Question

***Please code in Python

Write another code Newton (in double precision) implementing the Newton-Raphson Method
  (copy your Bisect code and modify).
  Evaluation of F(x) and F'(x) should be done in a subprogram FCN(x).
  The code should ask for input of: x0, TOL, maxIT (and should print output similar to Bisect code).
  Debug on a simple problem, like x²−3 = 0.
  Then use it to find root of F(x) in [1,2] with TOL=1.e-12.

Now consider the problem of finding zeros of
      G(x) = x−tan(x)   near x=99 (radians).

Are there any ? How many ? How do you know ?

Use your Newton code to find the zero of G(x) closest to x = 99 (radians) to 9 decimals ( use TOL=10⁻⁹,maxIT=20 ).
  Output your final approximate root, the residual, and how many iterations it took.

THIS IS MY CURRENT CODE FOR BISECTION:

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import fsolve
from scipy.misc import derivative

def func(x):
return x*x*x+2*x*x+10*x-20;
def bisection(a,b,tol,maxit):
  
if (func(a) * func(b) >= 0):
print("You have not assumed right a and b\n")
return

c = a
ct=0;
while ((b-a) >= tol):
ct=ct+1;
# Find middle point
c = (a+b)/2

# Check if middle point is root
if (func(c) == 0.0):
break

# Decide the side to repeat the steps
if (func(c)*func(a) < 0):
b = c
else:
a = c
print("%d"%ct,end='');
print("\t%f"%c,end='');
print("\t%e"%func(c),end='');
print("\t%e"%(b-a),end='\n');
if(ct==maxit):
break;
print("root=%f"%c,end="");
print(", residual=%e"%(b-a),end="");
print(", in%d"%ct,end="");
print(" iters",end="\n");
  
  
  
# Driver code
# Initial values assumed
a =float(input("Enter a: "))
b = float(input("Enter b: "));
tol=float(input("Enter tol: "));
maxit=int(input("Enter maxit: "));
bisection(a, b,tol,maxit)

Answer 1

def func( x ):

    return x * x * x - x * x + 2

  

# Derivative of the above function

# which is 3*x^x - 2*x

def derivFunc( x ):

    return 3 * x * x - 2 * x

  

# Function to find the root

def newtonRaphson( x ):

    h = func(x) / derivFunc(x)

    while abs(h) >= 0.0001:

        h = func(x)/derivFunc(x)

          

        # x(i+1) = x(i) - f(x) / f'(x)

        x = x - h

      

    print("The value of the root is : ",

                             "%.4f"% x)

  

# Driver program to test above

x0 = -20 # Initial values assumed

newtonRaphson(x0)