Consider the function $$ f(x) = \exp(x) - \frac{3}{2} - \arctan(x) $$
%config InlineBackend.figure_format = 'svg'
from numpy import linspace,abs
from sympy import Symbol,exp,atan,diff,lambdify
from matplotlib.pyplot import figure,subplot,plot,grid,title
We define the function using sympy and automatically compute its derivative. Then we make functions out of these.
x = Symbol('x')
# Define the function
f = exp(x)-3.0/2.0-atan(x)
# Compute its derivative
df = diff(f,x)
print("f = ", f)
print("df= ", df)
# Make functions
F = lambdify(x, f, modules=['numpy'])
DF = lambdify(x, df, modules=['numpy'])
# Plot the functions
X = linspace(-1,1,100)
figure(figsize=(9,4))
subplot(1,2,1)
plot(X,F(X))
grid(True)
title('Function')
subplot(1,2,2)
plot(X,DF(X))
grid(True)
title('Derivative');
f = exp(x) - atan(x) - 1.5 df= exp(x) - 1/(x**2 + 1)
Now we implement the Newton method. We have to specify the maximum number of iterations, an initial guess and a tolerance to decide when to stop.
M = 20 # maximum iterations
x = 0.5 # initial guess
eps = 1e-14 # relative tolerance on root
f = F(x)
for i in range(M):
df = DF(x)
dx = -f/df
x = x + dx
e = abs(dx)
f = F(x)
print("%6d %22.14e %22.14e %22.14e" % (i,x,e,abs(f)))
if e < eps * abs(x):
break
0 8.71059792151655e-01 3.71059792151655e-01 1.72847808769943e-01
1 7.76133043351671e-01 9.49267487999836e-02 1.30353403123729e-02
2 7.67717620302437e-01 8.41542304923425e-03 9.81774270505387e-05
3 7.67653269950858e-01 6.43503515784806e-05 5.71995606435394e-09
4 7.67653266201279e-01 3.74957960683442e-09 2.22044604925031e-16
5 7.67653266201279e-01 1.45556000556775e-16 0.00000000000000e+00
Modify the above Newton method implemention to use while loop.