,ually Pfoouco , an .,,0< mo ....ge Or unex""",,d OIIlpul For .,ample. sec .. h.1 h>ppen, ,f you lype ai n l ' p i' ), Ihc ."or me.
cI."
To ,~,.. .11 dtfi ...d , ·ariable•. Iype ,,1 . ... or "ha . . . ll You can ."" typ<'. roonamp~. ,,1...... Y 10 e~... only .. 100 y II" J,lfC\110 c~;u ,.,..,able, beforr stan
3.7
Sol ving [ (Illations
lIefore you wive di«... nll.1 .qualion< wtlh M ...Tl.... II, 11" hrlpful 10 !tam how 10 wh'. alg.bruie oquat'''''I. To wlv. lho quadnlllC <'<1031100 r' + 2.. · 1 O.ln":
H
.o1v.C'''"~
"."
~-nl2l 1-~-
• 2 · .. - • • 0'1
1
(112)
II, .. lho equallonl" be ""h-etJ i. '"",,Ificd IS I ' Inn" , , .• ,I IS ~UfTO\lndetl by , in,l. qU01r~. The InP' " 10 101ve can al"" b< a ~ymbol ic .,prr~!ion. bullhcn MATLAB m[Ulrt~ th'l the ",hi, hand "dt of lho equaIJon b< O. :ond ,n floCt lhc .ynll> roo- ",,1"log 3.. 7
r'
"H
I~
,./ l"l v.(,. -2 -
J.,. •
7)
"." 3/2.1/2 ' i ·19 - ( 1/2 ) 3/2-1/2 -1-19 - ( 1/ 2) Tho 31"''''' COMi\11 of lho uacl (lymbolic) laMi",, ! (3 ± V19, )/2 (compl.x numbers. who,. lho leller i in the anl""Cr , 1andl foo- lho irnagry unil Al· To gel nUnlencnl ..,lulI"" •. I)'PC 40ubh 1an.l . 0< "PI« ani) 10 dl§l>by more d'III'.
sollllion~. MATLAB Irpot1$ "S ",sui" by a,,,ng the 1110'0 . ThU$ the fin\$Olulion cons,rn of lhe fin l fIn\ .... ue of~. You can ow"'1 the", value! by typing" t 1)
,
prea:dina U'" ohhe .oh'. romm.nd, we M~'Knod Ihe oUlpul to the .oh'. on a sY$tom of 0GuMiO/lI withOltI .'$igning the oulput lutoma\irally d'$p!ay lhe ••!ue; of the wlution .
.o1"l'a + y- :a • 2 ',
."" -,
"'*" 01 sandI/ ~.1l ...
'I' - l O" . 7')
of tho iOIutton, typo .01." and .01.1' To "", the P '+ !._L_ • . . (1 ), ... .• . yt 11, n~_ ."_'. _!)'pC .... • u.equfoIionI c~ bf ""hod .ymbolically. ~nd ,n lheseU", ' aolv. tnM 10 find MhiMld _~r. fof o>ample: IbI
!
-
.... _ftl'oon'x, • • " ]'08513321516064165531208767387 IMWIimel tilt", i. mote Ihan on. ,0Minn, and you m.y 1\01 gel ,,·hol you • • poeted
IN. Orupillc. 1l>e an" .... II ~ comple, numbe •. 'I1Ioua,h ir " . ,·.1111 wlu{lon of !he equation. the.e are al", ...1 number ",luIIOl'l' 1l>e gaph_ of ~xp( - .I) and ,ill{.I)lR .shown in MZu" 3.1; .:ach
ml• •,""IIO<\ of !he r.. ·o CU"l:S rcp .... nrs • I.OlulIOII of !he equa{lon • -.
siu(.I) .
.. ,
••
,
..
v. . . u",
•
''''''''n on
nUIIl
»
h •
>~
f •• ro(h. 0.5)
,.,
8-(x) ."PC-x)
dn(x)1
0.5885 KeplKe O. 5 with l10 find !he .....1 solmioo. and w fonh.
3.8
Graphics
In thIS J,CC{IOI'I. ". inlroduce MATLAa'S!).uK p101ltnl rontnWId, and
3.S.1
nl(
Grnphinll wilh ezplot
Tile "",pio'l way {O gr-.ph a funChon of 0"'" ",,,,.ble i. wllh •• ploe , which Upt'CIS. ,{ring. a symbolic "ion, or a funchon ""p"'",!{tlng lhe f""",hon 10 bc ploncd. For uample, {O glllph 'i ' 00 the ;nlr,,,a] I to Z (U';1I8 lbe ,tring form of •• plot). ()'pC: H eaplotl'x - 2'. (-12]) 1llo ,,,,ph wIll appear on {he in a"",w wIndow bhel.d "F,gnn: I"
.'p..
"'fCC"
ClIJplct.l DoIng M.lhtn)3licllI'ilh MATlJI.8
,
.. Pia- )2- The P",.bol. II
"
,
.r l on lite In,. ".1 [. \, 21
:::'~;;::::~:,:~~:~~~:;,: the string "'i~mcn\ or a 'ymbohc eXI'.nsion ~Iaph In h lU'" 3.2 wllh the follow;ng
:
• "umbe, "fway ' _ You Can <'hange (he 1ill. above the graph in th<
I
in tho fi gu,", ..;!>dow by ~l«lIng An" l'roper1 in... ,,-indow (JuII Iypt lho new I,ll. In !be bo~ ,·.,ti.al -..il " ,m yl&b
_ _ ,, - I 2 0 31) n.1lnI two numbe....'" lh...~n'c uf lilt hm;lom.1 u,,; Il0l11 ring" mull be i""'l"dcd.
,0.
:-'.!!"
'HhaOMed. oh.1'" or the graph Iqunrc. Iy"" ..ti. Oqul"., 11111 81'>0 makes Ihe ..cale .;.;~; l;';" -;;~; ,I the r .od V ronge' ha'-' cq ""llcnglh For T1Ingc< uf any lengths. the .. me ",.Ie on bmh ..... 5 withoul changlll, Ike ~h.pc by typing I III .. command will upand one or ,'''' ",",eo &1 "",,01 rOlIIOlIO. by MATLAI:I. _lUI nn~' ",'h an axI. COOmJ.1r.d hl~ 1110 one al>o\". or Iype axI.
..,.i.
l8, C;nophK'
"
".ph
tillbt to aUlomallcllly ""the f1I.I1g~s 10 ",dude the ~nti •• Type bdp .,.t. for mort JlO",bllitir\, fR~",,"mb<. to type _1' ..... fi ..."f you wanl 10,..,ad a sc,..,.nful "' a
"""""You c"" make "'''''" of the"" chan,e, dltttlly in the f'gu ... wllIdow as lOll .. ,II s« ,f you uplOl'" t~ plllI-do.. n meou\ on the figure "'indo!w', tool b.>r Ou •• ,peri.DC. II tha, doong !.O "II MIITLAB cummands in the command .. induw I'«" "I,,, """'" robuSlllo15, .specillly ,f you wanl 10 sa,'. you. command, in an M hie (s« Chlptel 4) In order '0 ... produ«'~ san", ,",ph lite, 00. To dOle the fig..,~, type c 10•• or clo.. .U, or ,imply click on li>c M io the Oppel nghl COmo. ofw ",,"dow, Se. S..,tion 8.5 for more ways to manipula'. gmph\.
3.B.3
G rup hlng wit h p l ot
The I)lot command work< 00 HCIO", of n"n"'nc~l d>la. The: .ynw. i, I)lot (X, Yl whe •• X WIll Y are ,''"''t(ll'$ of the.arne length I'ornampl. n
X
~
(123.1 y . . . ,
6' 5.1 1)1ot[X, Yl
.. .. , ..
, 0:,
"
"
",
.
.
"
"
. " "
,
..
,
Figure 3,3: P1Ollln~ I.",. S
z and lICoon!I03'.\ of succeS'''. poe"!> 00 a graph, and joim the poIOII "'th III'" "'gIIl(nll. So. In hgure 3.1. The plot command cooside", the ,",'cton X and Y to b< I" .. of tilt
MATLAIJ COIIDCCI' (l. ,I) 10 (2,6) 10 (3, !o), Th plot z' on ,he: the: inl.,,,,,1 from ·1 10 2 "e ',n.1 make I 1"1 X of z ,otluc>. and then Iype plot ex, X. " 2 I W• ....,d 10 """ enough .r: "olur, '0 cn,ure th.1 the re,ulLing gl'lph dl'l"'" by "COll"..,,,ng tilt dOl<" 100"-' smooth We' II u"" an ,ncremenl ofO, I. Thus a "'C'pe for gl'lphing the rarnoo1a i~: u H
_1,0.01,2/ plot ex, X. "21 X
~
The 1\:1"1, appe"'" in Figure 3.4. NOIC thaI W. ulord a ~micnloo '0 SUI'P"'I~ prinhng of 11>0 .11·e1en"'nl nctor x .
•
. I'i l""" H
I
,
•
Plot of. ParabOI.
flulll'lI Muldpk Cun-es Ibe old
.
1'101 ""d d..."" a DtW bolo!. on Thi~ command
""pili., .nd iliaw any new ,,,,phies 011 101' oftbt old.
(0 lOll (0 10]1
'.-pI-x) ud dDt .. )' ,n hau", 3.1 .",h.. in \hi . <1\2.1'1(( The
IIIfIIbllIhc ,eo;lnr 01 E coonl,nal(' mU~1 be ,pccihcd onto for cxh funell"" "'lOg pLQItcd.
3.8.5 Paramrlric Plots
Now CUlUidt' ...,,\her ",ample th~l ill".! ... t•• tl,. n.,ibllil~ uf pl ot The circle of ... d,us 1 c:enIIimtlt (II,U) can be "'-I',(ued in pum",mk funn as $ C09(:Zlr]), Y sin{2l11j, w!l&ft 11\10' r",m U 10 l. Though y IS nol .. prc'SI'd n. n fU'lCt;on of z, you can ca"ly
... gure 3_~: Poor Rend'tion of Un.! Cirde 1)"0'0 [muffieion' plot Poin ..
In'M 1'1<>. prodUCtd by tho .. command. (F'II.'. 3.", tho comers btl"«n ""'01"\,,,e Ii .... r.c,nltllt\ are e3liLy "i,ibl., d.-pile tho ,null ,,,,,,,,men' of 0.1. In .lOCh a S,'uation you ,llouLd "'Jl
-
,. .
•• "
I
"I
,l
I
••
". "
••
"
-
"
Figure 3,6: 1lte Unll Corelo
I"'lmotri<: plOIS w,ll bt "",d ,n Chaplet! 13 and 14.0 prodlOCe ph . .. port",',. of ,y.tcn,' of d,ffe",mial
..
ChaplCf J, /)wlllI M. ,ltem." e.
,,', mMATLA H
3"
,. -1'M pial of .. nl'ru,ion in IwO ''lIllab1c! " • I'~ of tit<' 1",'~I .. u"...s of] tit<' c.~' - un It.• Kit. of po,ntl ,n the ,r'1I pl.". ",11m: lit<' c~"",1S1OIl hll constant ,"3 ..... 1"Of IV' Ibc 1f>101 ClI\'\tl of the t~ptr,lion ,r' +II' are .,relu. and 110< '...... Is ""' tit<' square, 01.,[ of lit<' cin;lu, ConlOUr plou ..... prod..ced on MAT1.AU ",,'h __ b grid and id to R~'..ct a grid of pornt l ,n I .pec, r. rd .... clangular rc· CI "DIll', Flnl uK - -... -r '"-is u>cd hy cOlltOl1r 10 I"~ucc. _u' I I poa. with Ilptcified \JOK,ng. Thil grid con our P ot rn
!.:." ndu
tile apet;it1ed "'8ion.
Wecan ~. coolour 1'\01 of,rl + y' a. fon"",~, ~. ta YJ . . . .bgrld(R],O.Ol,], _3,0.0 1 ,]1 •
•~
aoat~(a.
Y. S. '3 • Y. '31
•• .ai• •qu ....
1
1
I)
-.
,
•
,
Figu .... ],1: COOlOOr Plot of .. '
8. + II'
1be plot 11 thown on hill"' 3.7 w~ Iwo.. u.ed _abgrid and MA1\.A H·. roIoo OOlallon 10 produce • p1d .. ,III .~in80.01 ,n both d,,« linn. We h,,,...1.0 u~d ....i . aquar. 10 fomr 1M _ ",a1c on boIh uti. You can >pKify how mllII)' le,'cl .. I. to 1'101 by oddrnl I n o!,lIonal ,nleger :ugu,,.,.,nl to a_tCOla. or .. h~1 JI,<,1 ..11 to plOl hy irw;luding an iI••khlloo~1 "~IOI ..gumenl Poreump1c. to plot lit<' ctrdn ofradiu. 1.,[2 and .13. Iype ..,. aOlltCNl'(s. Y. 1:. '1 . Y. ' l, (I 1 ]]) 10 dilltn,ui.h • ,'octor argument from n ~3lnr a' gument . the ,tttor a'Bu""ent ""u,1 contain M IeaII two .lfmc"\.<. Tho. if you "'anI to plot •• inl le 1",'01 S( I. yOlI should specify lhe _1eW.llwi.~_ I'orcUJnpk. 10 plol the d.cl" ofmd,u, 1. type ... _tCOlaIS, Y. 1: .'3 . Y. ' 3, (1111 RtpI';q [1 11 IIy 1 he............ Id g;>,<, you one
.1,9, Clku!u.
3,9
"
Cu lcuilis
Tht dt ff COl1lmand d,ff"ftnuatt. a symbolic npreu,oo. and lht int command perf....... onlcinmon, !'of u.unple. 10 d, ffol.miale ~", you could .ype:
»
.~
x n
H dirt l . · n, x l
"0" x'n'nlx The 3n'''W i1 <'Of lOt i. but is nOl writlen ,n!1Ie fOfm you nuoy be Pled 10, II..,n-' You can Mmplify lhe a""wel ,t< fOllows: ~ o BlnplitYI ... B)
"0" x· In-I) 'n To '''''gnll •. PIC.1Ie i"" command >0
in" l"o.lx), xl
"0"
1Ii,,(x)
flf .
ha,..,
b.on dNlared '0 bc a Iymbollc , ... abl. "iIIltel. you would IQ .ypo B~ X bor"",'he comINnd abo...,,) 1ltis i, an .~"",plc of on ,ndtfin," ,nl.g:ral. You.3tl also compute a definite ' 01.",,1 •• foil"",., •• i"tlx - 2°.xp l· xl, x, - 5, 5 ) h;o.d not
"0" l7'oxp(-5»17'expI5) SO,",,, illlCiral. c"nnol b. done symbolic ally. and .n tho", ca"" MATLA B ",'urnS 11,. une,"a,,,,,.d "" eg",l, .> h.r41nt.~r.1 • in"llogllox - 2jo.xpl · x · ll, K, 0, 1) Warning, Explicit int"9ra1 could not be found. ~ In B}'In. int at 58 hardinteqral int(logll'x-2I'exp)-x-2)),x 0.,1) In lhal UIC you can ulC MATLA B lOeomputt I nUI1"I<"",,1 .ppro~lInali"" 10 lht d<'6Me ,n· ttCl"I ,n """ of Iwo "'a)', F,nt. MATlA B "'" ''',0 bu,l! on nUl1"l<';"aI LDttg13lKK1 !OUlones. qu.d and qu.dl The """ " .• will PIC ...... " qu"n Thul. o. qu.dI C_Ix ) log Cl.x, ·l) .·.xp l· • • ·l) . O. 1)
"""0.15l9 Ilere i. lhe ~ood method u,ong d ou.bl. >. d o ubl.lhardln". gral)
Of
"Pa
"0"0.1539 In ,he -,«""d mrthod MAH AB USCI !he S)'mbo!i( Mmh TooIlHu to nu",
oft~n
be c\,aIUOIrd ' • l ..t(log(.U.n C><, 0, 11 integral c o u l d n ot be found.
o ..
11
'3J.0 Some TIps and Reminders W.bI¥II iIIlrOdIKe4 qui\( • few command, and ~onc~pls in Ih, . chaple •. h will w..c you """ 10 bccolflt ( omfortahle and proficient wilh them. In particular. the me of strings IDlll)'lllbolk u lftnion\ i~ oft. n quite ,ubllc and can trip up even npenen«d usc ... The 11M! WI)' 10 learn h tlllO\I&b "periencc and by USing online he lp (!Cc Sectio n 2 . ~) 10 ace,," MA11.AB docuIDCotah<m when DC«;"')'. KIep i11 l11lnd 11111 PK"iounfu~ Siale thaI wln,nu·rf.", I'IIIIn o;aJ,:ulltion. Thi. can of\(o be 1i.«I by 1yp,ol c h .... .11 10 cle'" all v;an:.bk
,"'H o.
.... f\IncDoa 4efinl"""'.
Chapter 4 Using M-files and M-books ]nlh" eh'pler we d.",rib( .ff.~li'. pf'{)Ccdu ••• for ,,'orklng wilh MArLAR. and for P'<'P",. 1Il8 Il!Id r ... ","un~ lhe r.\UI" of a MATI-AD "";(>11.
4.1
The MATLAn Desktop
I.:ffec1i,-. ",e of MATL/\II "'qui.... understand,ng lh.
component. of the MATI-AB [k;llop. While you can cmtomll.c )'0"' [),,,llop 10 ~Ul{ yflU' OW" " . .',I-. "'. will '''"11l< in Ihl< chap"', 1hailOO ""amed lho deflull c""figu .... lion. ,,-;th lilt WOliup"". ll rov,,,,, and Cu.rr"l [)ueclOfy Brov.'\O, In tilt uppt. Idl. lhe Command 111\1(1) '" u.. 10\<<< left. varil)U$
ha,..,
and lhe Command Window on lhe tight.
4.1.1
The Workspa ce
The Wo,hpacc Hrov."$C' d'lpl.),~ • labl. thai contain •. (or each cUr numerical ami)" of d." "double" (alld wm~ other d."", of variabl.,), you can .1..0 chan,e Ihe emri .. of the IllTIIY dir«tly in Ille edilor. w'II>I.m l uling the Cotmnand \\~nd{J\'''' u,inllilt monu il.ml in lilt Amly Edit ... loolbM, You can reroo,'. a ''2Ilable from lilt Worl,poore by Stl«tin, 'lIn Ille WOIhp.ou Brow~. and liItn p<"",nllilt '"Dekk'" 0011011, If you mUll ,nlelTllpl. St"ion. lou.at! ~\"O,d tho nttd 10 I'ttQmpule e".f}!hina lIlt. by savmM Ihe cu"..,nl WOl~\pace, Simply type •• v. JO;yUl. 10 » ... liIt , ..1"., of.lI ."""Illly d.flned variahks 'n • m. called my tile . fMt.. To '~"c ooly I"" \"1,, •• of Ihe \'ariabk, X .nd T. type » •• v. JO;yfil. X T When you ~lar1 • _ " ""011 .1Id 'un! 10 rttO\ C' the ,"1.1 ..... of lhmc 'lfl.b1c: •• ,,\C the lo.d command, For ulUllfIlc. Iyp,ng load II\YU1. re, lO'U Ille ".luc. of all lilt able. siored ,n tile me my f 110 . mat.
,'.ri·
Ch,tpr~r 4
V, ,,,8' "'.files mcJ M·boob
DIrtdOry lind MATJ.AIII'llth
ptabI,,.,. ~ foldor or d,m:rOf)' ... he ", I OU c:m ~rore aU YOIIr M AT· i, to u,e rhe work fo lder nrh",,,s. (IIO'. ;e'"C:r. I folder.l Some ptople I 'I
i
" When you aft work ing 0" n parli cul >r pl'oj«: •. you can
:
;;~;;;'" 'd' ;" ";."'S
ill cuncnl workong dn'-':Io!)'.
11l1S is the
=~~~~::':~!':":.';:'r,:dfrom unng Ih. yourr omOland ~"ion a rclines,"ed nl..c s4 you ify . U~(urile com_pec mand.
.
,
\IIII&lPI'U •• Idd. dt=lory 10 lhe fIIIlh Ii" by u\ in, . """paU" CI UIa&bhrll •• iii.. in • dll'«IOI)' if you .nhc r om. il the cum:nl ",Ql kinll Whe n ) ou add a dll«tory 10 the p;llh . tile liles il COll1ai ns . l1Ic potd the palh i. by """8 tile Com:m Brooa-WI" and Ih< ScI Path dialog bo, 1be Curnn l 1)ll«lory Ilmwsrr i. paf1 of J)rl Ii.. In tbt <1<1.011 y I"in~ 1o menu il<m Stl Palh ... Optro ..,," oflhe", 1.... 0 brow~'$ i. Iy If.."plllUlOry The !ltr'lIl! f>lATLAD palh IncludtJ .111he t oo l box folders for llllhe IOOIbo.~. ,n,ulll h 1'1 ,,1<, Ih.I ... hen you add • fold.. to ..... RlKh ""th. ~"" <,n r ilh~r J"'I .dd II hy II, cif or . 1", me: Ih" ,\(hl ... llh S"hfpldus ...
,...xn..
!he
-
1be Informal",n d"pl.)rd in tile Sr l l'alh dl al!>g 1><,. con .Iw "" obI.,ncd from l11e
To "'. Ihe CUllTnl .... or ~'"g dlI«I01)'. I)'P" pwd, To h" W fIIrs In lhe curretd chrr
C(lIDIIIIIId Itne
,.,.!he
4./ 1M MAnABll.. 'krop
4.1.3 Thl' Comnulnd History SUPPI''''' )OU Won l lO ...,he Problem 2 in Problem ~I A of thi~ book. In thai prohlem. )OU .'e ",led 10 calcu l'le Ihe Vllu.. of sin(O 1)/0.1, 5in(001)/0 01. ~nd 5io(0001 )/0.001 10 I~ digi". Such I .impl. problom can be worked dl
,
~~
Y • • in( x) / x
0.9984 Afrer compleling 3 calculol ion. you "" 111 ofl.n ,eali z. rIl . l !ht '-.lull IS no, ""hll you In· I.ndrd. In Ih ••",mple IboH ... e inrondtd ," ad I '''''IOC In<",., WIth thr« oomponrnl!. and ". gOl ooly. lingle number, I'urthc~. MATLAI:I d,,,,laynl ooly·1 dil"!. nor I.~. To display 11> dlgils. we noed the command f o ....,. l o ng. '" on. could .imply add Ih
,
0.9983506581]858 lIu' Ihi. il silil not .. h.1 "" w;utl; " ....·.m.d thltt numh<-" lnd "-' only got Of>< A~. mall., of btl. )·ou mighl gues. Ih.llho one we gor '~5in(0.l )/0.1. bul nen tha' II IIlCOrT«t .in(O.I)/O,) 0.99!i33 11 6&16S2~. Which diffOllo from Iho obo"e an,..-er ,l.1J1ing In (ho S,b do<:;11lI1 place. [n,l.ad. O.!l'.»I35065813.~;. the "Ie,ul "'Iu.are!'. 'ppro" mo.lioo 10. ··wl.(lon" of lhe .;mulla........ "'I.llions ¥ ,inCz)/ r. ",th r 0_1. 0_01. 11.001. "Tho m,,(a).c i, Ihar " .• usod lho wrong Iymbol / for d",sion. m,(nd of "'~-Ioriud" d"' i,;o" • /. 1I0,,·.an ..... fficienlly repeallhe cokulalioo (0 ge( rho COrT«I aM"'''? Of cour",. in Ihis ump1c "'Imple .. e oould jU'1 r~I)'pe d n CK) ./x. B., !fi)'pl"1 i. llmeI'd .arli" In • >l'h;on. or e,'en In a pte"iou! MATLAB
",,,;on l'or lhe", PUIJlOSC". you can rale a.:h"nl.se or Iho' oornrnand hIliOf)' fUll"" of MAT· l.AII. U'" lho up and do.... n arrow loy. in lho Command Window 10 ",roll through the command< (hal )'0" ha,'e utary. and lho ... I'ttU the E!-.'TrR or RUlIlr; key (0 lell MATLAll 10 .,... Ipal. lho ronull.1nd Allemall\ .• ly. you can toral. lho old command you ......110 copy or rr1O(IIfy in lho Command 11",0f)' "'Indow in rho I...... lofl OOm.. of lho Iksk,op nllJ~ .... ,....s....' "-III indudt oonlll1.II<1I from ,,/I of your P"',iOll' MATL4.B !.C",on,. allea'i unl, 1you ",;nm.hn lho comm...d h'l'Of)' h'1.)"Thon)ou can ling Iho' old comlll.,1d inlO lho Command Window .. ,th 1M mo ..e: and modify II,f )'OU Ilk •.
..
"
Command Window 10 c",a~ an M·fiLo:. ftlts and funclion M ·fik!
.. ordlIIIIY IeII iiI<' .. containing
MATLAIl cummandl You can c",'le . nd word proccu ur Ih.1 i. cap.ble of,;.av,og files as plain ,"om"",n" Ihal )UU u>c MATLAB 's buill · i.. M·fi1. editor. I 'Ii II ch«ks for conn'" MATLAII in !ptel_1 COIOTl I , m bill<'. and In You can ..
0 ......, .• opIlOII
&3.1 8crtpl M·ftJn; WI .. _ JOIIIIIO >how )00 how 10 CO!I'UUCI I K ripl M flic 10 .01,.• Problem 2 of " . .." Sll /It. U.e tbc EdIlOl"Jl)ebllU" 10 crtal•• me conll,nmg lhe followmg !oI\CS:
117,,1 ' , • • lO.i,
o.n.
0.0011/
.
~. I~I.,.I.
lilt I II with tbc n..... problem2 ,m in you. wo,~ing direclory. 01" in lOme d''''Clory - -,..- piIh. u.in, tbc Sal" M.,. "",nu ilem In lhe ~·II . "",nu of Ihe .dilOl", You can _Ibt ft~ ItI)' WIY you Ilk. f,ubjecllO lhe usual n.nuog •• miclion! O!I your opt",,,"g iJrT d),1rtIt \he m" lum. i, m,,,,,,lalory. Nol. lUI the ""'IOfIDe""",,' adds Ihis sum . ,"'1' ukl'll' Ywl;llllfil MATLAB to Nn (0' r.r~u',) thl! Kript by Iypm, probl.-2 ;n tile MAT. 1JtB0:cmar>d W, ......... I""" mu,IM/type the ....... ",en,ion ho"" MATI.AB aUl",,uI ' IaIb Id4I II ... hon 'un:hini for M·fi1e.j The OUlpul "'" noIllIe commands I~I prOOuce M
•
will he
...u, fOE
' t 1-. • • 10 . 1. G.O l , 0.001, O.OOO l },
'II . -'Illa, . / . ud \hen nrn Ihr lnodifl.d Kripl by lypinK p robl-'! Bo IU'" lOla." you. change5 10 pl'obl.1II2 .111 h.-.I; Olltrrw;amplc. Ill< proaram WOn ",II cau§c all fu~u", IICIIDHKAI OUtpull" be d"I'I.)<:<J
",lit ](,
..,.,'C"
"
~:choinl: Command •• "~lIlCnlioncd 01:>0'-0. !he c<.lf\'Ul\aJ>d, in. KOp! M·file will DOl .ul0rnalJ~a]Jy be d"pla)d In Iht Command Willdow. If)OII WaIlt the commands 10 be dl~rl.ycd along WIth the •• ,yIt., u'"' the • .,ho command.
leho on to~t
1 o",,,
" • (D.I, 0.01. 0.00111 y • • 1"'h.) ./x lebo ott /\ddlng CommtnlS. II i. 'nlportam!O include Cornrnonl. In a \Cnp! />I-file. Tht", comme"l< might uplain Wh.l" being done In lilt calculalion, or might int.rp~llhc ",.uils of the calculatIOn. Any lin. In. ocTi", M (,I" lhall>tg,n. " 'lib I I"'=nl"", t"l il 1.ealed b. conu"."! and " not nccutcd by MA1l.AH H,,,,,,. Hf<JOO of probln02 ....... ith I few commenh added. lebo on
r'I,,,.
% Bol uti o n to Probl..
..t
A, 'robl..
~
" Turn on 15 digit 4hphy fo ..... 10"11 (O.l. 0.01. Y _ dn(It) ./x
It.
0.00111
" Thl •• valu •• tllultr ••• tb. tlct that tha It-it of " aln(><)/x ••
It
approach •• 0
t. 1.
\I' " n" "'
lebo off
When addin, commen11 10 a script M· fil ......... mlxr to P"I I ",«
y""
m.
,hould mn< .uro mal • ""nlll M is ~If ,(oota",.d . StructurinK Scripl M·fltt5. ""affected by oln<-r "ariable, thnl )'W m,ght """. defir.rd e].ewhcro ,n lbe MARAB lC\lion. and un<:QmIpitd by leflO"er gnrhicI, Your ""npu should uluJJJy""'" wi!h 1hc "l.a r .11 comm;ond, "'hlCh .... IU ..... WI f'K"JOUs ""f,n,ilOO' of 'Mllh~1 do not alleet tlx "'lull<. Tho, "''''y. )'our output drpo:nds only on tlH: ~ucnce of command. in the M -file, f Oti , ,5 not always SIIiclly """"SIll/)'; ) 01.1 'tally only noed to clear 1110", .,riabk, Ih,t yOll u", m tlH: M file itlClf) You ~ld oJw .,1hc1 mdud< Ihr doa. all comtnIJId at 1hc bt.,nn,n. of an M-filo Ihl! .",11<1 gnp/lICO. to . 10..< III .rarhlCs ""ndoIJ,'s and ..., ",ith I clean slat •. or else open I""W /'JI"'" w,nd"", "'lIh filJUr. Ixfon:' 1'1<>1I,n,_ (Oihtrwisr. If""',.., i. an open g.... I"".1 "'indov-' ,,"h b ol c1 5Cllo "on:' ~ .. n your""w S"'phicJ Will bt
atkItd 10 1111' old fi,~ .. ,rIdow_ l II.,..,,, our .umple of. <:Omplc:l• . careful. .omm
M_••••• " l i.... In the co·_ ••rut window
" - YMtOI'
.t ...1"••
of the i .. CI.~n4.Dt variahl.
0.01. 0.001],
sal? . . . ,..1r.4 .. 1"••
• cbCe,.'.
E . .s ft1.... ill".tnt. the fe"t that ttl. Uait of .&IIC•• ' . . . . qlDroacpe 0 i . -.qya1 to 1-
• h
011
. " hDdlotI M·Bles ~ptll' 1"0<:(" ,.,'ern! 11m., for d.ffc
In nuny ,uuauoos, 110...... ,'.,. ,[ i. I1\OfC conwnicm. or .'''" , ;n, (cad of an aoooymous 01 .nlin<: funchon mil"'" runc,ioo, and M ·fil .. aLl allow yOll 10 .~'.nd the fO""liornohly of ..... n funclioo! .
1 ''lIlucs of ",n(x)l" for you"~nllOfiool"'".rnallr51
_n1,.\"",ofb. SUf'PD'C . ln . of. fOl' .. hKh ' ill! 10 ")/(10 -~) and ] av« 10 l{o dlg, l, II • •• I. a fm>cuoo M_file
1nepower .111 dr,;,ncd 10 wh'. lh" l'robkm ,. 's\1_ :r •• tn.pet .... (c) • I"''OWIA ~~~" ,in(x)/x fo .. x • lO" ( · b). _ ...r. b. 1 •..•• c. be ,t loq It • llCl' 1'. C.ia(lO. · ( · b)) ./ lO. '(-b))'1 LIb &lCnpl M file. a fu ....'lion M ,m. II. r b m le.\l file Om ~hould ~Iidt m yom MA1l.AB path The 1i,,1 II .... of ~ fil. \ l¥IS wllh lhe f"...c ~lOi'l command. whICh
J,2. M-Uk.
11
,&nll(,es the m. a. I fUDI',ioo M -file. NOIe "'.. the F..d'IO,II~oo"er colon Ihi. lpeciol word ,n 1:>1>"" n.. fin, Ii... of the M·file 'pte,fin ,he !\lime of the fUDI',ioo and deocnbe. both ," IOPU' arsumenl' lor pata1tIoOltl1).oo. ". outpu' .'al... , In 1/'" •• ample, the funclion IS calkd .1".pow.r 1bc fil< "",me (",thotn lhe .to .... n'ion) and the funcllon I\.1me .hould rn&lCh Wben yoo erl'~l. "'i.... w funtllon M·(,I. ,n an u""ded ednor ....·indow and !-tIN! S~' •• the Ild'lOrl\"koogger t - , 10 .... 11 the hi. 81 n .. p"",,~ . '" llH: funclion in OUr e.ample lal •• 0 ... ,npu' argumenl ... h,eh i. cIU.d c in.ide ,be M·fil •. It al", ",'um, one OU'PUI ,·.Iu., When Ihe fun.lion fin"he, ..«ulInK. Ih. ,..Iu. ofy w,1I be ~.,gn.d '0 an. (by comm.nd will aU1oma1icaily reui.,'. Ih" lI,fu,,"," on. Por ... rnpk: H h.lp .tn.pO. ... 10",-b), SIN~POW~R computes sin,xl/x for x ..,hor. b 1. ...• c. "The: ",nuinmg lincs of ,he M-fik define the fu"c in,ioSe the fUI\CliOll and.", un,daled 10 any "arilbl
COII"'''"g
",,,,Of
m
0.99833416646828 0.99998333341661 0.99999983333334 0.99999999833333 0.99999999998333 No ... of ,he ''Jlutl of b from I .0 ~ yieldl ,he
4.2.3
"","c, 10 the quoII,on ". JIOK"d lbo;n" by ')'ping
Cells
A new f(aluft In MATI.AB 7 aHows one ,0 d",de an M \tIc ,nl0 >llbun,IHalled crlls_ "l'hi. ,~(.ptcilily u .. rul,f you, M.fil. il loog 01 ,f )'''u..., gO'"1 to pub~hh iI, al npla,ned below in Se~lIon 4.4. 3. The easi." "'ay 10 d" ,de an M- file '"to cdl' " 10 ,n,.n a commenl as tbe ",itlo" Oflh. edllh .. foll"",'<) .. orting "'''h a doubl. pt",.n' hne ( .... hieh will "gn (%%) follo ..·.d by a ,pace. If )'ou open the M ftl. '" ,he Ildnorllkbugg~, and eh •• on !>:"able C,U Mode ,n the Cfl1 ntr"u. ,hen ",hen you d ... ~ 1()mO"'he", ,n the M.file. 'he ccl1lhll ~onla,n. ",.,Iocalion w,lI be hlghbghled in hgh. gr«n. rOIl cao .,..."UI. tha. cell
"0'''.
;:;.~:;~,,~:;~,;:r;~:~~.7 ". A_.
"En ' •••c.n.nd ' Cd l" iron • . If imn yoo ·-", _1"",. Ad .-arlC<'··
If
•
__
eh ... , . In JOl t 0<1" cell and do not w am to run .. , f Qn<e )'OU h~'. ". n.hIM cell m<>
you ,,'n......-
Dlndrr Olem in the C.II "",nu
• o:ommand Of group of comm.ntl, . oould ~ ",pe.led
to"'" a fOl" 't.lrmenl.
'air
.UI.".,.Bl
Sl'y.rnl tim.,"
Ik"' '' a Mmple eJ<.mple th.n
and end' ""ib the end .ta.eme nl. Tbe cOllUlW>ll
loW of "".. lime., or>ee for each value of n from '2 to ,uwrr~' inlem>
I
II col",. the coml1\~nd, for and end in bl ue. II t
Ii
in belwe'" (., we dId abo'-e) ; the Hdl ' type for in the Comm. nd Window. MAT.
I (
Iii
I
command. at which lime
new pfOmpl .
iiFhtNy.)OII want 10 !he ""ult. Qf you. M·files You may wam to !how 1D)'lIUI' <1 ...,....>1.' or ,ubout tlltm 10 YOII ' ,n"f1.IClor. You can I""'",nl rOUt ""u!ts ..,~aa1 nJ" y"",.n Ie, !he ",adcr "'" yuu. M· me. or )"" can I"""ide jusl !he ""' pIlL fIJaIly." "C ,h.n .. c in Section . 4.4 1 Ind 4 . ~ .4. )"" eM "publish·· )our M-file IacwPCdit Jour MATLAD oull"'l '010 a doc umrol, su<;h :as one prcp;lIcd " " h Word or IfJlIX When p"",nling !he final re,"\". you mu" re member Ihallhe: r~adcr is nOl ""..Iy ao fImlU., With !he M·M.", you i, i., you, r~spons 'I>,loly 10 provide gu,da'lCe, M MI ,mparIIIdIy. )'011 .hould include numerou s commeOIS. Vuur comment' .hou ld e~plam w hat is heI'II uleuilled ,,' Ihe reaMr can und. " 'lUld you, I'roc<:dUIC5 . nd Ilmiell'c!, You should IIJo inlc.".... u.. re,ull< of ,IK: cakulalion. Ohla imnll a numbe, or a gmph " flllcly an IIIiIrpIle 1/U"1'I 10. quc<1ion, )'OU mus, npl. in Wh>l lhe numbcn and II"lphs ",un, H JOIII' R'adtn "'" "mply 1I0ing to run you , M filu {all ,he: ""'y through ,n • sm... pull-lIItn lOU ,hould male Iobe",1 uS(' of 11K: pau . . command. Each ti me M AT-
at"
"
l.AU ",.. lIts a ~au •• rornm.1J\d. n "1IpS nec~lIn, .1It M· file un.il !he "Iott prt'!Iot! • Ley. I'au..e. should be plac.d aflc!. ca.·h Cl.lmmon., aflc! •• ""h ""ph. and commar>d • •ha. prodUCt Importanl OUlpul. Tbo:sc ~Ulot. allow lilt ...",et'l.I ",ad and undc-"WId your I.,ulh. Nu'~: An al.ema",·. tu u.ing pau ... whIch is benet if)'Ou a", guing to "publllh" llle M·m .... tl,l "'para ••• h. M-m. intu cells", explaIned In S.c.il,ln 4.2. 3. and k. yI,IUI I•• dc. Mep .h.ougll.he me nne ccll 01 a lime.
"")0'
4.4. 1
1· I't.'S~ nt ing
ar'et
G rll llhies
", 1II.lIeato«! In Chaplet J. grarh",s appc;u In ... parate .... ,ndow You can p.-in.!he CUrKnt figure hy .. Iecting!he I'Mnt •.• Ilc!m from !he Flle menu In !he pap/llC! WI!>dow_ Altcl... i,..ly. tilt pdnt cOfllfl1o&lld (",lhou. any argumen'l] caUIot,!he figure in !he currenl grllil/IIC\ w",dow to be pnnled 00 you. defouh p.-i •• er Slnc. you probably don'l "' '''''1,1 pnn.,hc glaphiC5 eycry lIme you 1\," a script. you should no. Include a b= pdnt cl,lmm"nd \0 an M·fiIe. In" •• d, YI.IU Ihou ld Ulot a f""" uf pdllt oomnuruJ lh>' !(nd~ !he: I,IUlpU"1.I a file. It ...IM> helpful tl,l gi,·. tt.wnabl. Jill., ,1,1 YI,IUI flilutt. and to in ...... pau.a cl,lmmands inll,l yuu. ,,"np' '" thlt viewets ha,..,. eh....,.,.o $U the 68Utt bcfl,l", the tt" I,If lite ",ril" execule, h" • • ample: xx • 0,G,2:2·pi, pletlxx, dlllxxl) _ PUt • titla en tha figura. titl.(' rigura A, lilla CUrva')
,h.
pa....
_ Ster. th. ~ra~h in th. fila figuraA ••p •• prillt -4.p. figuraA 11.. foml I,Ir.he priDt command used in this ",,;p' doe, nn,,..nd anything '0 the printer In' lead. 11 cau
..
ld', an.aI~ 't lIIis .ol.,ioo. We $WI by defining the ...."..h",. (a. fraeholu] muluplcl of .. fRlIIIO 10 2R) Tht figure CO/1\I1'WId optn. allC .. vaphic. window. and the bol" o il _!!lind Ie" MATLAB know thai we """"' 10 draw !.C\'cral ~u"~, on the WIle set of ues. Tbr Imu "',,, ... n f o r and eM COOSbMe .. for-loop.•, dncnbed abcro~. The IInpo"an! put of Ibr loop j. tb< p lot rorornmd . .... hICh pIocs (he du,1\'d "ne cur\"cJ. We ,nSnICd ,. . "0 oft: f~ SO thai Vie only oee lho loop command. once In ,he CQfIlffilInd Wmdow Fllwly..." lurn « hoing back 011 .. fit. uiton,lhe loop. lilt: b o ld o ff to relu", die anphto; .....irKlow. and t i t h \be fil ll'''_
4A.2 Prru)' Printing If. I•• ~ymbolic c~r", s'ign, ,he" Iyplll!! puttyCn) ""play • • In prmy pnm fnrml !. wllkb uw. multipl. lin.~ on your ~ru n to ;m;tote wriu"n Illathematlc,. The ,~sull is ofIm III!ft ruily ,nd than It.. lIef,ull one· lI ... OOlput fonnal An importanl future IIr DCtItl:J' .- thillt ".""" long e',,",$O iool III fll ,,·,t hm lbe narginl (80 (!.a'kt,," WIde) IIf • 'I M'''Il ·wud willllovo If yoor . ymbolic OUlpul lS Ioog e_gII '" u~nd pasI the nlP"
edg. of your "',!\dow. 11 probably ... ill be lrunealtd "hen you pom Y"'" ""Ipp
4.4.3
y"" can .Iso the "publish" icon on the cdlloolbar ("hich appc:m under Ih. regular .dllor loolb:u") in Ih. Edilo.lOebullliOl. The def"ull. "hich " .• hne found wor,,", bell. " 10 pUhli5h 10 html (i.r .. a .... b ",g.l. bul yoo can _1'0 publish 10 3 Wom docun .. nl or !'ollcr!"''''1 rr."'nlalion (on a PC) Or 10 • I~.x document (regard I." of the op all' prinlCd. foll""td by 1ht OUlpul (In a lighl", color), You can also fornllll ....... of Ihe ~>\, in"," OOlleltd hSII. riC.. ul mllhe I nsrr1 Tnt Markup lIem In 1ht Cell menu of Ihe EdjwrlOebuUtr. ( Remember 10 do thl5 bt-fo~ publlshlns; II wIll ha,·. no .ffect_) If you all' gomg 10 "publish" you. M-file. Ihe .cb.o COIl\rN.nd i5 unnece..ary. ,inee lhe lc.l the M· hie. color-<:odcd :u in the EdII(Knkbugg<"" .uloml1lcally appean. Also. lhe pall •• eon,mand IS unncttssary (e,,,n I nUlu~. ,,~ 11 Inl0
othe"""
or
Ihnl il
"-4.4
"1'1"'"" in a ""w WIndow. !\i· books
AOOIher sophl>11C81td wa y oJ pre5Cnunl M ATI.A~ OIIlpul . wlthoul """ins 10 prodoce an M .III. ('''1. MlCrosofl "hnJ documenl. " .. OOpolal ln. 1t~1. MATf.AO COfIIlllamh. ami gr.>phICli . (Th" option;s noIl>,"I. bIt 00 UNIX nlllChlllu . 001On l lOCh ronlpull. MATLAD commands ami pap/'llC. InIO' 11Tj.' documenl WIth 1ht help of Ihe 11ll;X cOinmand \iocll1dqu~ic. In r... l. lhal " how " e prodU«d the grap/'litl In Ihi ' boo~ l) A . ;ml'l. fi"l 'rprWCh is 10 p.. pa .. . !\oro docun.. nl \I Ilh nplan. lory C011U1lCnil. ami 10 p'~k: In ooe ', MATLAD cornmands (0"" can do Ih,. In oooU,.. f"nl) or ,nciOOc 0...,', M.m., (u''''8 .'II ~". froll\ \\'IJId's l"s.r1 n.. nu ). ,",nally. l '-'" can ", Ite in lile g ... phi.. us"'8 l'ir!u lT" ~'rum ~·lIr ... from the 1",.r1 "", nil '" \\\mJ You ~hould h,,·c fi!l;1 ",'.01 11K- graphic! In pog. t i H . Or e pa fonna' A "lOre roh"U,lapproach. which I<mi, 10 be !tIICu mhe"""", ;n lhe 1000g run. ;, 10 cnahlc M ·bools 011 you. mach,,,,,. An M·booL, IS. 1I'0 nt doc un .. n! \11th emhtdded <.teutlble
",n .
Ch'pt~r J
V.IIIS M-fi!~$ 4I,d AI.boob
mw 1nKn'!. "i ~ the mlc,lII" d,ary of Vi
Vou Command Window, or by , llf1'"11 : ami ~ Iecling /II-book . d ot a1 111O: l)ocullIc nl , dot does not ah eady um on you. compute" you I>eed '0
_You ..... y be prompted for the '..,Bion of II'Ofd ,ha. you are • Y''" mUSI enable eAccot;on of macros ,n \1'<>nJ n ns "Secunty Le,'.I." To do Ihi •• 00 the T...... 5 mtnu SKurlty'ab. Undo. "Macro 5(OnlY", c1 ic~ 1\IIor 1O 'ab,
w"""'",,'
,,",,'
,''''.''~m, \\'"n/ at the IOP_ I command and : ,:. ;;';;' ~~,< ~,"d ~'~'I F.'·8Iu n l~ Cell will .... Iu. , • •he commaoo and Sl'OO the output bad . •o 'i cells" (MATLA Il inpuI) in llrecn Coorie. OUlput) , I Iype. Vou h3v" "" optioo (whorl> 0ptl0Ds. .. ' Iem in the NoI.book mrnu) of ha~inll figure 1M PH Kt*'lIItly, or bivin, them appellr in til. M·book. Of I>oIh. 1D _ . IIFt"1. M-booll beha", I,le M,file.: you can mod,fy them and run th<m agli n Uyou fIDd you mi. l)'pcd a command or Wi ll. '0 change a ( Ommllnd, yoo c all ao_k lo!be ippiopi iatc input cd l, c h""~ It, and then re~vaJU3le It. The new wID I.... ' f tile old No one need. 10 !.:now you originally made a mi slakel
U5
Pnpariq Homework Solulloll$
;.;;;;' ; ;' U J' I , ihf:r lwak problcnu
f t 11II1UIWi~
the "cps \It.al yoo Can UK 10 prep.1fC the $
JWI-".OII IIIIlI8 MATl.AB, I
DrecIde IIrbtlht. you..., IOing to ,ubmil "" M.fik (for yoo. reWe" o run hu" .. U). a "pIIbIlIbect' M file, • wOld-proc.s, ing dotumtnl with p'''trd In MATI.AB code and .",1",• • or • M ,book Which of th<~ you t~ lTIIy depend on the cooligur.u,oo of )'O\Il lDIChinc IIlId the prefert ncc:5 of yoo. ,"lintel... or aud,c llCe ,
2. Reprdle .. of which m.thnd you choo.. , ~ su'" yOll ha,'e " ,pl amet! what you are
doinJ III comn"lCnl lines (if yoo produte III M· fiI.
or publ "hcd M·file) or lIC<:ompa. oyiJIa tel' (if you produce a word· procc5~d
l
Be SUR' ' 0 go bac l llld correct al l mon, Try some of the dc""gRlng ,ec:hnKllICs IiIIed below in StthOfl J.~, ¥ootfude. docs not _d 10 see falst: llam; delete th<m from)'DIU t"Hle or M·book
U you w.o' ' 0 Creat. a publi.hed docUlllO:nt f'OOlI """PI M· f,le, here ,I ,)", procedure hma.tlofiru1h:
I, C'~'I~ an M· f,l~ ,n you. CUrren, ""OIlln, d"':<:lory 10 trnld the ...,Iution. Kttp th( M file open on 1M Ed.tOl/I).""U.' "'hllc ~OU Ill. Nlllnl" (y"" might W1IIIIO incl""" echo 0 .. In )'our M-file ........ 'lOU can >ce . .ch ~ 100ttber with iu 001",,1.) Sclt<:, F.nable Cell M od. and on ... n cell d'."'OfII lOS nttded. ~ialJy if the file i, lon,OI . ",.r •• nlOft 1har1 one fi,u",_
2. ('onlln". NllmK aDd runnin, lhe M·flle u""1),00 arc ronfi
J . Add com men" 10 your M-fil. 10 .. plam th. m"lhod be:in, u... d to soly. rJr. proN.nI un" In intorp,e, ,ho ""u lts. llre!oe C()mmenl, con be: 'o,matted wIth the Cell menu Gi,·. 1I,lc, to yom
fr,"",,_
.,the.
4 Once you. M·filo i, .cady. publilh " in tbe: doco"",n' 'ype yoo de,i", WlrJr ,be: publhb command o' the ·..... bli.h'· P"'.i ...... th( ""all. and if _",. ""', .,Iber .d,llt.e documem If01 ",amplc. '" Illlln 1I,'OnI 01 an hta l N.tor) 01 else chanB" ,be: M· fil. aDd publi'" II apln
1<"" a.
To ,1I.mar. rbe: ""ul1< of rJri, procell. "''' ...... '.11 Problem 2 of Problem ScI A_ H.", " the ""uh of puhh"',ng (10 I5TDXI a ~np1 M frle rJr.t ...,11"" th( problem and add. a gnrrpillell pr• ...,nl~II"" of the .... u\". You cln nammc th( M· file ....., prodllo
http : / /www.math.umd.edu/undergraduate /.chol/ode/Hfileo/
Solution to I'roblcm Sct A, I'rohlcm 2 Co ntents • Numcncal Soluli"" • 0"'1"11<.1Soluti""
NUlIIericll1 Solul ion
fOl'fllll,t lon \l cie
[ 0.1 , 0 . 0 1. 0.001)
, compute t he doslred va luee Y aln (X) ./X , Theae va ill e s illuBtrate the fact that the limit o f , .In( x )l x ''~ x "ppro"chc~ 0 18 1.
. .<)99999833 ,134
9999833134 66
GnphiCM Solution
clon all azplotl'alnlx)/x', hold on plotlX, 'f. '0')
(00.1))
title 'The Limit of nlnlx)!x' hold off Tho LWM 01 oItI(.y,
,
~
,
.-.-
~
~
~
e_\
'-1
.-..-.- ... ... •• .M
•
~
~
•
•• •• •• .. •• •• ,
~
,
~
~
•
~
"
4.5
Debugging Your M-filc..o;
You can ffi'l you. M·lllfc ropc ... dly wmlu prod...:u 1M dt",.d ou'pu1. Generally. you ",,11 Nn llIe I<:,i", each lime you ffitllM filt. If 1M I"'lCr.rm "'0111 or 'n'"OI,'.' rolllpil. cllffi cakul'''0fI1 or l.aplllU. 11 could tak •• "h.1t nch lime. So you n«d. S!faIrIY for debulll,n •. lIer ..... somt JCncn.i lips: I.
The .. ngle moo, C(lII1!l1Of1 problem " ' 0 forlOI !hal vari.bl. or fuocliOfl dtfinlUons P"'"'' "nlil cle"",d. and can ;n,..r••• ""th ullsn of !he un>e variabl. name. If I COlllm.nd is glV'ng "nc~p"cl.d OUlpul. "y cl.ar all and rll
2, U", Ill< .oho on COIll'n:lOO ..,.Iy ... ell IS "df«I"
IiQ"'."",
lit
cooluucunl ..... Y· 3 If you ..... prod.."n. araphic1. U!.e rho b old command c..... fully. U.. hold otf I . SOQIIIS)'OU ..... done ",!h I figll .... SO thaI 1M ftCU fil"'" ,,Il0l rn!.enedon lop of " . and rn\Of! I paua. WllCmcnl or cdl d",de. '" thol rho craPhi.. rornrrt>Dd ""II nor oblil...,.!he currenl 0IlC before you Iuo'e IImc lO!« ,I
n.,,'
"
1)0
nor ,ncl""" bare print commands rn )'00. M filt •. 1"'1Cad. pnnllQ a file.
S, The l!dtlor/l)cbuga:c' ha ...
".raI lI",ful fUIII""
f()( debulll,"g M·fiko:
(I) Splliling lhe M·fiI. inlQ cell. mal'" n ]lQ<1,blo 10 d.bua one cell ar a III"c.
(h) '1'1,,, "'"n" ,I"m Cheo:k e'KI. "hI> M· I.lnl ,n Ill< T""I~ menu "'ill check yoo. M file for rommon jlO!cntial Cm'>n.• Ul.h .' platt. "Il<'" )"" forgm a !.em.' rol()ll Inri one " IIsually ,""omal)', Th<'" &I. 11<>1 .1" 'Y' 1<'1. .1 erron. bul il " worth recheckin, them. You.an .Iso pI an M hnl .. port 011 an !>I -fik by lyp,ng lIlint~t and lire file name ,n rho ('oounand W,ndow.
lit"'''
"b",upoin"" ,nlO rho file ""nl1hc Ik""l menu ()( rho ... d 001 ee) You.an icon [] on rho loolbar 11... ",11 .lOp . . eb"K menu. and II", ilop through lhe M.filc hoc by hllC "','h Sl~p from lhe Ikbul: "",nu. Allrmal"'ciy. 10 r.. ume the .~rcIlUOO of )'011. M file, I)'P" r.turn allhe 11.> >
PI'O",I"
" d1J. remember !hal you elll , top . runnIng M·fil. by typ,ng CTML+C, l111S IS u~ · I'u1 if . CIlcuIaIioo II taking too long. tf MATLAB IS 'p"wmg out ulKlc:,ired output, It: ".pau_ comrrw'Id. you ~ahLC th.l you waHl
to SlOP .. e<: ""0n comple tely
Chapter 5 Solutions of Differential Equations In !hi< ch~plor: ,"" ,bow how 10 soh. d,ffe""I"J ~ualiOr'l ...... ,th MA11.AB. For many difr"enuaL oq".uoo" 1M ('QI!tJn.1I'I(! d _ol v . prOO ...c.lhe seneal solution 10 \be d,frero.ull oqu~h"". l>r lhe 'pec,fic WIUlion 10 ... a'«1",,,,,, In (hi' Ibrory and appli •• lloo of d,ff... nlJ.1 "'Iuall"",. An uodt",W>d"" of them bdp" In ,n,c'l'l'tung and uling /'tsull, prodllCcd by MATlAB.
S.l
Finding Symbolic Solutions
Consider d,. d,rr",enli.1equ.tion
d,
"
/(1,\1),
(51)
Po r.oluhon 10 th" "qu,uoo is • dlfr... ntiable fu~lIOfllI(') of the illlI~ JrHUIw.. Of fonNula Jr>lur,OII hodln&. \)'mbohc >Oiulion i •• clltrally noc I wai,luf"",'3td wl. and IIOIlurpO~inlly. rorJlru'ation.1 alionthrru. fOf M>I",nl d,ff.... n',ll rquauons lymbohcllly arc ,mperf«" Nonc:thelu~. MATLAIl ' , I)'mbal'" d,ff.... ntlal equation ",h"Cr 4 .o1v. can comctly ..oh", 100)' of ,Ite d'ff.... ntial .qll.1'lOn l II!JI ran be ooh'c d with the .tand>
• nn .'rhoi' ",Iullon ,hal .' prr.~~ l' In .. mi' of .pee"] function! of I;
"
~rct
••h
i
5, SoM;""s of Ditfc",nliJ./ t:qua(;""f
'h .1 toIutioD \hi! RimS 1/ ond t .lgelnicaUy ... ,i.ho;MJr e~pn:I~mg 1/ as a func_
:'~I'!t.~"Za.n= triaonomclnc funClions and lhei. ~~~""I. and .~, comb',nanons
lJy SIH'~1U1 fimcuom we liliiii wriouI_.elemenlUy funclion ! Ihal m:orh<mahc,an. g,~en. I\ll"" 10. oflCn , 1liiy uiI!I' .. soIurionl of parljcul",ly in'por1anl d,ff.",nllal ~Iua"o", AI. jWali\;aI mal\t'r, I ",'ulion from 4'01.,. " mosr u~ful .. hen" UpresSd ,I. e~_ plidllJ ID 1IZIDI of buill-in MATLAB funcuon, of t: lhe ", ,oclude Ihe .'emenlary funcr,ons allllD1.prci11 functK>n1 In the following ~umple, ~ ,lIustrale how lO Ose alia 0IIIpUI iD !hi, U,,", S« Sec lion SA for eumplcs ,nvolvlng othe. lyl"" of ",lull""s JIll D)' pc 1riltl llbol_
" . . . ti
1..,..,1IIrcua:h ala~br&ic O(l(l1Itionl and com~iI,onj
"""ta1 wlution 10 thi,"equal ion in MATLAIJ by Iyping:
• • tbolft " lII'. t"2 • y ',
't')
e. 22"t-2"xpltl'Cl n.lIDlullon of rh< diffcf1:nhal .quolion il the .'p",uion folLowinlll"" equal "lIn NOI,ce dIM MAn.AB prWua:' !be "",.... in lenns of "" asb,'r&IY conSlanr C1 . ( I'or higher order eI(\IItlODJ, theR ... ,11 M I. nwry .l b'l",,), wn,tanlS .. \he order of lhe eqn. """.) You CIII obWn .po
.......
d,
... ~~
"
,,'+11,
-.11 • lIbol_ I'1l)' • t "2 .. y',
1/(0) • 3,
'ylOI • 3', 't')
.011 " t ·2-2 ·t-2'5· ~xp(t)
,i'..
ID IhileUII'l'Ic. "'( h"e n rile name &011 10 rile OUtput Nul. )'ou unr 10 plot lhe IOluuon or find ,IS ,,,1,,,, II . parl iCul
"'1'fiO$<'
.~, J, "mt/ms
»
,.,
SymbolIC Solurion.
,,,1,.(.011,
"
't', 1)
36.9453 (Unhle the "ample, ""ng .ub, 'n &ennn J,J.I. you munplll "ngle qt>Oits around t "''''. U"leu you hne pttvioole rh. ",I UIIOll m.ny Ii""", II nlaY hr ~on".m.n' 10 defme an anonymou~ ("""tion oof.tlpond,ng 10 Ih •• olulion, Pi)! Ihe cumnl ox.mple.),ou can f";1
"~ »
yl ~ ~(t) · t. " :a . lOt· 1 ° 5°.xplt) and lhen Iype VI (1) 10 eVlI"ale the ..,Iullon NOli"" Ih. period bef"", Ihe c...,I al",,'e. which allow. Vl to IC""P' "ec'\o< 'npul. AltcOlJ,I,,'dy. you can "'Old "'Iyplng the ",luIion hy defln,ng yl Ill> foll(M's: » yl . . . (t) .v.llv.ctorl~'(loll)) W. d,,",,u,, the commm,h .v.l and vICtorh.,n Chapl.. 8, We ofl.n "~nl to 'lI.dy a family of soIUliOll\ ool.." .. d by ''U}'in, the Inilial coodlUon H.", ,. a nalural ""y 10 do thi, in MA"TL<.B Bo"n by ooI"n& the dlffereoull ""ualin" wllh "I....... inmll ,,,)"", Foruamp\e: » .011a ~ dlolv'I'Dr" t "2 .. v', 'YIO) .. 0', 't')
80lla t "2- 2 "t-2."xp I t I "(2,cl MATLAII . 'pre.s.cs lhe solution formul. '" Icrm. of Ihe Inlllru ,lIue c. Now IUppos.c ..... wanl 10 pial Ill •• olurion cur...,. willi inili.1 'aluo, 11(0) 3, -, 2,. ,,3 on the ,"1.",,1 o 5: I 5: 2 One .pprooeh i, 10 type hold 0 .. Ind plOll1le eu"'c' one al a lime, To ,a"• ..,me IyrinK. you con ~'O:"le the.arne rommond. ,n AlOOJI, n.. commandl
»
fl""r./ bold on
»
.Y'U t
~>
for cval • ·1.1 •• plot(.ubl(eolle,
'c', cvel l ,
[0211
OM ~~
... t. tight titl. 'SOlutio... of Dr. t -2 • Y witb y(O) . . . ], ••• , l' n ,.laHl t, ylaHl V H hold off produce tho g... ph.n Pigu ... 3,1. 1'he , .. phiel opllons ... i I. title. daHI .""d ylabd .re e>pl .ined '" So:Iion ],8,2, W,.hool .,.le tight . IIIe rnn~ s.c l by Ihe la'l .,.plot command ,,"ould ~u l off P"" of (h~ ,u".." (ro~1 the PfC' '0'" comm3nd\, Alw. if ,,'( dld,,'1 ' ''rrly a IlIle. IIIe mlo from lhe lUI .,.plot com,nand ,,'ould ha,'e remaIned. 11""'"1 (mi,lu(hngly) only lIIe formul. for on<: of (lie .olullOOl,
»
I.e"'"
_.... .......
54
;.,
•
. . " .
!,,, ..
FigUf. S.l: Solu\ions of dJlldl
/1
,,
-
+ \I
5.1 Existence and Uniqueness n. f"ed'""'1l\aI c~"t.(K"t and uniqurnen !~rem for dire.",n! ,,] "'Iua1 ,oos guaranl«5 1IIIICWf)' initial c0n4J11on 1/(1,, ) ,. !OJ \ea,h 10 a unique 101ullOO ""If to. provided thai ~ rIJht hIIId 1I_ of !he dlff.... ntial cqll3lion (SI ) ,il a "nice" (unction. A \hcorem 11k. thIS q; 2In III "iI!ually nrr)' Inlbook On ordInary d'rr. ... nual cqult,oo"
11" ',_ 5.1 S"I'P'IJ' ,m" I rmd Of {Oil u .... COII""""U JI~"(/i()l" '" rht
n
fUlll ngl.
{(t,II) , " ::; I ., ~, c"!l S d ),
Md./wl 1M p,m. {t", ".,.) Ii., III /l
n,ro Int m,lial ""/,,. I ,robl,m
IIou 01 M,.,.,.. ."/M',," V(I ) .Mr r~i'" 0. Iras, II. /(lfIg PI lIS ''''IM s.ay. "'lIh'" R.
NoIICC' thot h~ "n.... of ""ing a ~lul'on. 1/(1).5 dIfferentiable (and hence C<,lf\linuoos) for
-Iont .. " o,i""
... d lh>l sincc
f ;•• O.II;OOO.... !10 IS d~/dl
OrapIuoTI)' point In /I ond INt the J.OlulIOIl CUI\'.' .1II11Q1 CI'O!S_ Thus an ,mual val ... probJcm IIVPI ha.""Clly 0'" solution , bul, 5\1>te the", a,e an ,nfim,e lIumber of flOS"bk laitial rondi'ion., a d,ftn.nl,alcqu, 1ioo hlU an ,nfinl1e "umbOT of SOlul,OIt5. ThIs prin_ dpk I, Imph,,1 in lh. I'O l ulll oolained above w"h d.olv. , when we "" 1101 51'ttify an IIIId.. coad,tion, 'h" ",1"lion depends on an a,hilf11f)' con"." I; wilen .... e !!",c,fy a" inill.1 condIllon, 1M wlu"o" i. cumplelely de,.ml\ ... d. II II Impoltllnl 10 ... m.mho, !blllhe . " llCIlC<: and uniqu .... ss III<: ..... m only gu."'''' •• ! lilt Wotenl;c of ..... Mi"" MQr th. initi.l poinl 1o. C(NlIitlrr the ,," ,ial val"" rlO!.>l.m
1/(0)
I
(52)
j ,J, Sr~bllny
of D'If~"'n""l:'quJ!I
To wive il'ymbolicilly. r~p<: ~) .01 . d.o lvel' Dy . y '2', ' y lO) . 1 ', ' t ')
"
.0> - l I l t- I)
To unrk!'l;ll r>d 10 lit", lho: wiullon '"m, .... ,"ph .01 j...., 1'0", ... S.2): » eaplot(eol, I -I lJ)
•
•
• hgu
1/ (1
I). "hl
We see t~t lhe gnph hal; 110.., bnlICllts. S.rott lht kfl hrmI,~1 d.'a poinl (0, I), ,I is ,'''' desired !.OlulIon. Funlltrmorr, .....c Ihl tho wlution ",,\11 from 00 10 I 001 docs 001 ..1000 beyond I In bel. "I>«omr. unboundrd a~ I approacho, I from !he Idl Thr righl ~h ofl"" ,"ph dep,tU ~ oflho: f"no:l,on 1/(1 1).""1 il i, nOI p;tn of lho IOlul,on of lho: 'nlll~1 "II~ prot>km
5,3
Stability of Differential E(IUalions
In addlllon to '''II.nee and uniq""nc ... ,he ..n~,~." (Y of III< IOlmion of an ,nllill .·at"" pfObkm 10 lhe milill ronM;on i, a funda"",nl.l '"'''' ,n lilt liltOf)' and applie.lIon of d,ff.",.,,"1 rqullions Whrn I d,ffmntlat rqUIUon i, U""" to modoll phY'iuI1lllrm. tho cuel ,mllaC cond",on II """rally u.l.roov
~,.
,(I)=t-'+(~,
l}r".
IhaI. fOf any fixN t. It>< ~1"'iOfi ,(I) ikpoends conl,n_sly on J.\I be the ~lullOll of the »me ~qulllOO. but .... ,th tho mlH.l =~-'+(y.,
·1 \ "
"\\
\
.'
l)t l'.So.".ha,'e
, '
'I ',
, ,
,
+ 2,
~,
• ~18U'"
,
.5.3: SoIullOn.of d,/dl
s pll ,.3 The lolulion ofltte inih~l.al"" problem
"
-le' , ,
~(O)
I) "+( I)l' belbesolUIi:":;th :"\,;_~gru,'n, ~(I) depend. conllnuOIlsly on ~ foo- fi.td I. Ullln/: •• "" '"" "" Uo. We 1« thot
Now .. 'e !.«' (MII/CI) , ••-ery Kruni.-. 10 change. in til( iRma!
"
,-:a)"" for I I...., and JIOSJtI~.
bul,OStO<"'" f"r I.""ga'''''' I ",, " . TheSt ~"'~llon, or •• oofirmed b, "." ~ " , .ap~o ~"' ,. I W U10m CI)tn;'[>OIly'"g !O "" ' ,.[ ,a1ue, ~'O) 0.'17, I. I.UJ for _2 <: I <:; 2.
, , ,
'I ,
'.
"
"
"
Figure 54. S<:>lul,on, of dU/dl
211
, It· r
Often, ,,-. an: pnmanly ,nt""'ed on "",il;" ,.1"". of t. I""h as "hen I com:,por>lem cff«li, .. ly. m. wlu[,oo for """h'" I J.hould he flirly in"",,mc to m. mlllal ,.,.1"" -i.,.• '1J\lI1! chan~1 on !he ,mual ,.. h.. lhould ~ad [0 ,null chan"" 10 til( .olulion for posit;ve llme- ror tile follow;ng A, We '''''"lIoncd 81 !he beginn' n, of !hi. !ieCIlOO. f'" a physicol \yl~m lhr ,n;II.1 ,-alue 1/11 '),poeally i. ~ koov.'n " ..cUy. When ,\,~ found by muw,."..,nl, lht i. iilIIlppro''''I.>le .'al"" Ji... llIcn. If ~(I) il the solution com:'pOndong 10 yo, and 1M ",lmion" "~ry KnI""'. 10 !he matal nl .... Ji(t) w,lI h8l·. huk .dlllon 10 lhe I(lual 01"'.1/(1) orlM 'y,l.m AI' inc ........ , An ,nmal ,.. I... problem "hose: $OlulIoo IS f..... y ,o ... n"I1'" 10 ,nutl 'WIt! in tho: ,nili al ,,,,Iue as I inc .. lUts is ,allffl Jl
""'00.
,.,uh
h:I,-.
M.",a rk 5.1 W~ con,'''''''''' .,.billly as' ;nc....... '. i r.. 'Iab,hly to Ille ,;ghl. W. can .1$0 cofllulcr ,Lab'lIty to tho: kft The", II. «toa"onl thalli. " .. ble ""'" to the loft and to Ih. righI, and «tual"ms "nltable boIh 10 the kfl and nght
th.,.",
The follOWing IheOf<'m ...·hieh b,hly.
w' ,La,. ,,"bOOI proof. '\ often u..ful in a.",,\ml: "a·
~ , .", 'Ur"'D~ ullpI
,
(
s
O( /JrffrronllaJ l:quar,oo!
to ~ t <: I,.
{(t.1I)
....., <: 11 <: ""},
", !IIId L SIKh ,N.U
1\.<~(!,Y) Sl" f o ,,,lIlf.,I ES wl.'i"," <>1 dy/ dl f{f, 1/) till rhr j,,,ull,1 tu :S I <: I I ,,·.,h 110 //lid /I(I"l = y." rt'Jpnl""'>~ ,h ...
) IJlr
1110 - \1111< " 1< I
"j
S 11I{t}
Ji(I}I <: I""
;"'111"
lillie/ !' .,l,
I,
o.lbtn Ihe "JII.-hand incq"lIil~ in lhc lMOKm Showi lmol
If £
)\1(1) -
WI)":: 11/1,
~ll,
for all tu < , < t"
n.. lliuolllliolll difftr by "" molt than 1M dlfroftm:< ,n It.: ,RIII.1",,'un.. and !... d,rr•• -
aIIII ecr,e!I(I8lllUb~ (at Ius' for tn S I 'S 1,)- Mort O. bul
.,41
not too l"'lIt ,
t II not 100 I• • \ben
II/(t) -1/(/)1 S ;lliJlu
M
.L,,,
wi,
(or all In S I S I"
'0)" a n;odenu<-s;ztd e"n\lan! 11wI ,"" CqU.!iO!L;s only mildly ~. . ." 10 tllanan in the initial \"31ue, and lhe equal,,,n i, ooly mildly un'lable . On ,he .rhln4.lf K > O.Ih~n the lef,-hand in(qUJ\;ly;1\ lhe Ihcol'I:n1 ~how. lhallhe solol, (III ICIIIitiw 10 tbanit. in the ;niu,! ,-a10 •.• 'JlCCially Qver II>IIS '"Ic,,·~b. We can bneny -"IIi"rUe!he" fuulh by !oayinS Ihal if OJ 10~ <: 0 in a rtllioo of the plane. then SQlulioos IbII ftJIoa 1ft do",IQSctho, a. I incroa!.<5. wh,1e ,f OJ IO~ > 0 in I resion of the pbne. IllalIOIuIirxII m tIIIt "'glOO grl farthe, apart II 1 mcrol!.<s, In partlCullf. If Of lOy $ 0 •• u,,,,bt.. e Ihtn tho d,ff... nh.1 «tuaI;OI1 ;\ stahk; but if Of IO~ > 0 e,..,ry,i,he... then the
"PYkln II """...... rll5.1 llII' nghp·""'r>d i"«tuahty in tht thwr,m II an rumple of I «>nil""""" ,I,. ,.,.,t........ "'11111. II tIl>! tlte ,o!uhoo
..to"",
"00. R." ntonK ,he cquaul)II on
"'" .ump!" in light of IIteSO' 0/)\(,' .... lbampIe52""v!
r~.rnne
111(/) we round .Ieo,'.
Wli
- 2~+e
1110 .. wle·', fOf ~1I1
' andOI/OIl'" 2.
2 tocoocludc thai ~ 0,
flOm Ih •• olulio" formul. l '1ou, lite "Iu ""on 'S , table. We can all by ROling IhOl Jf l Oll <: 0, Sim,l~rly. for the «tu.tlOO of E,empIr 5 3. J( I.11) 2., :Ie -, and ilJ l illl 2 > 0, SO lite cquatlOll i. un.table
jA J),ffrrrnr 1)tpcs (If Sym!>Olic So/ur,(IJlj We un abo ulldenl.nd the ,ub'/oly of the: d,ff... nllal ~U~I'O'" in Ihc:.. 1"-0 eumples by u.m,mn, II>< ",Iut"", (ormul~_ DUI 0/ I~ can he: cakulatood and ,IS .. gn and "'" foond ..... n ,(. wlUI'OO fonnu!a cannot he: (oond !'or ulmpk ..... ran ,onmcd,"lt:iy!ell I~lthe: d,ffer.:nloalequllioo dlll dl + I'll" <081 i, nlbl" bt<: ... 1e
.nd 0// 011 31' U' 0:; O. Velne,ther d.olve llOf any OIhc:' st:tndard l..:hniq .... enables u< 10 fond. fQfOlul'lOluhon 10 Ih" d,ffer.:ntialequ,uon /(1.11) cannot be d."ified I' mally. we nut. Ihot many "'Iu'lioo< of Ihe f\'lnn dul,11 s,,"ply aj ,Iahle or unSl.hle. bt<:ause ,~/ IOu may be neXlli,'e ot ",mc points and JlO'l"'" alothers, Nonelhel • .,. we may '1,11 be able to del.m",,,, wh 0 alOOl, the: wluhOOlcu,,'e Throoghoullhe: hook we ""II _ many uamplu thal illullnlle lhe: de~mlence (e,the, .. n"ti,·. or on.. n!ili'-e) of the: ",IU1l0n 10 an ,n,hal \'Ilue problem on the: inillal "alllO
5,4
DifTcnmt Types or Symbolic Solutions
In S«t"'" S, I. ". showed how to "';1Iu.lte and plot I 'ymbohc ",Iutioo ,"~n by 4101v. on lilt idr.I."", th", n outpulS an explicit ...Iuh"" ,n"elv,ng unly function, bu,11 into MATLAIl A. we mcnl,o .... d then:. many Olhc:,'y~, of OUtpuI In: JlO'lI'bk. and.".. mull "".1 w,lh lhem differenlly, Here we gi". add",o"~I ..ampl .. illu,mumg other JlO'l,ibilniet_ "~llml)le ~.4
Cons,der the diffcrcnlt.1 "'Iu'l!On
Thil equal'on 'I .... ,ther /O .... OJ nor .. parabl~. and 'Ii WIUlIOR) do no! ha'~ • formul. 'n 'ennl of elementlry funcl""". HQ\\'e"cr. 4101 v. t in find • fom>ula for the: IOlution,. To getlhe: ... Iptton .... nh V(Oj -3.,YP" » .01lt .. dlolv.('Dy" • + y " lt', ' y (O) . . . ) ' , ' . ' ) / n pretty(lolll)
",1/
10_,21),
, '" , • ,'" , , '",,' ., ,'" ,
IhyA111,
( ~ ...... (1I)1
-o~U!l1
, A! rvB1U.
"
-O_UI))
, ,'"
"
•
, ,'" '"
,'" , • 2 pi
A!,-yAl(-tl , Al,-y8 1l· tJ
,'"
"
(b.opIcr _~ Soluli"". of D.tr~",oriaJ EquarioflJ
die IIDIutiOn is so 1001. ,, ~ ulCd pretty 10 prinl ,I -
-
fuJIChoo" iii -
II lJIdE>. ampl.51 ,,,","umes
a """"' A_ h Y Al . and not; this key
n.
In
,,!he
_ _ fUIICllon, 1$ Ii Toolbo~ are balCd on softwa", called Maple MATLA B al/emph 10 Maple OUt· M.pk (U"'(;OO' R"!IlJln ulICon,-.ned. In th,~ (.) and Alrylli (x) are nul co,w.n.d !U th." and airy (2 •• ) ,\ s a ",~ult> moSl MATLAB cum_ l"I0II- MAn.A B
.",,,,.n
Howe.. r. SlOe' . uh . 's part of the Ml p1c nprcllion' .ueh •• • 012 Thu,. we rely E umple ~_l 't' 11 ~imply luh,hlute, 1 for t ,n Iix formula for .012 . ;;;.., nu:n.ncally. You can foo:e the outpUt 10 be numeric with
't', 111
• 010.0101, IT'bh(~.{.o12,
' t', tv.h )l l .....y 10 plo!. family of solulions As 10 b.llI1lple ~ I. SW1 by defining.
wld....... nc ,rnuJI condillOn .,(0)
c:
. . .d21. Ilaol. . {,Dy. t . y '2'. ' yI O) • Cl', 't·) 1b pIoIlbe lIOIulion. ...,th .,(0) - ,3.· 2,~, -2, _ .,0 on lhe ,nler..a] 0 ., I bold OIl
•• hold off Hae .." firsl . uh, titu(ed Ihe "~lOr of I ,.,.1".. inro th. w lutiOft.llccau .. these ~aluc' are ,he MIIIII fOleac h cu""" , (We did not u'" doubl. Qt ,hi. point. because nu",.ricalevaluauon IIlIOI ~'YI)'. (lornen )IOSlibk, unt,l a .,.Iue of r ,. ~ified.) The resul! i •• veelor of 1)'IDbo1h: npn- .. ion. w. tailed yval . lncn " .. slICCeui.,.ly iubmtured clICh of Iix
th.,
j.4 Different '/}p<'. of Symbolic SoIlIt,on,
61
R.mark S.J ... // oflho comllWld. IH u~ ,n tI".c.amr~ 11>0 ....,rl for E..mp~ S.I, and m any O!htr ~ • ..:' 4aoh'a ~relch an UpliClI ",Iullon !O. f,n' order Njl/.rion.
F. ... mpl. S.S The diff... nlial NjWllion d~
~'+I
dt
~"
i • ..:'parubl., Bnd M ...TL ...B can do 1M nc"c\l:UY inlc.IPal.o whc 'I' ~~ ao13 • 4aolv.('Dy • (y" • lI/y '5'1 ao 13
1/2 'y-2,l/2'atan(y-21'Cl
t
0
1I0"'."or, MATLAS cannOl J,o/,.. the ••"u/l'n, "l1I&I,on for ~ ,0 .. nn. of I, ..... i. QUI""" an ,mph,'I.olulIon, Wilhou. an npli<:i. formul. for ~ In I.rm, of mu,1 131, & d,fferenl aPflro;och 10 plull,n, and cI·III1.un, "",lUon •. hrll, "'C npre" 1M CODsWII of .nlegnlhon, ",111<11 M...n. ... 8 call. Cl. as. fUlIClIon I and II' » ao13.xpr • &oh'&(&oll. 'Cl') 8ol) expr
I,,,,,
or
_t.l / 2·y-2 _1/2·atan(y-2) The ",Iulion cu," •• a.c Ihe le,'el cun' •• oflhi. c~ptt,,;on, "h'th .. c male inlo an onony, mo"' funellOn., fo//ow" ao13flloo • • It. y) .vallvaotor1 •• taoll.xpr)) Then wo cln u\<: OOOtOl1O 10 1'101 $<:1'0,,11 solulionl. "Y for 0 '. I . ~ and 0 ~ ~ S :1: » IT YI • _abl1ri410,O.05.5, 0,0.05,3), .~ cootou olT, Y• • ollf .... cIT. TIl » dabel t. ylabel Y l1H: .O\U/I '\ ~'n ,0 Fillln: S5. Unfonunalcly, wlltn 4aol v. cannol fond an e,pl"'l1 ""Iullon. ,I I"'" an error me ... ge when you 11)1!O spec,fy an IIl11i.1 <'OIId'lion None1Mkll. you c... C:l\Lly find the ron,tan( of inl~gr.l1iOf\ fOf a Ii"," inirial <0,0(1111011 by pl ugtn' the "r.... opo ... ""h... of 1 and V inlO .011ft,lDC So, 10 plot the ",lullOII .. ,ill ,ml ..1 COf\o:hllOll ,( 0) Ion 1M ,nl,,,.1
>.
0< 1 ~ 5. lype: 0 • ao13flloc(O, 11/ » IT Y) . . . .bgdd{0,0_05.5, 0,0,05,'11 » oootou r(T, Y• •013funo(T, Yl, (e OIl The Je5uII IS shown in I~,"re 5.6. Remember th." " Iltn m.lln8 contour 1'1011, you ",UII ehOOI. ho1h the hori~o"lal and "erli<:al rang.. In lhe . .ahl1r 14 command; some Iri.1 and clTOl ma~ be n....' 31)110 find a 1·.I1ICI I nlnge 11..1" appropriale.o. panic"l •• wlurio" Th plOl I f.m,ly of soluhonl. say with 10,/..1 COnlllllOll' V(O) I >I ,2, I.-I , _ , 2, o,'cr lilt u
same ."""" Iype::
ChapIN.~ Sul"u..",J uf /)iJTrrcnual 1i'q"3Iions
•
•
I •
..
I
.. •
•
•
. . •
,
•
•• U. ' • • ~ll!UDC(O. 1,o.a ,all •• uua'..rC~. y. ;oI3!UDC(T. Y), eval;)
'lb evalualt. ptiticulOlr ",1"lioo 31 a pwicular "alue of I, ~ou ell! ~ubstl1"IC the appre.. . .' vallIe of l,nlO .01 UIlDC, ~I i. equal.o lhe cOO\I~ot of Intcgralioo you fouod from lilt IalIlII condJboo. and "'he IOf U. One way 10 do Ihls is as follows'
•• .,.. YI ;o1.-C;o13!IlDCI', y) - e , y) 1116689668Sl623181628896?2S841 2 11 689668S)62318l628896?288412 HeN u \, thf \'&l1lC .... prniou,ly compu.ed for .he: m;tial Contl"'OO 1/(0) I, and we I'Id for thf nIl>< ,,/ Y( ~)_ 11",,"'.,-••, a, 'H d,,,,u\~ In S«'IOO 3,1. solving algchraic
ha,..,
......101\.1. nolll ..·.) • .u-aighlf....ward. In Ibi' else, lhere are lWO solution s 10 .he cquallon rot, 4) BI>C'd on "'Kure 5 6. ,I I. d.ar lila. rhe POS'lrl'C soluliO!! IS the one we ,,'ani ,n IbiI cue Bul hoar.n mmtllllat !>QffitlimeJ iolv. docs IlOI fInd all the .\.OIUlIOOS of lI!
equIIIon. aftd .. !wn II ,II.' a single aM".', .hat may IlOI be: lhe solutIOn you ,.,anl II is lifer 1m ,..MnJt" ,'''ph lho soIutioo )'011 want ROO tllen usc ! nco 10 .\.01,.., numerically rot I/llIInl'" Inlll.l , ..." """don lho gNpl.•• ' •• roC'Cy) .ollfllDcl a. y) - c. 3) ~
3.1117
R_rk 5.4
The
LAB Iynla. "hen d•• ling wi.h Iymbolic npl~'$lons. WhIle II may h,,-e """n mo", IIIW' IIlWIOI)pt _olvtrt.oll!llDc(a, y) _ e, y) ,u\III,,,"equab,illnhe.e,,,sulls ln IDClTO! mt' ..... B. In'lead, ,,·c I)'p" an .'1"","00 .ol lfunc(a. y) - c for .olv. 10 let equal I" 1<1" Wh,le e " .Iready dcfinrd, y 15 undcfinrd.' IIIls PO'OI, SO ,,'e m~s' cltellll' It h'l>c: • ,),moolrc ,,,"able 10 a,'old an ffTOf mr'~gr. The funclion !a.ro
am
"
,
. " .. "
, "
" I~gu,e
S 6 SOI""OI, or d~/d!
"
(~L
,
..
-+ 1)/11' with y(lI)
1
f".,
"''1<>m:5 Ih input 3fj1umen,!O ~ a funmon, nO! a ,~mbohc n,,,.,\;on, II) "'. mlk lhc: ,.'p'< S>ion 10 be ..110 urn inlu an anonymou. funcllOll In (h,\ co .. , y ;1 I dummy """able In the: funchon lkfinilion. loo"...d not be dttllml IImboF", bc(
•:umplc 5,6 The d,rt.",nh. , .qu~lIon
" "
... + II
" "'parab le. lIS in lh. P""' IOUS c>ample, bU I ,hlS l,n", d .o l y . can'l do the II inlcgral: H . 0 14 .. ,S-" lv. i' Dr .. . >
le for '''''h equation" Hr ma rk 5.5 Althoogh ",e ha,.., focuw:d on flr~ order cqUJuon~ In IhOl _ lion , .s. 01.,. al
Ch2p1rr .~. SoM,"'" o( DiffcrcnuaJ Equalio.u MATL'o D. lIl c an) J.Ofl,,·~'" syMcm. can occ._ rtSull, As the foll""mg n amrle shows,. good the of wluhon~ '0 d,ffc ... nl,al «j .... lions is a ",Iuable (O!IIJIII~r.I!C'".r.II.d ... sull •• and Ihc ... r"", in ,dtnli f)',og SIIu.lUonS in h&I ~ a mj , I. :><\mg or i~1 ",'Ull
na''''''
,
/ 7'" Coaudr, the initial ,000oe I'rotolr m
d.
,
II(O)
" .... MIl . hol_ C'Dy . t. - 2 . y - 2', 'yIO) . 1 ' , 't ' )1 .. . abtl_15. 1_1 11 1, t. i tl. " ... 4 1 die IIIPb ta FiIU'" 5,1_ The solu\iO/l ,hould be conlin_II . bul the grnph has a ._1 0 A«m1i"l 10 the in,lial coodnion, lhe solUlion \hould be I 31 j O. bul s1Ml5. ' t', 0) gi" es the ans ...... NaN. for ''nnI a number"_ The ",ason asa .... MATtAB 1ub 0, nnd wh ile ils solo lio" soh'.' dlldlIfIiualilll equation bollt for! > 0 and 1 <: O. •1 IS diKonun llOUS "1 t O. .,... .
2
•
I .. . . .. .... . .. .... .
PIJure ',7: Tho "Solution"
of dU/ d!
,
"
,' + ,'. , (0)
' . A""ord ",g to " .olv.
Chapter 6 A Qualitative Approach to Differential Equations In (hi~ chapler. we di~"I' a q~~I"~"'... OppnlO< h (0 (he \u,",)' (If diff.rential «IUII;<'>I1', &lid o/II:un quahlll"'" infOf11Utonn Ibo
m. d,fferential "'IUllion.
Con,iM' the g.nero] fll'll onk. d'rr.... nl'al.qualion
ffr, ~).
(6 I)
'I<'" ,n,
We can oblllin qUllilllt". ,nfor'lt1.l'ion about the «>I .. noos I/(t) by (6.1) gcomoruically. Specifically... c can obtain (hIS ,"fonttation from the d,rt<;Mn hdd of (6.1). Re<.11 ,halm. d"":lion fIeld is obtained by draWlnS IhlU\lp ... h point in the (I. III-plane a .hon lino Ofgmtnl wl1h .lope I(r .1/). Solution •. or intcgnol corv... of (6. I) h.,'clhc prop:ny 1Il.1 a,each orlhe" points thcy .... IIngenl to the d"cuioo field althll point. and thc",f" .. thc SCnc!'2l q""'IIIIII>" na'uft of the wlUlion, ClfI he dc,cfTflil>«l from \be wm:uon field. DI' rection held, can be dni..'" by h;and for >Or1l<' IImrle d,ff.renua] cqll.lllOns. l:oJ( MATl.AB can dnlw lhem for MY firsl ordor "4U,'iOfl. w. ,IIUllr.lle 1M qu.hllh . . . ppro<>eh w,th I",,, example •.
6.1
Direction Field for a ."jrst Order Linear Equation
Cons,d.r the equa"on
"
"
(6.2)
MATLAB'I command for plo"'n, d,rncliOfl lidd . is qv.1.., .... U\N ,n <:orljun
"que"". of <'''OlOlaool:
C'JJIpkr 6, A Qu;1/,r~!"'C APr""'.,)' /0 D;lTc~nnal Equa,;ons _ltO.2 ,2)I
I), ... il tillht by 16 arid of unifOflTlly ipxW po,nl< in the r"Cctangle_ !he rigill-Iwtd "ok of 1M d,fkI'Cn"oI equl1ion al.he &n d f(r, 11)' al .ilt5C poinl~ The ,'<'(;1(,rs ( 1,3) have •• lilt d,ffeI'Cnl grid po'"Ii. The q.,o1. ...." • ,equiI'C~ foul i"ruI~: lilt array T of I_value •. IWO coml"'nenli of lhe di=lion field .,me .i/.e, Ihe C(Kie 0"" ( , i (8) ) S;/C" 8 In lhe la" command . ...1., "'''0" fIeld TIle ",suI! looks hke 'hi.:
z,
II I,
,
,
I
,"
,' "
I !< I , .. I "
I " I "
"
" , ;
,
." , "
"
•
,, . "
,• .." , ,
FIJW1' /1.\. Direclion Field for P.qual,on \6.1)
i.
•
"rofTtt,: il " """,,whal hard tu ,.ad because Qf the /IC! thai many ' On.e cart
g., a belle' I"'lUfe by (.snhng the DIOW$ ~ Ihal
e.ampl •. by d .. ,d,"g .."h ,-ector ( l , ~) by 11, length In ookr ' 0 do thIS In MA1' ..... R VI." mod,fy lhe precedinl ~""nc. of
. . cYi.-zI'l'. T. 1./L. S./L, 0.51. axi l tight .. II1U1oil 't', ylabll 'y' ." ~itl. 'DlrlctlO1l rll14 for dy/dt .. IXP(-tl - 2y' n..1IIh enlf)'. nA in the quiv,r comm.nd ..doc" . ,lie lenlllh of .he .'«.oes by half ... p!WWnllo !he am'" Mad, from ' ''' anow,"g up ,he 13,1, of "" ..by .'...:,on_ The ""ul! i, 4 &1110 f\suft to 2
6, I. /)i,ec/ioo h eld (or • Firs, Order Linear Ilqu~''''''
Flgu.. 6,), Dirtc, ion Field ar
i',nrdH Ii. l l"II",ue lhi' ,dra furthe. by d,rr. .. nl,.ung (6 2) . r
lion PO'"U 011 lhe solution cu"..,J ho .Ionll the cu"·.
~
:,.'
U
J)lreCtiOD
Field (or a Non-Linear Equation
.2 1-- t6.llcaD \It' whd c' piicilly bt<;a~<.C il IS I"",» Now lei uS COII,ilkr an ~~lLffiplc
dill rUlf \It' 1OIVft1 uphcilly in itml.l of oknltnl.., fUl>ClIO
(6.3)
,. U ..,d Ih.1 011 ",Iuli"", ewnlually b...o me po»l'''l'. Ftg"re 6.4 oIso I"UCSIS thaI I lUll"'" IJ'I!"'O'I"+'
u' + I ..... ~I •. ;tfId can \It rewritten al
6 I. Au!""""""" Equ.ui(WI' and Ihu. can I' I mU ll blow up In finil. I,me lIy the unique""" of 'olut,oo, and ~ &llIICe al the du«I,OO field ,n Filun: 6.4. il i,clear thaI "II solut,onl mUI' blow up •• ~"rlO Ume.
F. •• rdH 6.2 U", MATLAlll0 graph """" or lhe: dn",c1ioo fidd,
your le,lbook. If yOll can sol .. e ony of Ihe .qU>I,OOI •• plrc,lly. If)' sUl"'flmpo.inl .orne of Ihr solUlion cu,ve, On lOp of II'
6.3
In
Autonomous Equations
Equat,on' of In.: form HII) .
,..sr.
",hich rio 1>01 ,"" 01\-. t ,n Illt nihl·hand arc • • Iled aUl_"" tqlWion .. If. [>hY' .nl Iy"cm fo11ow. rule, of .'OluIIOO thaI do nOl chanie "'Ith "'1'1<'. then the o>'Ol"uoo of the system is go.erned by:tll aulOnomous C<ju.uon Such C<jual;nnl ill. particularly IJlICn3blc 10 qualit.l" C 10,lysi" Considt. ~uat;OfI (65) To I'uPpolC ftu) hal 1"0 ,.ro, y, and 11, . C31led erilir a/ ".,.IIIS of the diff... nual cqu:mOfl. Funht~. ,uPf'OIC the grapIl of fell) i, a, In !-ogu •• 6.5 , 1bcn by ronlldonni the proponlu of f l u). lhe d,rtttion licld ofl6,5)
.how.
FiIU'. 6,j; Righl.h.od S,d< of Equalion (6 5) can be Clslly uno:lcrstood . nd l polllb (I. 1/) w,th II :> 1/, e
ChJpI<16 A Q<J.III~I"'" AI~>W.1
poslllve and I/If;rta>n from 0 tu ""~, II ,ncrt.": ' f. om 11110 ""; finally, the ~Iope. F,,, horiwntalliM an: COfl~llIlt ("nc~ f ~~ no! dcptnd 011 1)TbI dirKI_ field chu. h.a~ th< Be,.,.".L al'P"~".ntt ,hown In F'Ku.e 6,6. NOIc 11»1 Apu 6.5 dtpi(t. f(lI) H_ II. ",hilc hKurt 6,6 lhovo~ II" !. Sc,..,,,,1 fOClS au suggts~d
+
,
.. ,
"
. , .,oJ" " , ,, ,," , "
' '
. ,
.. . , ,. ,"
tlncLlon lick!. n..... ly. cha" Ii) t~r~ a.~ 11'1'0 coo"aOl ..,LuhOll. 1/ • II, aod II 11'.: ..... IIIIt1I1i u/>oYt" II, I. nd to 00; and (IIi) aLL OIhc. M.)\ulioo. tcnd 10 II, as I - 00. W. fkrt ve IhcK prt>JXll i.. of the ~oL"'i"n. ~(I) dlf«,Ly from lhe graph of f(lI) a1
""""
I 'I'hI ron.I ...1 funcl;""' 1/(1} II, and 1/(1) I/j arc ",Iuhuns. caLLed the ~q"jJjbr",m 1III1IIj0fU' Tllty...,!he: uOlque ",Iullon. thaI UI;\fy 11", ,o;t;al cond, hons II{O) II, udll(O)
1I' . ...·I""
Con ...... . ",lull,>o 1/(1) .. "h 11(0) < II,. Tlltn. b«au~ of lbe uoiqueness of sol .. II""
"mtI I hr Ih.u 1/(1)'
1/. for I)) I and II'{I)
f{W) > ()
",11<'1' 11(1)" ... 'I>ing fu ....,lIoo S ...«;I" allo boundlOd obo>'o. ,lila, . hm,1 al
Bu! nt"a""o' (b,6 ) and (6,7) lay Ihat I/{ I) "WroachU a hu"wnlal ltSyrnplolc 01 1M ...... I" .... mil Il' ,top< i. "ptnll:",rnl1y" blHarr than a p"'1l"'e number, The ~ ..111111i """loa.,1idl(>tl guarnn'«1 Ih.t
lim /I{I) ,-,
/I,.
). If 1/(1) 1\ • ..,Iution ""Ih 1/(0) > Ill. liltn 1/(1) > y, for aliI, Also
/MI)) > O.
1/'(1)
lIoJICO 1/(1) is one. ",;un inerusml In fat!. 1/(1) _ "'" TIu, can bt """'" by u... d in pat'! fbI. Somoti""', .•, .. r lhall iN from the an argu"",nt ,im,lar to eumple Iltlow. I/(t) act ..lIy rra.cbtl.",. ,n hmtetime,
th.,
w. "-r mml 4 Now con"tlt. 1/, < 10\, < 1/" Then if 1/(1) i, I .oluliOIl ,,',th 1/(0) hll'el/' < I/(t) < 1/, for aliI AI",. 11'(1) /(1/(1) < 0 lienee 1/(1) is. docrr",,", fUJlChOO A. In pan 2. n" no( d,fficulllO ,how that lim, .,1/(1) 1/,. N",o that In \h" analym tilt propen;',
or 1/(1) .... tktomunod wltly from the ",0 of
/(1/). We c.U ~'"
,t, mlrumum
Ettrd~ Ii.J I..,t 1/ be the ,aluo bot"~n ~I and 1/, "herr / lIle, on that if /I < 1/(0) < Ill. thtn 1/(1) h" an mft«11"" POint "here 1/(1) 1/.
Show
63. 1 .:"lIl1lples of Autonom ous EqUlltions Now We nom,ne. 'f"'C,fic ... mrlo. tho t<j""tll'ln
,, ,
d,
"'
I.. dir«hOO fidd has tho ,oneral "f'I"'ar.>nco of I-'~urt 66 ... ,th 1/1 0 and 1/. I. Il ut ,,-••an actuaUy don'. "" nrhcn formul. soiUIIOll If II .,. 0 and 1/ .,. I ... -0 can sol,,, by sc~lmB
lO be Qiid for t il >'alu .. of LN, , MATLAB~ ... ,,,h" Ihi! .quatlon and pl()! 11>0 $Oluuoo cu,"c" Hrre;\ a $tqurnc. of
COIII/IIIIIdI lllat dot' w. " . UVlln, hoi'" 0", ,.,. .yu t 0 ,.,. '01 . dIlP1Ur (4I o lv,('Dr· r " l - r', pp I~
'riO) • c',
't'»,
a.al . _1. 0 . 1 5,2
'Ulot(IIIbI(l ol. c, c v d), p p JIlu.l 't·, rl aM l ' v '
(0.3 ) 1. ,Old
p p titl' ' aolutiool of ~ /"'t • r-2 - r' Filii" 6,1 tilt aclual $01u\,on cu,,'ol for lhe d,ff.... ntl.1 equauon-as Wa"'n by
• •
•
• "
"•
hgure 6.7: SoluIIOII CUm'! for l:qu.>t_ 16.8)
MATLAB N.... Ih., tllt $Olu""" eun'.! rom'!poodon8 to ~(O)
,-
;> I
go to inC. ",I)' on fonn.
!uh1 7 .... (.1 U .. formul. (6,9) 10 verify tho f'I'OI'O,,,cs of 1I(1) obta;,,,,d .I:>ove by lhe quahlah>'c mrlbod. Drt.nm". wh
Unuta u 1 ~ '" Ib) SUppoK 1/1, 'I. u ... formula (6.9) 10 show Ihat lI{t} _ "" on fiolle hOle hnd Ill" time ' " 11... 11,,11 lIIi' happr"., leI Show 1IIa1 1«1) 0 and vet) I arc lhe <'I1',hbnum solu tion., Dor:s formula (6,9) ylrld lilt.. W)U\;onl'
n Here is ~bcr cLlmpl. Tho Itfl paph '0 Fig.,", 6 8"",", •• rullC~"" f(JI) "'Ill ~
<
•
,
• < •
•
<
",gure 6 8: An AUl<,!DOmotIl equatIOn ,..ah 1 Cnuull'
""1'0<; Ihu, the aulonornou, diffcrtnlial rquation
look altbe crillc.1 poinl,n the
modd~_
JI
f(1I) hlI, thrtt cm.al point,
~l".
Solullons sl.lr1,nl helow b "",,. a posit;,. slop< . ...,
they IlX'fUSC' iov.:tnl b. SoIullon. I.an,ng above h ha,·. a tIC,.m". 'Iope. "" they dtc .. _ to.... ard b. Therefor •. "'C Up«l b 10 be :on asymptotically .table rqu,hbrium ooIulion. A wnila. analysi, of the ';lIm of f(1/1 .01lK•• t. IIlll the OIll
Th. ,ight graph in Hgore 6 8
,"(}W, a few of the ""luI ion
~.
,1i.Slno!c, the conclusion. Ih., "-e drew from OUt analy.is of the gnoph of /(11)-
thIS
,,,,ph
Chapter 7 Numerical Methods 10, lhi' book ) 00 ha,. ~" lh., moor diffe",ntial "'luII,"n. Ca/1be ",hN "pileili)' on I
of lh. elemenlMy fUl\ChonS of clk.iu. W. hno '«n in C'·, ' , . .olv. \0 I\0' ''''pcr ' 'uol "':lily equat,on; on lenm of 11\(\.1' fo"
(UDo;.
, "
I/(t)
i" 'd'+C-
but " is tn.....n thaI w\.<' inlcK •• 11 canoot be Ol'"',.aj on 'erms of tilt fUnc.hQfl' of ukulu, (MATlAS dou "'nl<: lh,\ in, •• ,,1 m dlMd form in ,..,111 of. 1"""lmll u f o 001 th" (unchon. called (he "error furltuon:' il u""~J by tho formula ~rf(z) To ",II. and cannot"" ,ew,;u.n in lerml of ',ponen'illo, Ingl, Ingo"""",tric (O""lIonl. tic- Th,," the: limplell nample of • Jp" ",/ f"n< IjOll :u d,!t
J.:,
.d.,
We ",II 10 solution! tha. Cltl be .'ploc"ly .. rin,," In lerm of clemenlV)' fOIKhun! Of ~f"Xlal fu,,",IIons u fo .... ula Of P,• ..bo/I<" "heth<. Ibty " ... obUI"",J , •• hand-c •.kuI3t1on Of ',a d.olve When ". ran/lOl soh., • d,rr... nhal ftjllillion ,n !,h,. "3y, Or if ,he fonnul. " '. find '1100 romph ••1
.o/w'"''''
!""
"
(lI~pl'"
7. Numcncal MCl/lod,<
Numerleal Solullons Using MATLAB 1"
. . _iiIIUulCd ,D &ndlna the ",Iulion 10 the imllal ,."Iut rroblem / (1. y).
b cooLlioiog t". ",,1.1 IUPJ>O'C Ihal we do not have _ f()Tioula for
[a.bI. Tho sub~nJll a on y. he found with
Su
I·
P W:muWtU 1ht u"" of
od.n,
slaoo. MATLAII·,
,
ode.' by consl
I
yeO)
" ,
I.
(1.1 )
bt_1O/ubOII11 ralily fouoo 10 he
yet)
,It' + i
sa. we haoe an U pllCit fonnul. for \hi . loOluI;on, ...t will he .ble 10 compare lhe .prrox,
I
lUIutlwl/. (I)a.ndthenKI1(Itul;on ~ ( t).
tbe lnl *p II to ddin<. MAnAB fuoct;on "'lualto the nalll·hand ,ide of the dlf. . . LUi'" eqn_ Tk~ an: two way. 10 do thi~ One way is to drfin< an """"J'mO
f •
_c~.
yl t ,/y
OIul. 1be dol iD fn;LQI or \be di,ision " in; the fu nclion mUll be abl. 10 ope."le on ,'«101"$.) 'lilt odta way ], ~II • funcbon M .m•. as In Chapte. ~ For OUr problem, the M ·file could. ""WI"', brulltd fune. m. and would be , ~laa,. tuAclt . y )
..
~ , /'"
WI nc++hIlcf1d thol you usc anonymous fuoclion, wM novcr pm,ibk:, ) lowever. M_file • ...... JOIIlo use mort !han on<}in< 10 Jmale w luuOI1 I/o (t) OI1lhe inlf.vaI [O. 2), lyp"
;r '
"' ,'dSU, 10 2), 1) Itf It liD liDooymou. funclion. o. " al,UC_tuAc, 10 2], 1 )
If func , 1lI jOin M ,tile.
n.c", IS sllll lOOlbe. p01llb,lIiy. namely 10 1""
lhe ddiolllOl' of
lbI_ymou. full
J'dSC_Ct, y ) t. / y , 1021. 1 J J\ ' II beIe 01\ ~ WIll ,",um~ lh.l you ha,'e drfin~d the nlhl.MOO "de of the d,fJucnllal squatiOIl lS IllllIDoymOU! funclion f. The M:cond Ifgulnc nl lo 0«., 5 sp«ifLes lhe inlnml (II whlcb 10 MLloe, and th~ Ihird ILfIlumeol give, the milial value of the rolutloo .1 Ibe lint poull ill Ihi. ,olrr,'al. Tho "'lull os shIM·o io Fiaull! 1.1. "The oi",1« indicale lhe
1./. NIl,"n;"'1II Soillr,ool 1111D1 MA11..A8 poinu ~r which "". 45 cumput.d tilt a"""", . rr' connected by 10"" segmo ml_ V
..
.. /' , """"
-
"
n., W I01>On; tho ""nl .... of ."_u",,,, errd.! an: '"w,
/
~
.
.
"
"
.
Figu", 71 NUm
EnN"is~ 7. 1 I)j'piay both V.(I) and the .... '" lol"hon V(t ) Can you dlltinguilh btr..,«n the 1"'0 curve\1
I
j1' +T "" Ihe ~m< ,",ph.
You can ~I'IO plot a faJl1lly of Ippro1"""" s.oiur'''''1 by u"n," ,.«'!Of of innw '21 .... as !he third argument 10 ""e' 5 r..". .~amrl •. 10 rial Ihe ",10"(1111 of dv/dt I/v .. ,ill 1/(0) 1,1.2,1.1, . _,3.lnle »"". ' 5 (t, (0 l], 1,0.2,3) This procedu", for plallin, D fannly of s.oiolOonI ..'oo t ; as 10011 II 001l<' of Iht 101ull"". h.. a singulanly m lhe lpc<:ifi1 ,'Il ... problem 0.21 btl"",' and Ihe following E>.<",i ... Stt Sttlloo 862 for 1 infoomalloo on 00. poinl. By defaulr. od. ' 5 ploohtc<s " plot of . n 'ppnl>lrmle IOlurl<)fl To <,1111<1 "0m<1icaI
add'''.....
,al u.. r ~pc
»
(t , ye l • ode'SU, (0 l], 111 The I ,·oJ ... , II "hich ode ' 5 CQlllPUles lhe "1'P1'I.I"m>l. s.oiuhon _ \torod in lhe "OXlor t . lAd lite ~~pond'ng IJII"'O,,,,,,,lelOlulloo .~I,," art IrorN in ilIo ,oxror ye llH: "'micolon IUpprel ... S prinling of the "("ClotS. Wilt" o>ph. It I'" pl o t ( t, Y. ) Nale WI pl o t (t, y.) prod~l • graph wilhoUI cireles To di'pl"y rile ....1... ' of r and I/. (r) in columnl lide.by-si"'. 1)'1'0 [t yet , Nolic( ilIOi lhe I , ... Iue~ are ~"""'n hy 04.4 5 and m>y rndllllr _ , .. Iuel. dcd I f)'ou~", 'nr~",,1cd onl)' ,n ilIo 11111< or 1Io{l) II rho.1III of tho In" .."lIl. ,n this ca", I 2. )'ou can rype )'& (.nd ), "hllmare >olo ilon 011 0,0,2,0 .4, ' ,2.
",.
It. ,.1 • l'dSlf. 0.0 .3,2. 11, . , . . . . . . . 1Ibk ohhe ,'allie' of I ~nd y.( I ). alonll with tho cOI"'ll"'no1'''1I ,'al~' of ............... ,U)-Jil t l .• )~ .... ~. ~1I:.-2 • 1), .... II .1: 1 _
.
-~.~ w. ..... fOZ 71:
1 _ ,0 ob.ain IS ·o1illi. ou,pu, in order.o:> 'co 11",
,,
Tabl. 1, I, Appro.imlle and 10""" Soluhon s of dyl dl
'Iy
Jzuteiptdipto.n &II e_l. The 04et~ command III. mpls ,o prodllee an appro"!Nte ..,lmicw""Ib""" I... \ban or tquallo 10-', thouJlo oomellintS th<: K<:uracy !NY be ben., or .... lit. !Ire SKtion~ 7,] IUId 1.4 for addi1ion ~\ ,nform.>ll{III on Kcurocy. 1lMI oc.tS conut\lmd w,1I only compu,e a solution on an inl.rv~1 wnh the inol,al con diIIoa Jiven II one of the endpoinn. ' 0, w. hay. nlready descn bo:d how 10 use 5 on .1mavIi ollhr fO!Tll (I" . b]; 10 Ule il on &/I im . ...... 1 of th<: foon In, 10]. cnl.r ,he ''\lIlies vlt Ill .... "in, ordc,. For u&mf'k. 10 pIol an appro~unalt J.OIUlIOI1 of 1hc: ,",1,,1 'alII<' pJ' m(7.J)on1hc: in ...... I ,O]. I~pc ..:. l17lSlf. 10 ,11. 11 1b.,ec1fy YalIIe. for t. use a nrB"'i>. inc.. mem , II 11\ O. ~ O. 21 ~ 1 , NOlI &uJIPO'C: you wanl 10 plO! lhe fam,l y of J.Olullon. of d~/dl t/~ ""th ",,"al nt 'lI(O) 1.I.2.U .... ,3onlhrinlC ....... I (··1,2] U,""g o<'l• ., ,,,,h 1- 121 u,he II uuf qtI/I1Itnl ... ould produce .... ;1IC0Iltt1 gnph. "",,au1C 1hc: ,n,".1 rond,"on ' WOllkl .. £f,1led zl t ·1 ,n\le~ of I O. H. R., . 1Cnc, of commallll. lila. produce, 1hc: ...... FiC'liFiIjl/I by compu,ing !he appro~'m~'e sol"lion. sepataltly on the 'nle ..... I. (- I, 0]
0<'1.'
a1:
1IIII10,2J.
.... [I:b d l • Pakl .0000."(f. ..:. ,1otCI:b..'. ybak )
(0 -11. 1,0.1,3),
7,2. Some Numern:aJ Mc/hods
» » » »
ho14 on ltfor , yforl ·04.'5tf, plotttfor, yfor } ho14 off 1l>e "'lull IS Ihown In Fllu,e 7.2.
" to 21, 110.2'])1
,
,
."
II:
•
,,'
.. -
"
"
Figu." 7.2: A Family of Num
n.
We hl". Illustnilcd 04.n willi the ml~a1 •.,,11Ie problem I), .. h"h c;\o <;l"I~ to< sol,.d e.pllellly. We C;\Jl )uII as rudily .Wly 04d5 10 any fiJ')I order inluJI \.1 ... pr0blem. Conlider. for eumple, dV . I + V'
"hleh cannoc be wh.. d
"
v{O)· I,
021
terms of elellWn,uy f."""on •. 1lIc commands » II' .. _tt. y) t • y. ·2 1 » 04e'5tll'. [0 I}, 1 11 » .... h({O 1 0 10011 will ,"'ph ~n appro"nuleloOlutlOll . In
•~ ....ds. 7.11l>e solutiQn
vet) 10 (7.21 II 11Il'''''''In' for t po"" ••. ;>nd the ... " . fini,e .,' vet) . 00: lbe Intc,,'al of u;,l, 1').
value /. ,ur ll Ih.1 Ill", Execul. lhe p",vioul Ihl'« commonds, Doe< 04.' 5 .., ...1 ,be ,'ab", of t "
7.2
Somc NUlIlcrical Methods
In order 10 8"•• onle idea abo"1 how 04.' 5 nkulDlcS IW(\).im.'e , .. Iocs, .. c art going 10 d,w:u," Ie'''''') nume.kal nlclhods. We "'111 apply each melhod 10 problom 0.1).
Chapl~r 1.
,,
Nu",..';"", M..1hOOs
; ; =j(t,I/),
1o < ', < IJ<'
.:~:':~'!:':'~. ~
b.
part Ml the u rne wIdth, or sup siu, /, I,., I,' AI
TIlt Baler Method iJ dOlt 10 Euln and il b.>~ 00 I"'" Wl&<,nl hne Govea the initial , al .. ~, ....., d.fi .... ~, Ittursi,..,ly by
1/1.+1 = II,
.
,)
+ h / (1,,11,),
1·0,1 , . . ,n
\.
11(1,) + hll'(t, l,
by Ihe tange nt hne apprOxl matlOll
11(' , ) + h/(!" II(I, }), II. + h/ (t " lI,j ,
u~lng
the d,ffell'nl i.lequnl lon
since 1/(/,) 'II,
1/1+1'
11IIq. wol"",ioru
... ..u.. 7
c'
~(I ,) '" II, should become he lle • • 00 hel\., as h is taken snun.r
7.1 CoIuidc. !he initial "al"" probkm (1.1): d,
,
" ,
11(0)
I
w. wiIh 10 .ppnnimatc 1/(0.3) uling !he Eul.r Method "'''h MOp .i", h dept.
~lInd
'••" " " "
,
0,
"
. 0.1,
"
Yo + h/( ta,~) . I/IJ 10\ 1/, + ht,/1I1
1.02911.
0.2,
"
+ hlo/flo
03
,
0.1 and thltt
7.1 . .'>0",,, Numerical Method$
Smcc 1I10.:!)
J'j ()'j
93
\.Q.lI030 __ ,"C fiod Ih.ot 1~(03) -11,1
Error Nc~1
U'" /.
001~2.
0.05 and (j "cps:
.,
I
"II>
IOI)2~,
V,
I 001~
V.
10119
1/.
I02l~
U.
103iO
Error . 11/(0_3)
-~,
00070.
If lho: ,n,1Il1 , ...1... problem is su fficicnlly sl1>l.1Ollo", then (Of the Eule. Method one Can Ulow .h,t.M error in "epp,ng from I" 10 any I, ,n the inknillu < I <: b ... li,~ u
Error
r
Ch,
.'i,·",
,t.
,,'hen: 'I a constanl tha' depends on f(I, V), part ..1 den> the ,",ll.' C'OI'\dJ~on. and the mten1l1. hul n01 on h. MOftO'lc'. ,t can be ~hown that the error il actually prop:lftiuoal to 10_ llecao>e of this, the Eul •• Method i, calkd a[orJI rm/"rn"/wd. Note that on Ihe example, clliling the step lite 10 half hld the effecl of CUIt,ng the error appro'i .... tely in half, a, "'rc:cl.d for a firsl order method. It is 8110 u\(!fullll knuw th.l""a/ ,T",T. SupJlOiC u(l) " the ",lu,ion of the d,rr... nuoJ '(lual'On r.alisfying u(I, ) V,. Tlltn lho: local error in It"W,nl! from I, to I, .. is ""fi... d
,,'"
,, . ,
1l{I,.d
V,,,:
i r ., lho: e"or made in ~ ".p assuming lhe w lul,OII , .. I.. at IJ
"1/,.
Usinl! Tayl<>f\
formula II i. ea. ,ly . hown thaI. fOf the Eulc. Melhod,
2I \I ", I, ,.' ' for lome I J € (I" I, .1l. Thus the local mOf " pruportlOllalto h'. B«au~ local c,:,"", pl1)vidu a " mplc C{Jmparison Ilf mtthods and ,I used ,n the """gn of numtr>od we d, IoCUI'. TlIe errOr d,~u,.w ," lite pn:yi;,os l'"rnll",plt is called g/,,/>III , nUT, ,n order 11> d''''"Bu"h it from loc~1
'v(I,+d"~'"
(w(I,.d
~(I,.,»)+(\I(I,.,) ~1")'
(bopre, 7, Numerical M erhods
side of Ihi. ~qualion is the local NTI)r. which is p"'1lOr' 1M doff. ",...:.. . 1 I, +1 of III< IWO $Olulion~ of lhe drffcrt nlLal nIuc$ lIll , ) UId lI, . 1 I, 1lIe ,ize of Ihi . Imn depend. on (he l i« of IIId 011 1M dllf~",nhal equalLQII. 'p« ificall y on .... lI
1,1 Icad. a
r'
UiCWI 10
. . . . . COIIlWIt
10 I,. ",e
m
.1TQf$.
rune. ,,/o J
b_10 I" 10'
_(b - lo} h ,
b , !be &lobaIfrror i. bounded by a eon, lanl hmes /, _ Nnl, .U~ the d!ffe",nual equallon" u/lltablt • .o Ihal ll(I , ~l ) II,., IDly be IarJer !han 11(1, I II, . 1lIen the global error al I}~ I "the Slim of Ihi . llUJCf quantity .00 1M '()Cal ~rror , I ,,., . Thu •. ,n Ihi' , iIIiOlion, tile llooal crror dtpt"\lb 00 the , tabilny of 1M dilf.",nllal .qualion . , ",ell as on lilt numerical melhod, and C""nOI be completely ,,"ued b)o the . implt cak ul.lIOn sl .rchcd I bm.. . II can .,,11 be pro' cd (h'l, as . Ialed OIItia'. IM llobalerror i, proportional (0 h. In Ib .. rf,ult !ilab,hly ,nnr>enrc, the COO"'anl of propIlI1ionabty: if the din ..onli al ~qU'I'OIl ;5 stable, lhe cOIl, ranl "of modernle li« , if IIn'lahle, 1M eon, r.arrt nuy he large. The d fee! of 5tab,\t(y and ," ' tabiir(y on numencal lOIutiOllt will be di..:u... d furth. r in Soclion 7 A 'I1It EUtrMrrtbod '" MATI ..... a . Tht Eult. M~thod can be implc .... n1~d usinB the fol · function M·llle. namc:d my "ule •. m
..wm,
faDatiaa It, yl •
~l.r ef.
t ini t , yini t , b , n l
Il1o aal_ q»zwl='ti _ f o .. i .. ithl value " .. obl_. Il1o tr/4t • fIt, y l , Yltini tl • yinit. Il1o Ai;' '".tlo... y el ) , .. . , y en ~ 1) . ... c . lc ul.t." . t Il1o ~ • • 1 point. t i l ) •...• te . . . 11 i n the int. ~. l III 1t:iaJ.t. b). ftII ript_ b ..... aid. of t lw dltf . ... ntial \II 1 eel_ 1. "eUDed •• at! anonyJaOU. f unc tion f.
* "'.-l.tloo
of
b
f .. o. ti .. i t.
b • • nd
...
t.1atti/D' t. aDd y •• l_lIth . . . i c ol"""" vec t o ....
t
"
. . . .r o a(D • 1. 1),
y .. . . roa ( n + I , 1 ) , , Ca l cul a t i o n of !)Ollie. tli ) aDd tIM eornapOAdlDg '" .pprox ~ t. valu •• yli ) frca the KUla ....thod f o r.ula. tI l )
.. t lnt t/
10'(1) .. r i nl t,
tor1. l , n
tli •
0.'
1 ) .. t l i ) • hI
y li • 1 ) .. Yli) + h*f(t(11. yei)),
To run Ihis M·file. def,ne an anonymous rUntl/on f oolTtlpondmg lo the riK/lUw'KI ,ide of Iht d,ffe,"n!lal ~uallon. choose appro",... ,...I"", r.,.. I~ la\) four "'1"""'''15. ond type
It . yl .. ~." l.r ( f. to, y O, b, D). lf lllcnJlu.lwld.;do"tk§n.ed <wI' Ii.. methods., ..... of ",h ... h l rc d,K U\\n!
ne~l_
'l"h. ,« lQr t w,lI ronlain the po,n" lu, I, . . .I ~ . IIId the '"tC\(lf Y ,,,,,II CIlrIlaln the roI'"" poodmll arrro~,mal'oos !/O, 10'" _ , 11. - TlIrw con he d"pby
,1 ",,',\
The 1l11liali~alion of t ~nd y as ~olumn '"«(OB Ii nol .".nllal. bu( tu'O P" " P""" l·i"l. {t yl ,\ in lobu lar form beel.le t and Y "f<' columnl, Sorond. in,\r l lilmg t and y " ~,,,e ~ the rrogram 10 ",n n\Orr ,melen(ly. To appn)\imDlC (he wl uliOli al a poinl I th.r " r1QI It,. "t would h3'" 10 in,upuldir. A SI mple "ay (0 Ihos is 10 "Conn«l1M do"" by r\r.I.wing 11ral&h,Ii... SCi"",nl, from .ach point (I" ~.) 10 1M 1lt~1 poinl (I. +,,1/.,,); tha, ~(I) woulC'rcd ""Y II> inrcrpolal< " dt5",ihed in S«lion 11.6.1 ,) To gtaIlIl ~( I ). Iyp< p lot (t. y l.
7.2.2
Thc Impro\'cd Euler !\Iclhod
We as" m ,(nM"nh lhe I""semline: "ppro,mlilion. rrpl..,mg the ~Iopc V(I.) by 1M a,'c ra," of die I\\'O >IOPOi V(t,) and y'(t ,+ I)' n", yidd,
y(t .. d "
", ,)+ h
V_(~'li,V(',.d f(t" ,(1.» + f (I,. ,,1/(1,. ill
y(I,)+h -
-2
(7.3)
, , _ _ 1bI.,.......imlhon 11(1,) - y, and Iht I'r.. iou~ liu k, apl'",,,m.:IUOIIII(I,. ,) .
IU ,. 1/,). Oi,'. n tho initi . 1...lut rJu. Ihi. analym
lead! to .he ",c ursi ,·. formula
, '0, 1, .. ,11
l.
II.. I
'''''1', 7.1 Wf ~ n ",""dr, I'robkm (7.1) and find lUI app",..,m. t;on to y(0 .3) with /I
0.1 We obWn
,
~1+1, i,+J(t"fJ\l+ hll'o)
"
fJ\l+~'2
('U+ +1,_) ~>
I/J
'''"
lit
1.Q..l ~o&l
~,
hlu/II<>
l.OI9~2~
0.000033. Note thut the erro< on the IonpfO" e<.I Euler Method 0.1 ;1 , uR, idr"Lbly thlUl the mo, in the I!uler Method with" 0.05 . (11(03) _ 1/,1
I."
Tht local rlTUf rm the Iml"0,.d Eu k r M.thod i. p",portionl ' 0 h'.
'7.1.3 The Runlf-Kulla Melhod
....
k,
J(I" 1/,)
l-,
1(1, +
k,
J(I , + II2,11.
k,
J(I"" y. + " ~'l)'
"
/, 2.1/, + 2kd
+ "2k,)
~:xlI"'l'l~ 7.3 ConsIder !'foblem 17.11 and .pin fiml an '. 03 and ,ale ju~ 0 ... ',ep, Thrn ljlpO,una~on'o //(0,3).
le,"
k,
1(0,1)
kl k, k,
1(0+015,1+0.15[0))
~I
1+ i.;-(O
0
1(015,1+0,15(0,15)) /(0,3,1
03
" bu,_
011(;699
+ 0,3(01-11>1>')1)))
+ 2(O,I~
OIS U21173!j,1
t D,J llJ(,'J'J)
+ 0,21;735,J)
\.044038"" U(0,3), !//(03) ~II ,0,000007 ,Weo« ilia, threrror in lh< RUDge,KullaMelhod wnll /, 1I,3 IS much Ie" !han Iho enm in !bo Eukl MetOOd. or e'en in thr Imp"""<'d I:uler Method. wllh II 'O.O~. 1bolocal elTOf for ,ho Runge-Ku".. Method 1\ proportionlilO h'.
So EIIOI
HrmMrk 7.1 Tho ,""Ihods we Iu."e dl!<:u.>(tl can ho appliN 'u lO)' fin, OI'drr .qu.mOll, l·urlhorn1<>r<:. !hey can be appl...:! ."ng d,ffe~nl Uep 'lrC ••• tach ...p
7.2.:1
Inside 04.45
SOfl"'D~ for thr numerical sol."on of ordinary d.irre~n1ill
O<jIUUoru ran be I»lo
",,,.ho,h w. h.;o,'. !'feson ..d. as ",ell .. on I>IlIcr .....!hod., Thr numerical sol.", <>deU combine •• fourth order ,""lhod DmI a fiflll order mr'OOd. belh of which .... imilar .0 ,ho fourlh order Runge . Ku"a MelllOO di""u\\.Od abo"e, II ,'aries the "ep size, choo"ng the "ep , il-<: 01 each ,top In Dn attempt 10 .chie"e lhc de,i~d "",uraey. All effie .. n' "'Oolem numerieol ODE '01"." use ,'ari.ble '''p ,i/'< n.. ,llOO., Tho soh'rr ooI.U IS IUI.ahl. for a WIde "3ricly of ini'ial value !'foble"," Thr ..,h'., od.4.S produ.r, .ppro.i ...... solulIOII ".1 .... al the '"'111<11« of poinl. de· lennin"d hy '" \.01«1ion of s,ep .i1.<'; ..'r 10 !bose: pom.s .. intemallime poinll, Surpo", you h.'-e Kque,led appro.i"", .. !I toI""ide ",nb lbe Intrm.11 lime poml>. aDd InlerpollllOO mus, be u~ 10 gd approum."" «>Iulion value. "' Iho po,"1> 0 I 0 • 2 12 1l\( InlrrpolallM IWIX.durr coo_IS !hr .""nUI' n""le soluMn '''lues .,!he imrmal lime poinll. tt<>I ",1lI .1m"'1 II .... (lmear poI)-oomills). IJ menlloned for !he Eule. Mrihod. bul ""Ib hlp, dtptt poI)'nomJlIl. 1l\( !\lm3I. solutioo "dues 11 lbe "'le",ollll1lC points ami !he intrrpol~"00 proce.. In ,uch a wly thallhe totalemll in bolh p'ocelS"1 doe. no1 ex"..d!he dr"rcd 1e,'.I. by def,ul, 10 'J, Fin.illy. "ben od.n i, \lied wi,houl poinl' being sprcified. ""'. for Cl omple. ""ilt oddS!f. (0 21. 11. "pptoXimnle 50lUl ion v.lucs "'" computed a. lhe mlt>n.llinlC poin'<' .• nd!hrn by in.er· pol41ion at lhru eq".lIy ~p.ced points bel"''''' e .. h p." of lhe se: PO'"'" In thr g,'''ph produced by Ihi. romOland • •hr "J'IIlfO"m.lc solu llOn .-.111<1 01 .n of these: pUlnll are Indl' c.,,,d by Circles. and.1Ie "rnlers of !he circles .... lhon COflIlCCIM "nit h ... "'pnlS (iff
,d.,
F,gu,.., 1.1 I.
,"I .....
".
. MATLAB 'I>I;lud., two mher numeric,l .oh'cI\, 04.21 and ode113 . lltat Un ~ U>N "i(h,he pnlhl.m, in lhi, book 1'0' .om" i"III,1 value problem' (.cf. lfC-d 10 OS "riff prohlrm,l. 1he ",1,<" o4e45. od.n, and odelll on: 10.5 "'li,r~l"'y 1h.n IllCll>Olh thll haH be.n "''''K'''''' 'l'<"'ificBlI)' 10 . oh '. ' liff probkm•. MAT1.A B 1II':I .....~ fout .uch ....1..... , 04e15 •. odell., ""e23. aoo ode13bt. Addlllooal inform.>hOtl on aU of lht-.e ",h." ran hc In tho online 11<'11' aoo '" 1M hrow~'. Rlr Oldtllllonal ,nfOnnalH)rl on nu meric.l me1h".h and lllti. \oft"''''o IInpicmrnl.hon', ... ref.. to 0 " aha"" •. C, Mok •. ""d S. No,b. Num.'ri<'llIM.lh",ls nnd s"nwn .... , l'relll,.o H. II, 'nc., 1989. E>1'C" 1 ~lly ....'ommtnMd arc I"" boo~' lhat (<XU! on "umcrical "",!hod s In MAT LAIl L Shamp,"". I GI.dwdl. and S Thornp""'. Sohlnl: O U&5 "hll MATLAI! . ('llIll>od6< Un"CI1iIlY Pn:". 200.1, aM (' Mok •. Numtrkal Compull"~ "Uh MATIA lI. SIAM. 2~ .•11.0 11.,lable al http I"'-'W. mathl
r"un"
7.2.5 Round·off Error
.'''B
.bo>.,
n.,.lyl'" or.rror d,,,u'lN il called diK tr';Co,,,,,, ~m:>I". and ,,'auld be preJ.Cnl 'f ono ,,,,,Id ",l;I,n "" Infin,t( number of d'giIS. In .dJnion. the:", 15 " ",oJ 'r>/! aror, "hlth am« """au,," Ih. '(lmpUlr, ulel • (.. e<J. Fmne numbe, or d' gits. I...,mn/l II. (r)
~.( I )
fll(11
~.(I})
+ (11.(1)
iI.(1})
])l\c",h,.:.lwnIOrrof " Round.off Error, Sin« """1 ",,,npule~ Ih.1 run MATI.AI1 ca"y IG d'glt l, lhe major pon ion of the enor will ho th. di'c"'I"~li"n rno<, Thul. fo, m,,,1 problem •. ,ound·off Con be lafely .gnored, I" • •1>" "'a\llll ". w,lI "'ll d'llIng""h bclwttn 11.(1) and II. (t).
."Of
7.3
Conlrolling the Error in ode45
A, indrcaled am", •. 04.'5 aU.rnpl. 10 I"ovidc a" ~ ppro.irn'le .ruurron ""Ih erro, ap· pm"""'lrl)" III " A 100... acCUr1Ile &pprox, mal• ..,1"1100 can he ob1."",d by "loIn,:ur <'f'I'oo,] f""nh aq!umenllo 04.'5 11111 "'Iumenl i. c",.1«1 usrnM 1hI' 04 •••• com, m.md ~or n ..... pl •. to com"" •• an "1'1"""01.11. IOlullOI1 to lhe rnmll 'aloe problem (7, I) . 1I "'q""I,ng 12·digit Iype U optio.al. 04 •• lt('AblTo1', l . - l:l, 'R.lTol', 1._ 1:l ), u It . YI] • od.,SU, (0 :ll, 1. optionl], ~> fo~t 101111'
1.
u
"""U'''''y.
ya(lN>d ]
1M'. ,'omnWld, rn>doe. (he OUIPUI 2 , 236067tl77,199~1. " h,eh al'l'ffl~tma't$ the: tUl,,1 ~4lut ,lr) I" 12 di~'l' Ab.Tol 'I"'"itl", ab'olul. erro,. and R.1 Tol . pec,fic$ rei "IIVe ("or, Wllh the !oC1I rng< 1.-0 for AblT01 and 1.-r fur JI.1 To1 . .. he", a and r ne pOMI"'e ifl l.¥(", od.'5 (oes
99 10 IPI'rox,m.:nc 1/(1) w,th uror less \han Ulax( 10 '.I~(llllO '). W,IbouI thr~ option, M"TI-All U ~~ the .Tollnd 1_ -3 fO/i Rd Tol, The
7.4
Reliability of Numerical Methods
ad_'
We have ~laimcd lhal 5, when uled .. ith lhe deflull opl,on., I..d. 10 apt>foxim.:l.lcly 3·,10 8il ""C Uf"dCY, MOle p",ci~ly, the defaull optoon. rn,o", thellJ
11,.,
I"'" pour .. , ulls.
!I '
3c
"
1/(0)
,
(HI
'TlM:uaclwlutiooi.y(l ) e-". Now "'. find 11M: numerical wlulIO/Il u"ng 04.45, and p~ il ("ith a dalhcd \,ne) "'g_the, ... ,Ib tilt '~lCt ...lul1on (.. ,th. soI,d I,nel: ~~ f • !J(t, yl y - 3 ° .xp(-l Ot), » [t, yall .04d5(f, (0 nl, III
» » »
plot(t, yal, '--'I hold on •• plot('axp(-l Ot)',
~~
hold ott axh( [0 12 _1 11)
» ~~
»
[01111
xlaHl ' t ' y lab_l 'y'
.~ tit l a "
f a1\d arcindlII1nK",$h3blefromOlolboul W. ~e<: f.om FiIUn: 7,3 thaI the: grapltl u y al ".I~I I! I 12, (Ot in'1aoxc, an: '-.ry O. hul lhen the cu~1 \Cparate slw»1y. ~If , _'lO 3 71 , ~ 10- "), dirr. ... nl(y&l(.nd) -1.29 . • 1\d ¥i ) ,
How ,-" III .. '''h.... hc .,pl.;""d' The w ll/llon !O our d,ff• •• nhal rquation w"h mnw CUDdlIIOII \lIU1
I I • " ~I t)
,
~'ni -."n.lI.l ,al".. I (D..S' to!>
III
'1
+ fO· I ,
H~u'" 1 ,4 ~'$ a plol of the ",Julions
10"'1, 1-(O,2~,
10 ~), I. I+ {02!':i" IO -~), I +
)
w..... Uu.l."'J>
I ""plAIC 'Cf) .Iw(IIl I,,'m 'he ,olmioo ",ilh ,n"i31 , 'o lue • • ~ lly I: ,110"" wnh imhal ,'alue, pt'ltfllh ... I "l'I'fIl""" I. nnc. on .uch a 'OI,,'iuo. the nurncrical 50lul iOll will follow it ur ...otht, I.. ,h ,,,llIun" Son,. til ... w l"lion, 'I'Pro:t\;h ""'. lhe nume.lCal soluli"" doc. I~' ..
(,,,,,1.1 ". h"", anllk following lhe: soIUlIOI1 III .ll.mrl\n~ In <.kubte . R«all aho Ihal an tquallOll d, / dl 1(1., ) is un
U I Alllh" '"~~<>I' <"uII OO "h. n dealmg ""Ih unll.bl. tquallo"" CaUl ioo " alwayJ monmmtnd Sup!'""" ",' ha,'c lnadt. cak.IMio" u"og I.... e rc3son.:oble
n...
,:01""
h.,.
" "
'"
I I
•
• •
-
• •
,
•
1
••
••
•
•
• •
•
conh
thai they Be both .... ural.; bu1 If lhey d,ff•• ,"MUlltWly. WD one .t.ould ~u'fI'''' th., the: fori! cakulahOll;1 lIOI .... unl•. 1..1', do Ih» WIth our uamplc. SprcifICaily..... cakula" and plot the n"IIIt,;,,"] ooIu· lion, Y.~. 'aI .. fy;"& rnon: ,tringenl lCturIC)' rtq",rrllltnll, lOj!
» » » » » »
1 and 4. W. dcscn bc """'" of the more ad,'~"".d futu, .. of MATlAB. focu'in, QIl MATLA" «Imm>nd...d l«:hnique, lila, art Him! "",(ul for ,md)lnll diffcrtn,ial ((juahOn,
8.1
Dahl Classes
"''Cry ,-.nlbl. you define in MATLAB. as .....11 u .,'(1)' '"PU' and OUtpOJllO a command. " an "rmy of dala belonging 10 • pMticulu rlaJl. In lIois boo1 "'( Ul.t primanly fourl)l'" of dar.' flontm, poi'" numbers .• ymbolic . ' prr."on,. ! for cach Ill" of d.>ta It. da.~ (II.. g"-(I' by the command ..hoa):md bow you clll crt.,< il.
Ty"" of dal. l~o.1in8
p
('rUled by
Cla« double
.,.
char function_handle
.,w...
'lpwi' numbor ""nB .". OI lyaoo IYP'"I I min, ,uidc .ingk qllOl ••
ul,na·
Vuu CM think of "" amoy as d"orn,;ooll '"" of dn.; MATI..AS IIleM'. am)1 !O h.,.. moxe Ih:on IwO bul ,,'e " "II IlOl "'" lh" f~~r"re, A IIIIgI. number (or .ymhohe up"",ion. or fu""tion handl.) " ... ,artltd by MATIAB 0$ a I.by-\ amy. '0",..,1,",1'1 called a Jml,", A I.by·" OITllY I. ~olled a""" In,,,r. ""d ... m-by-\ ;uny " cn lled a col","" Iwlr>r. (A llIing ,. ""Iually a row , ·",, 1Or of characters,) An m·by." =~y of n,IIlIl><:fS i, co iled a m""it;~. Stelion 8 4 !>elow You can sec \be dos. and amy'''''' of .,'ery ,-arlahl. you ha'" dtfincd by Iypmg ..hoa ("". SOCI;OO J6). 'The s." of ""mble dcfml,,,m< 500.. n hy ..hoa i. calk<.l y"m m,,""""c~.
<.1;""'""00'.
'OJ
""
In order to u>C MATLAII y "",mmas; often. )Orne argument. :orr opUonJI. Wh1'0 Section 2.S) for cach com'n~11d u,u.ll) (db" hal da,,,,, of input. lilt command up« 'I, as "'ell., "I'l>l tia>. of OIIIP"1 II
..,,,,,,,mn, SQ.,,.
""urn.
\bn~ romm~nd, all.",' more than OIIC'
of mpul, though somc:Umtl only oroe !bID d", "rncnllonrd In llIe onhne hdp. Th" fl.~illlhly (all hc • cOII'cmooce m!.Orne c."", Jj,d • rurali '" III"" .... For eumpk. a, we Sa'" III SeWoII J.S. a.plot "ilow! )OU 10 '1'«11)' tM funcuon 10 be: ploUed., a ~lrmK, a symoohc np"'sslOO, or. (u"cllOn hnndle. On 1h< othr. hand, \ul'(lO«: thai IOU "",'e prcvlOO,ly defined I< - ( 5 7 J, and now you all~mpt 10 ,ale thr den'all,. of JOII:r by lIpmg H 4ift (log!l<) I
'"' 0.3365 1lIC rcawn )'OU don', gel an emil nl<,\.3g. i, thaI 4itt IS the name of I funclion Ih~1 U>II1J'UICI no ..... n,·'1 d,ffe",,," ql.ouenu .. "ell I ... ymoohc den'~I"'o, Smtt '" I, I 't<.:l'>r, W L~ log (1<', and the no .........·.1 ,rJ~;on of 41 t t "apphed In ~ne.aI, "hen the OUIJ'UI uf a cOnlmand dOCSn'llool rilh, IQ you, and MATLAB I;"OS 00 error .... "ammg "",,'>g., )'OU ,hould cOII\ i
Mf mark 8, 1 1}pml h.lp 4ift only lholos YOUlhe hdp luI for lhe nurnencal ,-enion of 1110 ~umnlalld, bul cro,.. -",fe"'lICC 10 lhe Iymbohe ,','mon ill lhe bonom, If you "~nIIO \ott Ihe help Ie" for IIie .ymooh, "e"ion '"Ilead, type help eY1l/~itt
Ii'"' •
Someli"",\ lOU nred 10 coo,.n Ollt dala dass mlo anOlIle. in ordoer 10 prepare the OUlpul of {lfIt comnulld 1<1 \(n' •• , the '"pul fOf anoIller I'or nample, ". "",. u"!d doub1. 10 ~Of".n I) UP'U'UlII' 10 f\o;ollng'poonl numbc~ and a)'1ll0 con'ert numbr .. or \In"¥IIO ,ymbolic .'p"'\lioos. The cQfllmallds "UJIIlat .. alld at r ln. . con,'.rt bel",.cn rI"mbcA and sirin g', ",h,1e cha .. COO,crts a symoolic np",ssion 10 a \lnng, You can abo U,e vectori ze 10 000""11 a symbolic e~ pre~sion 10 a "CClonlCd slnng; II adds a , before ncr), · , I, and· m 1h< c~ pI"C~,ioo
mhoI,.
8.1.1
Sl ri ng M lIuilJulalion
Oflen 11 i, ulcful10 C(JncaICnalC 1"'0 lIT more UOII, keeping In 'mild '''''I a Mnng" a "row 'eelor" of characlC~ f ..... e-
""
Here ,\ a u\Cful al'l'h~~1JOn of .lnnl (Oio~n •• ion You ma, ._ , . (> "_, . _ u< lit a mini ,an.blt contDIn'"1 an e~prc'llon , .... , talel more than ~ 1, __ ,•• , ({ . ~.~ .~ '" pc. n molt meum-
'laolC~1 you car, COI~'lnll1: your mpul ontu.t.. next hoe by ' fpml '"
follow~d by ENltN
or RbTUMN, bullhll IS not allowed In lhe IlI,ddle of a .'n"8.) Breu lilt e~""'I'iOfl onto 'malier part~, "'1'1" a '-anable 10 nch pall. IfId COllCa~n.'e the .anables. as on: lb_ • 'laft hand .id. ot -.quatiOD'/ rba .. 'right "-'>lI aid. of aquat i OD', aqn .. (lb. ' .. ' rha]
~>
» »
."" left hand .1d8 of equation K. I.2
right hand aide of equation
Symbul1c und FloatinJ,: Point Numbers
We mon"oncd aoo.'c 1113. )'011 Can ",,,",,crt btl"'~n symbolIC numbtr! and IIooIJnl point numbers with double and.~ Jl;umbulO th.,)OU ')'pC' m. by derau] •. nOllUn. pClllll Hov.'."cr. If you mix .ymbohc and nOlI!!", l'O'nl numbt", ," an antlunthc ex~'''on. lho Hoollng p<:>,"1 nurnbc:r!i a", aulOlIlJI,ully con'·clted 10 Iymbollc. This ~~pl3Jnl ... hy )'00 can Iypc aY"U x and Ihen x -2 .... ,Iimul ha_ing 10 coo"er1110 a symbolic "umlle,. Ir.", j. anOlheruample: » a .. 2
•
,
u
b .. a/'Y>a(l)
o
'"
MATtA II wu ik,iH"",J so Ihal wme noaung [lOInl "umbel'$ .'" ftltoml to lht,r <\1<1 ,alues whotn cw'cl1cd 10 symbolic. Inlegcl'$. r.l1,oooi nu~', ... ,th lItull nUlIl<'ral<>n .nd iknom"'alon. loI'Iuare root> Qf .mal1 ,nlcl(rI. the numbe, 11. and certain combin."oo, of
lhese number; are liO re.lore
2.2361
.". »
aY>alc) ~
aqrt IS)
I'
bell
S"''''' n 'S d,lhculi 10 predlCi "hen MATI..AO ",II pre!ltf\" e,XI' ues., 'I ., ",ulIl<'nl 10 ap 1». (AClo"n, II In ,ul'f'!'Cn Ihc noah"g poinl c, .. lual'OO '" a nulll<'nc '" II ""~ 1'1 . aq r t (5)') ('" "'" on I'ngle 'Iuole_ 10 male ,I 3 Ilnng. t.g·. ap . nnOlloer wily on whkh single quolC$
IU .3
Structur('S
II Ie' Anothc:r data clau Ihll )·00 ... ,11 OCCl
In
"'ATLAS is • J,rve, .....
""
" 101 _ lolveC'x. Y"lI _ 2'. 1,,1
x' Y If lOU
'y - 3· .. • " )
~Yf
~
yml 1:108 801, lou \.CC WI t~ dlt. da~. of aol " ·'.truet am) ". 1\ structure ~an ~"n\aln an) numl>cr of named '"allablt~. called '1i~lds", \~tlh<-r "uh their .aluo:s. In Ih,. ~a", .•01 ba, 1\'" field., x aoo y, and cach " II ,«to, wntai",nll'w" symbolic numl>o:"" To .... ,.~" a 1'3",<,,13' r,dd. \H~ 11>0: n3n1e of 11>0: '1"",lure fllll""ed by the name of lhe I,dd, wnh a pcnod in 1>o:',,"cco. Thus. 801.x " a ,'ecl", wnla,mng the;r ,'aluc$ of Ihe two ,,,IUIIM' of lh~ 1)" '0'" of etju'lions li,'cn III .olve, and 101 ... ( 1) is the first
')I'''' ..
",11
""milt, in Ih" ,'""I", 1\1".., ,.nerally. a \\,ue,,,", can cOIllam lllulllplc dala cla'W'I.
l'Of cnmrlc. one field ~;I!I 1>0:. 'lnn~ aod "'I>lIIt, can br an .m>y orllo>li"1 POint numb",,!;, or. fUIIClIOfI handle. A tield can nen l>o: '1K'II1>er mudu"" An unmplc of II mo ... (»uplc. \Iruet""' like 1111, IS gi,eo b) od •• 5 ... I>ro II~ OUlpUl IS I>sIgned to a sioglr ''3I1abl.; \Ce Seclion 8.6.1 ~Iow. Ahhou~h In th" I!oo». .... "ill u\C ~trllCtPn:~ only "hro!he) occur u MATLAB output, lOU ,-.n ddiO<' yoo. O\\n \UU<;' un:\ with the conunand nruct, ~ the onlonello:lp for
.",mpl¢,.
8.2 II'.
Functions and Expressions
h"" u,.d Ihe leml' rll'"nrotr andfimcrrOll ,,"hQUl carefully dl§cu
""'W""O IhrlWO. Slriclly .pea long. if we define !(z) zl I I. Ihen! (wnllen wilhoul any p;onl
courM', " C oflen blu. thIS dl~lIocllon by calhng !(I) D. zl + I a fun<"oo, 001 In MATLAll 11M: d,lT.rc""e fuocllOl" and .'p.c.,ion. is impollaol. In MAl LAll. an nrre'\loo Un belong to enl>t, II>t min, o. 'ymoohc dass of data ('oo,,,k. tbe [un"",ngrumrle » t • ,,,All . 1'1 fIJ)
"",,,un
,'"' TIt" ... ull mOl)' ht puullng If IOU are .1pe<;lIng f 10 ""I 101•• funcllOl. SInce r
,~a
string. t(31 theth"'l char.>lIn8 of onc charac,.r,) TIping r ( 5) '" ,11110.1 yield . and t ( ~ 1) woold p.oduce an eftor "'..'ngc. 0"" way [[I ,e, Ihr YI>U ha'"c 11'1" ,igncd 10 '~fI~hl~,·
»
'"
x •
3/ .".l(f)
S. I.
M,,(~
IIbour M·fjles
'"
T)'pm, f rut .... , than eval If) abo,-c " 'ould Iu,·c "mply OUtput tile e.prts~ion f w.thout ~ub\IL1UHnll 3 for .. aoo e"31uahng tIM: ~Iull 1"Qf a .I11ful;u ~a.on .... used eval .n Cl\Jpl~r ~ IU define I funclioo from tile OUlpul (If. symbolic ton'IIMnd_ You eM 11<00 \uh,tilule ••'a1"" mlO an urression uI.n. auba, II dtlCnbed In S«tion 13.1. You h~"e lu"",d Iwo ways 10 dtfiroe YIJU. O\O-'n functions: 00 \be rornrnand II .... ulm, e Ill' ,~~Ic In "anonymous function", Of bUD. (Ke Secllon 3.5); and by c:~.un, a /unclll>n M hk (see 5«1100 4.2.2)_ Anonymous &n.d 'nllne fu .... tlOOs are ~ u!.eful f(lf defining "mple funel.ons Ihal can ~ ~\pn:sJ.td In ~ hne and f(lf IUm'"l the IJUtpul of . ~yrn""li~ ~ommand inl(l a luncl,,'" Funchon M. llI01 "'e u!.elul lor dtfin'"l IUlICliool Ihal '~IUI"" sr>"cl1t l inlcnncdi~le commandllO c()mrUI~ lhe oolpul Mo.1 MATLAB comf1Wldl arc aclually M II le\, aml you can reru~ rhein for ;den 10 Ulc in )'011' ow" M. r.1e s- Io ~ Ihe M !ile fu., ' 8Y. lhe rommand _an you can .nl.1 typa . . an. Sre 1110 Seclion HJ below ,~"IlIC: c(~nn,"nd •.
loch as cx'I •• S. r'flJUIf. lhelr fiul UJu .... nt 10 be a funcllOll II you lyre an .~~I\ion ,"11.00. YOll ""II 1.1 an error .... IU~. On rhe otht. hand. rTI(lIt symbolic commands. such as .01 .... or ,boh·e . requIre their IiBI arxumonllo be .,the. a ,'ring or a symbolic uJlRl$ion. and...". luncllOO An Importanl dlffe~fICe ber,,-ecn lInn,s and I)'moohc uprtmonl II thaI MA1LAB aUlom:llluJly ~ub~lItu~~ u~.-defined funcliorll and "ari-,>lcs ,nlO ~ymbolic c>p.tlsioo thaI you I)'re. bul 001 InIO mings . (OIi l il anoth.1 ~n\C In "hlch lhe ~Ioglc '1~lllO t)')lC around I "nng ~urrre~~ C,,,,IU3Iion, ) Cooslder lhe fulk~""ni: cumplc: » h . elt) t. " 31 intl'hlt)', ·t·)
Warning: Explicit integral could not be found. >- In Bytn. inl; at 58 In "hilt .. in!;. at 9
'" int(hlt). t)
MATtA B ~anOO\ culuate the mtegol bc:cau!.e .. ,thm a Itnn,. II. IS regarded as an uni.nO'll·n
fU",,11on II you make (he 'npul to iDt IpnOObe.l.hen MAT1.AB ~ubslilule~ the 1""'-'011' defimlion of II. hefo~ rerfOfuun, tile mleg"'''OO' » _ya8 t/ int(h{t), t)
'" 1/4't·4 8.3
More about M-liles
II 1M fil., Thel"C'~ twO kind, of M-lllc, h ks C()nlUlIlIllW ~lATI.AI:I ~a(c",.nt' are ca ot rod~ 0011"'1. and ~cripl ",-filn. wh'''h M ·filn "h1Ch :>CC'<'J>l aIgo>nent, and P d . ' d'." ""'s In Ihi' , E",I":I"~ C~3 1. anu 010 "..... ~-. uec:utc a >o!rlCl of MAT1.AB lt3lCn,.n I. -I« hon "e I""ICIII IMkhu()nal ,of""""I100 00 M liles.
j,,,,,,,,,,,
,os 8.3.1 Vurinhtcs hI
~ rillt ~ l·files
W~n)oo run I Knp! M fik. the ''''';3bl~s yoo u,," and dell"" ""long 10 your Worhpace; i ~. IMy talc on any ,aluts you ass"nnl urhCI on you. MATLAB ~s;on. and lhey ptr.m af~f the scnpl finllhrs c..uut10, Conmlel 1M foll(M,ng loCrirl \I.fi~. cal1M
scriptexl.m" ,,_[123t)1
1)'1"118 ecripteKl a\~lgnl the g"'~n ,ector 10 u bill d"pIJy' 110 output N"", cOnl,der anolhrl SCflpl . called scr! ptex2 . III: l ength!u)
If you II". nOI p",vioully defined u. then 'YP"'i sc ript .x 2 .... ill prooJucc an CITO< meso sage. 1l 0J'0"evcf, if you type scriptex2 .r,c. runmnll scriptexl , then the dcfinltion 0(" from the fir)1 \Cnpl ",II be uKd ,n lh< _OM .... "1'1 ~nd the outrut an• • 'w,1I be " " played If )'00 don' , " anllhe OUtput of a !.Cop! M file to depend 011 any earlte. cornplllaliOlll In )'00' MATI.AII ""\'011. pul the 1m. c l •• r all ,ltal' lhe btginmng of the M.fil., a~ 'H JU"'~IM In S«lion" 2. I, s.om.U~1 you
IIUghl "anI your ",npt M·filc to IOC'a:pt mput! mlcract"cly, You can anani~ !hll " nh I"" command iDput kif .umr~. ,f you mdlklc lhe 1m. v a .. .. i npu t (' Input var h.ra, ' I ,n lour ~nl'l. ",h.n MATI.AH setl 10 lhal polnl ,I ",II J'Iln , "Input vllr here: " aDd paUl( ",h,1e )'00 type ,"" "aluc 10 be D\~i8""d III v a .. YII\I can 41<0 fOfmal )'OIIr 01111'' '' in a more de<.<:npt"'c manner than an" .. 40 by 1I~,"g dh" n d h ,, ('The l.ng th of II h " I , dh,,(l.ngth ( u)) The length of II i8:
•
linc. yoo Can coo,'cn lhe numeriC OUlpul 10 a ~lrinl w,lh ..ua2.t r and cone_lena", the ,mngl:
Til 1<1 lh< outpul all on
0I>e
» d h" I I 'Th. leng th of II i . The length of u 1.. 4.
8.3.2
' nual.t .. U . ngtb lll) )
' . '])
VHr lllhles in ,,' unc t ion ~ 1 ·liI cs
~ ,'lIri . hl .. UloC'd ,n a funrlioo M·fi1e ore loml, mcan,ng lhallll<=y arc unaffected I>y. and ha, . nOd fN ' QI', the "annbl« '" ) OUr WOI ~ 'P=' Con,ider Ihe rollov.-mg funcilon M.m., called uQ.m: fu.tlc t1 o n a .. . q (><) " .q (><) ... t u rn. th • •Q\lar. o f x
a .. IC . oa,
l'yp,nl . <1 t )) prod""Cl 111<= ani ..... 9 . """Iher
W", ~~. a,id dOC'~ "III chanl<' I""i, defin,"ons
Of not
x or a 'I already defined ,n )"'"
8.J M"'ricc;
8.3.3
109
Siructure of Funclion M·fiIcs
TM fil'>l 1m.. In ... fUn<;lion M·file I< tailed ~ /lmo,,,,, Ikforlllon /"'t; II defines lho fullC'ion name. U' "'fil ft • • he numMr and order of mpul and OUlpul arguments Folloo>m, lho functlO" dehnltlon Ime. the.., tan M se.'era! commc:n! hllt\ th..1 """n ,,-nh _ pem:n, s'gn I"'). TM!.C Irnes arc called /rtlp It..:, and are d".p1a)ed In "'IPOOse 10 lho Hlp CQIIIIIWId In 'he 1\1 file sq .m .lxy.e. the,e I; onlyQfIC Irllt of hell' leAl. It II dl,n the MATI.AB ".temen" tha, cakulal. the fuoclion .'alw:s. In addition. the.., un be COIrUIlrnl lines (1i1lO1 bc,inning with ') any"here in an M·file. AlIs'aIC!lk'nl! In _ function M·file Ihar DOI'fIlally plod,,,e OUlfllit Ihoulll end wilh a semicolon to SUpr"'lllhe ""'Ipur , FUI...,lion M ·m.. can hn.·. mulliple iopul nnd OIIlpul Wjumc:nl<. Here is 10 cArunplc. called polarconve rt. III, with IWO mput and IWO """pul ar,umc:ntl, funct ion [r, thata, • pola",conva"'t(", )'1 , polarconvartl", Y) ",aturns tha poll'" coo",,,,l,.ata • ... of tha point with ",..,tanlJ\lll", eoo","'illlta. Ix, y) r . aq"'t(".~2. y. ' 2)1 theta • atanl(y, "II If you Iypc pol.",conva"'t (l,
,
,
th et a
0.9213 If you a'''go lhe I)Ulpul to only one >anable. II Will rt«l\'C lhe firs. OOlput "'lIu ..... "I . ..,,ardlc.\S of ,he ~ariablc: 's name; tYI',n, tHta • po l . "'CODva "'t (l, a".l>le~ "nh lhe SlIfIlC na ..... in )'oor "orklpxe.)
8.4
Matrices
..... R nd ro/U,M .-celon; ",hiel>"~ d,~usw.J A "'illm IS a rcelangular aml) 0 r ""m""B. f1W a . . In ScCliOO 3. 4. af1' nnmpl~s of malnn:s Co.mdrr lho 3 x 1 matn>
A
",til
,
3
,
7
to II
:)
"
It can he ~ntc.-.:d In MATLAB !he ronlllund ~~ ,. • 11.1,3, " 5,5.7,81 ,,10,11,13 )
'" ,
,
,
,
, 3
•
9 10 11 12 NOI~ (hall"" ",MO' ,I,m.mt in any row nil' ~paralcd by ~Onllll~$. and tl>c row. arc .><:pa",ted hy ",mieoloo•. lllc ~ltmcnl' in a row can al
.,00' of • main. CM be found" uh .1". ~>
