def bisection(f,a,b,eps=None,N=None,exactval=None):
    if N is None:
        N=60
    if eps is None:
        eps=10**(-10)
    i=0
    m=(a+b)/2.0
    abserr=(b-a)/2.0
    while i<N and abserr>eps*abs(m):
        if f(m)==0:
            a=b=m
        if f(a)*f(m)<0:
            b=m
        else:
            a=m
        m=(a+b)/2
        abserr=(b-a)/2
        i=i+1
        print 'Iteration: ', i, 'Estimate: ', m
        print '  Estimated relative error: ', abserr/abs(m)
        if not exactval is None:
            print '  Exact relative error: ', abs((m-exactval)/exactval)
    return m

def secant(f,x0,x1,eps=None,N=None,exactval=None):
    if N is None:
        N=60
    if eps is None:
        eps=10**(-10)
    xpp=float(x0)
    xp=z=float(x1)
    abserr=abs(z)
    i=0
    while i<N and abserr>=eps*abs(z):
        z=xp-f(xp)*(xp-xpp)/(f(xp)-f(xpp))
        abserr=abs(z-xp)
        xpp=xp
        xp=z
        i=i+1
        print 'Iteration: ', i, 'Estimate: ', xp
        print '  Estimated relative error: ', abserr/abs(xp)
        if not exactval is None:
            print '  Exact relative error: ', abs((xp-exactval)/exactval)
    return xp

def newton(f,df,x0,eps=None,N=None,exactval=None):
    if N is None:
        N=60
    if eps is None:
        eps=10**(-10)
    xp=z=float(x0)
    abserr=abs(z)
    i=0
    while i<N and abserr>=eps*abs(z):
        z=xp-float(f(xp))/df(xp)
        abserr=abs(z-xp)
        xp=z
        i=i+1
        print 'Iteration: ', i, 'Estimate: ', xp
        print '  Estimated relative error: ', abserr/abs(xp)
        if not exactval is None:
            print '  Exact relative error: ', abs((xp-exactval)/exactval)
    return xp