Polinómios de Chebyshev

> with(plots):

> n:=4;
fX:=vector(n,0);
X:=vector(n,i->evalf(cos(((2*i-1)*Pi)/(2*n))));
A:=pointplot({[X[1],fX[1]],[X[2],fX[2]],[X[3],fX[3]],[X[4],fX[4]]},axes=BOXED,symbol=circle,title="Polinómio de Chebyshev de grau 4"):
f:=x->0:
B:=plot(f(x),x=-1..1):
g:=x->8*x^4-8* x^2+1;
C:=plot(g(x),x=-1..1):
display(A,B,C);

> n:=7;
fX:=vector(n,0);
X:=vector(n,k->evalf(cos(((2*k-1)*Pi)/(2*n))));
A:=pointplot({[X[1],fX[1]],[X[2],fX[2]],[X[3],fX[3]],[X[4],fX[4]],[X[5],fX[5]],[X[6],fX[6]],[X[7],fX[7]]},axes=BOXED,symbol=circle,title="Zeros do polinómio de Chebyshev de grau 7 no intervalo [-1,1]"):
f:=x->0:
B:=plot(f(x),x=-1..1,y=-0.1..1):
display(A,B);

> a:=0;
b:=1;
n:=7;
fX:=vector(n,0);
X:=vector(n,k->evalf((a+b)/2+(b-a)/2*cos(((2*k-1)*Pi)/(2*n))));
A:=pointplot({[X[1],fX[1]],[X[2],fX[2]],[X[3],fX[3]],[X[4],fX[4]],[X[5],fX[5]],[X[6],fX[6]],[X[7],fX[7]]},axes=BOXED,symbol=circle,title="Zeros do polinómio de Chebyshev de grau 7 no intervalo [0,1]"):
f:=x->0:
B:=plot(f(x),x=a..b,y=-0.1..1):
display(A,B);

>

Exercício 4.13

Função de Runge e nós de Chebyshev

Polinómios de Lagrange L_i

Lagrange-Chebyshev