MS-E1999 Matemaattiset ohjelmistot
Maaliskuu 2014
HA
Document mode vs. worksheet mode.
Document: Uudempi,tavoitteena tyylik"as matemaattinen dokumentti/laskenta-arkki
Worksheet: Korostaa enemm"an laskentaa. Tyylik"as sekin matemaattisine kaava-edtointeineen.
Varsinainen toiminta alkaa kohdasta 6 , lue kuitenkin ainakin 1 l\303\244pi huolella.
<Text-field style="Heading 1" layout="Heading 1">Maple 17</Text-field>
Maple 18 julkistus viime viikolla.
Oppilaat saavat asentaa omalle koneelleen, lisenssi voimassa vuoden loppuun.
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">1. Ty\303\266skentelyohjeita, ty\303\266arkin k\303\244sittelyn perusasiat</Font></Text-field>
Muista tallettaa aika ajoin (CTR-S) tai File-valikko: save [Mac-ty\303\266skentelyn yleisohje: CTR -> CMD] Talletus on joka tapauksessa syyt\303\244 tehd\303\244 aina ennen jotain potentiaalisesti isoa laskentaa. T\303\244m\303\244 ty\303\266arkki sis\303\244lt\303\244\303\244 ohjeita ja ohjeisiin liittyvi\303\244 teht\303\244vi\303\244, joita voi samantien ryhty\303\244 kokeilemaan. Voit tehd\303\244 muistiinpanoja ty\303\266arkille ja tallettaa sen itsellesi. Suositus: Selaa ty\303\266arkkia jonkin matkaa, totuttele heti helpin k\303\244ytt\303\266\303\266n. Harjoitusteht\303\244v\303\244dokumentti: Kannattaa aloittaa INSERT-valikon SECTION-valinnalla ja napsauttaa muutama sektio heti k\303\244ttelyss\303\244 arkille. Huomaa, ett\303\244 ty\303\266arkin voi julkaista (export) my"os pdf tai HTML-muodossa. Edellinen siistimpi, j\303\244lkimm\303\244isess"a toimii my"os animaatiot. M T\303\204LL\303\204 KURSSILLA pdf (tottakai). El\303\244m\303\244nohje: Maplen filosofiaa kannattaa opetella senverran, ett\303\244 ty\303\266skentely k\303\244y nautittavaksi. (Sama p\303\244tee Matlabiin.) Tavoitteita: - Ty\303\266arkin k\303\244sittely - Peruslaskutoimitukset ja sievennykset
- Grafiikka
- Vapaat ja sidotut muuttujat
- Lauseke vs. funktio
- Matriisit
- Perustietorakenteet, erit. jonot, joukot, listat.
- V ertailu Matlab-ty\303\266skentelyyn
Katsotaan ensin ty\303\266arkkiin (worksheet) liittyvi\303\244 asioita. - Uuden (laskenta)kehotteen saa hiirell\303\244 ty\303\266kalunauhasta [ > ] tai CTR-J (j\303\244lkeen) tai CTR-K (ennen) - Tekstitila: T ( edellisen vasemmalla puolella) tai CTR-t (Mac: CMD-t)
- F5 Math/Input ja text/math - Math-tilassa matemaattista kaavaa voi editoida k"atev"asti. Nuoli oikealla -> vie takaisin perustasolle.
T"ass"a F5 ja sitten vain editoidaan kaavaa, kokeile!
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- Ei tarvitse menn"a rivin loppuun ennen ENTER:ia.
- CTR-l (label) liitt"a"a viittauksen aiemman tuloksen kaavanumeroon.
Yleisia ohjeita: Leikkaa/liimaa: UNIX/X:ss\303\244 oikein k\303\244tev\303\244, eli kuten aina X:ss\303\244 - Maalataan hiiren vasemmalla
- vied\303\244\303\244n kursori haluttuun kohtaan - liimataan keskimm\303\244isell\303\244 (Windows:ssa maalaus, CTR-C, vienti, CTR-V, Mac: CTR <-> CMD)
Ennen "lopullista" tallettamista kannattaa valita EDIT-valikosta remove output => tiedosto pieneksi.
K\303\244\303\244nteinen toimenpide on EDIT-valikon edellinen valinta: execute worksheet.
Matemaattista tekstink\303\244sittely\303\244 Maplella Versioissa >=15 varsin kehittynyt. Ideaali: Matemaattisin kaavoin, kuvin ym. varustettu dokumentti, joka suorittaa itsens\303\244. Toki LaTeX on matemaattisen tekstink\303\244sittelyn ykk\303\266nen. Maplella k"atevaa generoida LaTeX-kaavoja. komennolla latex .
Omia kokeiluja ... tekstia, matem. kaavoja...
Seuraavissa voit kokeilla laskentaa. Tee lis"a"a kehotteita CTR-J.
JSFH
JSFH
<Text-field style="Heading 1" layout="Heading 1">2. Viitteit"a, ty"oarkkeja, koodeja</Text-field>
Samat linki ovat luentosivulla.
[HAM] Apiola: Symb ja num. lask Maple-ohjelmalla, Otatieto 588
T\303\244m\303\244 teos ilmeisine puutteineenkin lienee hy\303\266dyllinen. Perustuu versioon 5. T\303\244ydennyst\303\244 vaatii erityisesti matriisilaskentaosuus, sill\303\244 uusi LinearAlgebra -tyyli tuli vasta versioon 6. Kirjastossa useita , minulla muutama j\303\244ljell\303\244, lainaan kurssin omiani ajaksi .
[SOL] Solmun Maple-kirjoitus http://www.math.helsinki.fi/Solmu/solmu12/apiola
[KOF] M. Kofler: Maple, an introduction and reference, Addison Wesley 1997 www.awl-he.com, eritt\303\244in hyv\303\244 Maple-yleiskirja. [LYN] Lynch: Dynamical systems with Applications using Maple, Birkh\303\244user (Mat-kirjastossa) (Maple-tutoriaalit ja paljon diffyht\303\244l\303\266koodeja) www.maplesoft.com/apps/
[HECK] Heck: An Introductio to Maple (Springer), t\303\244ydellisin ja kattavin Maple-kirja
[ISR] Israel: Calculus with Maple (pari ylim. kappaletta kirjastossa)
[Coomb] Coombes-Hunt-Lipsman-Osborn-Stuck: Diff. Equations with Maple (ainakin 1 kirjastossa)
Kurssia varten kirjoitettuja/tehtavia Maple-ty\303\266arkkeja, kuten LA, ominaisarvot, lindys, fourier, numint, ... maple/ns05.mpl
<Text-field bookmark="Laskentoa symboleilla ja numeroilla" style="Text" layout="Normal"><Font bold="true" size="18">6. Laskentoa symboleilla ja numeroilla</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">Vapaat ja sidotut muuttujat</Text-field>
Lue
[SOL] s 15-16 Symbolien hallintaa ja periaatteita
[HAM] s. 70 - 71 (korjaus: http://www.math.hut.fi/~apiola/maple/opas/eval.html )
(see Heck Ch 3 Variables and names p. 65 -..
The secret behind success of CA : free (unbound) variables . )
On hyv\303\244 totutella laittamaan restart ty\303\266arkin alkuun (ja muihinkin heng\303\244hdyspaikkoihin) Painele nyt ENTER:i\303\244 ja palaa ohjeen mukaan takaisin.
restart:
lauseke:=(a+b)^2; # Muuttujat a ja b ovat "vapaat"
KiQpLCZJImFHNiIiIiJJImJHRiZGJyIiI0Yn
lauseke:=expand(%);
LCgqJClJImFHNiIiIiMiIiJGKCooRidGKEYlRihJImJHRiZGKEYoKiQpRipGJ0YoRig=
a:=x^2 :b:=exp(c*z):
lauseke;
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x:=1:
lauseke;
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T\303\244m\303\244 on ensimm\303\244inen merkitt\303\244v\303\244 havainto: Kun x muuttuu, niin lauseke muuttuu, koska yll\303\244 olevassa sijoituslauseessa muuttujat a ja b olivat vapaat. T"aysevaluaatio
JSFH
Jos haluat vapauttaa vain joitakin valittuja muuttujia, niin:
Muuttujien vapauttaminen arvostaan:
Joko a:='a'; tai unassign('a'):
a:='a': # Vapautetaan a
lauseke;
b:='b': lauseke;
Filosofiesimerkki t\303\244ysevaluaation havainnollistukseen:
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Jos ihminen on sidottu, kun Sokrateelle annetaan arvo, niin Sokrateen arvoksi tulee ihmisen arvo (ei ihminen).
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
SSxrdW9sZXZhaW5lbkc2Ig==
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
SStrdW9sZW1hdG9uRzYi
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
SSxrdW9sZXZhaW5lbkc2Ig==
Tietysti Sokrates muuttuu, jos 'kuolevainen' muttuu.
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
SSNQaUclKnByb3RlY3RlZEc=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEpU29rcmF0ZXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2V4ZWN1dGFibGVHRj1GOQ==
SSNQaUclKnByb3RlY3RlZEc=
T\303\244ysevaluaatio on yksi t\303\244rke\303\244 ominaisuus symbolilaskennan polulla.
JSFH
Vakiot Pi ja I
Pi,I;
NiRJI1BpRyUqcHJvdGVjdGVkR14jIiIi
evalf(Pi);
JCIrYUVmVEohIio=
evalf(Pi,30);
JCI/RyRRVkVZUUt6KmVgRWZUSiEjSA==
2/3+4/5;evalf(%);
IyIjQSIjOg==
JCIrbm1tbTkhIio=
evalf(pi);
SSNwaUc2Ig==
Huomaa t\303\244m\303\244 : Maple osaa vain "tekstink\303\244sitell\303\244" pi :t\303\244, laskentaan tarvitaan Pi T\303\244ss\303\244 tulee usein virheit\303\244 alkavalle Maple-urheilijalle. (Voidaan kysy"a, miksi Maplesoft haluaa t"all"a kohden kiusata k"ayttajaa.) Virhetilanteista yleens\303\244: [HAM] Liite A s. 191 -198, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElJnBpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== -problematiikka s. 192
Kompleksilukuja, imag. yksikk\303\266 I Kompleksisievennys , erityisesti LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmZXZhbGNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn
(1+2*I)/(2-3*I); # Numeerinen kompleksiluku muuntuu automaattisesti muotoon x+I*y .
Jos kompleksiluku annetaan symbolisessa muodossa, Maple olettaa a :n ja b :n reaalisiksi esityksess\303\244 a+I*b .
z1:=a1+I*a2;z2:=a2+I*b2;
z1/z2;
Komento evalc on varsin tehokas, kannattaa muistaa kompleksisievennyksiss\303\244.
evalc(%);
.
abs ja argument:
z:=((1+2*I)/(3-4*I));
r:=abs(z);
phi:=argument(z);
r*exp(I*phi);
evalc(%);
Esim: 1:n juuret
restart:
w:=exp(I*2*k*Pi/n);
LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqKl4jIiIjIiIiSSJrR0YnRixJI1BpR0YlRixJIm5HRichIiI=
with(plots): # Ladataan plots-kirjastopakkaus.
setoptions(scaling=constrained,axes=framed): # Sama skaala akseleilla
n:=5: ykkosen5juuret:=seq(w,k=0..n-1);
NiciIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomXiMjIiIjIiImRiNJI1BpR0YnRiMtRiU2IyomXiMjIiIlRi9GI0YwRiMtRiU2IyomXiMjISIlRi9GI0YwRiMtRiU2IyomXiMjISIjRi9GI0YwRiM=
rinkulat:=complexplot([ykkosen5juuret],style=point,symbol=circle,symbolsize=20):
yksikkoymp:=complexplot(exp(I*Pi*t),t=0..2*Pi,color=blue):
Grafiikat saadaan samaan kuvaan funktiolla display (joka asustaa plots -pakkauksessa), kts alempana.
display(rinkulat,yksikkoymp);
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Huom: complexplot tuli vasta versiossa 7, [HAM]:ssa esitett\303\244v\303\244t pikku konversiotemput eiv\303\244t ole en\303\244\303\244 tarpeen kompleksipiirron yhteydess\303\244.
Vrt: Matlab: n=5;
K=0:n-1;
w=exp(2*K*i*pi/n);
plot(w,'or')
hold on
t=linspace(0,2*pi); plot(exp(i*t),'b'); axis equal
<Text-field bookmark="Matemaattista tekstink\303\244sittely\303\244" style="Heading 1" layout="Heading 1">7. Jono, joukko, lista, taulu(kko),vektori</Text-field>
Lue: [HAM] s. 30 ja ss. 76 - 84
<Text-field style="Heading 2" layout="Heading 2">Joukot { }, jonot, listat [ ]</Text-field>
Kolmet erilaiset sulut:
"( )","{ }", "[ ]"
- Kaarisulut ( ) : Ryhmittely, funktion argumentti. factor(x^2-1), sin(Pi).
- Hakasulut [ ] : .
a) Indeksoidaan listaa, vektoria, matriisia. v[2], A[1,2], ..
b) Listan muodostaminen
- Aaltosulut { } : Joukko: {a,b,c}
Jono , esim:
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
Lista, esim: > lista:=[a,b,c];
> op(lista); # Listan operandi on sis\303\244lt\303\266n\303\244 oleva jono Joukko , esim: > joukko:={a,b,c}; > op(joukko); # Vastaavasti joukon operandi
Huom : Listan j\303\244rjestys on k\303\244ytt\303\244j\303\244n hallinnassa, joukon ei.
Joukon hy\303\266ty on ainakin se, ett\303\244 Maple osaa poistaa toistot ja joukko-operaatiot toimivat. map - funktion soveltaminen listan (tai muun rakenteen) jokaiseen alkioon (osaan)
JSFH
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
NiUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJjR0YpRiNJInpHRilGI0Ys
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
NyUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJjR0YpRiNJInpHRilGI0Ys
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
NiUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJjR0YpRiNJInpHRilGI0Ys
IiIk
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEnam91a2tvRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtZmVuY2VkR0YkNiYtRiM2Ki1GLDYlUSJhRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRi9GMkZXLUYsNiVRImNGJ0YvRjIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGPUY5RjkvJSVvcGVuR1EifGZyRicvJSZjbG9zZUdRInxockYnLUY2Ni1RIjtGJ0Y5RjtGWkZARkJGREZGRkhGZW5GTUZfb0Zib0Y5
PCUiIiJJImNHNiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomRiRGI0kiekdGJUYj
lista:=[a,b,c];map(f,lista);
NyUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJjR0YpRiNJInpHRilGI0Ys
NyUtSSJmRzYiNiMiIiItRiQ2Iy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJImNHRiVGJ0kiekdGJUYnLUYkNiNGMQ==
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
NyUtSSJmRzYiNiMiIiItRiQ2Iy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJImNHRiVGJ0kiekdGJUYnLUYkNiNGMQ==
jono:=a,b,c;lista:=[a,b,c];op(lista);joukko:={a,b,c}; op(joukko);
NiUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJjR0YpRiNJInpHRilGI0Ys
NyUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJjR0YpRiNJInpHRilGI0Ys
NiUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJjR0YpRiNJInpHRilGI0Ys
PCUiIiJJImNHNiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomRiRGI0kiekdGJUYj
NiUiIiJJImNHNiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomRiRGI0kiekdGJUYj
map(f,jono);f(jono);
LUkiZkc2IjYlIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMqJkkiY0dGJEYmSSJ6R0YkRiZGLg==
LUkiZkc2IjYlIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMqJkkiY0dGJEYmSSJ6R0YkRiZGLg==
Huomaa, ett\303\244 jono ei ole samanlainen "paketti" kuin lista tai joukko. map-esimerkki osoittaa, ett\303\244 voidaksemme soveltaa t\303\244llaista operaatiota jonoon, se on ensin ymp\303\244r\303\266it\303\244v\303\244 listasuluilla [] ja lopuksi riisuttava sulut pois tuloksesta op -komennolla:
op(map(f,[jono]));
NiUtSSJmRzYiNiMiIiItRiQ2Iy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJImNHRiVGJ0kiekdGJUYnLUYkNiNGMQ==
<Text-field style="Heading 2" layout="Heading 2">Funktion soveltaminen listan tms. kaikkiin alkioihin, map</Text-field>
<Text-field style="Heading 3" layout="Heading 3"></Text-field>
Matlabissa kaikki ns. "skaalaarifunktiot", kuten kaikki matemaattiset 1:n muuttujan funktiot toimivat automaattisesti alkioittain
vektoreihin ja matriiseihin. Siten sin([1,2,3]) antaa [sin(1),sin(2),sin(3)].
Maplessa n\303\244in ei ole, mutta se saadaan aikaan " map" :lla:
x:=[a,b,c]; fx:=map(f,x);
NyUiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJjR0YpRiNJInpHRilGI0Ys
NyUtSSJmRzYiNiMiIiItRiQ2Iy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJImNHRiVGJ0kiekdGJUYnLUYkNiNGMQ==
restart:
M:=Matrix([seq([seq((x+y)^(i+j),j=1..3)],i=1..2)]);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtYWN0aW9uR0YkNiUtSShtZmVuY2VkR0YkNigtRiM2JS1JJ210YWJsZUdGJDY2LUkkbXRyR0YkNigtSSRtdGRHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUZTNiQtRiM2Ji1GLDYlUSJ4RidGL0YyLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5Gam8tRiw2JVEieUYnRi9GMkY5RjktSSNtbkdGJDYkUSIyRidGOS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGOS8lKXJvd2FsaWduR1EhRicvJSxjb2x1bW5hbGlnbkdGaHAvJStncm91cGFsaWduR0ZocC8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZfcS1GaG42KC1GIzYkLUZdbzYlRl9vLUZgcDYkUSIzRidGOUZjcEY5RmZwRmlwRltxRl1xRmBxLUZobjYoLUYjNiQtRl1vNiVGX28tRmBwNiRRIjRGJ0Y5RmNwRjlGZnBGaXBGW3FGXXFGYHFGZnBGaXBGW3EtRmVuNihGYnFGW3ItRmhuNigtRiM2JC1GXW82JUZfby1GYHA2JFEiNUYnRjlGY3BGOUZmcEZpcEZbcUZdcUZgcUZmcEZpcEZbcS8lJmFsaWduR1ElYXhpc0YnL0ZncFEpYmFzZWxpbmVGJy9GanBRJ2NlbnRlckYnL0ZccVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjEvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0ZcdC8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0ZndC8lJmZyYW1lR0ZndC8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRj0vJS1lcXVhbGNvbHVtbnNHRj0vJS1kaXNwbGF5c3R5bGVHRj0vJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGZHQvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnRjlGOS9JK21zZW1hbnRpY3NHRiRRJ01hdHJpeEYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRl12LyUrYWN0aW9udHlwZUdRLnJ0YWJsZWFkZHJlc3NGJy8lKXJ0YWJsZWlkR1E1MTg0NDY4ODQyMzQ2OTg0MzY2MDZGJ0ZqdUY5
map(expand,M);
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
expand~(M);
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
<Text-field style="Heading 2" layout="Heading 2">Hiukan perusteellisemmin (osittain toistaen)</Text-field>
Lue: [HAM] s. 30 ja ss. 76 - 84
restart: with(LinearAlgebra):
jono1:=a,b,c; jono2:=seq(x^i,i=-3..3);
NiVJImFHNiJJImJHRiRJImNHRiQ=
NikqJClJInhHNiIiIiQhIiIqJClGJSIiI0YoKiRGJUYoIiIiRiUqJEYqRi0qJEYkRi0=
lista1:=[jono1]; lista2:=[jono2];
NyVJImFHNiJJImJHRiRJImNHRiQ=
NykqJClJInhHNiIiIiQhIiIqJClGJSIiI0YoKiRGJUYoIiIiRiUqJEYqRi0qJEYkRi0=
joukko1:={jono1}; joukko2:={jono2}; # Alkioiden j\303\244rjestys on "satunnainen"
PCVJImFHNiJJImJHRiRJImNHRiQ=
PCkiIiJJInhHNiIqJClGJCIiJCEiIiokKUYkIiIjRikqJEYkRikqJEYrRiMqJEYnRiM=
joukko1 union {a,b,-1,1};
PCchIiIiIiJJImFHNiJJImJHRiZJImNHRiY=
[op(lista1),op(lista2)]; # Listat liitet\303\244\303\244n liitt\303\244m\303\244ll\303\244 operandijonot ja ymp\303\244r\303\266im\303\244ll\303\244 []:lla
NyxJImFHNiJJImJHRiRJImNHRiQqJClJInhHRiQiIiQhIiIqJClGKSIiI0YrKiRGKUYrIiIiRikqJEYtRjAqJEYoRjA=
logaritmit:=seq([i,log(i)],i=1..10);
Niw3JCIiIiIiITckIiIjLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0YnNyQiIiQtRik2I0YwNyQiIiUsJComRidGJEYoRiRGJDckIiImLUYpNiNGODckIiInLUYpNiNGPDckIiIoLUYpNiNGQDckIiIpLCQqJkYwRiRGKEYkRiQ3JCIiKiwkKiZGJ0YkRjFGJEYkNyQiIzUtRik2I0ZM
Maplessa vaakavektoreiden lista ei ole sama kuin matriisi (kuten Mathematicassa). Se voidaan muuntaa Matrix:lla, takaisin saadaan convert: lla .
Esitys listojen listana on hy\303\266dyllinen esimerkiksi piirrett\303\244ess\303\244, se muoto kelpaa suoraan plot :lle.
logtaulu:=Transpose(Matrix([logaritmit]));
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
plot([logaritmit]);
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
convert(logtaulu,listlist);
NyQ3LCIiIiIiIyIiJCIiJSIiJiIiJyIiKCIiKSIiKiIjNTcsIiIhLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0YlLUYxNiNGJiwkKiZGJUYkRjBGJEYkLUYxNiNGKC1GMTYjRiktRjE2I0YqLCQqJkYmRiRGMEYkRiQsJComRiVGJEY3RiRGJC1GMTYjRi0=
convert(Transpose(logtaulu),listlist);
Nyw3JCIiIiIiITckIiIjLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0YnNyQiIiQtRik2I0YwNyQiIiUsJComRidGJEYoRiRGJDckIiImLUYpNiNGODckIiInLUYpNiNGPDckIiIoLUYpNiNGQDckIiIpLCQqJkYwRiRGKEYkRiQ3JCIiKiwkKiZGJ0YkRjFGJEYkNyQiIzUtRik2I0ZM
Vektorit ja matriisit:
restart:
pysty:=<a,b,c>; vaaka:=<1 | 2 | 3>;
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
LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCI1WW9WKXBNVSlvVz0=
M:=<<1,2,3>|<b,e,x>|<u,v,w>>; # Sarakkeittain.
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
Mt:=<<1|2|3>,<b|e|x>,<u|v|w>>; # Riveitt\303\244in
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
M.Mt; # Matriisikertolasku
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
convert(vaaka,list);
NyUiIiIiIiMiIiQ=
convert(pysty,list);
convert(M.Mt,listlist);
NyVJImFHNiJJImJHRiRJImNHRiQ=
NyU3JSwoKiQpSSJiRzYiIiIjIiIiRioqJClJInVHRihGKUYqRipGKkYqLCgqJkYnRipJImVHRihGKkYqKiZGLUYqSSJ2R0YoRipGKkYpRiosKComRidGKkkieEdGKEYqRioqJkYtRipJIndHRihGKkYqIiIkRio3JUYuLCgqJClGMEYpRipGKiokKUYyRilGKkYqIiIlRiosKComRjBGKkY1RipGKiomRjJGKkY3RipGKiIiJ0YqNyVGM0ZALCgqJClGN0YpRipGKiokKUY1RilGKkYqIiIqRio=
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Teht\303\244vi\303\244</Font></Text-field>
Miten poistaisit listasta saman alkion toistot?
<Text-field bookmark="Defining Variables and Functions" style="Heading 1" layout="Heading 1">8. Symbolinen lauseke vs. funktio</Text-field>
T\303\244m\303\244 on t\303\244rke\303\244 asia ymm\303\244rt\303\244\303\244 kunnolla. [SOLmu] s. 17 Ohjelmointi
[HAM] 2.4 Matemaattiset funktiot s. 60 - 63
<Text-field style="Heading 2" layout="Heading 2">Pohdiskelua</Text-field>
Matemaattisen funktion k\303\244sittely Maple-lausekkeena
f := x^2;
M\303\244\303\244rittelemme muuttujan f, jota k\303\244yt\303\244mme matemaattisen funktion tavoin. Maplen kannalta f on muuttuja , jonka arvoksi olemme sijoittaneet lausekkeen x^2Jos haluamme laskea lausekkeen arvon eri x:n arvoilla, joudumme k\303\244ytt\303\244m\303\244\303\244n subs-komentoa.
subs(x=5,f);
eval(f,x=5); # N\303\244inkin voidaan, t\303\244t\303\244 tapaa k\303\244ytetty gsg:ss\303\244.
Jos haluat seurata lauseke/funktio-juonta, hypp\303\244\303\244 t\303\244ss\303\244 kohdassa seq- kometojen yli.
seq(subs(x=k, f),k=0..10);
a:=-4: b:=4: N:=10: h:=(b-a)/N: arvojono:=seq(subs(x=a+k*h, f),k=0..N);
evalf(arvojono);
VAROITUS: Kun matemaattista funktiota k\303\244sitell\303\244\303\244n lausekkeena, on oltava johdonmukainen. Siit\303\244 ei saa v\303\244lill\303\244 k\303\244ytt\303\244\303\244 merkint\303\244\303\244 f(x), tietenk\303\244\303\244n!
Kokeile vaikka:
f(x); f(5);
Matemaattisen funktion k\303\244sittely Maple-funktiona:
f := x -> x^2;
f(x),f(y),f(5);
Tyytyk\303\244\303\244mme ensialkuun t\303\244h\303\244n. Lue kuitenkin jossain vaiheessa loppukin.
unapply on toinen funktion m\303\244\303\244rittelytapa.
g := x^3;
g(5); # T\303\244m\303\244 on tuhoon tuomittu.
Huomaamme nyt, ett\303\244 olis ollut mukavampi k\303\244sitell\303\244 funktiona. No seh\303\244n k\303\244y:
g := unapply(g, x);
g(5);
What is the difference unapply vs -> ?
Nuolim\303\244\303\244rittely evaluoi (vasta) suoritusaikana. unapply evaluoi (jo) m\303\244\303\244rittelyaikana.
a:=x^2: f:=x->a;g:=unapply(a,x);
f(3);
g(3);
a:=Pi;
f(3);
g(3);
T\303\244ss\303\244 tilanteessa siis nuolim\303\244\303\244ritys -> tomii "omituisesti", kun taas #unapply toimi "odotetusti".
Siniset m\303\244\303\244ritysrivit yll\303\244 kertovat kyll\303\244 asian hyvin, niit\303\244 kannattaa seurailla.
Jatketaan t\303\244st\303\244 (harj0): Varoitus: Hengenvaarallista: \303\204l\303\244 koskaan m\303\244\303\244rittele funktiota tyyliin F(x):=lauseke;
T\303\244ll\303\266in tulee m\303\244\303\244ritellyksi funktion F arvo yhdess\303\244 pisteess\303\244 x, muilla symboleilla funktio f on m\303\244\303\244rittelem\303\244t\303\266n. Poikkeus : Funktion m\303\244\303\244rittely\303\244 t\303\244ydent\303\244vien poikkeusarvojen m\303\244\303\244rittelyss\303\244 t\303\244m\303\244 on hy\303\266dyllist\303\244, esimerkki otetaan kohta.) Palataan nyt viel\303\244 tuohon hengenvaaraan:
restart:
F(x):=x^3; # kokeillaan kielletty\303\244 leikki\303\244.
KiQpSSJ4RzYiIiIkIiIi
Tuo on syntaksiltaan oikein, mutta hyvin h\303\244m\303\244\303\244v\303\244\303\244. Mit\303\244 on F(x)
F(x);
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
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQpRiQiIiMiIiJGJUYlRiU=
N\303\244ytt\303\244\303\244, kuin toimisi oikein. Vaan eipas toimi muilla kuin x:ll\303\244.
F(x),F(3),F(a),F(sin(z));
Toisin sanoen funktio on m\303\244\303\244ritelty vain, kun argumenttina on symboli x
Esimerkki: Funktion m\303\244\303\244rittelyn t\303\244ydent\303\244minen poikkeusarvoilla .
Haluamme m\303\244\303\244ritell\303\244 funktion, jota joskus kutsutaan nimell\303\244 sinc, olkoon t\303\244ss\303\244 vain F
F:=x->sin(x)/x;
F(0);
Yleinen s\303\244\303\244nt\303\266 ei m\303\244\303\244rittele F-funktiota 0:ssa, koska siin\303\244 tulee 0:lla jako. Annamme siksi funktiolle 0:ssa sopivan arvon suoralla asetuksella:
F(0):=1;
F(a),F(1/2),F(0),F(sin(x));
seq(F(x),x=-2..2);evalf(%);
Remember: # evalf means: eval uate f loating point.
Lue lis\303\244\303\244: [HAM] ss. 62-63
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Harjoitusteht\303\244v\303\244</Font></Text-field>
M\303\244\303\244rittele polynomifunktio 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 . M\303\244\303\244rit\303\244 nollakohdat ja paikalliset minimit sek\303\244 maksimit. Piirr\303\244 funktio ja sen derivaatta. Tarkista laskemalla funktion arvot, ett\303\244 nollakohdat ovat nollakohtia. K\303\244sittele polynomia ensin lausekkeena ja sitten funktiona. Mitk\303\244 ovat kunkin tavan hyv\303\244t/huonot puolet. Derivaattalauseke -- diff
Derivaattafunktio -- D (K\303\244yttele helppi\303\244 ) ?plot, ?solve,?fsolve ?diff, ?D T\303\244m\303\244ntyyppisten teht\303\244vien, niin yksinkertaisia matemaattisesti kuin ovatkin, sujuva hallinta tekee Maple-ty\303\266skentelyst\303\244 nautittavaa puuhaa ja auttaa pitk\303\244lle.
<Text-field bookmark="Algebra" style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">11. Sievennyst\303\244 ja yht\303\244l\303\266n ratkaisua</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">Pohdiskelua
<Font style="Text"> simplify</Font><Font encoding="UTF-8"> - yleissievent\303\244j\303\244, ensimm\303\244iseksi tarjoiltava.</Font></Text-field>
expand - kertoo auki
collect - kokoaa
? collect ja kokeile joitakin esimerkkej\303\244. Varsin hyv\303\244 komento monessa paikassa.
factor - Jakaa tekij\303\266ihin.
convert - ... Monenmoisiin konversioihin sievennyksess\303\244 esim convert(lauseke,parfrac,muuttuja); Huomaa, ett\303\244 lauseke ei muutu, uusi tulos palautetaan. Jos halutaan p\303\244ivitt\303\244\303\244 lauseke, on komennetava lauseke:=convert(lauseke,parfrac,muuttuja);
solve - ratkaisee yht\303\244\303\266n tai systeemin fsolve - ratkaisee numeerisesti
(x^3+1)/(x^2-x+1);simplify(%);
expand((x^2-4)*(x+1)*(x-2)*(x^2+x+1));
factor(%);
Yht\303\244l\303\266ist\303\244 [SOL] s. 11-12
[HAM] s. Luku 6 s. 133 - 142
yhtalot:={2*x-5*y=12, 12*x+4*y=17}; # Kyseess\303\244 on joukko
ratk:= solve(yhtalot, {x,y});
Maple ei sijoita ratkaisuja muuttujille arvoiksi, vaan palauttaa sijoituss\303\244\303\244nn\303\266t. Ne voidaan antaa suoraan #subs -komennolle. Mieti seuraavan komentojonon logiikkaa!
subs(ratk,x),subs(ratk,y);
X:=subs(ratk,x); Y:=subs(ratk,y);
Tarkistaminen k\303\244y vaivattomasti:
subs(ratk,yhtalot);
Voidaan my\303\266s k\303\244ytt\303\244\303\244 #lhs (left hand side) ja rhs (right hand side)-tapaa, mutta se on hiukan v\303\244hemm\303\244n elegantti ja my\303\266s altis Maplen harrastamalle j\303\244rjestyksen vaihtumiselle ratkaisujoukossa. Niinp\303\244 t\303\244t\303\244 tyyli\303\244 ei voida ajaa automaattisesti, vaan tuloksiin on puututtava interaktiivisesti, mik\303\244 voi olla kiusallista, jos on "vakavahenkisest\303\244" dokumentista kyse. Joka tapauksessa t\303\244m\303\244kin tyyli kannattaa omaksua.
rhs(ratk[1]);lhs(ratk[1]);
lhs(ratk[2])=rhs(ratk[2]);
Ratkaisun sijoittaminen muuttujan arvoksi
T\303\244ss\303\244 nyt on toisin sanoin selitetty samaa, pyyhi omasta dokustasi pois tai j\303\244rjest\303\244 paremmin. Tietysti n\303\244in:
X:=rhs(ratk[1]);Y:=rhs(ratk[2]);
Turvallisempi ja elegantimpi tapa (jota edell\303\244 mainostettiin): Suorita ratk-yht\303\244l\303\266iden ilmaisemat korvaamiset lausekkeessa (pelkk\303\244) x ja
suorita ratk-yht\303\244l\303\266iden ilmaisemat korvaamiset lausekkeessa (pelkk\303\244) y .
Edellisess\303\244 y-yht\303\244l\303\266 j\303\244\303\244 vaille k\303\244ytt\303\266\303\244 (kun pelkk\303\244 x ei sis\303\244ll\303\244 y:t\303\244) ja j\303\244lkimm\303\244isella vastaavasta syyst\303\244 x-yht\303\244l\303\266. ratk -joukon j\303\244rjestys ei n\303\244yttele mit\303\244\303\244n osaa.
X:=subs(ratk,x);Y:=subs(ratk,y);
Tehokkain, mutta turvattomin tapa, joskus kyll\303\244kin tosi k\303\244tev\303\244: assign(ratk);
assign(ratk);
x;y;yhtalot;
solve(yhtalot,{x,y}); # assign-komennon j\303\244lkeen x ja y eiv\303\244t ole vapaita muuttujia, josta syyst\303\244 solve menee virheeseen.
assign(ratk); toimii ik\303\244\303\244nkuin ratk-yht\303\244l\303\266iss\303\244 yht\303\244l\303\266merkki (=) vaihdettaisiin sijoitusmerkkin (:=) Yht\303\244l\303\266iden toteutuminen saadaan hyvin k\303\244tev\303\244sti tarkistetuksi kirjoittamalla vain yhtalot; Sensijaan yht\303\244l\303\266iden ratkaisua ei voida toistaa, ennekuin muuttujat x ja y on vapautettu arvoistaan.
x:='x':y:='y':solve(yhtalot,{x,y});
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field bookmark="Piirtoa" style="Heading 1" layout="Heading 1">12. Piirtoa</Text-field>
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEiP0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEuUGxvdHRpbmdHdWlkZUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2ZvcmVncm91bmRHUSpbMjU1LDAsMF1GJy8lK2V4ZWN1dGFibGVHRkwvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRi8= [SOL] ss. 10 - 11
[HAM] Luku 3 ss. 89 - 100
<Text-field style="Heading 2" layout="Heading 2">Parametrimuotoinen piirto, matrixplot(logtaulu);
</Text-field>
Syntaksi poikkeaa aika v\303\244h\303\244n, t\303\244m\303\244 t\303\244ytyy vain oppia (tai katsoa aina uudelleen helpist\303\244). Aina ei ole hauskaa joutua valitsemaan hirell\303\244, samaskaalaiisuus (ja monia muita) voidaan antaa plot-komennon tarkentimena.
plot([cos(t),sin(t),t=0..2*Pi],scaling=constrained);
Pisteet voidaan yht\303\244 hyvin ajatella kompleksitason pistein\303\244. complexplot osaa k\303\244sitell\303\244 suoraan reaalimuuttujan kompleksiarvoista funktiota. T\303\244ss\303\244 on sekin mukava piirre, ett\303\244 syntaksi on luonnollisempi ja siten helpompi muistaa kuin tuossa parametrimuotoisessa R^2-piirrossa.
with(plots):
complexplot(cos(t)+I*sin(t),t=0..2*Pi,scaling=constrained,color=blue);
Edellinen voidaan ilmaista lyhyemmin eksponenttifunktion (Eulerin kaavan) avulla. Piirret\303\244\303\244n vaihteeksi pelk\303\244t pisteet.
complexplot(exp(I*t),t=0..2*Pi,scaling=constrained,color=cyan,style=point);
<Text-field style="Heading 2" layout="Heading 2">Taulukoidun datan piirto ("listaplotti")</Text-field>
Sometimes we will be interested in functions defined in terms of a discrete
table of values rather than a formula. For example, consider the following table of
temperatures recorded at various times on a spring day in Raleigh.
Time: | 6:00 a.m. | 10:00 a.m. | 12:00 p.m. | 4:00 p.m. | 5:00 p.m.
----------------------------------------------
Temp: | 45 deg. | 57 deg. | 65 deg. | 67 deg. | 66 deg.
Table of Temperatures on a Spring Day in Degrees Fahrenheit
To plot this data you must first define the points as a list. Since the time is recorded in a cyclic fashion, we will plot the times in the so-called "military style", i.e. 6:00 a.m. is 0600 and 6:00 p.m. is 1800.
ListOfPoints := [[600,45], [1000,57], [1200,65], [1600,67], [1700,66]];
Nyc3JCIkKyciI1g3JCIlKzUiI2Q3JCIlKzciI2w3JCIlKzsiI243JCIlKzwiI20=
The plot command allows you to plot a list of points and to connect them with straight lines (unless you choose another option). This is illustrated in the next command. Notice also the option which produces a title on the plot.
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
Zio2I0klcGFyaUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlJkYkNiMiIiJGJUYlRiU=
Zio2I0klcGFyaUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlJkYkNiMiIiNGJUYlRiU=
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
NyciJCsnIiUrNSIlKzciJSs7IiUrPA==
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
NyciI1giI2QiI2wiI24iI20=
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
plot(ListOfPoints, title="Temperatures at Various Times");
While the data can be plotted in this way, because the data is given for a discrete set of points you may prefer the style=POINT option as illustrated in the next command. Note also the effect of the symbol=CIRCLE option.
plot(ListOfPoints, style=POINT, symbol=CIRCLE,symbolsize=15, title=`Temperatures at Various Times`);
<Text-field style="Heading 3" layout="Heading 3">Exercise</Text-field>
1. Plot the graph of 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 over the interval [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUYsNiVRJSZwaTtGJy8lJ2l0YWxpY0dGOkY1RjUtRjI2LVEiLEYnRjVGOC9GPFEldHJ1ZUYnRj1GP0ZBRkNGRS9GSFEmMC4wZW1GJy9GS1EsMC4zMzMzMzMzZW1GJ0ZMRjU= ] .
2. Make a plot of the points in the following table:
X: | -1 | 0 | 1 | 2 | 3
-------------------------------------------
Y: | 2.72 | 1 | 0.368 | 0.135 | 0.0498
Make two plots, one with the points connected and the other with only the data points.
Miten saadaan k\303\244tev\303\244sti koordinaattiparien lista [[x1,y1],[x2,t2],...] ? Olkoon annettu numeeriset listat (Matlabissa sanoisimme vektorit) X ja Y. Matlabin plot toimii tyyliin plot(x,y);
Maplessa t\303\244ytyy muodostaa parien lista vaikkapa t\303\244h\303\244n tapaan:
a:=-Pi:b:=Pi:N:=20:h:=(b-a)/N:x:=seq(evalf(a+i*h),i=0..N);
Olkoot y-arvot vaikkapa sin-funktion arvoja x-pisteiss\303\244. Parijono saataisiin nyt n\303\244in:
xy:=[seq([x[i],sin(x[i])],i=1..N+1)];
plot(xy);plot(xy,style=point,symbol=circle);
Huom: plots -pakkauksessa on listplot ja pointplot . Maplelle luonteenomaista on, ett\303\244 siin\303\244 on joukko redundantteja funktioita. Mielest\303\244ni on parempi oppia k\303\244ytt\303\244m\303\244\303\244n harvempaa ydinfunktiojoukkoa kuin omaksua monenlaisia synonyymej\303\244. Niiss\303\244 voi toki olla joitakin uusia mahdollisuuksia, mutta esim. n\303\244iss\303\244 ei v\303\244ltt\303\244m\303\244tt\303\244 ole (kuka noita kaikkia tuhansia ehtii penkoa).
Taulukointi
T\303\244ss\303\244 teimme itse asiassa taulukon, joka on havainnollisempi usein matriisina.
txy:matrix(xy):linalg[transpose](xy):
Tilan s\303\244\303\244st\303\244miseksi otetaan v\303\244hemm\303\244n dataa esimerkkiimme. J\303\244tet\303\244\303\244np\303\244 my\303\266s evalf pois.
a:=-Pi:b:=Pi:N:=8:h:=(b-a)/N:x:=seq(a+i*h,i=0..N);
xy:=[seq([x[i],sin(x[i])],i=1..N+1)];
matrix(xy);
linalg[transpose](xy);
array(xy); # Taas synonyymi, mutta tiettyj\303\244 erojakin yleisyydessa / matriisilaskukyvyiss\303\244.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Grafiikoiden yhdist\303\244minen, plots[display] ja textplot</Font></Text-field>
T\303\244ss\303\244 tarvitaan lis\303\244grafiikkapakkaus plots
restart:with(plots):
Selvit\303\244 itsellesi, mit\303\244 seuraavassa tehd\303\244\303\244n. Mieti tarkkaan, miksi jossain pit\303\244\303\244 antaa plot:lle argumentiksi f(x) ja miksi taas jossain esim. tang. Voit sitten huvitella vaihtelemalla funktion m\303\244\303\244ritelm\303\244\303\244 ja / tai pistett\303\244 x0. Kirjoitetaan pieni malliskripti, jonkalaisia voit tehd\303\244 moninaisissa yhteyksiss\303\244.
x0:=1: f:=x->x^3: # Vaihtuva sy\303\266te
df:=diff(f(x),x): kk:=subs(x=1,df);
IiIk
x0:=1: tang:=f(x0)+kk*(x-x0);
LCYiIiMhIiIqJiIiJCIiIkkieEc2IkYnRic=
fkuva:=plot(f(x),x=0..2,color=red): tangkuva:=plot(tang,x=0..2,color=blue): p0kuva:=plot([[x0,f(x0)]],style=point,symbol=circle,symbolsize=15,color=black):
display([fkuva,tangkuva,p0kuva]);
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Kokeile (mutta vain kerran el\303\244m\303\244ss\303\244si, silloinkin syv\303\244sti katuen, ett\303\244 olit yllytyshullu), mit\303\244 tapahtuu, kun vaihdat (:) -> (;) vaikkapa fkuva:=; . .. yll\303\244. PLOT-tietorakenne paljastuu. - No, mene edit-valikkoon ja -> remove output, kyll\303\244 se siit\303\244.
Sometimes it will be desirable to print text on a plot. The plots procedure textplot will allow this. The following commands will serve as examples of the textplo t and display routines. Make sure you notice where colons and semicolons are used in these commands.
In the next commands it is important to use colons to punctuate the first two statements. Otherwise Maple will ouput an entire page of text describing the plot structure rather than the graph. The textplot in the second command causes the text x=0.6356,y=0.5000 to be printed at the point (.6356,.5000). The statement align=RIGHT aligns this text to the right of the point.
f := x -> exp(-x)*sin(3*x);
plot1 := plot(f, 0..3):# tai plot1:=plot(f(x),x=0..3):
plot2 := plots[textplot]([0.6356, 0.5000, "x=.6356, y=.5000"], align=RIGHT):
plots[display]({plot1, plot2});
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Huom:
plots-pakkauksen (kuten muidenkin pakkausten) funktioita voidaan k\303\244ytt\303\244\303\244 my\303\266s lataamatta koko pakkausta, t\303\244ll\303\266in pakkauksen nime\303\244 ik\303\244\303\244nkuin indeksoidaan ao. funktion nimell\303\244 tyyliin plots[display]
<Text-field style="Heading 2" layout="Heading 2">3d-grafiikkaa, animaatiot</Text-field>
Otetaan l\303\244mm\303\266njohtumisesimerkki, jossa valaistaan 3d-pinnan piirtoa
Animaatiota
3d-kuvan projektiok\303\244yr\303\244parven piirtoa Tarkastellaan sivuiltaan l\303\244mp\303\266eristetty\303\244 sauvaa, jonka p\303\244\303\244t upotetaan hetkell\303\244 t=0 j\303\244\303\244vesis\303\244ili\303\266ihin (0 astetta) ja jonka
alkul\303\244mp\303\266tilajakauma on
f:=x->100*sin(Pi*x/80);
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqJiIkKyIiIiItSSRzaW5HRiU2IywkKigjRiwiIyEpRixJI1BpRyUqcHJvdGVjdGVkR0YsRiRGLEYsRixGLEYlRiVGJQ==
Huomaa funktiom\303\244\303\244ritys .
Olkoon sauvan pituus L=80.
T\303\244ss\303\244 tapauksessa l\303\244mp\303\266yht\303\244l\303\266n ratkaisuna olevasta Fourier-sarjasta tulee vain yksi termi. Ratkaisu on (sopivalla l\303\244mm\303\266njohtumiskertoimella)
u:=(x,t)->100*sin((Pi*x)/80)*exp(-0.001785*t);
Zio2JEkieEc2IkkidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKCIkKyIiIiItSSRzaW5HRiU2IywkKigjRi0iIyEpRi1JI1BpRyUqcHJvdGVjdGVkR0YtRiRGLUYtRi0tSSRleHBHRiU2IywkKiYkIiUmeSIhIidGLUYmRi0hIiJGLUYtRiVGJUYl
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">L\303\244mp\303\266tilaprofiilit ja korkeusk\303\244yr\303\244t</Font></Text-field>
plot({seq(u
(x,t),t=[0,10,50,100,200,300,400])},x=0..80);
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contourplot(u(x,t),x=0..80,t=0..400);
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<Text-field style="Heading 1" layout="Heading 1">13. Diffyht"al"ot</Text-field>
dsolve