CoCalc Public FilesLab 03 / Lab03-turnin.sagews
Author: Michael Le
Views : 114
# Lab 3: Programming Euler's Method
# Name: Michael Le
# Section: 1K
# I worked on this code with: Calvin Tong
# This code is identical to that turned in by: Calvin Tong

# PLEASE MAKE SURE TO COMMENT ON YOUR CODE ENOUGH TO EXPLAIN WHAT YOU'RE DOING AND WHY, PARTICULARLY FOR COMPLICATED PIECES!


#Exercise1
#Let N = 10 + at where 'Nprime' is equal to N' and 't' is time.
#The 10 comes from the fact that the initial solution is 10 [n(0)=10 is given]
#If N(0)=10, then N' = 0.2(10)(1-(10)/100)= 1.8
#Let Nnew denote the value after one iteration of Euler's, N = Nold or Ninitial
Nprime(n) = 0.2*n*(1-n/100)
Nprime(10)
N(t) = 10 + Nprime(n)*t
N(0)


1.80000000000000 10.0000000000000
#Stepsize is given at 0.1
#Nnew = Nold + Nprime * stepsize
#Nprime is evaluated at 10 because the differential equation is a function of N not of time like N(t) is.
#N(t) is evaluated as N(t-0.1) because our old function is 0.1 time units behind the new function.
Nnew(t) = N(t-0.1) + Nprime(10) * t
Nnew(0.1)

10.1800000000000
#Nprime(n) = 0.2*n*(1-n/100)
#N(t) = 10 + Nprime(n)*t
#Nnew(t) = N(t-0.1) + Nprime(10) * 0.1
#Nnew(0.1)
#Exercise 2
vals=[10] #We set the initial value to N = 10
numIter=1000 #Iterate Euler's Method 1000 times
for i in srange(0,numIter): #assigning the variable i from 0 to 1000
vals.append(vals[i] + 0.1 * Nprime(vals[i])) #appending additional values into list named vals; instead of Xnew = Xold + deltaX * step size, we rewrote the function according to the previous value... aka vals[i] <- Xold + 0.1 <--- Step Size * Nprime(vals[i])) <-- Nprime(x(old))
vals[1000] #Calls the 1000th number (not the 100th because our time values are 0.1


99.9999984129133
#Exercise3
vals=[10]
numIter=1000
for i in srange(0,numIter):
vals.append(vals[i] + 0.1 * Nprime(vals[i]))
list_plot(vals, size=1, color="red") #The graph models that of a logistic growth curve; As N approaches 100, the value from Nprime (1-N/100) approaches 0. Therefore as N increases, the change of our function becomes minimal and this is reflected in the plot's plateau.

#Exercise4
vals=[30]
numIter=1000
for i in srange(0,numIter):
vals.append(vals[i] + 0.1 * Nprime(vals[i]))
list_plot(vals, size=1, color="red")
vals=[70]
numIter=1000
for i in srange(0,numIter):
vals.append(vals[i] + 0.1 * Nprime(vals[i]))
list_plot(vals, size=1, color="blue")
vals=[101]
numIter=1000
for i in srange(0,numIter):
vals.append(vals[i] + 0.1 * Nprime(vals[i]))
list_plot(vals, size=1, color="green")
#Adjusting the intial values from 10 to 30 or 70 just raised the y-intercept value. The function reached 100 in less time as a result. Once N hits 100 the value for (1-N/100) becomes 0 and the graphs plateau. As for the 3rd graph where N(0) > K, we saw the graph decrease gradually to approach 100. The same phenomenon occured where as N hit 100, the value for (1-N/100) becomes 0 and the graph plateaus.

#Exercise5
time=[t for t in srange(0,100.05,.1)] #Creates a list for time from 0 to 100 in increments of 0.1 (to match the 1000 values in our list for the y-values)
vals=[10]
numIter=1000
for i in srange(0,numIter):
vals.append(vals[i] + 0.1 * Nprime(vals[i]))
list_plot(zip(time,vals), size=1) #Zips the list vals and time, combining time and vals into order pairs to be plotted
#Without zipping the two lists together, the graph will just plot in terms of N again. Zipping it plots N as a functino of t.

#Exercise6
def Eulersvalue(Nprimegen, Ninitial, StepSize, NumSteps): #Define a function named Eulers with arguments Nprimegen, Ninitial, StepSize, NumSteps
universal =[Ninitial] #Sets universal as the list beginning with the value Ninitial
numIter=NumSteps #Iterate the function however many steps are given
for i in srange(0,numIter): #Assign i to the list from 0 to numIter
universal.append(universal[i] + StepSize * Nprimegen(universal[i])) #Append the values from Xnew=Xold + DeltaX * StepSize into the list universal
return universal #Return function

Eulersvalue(Nprime, 10, 0.1, 1000) #Try the generic script with our previous function and the same list is generated

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99.9999824330009, 99.9999827843408, 99.9999831286539, 99.9999834660808, 99.9999837967591, 99.9999841208239, 99.9999844384074, 99.9999847496392, 99.9999850546463, 99.9999853535534, 99.9999856464822, 99.9999859335526, 99.9999862148815, 99.9999864905838, 99.9999867607721, 99.9999870255566, 99.9999872850454, 99.9999875393445, 99.9999877885575, 99.9999880327864, 99.9999882721306, 99.9999885066880, 99.9999887365542, 99.9999889618231, 99.9999891825866, 99.9999893989348, 99.9999896109561, 99.9999898187370, 99.9999900223622, 99.9999902219149, 99.9999904174766, 99.9999906091271, 99.9999907969445, 99.9999909810056, 99.9999911613855, 99.9999913381577, 99.9999915113946, 99.9999916811666, 99.9999918475433, 99.9999920105924, 99.9999921703806, 99.9999923269729, 99.9999924804335, 99.9999926308248, 99.9999927782083, 99.9999929226441, 99.9999930641912, 99.9999932029074, 99.9999933388492, 99.9999934720722, 99.9999936026308, 99.9999937305782, 99.9999938559666, 99.9999939788472, 99.9999940992703, 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99.9999980964426, 99.9999981345138, 99.9999981718235, 99.9999982083870, 99.9999982442193, 99.9999982793349, 99.9999983137482, 99.9999983474732, 99.9999983805238, 99.9999984129133]
#Exercise7
def Eulers(Nprimegen, Ninitial, StepSize, NumSteps):
universal =[Ninitial]
numIter=NumSteps
for i in srange(0,numIter):
universal.append(universal[i] + StepSize * Nprimegen(universal[i]))
time = [t for t in srange(0,NumSteps*StepSize + StepSize*0.5,StepSize)] #Creates a list of time that starts with 0 and goes up to the number of steps multiplied by the step size (remember we had to multiply 1000 by 0.1 to get 100 time units) and then adds the step size * 1/2 to account for the fact that srange doesn't include the last value. The numbers are counted by a generic stepsize.
Pairs = zip(time,universal) #Pairs the time and universal lists as an ordered pair
return Pairs #Returns the ordered pairs
Eulers(Nprime, 10, 0.1, 1000) #Test Run


[(0.000000000000000, 10), (0.100000000000000, 10.1800000000000), (0.200000000000000, 10.3628735200000), (0.300000000000000, 10.5486531608817), (0.400000000000000, 10.7373714073976), (0.500000000000000, 10.9290606065975), (0.600000000000000, 11.1237529455809), (0.700000000000000, 11.3214804285737), (0.800000000000000, 11.5222748533262), (0.900000000000000, 11.7261677868336), (1.00000000000000, 11.9331905403773), (1.10000000000000, 12.1433741438903), (1.20000000000000, 12.3567493196484), (1.30000000000000, 12.5733464552916), (1.40000000000000, 12.7931955761805), (1.50000000000000, 13.0163263170940), (1.60000000000000, 13.2427678932773), (1.70000000000000, 13.4725490708478), (1.80000000000000, 13.7056981365714), (1.90000000000000, 13.9422428670207), (2.00000000000000, 14.1822104971285), (2.10000000000000, 14.4256276881541), (2.20000000000000, 14.6725204950778), (2.30000000000000, 14.9229143334436), (2.40000000000000, 15.1768339456718), (2.50000000000000, 15.4343033668624), (2.60000000000000, 15.6953458901155), (2.70000000000000, 15.9599840313958), (2.80000000000000, 16.2282394939672), (2.90000000000000, 16.5001331324319), (3.00000000000000, 16.7756849164029), (3.10000000000000, 17.0549138938481), (3.20000000000000, 17.3378381541397), (3.30000000000000, 17.6244747908507), (3.40000000000000, 17.9148398643370), (3.50000000000000, 18.2089483641508), (3.60000000000000, 18.5068141713282), (3.70000000000000, 18.8084500206003), (3.80000000000000, 19.1138674625768), (3.90000000000000, 19.4230768259530), (4.00000000000000, 19.7360871797947), (4.10000000000000, 20.0529062959569), (4.20000000000000, 20.3735406116931), (4.30000000000000, 20.6979951925157), (4.40000000000000, 21.0262736953681), (4.50000000000000, 21.3583783321730), (4.60000000000000, 21.6943098338204), (4.70000000000000, 22.0340674146637), (4.80000000000000, 22.3776487375901), (4.90000000000000, 22.7250498797374), (5.00000000000000, 23.0762652989248), (5.10000000000000, 23.4312878008740), (5.20000000000000, 23.7901085072900), (5.30000000000000, 24.1527168248781), (5.40000000000000, 24.5191004153711), (5.50000000000000, 24.8892451666427), (5.60000000000000, 25.2631351649825), (5.70000000000000, 25.6407526686093), (5.80000000000000, 26.0220780824990), (5.90000000000000, 26.4070899346026), (5.99999999999999, 26.7957648535318), (6.09999999999999, 27.1880775477853), (6.19999999999999, 27.5840007865921), (6.29999999999999, 27.9835053824450), (6.39999999999999, 28.3865601753961), (6.49999999999999, 28.7931320191857), (6.59999999999999, 29.2031857692746), (6.69999999999999, 29.6166842728451), (6.79999999999999, 30.0335883608386), (6.89999999999999, 30.4538568420897), (6.99999999999999, 30.8774464996198), (7.09999999999999, 31.3043120891448), (7.19999999999999, 31.7344063398528), (7.29999999999999, 32.1676799575012), (7.39999999999999, 32.6040816298816), (7.49999999999999, 33.0435580346936), (7.59999999999999, 33.4860538498691), (7.69999999999999, 33.9315117663792), (7.79999999999999, 34.3798725035564), (7.89999999999999, 34.8310748269554), (7.99999999999999, 35.2850555687743), (8.09999999999999, 35.7417496508515), (8.19999999999999, 36.2010901102477), (8.29999999999999, 36.6630081274186), (8.39999999999999, 37.1274330569767), (8.49999999999999, 37.5942924610362), (8.59999999999999, 38.0635121451277), (8.69999999999999, 38.5350161966658), (8.79999999999998, 39.0087270259437), (8.89999999999998, 39.4845654096256), (8.99999999999998, 39.9624505367008), (9.09999999999998, 40.4423000568551), (9.19999999999998, 40.9240301312145), (9.29999999999998, 41.4075554854027), (9.39999999999998, 41.8927894648554), (9.49999999999998, 42.3796440923231), (9.59999999999998, 42.8680301274912), (9.69999999999998, 43.3578571286387), (9.79999999999998, 43.8490335162540), (9.89999999999998, 44.3414666385172), (9.99999999999998, 44.8350628385566), (10.1000000000000, 45.3297275233803), (10.2000000000000, 45.8253652343791), (10.3000000000000, 46.3218797192938), (10.4000000000000, 46.8191740055340), (10.5000000000000, 47.3171504747325), (10.6000000000000, 47.8157109384175), (10.7000000000000, 48.3147567146766), (10.8000000000000, 48.8141887056904), (10.9000000000000, 49.3139074760053), (11.0000000000000, 49.8138133314150), (11.1000000000000, 50.3138063983199), (11.2000000000000, 50.8137867034288), (11.3000000000000, 51.3136542536690), (11.4000000000000, 51.8133091161694), (11.5000000000000, 52.3126514981792), (11.6000000000000, 52.8115818267888), (11.7000000000000, 53.3100008283151), (11.8000000000000, 53.8078096072184), (11.9000000000000, 54.3049097244174), (12.0000000000000, 54.8012032748703), (12.1000000000000, 55.2965929642930), (12.2000000000000, 55.7909821848871), (12.3000000000000, 56.2842750899540), (12.4000000000000, 56.7763766672728), (12.5000000000000, 57.2671928111254), (12.6000000000000, 57.7566303928546), (12.7000000000000, 58.2445973298443), (12.8000000000000, 58.7310026528181), (12.9000000000000, 59.2157565713534), (13.0000000000000, 59.6987705375169), (13.1000000000000, 60.1799573075290), (13.2000000000000, 60.6592310013724), (13.3000000000000, 61.1365071602642), (13.4000000000000, 61.6117028019181), (13.5000000000000, 62.0847364735261), (13.6000000000000, 62.5555283023992), (13.7000000000000, 63.0240000442087), (13.8000000000000, 63.4900751287784), (13.9000000000000, 63.9536787033824), (14.0000000000000, 64.4147376735109), (14.1000000000000, 64.8731807410717), (14.2000000000000, 65.3289384400004), (14.3000000000000, 65.7819431692609), (14.4000000000000, 66.2321292232214), (14.5000000000000, 66.6794328193975), (14.6000000000000, 67.1237921235622), (14.7000000000000, 67.5651472722240), (14.8000000000000, 68.0034403924850), (14.9000000000000, 68.4386156192918), (15.0000000000000, 68.8706191101006), (15.1000000000000, 69.2993990569809), (15.2000000000000, 69.7249056961888), (15.3000000000000, 70.1470913152441), (15.4000000000000, 70.5659102575512), (15.5000000000000, 70.9813189246068), (15.6000000000000, 71.3932757758436), (15.7000000000000, 71.8017413261593), (15.8000000000000, 72.2066781411888), (15.9000000000000, 72.6080508303755), (16.0000000000000, 73.0058260379057), (16.1000000000000, 73.3999724315685), (16.2000000000000, 73.7904606896088), (16.3000000000000, 74.1772634856441), (16.4000000000000, 74.5603554717132), (16.5000000000000, 74.9397132595338), (16.6000000000000, 75.3153154000403), (16.7000000000000, 75.6871423612796), (16.8000000000000, 76.0551765047419), (16.9000000000000, 76.4194020602032), (17.0000000000000, 76.7798050991595), (17.1000000000000, 77.1363735069297), (17.2000000000000, 77.4890969535082), (17.3000000000000, 77.8379668632443), (17.4000000000000, 78.1829763834285), (17.5000000000000, 78.5241203518627), (17.6000000000000, 78.8613952634932), (17.7000000000000, 79.1947992361820), (17.8000000000000, 79.5243319756939), (17.9000000000000, 79.8499947399717), (18.0000000000000, 80.1717903027764), (18.1000000000000, 80.4897229167615), (18.2000000000000, 80.8037982760533), (18.3000000000000, 81.1140234784069), (18.4000000000000, 81.4204069870040), (18.5000000000000, 81.7229585919582), (18.6000000000000, 82.0216893715927), (18.7000000000000, 82.3166116535506), (18.8000000000000, 82.6077389757973), (18.9000000000000, 82.8950860475746), (19.0000000000000, 83.1786687103591), (19.1000000000000, 83.4585038988807), (19.2000000000000, 83.7346096022505), (19.3000000000000, 84.0070048252472), (19.4000000000000, 84.2757095498103), (19.5000000000000, 84.5407446967817), (19.6000000000000, 84.8021320879401), (19.7000000000000, 85.0598944083668), (19.8000000000000, 85.3140551691816), (19.9000000000000, 85.5646386706832), (20.0000000000000, 85.8116699659280), (20.1000000000000, 86.0551748247783), (20.2000000000000, 86.2951796984492), (20.3000000000000, 86.5317116845807), (20.4000000000000, 86.7647984928596), (20.5000000000000, 86.9944684112155), (20.6000000000000, 87.2207502726098), (20.7000000000000, 87.4436734224386), (20.8000000000000, 87.6632676865654), (20.9000000000000, 87.8795633399994), (21.0000000000000, 88.0925910762336), (21.1000000000000, 88.3023819772534), (21.2000000000000, 88.5089674842271), (21.3000000000000, 88.7123793688868), (21.4000000000000, 88.9126497056067), (21.5000000000000, 89.1098108441845), (21.6000000000000, 89.3038953833309), (21.7000000000000, 89.4949361448701), (21.8000000000000, 89.6829661486526), (21.9000000000000, 89.8680185881816), (22.0000000000000, 90.0501268069521), (22.1000000000000, 90.2293242755015), (22.2000000000000, 90.4056445691688), (22.3000000000000, 90.5791213465588), (22.4000000000000, 90.7497883287071), (22.5000000000000, 90.9176792789402), (22.6000000000001, 91.0828279834253), (22.7000000000001, 91.2452682324022), (22.8000000000001, 91.4050338020896), (22.9000000000001, 91.5621584372592), (23.0000000000001, 91.7166758344664), (23.1000000000001, 91.8686196259308), (23.2000000000001, 92.0180233640546), (23.3000000000001, 92.1649205065702), (23.4000000000001, 92.3093444023051), (23.5000000000001, 92.4513282775545), (23.6000000000001, 92.5909052230488), (23.7000000000001, 92.7281081815050), (23.8000000000001, 92.8629699357510), (23.9000000000001, 92.9955230974084), (24.0000000000001, 93.1258000961244), (24.1000000000001, 93.2538331693382), (24.2000000000001, 93.3796543525700), (24.3000000000001, 93.5032954702203), (24.4000000000001, 93.6247881268665), (24.5000000000001, 93.7441636990437), (24.6000000000001, 93.8614533274979), (24.7000000000001, 93.9766879098979), (24.8000000000001, 94.0898980939930), (24.9000000000001, 94.2011142712052), (25.0000000000001, 94.3103665706420), (25.1000000000001, 94.4176848535171), (25.2000000000001, 94.5230987079678), (25.3000000000001, 94.6266374442559), (25.4000000000001, 94.7283300903397), (25.5000000000001, 94.8282053878057), (25.6000000000001, 94.9262917881474), (25.7000000000001, 95.0226174493806), (25.8000000000001, 95.1172102329820), (25.9000000000001, 95.2100977011406), 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(77.9999999999998, 99.9998648352599), (78.0999999999998, 99.9998675385511), (78.1999999999998, 99.9998701877766), (78.2999999999998, 99.9998727840177), (78.3999999999998, 99.9998753283341), (78.4999999999998, 99.9998778217643), (78.5999999999998, 99.9998802653260), (78.6999999999998, 99.9998826600166), (78.7999999999998, 99.9998850068135), (78.8999999999998, 99.9998873066746), (78.9999999999998, 99.9998895605386), (79.0999999999998, 99.9998917693254), (79.1999999999998, 99.9998939339365), (79.2999999999998, 99.9998960552555), (79.3999999999998, 99.9998981341483), (79.4999999999998, 99.9999001714632), (79.5999999999998, 99.9999021680320), (79.6999999999997, 99.9999041246694), (79.7999999999997, 99.9999060421742), (79.8999999999997, 99.9999079213289), (79.9999999999997, 99.9999097629007), (80.0999999999997, 99.9999115676410), (80.1999999999997, 99.9999133362867), (80.2999999999997, 99.9999150695594), (80.3999999999997, 99.9999167681668), (80.4999999999997, 99.9999184328021), (80.5999999999997, 99.9999200641447), (80.6999999999997, 99.9999216628605), (80.7999999999997, 99.9999232296021), (80.8999999999997, 99.9999247650089), (80.9999999999997, 99.9999262697075), (81.0999999999997, 99.9999277443123), (81.1999999999997, 99.9999291894250), (81.2999999999997, 99.9999306056355), (81.3999999999997, 99.9999319935218), (81.4999999999996, 99.9999333536505), (81.5999999999996, 99.9999346865766), (81.6999999999996, 99.9999359928442), (81.7999999999996, 99.9999372729865), (81.8999999999996, 99.9999385275260), (81.9999999999996, 99.9999397569747), (82.0999999999996, 99.9999409618345), (82.1999999999996, 99.9999421425971), (82.2999999999996, 99.9999432997445), (82.3999999999996, 99.9999444337490), (82.4999999999996, 99.9999455450734), (82.5999999999996, 99.9999466341713), (82.6999999999996, 99.9999477014873), (82.7999999999996, 99.9999487474570), (82.8999999999996, 99.9999497725074), (82.9999999999996, 99.9999507770567), (83.0999999999996, 99.9999517615151), (83.1999999999995, 99.9999527262843), (83.2999999999995, 99.9999536717582), (83.3999999999995, 99.9999545983226), (83.4999999999995, 99.9999555063557), (83.5999999999995, 99.9999563962282), (83.6999999999995, 99.9999572683033), (83.7999999999995, 99.9999581229368), (83.8999999999995, 99.9999589604778), (83.9999999999995, 99.9999597812679), (84.0999999999995, 99.9999605856422), (84.1999999999995, 99.9999613739290), (84.2999999999995, 99.9999621464501), (84.3999999999995, 99.9999629035209), (84.4999999999995, 99.9999636454502), (84.5999999999995, 99.9999643725409), (84.6999999999995, 99.9999650850898), (84.7999999999995, 99.9999657833878), (84.8999999999995, 99.9999664677198), (84.9999999999994, 99.9999671383652), (85.0999999999994, 99.9999677955976), (85.1999999999994, 99.9999684396855), (85.2999999999994, 99.9999690708916), (85.3999999999994, 99.9999696894735), (85.4999999999994, 99.9999702956839), (85.5999999999994, 99.9999708897700), (85.6999999999994, 99.9999714719745), (85.7999999999994, 99.9999720425348), (85.8999999999994, 99.9999726016840), (85.9999999999994, 99.9999731496501), (86.0999999999994, 99.9999736866570), (86.1999999999994, 99.9999742129237), (86.2999999999994, 99.9999747286651), (86.3999999999994, 99.9999752340917), (86.4999999999994, 99.9999757294097), (86.5999999999994, 99.9999762148214), (86.6999999999993, 99.9999766905249), (86.7999999999993, 99.9999771567143), (86.8999999999993, 99.9999776135799), (86.9999999999993, 99.9999780613082), (87.0999999999993, 99.9999785000819), (87.1999999999993, 99.9999789300802), (87.2999999999993, 99.9999793514785), (87.3999999999993, 99.9999797644488), (87.4999999999993, 99.9999801691598), (87.5999999999993, 99.9999805657765), (87.6999999999993, 99.9999809544609), (87.7999999999993, 99.9999813353716), (87.8999999999993, 99.9999817086641), (87.9999999999993, 99.9999820744907), (88.0999999999993, 99.9999824330009), (88.1999999999993, 99.9999827843408), (88.2999999999993, 99.9999831286539), (88.3999999999993, 99.9999834660808), (88.4999999999992, 99.9999837967591), (88.5999999999992, 99.9999841208239), (88.6999999999992, 99.9999844384074), (88.7999999999992, 99.9999847496392), (88.8999999999992, 99.9999850546463), (88.9999999999992, 99.9999853535534), (89.0999999999992, 99.9999856464822), (89.1999999999992, 99.9999859335526), (89.2999999999992, 99.9999862148815), (89.3999999999992, 99.9999864905838), (89.4999999999992, 99.9999867607721), (89.5999999999992, 99.9999870255566), (89.6999999999992, 99.9999872850454), (89.7999999999992, 99.9999875393445), (89.8999999999992, 99.9999877885575), (89.9999999999992, 99.9999880327864), (90.0999999999992, 99.9999882721306), (90.1999999999992, 99.9999885066880), (90.2999999999991, 99.9999887365542), (90.3999999999991, 99.9999889618231), (90.4999999999991, 99.9999891825866), (90.5999999999991, 99.9999893989348), (90.6999999999991, 99.9999896109561), (90.7999999999991, 99.9999898187370), (90.8999999999991, 99.9999900223622), (90.9999999999991, 99.9999902219149), (91.0999999999991, 99.9999904174766), (91.1999999999991, 99.9999906091271), (91.2999999999991, 99.9999907969445), (91.3999999999991, 99.9999909810056), (91.4999999999991, 99.9999911613855), (91.5999999999991, 99.9999913381577), (91.6999999999991, 99.9999915113946), (91.7999999999991, 99.9999916811666), (91.8999999999991, 99.9999918475433), (91.9999999999990, 99.9999920105924), (92.0999999999990, 99.9999921703806), (92.1999999999990, 99.9999923269729), (92.2999999999990, 99.9999924804335), (92.3999999999990, 99.9999926308248), (92.4999999999990, 99.9999927782083), (92.5999999999990, 99.9999929226441), (92.6999999999990, 99.9999930641912), (92.7999999999990, 99.9999932029074), (92.8999999999990, 99.9999933388492), (92.9999999999990, 99.9999934720722), (93.0999999999990, 99.9999936026308), (93.1999999999990, 99.9999937305782), (93.2999999999990, 99.9999938559666), (93.3999999999990, 99.9999939788472), (93.4999999999990, 99.9999940992703), (93.5999999999990, 99.9999942172849), (93.6999999999990, 99.9999943329392), (93.7999999999989, 99.9999944462804), (93.8999999999989, 99.9999945573548), (93.9999999999989, 99.9999946662077), (94.0999999999989, 99.9999947728835), (94.1999999999989, 99.9999948774258), (94.2999999999989, 99.9999949798773), (94.3999999999989, 99.9999950802798), (94.4999999999989, 99.9999951786742), (94.5999999999989, 99.9999952751007), (94.6999999999989, 99.9999953695987), (94.7999999999989, 99.9999954622067), (94.8999999999989, 99.9999955529625), (94.9999999999989, 99.9999956419033), (95.0999999999989, 99.9999957290652), (95.1999999999989, 99.9999958144839), (95.2999999999989, 99.9999958981942), (95.3999999999989, 99.9999959802304), (95.4999999999988, 99.9999960606257), (95.5999999999988, 99.9999961394132), (95.6999999999988, 99.9999962166250), (95.7999999999988, 99.9999962922925), (95.8999999999988, 99.9999963664466), (95.9999999999988, 99.9999964391177), (96.0999999999988, 99.9999965103353), (96.1999999999988, 99.9999965801286), (96.2999999999988, 99.9999966485260), (96.3999999999988, 99.9999967155555), (96.4999999999988, 99.9999967812444), (96.5999999999988, 99.9999968456195), (96.6999999999988, 99.9999969087071), (96.7999999999988, 99.9999969705330), (96.8999999999988, 99.9999970311223), (96.9999999999988, 99.9999970904999), (97.0999999999988, 99.9999971486899), (97.1999999999988, 99.9999972057161), (97.2999999999987, 99.9999972616018), (97.3999999999987, 99.9999973163697), (97.4999999999987, 99.9999973700423), (97.5999999999987, 99.9999974226415), (97.6999999999987, 99.9999974741887), (97.7999999999987, 99.9999975247049), (97.8999999999987, 99.9999975742108), (97.9999999999987, 99.9999976227266), (98.0999999999987, 99.9999976702720), (98.1999999999987, 99.9999977168666), (98.2999999999987, 99.9999977625293), (98.3999999999987, 99.9999978072787), (98.4999999999987, 99.9999978511331), (98.5999999999987, 99.9999978941105), (98.6999999999987, 99.9999979362282), (98.7999999999987, 99.9999979775037), (98.8999999999987, 99.9999980179536), (98.9999999999986, 99.9999980575945), (99.0999999999986, 99.9999980964426), (99.1999999999986, 99.9999981345138), (99.2999999999986, 99.9999981718235), (99.3999999999986, 99.9999982083870), (99.4999999999986, 99.9999982442193), (99.5999999999986, 99.9999982793349), (99.6999999999986, 99.9999983137482), (99.7999999999986, 99.9999983474732), (99.8999999999986, 99.9999983805238), (99.9999999999986, 99.9999984129133)]
Nprime1(x)=0.5*x #Exercise8
x(t)=2*e^(0.5*t)
x(0)
list_plot(Eulers(Nprime1, x(0), 1, 100), size=15, color="red", xmax=100) + list_plot(Eulers(Nprime1, x(0), 0.1, 1000), size=5, color="blue", xmax=100) + list_plot(Eulers(Nprime1, x(0), 0.01, 10000), size=1, color="green", ymax=8e17, xmax=100) + plot(2*e^(0.5*t), color="black", ymax = 8e17, xmax = 100) #As we decrease our step size we see that the graphs become closer and closer to the true value (black). This trend is obvious from red -> blue -> green -> black. As delta T approaches 0, we get a graph that comes closer to resembling the true graph.


2
#Exercise 9
a=Eulersvalue(Nprime1, x(0), 1, 100)
b=[]
errorstepsize1=[]
for t in srange(0,101):
b.append(2*e^(0.5*t))
for hh in srange(0,101):
errorstepsize1.append(abs(a[hh]-b[hh])/(b[hh]))

errorstepsize1
mean(errorstepsize1)


[0.000000000000000, 0.0902040104310499, 0.172271257364255, 0.246935709499049, 0.314865128614648, 0.376667041699769, 0.432894174372300, 0.484049394182730, 0.530590208011777, 0.572932853784720, 0.611456023096682, 0.646504248042190, 0.678390982539124, 0.707401405704885, 0.733794972356792, 0.757807733447118, 0.779654447185574, 0.799530499730082, 0.817613652623537, 0.834065632604764, 0.849033578012154, 0.862651354715884, 0.875040753347783, 0.886312578536255, 0.896567639887850, 0.905897653578315, 0.914386062616523, 0.922108783117305, 0.929134883257479, 0.935527200987319, 0.941342906021978, 0.946634011139026, 0.951447837354905, 0.955827437140593, 0.959811979461529, 0.963437100085385, 0.966735220290672, 0.969735836826559, 0.972465785717143, 0.974949482269525, 0.977209139432188, 0.979264966456579, 0.981135349638618, 0.982837016756594, 0.984385186676110, 0.985793705460057, 0.987075170200925, 0.988241041682940, 0.989301746881631, 0.990266772217514, 0.991144748397933, 0.991943527605816, 0.992670253725698, 0.993331426235082, 0.993932958332533, 0.994480229822391, 0.994978135229069, 0.995431127571249, 0.995843258187470, 0.996218212969287, 0.996559345326053, 0.996869706176152, 0.997152071232890, 0.997408965829106, 0.997642687502484, 0.997855326543599, 0.998048784690432, 0.998224792136569, 0.998384923005199, 0.998530609427285, 0.998663154349834, 0.998783743188806, 0.998893454430890, 0.998993269278948, 0.999084080427411, 0.999166700046091, 0.999241867043826, 0.999310253676913, 0.999372471561435, 0.999429077143253, 0.999480576674579, 0.999527430741643, 0.999570058383953, 0.999608840841972, 0.999644124966743, 0.999676226321955, 0.999705432006187, 0.999732003220573, 0.999756177604860, 0.999778171362735, 0.999798181195444, 0.999816386060996, 0.999832948774665, 0.999848017465138, 0.999861726899298, 0.999874199687516, 0.999885547380215, 0.999895871465524, 0.999905264276934, 0.999913809819086, 0.999921584519064] 0.890245636592289
[(0.000000000000000, 2), (1.00000000000000, 3.00000000000000), (2.00000000000000, 4.50000000000000), (3.00000000000000, 6.75000000000000), (4.00000000000000, 10.1250000000000), (5.00000000000000, 15.1875000000000), (6.00000000000000, 22.7812500000000), (7.00000000000000, 34.1718750000000), (8.00000000000000, 51.2578125000000), (9.00000000000000, 76.8867187500000), (10.0000000000000, 115.330078125000), (11.0000000000000, 172.995117187500), (12.0000000000000, 259.492675781250), (13.0000000000000, 389.239013671875), (14.0000000000000, 583.858520507812), (15.0000000000000, 875.787780761719), (16.0000000000000, 1313.68167114258), (17.0000000000000, 1970.52250671387), (18.0000000000000, 2955.78376007080), (19.0000000000000, 4433.67564010620), (20.0000000000000, 6650.51346015930), (21.0000000000000, 9975.77019023895), (22.0000000000000, 14963.6552853584), (23.0000000000000, 22445.4829280376), (24.0000000000000, 33668.2243920565), (25.0000000000000, 50502.3365880847), (26.0000000000000, 75753.5048821270), (27.0000000000000, 113630.257323191), (28.0000000000000, 170445.385984786), (29.0000000000000, 255668.078977179), (30.0000000000000, 383502.118465768), (31.0000000000000, 575253.177698652), (32.0000000000000, 862879.766547978), (33.0000000000000, 1.29431964982197e6), (34.0000000000000, 1.94147947473295e6), (35.0000000000000, 2.91221921209943e6), (36.0000000000000, 4.36832881814914e6), (37.0000000000000, 6.55249322722371e6), (38.0000000000000, 9.82873984083557e6), (39.0000000000000, 1.47431097612533e7), (40.0000000000000, 2.21146646418800e7), (41.0000000000000, 3.31719969628200e7), (42.0000000000000, 4.97579954442301e7), (43.0000000000000, 7.46369931663451e7), (44.0000000000000, 1.11955489749518e8), (45.0000000000000, 1.67933234624276e8), (46.0000000000000, 2.51899851936415e8), (47.0000000000000, 3.77849777904622e8), (48.0000000000000, 5.66774666856933e8), (49.0000000000000, 8.50162000285399e8), (50.0000000000000, 1.27524300042810e9), (51.0000000000000, 1.91286450064215e9), (52.0000000000000, 2.86929675096322e9), (53.0000000000000, 4.30394512644483e9), (54.0000000000000, 6.45591768966725e9), (55.0000000000000, 9.68387653450088e9), (56.0000000000000, 1.45258148017513e10), (57.0000000000000, 2.17887222026270e10), (58.0000000000000, 3.26830833039405e10), (59.0000000000000, 4.90246249559107e10), (60.0000000000000, 7.35369374338660e10), (61.0000000000000, 1.10305406150799e11), (62.0000000000000, 1.65458109226199e11), (63.0000000000000, 2.48187163839298e11), (64.0000000000000, 3.72280745758947e11), (65.0000000000000, 5.58421118638420e11), (66.0000000000000, 8.37631677957631e11), (67.0000000000000, 1.25644751693645e12), (68.0000000000000, 1.88467127540467e12), (69.0000000000000, 2.82700691310700e12), (70.0000000000000, 4.24051036966050e12), (71.0000000000000, 6.36076555449076e12), (72.0000000000000, 9.54114833173613e12), (73.0000000000000, 1.43117224976042e13), (74.0000000000000, 2.14675837464063e13), (75.0000000000000, 3.22013756196095e13), (76.0000000000000, 4.83020634294142e13), (77.0000000000000, 7.24530951441213e13), (78.0000000000000, 1.08679642716182e14), (79.0000000000000, 1.63019464074273e14), (80.0000000000000, 2.44529196111409e14), (81.0000000000000, 3.66793794167114e14), (82.0000000000000, 5.50190691250671e14), (83.0000000000000, 8.25286036876006e14), (84.0000000000000, 1.23792905531401e15), (85.0000000000000, 1.85689358297101e15), (86.0000000000000, 2.78534037445652e15), (87.0000000000000, 4.17801056168478e15), (88.0000000000000, 6.26701584252717e15), (89.0000000000000, 9.40052376379076e15), (90.0000000000000, 1.41007856456861e16), (91.0000000000000, 2.11511784685292e16), (92.0000000000000, 3.17267677027938e16), (93.0000000000000, 4.75901515541907e16), (94.0000000000000, 7.13852273312861e16), (95.0000000000000, 1.07077840996929e17), (96.0000000000000, 1.60616761495394e17), (97.0000000000000, 2.40925142243090e17), (98.0000000000000, 3.61387713364636e17), (99.0000000000000, 5.42081570046954e17), (100.000000000000, 8.13122355070430e17)]
c=Eulersvalue(Nprime1, x(0), .1, 1000)
errorstepsize2=[]
b=[]
errorstepsize1=[]
for kk in srange(0,1001):
b.append(2*e^(0.5*(kk*0.1)))
for ii in srange(0,1001):
errorstepsize2.append(abs(c[ii]-b[ii])/(b[ii]))

errorstepsize2
mean(errorstepsize2)


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d=Eulersvalue(Nprime1, x(0), .01, 10000)
errorstepsize3=[]
b=[]
errorstepsize1=[]
for zzz in srange(0,10001):
b.append(2*e^(0.5*(zzz*0.01)))
for jj in srange(0,10001):
errorstepsize3.append(abs(d[jj]-b[jj])/(b[jj]))

errorstepsize3
mean(errorstepsize3)

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0.0597361667438490, 0.0597478809374677, 0.0597595949851429, 0.0597713088868795, 0.0597830226426828, 0.0597947362525481, 0.0598064497164804, 0.0598181630344815, 0.0598298762065567, 0.0598415892327010, 0.0598533021129195, 0.0598[...] 0.0597840295811412
#We see that as we decrease the step size and consequently increase the number of steps, the mean of the errors start becoming closer and closer to 0. Over time, the values of error increases as the estimation becomes farther and farther away from the true curve.

Nprime1

x |--> 0.500000000000000*x


Eulersvalue(Nprime1, x(0), .1, 1)

[2, 2.10000000000000]

Eulersvalue(Nprime1, x(0), .01, 1)

[2, 2.01000000000000]
b

[2.00000000000000, 3.29744254140026]

2*e^(0.5*1)

3.29744254140026