# let approximate the 4th root of 3
f(x) = x^4-3
g(x) = 4*x^3
#bisection
a=1
b=1.5
errBi=[]
c=1.5
while abs(a-b)>.0000001:
    c = (a+b)/2
    if f(c)*f(a)>0:
        a=c
    else:
        b=c
    errBi+=[3^.25-c]
print('ans is ',c)
print('error = ',errBi)
print('error ratio(d=1) = ',[abs(errBi[j+1])/abs(errBi[j])^1 for j in range(len(errBi)-1)])
print('error ratio(d=1.5) = ',[abs(errBi[j+1])/abs(errBi[j])^1.5 for j in range(len(errBi)-1)])
print('error ratio(d=2) = ',[abs(errBi[j+1])/abs(errBi[j])^2 for j in range(len(errBi)-1)])

# secant
a=1
b=1.5
errBi=[]
c=1.5
while abs(a-b)>.0000001:
    c = (a*f(b)-b*f(a))/(f(b)-f(a))
    a=b
    b=c
    errBi+=[3^.25-c]
print('ans is ',c)
print('error = ',errBi)
print('error ratio(d=1) = ',[abs(errBi[j+1])/abs(errBi[j])^1 for j in range(len(errBi)-1)])
print('error ratio(d=1.5) = ',[abs(errBi[j+1])/abs(errBi[j])^1.5 for j in range(len(errBi)-1)])
print('error ratio(d=2) = ',[abs(errBi[j+1])/abs(errBi[j])^2 for j in range(len(errBi)-1)])

# secant
a=1
b=1.5
errBi=[]
c=1.5
while abs(a-b)>.0000001:
    c = b-f(b)/g(b)
    a=b
    b=c
    errBi+=[3^.25-c]
print('ans is ',c)
print('error = ',errBi)
print('error ratio(d=1) = ',[abs(errBi[j+1])/abs(errBi[j])^1 for j in range(len(errBi)-1)])
print('error ratio(d=1.5) = ',[abs(errBi[j+1])/abs(errBi[j])^1.5 for j in range(len(errBi)-1)])
print('error ratio(d=2) = ',[abs(errBi[j+1])/abs(errBi[j])^2 for j in range(len(errBi)-1)])

# conclusion: Bisection has order 1, secant method has order >=1.5, <2, 
#NR has order >=2