Homework 1

passed 100.00 out of 100.00

homework1.py

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+# Homework 1 for Principles of Computing class, by k., 06/24/2014
+
+import math
+import pylab
+#import simpleplot
+#import codeskulptor
+#codeskulptor.set_timeout(20)
+
+
+# Question 1
+
+'''
+Simulator for resource generation with upgrades
+'''
+
+def resources_vs_time(upgrade_cost_increment, num_upgrade):
+    '''
+    build function that performs unit upgrades with specified cost increments;
+    return a list whose length is num_upgrades and whose entries are pairs of the form:
+    [current_time, total_resources_generated]
+    '''
+    resources = []
+    total_resources_generated = 0.0
+    current_time = 0.0
+    current_resources = 1.0
+    growth = 1.0
+
+    for item in xrange(num_upgrade):
+        current_time += current_resources / growth # incremental time to achieve upgrade
+        total_resources_generated += growth * (current_resources / growth)
+        current_resources += upgrade_cost_increment
+        growth += 1
+        resources.append([current_time, total_resources_generated])
+
+    return resources
+
+# sample output from the print statements for data1 and data2
+#[[1.0, 1], [1.75, 2.5], [2.41666666667, 4.5], [3.04166666667, 7.0], [3.64166666667, 10.0], [4.225, 13.5], [4.79642857143, 17.5], [5.35892857143, 22.0], [5.91448412698, 27.0], [6.46448412698, 32.5], [7.00993867244, 38.5], [7.55160533911, 45.0], [8.09006687757, 52.0], [8.62578116328, 59.5], [9.15911449661, 67.5], [9.69036449661, 76.0], [10.2197762613, 85.0], [10.7475540391, 94.5], [11.2738698286, 104.5], [11.7988698286, 115.0]]
+#[[1.0, 1], [2.25, 3.5], [3.58333333333, 7.5], [4.95833333333, 13.0], [6.35833333333, 20.0], [7.775, 28.5], [9.20357142857, 38.5], [10.6410714286, 50.0], [12.085515873, 63.0], [13.535515873, 77.5]]
+
+# quick test
+#print resources_vs_time(0.5, 20)
+#print resources_vs_time(1.5, 10)
+
+# results for question
+#print resources_vs_time(0.0, 10)
+#print
+#print resources_vs_time(1.0, 10)
+
+def plot_it():
+    '''
+    helper function to gain insight on provided data sets background,
+    using pylab
+    '''
+    data1 = [[1.0, 1], [2.25, 3.5], [3.58333333333, 7.5], [4.95833333333, 13.0], [6.35833333333, 20.0], [7.775, 28.5], [9.20357142857, 38.5], [10.6410714286, 50.0], [12.085515873, 63.0], [13.535515873, 77.5]]
+    data2 = [[1.0, 1], [1.75, 2.5], [2.41666666667, 4.5], [3.04166666667, 7.0], [3.64166666667, 10.0], [4.225, 13.5], [4.79642857143, 17.5], [5.35892857143, 22.0], [5.91448412698, 27.0], [6.46448412698, 32.5], [7.00993867244, 38.5], [7.55160533911, 45.0], [8.09006687757, 52.0], [8.62578116328, 59.5], [9.15911449661, 67.5], [9.69036449661, 76.0], [10.2197762613, 85.0], [10.7475540391, 94.5], [11.2738698286, 104.5], [11.7988698286, 115.0]]
+    time1 = [item[0] for item in data1]
+    resource1 = [item[1] for item in data1]
+    time2 = [item[0] for item in data2]
+    resource2 = [item[1] for item in data2]
+
+    # plot in pylab (total resources over time)
+    pylab.plot(time1, resource1, 'o')
+    pylab.plot(time2, resource2, 'o')
+    pylab.title('Silly Homework')
+    pylab.legend(('Data Set no.1', 'Data Set no.2'))
+    pylab.xlabel('Current Time')
+    pylab.ylabel('Total Resources Generated')
+    pylab.show()
+
+#plot_it()    
+
+def test():
+    '''
+    testing code for resources_vs_time,
+    '''
+    data1 = resources_vs_time(0.5, 20)
+    data2 = resources_vs_time(1.5, 10)
+    print data1
+    print data2
+    simpleplot.plot_lines("Growth", 600, 600, "time", "total resources", [data1, data2])
+
+# using silly simpleplot from the class, don't bother!
+#test()
+
+
+# Question 2
+
+def plot_question2():
+    '''
+    graph of total resources generated as a function of time,
+    for four various upgrade_cost_increment values
+    '''
+    for upgrade_cost_increment in [0.0, 0.5, 1.0, 2.0]:
+        data = resources_vs_time(upgrade_cost_increment, 5)
+        time = [item[0] for item in data]
+        resource = [item[1] for item in data]
+
+        # plot in pylab (total resources over time for each constant)
+        pylab.plot(time, resource, 'o')
+
+    pylab.title('Silly Homework')
+    pylab.legend(('0.0', '0.5', '1.0', '2.0'))
+    pylab.xlabel('Current Time')
+    pylab.ylabel('Total Resources Generated')
+    pylab.show()
+
+#plot_question2()   
+
+
+# Question 3
+
+def plot_question3():
+    '''
+    graph of total resources generated as a function of time;
+    for upgrade_cost_increment == 0
+    '''
+    data = resources_vs_time(0.0, 100)
+    time = [item[0] for item in data]
+    resource = [item[1] for item in data]
+
+    # plot in pylab on logarithmic scale (total resources over time for upgrade growth 0.0)
+    pylab.loglog(time, resource)
+
+    pylab.title('Silly Homework')
+    pylab.legend('0.0')
+    pylab.xlabel('Current Time')
+    pylab.ylabel('Total Resources Generated')
+    pylab.show()
+
+#plot_question3()
+
+
+# Question 4
+
+def arithmetic_sum_model():
+    '''
+    find what arithmetic sum models the value of current_time after n upgrades,
+    for upgrade_cost_increment == 0
+    '''
+    print 'Incremental time to achieve upgrade / Incremental resources'
+    data = resources_vs_time(0.0, 10)
+    time = [item[0] for item in data]
+    resource = [item[1] for item in data]
+    for index in xrange(len(time) - 1):
+        delta1 = time[index + 1] - time[index]
+        delta2 = resource[index + 1] - resource[index]
+        print delta1, '\t', delta2
+
+#arithmetic_sum_model()
+
+
+# Question 5
+
+def total_sum_model():
+    '''
+    find what function f(n) models the total value of the sum most accurately,
+    for upgrade_cost_increment == 0
+    '''
+    num_upgrade = 1000.0
+    deltas = []
+
+    data = resources_vs_time(0.0, int(num_upgrade))
+    resource = [item[0] for item in data]
+    for index in xrange(len(resource) - 1):
+        deltas.append(resource[index + 1] - resource[index])
+
+    print 'n number of upgrades: ', num_upgrade
+    print 'Note that number in question will differ by a small constant as grows large.'
+    print 'The total value of the sum: ', sum(deltas)
+    print 'Modeled value (1/2n(n+1)) of the sum: ', 1.0 / 2 * num_upgrade * (num_upgrade + 1.0)
+    print 'Modeled value (n) of the sum: ', num_upgrade
+    print 'Modeled value (2) of the sum: ', 2.0
+    print 'Modeled value (log(n)) of the sum: ', math.log10(num_upgrade)
+
+#total_sum_model()
+
+
+# Question 6
+
+def time_upgrades_relationship_0():
+    '''
+    helper function to show relationship between time and number of upgrades,
+    for upgrade_cost_increment == 0.0
+    '''
+    print 'Incremental time to achieve upgrade'
+    data = resources_vs_time(0.0, 11)
+    time = [item[0] for item in data]
+    resource = [item[1] for item in data]
+    for index in xrange(len(time) - 1):
+        delta1 = time[index + 1] - time[index]
+        delta2 = resource[index + 1] - resource[index]
+        print delta1, '\t\t', delta2
+    print '\nsum is called a harmonic sum and has only an approximate solution: log(t) + constant;'
+    print 'question 6 asks for "...we seek the inverse function g for f..." thus e**(t) in form E^t'
+
+#time_upgrades_relationship_0()
+
+
+# Question 7
+
+def polyfitting():
+    '''
+    helper function to play around with polyfit from:
+    http://www.wired.com/2011/01/linear-regression-with-pylab/
+    '''
+    x = [0.2, 1.3, 2.1, 2.9, 3.3]
+    y = [3.3, 3.9, 4.8, 5.5, 6.9]
+    slope, intercept = pylab.polyfit(x, y, 1)
+    print 'slope:', slope, 'intercept:', intercept
+
+    yp = pylab.polyval([slope, intercept], x)
+    pylab.plot(x, yp)
+    pylab.scatter(x, y)
+    pylab.show()
+
+#polyfitting()
+
+def fitting_easy():
+    '''
+    helper function to check slot, the slope is 4.0
+    '''
+    x = [0.0, 7.0, 14.0]
+    y = [0.0, 28.0, 56.0]
+    print round((y[1] - y[0]) / (x[1] - x[0]))
+
+#fitting_easy()
+
+def plot_question7():
+    '''
+    graph of total resources generated as a function of time,
+    for upgrade_cost_increment == 1
+    '''
+    data = resources_vs_time(1.0, 50)
+    time = [item[0] for item in data]
+    resource = [item[1] for item in data]
+    a, b, c = pylab.polyfit(time, resource, 2)
+    print 'polyfit with argument \'2\' fits the data, thus the degree of the polynomial is 2 (quadratic)'
+
+    # plot in pylab on logarithmic scale (total resources over time for upgrade growth 0.0)
+    #pylab.loglog(time, resource, 'o')
+
+    # plot fitting function
+    yp = pylab.polyval([a, b, c], time)
+    pylab.plot(time, yp)
+    pylab.scatter(time, resource)
+    pylab.title('Silly Homework, Question 7')
+    pylab.legend(('Resources for increment 1', 'Fitting function' + ', slope: ' + str(a)))
+    pylab.xlabel('Current Time')
+    pylab.ylabel('Total Resources Generated')
+    pylab.grid()
+    pylab.show()
+
+plot_question7()
+
+
+# Question 8
+
+def arithmetic_sum_model_resources():
+    '''
+    find what arithmetic sum models the value of current_time after n upgrades,
+    for upgrade_cost_increment == 1.0
+    '''
+    print 'Incremental time to achieve upgrade / Incremental resources'
+    data = resources_vs_time(1.0, 10)
+    time = [item[0] for item in data]
+    resource = [item[1] for item in data]
+    for index in xrange(len(time) - 1):
+        delta1 = time[index + 1] - time[index]
+        delta2 = resource[index + 1] - resource[index]
+        print delta1, '\t', delta2
+
+#arithmetic_sum_model_resources()
+
+
+# Question 9
+
+def time_upgrades_relationship():
+    '''
+    helper function to show relationship between time and number of upgrades,
+    for upgrade_cost_increment == 1.0
+    '''
+    print 'Time unit / Incremental resources'
+    data = resources_vs_time(1.0, 10)
+    time = [item[0] for item in data]
+    resource = [item[1] for item in data]
+    for index in xrange(len(time) - 1):
+        delta = resource[index + 1] - resource[index]
+        print time[index], '\t', delta
+    print 'sum is known as a triangular sum, 1/2(n + 1)n; for t it\'s 1/2(t + 1)t'    
+
+#time_upgrades_relationship()
+
+
+# Question 10
+
+def upgrade_cost(number):
+    '''
+    helper function for expression that models the cost of the nth upgrade
+    '''
+    accumulator = []
+    for number in xrange(1, number + 1):
+        cost = 1.15 ** (number - 1)
+        accumulator.append([number, cost])
+    return accumulator
+
+#print upgrade_cost(10)
+
+
+# Question 11
+
+def plot_question11():
+    '''
+    graph of total resources generated as a function of time;
+    for upgrade_cost_increment == 1
+    '''
+    num_upgrade = 10.0
+    # function from question 1
+    data1 = resources_vs_time(1.0, int(num_upgrade))
+    time1 = [item[0] for item in data1]
+    resource1 = [item[1] for item in data1]
+    # function from question 10
+    data2 = upgrade_cost(int(num_upgrade))
+    time2 = [item[0] for item in data2]
+    resource2 = [item[1] for item in data2]
+
+    # plot in pylab (to see how fast both functions grow)
+    pylab.plot(time1, resource1)
+    pylab.plot(time2, resource2)
+
+    pylab.title('Silly Homework')
+    pylab.legend(('quadratic function of time', 'upgrade costs 15% more'))
+    pylab.xlabel('Current Time')
+    pylab.ylabel('Total Resources Generated')
+    pylab.show()
+
+#plot_question11()