Implementation of Newton-Raphson method (#650)

Implemented Newton-Raphson method using pure python. Third party library is used only for visualizing error variation with each iteration.

maths/newton_raphson.py

@@ -0,0 +1,50 @@
+'''
+    Author: P Shreyas Shetty
+    Implementation of Newton-Raphson method for solving equations of kind
+    f(x) = 0. It is an iterative method where solution is found by the expression
+        x[n+1] = x[n] + f(x[n])/f'(x[n])
+    If no solution exists, then either the solution will not be found when iteration
+    limit is reached or the gradient f'(x[n]) approaches zero. In both cases, exception
+    is raised. If iteration limit is reached, try increasing maxiter.
+    '''
+
+import math as m
+
+def calc_derivative(f, a, h=0.001):
+    '''
+     Calculates derivative at point a for function f using finite difference
+     method
+    '''
+    return (f(a+h)-f(a-h))/(2*h)
+
+def newton_raphson(f, x0=0, maxiter=100, step=0.0001, maxerror=1e-6,logsteps=False):
+    
+    a = x0 #set the initial guess
+    steps = [a]
+    error = abs(f(a))
+    f1 = lambda x:calc_derivative(f, x, h=step) #Derivative of f(x)
+    for _ in range(maxiter):
+        if f1(a) == 0:
+            raise ValueError("No converging solution found")
+        a = a - f(a)/f1(a) #Calculate the next estimate
+        if logsteps:
+            steps.append(a)
+        error = abs(f(a))
+        if error < maxerror:
+            break
+    else:
+        raise ValueError("Itheration limit reached, no converging solution found")
+    if logsteps:
+        #If logstep is true, then log intermediate steps
+        return a, error, steps
+    return a, error
+ 
+if __name__ == '__main__':
+    import matplotlib.pyplot as plt
+    f = lambda x:m.tanh(x)**2-m.exp(3*x)
+    solution, error, steps = newton_raphson(f, x0=10, maxiter=1000, step=1e-6, logsteps=True)
+    plt.plot([abs(f(x)) for x in steps])
+    plt.xlabel("step")
+    plt.ylabel("error")
+    plt.show()
+    print("solution = {%f}, error = {%f}" % (solution, error))