# Decomposição QR
QRGram<-function(A){
n=length(A[1,])
m=length(A[,1])
V=A
Q=0*A
R=matrix(0,n,n)

for (i in 1:n){
R[i,i]=sqrt(V[,i]%*%V[,i])
Q[,i]=V[,i]/R[i,i]

for (j in (i+1):n){ 
if (j!=(n+1)){
R[i,j]=Q[,i]%*%V[,j]
V[,j]=V[,j]-R[i,j]*Q[,i]
}
}}
return(list(Q=Q,R=R))
}


#-------------------------------------------------------
# Resolução de sistema triangular superior

TSup<-function(R,v){ # Retorna u tal que Ru=v
n=length(v);u=0*v
u[n]=v[n]/R[n,n]
for ( i in (n-1):1){
p=v[i]
for (j in (i+1):n){
p=p-u[j]*R[i,j]
}
u[i]=p/R[i,i]
}
u
}

#---------------------------------------------------------------------------
# Problema de interpolação polinomial

n=41
a=-5;b=5;x=0*1:n

for ( k in 1:n){  # Nós de Chebyshev reescalados.
x[k]=(a+b)/2+(b-a)*cos((2*k-1)/(2*n)*pi )/2}

y=1/(1+x^2)

n=length(x)
plot(x,y,col="red")

n=length(x)
A=matrix(0,n,n); 
for ( j in 1:n){A[,j]=x^(j-1)}


B=QRGram(A)

Q=B$Q;Q
R=B$R; R

v=t(Q)%*%y; v


u=TSup(R,v); u
print("Teste de erro"); A%*%u-y


poli<-function(s){
    p=u[1]
    for ( i in 2:n){p=p+u[i]*(s)^(i-1)}
    return(p)
} 

curve(poli,-5,5,col="blue",ylim=c(0,1))
points(x,y,col="red")        # Teste de ajuste.


Erro<-function(s){1/(1+s^2)-poli(s)}
curve(Erro,-5,5)