3513.sagews

#f(x)= (x^2)-ln(1+x)#
#fp(x)=derivative(f(x))#
#phi1(x)=x-(f(x)/fp(x))#

#méthode de newton
def f(x):
    return ((x**2)-ln(1+x))
def fp(x):
    return (2*x-1/(x+1))
def phi1(x):             #relation de récurence
    return(x-(f(x)/fp(x)))
def newton1(x_start,eps):  # eps :epsilon
    x=x_start
    x_old=x+10*eps
    while(abs(x-x_old)>eps):
        x_old=x
        x=phi1(x)
    return(x)

plot(f(x),(x,0,1))

newton1((0.1),(1.e-7))

-5.38956478861568e-17

def fospos(x0,x1,eps):
    xold=x0
    x=x1
    while(abs(x-xold)>eps):
        xold2=xold
        xold=x
        x=phifp(x,xold2)
    return(x)

def phifp(xn,xn1):
    res=xn-(f(xn)*((xn-xn1)/(f(xn)-f(xn1))))
    return(res)

phifp(0.1,2.1)

0.150720874692097

fospos(0.1,2.1,1.e-7)

0.150720874692097 0.218258268955401 -0.0967512581539759 0.0374103980757303 0.00504579648280168 -0.000299464926141325 2.28039407593010e-6 1.02395815757518e-9 -3.59650406559766e-15 -3.59650406559766e-15

fp(x)

2*x - 1/(x + 1)