Maple Tutoriaali
<Text-field style="Heading 1" layout="Heading 1">Intro</Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">T\303\244st\303\244 tutoriaalista</Font></Text-field>T\303\244m\303\244 tutoriaali on tarkoitettu interaktiiviseksi oppimisty\303\266kaluksi. Se kirjoitettiin 2013 j\303\244rjestetty\303\244 "Matemaattiset ohjelmistot" kurssia varten luentojen tueksi ja ensikosketukseksi Mapleen mukaillen Mark S. Gockenbachin laatimia tutoriaaleja.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Mik\303\244 on </Font><Font italic="true">Maple</Font></Text-field>Maple on MapleWorks-yhti\303\266n kehitt\303\244m\303\244 symbolinen laskentaohjelmisto.
<Text-field style="Heading 2" layout="Heading 2"><Font bold="true" style="Text" italic="true" size="14">Maple</Font><Font encoding="UTF-8"> ty\303\266arkki</Font></Text-field>Maplen ty\303\266arkki on dokumenttimuoto, jota esimerkiksi parhaimmillaan luet. Ty\303\266arkki on interaktiivinen objekti: se voi sis\303\244lt\303\244\303\244 grafiikkaa, symbolisia ja numeerisia laskuja, sek\303\244 ihan puhdasta ohjelmointia.
<Text-field style="Heading 2" layout="Heading 2">Kuinka saan apua nopeasti?</Text-field>Jos tarvitset v\303\244lit\303\266nt\303\244 apua siit\303\244, kuinka jokin asia Maplessa toimii, kirjoita kehotteeseen esim:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEiP0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEkaW50RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2V4ZWN1dGFibGVHRjRGLw==Edellinen komento avaa Maplen integraalifunktion apusivun
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Alkuunp\303\244\303\244sy</Font></Text-field>Yksinkertaisimmillaan Maple on vain komentotulkki, joka toteuttaa oikeaa syntaksia noudattavia k\303\244skyj\303\244:2+2;Symboli ">" on Maplen kehote; t\303\244ll\303\266in Maple-komentoja voidaan sy\303\266tt\303\244\303\244 ja suorittaa. Komento voi jakautua usealle riville, sen p\303\244\303\244ttyminen osoitetaan puolipisteell\303\244 (uusimmat versiot antavat t\303\244ss\303\244 anteeksi enemm\303\244n, ei en\303\244\303\244 v\303\244ltt\303\244m\303\244t\303\266n). Komento suoritetaan, kun sy\303\266t\303\244t puolipisteeseen p\303\244\303\244ttyv\303\244n rivin, tai rivej\303\244 ja painat Enteri\303\244. Jos tarvitset useamman rivin sy\303\266tteellesi, yhdistelm\303\244 Shift-Enter antaa uuden rivin. Jos haluat sy\303\266tt\303\244\303\244 Maple-komennon, ja kehotetta ei ole n\303\244kyvill\303\244, sen saa esille "Insert" menun kohdasta "Execution Group". N\303\244ille l\303\266ytyy my\303\266s pikakomennot : Control-j lis\303\244\303\244 kehotteen nykyisen kursorin alle, control-k lis\303\244\303\244 kehotteen kursorin yl\303\244puolelle.Painettuasi enteria, Maple evaluoi annetun komennon, ja tulostaa mahdollisen tuloksen kehotteen alapuolelle. Jos haluat, ett\303\244 tulosta ei tulosteta, p\303\244\303\244t\303\244 komento kaksoispisteeseen.K\303\244yd\303\244\303\244n viel\303\244 nopeasti l\303\244pi Maplen perusmatematiikka kyvyt. Ensiksi Maple osaa aritmetiikan sek\303\244 kokonais-, ett\303\244 rationaaliluvuilla. 123^45;115/39+727/119;Maple tuntee kaikki standardit alkeisfunktiot, kuten trigonometriset funktiot, juurifunktiot, logaritmit ja eksponentint. Se tuntee my\303\266s joitain vakioita, kuten esimerkiksi vakion \317\200. Esimerkiksi:sin(Pi/4);Huomaa, ett\303\244 vakio \317\200 kirjoitettiin isolla alkukirjaimella (Pi): pienell\303\244 kirjoitettuna se tulkitaan vain tavalliseksi muuttujaksi, ei merkitsev\303\244ksi vakioksi. sin(pi/4);Huomaa, ett\303\244 Maple ei evaluoi sin(pi/4), koska se ei tunne arvoa "pi".Toinen huomattava asia on, ett\303\244 Maple tiet\303\244\303\244 lausekkeen 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 arvon: 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 Maple ei palauta estimaattia, kuten 0.70710678, niinkuin tavallinen laskin saattaisi. Lasketaan sitten 2:n neli\303\266juuri kahdella tavalla. sqrt(2);sqrt(2.0);Ensimm\303\244isess\303\244 esimerkiss\303\244 Maple palauttaa symbolisen arvon, koska sy\303\266tearvo oli symbolinen (2). J\303\244lkimm\303\244isess\303\244 esimerkiss\303\244 palautetaan numeerinen arvo, koska sy\303\266te oli numeerinen; Maple erottelee symbolisten ja numeeristen sy\303\266tteiden v\303\244lill\303\244 desimaalierottimen, eli pisteen avulla: jos sy\303\266tteess\303\244 on piste (2.0) se on numeerinen, jos ei, se on symbolinen. My\303\266hemmin kerrotaan, kuinka symboliset tulokset voidaan muuntaa numeerisiksi.
sin(Pi/4.0);Huomaa kuinka lauseke \317\200/4.0 on osittain sievennetty: 1/4.0 on sievennetty, mutta \317\200 ei evaluoidu numeerisesti. My\303\266s, koska liukulukuna 1/4 \342\211\240 1/4.0, tarkkaa symbolista arvoa ei palauteta. (100-9)*(100+9);(-5+sqrt(5^2-4*1*4))/2;(-1)^2+5*(-1)+4;Hattusymboli "^" merkitsee potenssiinkorotusta, "*" kertolaskua ja "/" jakolaskua.Maplessa voidaan viitata edelliseen tulosteeseen prosenttimerkill\303\244 "%". Maple-puheessa t\303\244st\303\244 k\303\244ytet\303\244\303\244n nimityst\303\244 ditto-operaattori. 42^2;%-20;Edellinen laajeenee kauempiin tulosteisiin: k\303\244yt\303\244 "%%" viittaamaan toiseksi viimeiseen tulokseen, "%%%" kolmanneksi viimeiseen. T\303\244t\303\244 kauemmaksi ei p\303\244\303\244se (t\303\244ll\303\244 notaatiolla, muita keinoja on). Jos kuitenkin tied\303\244t, ett\303\244 jotain arvoa tarvitaan viel\303\244 tulevaisuudessa, kannattaa se sijoittaa muuttujaan, jolloin siihen voi viitata muuttujan nimell\303\244. Maplessa muuttujaan sijoitusta merkit\303\244\303\244n ":=" :ll\303\244.a:=23;a^2+2*a-1;Huomio: Sijoitusoperaattori on ":=", eik\303\244 "=" niinkuin useissa ohjelmointikieliss\303\244. Yht\303\244suuruusmerkki tarkoittaa Maplessa nimenomaan yht\303\244suuruutta, ja sit\303\244 k\303\244ytet\303\244\303\244n m\303\244\303\244rittelem\303\244\303\244n yht\303\244l\303\266it\303\244. On yksi yleisimmist\303\244 virheist\303\244 kirjoittaa "=" sen sijaan ett\303\244 kirjoittaisi ":= ".Jos haluat saada numeerisen arvon symboliselle tulokselle, k\303\244yt\303\244 Maplen sis\303\244\303\244nrakennettua evalf-funktiota (eval to float):evalf(Pi);Oletuksena kaikki numeeriset operaatiot lasketaan 10 desimaalin tarkkuudelle. Funktion evalf-tuottaa haluttaessa tarkempia tuloksia: muoto "evalf[n](x)" tuottaa n lukua x:n desimaalikehitelm\303\244st\303\244:evalf[30](Pi);Tarkkuutta voi s\303\244\303\244t\303\244\303\244 koko ty\303\266arkin laajuisesti muuttamalla sis\303\244ist\303\244 vakiota "Digits":Digits;Digits:=100;evalf(Pi);sqrt(2.0);T\303\244ss\303\244 tutoriaalissa Digits kiinnitet\303\244\303\244n 10:een. Digits:=10;
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Ty\303\266arkkien tallentaminen ja avaaminen</Font></Text-field>Maple-ty\303\266arkkeja voidaan tallentaa aivan kuin mit\303\244 tahansa dokumentteja: valitse File, Save (as) ja kirjoita haluamasi tiedostonimi. Maple-ty\303\266arkkeihin liitty vain muutamia asioita, jotka tulee huomioida: ensiksi, ty\303\266arkki saattaa sis\303\244lt\303\244\303\244 sek\303\244 laskutoimitukset, ett\303\244 tulosteet, tai vain laskutoimitukset. Avatessasi ty\303\266arkin, Maple ei ole tietoinen mist\303\244\303\244n siihen liittyvist\303\244 komennoista. Erityisesti mill\303\244\303\244n muuttujilla ei ole arvoja. Jotta ty\303\266arkki toimisi niinkuin sen halutaan toimivan, tulee kaikki sen sis\303\244lt\303\244m\303\244t lausekkeet evaluoida ensin.
<Text-field style="Heading 1" layout="Heading 1">Osa 1:Ensiaskeleet</Text-field>Maple osaa perusalgebran lis\303\244ksi m\303\244\303\244ritell\303\244 funktioita, ja laskea niille derivaattoja. N\303\244iden avulla on helppo (esimerkiksi) etsi\303\244 ratkaisuja differentiaaliyht\303\244l\303\266ille.
<Text-field style="Heading 2" layout="Heading 2">Esimerkki</Text-field>Halutaan varmistua, ett\303\244 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 differentiaaliyht\303\244l\303\266n 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Aluksi m\303\244\303\244rittele funktio u:u:=t->exp(a*t);Kuten syntaksi paljastaa, funktio m\303\244\303\244ritell\303\244\303\244n kuvausmuodossa: t->exp(a*t)tarkoittaa, ett\303\244 sy\303\266temuuttuja t kuvataan tulostearvolle exp(a*t). Huomaa my\303\266s, ett\303\244 eksponenttifunktio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInhGJ0YyRjVGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy9GNlEnbm9ybWFsRic= kirjoitetaan muodossa exp(x).Kun funktio on m\303\244\303\244ritelty, voidaan sit\303\244 manipuloida useilla tavoilla. Ensimm\303\244inen mieleen tuleva lienee funktion arvojen laskeminen halutussa pisteess\303\244. Funktio evaluoidaan totutulla tavalla:u(1);Oletettu tulos on 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. Mist\303\244 23 tuli? T\303\244ss\303\244 on tehty toinen Maplessa eritt\303\244in yleinen virhe: aikaisemmin ty\303\266arkissa m\303\244\303\244riteltiin muuttuja a:=23. Nyt kun sit\303\244 on k\303\244ytetty, haluttaisiin ett\303\244 a on vain m\303\244\303\244rittelem\303\244t\303\266 avoin parametri. T\303\244t\303\244 varten siivotaan a:n arvo. Arvojen siivoamatta j\303\244tt\303\244minen on yleisin virheiden l\303\244hde Maplessa. Aloita ty\303\266kirja tai luku aina komennolla restart, joka putsaa ty\303\266tilan kaikista muuttujista. T\303\244ss\303\244 on a:n arvo:a;Muuttujan arvo putsataan asettamalla muuttujan arvoksi sen oma nimi:a:='a';Nyt a:ta voi k\303\244ytt\303\244\303\244 avoimena parametrina:a;Funktiota ei tarvitse m\303\244\303\244ritell\303\244 uudelleen, ett\303\244 muutos vaikuttaisi siihen. u(1);T\303\244ss\303\244 esimerkiss\303\244 n\303\244hd\303\244\303\244n t\303\244rke\303\244 Maplen ominaisuus: kun m\303\244\303\244rittelet funktion Maplessa, sit\303\244 ei sievennet\303\244 tai evaluida (ns. laiska evaluaatio). Funktio ottaa kaikkien parametriens\303\244 arvoiksi ne mitk\303\244 ty\303\266tilassa kulloinkin ovat (vrt. MATLAB funktiokahvat, jotka toimivat t\303\244sm\303\244lleen toisin).Toinen tapa putsata muuttujia on "unassign" komento, joka on monella tapaa k\303\244yt\303\244nn\303\266llisempi, se kun antaa putsata usean muuttujan kerralla. Esimerkiksi:x:=1;y:=2;unassign('x','y');x;y;Palataan esimerkkiin. Halutaan varmistaa, ett\303\244 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 ratkaisu differentiaaliyht\303\244l\303\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Komento diff laskee derivaatan symbolisesti:diff(u(t),t)-a*u(t);Koska tulos on nolla, annettu funktio u on ratkaisu differentiaaliyht\303\244l\303\266lle. Komennon diff syntaksi on seuraava: "diff(expr,var)" derivoi lausekkeen "expr" muuttujan "var" suhteen. T\303\244ss\303\244 esimerkkej\303\244:diff(x^3,x);diff(x^3*y^2,y);Niinkuin n\303\244hd\303\244\303\244n, diff osaa laskea my\303\266s osittaisderivaattoja. Tutkitaan seuraavaa kahden muuttujan funktiota w:unassign('x','y','a');w:=(x,y)->sin(y-a*x);Lasketaan derivaatat:diff(w(x,y),x)+a*diff(w(x,y),y);N\303\244inollen w on ratkaisu osittaisdifferentiaaliyht\303\244l\303\266\303\266n LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkmbWZyYWNHRiQ2KC1GIzYpLUkjbW9HRiQ2LlErJlBhcnRpYWxEO0YnLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPC8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUkjbWlHRiQ2JlEidUYnLyUnaXRhbGljR0Y2RjQvRjhRJ2l0YWxpY0YnRlIvJStmb3JlZ3JvdW5kR1ErWzAsMTYwLDgwXUYnLyUscGxhY2Vob2xkZXJHRjZGNEZULUYjNigtRjE2LUYzRjdGOkY9Rj9GQUZDRkVGR0ZJRkwtRk82JVEieEYnRlJGVEZSL0ZXUSxbMjAwLDAsMjAwXUYnRllGVC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGY28vJSliZXZlbGxlZEdGPC1GMTYtUSIrRidGN0Y6Rj1GP0ZBRkNGRUZHL0ZKUSwwLjIyMjIyMjJlbUYnL0ZNRlxwLUZPNiVRImFGJ0ZSRlQtRiw2KEYuLUYjNihGZ24tRk82JVEieUYnRlJGVEZSRlxvRllGVEZeb0Zhb0Zkb0Zmby1GMTYtUSI9RidGN0Y6Rj1GP0ZBRkNGRUZHL0ZKUSwwLjI3Nzc3NzhlbUYnL0ZNRlxxLUkjbW5HRiQ2JFEjMC5GJ0Y3Rjc=JSFHOnko sama funktio ratkaisu osittaisdifferentiaaliyht\303\244l\303\266\303\266n (Laplace-yht\303\244l\303\266)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?T\303\244m\303\244n selvitt\303\244miseksi meid\303\244n tulee laskea korkeamman kertaluvun derivaattoja. Yksi tapa on iteroida diff komentoa:diff(diff(w(x,y),x),x);Yksinkertaisempiakin tapoja on:diff(w(x,y),x,x);Kolmas tapa on k\303\244ytt\303\244\303\244 $ operaattoria. Seuraava esimerkki laskee w:n viidennen derivaatan: kahdesti x:n suhteen ja kolmesti y:n diff(w(x,y),x$2,y$3);N\303\244ill\303\244 keinoin saadaan vastaus viimeisimp\303\244\303\244n kysymykseen: lasketaan w(x,y) derivaatat ja lasketaan yhteen ja saadaan:diff(w(x,y),x$2)-a^2*diff(w(x,y),y$2);Since the result is zero, w also solves the above second-order PDE.
<Text-field style="Heading 2" layout="Heading 2">Funktioista ja derivaatoista</Text-field>On t\303\244rke\303\244\303\244 ymm\303\244rt\303\244\303\244 lausekkeen ja funktion ero. Edellisess\303\244 esimerkiss\303\244 w(x,y) on lauseke (funktio evaluoituna pisteess\303\244 (x,y)), kun taas w on funktio; diff manipuloi lausekkeita, eik\303\244 sen k\303\244ytt\303\244miseksi tarvitse m\303\244\303\244ritell\303\244 funktiota.Toisaalta, jos on m\303\244\303\244ritelty funktio, sit\303\244 ei voi suoraan antaa diff:ille k\303\244sittelyyn: se on evaluoitava lausekkeeksi. Muuten k\303\244y n\303\244in:unassign('x');f:=x->x*sin(x);diff(f,x);Tulos on nolla, koska muuttujaa x ei esiinny lauskkeessa "f".Toisaalta t\303\244m\303\244 on varmaan se haluttu tulos:diff(f(x),x);Funktion f derivaatta on toinen funktio, jonka Maple kykeneen laskemaan D operaattorin avulla. T\303\244ss\303\244 on f:n derivaattafunktio:D(f);Huomaa, ett\303\244 f esitett\303\244\303\244n kuvauksena. T\303\244m\303\244 johtuu siit\303\244, ett\303\244 f:n derivaatta on funktio, ei lauseke. Toisaalta, f:n derivaattafunktio voidaan evaluoida pisteessa x ja saada aikaan lauseke:D(f)(x);D operattori osaa my\303\266s laskea osittaisderivaattoja. Esimerkiksi tutkitaan seuraavaa kahden muuttujan funktiota.unassign('x','y');w:=(x,y)->x^4*y^4+x*y^2;T\303\244ss\303\244 osittaisderivaatta ensimm\303\244isen sy\303\266temuuttujan suhteen (x):D[1](w);T\303\244ss\303\244 osittaisderivaatta toisen sy\303\266temuuttujan suhteen (y):D[2](w);Laskeaksemme korkeamman kertaluvun osittaisderivaattoja, muuttujien indeksit voidaan listata hakasuluissa. Esimerkiksi t\303\244ss\303\244 derivoidaan kahdesti y:n suhteen:D[2,2](w);Derivoidaksemme toistuvasti annetun muuttujan suhteen voidaan k\303\244ytt\303\244\303\244n "k$n" notaatiota, esimerkiksi D[1$4](f) laskee f:n nelj\303\244nnen osittaisderivaatan ensimm\303\244isen vapaan muuttujan suhteen. T\303\244ss\303\244 edellinen komento uudella syntaksilla: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Seuraava komento laskee 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Jos et viel\303\244 ole k\303\244ytt\303\244nyt Maplen help-toimintoa, nyt on hyv\303\244 hetki tutkia, kuinka edell\303\244 esitetyt kaksi derivointikomentoa toimivat.
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Osa 2:Haparoidaan eteenp\303\244in </Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">Integraaleja</Text-field>Maple pystyy laskemaan sek\303\244 m\303\244\303\244r\303\244\303\244ttyj\303\244 ett\303\244 m\303\244\303\244r\303\244\303\244m\303\244tt\303\266mi\303\244 integraaleja. Komento integraalin laskemiseen kummassakin tapauksessa on sama. Lasketaan integraali
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int(sin(x),x);Komento noudattelee jo opittua kaavaa: integraalioperaattori int("expr","var") ottaa sis\303\244\303\244ns\303\244 integroitavan lausekkeen (t\303\244h\303\244n palataan) "expr" ja muuttujan "var" jonka suhteen integroidaan. Kuten esimerkki n\303\244ytt\303\244\303\244, Maple ei automaattisesti lis\303\244\303\244 "integrointivakiota"; se vain yksinkertaisesti palauttaa lausekkeen antiderivaatan, jos mahdollista. Jos integrandi on liian monimutkainen, integraali palautetaan evaluoimatta:int(exp(cos(x)),x);(Muista, ett\303\244 on olemassa alkeisfunktioiden yhdistelmi\303\244, jotka eiv\303\244t ole mink\303\244\303\244n alkeisfunktion derivaattoja.)Laskeaksemme m\303\244\303\244r\303\244tyn integraalin 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,meid\303\244n tulee antaa edelliseen komentoon viel\303\244 integraalirajat. Maple k\303\244ytt\303\244\303\244 erityist\303\244 notaatiota integraalirajojen ilmaisemiseen:int(sin(x),x=0..1);(Muista, ett\303\244 int ja diff toimivat lausekkeille, ei funktioille)Esimerkkin\303\244 testataan lausetta, jonka mukaan tiettyjen ehtojen ollessa 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.T\303\244ss\303\244 lauseen spesifi instanssiF:=(x,y)->x*y^3+x^2*y;diff(int(F(x,y),y=c..d),x);int(diff(F(x,y),x),y=c..d);Tulosten tulisi olla samat.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Esimerkki: ratkaistaan reuna-arvoteht\303\244v\303\244 integroimalla</Font></Text-field>Ratkaistaan seuraava reuna-arvoteht\303\244v\303\244 (RAT):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiUtSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSy1JKG1mZW5jZWRHRiQ2JC1GIzYmRistRiM2JUYrLUkmbWZyYWNHRiQ2KC1GIzYpLUklbXN1cEdGJDYlLUYsNiZRImRGJy8lJ2l0YWxpY0dGPC8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJ0Y3LUYjNictSSNtbkdGJDYkUSIyRidGNy9GW29GXm8vJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Zeby9GOFEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYsNiVRInVGJ0Zlb0ZbcEZlby9GZ29RK1swLDE2MCw4MF1GJ0Zpb0Zcb0ZbcC1GIzYnLUZlbjYlLUYsNiVRI2R4RidGZW9GW3BGX29GXXBGZW9GZm9GaW9GW3AvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmFxLyUpYmV2ZWxsZWRHRjxGN0YrRjdGN0Y3LUY0Ni1RIj1GJ0Y3RjpGPUY/RkFGQ0ZFRkcvRkpRLDAuMjc3Nzc3OGVtRicvRk1GanEtRiM2Ji1GYm82JEZecUY3LUY0Ni1RIitGJ0Y3RjpGPUY/RkFGQ0ZFRkdGSUZMLUYsNiVRInhGJ0Zlb0ZbcEY3RitGN0YrRjc=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4=,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiYtRiw2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIwRidGQEZARkBGQC1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRl1vRmVuRkBGK0ZA, 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.Ratkaisu saadaan integroimalla puolittain kahdesti. Kuten aiemmin todettiin, Maple ei lis\303\244\303\244 integraaleihin vakiota, joten se on teht\303\244v\303\244 eksplisiittisesti (Aloitetaan siivoamalla k\303\244ytett\303\244v\303\244t muuttujat, jotta kaikki sujuu oikein.) unassign('x','u','C1','C2');Ensimm\303\244inen integraali (miinus kerrottuna oikealle)int(-(1+x),x)+C1;Integroidaan toisen 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(% viittaa edelliseen tulokseen)Meill\303\244 on nyt tilanne, jossa haluttaisiin muuttaa saatu lauseke funktioksi. T\303\244m\303\244n voi toki tehd\303\244 k\303\244sinkin, mutta Maple tarjoaa my\303\266s valmiin ty\303\266kalun t\303\244h\303\244n: funktio unapply tekee lausekkeesta funktion:u:=unapply(%,x); Unapply ottaa lausekkeen ja yhden tai useamman vapaan muuttujan, ja luo funktion, joka kuvaa muuttuja(t) lausekkeelle.Toinen esimerkki:f:=unapply(x*y^2,x,y);f(2,3);Nyt meill\303\244 on kandidaatti ratkaisulle: u. Ainoastaan vakiot C1 ja C2 puuttuvat. Etsit\303\244\303\244n niille arvot Maplen ratkaisinta k\303\244ytt\303\244en. Kuten muistetaan, "=" m\303\244\303\244rittelee yht\303\244l\303\266n Maplessa; funktio solve ottaa sis\303\244\303\244ns\303\244 yht\303\244l\303\266n tai joukon yht\303\244l\303\266it\303\244, sek\303\244 muuttujan tai useamman muuttujan, ja palauttaa (jos osaa) sellaiset muuttujan arvot, jotka toteuttavat annetut yht\303\244l\303\266t. Useampi yht\303\244l\303\266 tai muuttuja ilmoitetaan haka- tai kaarisulkujen avulla. sols:=solve([u(0)=0,u(1)=0],[C1,C2]);Komennon solve yleinen muoto on "solve(eqns,vars)", miss\303\244 eqns on yht\303\244l\303\266 tai joukko yht\303\244l\303\266it\303\244, ja vars on tuntematon muuttuja tai joukko tuntemattomia muuttujia. Huomaa, ett\303\244 Maple k\303\244ytt\303\244\303\244 yht\303\244l\303\266iden m\303\244\303\244rittelyyn merkki\303\244 "=".Edellinen tulos ei vaikuta muuttujien arvoon mill\303\244\303\244n tavalla, ts. solve ei aiheuta sivuvaikutuksia. C1;C2;T\303\244m\303\244n voi tehd\303\244 k\303\244sinkin, mutta j\303\244lleen Maplella on valmis komento jolla saa ratkaisun tulokset asetettua oikeille muuttujille: assign funktio tekee sen mit\303\244 halutaan:assign(sols);C1;C2;u(x);Meill\303\244 on nyt ratkaisu annetulle yht\303\244l\303\266lle, tarkistetaan se viel\303\244.-diff(u(x),x$2);u(0);u(1);Sek\303\244 differentiaaliyht\303\244l\303\266, ett\303\244 kummatkin reuna-arvot t\303\244ytet\303\244\303\244n.As another example, I will solve a BVP with a nonconstant coefficient: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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4=,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, 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.Ensimm\303\244inen integraali menee p\303\244\303\244ss\303\244 (ja jollei mene niin nyt on kyn\303\244+paperi aika):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictSSZtZnJhY0dGJDYoLUYjNiQtSSNtb0dGJDYtUTAmRGlmZmVyZW50aWFsRDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUEY8LUYjNiZGKy1GIzYlRjgtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y9USdpdGFsaWNGJ0Y8RitGPC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXm8vJSliZXZlbGxlZEdGQS1GOTYtUTEmSW52aXNpYmxlVGltZXM7RidGPEY/RkJGREZGRkhGSkZMRk5GUS1GIzYmLUYsNiVRInVGJ0ZaRmduLUY5Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjxGP0ZCRkRGRkZIRkpGTEZORlEtSShtZmVuY2VkR0YkNiQtRiM2JEZXRjxGPEY8RitGPC1GOTYtUSI9RidGPEY/RkJGREZGRkhGSkZML0ZPUSwwLjI3Nzc3NzhlbUYnL0ZSRmdwLUY0NigtRiM2JC1GLDYlUSNDMUYnRlpGZ25GPC1GIzYmRistRiM2Ji1JI21uR0YkNiRGW29GPC1GOTYtUSIrRidGPEY/RkJGREZGRkhGSkZML0ZPUSwwLjIyMjIyMjJlbUYnL0ZSRltyLUY0NihGYXAtRiM2JC1GZXE2JFEiMkYnRjxGPEZpbkZcb0Zfb0Zhb0Y8RitGPEZpbkZcb0Zfb0Zhb0Y8RitGPA==.Toinen integraali tuottaa:unassign('C1','C2','x');int(C1/(1+x/2),x)+C2;u:=unapply(%,x);Ratkaistaan integrointivakiot:solve({u(0)=20,u(1)=25},{C1,C2});assign(%);u(x);Tarkistetaan-diff((1+x/2)*diff(u(x),x),x);u(0);u(1);N\303\244in u on haluttu ratkaisu.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Nopeasti piirt\303\244misest\303\244</Font></Text-field>Yksi hy\303\266dyllisimmist\303\244 Maplen ominaisuuksista on kyky piirt\303\244\303\244 monenlaisia kuvaajia. Seuraavassa esimerkiss\303\244 piirret\303\244\303\244n yhden muuttujan funktion kuvaaja v\303\244lill\303\244 [0,1]. Komennon nimi on plot(expr,var=a..b), miss\303\244 "expr" on piirrett\303\244v\303\244n lausekkeen kuva, "var" on muuttuja jonka suhteen piirret\303\244\303\244n, ja viimeiseksi a ja b ovat piirtov\303\244lin m\303\244\303\244rittelev\303\244t luvut:plot(cos(x),x=0..1);Pakollisten osien lis\303\244ksi plotilla on useita vapaaehtoisia optioita, esimerkiksi otsikko:plot(sin(x),x=0..1,title="sinik\303\244yr\303\244");Toinen esimerkki vapaaehtoisesta optiosta on viivan paksuus: komentamalla"thickness=n" saadaan viiva paksummaksi. Oletusarvo on n=0, ja n:n tulee olla kokonaisluku 0 ja 15 v\303\244lill\303\244. plot(tan(x),x=0..1,thickness=3);For more details about options to a plot, see ?plot[options].Here is a graph of the two solutions:plot([sin(x),cos(x)],x=0..1,title="Sini ja kosini",thickness=3);It is frequently useful to compare two functions by plotting their difference:plot(u(x)-v(x),x=0..1,title="The difference between the two solutions",thickness=3);
<Text-field style="Heading 3" layout="Heading 3">Yleinen virhe</Text-field>Lauseketta, jossa esiintyy m\303\244\303\244rittelem\303\244tt\303\266mi\303\244 parametreja, ei voida piirt\303\244\303\244. Esimerkki: unassign('x');f:=x->sin(pi*x);plot(f(x),x=0..1);Koska Maple ei pysty evaluoimaan funktiota f miss\303\244\303\244n pisteess\303\244 realiselle arvolle, ei tulosta my\303\266sk\303\244\303\244n voida piirt\303\244\303\244. f(1.0);Ongelma on, ett\303\244 edell\303\244 ei muistettu ett\303\244 \317\200 pit\303\244\303\244 kirjoittaa isolla kirjaimella, jotta sill\303\244 olisi syntaktinen merkitys. T\303\244ss\303\244 edellinen korjattuna. f:=x->sin(Pi*x);plot(f(x),x=0..1);
<Text-field style="Heading 1" layout="Heading 1">Osa 3:Suoria! Matriiseja! Vektoreita!</Text-field>
<Text-field style="Heading 2" layout="Heading 2">Perusteet: matriisit, vektorit ja operaatiot</Text-field>Perusalgebran lis\303\244ksi Maple osaa manipuloida vektoreita ja matriiseja, ja suorittaa lineaarialgebran tavallisia operaatioita -- sek\303\244 numeerisesti, ett\303\244 symbolisesti. Ongelmien luenteesta johtuen symbolinen ratkaisu ei yleens\303\244 ole kannattavaa. Jotta p\303\244\303\244set k\303\244siksi Maplen lineaarialgebraan, sinun on ensin ladattava tarvittavat kirjastot. Maplessa t\303\244m\303\244 tapahtuu kirjoittamalla with(TarvittavaKirjasto). T\303\244ll\303\244 kerralla kirjasto on nimelt\303\244\303\244n LinearAlgebra. (HUOM. Maplessa on my\303\266s vanhempi versio samasta kirjastosta, nimelt\303\244ns\303\244 Linalg. \303\204L\303\204 k\303\244yt\303\244 sit\303\244, ellei aivan v\303\244ltt\303\244m\303\244t\303\266nt\303\244.)with(LinearAlgebra):Huomaa kaksoispiste komennon per\303\244ss\303\244. Aina kun annat Mapleen komennon, se automaattisesti tulostaa tuloksen ruudulle, ja with komennon tapauksessa tuloste on kaikkien niiden funktioiden nimet, jotka nyt ovat k\303\244yt\303\266ss\303\244. Lineaarialgebran tapauksessa niit\303\244 on paljon, joten olen vaimentanut tulostumisen k\303\244ytt\303\244m\303\244ll\303\244 kaksoispistett\303\244.Aloitetaan m\303\244\303\244rittelemell\303\244 vektorix:=Vector([1,0,-3]);Vektorin alkioita indeks\303\266id\303\244\303\244n hakasuluilla: (indeksi alkaa ykk\303\266sest\303\244)x[2];Matriisi on lista vektoreita (huom, vaakavektoreita t\303\244ll\303\244kertaa. Opimme lis\303\244\303\244 aivan pian).A:=Matrix([[1,2,3],[4,5,6],[7,8,9]]);Huomaa kaksoissulut m\303\244\303\244rittelyss\303\244. Matriisi on todellakin vain lista listoja. Lasketaan matriisin transpoosi:Transpose(A);Matriisia indeks\303\266id\303\244\303\244n my\303\266s hakasuluilla, ensin rivi, sitten sarake. Muista, ett\303\244 aloitamme ykk\303\266sest\303\244.A[1,3];A[3,3];Matriisi-vektori ja matriisi-matriisi tuloa merkit\303\244\303\244n "." operaattorillaA.x;A.A;Luonnollisesti tulon laskeminen onnistuu vain, jos koot ovat yhteensopivia. B:=Matrix([[1,2],[3,4],[5,6]]);A.B;B.A;(Symboli "<>" joka esiintyy virheilmoituksessa tarkoittaa "erisuuri").Jos vektorin alkoita voidaan ilmaista funktion avulla, Maple tarjoaa erityiskeinoja vektorin luontiin. Tarkastellaan esimerkiksi vektoria, jonka i:s alkio on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= :f:=i->i^2;y:=Vector(8,f);T\303\244ss\303\244 Vector- syntaksi vaatii vektorin alkioiden m\303\244\303\244r\303\244n, ja funktion joka m\303\244\303\244r\303\244\303\244 vektorin i:en komponentin arvon. Komento "Vector(n,f)" luo vektorin, jossa on n alkioita, joiden arvot ovat 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, ..., 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.Funktion voi antaa argumentiksi my\303\266s suoraan, ilman m\303\244\303\244rittely\303\244:z:=Vector(8,i->i^2);Sama tekniikka toimii my\303\266s matriiseilla, nyt tarvitaan vain kaksi dimensiota yhden sijaan. T\303\244ss\303\244 esimerkkin\303\244 kuuluisa Hilbertin matriisi. (Koeta ratkaista siihen liittyv\303\244 yht\303\244l\303\266ryhm\303\244 H.x = b)H:=Matrix(5,5,(i,j)->1/(i+j-1));
<Text-field style="Heading 3" layout="Heading 3">Vaihtoehtoinen notaatio matriiseille ja vektoreille</Text-field>Vektoreille on my\303\266s toinen, yksinkertaisempi notaatio kulmasulkujen avulla:x:=<1,2,3>;K\303\244ytt\303\244m\303\244ll\303\244 t\303\244t\303\244 notaatiota Mapleen voidaan sy\303\266tt\303\244\303\244 matriiseja my\303\266s sarakkeittain:A:=<<1,2>|<3,4>>;Huomaa, ett\303\244 erokeviiva erottaa matriisin sarakkeet. Alla sama matriis vanhalla syntaksilla. Huomaa erot. A:=Matrix([[1,3],[2,4]]);
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Ax=b: ratkaisut, olemassaolo ja yksik\303\244sitteisyys</Font></Text-field>Maple osaa l\303\266yt\303\244\303\244 annetun matriisin nolla-avaruudet (alkioit joille A.x = 0)A:=Matrix([[1,2,3],[4,5,6],[7,8,9]]);NullSpace(A);T\303\244ss\303\244 esimerkiss\303\244 nolla-avaruus on yksi-ulotteinen, joten kanta koostuu yhdest\303\244 vektorista. Huomaa my\303\266s, NullSpace antaa tulokseksi joukon vektoreita. Jos halutaa p\303\244\303\244st\303\244 k\303\244siksi vektoreihin, tarvitaan temppuilua (s.o. indeks\303\266inti\303\244. Kuinkas se menik\303\244\303\244n?)y:=%[1];Tarkastetaan viel\303\244, ett\303\244 tulos on p\303\244tev\303\244 (mit\303\244 pit\303\244\303\244 tulla, jos y on A:n nolla-avaruudessa?)A.y;Jotta matriisi olisi ei-singulaarinen, sen nolla-avaruuden on oltava triviaali (koostuu vain nollavektorista), ja n\303\244in ollen sill\303\244 ei voi olla kantaa. Jos n\303\244in on, NullSpace palauttaa tyhj\303\244n joukon. A:=Matrix([[1,2,1],[-3,0,0],[1,2,2]]);NullSpace(A);Esimerkki hiukan monimutkaisemmasta tilanteesta:C:=Matrix([[1,3,-1,2],[0,1,4,2],[2,7,2,6],[1,4,3,4]]);NullSpace(C);Jos matriisi ei ole singulaarinen, sill\303\244 on k\303\244\303\244nteismatriisi. Maple osaa laskea matriisin k\303\244\303\244nteismatriisin, jos sellainen on olemassa (varauksella tosin, esim. edell\303\244 mainittu Hilbertin matriisi on k\303\244\303\244ntyv\303\244, mutta sen k\303\244\303\244nteismatriisin m\303\244\303\244r\303\244minen saattaa olla hankalaa):MatrixInverse(A);C:=Matrix([[1,0,1],[1,2,1],[0,1,0]]);MatrixInverse(C);K\303\244\303\244nteismatriisia voidaan k\303\244ytt\303\244\303\244 ratkaisemaan systeemi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSNBeEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJiRidGNEY3LyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnRj5GK0ZXRj4=, koska ratkaisu on, (olettaen ett\303\244 A on k\303\244\303\244ntyv\303\244) 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGJy9GM1Enbm9ybWFsRic=. Kuitenkin, kuten MATLAB-osiossa jo opimme, on paljon tehokkaampaa ratkaista systeemi, kuin laskea eksplisiittist\303\244 k\303\244\303\244nteismatriisia:A:=Matrix([[1,0,1],[0,1,1],[1,1,1]]);b:=<1,0,2>;x:=LinearSolve(A,b);Kaikki k\303\244\303\244ntyv\303\244t matriisit kerrottuna k\303\244\303\244nteismatriisillaan tuottavat identiteettimatriisin:A.MatrixInverse(A);Maple osaa luoda identiteettimatriisin (vrt. MATLAB-eye())Id:=IdentityMatrix(3);HUOM \303\244l\303\244 nime\303\244 identiteettimatriisia I:ksi, koska Maplessa I on varattu yksikk\303\266imagin\303\244\303\244riksi:I^2;
<Text-field style="Heading 2" layout="Heading 2">Kanta ja dimensio</Text-field>
<Text-field style="Heading 3" layout="Heading 3">Esimerkki 1</Text-field>Varmista, ett\303\244 alla m\303\244\303\244ritetyt vektorit v1,v2 ja v3 muodostava kannan LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4=ssa:v1:=<1/sqrt(3),1/sqrt(3),1/sqrt(3)>;v2:=<1/sqrt(2),0,-1/sqrt(2)>;v3:=<1/sqrt(6),-2/sqrt(6),1/sqrt(6)>;Vektorien ja matriisien kertomiseen k\303\244ytet\303\244\303\244n operaattoria ".". Tavallinen t\303\244htioperaattori tekee alkioittaisia operaatioita. v1.v2;v1.v3;v2.v3;
<Text-field style="Heading 3" layout="Heading 3">Esimerkki 2</Text-field>Osoitetaan, ett\303\244 kolme kvadraattista polynomia muodostavat kannan LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+lle, eli kaikkien korkeintaan toisen asteen polynomeille.unassign('x');p1:=x->1;p2:=x->x-1/2;p3:=x->x^2-x+1/6;Oletetaan, ett\303\244 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 on mielivaltainen korkeintaan toisen asteen polynomi. unassign('x','a','b','c');q:=x->a*x^2+b*x+c;Kirjoitetaan nyt q polynomien p1,p2 ja p3 lineaarikombinaationa:unassign('c1','c2','c3');q(x)-(c1*p1(x)+c2*p2(x)+c3*p3(x));Seuraavaksi halutaan etsi\303\244 yhteisi\303\244 tekij\303\266it\303\244: tavallisesti paperilla teht\303\244v\303\244 asia hoituu yhdell\303\244 komennolla: collect collect(%,x);Sitten eritell\303\244\303\244n lausekkeesta kertoimet, ja ratkaistaan arvot c1,c2 ja c3 kun kertoimet on asetettu nolliksi:solve({coeff(%,x,0)=0,coeff(%,x,1)=0,coeff(%,x,2)=0},{c1,c2,c3});Koska yht\303\244l\303\266ryhm\303\244lle on yksik\303\244sitteinen ratkaisu c1,c2,c3 kaikilla a,b,c niin t\303\244m\303\244 implikoi, ett\303\244 mielivaltainen polynomi q voidaan kirjoitaa lineaarikombinaationa polynomeista {p1,p2,p3}, eli toisin sanoin {p1,p2,p3} on kanta avaruudelle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+
<Text-field style="Heading 3" layout="Heading 3">Esimerkki 3</Text-field>Olkoot u1,u2 ja u3 vektoreita LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiM0YnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE=ssau1:=<1,0,2>:u2:=<0,1,1>:u3:=<1,2,-1>:Todistetaan, ett\303\244 {u1,u2,u3} on kanta LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiM0YnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE=lle, ja kirjoitetaan vektori x:=<8,2,-4>;t\303\244ss\303\244 kannassa. Kuten tied\303\244mme (toivottavati) {u1,u2,u3} ovat lineaarisesti riippumattomia, jos ja vain jos matriisi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= jonka sarakkeet ovat u1, u2 ja u3 ei ole singulaarinen. Luodaan matriisi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEY5A:=<u1|u2|u3>;Jos matriisi on nonsingulaarinen, sen determinantti on nollasta poikkeava. Tarkistetaan:Determinant(A);Yleisesti determinantin k\303\244ytt\303\244minen matriisin singulaarisuuden tutkimiseen ei ole hyv\303\244 idea: suurilla matriiseilla liukulukujen aiheuttamat virheet tekev\303\244t tuloksesta usein todella harhaanjohtavan. Paljon parempi tapa on tarkastella matriisin rankkia (astetta), jota varten on kehitetty funktio Rank: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 matriisilla A on t\303\244ysi aste, se on ep\303\244singulaarinen.Nyt voidaan kirjoittaa x u1, u2 ja u3:n lineaarikombinaationa. Etsit\303\244\303\244n kertoimet c1,c2,c3 ratkaisemalla LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSNBY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GLDYlUSJ4RidGNEY3Rj5GK0Y+:c:=LinearSolve(A,x);Lopuksi tarkistetaan tulos:c[1]*u1+c[2]*u2+c[3]*u3;x;
<Text-field style="Heading 2" layout="Heading 2">Ortogonaalit kannat ja projektiot</Text-field>Edell\303\244 k\303\244ytettiin sis\303\244tuloa n\303\244ytt\303\244m\303\244\303\244n, ett\303\244 vektorit ovat ortogonaalisia. Esimerkiksi seuraaravat vektoritv1:=<1/sqrt(3),1/sqrt(3),1/sqrt(3)>:v2:=<1/sqrt(2),0,-1/sqrt(2)>:v3:=<1/sqrt(6),-2/sqrt(6),1/sqrt(6)>:ovat ortogonaalisia.v1.v2;v1.v3;v2.v3;Ne ovat my\303\266s normalisoituja (pituus on 1 )v1.v1;v2.v2;v3.v3;Koska edell\303\244mainitut vektorit ovat kanta LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjNGJy9GNlEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiNRIUYnLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnRjw=ssa, voidaan mik\303\244 tahansa LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjNGJy9GNlEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiNRIUYnLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnRjw=n vektori kirjoittaa lineaarikombinaationa n\303\244ist\303\244 vektoreista:unassign('a','b','c');x:=<a,b,c>;y:=(v1.x)*v1+(v2.x)*v2+(v3.x)*v3;Huomaa, ett\303\244 Maple ei tuottanut vektoria x kuten odotettiin. Kuitenkin on my\303\266s ep\303\244todenn\303\244k\303\266ist\303\244, ett\303\244 Maple olisi tehnyt virheen, joten kokeillaan sievent\303\244mist\303\244:simplify(%);Komento simplify kehottaa Maplea kokeilemaan erilaisia algebrallisia muunnoksia jotta saataisiin mahdollisimman selke\303\244 tulos. Aina t\303\244m\303\244 ei onnistu, mutta usein kyll\303\244.
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">Pieni harppaus \303\244\303\244ret\303\266nulotteiseen lineaarialgebraan</Font></Text-field>Pistetulo ei ole ainoa m\303\244\303\244ritelty sis\303\244tulo, mutta se on ainoa joka on m\303\244\303\244ritelty Maplessa. Ty\303\266skennelt\303\244ess\303\244 kaikkien neli\303\266integroituvien funktioiden avaruudesssa (v\303\244lill\303\244 [0,1]) sis\303\244tulokso m\303\244\303\244ritell\303\244\303\244n usein 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Koska t\303\244t\303\244 ei Maplesta l\303\266ydy valmiina, m\303\244\303\244ritell\303\244\303\244n se:l2ip:=(f,g)->int(f(x)*g(x),x=0..1);K\303\244yt\303\244nn\303\266llisyyden nimiss\303\244 m\303\244\303\244ritell\303\244\303\244n my\303\266s funktion LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJMRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGQUYrRkZGQQ== normi:l2norm:=f->sqrt(l2ip(f,f));Tarkoitus ei ole pelotella vaikeilla asioilla tai tehd\303\244 t\303\244st\303\244 introa funktionaalianalyysiin: t\303\244rke\303\244mp\303\244\303\244 on oppia periaatteet, joilla voidaan m\303\244\303\244ritell\303\244 omia normeja ja sis\303\244tuloja, koska usko pois, valmiiksi tarjotut eiv\303\244t riit\303\244 pitk\303\244lle.
<Text-field style="Heading 3" layout="Heading 3">Esimerkki 1</Text-field>Tarkastellaan kahta funktiota:unassign('x');f:=x->x*(1-x):g:=x->8/Pi^3*sin(Pi*x):Piirrett\303\244ess\303\244 n\303\244m\303\244 n\303\244ytt\303\244v\303\244t hyvin samankaltaisiltaplot({f(x),g(x)},x=0..1,thickness=3);Kuinka mitata "funktioiden samankaltaisuutta"? Kuinka paljon ne oikestaan eroavat? Yksi vaihtoehto on tarkastella funktioiden suhteellista erotusta LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GM1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYn-mieless\303\244.l2norm(f-g)/l2norm(f);evalf(%);Funktiot eroavat toisistaan v\303\244hemm\303\244n kuin 4%.
<Text-field style="Heading 3" layout="Heading 3">Esimerkki 2</Text-field>T\303\244m\303\244n esimerkin data annetaan kahdessa vektorissa (koordinaattidataa)x:=<0.1,0.3,0.4,0.75,0.9>:y:=<1.7805,2.2285,2.3941,3.2226,3.5697>:On hy\303\266dyllist\303\244 piirt\303\244\303\244 data ennenkuin aloitetaan tarkempi analyysi. Pisteitt\303\244inen data voidaan piirt\303\244\303\244 komennolla pointplot. Se l\303\266ytyy plots kirjastosta. with(plots):Piirt\303\244mist\303\244 varten tarvitaan my\303\266s lista datapisteist\303\244: yksi tapa on luoda matriisi, jonka sarakkeina ovat vektorit x ja y. d:=<x|y>;T\303\244ss\303\244 on pointplot. pointplot(d,symbolsize=15);Datapisteet vaikuttaisivat sijaitsevan suoralla: sijoitetaan pisteisiin suoram muotoa y(x)=mx+b, joka parhaiten sopii dataan pienimm\303\244n neli\303\266summan mieless\303\244. Kuten MATLAB-osasta muistamme, lasketaan 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\303\244 c-vektorissa ovat kertoimet m ja b, ja matriisi A muodostetaan seuraavasti:e:=<1,1,1,1,1>;A:=<x|e>;Nyt voidaan ratkaista suoran kertoimille parhaat mahdolliset approksimaatiot pienimm\303\244n neli\303\266summan mieless\303\244:c:=LinearSolve(Transpose(A).A,Transpose(A).y);Lopullinen ratkaisu on l:=s->c[1]*s+c[2];Here is a graph of the best fit polynomial:plot(l(s),s=0..1,thickness=3);Jos halutaan katsoa datapisteit\303\244 ja suoraa samassa kuvassa, tarvitaan display komentoa, joka n\303\244ytt\303\244\303\244 useampia kuvia yht\303\244aikaa p\303\244\303\244llekk\303\244in. Se on osa plots kirjastoaT\303\244m\303\244n j\303\244lkeen piirret\303\244\303\244n kuvat uusiksi, t\303\244ll\303\244 kertaa vain tallettaen kuvat muuttujille plot1 ja plot2. Koska kuvat on jo piirretty kerran, vaimennetaan tulostus kaksoispisteell\303\244. plot1:=pointplot(d,symbolsize=15):plot2:=plot(l(s),s=0..1,thickness=3):Nyt voidaan k\303\244ytt\303\244\303\244 display-komentoadisplay([plot1,plot2]);Sovitus on hyv\303\244.
<Text-field style="Heading 3" layout="Heading 3">kesken</Text-field>T\303\244ss\303\244 esimerkiss\303\244 approksimoidaan funktiota 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 kvadraattisella funktiolla v\303\244lill\303\244 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNictSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMUYnRjQvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGNEY0LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRlRGNA==. Aloitetaan m\303\244\303\244ritteIn this example, I will compute the best quadratic approximation to the function 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 on the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMUYnRjRGNEY0LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRjQ=. Here are the standard basis functions for the space LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJQRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE=:unassign('x');p1:=x->1:p2:=x->x:p3:=x->x^2:I will now compute the Gram matrix and the right hand side of the normal equations. Notice that the function 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 is named exp.G:=Matrix([[l2ip(p1,p1),l2ip(p2,p1),l2ip(p3,p1)],[l2ip(p1,p2),l2ip(p2,p2),l2ip(p3,p2)],[l2ip(p1,p3),l2ip(p2,p3),l2ip(p3,p3)]]);b:=<l2ip(p1,exp),l2ip(p2,exp),l2ip(p3,exp)>;I now solve the normal equations and find the best quadratic approximation:c:=LinearSolve(G,b);q:=x->c[1]*p1(x)+c[2]*p2(x)+c[3]*p3(x);One way to judge the goodness of fit is with a plot:plot({exp(x),q(x)},x=0..1,thickness=3);The fit is quite good, so it is more informative to graph the error:plot(exp(x)-q(x),x=0..1,thickness=3);We can also judge the fit by the relative error in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJMRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE=:evalf(l2norm(exp-q)/l2norm(exp));The error is less than 0.3%.Particularly if you wish to use a subspace with a larger dimension, typing in the formulas for the Gram matrix and the right-hand side of the normal equations can be quite monotonous. One can avoid the repetitive typing, but only by representing the basis functions differently. Here is an example:unassign('x');p[1]:=x->1:p[2]:=x->x:p[3]:=x->x^2:Now, for each 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, p[i] is one of the basis functions. Here is the third, for example:p[3](x);I can now define the Gram matrix using the alternate version of the Matrix command:G:=Matrix(3,3,(i,j)->l2ip(p[j],p[i]));I define the right-hand side similarly:b:=Vector(3,i->l2ip(p[i],exp));Just for fun, I will now repeat the above computation, but with sixth-degree polynomials. I will define the basis functions themselves in a vector:p:=Vector(7,i->(x->x^(i-1)));Now I will compute the Gram matrix and the right-hand side, and solve the normal equations:G:=Matrix(7,7,(i,j)->l2ip(p[j],p[i])):b:=Vector(7,i->l2ip(p[i],exp)):c:=LinearSolve(G,b):Now, the best sixth-degree approximation 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.To form this expression without an undue amount of typing, I can use the add command:unassign('x');q:=x->add(c[i]*p[i](x),i=1..7);Now I will plot the approximation and the error:plot(q(x),x=0..1,thickness=3);plot(exp(x)-q(x),x=0..1,thickness=3);The fit is now very good.
<Text-field style="Heading 2" layout="Heading 2">Matriisin ominaisarvot</Text-field>Maple osaa laskea neli\303\266matriisin ominaisarvot (osa LinearAlgebra-kirjastoa. Muista ladata se).A:=Matrix([[1,2,-1],[4,0,1],[-7,-2,3]]);res:=Eigenvectors(A);Tuloksen res ensimm\303\244inen alkio on vektori, jossa ovat matriisin ominaisarvot, toinen vektori, jonka sarakkeet ovat ominaisvektorit. l:=res[1];V:=res[2];Seuraavalla syntaksilla p\303\244\303\244st\303\244\303\244n k\303\244siksi matriisin sarakkeisiin (t.s. ominaisvektoreihin, t\303\244ss\303\244 tapauksessa. ) Hyvin MATLAB-henkinen syntaksi.V[1..3,1];Tarkastetaan ominaisarvo-ominaisvektori parit. A.V[1..3,1]-l[1]*V[1..3,1];A.V[1..3,2]-l[2]*V[1..3,2];A.V[1..3,3]-l[3]*V[1..3,3];Jos kaikki ei vaikuta siistilt\303\244, niin ei h\303\244t\303\244\303\244, simplify auttaa kotona ja puutarhassa. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmdGltZXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlNGK0Y1Rjk=matriisin ominaisarvojen l\303\266yt\303\244mienen on ekvivalentti ongelma n-asteisen polynomin juurien l\303\266yt\303\244misen kanssa. Yhden 1800-luvun kuuluisimman lauseen mukaan on mahdotonta l\303\266yt\303\244\303\244 kaavaa mielivaltaisen polynomin juurien ilmaisemiseen, jos polynomin aste on isompi kuin 5. Niinp\303\244 ominaisarvojen symbolinen (tarkka) m\303\244\303\244r\303\244\303\244minen suurelle matriisille johtaa todenn\303\244k\303\266isesti ongelmiin:B:=Matrix(5,5,(i,j)->1/(i+j-1));Eigenvectors(B):Tulostus on vaimennettu, koska se on useita sivuja pitk\303\244. Useat ominaisvektorit ilmoitetaan karakteristisen polynomin tuntemattomina juurina. N\303\244it\303\244 on mahdollista manipuloida, mutta se ylitt\303\244\303\244 t\303\244m\303\244n tutoriaalin taitotason moninkertaisesti. Edell\303\244 ilmenneen ongelman takia on yleens\303\244 hyv\303\244 idea laskea ominaisarvot numeerisesti (MATLAB...). Koska Maple on syd\303\244melt\303\244\303\244n symbolinen ohjelma, t\303\244m\303\244 vaatii hieman temppuilua. res1:=Eigenvectors(evalf(B)):l1:=res1[1]:V1:=res1[2]:Tarkastetaan tulokset:B.V1[1..5,1]-l1[1]*V1[1..5,1];Huomaa, ett\303\244 tulos ei ole en\303\244\303\244 identtisesti nollaa, vaan sis\303\244lt\303\244\303\244 liukulukuoperaatioihin aina sis\303\244ltyv\303\244n py\303\266ristysvirheen.Yll\303\244olevat ominaisparit muodollisesti sis\303\244lt\303\244v\303\244t kompleksilukuja, vaikka kompleksiosat ovat kaikki nollia; itse asiassa, B on symmetrinen, ja symmetrisen matriisin ominaisarvot ovat aina realisia. T\303\244llaisille erikoistapauksille on oma optionsa:B:=Matrix(5,5,(i,j)->1/(i+j-1),shape=symmetric);res1:=Eigenvectors(evalf(B));
<Text-field style="Heading 3" layout="Heading 3">Esimerkki </Text-field>Ratkaistaan yht\303\244l\303\266ryhm\303\244 Ax=b spektraalimetodillaA:=Matrix([[11,-4,-1],[-4,14,-4],[-1,-4,11]],shape=symmetric);b:=<1,2,1>;Lasketaan A:n ominaisarvo-vektori parit. Koska A on pieni, symbolinen laskenta ei johda ongelmiin. res:=Eigenvectors(A);l:=res[1];V:=res[2];Toisin kuin MATLAB, Maple ei automaattisesti normalisoi ominaisvektoreita. Ominaisvektorit ovat m\303\244\303\244ritelm\303\244ns\303\244 nojalla ortogonaalisia (A on symmetrinen), ja ominaisarvot ovat erilliset, mutta ominaisvektorit on normitettava k\303\244sin. u1:=V[1..3,1]/sqrt(V[1..3,1].V[1..3,1]);u2:=V[1..3,2]/sqrt(V[1..3,2].V[1..3,2]);u3:=V[1..3,3]/sqrt(V[1..3,3].V[1..3,3]);Ratkaisu Ax=b:lle on x:=(u1.b/l[1])*u1+(u2.b/l[2])*u2+(u3.b/l[3])*u3;Tarkista:A.x-b;
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Osa 4:(Ep\303\244)tavallisia differentiaaliyht\303\244l\303\266it\303\244 </Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">Section 4.2: Solutions to some simple ODEs</Text-field>
<Text-field style="Heading 3" layout="Heading 3">Second-order linear homogeneous ODEs with constant coefficients</Text-field>Suppose we wish to solve the following IVP: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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=,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,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The characteristic polynomial is 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, which has the following roots:r:=solve(r^2+4*r-3=0,r);The general solution isunassign('c1','c2','t');u:=t->c1*exp(r[1]*t)+c2*exp(r[2]*t);We can now solve for the unknowns c1 and c2:sols:=solve({u(0)=1,D(u)(0)=0},{c1,c2});assign(sols);Here is the solution,u(t);and here is its graph:plot(u(t),t=0..5,thickness=3);
<Text-field style="Heading 3" layout="Heading 3">A special inhomogeneous second-order linear ODE</Text-field>Tutkitaan alkuarvoteht\303\244v\303\244\303\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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=,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,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNidGKy1GIzYmLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW9HRiQ2LVErJlBhcnRpYWxEO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQy8lKXN0cmV0Y2h5R0ZDLyUqc3ltbWV0cmljR0ZDLyUobGFyZ2VvcEdGQy8lLm1vdmFibGVsaW1pdHNHRkMvJSdhY2NlbnRHRkMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZSRj4tRiM2JkYrLUYjNiVGOi1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvRj9RJ2l0YWxpY0YnRj5GK0Y+LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zgby8lKWJldmVsbGVkR0ZDLUY7Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y+RkFGREZGRkhGSkZMRk5GUEZTLUYsNiVRInVGJ0ZmbkZpbkY+LUY7Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRj5GQUZERkZGSEZKRkxGTkZQRlMtSShtZmVuY2VkR0YkNiQtRiM2JC1JI21uR0YkNiRRIjBGJ0Y+Rj5GPkY+LUY7Ni1RIj1GJ0Y+RkFGREZGRkhGSkZMRk4vRlFRLDAuMjc3Nzc3OGVtRicvRlRGW3EtRmRwNiRRIzAuRidGPkY+RitGPg==The solution, as given in Section 4.2.3 of the text, isunassign('t','s');(1/2)*int(sin(2*(t-s))*sin(Pi*s),s=0..t);u:=unapply(%,t);Check:diff(u(t),t$2)+4*u(t);simplify(%);u(0);D(u)(0);
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">Ensimm\303\244isen kertaluvun lineaarinen alkuarvoteht\303\244v\303\244</Font></Text-field>Tutkitaan seuraavaa alkuarvoteht\303\244v\303\244\303\244 (AAT)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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=,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.Ratkaistaan differentiaaliyht\303\244l\303\266:QyQ+SSRlcW5HNiIvLCYtLUkiREdGJTYjSSJ1R0YlNiNJInRHRiUiIiItRixGLSMhIiIiIiMsJEYuRjJGLw==QyQtSSdkc29sdmVHNiI2JDwkSSRlcW5HRiUvLUkidUdGJTYjIiIhIiIiLUYrNiNJInRHRiVGLg==assign(%);Piirret\303\244\303\244n kuva ratkaisustaplot(u(t),t=0..3,thickness=3);Lasketaan ut(t):n nollakohtasolve(u(t)=0,t);Funktio solve palauttaaa kaksi ratkaisua, ilmaistuna LambertW funktioiden avulla. T\303\244m\303\244 ei ole harvinaista differentiaaliyht\303\244l\303\266iden tutkimisessa: tuloksia ei yleens\303\244 voida esitt\303\244\303\244 standardien alkeisfunktioiden avulla. Tulokset voidaan toki esitt\303\244\303\244 numeerisesti:evalf(%);J\303\244lkimm\303\244inen tulos on se haluttu.Ratkaisuun voidaan k\303\244ytt\303\244\303\244 my\303\266s numeerista etsint\303\244 fsolve, joka etsii parhaan liukulukuapproksimaation juurelle annetulla v\303\244lill\303\244. fsolve(u(t)=0,t=1..2);Numeerisisissa tilanteissa on yleens\303\244 parempi k\303\244ytt\303\244\303\244 suoraan fsolvea.
<Text-field style="Heading 2" layout="Heading 2">Section 4.3: Linear systems with constant coefficients</Text-field>Since Maple can compute eigenvalues and eigenvectors (either numerically or, when possible, symbolically), it can be used to solve linear systems with constant coefficients. I will begin with a simple example, solving the homogeneous 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=,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.I will take the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and the initial vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj1GPg== to be as follows:A:=Matrix([[1,2],[3,4]]);x0:=<4,1>;The first step is to find the eigenpairs of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=:res:=Eigenvectors(A);l:=res[1];V:=res[2];Next, the initial vector must be expressed in terms of the eigenvectors of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=:c:=LinearSolve(V,x0);I can now write down the desired solution:unassign('t');x:=t->c[1]*exp(l[1]*t)*V[1..2,1]+c[2]*exp(l[2]*t)*V[1..2,2];Notice that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg== is a vector (that is, the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is vector-valued). For example, the first component isx(t)[1];Here is its graph:plot(x(t)[1],t=0..1,thickness=3);(The solution is dominated by a rapidly growing exponential.)Before I leave this example, I would like to check that the solution is correct. This requires differentiating the vector-valued function x(t):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The diff command cannot be applied directly to a vector-valued function. There are two alternatives. First, D works fine: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Alternatively, we can apply diff to each component of x(t) by using map: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(0)-x0;simplify(%);
<Text-field style="Heading 3" layout="Heading 3">Inhomogeneous systems and variation of parameters</Text-field>I will now show how to use Maple to solve inhomogeneous systems of the 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiYtRiw2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtbkdGJDYkUSIwRidGQEZARkBGQC1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRl1vLUklbXN1YkdGJDYlRjNGWS8lL3N1YnNjcmlwdHNoaWZ0R0ZobkZARitGQA==.Consider the matrixA:=Matrix([[1,2],[2,1]],shape=symmetric);Letx0:=<0,1>;unassign('t');f:=t-><sin(t),0>;I will solve this problem numerically. Here are the eigenpairs of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=:res:=Eigenvectors(evalf(A));l:=res[1];V:=res[2];Maple automatically normalized numerical eigenvectors, and the eigenvectors are necessarily orthogonal since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is symmetric.u1:=V[1..2,1];u2:=V[1..2,2];Now, we have 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 and 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, wherec1:=unapply(u1.f(t),t);c2:=unapply(u2.f(t),t);b1:=u1.x0;b2:=u2.x0;We then solve the decoupled IVPsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiYtSSZtZnJhY0dGJDYoLUYjNiQtSSNtb0dGJDYtUSsmUGFydGlhbEQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZBLyUpc3RyZXRjaHlHRkEvJSpzeW1tZXRyaWNHRkEvJShsYXJnZW9wR0ZBLyUubW92YWJsZWxpbWl0c0dGQS8lJ2FjY2VudEdGQS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlBGPC1GIzYmRistRiM2JUY4LUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GPVEnaXRhbGljRidGPEYrRjwvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl5vLyUpYmV2ZWxsZWRHRkEtRjk2LVExJkludmlzaWJsZVRpbWVzO0YnRjxGP0ZCRkRGRkZIRkpGTEZORlEtSSVtc3ViR0YkNiUtRiw2JVEiYUYnRlpGZ24tRiM2JC1JI21uR0YkNiRGW29GPEY8LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPC1GOTYtUSI9RidGPEY/RkJGREZGRkhGSkZML0ZPUSwwLjI3Nzc3NzhlbUYnL0ZSRmhwLUYjNihGKy1GIzYmLUZnbzYlLUYsNiVRImxGJ0ZaRmduRlxwRmFwRmNvRmZvRjwtRjk2LVEiK0YnRjxGP0ZCRkRGRkZIRkpGTC9GT1EsMC4yMjIyMjIyZW1GJy9GUkZncS1GIzYmLUZnbzYlLUYsNiVRImNGJ0ZaRmduRlxwRmFwLUY5Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRjxGP0ZCRkRGRkZIRkpGTEZORlEtSShtZmVuY2VkR0YkNiQtRiM2JEZXRjxGPEY8RitGPEYrRjxGK0Y8, 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,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, 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.The solutions, using the techniques of Section 4.3, 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*exp(l[2]*t)+int(exp(l[2]*(t-s))*c2(s),s=0..t);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 solution to the original system is thenx:=t->a1(t)*u1+a2(t)*u2;Check:D(x)(t)-A.x(t)-f(t);x(0)-x0;The answer is correct, up to roundoff error.
<Text-field style="Heading 2" layout="Heading 2">Section 4.4: Numerical methods for initial value problems</Text-field>When we turn to numerical methods for IVPs in ODEs, we naturally wish to write programs to implement the methods. Since time-stepping methods involve repeated steps of the same form, it would be quite tedious to apply the method manually. Here is another strength of Maple: not only does it integrate symbolic and numerical computations with graphics, it also provides a programming environment. I will explain the basics of programming in Maple.
<Text-field style="Heading 3" layout="Heading 3">Interactive commands</Text-field>Maple supports the usual programming constructs, such as loops and conditionals, which can be used both interactively and in programs. For example, suppose we wish to apply Euler's method to estimate the solution 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=,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on the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMUYnRjRGNEY0LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRjQ=. The exact solution isunassign('t');v:=t->exp(arctan(t));I will use the exact solution to test my results below. In preparation for applying Euler's method, I define the right-hand side of the ODE as a function:unassign('u');f:=(t,u)->u/(1+t^2);I now apply Euler's method with a step length of 0.1. The Maple command we need is the "repetition statement," a form of the familiar for-loop (in C programming) or do-loop (in Fortran programming). First, some initialization:dt:=0.1;n:=10;u:=1.0;tt:=0.0;(Here is an important but subtle point. I definitely want to do numeric computations, not symbolic. Therefore, I assign the quantities dt, u, and t to have floating point values (0.1, 1.0, and 0.0) rather than exact values (1/10, 1, and 0). If I do not do this, then the symbolic computation becomes quite involved---all the quantities involved will be rational numbers, and the number of digits will explode. The results are not bad for 10 steps, but you would not want to do a symbolic computation for hundreds of steps.)The following loop executes 10 steps of Euler's method (notice that I do not save the intermediate values of the computed solution):for i from 1 to ndo u:=u+dt*f(tt,u); tt:=tt+dt;end do;(If I do not wish to see the intermediate results, I can end the "end do" statement with a colon instead of a semicolon.)Here is the relative error in the computed solution (at t=1.0):abs(v(1.0)-u)/abs(v(1.0));The result is not bad.If I want to save the intermediate results (for plotting purposes, say), I can put them in a matrix. The following command creates an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiK0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= by 2 matrix filled with zeros.U:=Matrix(n+1,2);Here is another version of the calculation:U[1,1]:=0.0:U[1,2]:=1.0:for i from 1 to ndo U[i+1,2]:=U[i,2]+dt*f(U[i,1],U[i,2]): U[i+1,1]:=U[i,1]+dt:end do:I can now look at the results using pointplot:with(plots,pointplot):pointplot(U,symbolsize=15);plot1:=%:Here is the exact solution:plot(v(t),t=0..1,thickness=3);plot2:=%:Here are the exacted and computed solution:with(plots,display):display(plot1,plot2);I can define a function representing the computed solution using interpolation. One way to do this is to use the Spline function (which is part of the CurveFitting package). Spline (by default) produces a piecewise cubic function agreeing with the data points, and it is simple to use, as the following example shows. (I recommend ending the command with a colon, as the output is quite complicated.)with(CurveFitting,Spline):Spline(U,t):(The reason for the argument "t" is that Spline produces an expression in the given variable.)ui:=unapply(%,t):plot(ui(t),t=0..1,thickness=3);Now we can compare the computed solution directly with the exact solution:plot(v(t)-ui(t),t=0..1,thickness=3);
<Text-field style="Heading 3" layout="Heading 3">Creating new <Font bold="true" style="Text">Maple</Font> commands</Text-field>For convenience, we can write a program to implement Euler's method. In Maple, writing a program really implies creating a new Maple command. A reasonable command for applying Euler's method to the 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSI8RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZNLUYsNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU=,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would take as inputs LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj1GPg==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj1GPg==, as well as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (the final time) and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (the number of time steps), and would produce the table of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImlGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlLUYvNiVRInVGJ0YyRjVGOEY/Rj0= values (like the result U above). In Maple, a command is implemented as a procedure:myeuler:=proc(f,t0,u0,T,n)local i,U,dt;U:=Matrix(n+1,2);dt:=evalf(T/n);
U[1,1]:=t0:U[1,2]:=u0:for i from 1 to ndo U[i+1,1]:=U[i,1]+dt: U[i+1,2]:=U[i,2]+dt*f(U[i,1],U[i,2]):end do:Uend proc;The last statement in the above procedure just echos the value of U, since the return value of the procedure is the last computed value.I can now use the myeuler procedure instead of typing out the loop each time I want to apply Euler's method. The following command repeats the above computation:U:=myeuler(f,0.0,1.0,1.0,10);Here is a graph of the computed solution:pointplot(U,symbolsize=15);Notice that, as implemented above, the myeuler command uses local variables i, U, dt. When the command is invoked, the variables are known only during the execution of the command. Also notice that the return value of the procedure is the last result generated by it; thus, the desired output value U is echoed in the final statement.As written, there is nothing indicating to the myeuler command that the IVP to be solved is scalar. Therefore, the same command will solve a system, such 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, 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,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,whose solution isunassign('t');x:=t-><cos(t),sin(t)>;Here is how I would apply myeuler to the system:unassign('u');f:=(t,u)-><-u[2],u[1]>;n:=20;U1:=myeuler(f,0.0,<1.0,0.0>,10.0,n);Now I have computed the solution, but it is not stored in a very convenient fashion. The second column of U1 consists of 2-vectors; for instance, here is the last computed value:U1[n+1,2];I can display the components of the solution by first extracting them, along with the time grid:tt:=U1[1..n+1,1];U11:=Vector(n+1,i->U1[i,2][1]);U12:=Vector(n+1,i->U1[i,2][2]);Now I can plot the first component of the solution using the pointplot command:pointplot(<tt|U11>,symbolsize=15);plot1:=%:plot(x(t)[1],t=0..10,thickness=3);plot2:=%:display(plot1,plot2);Obviously the computed solution is not very accurate. I leave it to you to verify that, if the number of steps is increased, the error decreases in a satisfactory way.
<Text-field style="Heading 3" layout="Heading 3">Storing programs in files</Text-field>Having decided that a certain algorithm is useful enough to be made into a Maple command, you will probably wish to save it in a file, so that it can be loaded (rather than type anew) whenever it is needed in a new Maple session. Doing this is simple enough: the Maple commands defining the function are typed into a plain text file and read into Maple using the read command. For example, I created a text file containing the myeuler command defined above (the file is also called "myeuler"). I read it in as follows:read myeuler;I can now use myeuler as before.When working with user-defined files, it is important that Maple be able to find the files. By default, Maple looks in the current directory. You can determine the current directory with the currentdir command:currentdir();The most recent read command may have failed when you ran it; this would happen if the file "myeuler" is not stored (on your computer) in the current directory. You can change the current directory with the same currentdir command; the syntax is "currentdir(new_dir)", where "new_dir" is the name of the new directory (use the full path name). If the directory name contains special characters (like spaces), enclose it in quotes.If necessary, you can give the full path name of the file. Here is an alternate version of the above command:read "/home/msgocken/books/pdebook2/tutorial/maple/myeuler";(On a Windows-based machine, the directory separator (the backslash) must be repeated; also, the file name must be enclosed in quotes if it contains spaces.) The above command is likely to fail on your computer, unless you happen to have the same directory structure as I have.