newton+lagrange

source("SPLINES.R")

main <- function()
{
    #X <- c(1,2,4,6,7)
    #Y <- c(2,4,1,3,3)
    X <- c(3,4.5,7,9)
    Y <- c(2.5,1,2.5,0.5)
    xis <- seq(from = 3.2, to = 8.7, length.out = 10000)
    #xis <- c(1.2,2.9,5.2,6.7)

    sp3 <- splines3(4,X,Y,xis,0)
    sp31 <- splines3(4,X,Y,xis,1)
    sp2 <- splines(4,X,Y,xis)

    new <- newton(X,Y,xis)
    lagg <- lagrange(X,Y,xis)

    #print(sp2)
    #print(sp3)

    plot(sp2,type = "o",col = "purple",pch = 20) #Desenha
    lines(sp3,type = "o",col = "red",pch = 20) #Desenha
    lines(new,type = "o",col = "green",pch = 20) #Desenha
    lines(lagg,type = "o",col = "blue",pch = 20) #Desenha
    lines(X,Y,type = "p",pch = 8)
    legend("top", xpd=TRUE, ncol=2, legend=c("SP", "SP3","Newton","LaGrange"),
       fill=c("purple", "red","green","blue"), bty="n")

    #natural vs not a knot:
    #plot(sp3,type = "o",col = "red") #Desenha
    #lines(sp31,type = "o",col = "blue") #Desenha
    #legend("top", xpd=TRUE, ncol=2, legend=c("Natural", "Not a Knot"),
    #   fill=c("red", "blue"), bty="n")
    #-----------------------------
    #Newton vs Lagrange:
    #plot(new,type = "o",col = "green") #Desenha
    #lines(lagg,type = "o",col = "blue") #Desenha
    #legend("top", xpd=TRUE, ncol=2, legend=c("Newton", "Lagrange"),
    #   fill=c("green", "blue"), bty="n")
    #-----------------------------
    # sp vs sp3
    #plot(sp2,type = "o",col = "green") #Desenha
    #lines(sp3,type = "o",col = "blue") #Desenha
    #legend("top", xpd=TRUE, ncol=2, legend=c("Splines 2", "Splines 3"),
    #   fill=c("green", "blue"), bty="n")

}

newton <- function (X,Y,x)
{
    n <- length(X)
    V <- Y  #v[i] = y[i]

    for(k in 1:(n-1) ){
        for(i in seq(from = n, to = (k +1))){
            V[i] = (V[i] - V[i-1]) / (X[i] - X[i-k]) }}


    R <- array(dim = length(x)) #Criar array com as respostas

    for(j in 1:length(x)) #Percorre o array x e salva as respostas no array R
    {
        R[j] <- V[n]
        for(i in (n-1):1)
        {
            R[j] <- R[j] * (x[j] - X[i]) + V[i]
        }
    }

    X <- c(X, x) #Juntando os dados
    Y <- c(Y, R)

    dataF <- data.frame(X,Y)
    ord <- dataF[order(X) , ] #Ordenando pela abscissa

    #plot(ord,type = "b") #Desenha

    return(ord)

}

# x = c(1.1,1.4,1.9,2.1,2.5,3.0,3.2)
# y = c(0.6942,0.6952,1.1759,1.6562,3.4325,8.0855,11.0925)
#seq(from = -1, to = 1, length.out = 50)

#f(x) = 1/ 1+25x^2
fx <- function(limite1,limite2,qtd)
{
    X <- seq(from = limite1, to = limite2, length.out = qtd)
    Y <- array(dim = qtd)
    for(i in 1:qtd)
    {
        Y[i] = 1 / ( 1 + (25 * (X[i]^2)  ))
    }
}

lagrange <- function(X,Y,x)
{
    n <- length(X)

    Px2 <- array(dim = length(x)) #Criar array com as respostas

    for(k in 1:length(x))
    {
        Px <- 0
        for(i in 1:n)
        {
            P <- 1
            for(j in 1:n)
            {
                if(j != i )
                {
                    P <- P * ( (x[k] - X[j]) / (X[i] - X[j]) )  #prod
                }
            }
            Px <- (P * Y[i]) + Px #Somatorio
        }
        Px2[k] = Px
    }

    X <- c(X, x) #Juntando os dados
    Y <- c(Y, Px2)

    dataF <- data.frame(X,Y)
    ord <- dataF[order(X) , ] #Ordenando pela abscissa

    #plot(ord,type = "b") #Desenha

    return(ord)

}