{VERSION 2 3 "IBM INTEL OS2" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 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 0 0 0 0 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 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 }1 1 0 0 0 0 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 }3 1 0 0 12 12 3 0 3 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 33 "P\370\355mka proch\341zej \355c\355 zadan\375mi body" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A:=[0.5,1];\nB:=[1,7.5];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "primka:=y=p*x+q;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 111 "rceA:=subs(x=A[1],y=A[2],primka);\nrceB:=subs(x=B[ 1],y=B[2],primka);\nparam:=solve(\{rceA,rceB\},\{p,q\});\nop(param);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "primka:=subs(op(param),pr imka);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "primka:=rhs(primk a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p:=unapply(primka,x) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p(A[1])=A[2];p(B[1])=B[2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p(.75);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "with(plots):\npointplot(\{[0,1],[1,-1],[3,0],[4,-3]\},axes=BOXED, \ncolor=RED,symbol=CROSS);\nA;B;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "plotA:=\npointplot(\{A,B\},color=RED,symbol=CROSS):\nplotB:=\np lot(primka,x=A[1]-.5..B[1]+.5,color=NAVY):\ndisplay(plotA,plotB);\n" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "pointplot([A,B],color=RED, symbol=CROSS,symbolsize=100,connect=true);" }}}{EXCHG {PARA 18 "" 0 " " {TEXT -1 67 "Funkce, kter\341 dv\276ma zadan\354m bod\364m p\235i \235ad\355 p\235\355mku, je\236 jimi proch\341z\355" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "line:=proc(A,B)\nlocal x;\nif A[1]=B[1] \+ then \n print(`primka je svisla, nejde o funkci`)\n else\nunapply ( (B[2]-A[2])/(B[1]-A[1])*x-(-A[2]*B[1]+A[1]*B[2])/(B[1]-A[1]),x ); \n fi;\nend;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x:='x';\nline( [1,0],[2,1])(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "line([1 ,0],[2,1])(1.5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "#X:=Bo dy;\n#Y:=Hodnoty;\nPlin:=proc(X,Y)\nlocal N,i,x;\nN:=nops(X);\nparam:= x " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(xxx,x=0..30);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "i:='i';\nBody:= [i $i=1..27];\nHodnoty:=[evalf(sin(i*1.)) $i=1..27]; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Plin(Body,Hodnoty);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 18 "" 0 " " {TEXT -1 18 "Lagrangeuv polynom" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "n:=5;\nfor i from 1 to n do\nX[i]:=i;\nY[i]:=evalf(ln (i));\nod;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "i:='i';j:='j';\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for j f rom 1 to n do\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "citatel[j]:=simp lify(product(x-X[i],i=1..n)/(x-X[j]));\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "jmenovatel[j]:=subs(x=X[j],citatel[j]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "clen[j]:=citatel[j]/jmenovatel[j]*Y[j];\nod; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Lagrange:=simplify(sum( clen[i],i=1..n));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(p lots);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "i:='i';\nbody:=[ [X[i],Y[i]] $i=1..n];\nAplot:=pointplot(body):\nBplot:=plot(ln(x),x=.5 ..7.5):\nCplot:=plot(Lagrange,x=.5..7.5,color=BLACK):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display(Aplot,Bplot,Cplot);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Obecn\276:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "Lagrange:=proc(X,Y)\nlocal C,J,N,x,i,j,substitu t,xxx;\ni:='i';\nXpom:=convert(X,list);\nN:=nops(Xpom);\nfor j from 1 \+ to N do\nC[j]:=product((x-X[i]),i=1..N)/(x-X[j]);\nJ[j]:=subs(x=X[j],C [j]);\n#print(C[j],J[j]);\nod;\ni:='i';\nxxx:=sum(C[i]/J[i]*Y[i],i=1.. N);\nunapply(xxx,x)\nend;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "xxx:=Lagrange(X,Y);\nLagrange(X,Y)(3.1)=ln(3.1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "whattype(X);print(X);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "yyy:=interp(convert(X,list),convert(Y,list) , x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(xxx(x)-yy y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot(\{sin(x),Lagran ge([1,3,5,7],[sin(1),sin(3),sin(5),sin(7)])(x)\},x=-1..8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A[1]:=7;A[2]:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A:=convert(A,l ist);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "whattype(A);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "A:='A';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "xxx:=x^2-1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs (x=2,xxx);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=unapply(xx x,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "g:=t->t-1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x);" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 15 "Fourierova \235ada" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(sin(x),x=0..Pi/4)=int (sin(x),x=0..Pi/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "fou rier:=proc(f,n)\nlocal i,N;\nif n mod 2 = 1 then N:=(n+1)/2\n else\n \+ N:=n/2; \nfi;\nfor i from 0 to N do\n a[i]:=evalf(1/Pi*int(f(x)*co s(i*x),x=-Pi..Pi));\n b[i]:=evalf(1/Pi*int(f(x)*sin(i*x),x=-Pi..Pi)); \nod;\ni:='i';\nif n mod 2 = 1 then\n unapply(a[0]/2+sum(a[i]*cos(i*x )+b[i]*sin(i*x),i=1..N-1),x)\nelse\n unapply(a[0]/2+sum(a[i]*cos(i*x) +b[i]*sin(i*x),i=1..N-1)+a[N]*cos(N*x),x);\nfi\nend;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f: =x->x^2-2^x;\n\n\nfor i from 1 to 2 do\nprint(i,fourier(f,i));\nod;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{f(x),fourier(f,15)( x)\},x=-Pi/2..Pi/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "f: =x->abs(x);\nK:=21;\nll:=t<1,(t)*fourier(f,1);\nfor i from 1 to K do\n ll:=ll, t " 0 "" {MPLTEXT 1 0 52 "animate( \{G(t)(x),f(x)\},x=-Pi..Pi,t=1..K,frames=K+1);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "plot(\{G(12.5)(x),f(x)\},x=-Pi..Pi);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1025 "Fourier:=proc(f,n,beta)\nl ocal i,N;\nif n mod 2 = 1 then N:=(n+1)/2\n else\n N:=n/2; \nfi;\n for i from 0 to N do\n a[i]:=evalf(2/beta*evalf(int(f(x)*cos(2*i*Pi*x /beta),x=0..beta)));\n b[i]:=evalf(2/beta*evalf(int(f(x)*sin(2*i*Pi*x /beta),x=0..beta)));\nod;\ni:='i';\nif n mod 2 = 1 then\n unapply(a[0 ]/2+sum(a[i]*cos(2*i*Pi*x)+b[i]*sin(2*i*Pi*xx),i=1..N-1),x)\nelse\n u napply(a[0]/2+sum(a[i]*cos(2*i*Pi*x)+b[i]*sin(2*i*Pi*x),i=1..N-1)+a[N] *cos(N*2*Pi*x),x);\nfi\nend;\n\nAproximace:=proc(X,Y,F) #vstup: pole b odu, pokle hodnot, pole funkci. Vystup: linearni kombinace zadanych fu nkci, ktera nejlepe aproximuje zadane hodnoty.\nglobal MaticeSoustavy, PraveStrany;\nlocal i,j,PX,PF,v\n;\nPF:=nops(F);PX:=nops(X);\ni:='i';j :='j';\nv:=evalf\n([ unapply(F[i],x)(X[j]) $j=1..PX] $i=1..PF);\n\n#p rint(v);\nMaticeSoustavy:=\nmatrix(PF,PF,\n[[sum((v[i][k]*v[j][k]),k=1 ..PX) $j=1..PF] $i=1..PF]\n);\nPraveStrany:=vector([sum(Y[j]*v[i][j],j =1..PX) $i=1..PF]);\n\nKoeficienty:=linsolve(MaticeSoustavy,PraveStran y):\nunapply(sum(Koeficienty[i]*F[i],i=1..PF),x);\nend;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Fourier(exp(x),4,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "AZ:=Fourier(xxx,2,10);\nplot(AZ(x), x=0..30);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(\{Fourier (exp(x),6,1)(x),exp(x)\},x=-0.1..1.1);" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 66 "Legenderovy polynomy tvo\235\355 \372pln\354 ortonorm\341 ln\355 syst\351m na intervalu " }{MPLTEXT 0 21 7 "<-1,1>." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Legender:=n->sqrt(n+1/2)*`if`(n=0,1 ,simplify(diff(1/(2^n*n!)*(x^2-1)^n,x$n)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "for i from 0 to 5 do\nprint(i,Legender(i));\nod;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 25 "Metoda nejmensich ctvercu" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "X:=[1,2,3];Y:=[2,4,6];\nRegresniPrimka:=proc(X,Y)\nl ocal Primka,x,p,q;\nPrimka:=x->p*x+q;\nSoucetCtvercu:=sum(((Y[i]-Primk a(X[i]))^2),i=1..nops(X));\nRce:=diff(SoucetCtvercu,p)=0,diff(SoucetCt vercu,q)=0;\nParam:=solve(\{Rce\},\{p,q\});\nunapply(subs(Param,Primka (x)),x);\nend;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "RegresniPrimka([i $i=1..17],[2*i+sin(i) $ i=1..17]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "with (plots) :\nObr:=proc(X,Y)\nA:=pointplot(\{[X[i],Y[i]] $i=1..nops(X)\});\nB:=pl ot(RegresniPrimka(X,Y)(x),x=0..X[nops(X)]);\ndisplay(A,B)\nend;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Obr([i $i=1..17],[2*i+sin(i) $i=1..17]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "with(stats) :\nfit[leastsquare[[x,y]]]([[1,2,3,5],[2,4,6,8]]);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Regre sniPrimka([1,2,3,5],[2,4,6,8]);" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 66 "Aproximace namerenych hodnot linearni kombinbaci zvolenych funkci: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 434 "Aproximace:=proc(X,Y,F)\ngloba l MaticeSoustavy,PraveStrany;\nlocal i,j,PX,PF,v\n;\nPF:=nops(F);PX:=n ops(X);\ni:='i';j:='j';\nv:=evalf\n([ unapply(F[i],x)(X[j]) $j=1..PX] $i=1..PF);\n\n#print(v);\nMaticeSoustavy:=\nmatrix(PF,PF,\n[[sum((v[i ][k]*v[j][k]),k=1..PX) $j=1..PF] $i=1..PF]\n);\nPraveStrany:=vector([s um(Y[j]*v[i][j],j=1..PX) $i=1..PF]);\n\nKoeficienty:=linsolve(MaticeSo ustavy,PraveStrany):\nunapply(sum(Koeficienty[i]*F[i],i=1..PF),x);\nen d;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "i:='i';\nX:=[i $i=1..5];\nY:=[sin(i)-cos(i)+(sin(cos (i))) $i=1..5];\n(Aproximace(X,Y,[sin(x),cos(x)]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "print(MaticeSoustavy);\nprint(PraveStrany);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "linsolve(MaticeSoustavy,PraveStrany);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 7 "" 1 "" {TEXT -1 72 "Warning, `fce` is implicitly declare d local to procedure `PrumetFunkce`\n" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }