SharedLab 03 / Lab03-turnin.sagewsOpen in CoCalc
Author: Michael Le
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# 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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#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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#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