clear
for id=winsid() close(scf(id)) end

//exercice préliminaire
function U=maSuite(n)
    if n<0 then
        U=null
    elseif n == 0 then
		U = 1
	else
		U = maSuite(n - 1) + 1
	end
endfunction

function U=fibonacci(n)
    if n<0 then
        U=null
    elseif n == 0 | n == 1 then
		U = 1
	else
		U = fibonacci(n-1) + fibonacci(n-2)
	end
endfunction

function F=fctSuite(n, x)
    if n<0 then
        F=null
    elseif n == 0 then
        F=x
    else
        F=x.*fctSuite(n-1, x)
    end
endfunction

n=50
x=linspace(-1,1,n)
title("Exercice préliminaire")
plot(x, fctSuite(0, x), 'r')
plot(x, fctSuite(1, x), 'g')
plot(x, fctSuite(2, x), 'b')

//Q1
function y=LegendreP(n, x)
    if n<0 then
        y=null
    elseif n==0 then
        y=ones(1, size(x,2))
    elseif n==1 then
        y=x
    else
        y=(2-1/n).*x.*LegendreP(n-1,x)-(1-1/n).*LegendreP(n-2,x)
    end
endfunction

a_0=sqrt(2)*sinh(1),
a_1=-sqrt(6)/exp(1)
a_2=(exp(2)-7)/exp(1)*sqrt(5/2)
a_3=(5*exp(2)-37)/exp(1)*sqrt(7/2)
a_4=3*sqrt(2)*(18*exp(2)-133)/exp(1)
a_5=(329*exp(2)-2431)/exp(1)*sqrt(11/2)
a_6=(3655*exp(2)-27007)/exp(1)*sqrt(13/2)
A=[a_0, a_1, a_2, a_3, a_4, a_5, a_6]

//Q2
f2=sinh(2)
disp(f2-sum(A^2))

//Q3
n=400
x=linspace(-1,1,n)

scf()
subplot(3, 1, 1)
title("Les 7 premiers polynômes de Legendre")
for i=0:7
    plot(x, LegendreP(i, x))
end

subplot(3, 1, 2)
title("exp(-x) et son approximation")
plot(x, exp(-x), "g")

function y=norm_LegendreP(i)
    y=sqrt(2/(2*i+1))
endfunction

function y=pif(a, n, x)
    y=0
    for i=0:n
        y=y+a(i).*LegendreP(i, x)/norm_LegendreP(i)
    end
endfunction

function a=a_exp(k)
    a=A(k+1)
endfunction

plot(x, pif(a_exp, 6, x), ":")
legend("exp(-x)", "approximation au rang 6")

subplot(3, 1, 3)
title("Erreur d''approximation au rang 6")
plot(x, exp(-x)-pif(a_exp, 6,x))

//Q4
function a = a_abs(k)
    if k<0 then
        a=null
    elseif pmodulo(k,2)==1 then
        a=0
    else
        a=sqrt(4*k+2)*integrate("x*LegendreP(k, x)", "x", 0, 1)
    end
endfunction

scf()
//remarque: il serait plus efficace de "cacher" le résultat de pif(...)
subplot(2,1,1)
plot(x, pif(a_abs, 2, x),"r")
plot(x, pif(a_abs, 4, x),"g")
plot(x, pif(a_abs, 6, x),"b")
legend(["degré 2", "degré 4", "degré 6"])
title("Approximations de |x| par des polynômes de Legendre")

subplot(2,1,2)
plot(x, abs(x) - pif(a_abs, 2, x),"r")
plot(x, abs(x) - pif(a_abs, 4, x),"g")
plot(x, abs(x) - pif(a_abs, 6, x),"b")
legend(["degré 2", "degré 4", "degré 6"])
title("Erreur d''approximations de |x| par des polynômes de Legendre")