Pienimm"an neli"osumman sovitus V2 19.3. 2002, V3 lokakuu -02 2013kevat/maple/

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== with(LinearAlgebra):alias(Tr=Transpose): LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRJnBsb3RzRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjpGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQ= xd:=[-1.3,-0.1,0.2,1.3]; yd:=[0.103,1.099,0.808,1.897]; n:=nops(xd); plot([seq([xd[i],yd[i]],i=1..n)],style=point,symbol=circle,symbolsize=15); datakuva:=%: C:=<<1,1,1,1>|<op(xd)>>; M:=Tr(C).C; B:=Tr(C).Vector(yd); ab:=LinearSolve(M,B); suora:=ab[1]+ab[2]*x; display(plot(suora,x=-2..2,color=blue),datakuva);

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

Muistellaan aluksi interpolaatioteht. Esim:

xd:=[$1..10];yd:=[1.3,3.5,4.2,5.0,7.0,8.8,10.1,12.5,13.0,15.6]; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRI3hkRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSN5ZEYnRi9GMkZBLUY+Ni1RIjtGJ0ZBRkNGRkZIRkpGTEZORlBGUi9GVlEsMC4yNzc3Nzc4ZW1GJw== 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 plot([seq([xd[i],yd[i]],i=1..10)],style=point,symbol=cross,symbolsize=18); # Vaihtoehtoinen tapa: pisteiden lista. p:=interp(xd,yd,x); # Ei tarvitse ladata. display(datakuva,plot(p,x=0..10,color=blue)); Interpolaatio pakottaa polynomin usein voimakkaaseen heilahteluun, jolloin "mets"a katoaa n"akyvist"a puilta". Luovutaan vaatimuksesta, ett"a polynomin tulisi kulkea datapisteiden kautta. Valitaan alempiasteinen polynomi, joka sopivassa mieless"a approksimoi parhaiten annettua dataa. Kriteeriksi valitaan usein funktiomallin (t"ass polynomi) ja datapisteiss"a laskettujen erotuksten neli"osumman minimointi. T"at"a kautta menetelm"a liittyy funktioiden "a"ariarvoteemaan. Polynomi-PNS-teht"av"ass"a on annettu datapisteit"a ( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImlGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlLUYvNiVRInlGJ0YyRjVGOEY/Rj0=) (paljon) ja teht"av"an"a on sovittaa esim. polynomi 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 annettuun dataan niin, ett"a 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 minimoituu. Yleens"a datapisteiden lkm =n on paljon suurempi kuin polynomin asteluku m (t"ass"a 3). Jos A on matriisi , jonka alkiot ovat 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 , i=1..n, j=0..3, saadaan b-kertoimien m"a"ar"a"amiseksi ylim"a"ar"aytyv"a yht"al"osysteemi A b = y, miss"a siis A on n x 3 - matriisi. PNS-ratkaisuun p"a"ast"a"an kertomalla t"am"a systeemi A:n transpoosilla, jolloin ratkaistavaksi tulee systeemi (1) (A^T)A b = A^T y (miss"a A^T tarkoittaa A:n transpoosia) (T"am"a johdetaan joko osittaisderivoimalla tai ortogonaaliprojektioon perustuvalla geometrisluonteisella argumentilla.) Jos tuntemattomia kertoimia on yht"a monta kuin datapisteit"a (polynomin asteluku = n-1), niin systeemill"a A b = y on yksik"asitteinen ratkaisu, mik"ali x-pisteet ovat erillisi"a. T"all"oin on kyse interpolaatiopolynomista. Yll"a oleva ratkaisu voidaan Maplella tehd"a rakentamalla matriisi A ja ratkaisemalla systeemi (1) . Maplen LinearAlgebra-kirjastossa on my"os LeastSquares- funktio, jolle annetaan pelkk"a A - matriisi ja datapisteet, joka siis ratkaisee ylim"a"ar"aytyv"an systeemin A b = y PNS-mieless"a. Jotta valinta ei olisi liian helppoa, Maplessa on my"os kirjasto CurveFitting ja siell"a funktio LeastSquares , joka suorittaa sovituksen suoraan datan perusteella, ja k"aytt"aj"all"a on valittavanaan eri menetelmi"a. restart:with(LinearAlgebra): #with(linalg): with(plots):#with(stats): alias(Tr=Transpose,Van=VandermondeMatrix):

Sovita 2. asteen polynomi annettuun dataan xd:=[8,10,12,16,20,30,40,60,100]; yd:=[0.88,1.22,1.64,2.72,3.96,7.66,11.96,21.56,43.16]; n:=nops(xd); C:=Van(xd,n,3); # tai # Matrix((i,j)->xd[i]^(j-1),n,3); M:=Tr(C).C; b:=Tr(C).Vector(yd); a:=LinearSolve(M,b); p:=add(a[i]*x^(i-1),i=1..3); datakuva:=plot([seq([xd[i],yd[i]],i=1..n)],style=point,symbol=cross,symbolsize=18): display(plot(p,x=7..100,color=blue),datakuva); Tarkistetaan LinearAlgebra- ja CurveFitting-pakkauksten LeastSquares-funktioilla. Huomaa, ett"a samannimisten funktioiden tapauksessa on syyt"a k"aytt"a"a pitk"a"a muotoa, muuten toiminta riipuu kirjastojen latausj"arjestyksest"a (onneksi menee virheeseen, eik"a laske jotain p"a"at"ont"a). QyQtJkkuTGluZWFyQWxnZWJyYUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSS1MZWFzdFNxdWFyZXNHRik2JEkiQ0dGKS1JJ1ZlY3RvckdGJjYjSSN5ZEdGKSIiIg== LSZJLUN1cnZlRml0dGluZ0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSS1MZWFzdFNxdWFyZXNHRig2JkkjeGRHRihJI3lkR0YoSSJ4R0YoL0kmY3VydmVHRigsKComSSNjMEdGKCIiIkYuIiIjRjQqJkkjYzFHRihGNEYuRjRGNEkjYzJHRihGNA== JSFH

LUkmYWxpYXNHJSpwcm90ZWN0ZWRHNiMvSSVjb25kRzYiSTBDb25kaXRpb25OdW1iZXJHRig= n:=5: m:=n-2:x:=[$1..n]; C:=Van(x,n,m);cond(Tr(C).C): evalf(%); n:=6: m:=n-2:x:=[$1..n]:C:=Van(x,n,m):cond(Tr(C).C): evalf(%); n:=10: m:=n-2:x:=[$1..n]:C:=Van(x,n,m):cond(Tr(C).C): evalf(%); H"airi"oluku cond mittaa sensitiivisyytt"a virheille (datan virheille ja py"oristysvirheille). Normaaliyht"al"oratkaisun varjopuoli on suuri h"ari"oalttius, jos teht"av"an koko on v"ah"ank"a"an suuri. n:=6: m:=2:x:=[$1..n]:C:=Van(x,n,m):cond(Tr(C).C): evalf(%); n:=20: m:=2:x:=[$1..n]:C:=Van(x,n,m):cond(Tr(C).C): evalf(%); n:=20: m:=3:x:=[$1..n]:C:=Van(x,n,m):cond(Tr(C).C): evalf(%); Paremmat menetelm"at perustuvat ortogonalisointiin tai ns. singulaariarvohajoitelmaan.

Sovita a) 2. asteen PNS-polynomi ja b) interpolaatiopolynomi P:=[[1.4,7400],[1.8,7500],[2.4,7600],[3,7500],[4,7200]];

Nyc3JCQiIzkhIiIiJSt1NyQkIiM9RiYiJSt2NyQkIiNDRiYiJSt3NyQiIiRGKzckIiIlIiUrcw==

xd:=map(lis->lis[1],P); yd:=map(lis->lis[2],P);

NyckIiM5ISIiJCIjPUYlJCIjQ0YlIiIkIiIl

NyciJSt1IiUrdiIlK3dGJCIlK3M=

Jos tuo n"aytt"a"a oudolta, voitaisiin tehd"a funktiot poimieka:=x->x[1]; poimitoka:=x->x[2]; ja suorittaa xd:=map(poimieka,P);yd:=map(poimitoka,P); T"ass"a on k"atev"a ja yleisp"atev"a tapa muodostaa matriisi. C:=Matrix(5,3,(i,j)->xd[i]^(j-1));

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNXknKW8sInlTdVklPQ==

Tai, kuten edell"a, suoraan LinearAlgebra-kirjaston funktiolla (joka yll"a "aliasoitiinkin" nimelle Van). VandermondeMatrix(xd):%[1..5,1..3]; #

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNT0qKW8sInlTdVklPQ==

K"aytet"a"an nyt sit"a keskimm"aist"a "harmaata laatikkoa" kertoimet:=LinearAlgebra[LeastSquares](C,Vector(yd));

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1US1wLCJ5U3VZJT0=

PkkieEc2Ii5GIw==

SSJ4RzYi

poly2:=Tr(kertoimet).<1,x,x^2>;

LChdNDBCOUVGOTczRjE3MThDNyIiIkkieEc2Il00MDg3RDczQTc2Rjg1MzY4KiRGJSIiI11DMDYzODBCMTY2OUM1REQ3

display(plot(P,style=point,symbol=circle,color=black),plot(poly2,x=1.3..4.2));

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b) Nyt haetaan interpolaatioplynomia. Sen asteluku = datapisteiden lkm -1 = 4. No siihen l"oytyy taas ihan valmiskin funktio: (Jos ratkaistaisiin LSQ-teht"av"an"a, matriisina olisi koko Vandermonde (transposilla kertominen voitaisiin "supistaa pois").) intpol:=interp(xd,yd,x);

LCwqJEkieEc2IiIiJSQiKy1QX3QhKSEiKSokRiQiIiQkIStoIjMmZSIpISIoKiRGJCIiIyQiK0FPMnhGISInRiQkIStNO3EibyRGMyQiK1QsJylSISpGMyIiIg==

display(plot(intpol,x=1.3..4.2,color=blue),plot(P,style=point,symbol=circle,color=black),plot(poly2,x=1.3..4.2,color=gray));

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Hyv"a tiet"a"a: K"aytt"aj"a voi valita menetelm"an CurveFitting-pakkauksen LeastSquares-funktiossa.

Polynomin asteluvun nostaminen lis"a"a dramaattisesti h"airi"oalttiutta.

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