{VERSION 4 0 "IBM INTEL LINUX22" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1 " -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 37 "Numeerista integrointia, \+ L/numint.mws" }}{PARA 19 "" 0 "" {TEXT -1 30 "V2/2002, viimeinen viikk o HA" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Alustukset" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "#read(\"c:\\\\opetus\\\\maple\\\\v202.mpl\"):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "#read(\"/p/edu/mat-1.414/map le/v202.mpl\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "read(\"/ home/apiola/opetus/peruskurssi/v2-3/maple/v202.mpl\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#op(L);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "op(L1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(L1(i,[x1,x2,x3],x),i=1..3);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "Kertausta: Lagrangen interpolaatio" }}{PARA 0 "" 0 "" {TEXT -1 124 "Lagrangen interpolaatiomenetelm\344ss\344 muodostetaan a nnettuun xdataan [x0,...,xn] liittyv\344t polynomit L[0],...,L[n], sit en ett\344" }}{PARA 0 "" 0 "" {TEXT -1 21 "L[i](x[j])=delta[i,j]" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "L:='L': L[i](x[j])=delta[i,j ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xd:=[x0,x1,x2,x3];" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "read(\"/home/apiola/opetus /peruskurssi/v2-3/maple/v202.mpl\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "L(0,xd,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "L(1,xd,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "jne. Yleisesti : KRE s. 849 Lagrange interpolation." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Lagrangen (kertoja)polynomit on valmiina koodattu tiedostossamm e " }{TEXT 261 8 "v202.mpl" }{TEXT -1 49 ", joka luettiin yll\344. Sik si nuo kaavatkin tulivat" }}{PARA 0 "" 0 "" {TEXT -1 51 "yll\344 ihan \+ oikein, kun vaan tarjottiin L-kirjainta! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Lagrangen polynomien avulla voidaan kirjoittaa suoraan yl einen interpolaatioteht\344v\344n ratkaisu:" }}{PARA 0 "" 0 "" {TEXT 256 10 "Teht\344v\344: " }}{PARA 0 "" 0 "" {TEXT -1 119 " Annettu x data=[x0,...,.xn] ja ydata=[y0,..,yn]. M\344\344r\344tt\344v\344 korke intaan astetta n oleva polynomi n\344iden datapisteiden" }}{PARA 0 "" 0 "" {TEXT -1 11 " kautta. " }}{PARA 0 "" 0 "" {TEXT -1 230 " Tied \344mme, ett\344 polynomin kertoimet saataisiin ratkaisemalla lineaari nen yht\344l\366systeemi, jonka kerroinmatiirina on Vandermonden ma triiisi. Kuten PNS:n yhteydest\344 muistamme, Vandermonde on ei-singul aarinen ja t\344ss\344 tapauksessa" }}{PARA 0 "" 0 "" {TEXT -1 42 " \+ neli\366matriisi, joten 1-k\344s ratkaisu on." }}{PARA 0 "" 0 "" {TEXT -1 117 " T\344t\344 tietoa emme tarvitse Lagrangen menetelm \344ss\344, emme my\366sk\344\344n joudu ratkaisemaan yht\344l\366syst eemi\344, vaan nerokkuus" }}{PARA 0 "" 0 "" {TEXT -1 85 " piilee Lag rangen polynomien ortogonaalisuustyyppisess\344 ominaisuudessa (kts. y ll\344)." }}{PARA 0 "" 0 "" {TEXT 257 9 "Ratkaisu:" }{TEXT -1 1 " " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "L:='L':" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "'p(x)'=sum(y[i]*L[i](x),i=0..n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "J\344\344nn\366stermille voidaan johtaa l auseke:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "R=product((x-x[i ]),i=0..n)*D[n+1](f)(xi)/n!;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 " T\344ll\366in k\344ytett\344viss\344 on funktio (n+1) kertaa derivoitu va funktio f, jonka arvot x-datapisteiss\344 ovat y-data-arvoja:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "y[i]=f(x[i]), i=0..n;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 44 "Kertausta : Gaussin integrointi 1-dim tapaus" }}{PARA 0 "" 0 "" {TEXT -1 95 "Erilaisia integrointis\344\344nt\366j\344 \"qua drature rules\" saadaan integroimalla interpolaatiopolynomeja." }} {PARA 0 "" 0 "" {TEXT -1 122 "Ensimm\344iseksi mieleen juolahtava ajat us on jakaa v\344li tasav\344lisesti ja muodostaa interpolaatiopolynom i, joka integroidaan." }}{PARA 0 "" 0 "" {TEXT -1 92 "N\344in saadaan \+ ns. Newton-Cotes-kaavat. Kaksi alinta astetta olevaa Newton-Cotes-kaa vaa ovat " }{TEXT 262 15 "trapetsis\344\344nt\366 " }{TEXT -1 21 "(puo lisuunnikas-) ja" }}{PARA 0 "" 0 "" {TEXT 263 10 "Simpsonin " }{TEXT -1 117 "s\344\344nt\366. Niiss\344 approksimidaan pinta-alaa puolisuu nnikkaalla ja vastaavasti 2. asteen polynomin reunustamalla alalla." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 19 "Yhdiste tyt s\344\344nn\366t:" }{TEXT -1 20 " (\"Composite rules\")" }}{PARA 0 "" 0 "" {TEXT -1 160 "Alhaista kertalukua olevilla kaavoilla laskett aessa jaetaan koko integroimisalue osiin, joilla kullakin sovelletaan \+ ao. s\344\344nt\366\344. Koko integraali on osien summa." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 47 "Optimaalinen solm ujen valinta, Gaussin s\344\344nn\366t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Yleisemmin voidaan valita mik\344 t ahansa (ei v\344ltt\344m\344tt\344 tasav\344linen) pisteist\366 x1,... ,xn integrointiv\344lilt\344 [a,b]." }}{PARA 0 "" 0 "" {TEXT -1 123 "H uom! Vaihdoimme indeksoinnin alkamaan 1:st\344, t\344m\344 korostaa si t\344, ett\344 johtamisen alaisena olevat kaavat ovat yleens\344 ns. \+ \"" }{TEXT 266 16 "avoimia kaavoja\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 8 "joissa s" }{TEXT 267 41 "olmupisteet x[i] ovat v\344lin \+ sis\344pisteit\344." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "L:='L':n:='n':p:=sum(y[i]*L[i](x),i=1..n); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "intsaanto:=sum(f(x[i])* Int(L[i](x),x=a..b),i=1..n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Me rk: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "w[i]=Int(L[i](x),x= a..b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "intsaanto:=sum(w[ i]*f(x[i]),i=1..n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Yleisesti \+ saadaan painotettu summa funktion arvoista. " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 258 75 "Gaussin idea: Annetaan solmujen x[i] olla m uuttujina optimointiteht\344v\344ss\344:" }}{PARA 0 "" 0 "" {TEXT -1 129 " Valitse solmut x[i] ja painot w[i] siten, ett\344 integro intis\344\344nt\366 on tarkka mahdollisimman korkea-asteisilla polynom eilla." }}{PARA 0 "" 0 "" {TEXT -1 114 " Korkein asteluku, joka \+ voidaan toivoa saavutettavaksi, on 2n-1 (2n tuntematonta, 2n m\344\344 r\344tt\344v\344\344 kerrointa)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 259 15 "Ratkaisutapa 1:" } {TEXT -1 83 " Muodostetaan (ep\344lineaarinen) yht\344l\366systeemi in tegroimalla monomeja 1,x,x^2, ... ." }}{PARA 0 "" 0 "" {TEXT -1 113 " \+ T\344ll\344 saadaan alhaista kertalukua olevia (ainakin n=2). Korkeam pia voidaan yritt\344\344 vaikka Newtonin menetelm\344ll\344" }}{PARA 0 "" 0 "" {TEXT -1 40 " ratkaista, mutta se ei johda pitk\344lle." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 260 16 "Ratkaisutapa 2: " }{TEXT -1 99 " Normeerataan integrointi v \344lille [-1,1]. Osoittautuu, ett\344 solmut x1,...,xn saadaan ns. Le gendren " }}{PARA 0 "" 0 "" {TEXT -1 89 " polynomien nollakohtina. Kun solmut tunnetaan, saadaan painot yksinkertaisesti kaavasta:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "w [i]=Int(L[i](x),x=-1..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Int egrointi yleisen v\344lin [a,b] yli hoituu yksinkertaisella lineaarise lla muuttujanvaihdolla. (Nokkelia kun ollaan," }}{PARA 0 "" 0 "" {TEXT -1 93 "niin muodostetaan 1. asteen interpolaatiopolynomi, joka a :ssa saa arvon -1 ja b:ss\344 arvon 1.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "read(\"/home/apiola/opetus/peruskurssi/v2-3/maple/v20 2.mpl\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "(-1)*L(0,[a,b] ,x)+1*L(1,[a,b],x);simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "tyht:=t=% %;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "xx:=solve(%,x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dx:=diff(xx,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Saadaan integrointikaava:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(f(x),x=a..b)=Int(f(xx)*dx,x=-1. .1);" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Maple-toteutus" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(orthopoly);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "#read(\"c:\\\\opetus\\\\maple\\\\v2 02.mpl\"):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Varusta read-k\344s ky kommentilla ja mene alkuun suorittamaan asianmukainen read." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "read(\"/home/apiola/opetus/ peruskurssi/v2-3/maple/v202.mpl\"): # Nimikonflikti on l\344hell\344, \+ emme kuitenkaan tarvitse polynomia L - Laguerren polynomi, vaan Legenr en polynomia, jonka Maplen orthopoly nime\344\344 P:ksi." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "( L-matemaatikoita riitt\344\344: Lagrang e, Legendre, Laguerre, Laplace, ... )" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 21 "Gaussin integroinnin " }{TEXT -1 92 "yhteydess\344 on ta pana indeksoida Lagrangen polynomit (siis solmut) 1..n. Siis datapiste it\344 on" }}{PARA 0 "" 0 "" {TEXT -1 91 "n kpl, jolloin interpoloidaa n n-1 asteisella polynomilla. Tavoitteena siis kaava, joka on " } {TEXT 269 22 "tarkka asteeseen 2n-1 " }{TEXT -1 7 "saakka." }}{PARA 0 "" 0 "" {TEXT -1 84 "Jatkon indeksipeli\344 helpottaa, kun kirjoitamme Lagrangen kertojapolynomista version " }{TEXT 270 3 "L1," }{TEXT -1 24 " jossa indeksointi menee" }}{PARA 0 "" 0 "" {TEXT -1 44 "1..n. (Ko odikin on hiukan yksinkertaisempi.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "op(L1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "P(10,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=15:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "fsolve(P(10,x)=0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "gausssolmut:=seq([fsolve(P(k,x)=0,x)],k=1..5);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "%[2,1];" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 160 "N\344in voitaisiin muodostaa riitt\344v\344n iso \+ taulukko, josta haettaisiin solmut. J\344rkev\344\344 olisi kirjoittaa taulukko tiedostoon, josta solmut luettaisin, ainakin, jos" }}{PARA 0 "" 0 "" {TEXT -1 40 "haluttaisiin varautua isoon taulukkoon. " }} {PARA 0 "" 0 "" {TEXT -1 113 "Ohjelmoinnin kannalta on k\344tev\344mp \344\344 tehd\344 pari pikku funktiota, joista kasataan varsinainen in tegrointifunktio. " }}{PARA 0 "" 0 "" {TEXT -1 151 "Joka tapauksessa \+ kannattaa tehd\344 erikseen solmujen ja painojen laskenta, jotta niit \344 ei tarvitse laskea kuin kerran tietyn kertaluvun menetelm\344\344 kohti." }}{PARA 0 "" 0 "" {TEXT -1 43 "Ohjelmointi on \344\344rimm \344isen yksinkerttaista." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "gaussolmut:=k->fsolve(P(k,x)=0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "L1(3,[a,b,c],x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "gausspainot:=n->seq(int(L1(i,[gaussolmut(n)],x),x=-1. .1),i=1..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gausspainot (3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "gausstaulukko:=n->[ [gaussolmut(n)],[gausspainot(n)]];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gausstaulukko(10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sw:=gausstaulukko(4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "gaussint:=proc(n,f)\n# Suorita ensin gtaulu:=gaussta ulukko(n);\nglobal gtaulu;\nlocal s,w,i;\ns:=gtaulu[1]; w:=gtaulu[2]; \nsum(w[i]*f(s[i]),i=1..n);\nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gtaulu:=gausstaulukko( 3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "gaussint(3,x->x^4),i nt(x^4,x=-1..1); evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "n:=3: gtaulu:=gausstaulukko(n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "seq([gaussint(3,x->x^k),evalf(int(x^k,x=-1..1))],k=1. .2*n-1); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Astetta 5=2n-1 olev aan saakka kaava integroi polynomit tarkkaan (py\366ristyst\344 vaille ). Arvolla 2n tulee jo selke\344 ero." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "gaussint(3,x->x^(2*n)),evalf(int(x^(2*n),x=-1..1)); \+ " }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Integrointi yli neli\366n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "gaussintnelio:=proc(m,n,f)\n# Suorita ensin gtaulux:=gausstau lukko(m);\n# gtauluy:=gausstaulukko(n);\nglobal gtaulux,gtauluy;\nloca l sx,wx,sy,wy,i,j;\nsx:=gtaulux[1]; wx:=gtaulux[2];\nsy:=gtauluy[1]; w y:=gtauluy[2];\nsum(wx[i]*sum(wy[j]*f(sx[i],sy[j]),j=1..n),i=1..m);\ne nd:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "gtaulux:=gausstaulukko(3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "gtauluy:=gtaulux:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "gaussintnelio(3,3,(x,y)->x^4 *y^4),int(int(x^4*y^4,x=-1..1),y=-1..1); evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sx:=gtaulux[1]:sy:=gtauluy[1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plottools): with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=3:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "nelio:=polygon([[1,1],[-1,1],[-1,-1],[1,-1]]): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ss:=seq(seq([sx[i],sy[j ]],j=1..n),i=1..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "disp lay(nelio,plot([ss],style=point),axes=frame);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }}}}{MARK "7" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }