[2022 Fall] Assignment1¶

Course: AP3021

In [1]:

import math

import math

Build the factorial mechanism

In [2]:

def factorial(n) :
    sum = 1

    for i in range(2, n + 1) :
        sum *= i

    return sum

def factorial(n) : sum = 1 for i in range(2, n + 1) : sum *= i return sum

Build the error criterion : εs

In [3]:

def countErrorCriterion(n) :
    errorCriterion = (0.5 * 10 ** (2 - n)) * 100

    return errorCriterion

def countErrorCriterion(n) : errorCriterion = (0.5 * 10 ** (2 - n)) * 100 return errorCriterion

Build the approximate estimate of the error : εa

In [4]:

def countApproximateEstimateError(currentApproximation, previousApproximation) :
    approximateError = currentApproximation - previousApproximation

    if currentApproximation == 0 :    # The first and second value of cosine tylor polynomial is 0.0 and 1.0; therefore we have to pass them.
        return 999
    else :
        approximateEstimateError = abs((approximateError / currentApproximation) * 100)

    return approximateEstimateError

def countApproximateEstimateError(currentApproximation, previousApproximation) : approximateError = currentApproximation - previousApproximation if currentApproximation == 0 : # The first and second value of cosine tylor polynomial is 0.0 and 1.0; therefore we have to pass them. return 999 else : approximateEstimateError = abs((approximateError / currentApproximation) * 100) return approximateEstimateError

Build the percent relative error : εt

In [5]:

def countPercentRelativeError(trueValue, approximation) :
    trueError = trueValue - approximation
    percentRelativeError = (trueError / trueValue) * 100

    return percentRelativeError

def countPercentRelativeError(trueValue, approximation) : trueError = trueValue - approximation percentRelativeError = (trueError / trueValue) * 100 return percentRelativeError

Build cosine tylor polynomial

In [6]:

def tylorPolynomial(n, x) :
    sum = 0

    for i in range(n) : 
        sum = sum + ((-1) ** i * x ** (2 * i)) / factorial(2 * i)

    return float(sum)

def tylorPolynomial(n, x) : sum = 0 for i in range(n) : sum = sum + ((-1) ** i * x ** (2 * i)) / factorial(2 * i) return float(sum)

Announce the variable.

In [7]:

def main():

    PI = math.pi
    n = 5
    errorCriterion = countErrorCriterion(n)
    loopCount = 0
    currentApproximate = 0.0
    previousApproximate = 0.0
    trueValue = math.cos(2 * PI)
    print(f"True value of cos(0): {trueValue}.")

    # print((1.00000000000000 - 0.999978232974615 / 1.00000000000000) * 100)
    print(f"The error criterion is: {errorCriterion}, and n = {n}.\n\nStart the estimate: \n")

    while True :
        previousApproximate = currentApproximate
        currentValue = tylorPolynomial(loopCount, 2 * PI)
        currentApproximate = currentValue
        approximateEstimateError = countApproximateEstimateError(currentApproximate, previousApproximate)
        percentRelativeError = countPercentRelativeError(trueValue, currentValue)

        if (approximateEstimateError < errorCriterion) :
            loopCount += 1

            print(f"Times: {loopCount}, \ncurrent value: {currentValue}, \tapproximate estimate error: {approximateEstimateError} \tpercent relative error: {percentRelativeError}%.\n")
            print(f"Here is the error that we accept!!!")
            print(f"Total use {loopCount} times to get the result we want!")

            break
        else :
            loopCount += 1

        print(f"Times: {loopCount}, \ncurrent value: {currentValue}, \tapproximate estimate error: {approximateEstimateError} \tpercent relative error: {percentRelativeError}%.")

def main(): PI = math.pi n = 5 errorCriterion = countErrorCriterion(n) loopCount = 0 currentApproximate = 0.0 previousApproximate = 0.0 trueValue = math.cos(2 * PI) print(f"True value of cos(0): {trueValue}.") # print((1.00000000000000 - 0.999978232974615 / 1.00000000000000) * 100) print(f"The error criterion is: {errorCriterion}, and n = {n}.\n\nStart the estimate: \n") while True : previousApproximate = currentApproximate currentValue = tylorPolynomial(loopCount, 2 * PI) currentApproximate = currentValue approximateEstimateError = countApproximateEstimateError(currentApproximate, previousApproximate) percentRelativeError = countPercentRelativeError(trueValue, currentValue) if (approximateEstimateError < errorCriterion) : loopCount += 1 print(f"Times: {loopCount}, \ncurrent value: {currentValue}, \tapproximate estimate error: {approximateEstimateError} \tpercent relative error: {percentRelativeError}%.\n") print(f"Here is the error that we accept!!!") print(f"Total use {loopCount} times to get the result we want!") break else : loopCount += 1 print(f"Times: {loopCount}, \ncurrent value: {currentValue}, \tapproximate estimate error: {approximateEstimateError} \tpercent relative error: {percentRelativeError}%.")

In [8]:

if __name__ == '__main__':
    main()

if __name__ == '__main__': main()

True value of cos(0): 1.0.
The error criterion is: 0.05, and n = 5.

Start the estimate: 

Times: 1, 
current value: 0.0, 	approximate estimate error: 999 	percent relative error: 100.0%.
Times: 2, 
current value: 1.0, 	approximate estimate error: 100.0 	percent relative error: 0.0%.
Times: 3, 
current value: -18.739208802178716, 	approximate estimate error: 105.33640459720868 	percent relative error: 1973.9208802178716%.
Times: 4, 
current value: 46.200185220489566, 	approximate estimate error: 140.560895400627 	percent relative error: -4520.018522048957%.
Times: 5, 
current value: -39.25663198620415, 	approximate estimate error: 217.6875928549489 	percent relative error: 4025.6631986204147%.
Times: 6, 
current value: 20.98800938567249, 	approximate estimate error: 287.0431409898395 	percent relative error: -1998.8009385672492%.
Times: 7, 
current value: -5.438247397701897, 	approximate estimate error: 485.9333320242407 	percent relative error: 643.8247397701897%.
Times: 8, 
current value: 2.465288973616568, 	approximate estimate error: 320.59269545687425 	percent relative error: -146.5288973616568%.
Times: 9, 
current value: 0.7508982625278968, 	approximate estimate error: 228.31198268020754 	percent relative error: 24.91017374721032%.
Times: 10, 
current value: 1.0329042309836878, 	approximate estimate error: 27.302237709610523 	percent relative error: -3.2904230983687777%.
Times: 11, 
current value: 0.9965213898411421, 	approximate estimate error: 3.650984465907502 	percent relative error: 0.34786101588578644%.
Times: 12, 
current value: 1.000301224041822, 	approximate estimate error: 0.3778695966608076 	percent relative error: -0.030122404182209017%.
Times: 13, 
current value: 0.999978232974615, 	approximate estimate error: 0.032299809791492 	percent relative error: 0.002176702538503328%.

Here is the error that we accept!!!
Total use 13 times to get the result we want!

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Last update: 2024-04-27