[2022 Fall] Assignment4-2¶

Course: AP3021

In [1]:

import matplotlib.pyplot as plt
import math
import numpy as np
import pandas as pd
import os

import matplotlib.pyplot as plt import math import numpy as np import pandas as pd import os

4-2-1¶

How many bisection iterations would be required to determine temperature to an absolute error of 0.05°C? (𝑥𝑙 = 0°C, 𝑥𝑢 = 40°C)

4-2-2¶

延續(1). Bisection program. (𝑂𝑠𝑓 = 8, 10, 𝑎𝑛𝑑 12𝑚𝑔/𝐿) (Python) PS : 溫度請帶絕對溫標

In [2]:

def f(temperature, osf) :
    absolute_temperature = temperature + 273.15 # Ta

    ans = ((-8.621949 * 10 ** 11) / absolute_temperature ** 4) \
        + ((1.243800  * 10 ** 10) / absolute_temperature ** 3) \
        + ((-6.642308 * 10 **  7) / absolute_temperature ** 2) \
        + ((1.575701  * 10 **  5) / absolute_temperature)      \
        - 139.34411 - math.log(osf)
    
    return ans

def f(temperature, osf) : absolute_temperature = temperature + 273.15 # Ta ans = ((-8.621949 * 10 ** 11) / absolute_temperature ** 4) \ + ((1.243800 * 10 ** 10) / absolute_temperature ** 3) \ + ((-6.642308 * 10 ** 7) / absolute_temperature ** 2) \ + ((1.575701 * 10 ** 5) / absolute_temperature) \ - 139.34411 - math.log(osf) return ans

In [3]:

def count_et(true_value, approximation) :
    true_error = true_value - approximation
    et = (true_error / true_value) * 100

    return et

def count_et(true_value, approximation) : true_error = true_value - approximation et = (true_error / true_value) * 100 return et

In [4]:

def count_ea(new_x_root, old_x_root) :

    if (old_x_root == -1) : # jump out the first data.
        return 9999
    else :
        ea = abs((new_x_root - old_x_root) / new_x_root)
        ea = ea * 100   # turn into percent
        
        return ea

def count_ea(new_x_root, old_x_root) : if (old_x_root == -1) : # jump out the first data. return 9999 else : ea = abs((new_x_root - old_x_root) / new_x_root) ea = ea * 100 # turn into percent return ea

In [5]:

def count_iter_times(x_lowwer, x_upper, Ead) :
    iter_times = math.log(((x_upper - x_lowwer) / Ead), 2)

    return iter_times

def count_iter_times(x_lowwer, x_upper, Ead) : iter_times = math.log(((x_upper - x_lowwer) / Ead), 2) return iter_times

In [6]:

def bisection(x_lowwer, x_upper, Ead, osf, iter_max, iter_count) :
    iter_times = count_iter_times(x_lowwer, x_upper, Ead)
    print("Iterator at least:", iter_times, "times.")
    x_root = -1
    
    while True :
        last_x_root = x_root
        x_root = (x_lowwer + x_upper) / 2
        iter_count += 1
        temp = f(x_lowwer, osf) * f(x_root, osf)
        # print(temp)

        if (temp < 0) :
            x_upper = x_root
            # print("here")
        elif(temp > 0) :
            x_lowwer = x_root
            # print("here2")
        else :
            return x_root

        # how to get the true_value?
        # true_value = 

        # et = (count_et(true_value, x_root))

        ea = count_ea(x_root, last_x_root)

        print("count:", iter_count, "root:", x_root, "ea", ea)
        # print(x_lowwer, x_upper)
        
        if iter_count >= iter_times or iter_count >= iter_max:
            temperature = x_root + 273.15
            
            print(f"if the os is {osf}\nI iterate {iter_count} times\nThe temperature is {temperature}K")
            return x_root

def bisection(x_lowwer, x_upper, Ead, osf, iter_max, iter_count) : iter_times = count_iter_times(x_lowwer, x_upper, Ead) print("Iterator at least:", iter_times, "times.") x_root = -1 while True : last_x_root = x_root x_root = (x_lowwer + x_upper) / 2 iter_count += 1 temp = f(x_lowwer, osf) * f(x_root, osf) # print(temp) if (temp < 0) : x_upper = x_root # print("here") elif(temp > 0) : x_lowwer = x_root # print("here2") else : return x_root # how to get the true_value? # true_value = # et = (count_et(true_value, x_root)) ea = count_ea(x_root, last_x_root) print("count:", iter_count, "root:", x_root, "ea", ea) # print(x_lowwer, x_upper) if iter_count >= iter_times or iter_count >= iter_max: temperature = x_root + 273.15 print(f"if the os is {osf}\nI iterate {iter_count} times\nThe temperature is {temperature}K") return x_root

Osf = 8¶

In [7]:

x_lowwer = 0
x_upper = 40
Ead = 0.05
osf = 8
iter_max = 500
iter_count = 0

ans = bisection(x_lowwer, x_upper, Ead, osf, iter_max, iter_count)

x_lowwer = 0 x_upper = 40 Ead = 0.05 osf = 8 iter_max = 500 iter_count = 0 ans = bisection(x_lowwer, x_upper, Ead, osf, iter_max, iter_count)

Iterator at least: 9.643856189774725 times.
count: 1 root: 20.0 ea 9999
count: 2 root: 30.0 ea 33.33333333333333
count: 3 root: 25.0 ea 20.0
count: 4 root: 27.5 ea 9.090909090909092
count: 5 root: 26.25 ea 4.761904761904762
count: 6 root: 26.875 ea 2.3255813953488373
count: 7 root: 26.5625 ea 1.1764705882352942
count: 8 root: 26.71875 ea 0.5847953216374269
count: 9 root: 26.796875 ea 0.2915451895043732
count: 10 root: 26.7578125 ea 0.145985401459854
if the os is 8
I iterate 10 times
The temperature is 299.9078125K

Osf = 10¶

In [8]:

x_lowwer = 0
x_upper = 40
Ead = 0.05
osf = 10
iter_max = 500
iter_count = 0

print("Osf:", osf)
ans = bisection(x_lowwer, x_upper, Ead, osf, iter_max, iter_count)
print(ans)

x_lowwer = 0 x_upper = 40 Ead = 0.05 osf = 10 iter_max = 500 iter_count = 0 print("Osf:", osf) ans = bisection(x_lowwer, x_upper, Ead, osf, iter_max, iter_count) print(ans)

Osf: 10
Iterator at least: 9.643856189774725 times.
count: 1 root: 20.0 ea 9999
count: 2 root: 10.0 ea 100.0
count: 3 root: 15.0 ea 33.33333333333333
count: 4 root: 17.5 ea 14.285714285714285
count: 5 root: 16.25 ea 7.6923076923076925
count: 6 root: 15.625 ea 4.0
count: 7 root: 15.3125 ea 2.0408163265306123
count: 8 root: 15.46875 ea 1.0101010101010102
count: 9 root: 15.390625 ea 0.5076142131979695
count: 10 root: 15.3515625 ea 0.2544529262086514
if the os is 10
I iterate 10 times
The temperature is 288.5015625K
15.3515625

Osf = 12¶

In [9]:

x_lowwer = 0
x_upper = 40
Ead = 0.05
osf = 12
iter_max = 500
iter_count = 0

print("Osf:", osf)
ans = bisection(x_lowwer, x_upper, Ead, osf, iter_max, iter_count)
print(ans)

x_lowwer = 0 x_upper = 40 Ead = 0.05 osf = 12 iter_max = 500 iter_count = 0 print("Osf:", osf) ans = bisection(x_lowwer, x_upper, Ead, osf, iter_max, iter_count) print(ans)

Osf: 12
Iterator at least: 9.643856189774725 times.
count: 1 root: 20.0 ea 9999
count: 2 root: 10.0 ea 100.0
count: 3 root: 5.0 ea 100.0
count: 4 root: 7.5 ea 33.33333333333333
count: 5 root: 6.25 ea 20.0
count: 6 root: 6.875 ea 9.090909090909092
count: 7 root: 7.1875 ea 4.3478260869565215
count: 8 root: 7.34375 ea 2.127659574468085
count: 9 root: 7.421875 ea 1.0526315789473684
count: 10 root: 7.4609375 ea 0.5235602094240838
if the os is 12
I iterate 10 times
The temperature is 280.6109375K
7.4609375

In [ ]:

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Last update: 2024-04-27