f(x) = x^2-35
# roots of this are +sqrt(35) and -sqrt(35)
# We know there is a root between x=5 (f is negative) and x=6 (f is positive)
a=5
b=6
# bisection
n_bisection = 0
while b-a>.000000001:
    n_bisection += 1
    c = (a+b)/2
    if f(a)*f(c)>0:
        a=c
    else:
        b=c
print('Root is between ',a.n(),' and ',b.n())
print(sqrt(35).n())
print('Number of steps for bisection: ',n_bisection)
print('Error bound estimate: ', n(b-a))


# Regula Falsi Method
a=5.9
b=6
n_RF = 0
while abs(b-a)>.000000001:
    n_RF +=1
    c = (a*f(b)-b*f(a))/(f(b)-f(a))
    if f(c)*f(a)>0:
        a=c
    else:
        b=c
print('Root is approximately ',b.n())
print(sqrt(35).n())
print('Number of steps for RF: ',n_RF)
print('Error bound estimate: ', abs(b-a).n())

# Secant Method
a=5
b=6
n_RF = 0
while abs(b-a)>.000000001:
    n_RF +=1
    c = (a*f(b)-b*f(a))/(f(b)-f(a))
    a=b
    b=c
print('Root is approximately ',b.n())
print(sqrt(35).n())
print('Number of steps for Secant: ',n_RF)
print('Error bound estimate: ', abs(b-a).n())

## Newton-Raphson
a=5
b=6
n_RF = 0
while abs(b-a)>.000000001:
    n_RF +=1
    c = b - f(b)/(2*b) # 2c is the derivative
    a=b
    b=c   
print('Root is approximately ',b.n())
print(sqrt(35).n())
print('Number of steps for Newton: ',n_RF)
print('Error bound estimate: ', abs(b-a).n())