Přímka procházející zadanými body

   > A:=[0.5,1];

   B:=[1,7.5];

   >

   :

   > primka:=y=p*x+q;

   > rceA:=subs(x=A[1],y=A[2],primka);

   rceB:=subs(x=B[1],y=B[2],primka);

   param:=solve({rceA,rceB},{p,q});

   op(param);

   > primka:=subs(op(param),primka);

   > primka:=rhs(primka);

   > p:=unapply(primka,x);

   >

   > p(A[1])=A[2];p(B[1])=B[2];

   > p(.75);

   > with(plots):

   pointplot({[0,1],[1,-1],[3,0],[4,-3]},axes=BOXED,

   color=RED,symbol=CROSS);

   A;B;

   

   > with(plots):

   > plotA:=

   pointplot({A,B},color=RED,symbol=CROSS):

   plotB:=

   plot(primka,x=A[1]-.5..B[1]+.5,color=NAVY):

   display(plotA,plotB);

   

   > pointplot([A,B],color=RED,symbol=CROSS,symbolsize=100,connect=true);

   Funkce, která dvµma zadaněm bodôm p»i»adí p»ímku, jeľ jimi prochází

   > line:=proc(A,B)

   local x;

   if A[1]=B[1] then

        print(`primka je svisla, nejde o funkci`)

     else

   unapply( (B[2]-A[2])/(B[1]-A[1])*x-(-A[2]*B[1]+A[1]*B[2])/(B[1]-A[1]),x );

   fi;

   end;

   > x:='x';

   line([1,0],[2,1])(x);

   > line([1,0],[2,1])(1.5);

   > #X:=Body;

   #Y:=Hodnoty;

   Plin:=proc(X,Y)

   local N,i,x;

   N:=nops(X);

   param:=x<X[1],Y[1];

   for i from 1 to N-1 do

     param:=param, x<X[i+1],line([X[i],Y[i]],[X[i+1],Y[i+1]])(x);

   od;

     param:=param, Y[N];

   unapply(piecewise(param),x);

   end;

   

   xxx:=Plin(Body,Hodnoty)(x);

   

   

   >

   > plot(xxx,x=0..30);

   > i:='i';

   Body:= [i $i=1..27];

   Hodnoty:=[evalf(sin(i*1.)) $i=1..27];

   > Plin(Body,Hodnoty);

   >

   Lagrangeuv polynom

   > n:=5;

   for i from 1 to n do

   X[i]:=i;

   Y[i]:=evalf(ln(i));

   od;

   

   >

   > i:='i';j:='j';

   

   >

   > for j from 1 to n do

   

   > citatel[j]:=simplify(product(x-X[i],i=1..n)/(x-X[j]));

   

   > jmenovatel[j]:=subs(x=X[j],citatel[j]);

   > clen[j]:=citatel[j]/jmenovatel[j]*Y[j];

   od;

   > Lagrange:=simplify(sum(clen[i],i=1..n));

   > with(plots);

   Warning, the name changecoords has been redefined

   

   > i:='i';

   body:=[[X[i],Y[i]] $i=1..n];

   Aplot:=pointplot(body):

   Bplot:=plot(ln(x),x=.5..7.5):

   Cplot:=plot(Lagrange,x=.5..7.5,color=BLACK):

   

   > display(Aplot,Bplot,Cplot);

   Obecnµ:

   > Lagrange:=proc(X,Y)

   local C,J,N,x,i,j,substitut,xxx;

   i:='i';

   Xpom:=convert(X,list);

   N:=nops(Xpom);

   for j from 1 to N do

   C[j]:=product((x-X[i]),i=1..N)/(x-X[j]);

   J[j]:=subs(x=X[j],C[j]);

   #print(C[j],J[j]);

   od;

   i:='i';

   xxx:=sum(C[i]/J[i]*Y[i],i=1..N);

   unapply(xxx,x)

   end;

   

   Warning, `Xpom` is implicitly declared local to procedure `Lagrange`

   

   > xxx:=Lagrange(X,Y);

   Lagrange(X,Y)(3.1)=ln(3.1);

   > whattype(X);print(X);

   > yyy:=interp(convert(X,list),convert(Y,list), x);

   > simplify(xxx(x)-yyy);

   > plot({sin(x),Lagrange([1,3,5,7],[sin(1),sin(3),sin(5),sin(7)])(x)},x=-1..8);

   >

   > A[1]:=7;A[2]:=2;

   > print(A);

   > nops(A);

   > A:=convert(A,list);

   > A[1];

   > whattype(A);

   > A:='A';

   > xxx:=x^2-1;

   > subs(x=2,xxx);

   > f:=unapply(xxx,x);

   > f(2);

   > g:=t->t-1;

   > g(2);

   > g(x);

   Fourierova »ada

   > Int(sin(x),x=0..Pi/4)=int(sin(x),x=0..Pi/4);

   > fourier:=proc(f,n)

   local i,N;

   if n mod 2 = 1 then N:=(n+1)/2

     else

       N:=n/2;

   fi;

   for i from 0 to N do

     a[i]:=evalf(1/Pi*int(f(x)*cos(i*x),x=-Pi..Pi));

     b[i]:=evalf(1/Pi*int(f(x)*sin(i*x),x=-Pi..Pi));

   od;

   i:='i';

   if n mod 2 = 1 then

     unapply(a[0]/2+sum(a[i]*cos(i*x)+b[i]*sin(i*x),i=1..N-1),x)

   else

     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);

   fi

   end;

   >

   > f:=x->x^2-2^x;

   

   

   for i from 1 to 2 do

   print(i,fourier(f,i));

   od;

   > plot({f(x),fourier(f,15)(x)},x=-Pi/2..Pi/2);

   > f:=x->abs(x);

   K:=21;

   ll:=t<1,(t)*fourier(f,1);

   for i from 1 to K do

   ll:=ll, t<i+1 ,(1-(t-i))*fourier(f,i)+(t-i)*fourier(f,i+1);

   print(i);

   od:

   F:=piecewise(ll):

   G:=unapply(F,t):

   #subs();

   

   > animate({G(t)(x),f(x)},x=-Pi..Pi,t=1..K,frames=K+1);

   > plot({G(12.5)(x),f(x)},x=-Pi..Pi);

   > Fourier:=proc(f,n,beta)

   local i,N;

   if n mod 2 = 1 then N:=(n+1)/2

     else

       N:=n/2;

   fi;

   for i from 0 to N do

     a[i]:=evalf(2/beta*evalf(int(f(x)*cos(2*i*Pi*x/beta),x=0..beta)));

     b[i]:=evalf(2/beta*evalf(int(f(x)*sin(2*i*Pi*x/beta),x=0..beta)));

   od;

   i:='i';

   if n mod 2 = 1 then

     unapply(a[0]/2+sum(a[i]*cos(2*i*Pi*x)+b[i]*sin(2*i*Pi*xx),i=1..N-1),x)

   else

     unapply(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);

   fi

   end;

   

   Aproximace:=proc(X,Y,F) #vstup: pole bodu, pokle hodnot, pole funkci. Vystup: linearni
   kombinace zadanych funkci, ktera nejlepe aproximuje zadane hodnoty.

   global MaticeSoustavy,PraveStrany;

   local i,j,PX,PF,v

   ;

   PF:=nops(F);PX:=nops(X);

   i:='i';j:='j';

   v:=evalf

   ([ unapply(F[i],x)(X[j])  $j=1..PX] $i=1..PF);

   

   #print(v);

   MaticeSoustavy:=

   matrix(PF,PF,

   [[sum((v[i][k]*v[j][k]),k=1..PX) $j=1..PF] $i=1..PF]

   );

   PraveStrany:=vector([sum(Y[j]*v[i][j],j=1..PX) $i=1..PF]);

   

   Koeficienty:=linsolve(MaticeSoustavy,PraveStrany):

   unapply(sum(Koeficienty[i]*F[i],i=1..PF),x);

   end;

   > Fourier(exp(x),4,1);

   > AZ:=Fourier(xxx,2,10);

   plot(AZ(x),x=0..30);

   > plot({Fourier(exp(x),6,1)(x),exp(x)},x=-0.1..1.1);

   Legenderovy polynomy tvo»í úplně ortonormální systém na intervalu <-1,1>.

   > Legender:=n->sqrt(n+1/2)*`if`(n=0,1,simplify(diff(1/(2^n*n!)*(x^2-1)^n,x$n)));

   > for i from 0 to 5 do

   print(i,Legender(i));

   od;

   >

   Metoda nejmensich ctvercu

   > X:=[1,2,3];Y:=[2,4,6];

   RegresniPrimka:=proc(X,Y)

   local Primka,x,p,q;

   Primka:=x->p*x+q;

   SoucetCtvercu:=sum(((Y[i]-Primka(X[i]))^2),i=1..nops(X));

   Rce:=diff(SoucetCtvercu,p)=0,diff(SoucetCtvercu,q)=0;

   Param:=solve({Rce},{p,q});

   unapply(subs(Param,Primka(x)),x);

   end;

   >

   > RegresniPrimka([i $i=1..17],[2*i+sin(i) $i=1..17]);

   > with (plots):

   Obr:=proc(X,Y)

   A:=pointplot({[X[i],Y[i]] $i=1..nops(X)});

   B:=plot(RegresniPrimka(X,Y)(x),x=0..X[nops(X)]);

   display(A,B)

   end;

   > Obr([i $i=1..17],[2*i+sin(i) $i=1..17]);

   > with(stats):

   fit[leastsquare[[x,y]]]([[1,2,3,5],[2,4,6,8]]);

   >

   > RegresniPrimka([1,2,3,5],[2,4,6,8]);

   Aproximace namerenych hodnot linearni kombinbaci zvolenych funkci:

   >

   >

   

   > with(linalg):

   > Aproximace:=proc(X,Y,F)

   global MaticeSoustavy,PraveStrany;

   local i,j,PX,PF,v

   ;

   PF:=nops(F);PX:=nops(X);

   i:='i';j:='j';

   v:=evalf

   ([ unapply(F[i],x)(X[j])  $j=1..PX] $i=1..PF);

   

   #print(v);

   MaticeSoustavy:=

   matrix(PF,PF,

   [[sum((v[i][k]*v[j][k]),k=1..PX) $j=1..PF] $i=1..PF]

   );

   PraveStrany:=vector([sum(Y[j]*v[i][j],j=1..PX) $i=1..PF]);

   

   Koeficienty:=linsolve(MaticeSoustavy,PraveStrany):

   unapply(sum(Koeficienty[i]*F[i],i=1..PF),x);

   end;

   >

   > i:='i';

   X:=[i $i=1..5];

   Y:=[sin(i)-cos(i)+(sin(cos(i))) $i=1..5];

   (Aproximace(X,Y,[sin(x),cos(x)]));

   >

   > print(MaticeSoustavy);

   print(PraveStrany);

   > linsolve(MaticeSoustavy,PraveStrany);

   >

   >

   Warning, `fce` is implicitly declared local to procedure `PrumetFunkce`