[2022 Fall] Assignment4-6¶

Course: AP3021

4-6-1¶

If R = 3 m, what depth must the tank be filled to so that it holds 30 𝑚3?

4-6-2¶

Newton-Raphson method (3 iterations; determine relative error after each iterations)(Python)

In [1]:

import math

import math

In [2]:

def f(h, R, V) :
    PI = math.pi
    ans = PI * (h ** 2) * (((3 * R) - h) / 3) - V

    return ans

def f(h, R, V) : PI = math.pi ans = PI * (h ** 2) * (((3 * R) - h) / 3) - V return ans

In [3]:

def f_prime(h, R) :
    PI = math.pi
    ans = (2 * PI * h * R) - (PI * (h ** 2))

    return ans

def f_prime(h, R) : PI = math.pi ans = (2 * PI * h * R) - (PI * (h ** 2)) return ans

In [4]:

def newton_raphson(x0, es, iter_max, R, V) :
    iter_count = 0
    iter_count_list = []
    x_root = x0
    print("x0 =", x0)
    print()

    while True :
        last_x_root = x_root

        try :
            x_root = last_x_root - (f(x_root, R, V) / f_prime(x_root, R))
        except :
            print("total use", iter_count, "times.")
            return "Divergence"

        iter_count += 1
        iter_count_list.append(iter_count)
        if x_root != 0 :
            ea = abs((x_root - last_x_root) / x_root) * 100        

        print("iter time:", iter_count, ",ea =", ea, "root:", x_root)

        if (ea < es or iter_count >= iter_max) :
            print("total use", iter_count, "times.")
            return x_root

def newton_raphson(x0, es, iter_max, R, V) : iter_count = 0 iter_count_list = [] x_root = x0 print("x0 =", x0) print() while True : last_x_root = x_root try : x_root = last_x_root - (f(x_root, R, V) / f_prime(x_root, R)) except : print("total use", iter_count, "times.") return "Divergence" iter_count += 1 iter_count_list.append(iter_count) if x_root != 0 : ea = abs((x_root - last_x_root) / x_root) * 100 print("iter time:", iter_count, ",ea =", ea, "root:", x_root) if (ea < es or iter_count >= iter_max) : print("total use", iter_count, "times.") return x_root

In [5]:

R = 3
V = 30
x0 = R
es = 0.01
iter_max = 3

print("\nThe constrains of h:", newton_raphson(x0, es, iter_max, R, V))

R = 3 V = 30 x0 = R es = 0.01 iter_max = 3 print("\nThe constrains of h:", newton_raphson(x0, es, iter_max, R, V))

x0 = 3

iter time: 1 ,ea = 45.55808019742351 root: 2.061032953945969
iter time: 2 ,ea = 1.6768713794196717 root: 2.0270420656974903
iter time: 3 ,ea = 0.0067262694875214685 root: 2.026905730555795
total use 3 times.

The constrains of h: 2.026905730555795

4-6-3¶

What are the constraints of h?

In [6]:

print("\nThe constrains of h:", newton_raphson(x0, es, iter_max, R, V))

print("\nThe constrains of h:", newton_raphson(x0, es, iter_max, R, V))

x0 = 3

iter time: 1 ,ea = 45.55808019742351 root: 2.061032953945969
iter time: 2 ,ea = 1.6768713794196717 root: 2.0270420656974903
iter time: 3 ,ea = 0.0067262694875214685 root: 2.026905730555795
total use 3 times.

The constrains of h: 2.026905730555795

In [ ]:

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Last update: 2024-04-27