> restart:
# Hörner
> Horner:=proc(P,x)
> local k,H;
> H:=0;
> for k from degree(P) by -1 to 0 do 
> H:=H*x+coeff(P,x,k);
> od:
> H;
> end:
> Q:=3*x^5-2*x^4+7*x^3+2*x^2+5*x-3:
> readlib(cost):cost(Q);
5*additions+15*multiplications
> cost(Horner(Q,x));
5*additions+5*multiplications
> 
> 
# Développements limités
# 
> dl:=proc(f,x,n)
> convert(taylor(f(x),x=0,n),polynom);
> end:
> dl(tan,x,10);
x+1/3*x^3+2/15*x^5+17/315*x^7+62/2835*x^9
> graph:=proc(f,xmax,ymax,p)
> local s,i:
> s:=seq(plot(dl(f,x,i),x=-xmax..xmax),i=1..p),plot(f(x),x=-xmax..xmax,c
> olor=blue):
> plots[display](s,insequence=true,view=[-xmax..xmax,-ymax..ymax]);
> end:
> graph(cos,15,1,50);

> restart:
# Lagrange
> 
> Lagrange:=proc(X,Y,t)
> local k,L,P;
> P:=0:
> for k to nops(X) do
> L:=simplify(product(t-X[i],i=1..nops(X))/(t-X[k])):
> L:=L/subs(t=X[k],L):
> P:=P+Y[k]*L:
> od:
> sort(expand(P));
> end:
> Lagrange([1,2,3,4,5],[7,-8,9,-10,11],x);
6*x^4-214/3*x^3+294*x^2-1463/3*x+266
> 
> LagrangeFonction:=proc(f,X,x)
> local Y,k;
> Y:=[seq(f(X[k]),k=1..nops(X))]:
> Lagrange(X,Y,x);
> end:
> LagrangeFonction(t->1/(1+t^2),[1,2,3,4,5],x);
19/4420*x^4-147/2210*x^3+1731/4420*x^2-2373/2210*x+275/221
> 
> 
> 
> 
> LagrangeGraphe:=proc(g,n,a,b,t,xmin,xmax,ymin,ymax)
> local i,s,p;
> s:=seq( plots[display]( plot (
> LagrangeFonction(g,[a+(b-a)*i/p$i=0..p],t),t=xmin..xmax),plot(g(t),t=x
> min..xmax,color=blue)),p=1..n):
> plots[display](s,insequence=true,view=[xmin..xmax,ymin..ymax]);
> end:
> LagrangeGraphe(t->1/(1+t^2),40,-5,5,t,-5,5,-0.2,1.2);

> LagrangeGraphe(t->ln(1+t),20,0,5,t,0,5,-2,3);

# Tchebychev
> with(orthopoly):
> T(5,x);
16*x^5-20*x^3+5*x
> S:=solve(T(5,5*x)=0,x);
S := 0, 1/20*(10+2*5^(1/2))^(1/2), -1/20*(10+2*5^(1/2))^(1/2),
1/20*(10-2*5^(1/2))^(1/2), -1/20*(10-2*5^(1/2))^(1/2)
> [evalf(S)];
Error, invalid types in sum
> LagrangeTcheb:=proc(g,n,t,ymin,ymax)
> local i,s,p;
> s:=seq( plots[display]( plot (
> LagrangeFonction(g,[evalf(solve(T(p,x)=0,x))],t),t=-1..1),plot(g(t),t=
> -1..1,color=blue)),p=1..n):
> plots[display](s,insequence=true,view=[-1..1,ymin..ymax]);
> end:
> LagrangeTcheb(t->1/(1+t^2),10,t,-0.2,1.2);

>