## NEWTON RAPHSON METHOD

OBJECTIVE: To find minimum pressure using Newton-Raphson method

PROGRAMMING EXPLAINED:

1. Two functions are defined. f and fprime. These function are defined to calculate next iteration value using NR method.
2. To calculate pressure values for different thickness, for and while loop was used.In while loop absolute value of function, for given thickness was compared with tolerance 0.0001
3. .
4. Pressure and iterations values was appended when function value was less than tolerance.
5. Imported PrettyTable and used for loop for adding values to table.
6. For second part of challenge,linspace was used for gettting different values of relaxation factor. Thickness of ice was fixed to 0.6 units.
7. Newton raphson method was used to calculate pressure.
8. Here only iterations values was appended.
9. relaxation factor was plotted against iterations values to find best realaxation factor.

#NEWTON RAPHSON METHOD
import math
import matplotlib.pyplot as plt
from prettytable import PrettyTable
import numpy as np

#INPUTS
r = 40
B = 0.5
sigma = 150*144
h = [0.6,1.2,1.8,2.4,3,3.6,4.2]
#Defining function to evaluate pressure
def f(p,h):
t1 = pow(p,3)*(1-pow(B,2))
t2 = (0.4*h*(B**2)-(sigma*(h**2))/(r**2))*(p**2)
t3 = ((sigma**2)*(h**4)*p)/(3*(r**4))
t4 = pow((sigma*(h**2))/(3*(r**2)),3)
F = t1 + t2 + t3 -t4
return F

#Differentiating first fumction for iteration in while loop
def fprime(p,h):
t_1 = 3*pow(p,2)*(1-pow(B,2))
t_2 = (0.4*h*(B**2)-(sigma*(h**2))/(r**2))*(2*p)
t_3 = ((sigma**2)*(h**4))/(3*(r**4))
Fp = t_1 + t_2 + t_3
return Fp

P = []
it = []
xp = 100
tol = 1e-4
alpha = 1
for a in h:
i = 1

while(abs(f(xp,a))>tol):
xp = xp - (alpha*(f(xp,a)/fprime(xp,a)))

i = i+1
it.append(i)
P.append(xp)

table = PrettyTable(['iteration','thickness','pressure'])
for x in range(0,7):
print(table)

plt.plot(h,P)
plt.xlabel('thickness')
plt.ylabel('Pressure')
plt.show()
alpha__= np.linspace(0.1,1.9,500)
#print(alpha__)
h = 0.6
tol = 1e-4
iterations = []

for j in alpha__:
i = 1
xp = 100

while(abs(f(xp,0.6))>tol):
xp = xp - (j*(f(xp,0.6)/fprime(xp,0.6)))

i = i+1
iterations.append(i)

print(iterations)
print('the number of iterations ('+str(min(iterations))+') is least at: ')
for i in range(len(iterations)):

if iterations [i]==min(iterations):
print(alpha__[i])

plt.plot(alpha__,iterations)
plt.xlabel('aplha')
plt.ylabel('iterations')
plt.title('plot showing relaxation factor')

plt.show()



CONCLUSION:

Optimum relaxation factor was around 1.1 and we can see from results it takes less iterations to get the answer.

If we use relaxation factor less than 1, it will take more iterations to converge to answer.

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