CoCalc Shared FilesLab 4 / Lab4-turnin.sagewsOpen in CoCalc with one click!
Authors: Tarek Abushamma, Qi Wu
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# Lab 1: # Name: Tarek Abushamma # I worked on this code with: # Please do all of your work for this week's lab in this worksheet. If # you wish to create other worksheets for scratch work, you can, but # this is the one that will be graded. You do not need to do anything # to turn in your lab. It will be collected by your TA at the beginning # of (or right before) next week’s lab. # Be sure to clearly label which question you are answering as you go and to # use enough comments that you and the grader can understand your code.1
#1 # need input for function, step size, number of steps and initial value # make list to track all the values that we obtain # loop through list as many times as number of steps # using initial value, iterate using euler method and update list with new values # output values #2.1 Nprime(N) = 0.2*N*((1-N)/100) initial_N = 10 stepsize = 0.1 new_N = initial_N + Nprime(initial_N)*stepsize new_N
9.98200000000000
#2.2 Nprime(N) = 0.2*N*((1-N)/100) initial_N = [10] stepsize = 0.1 final_t = 100 num_iter = ((final_t)/stepsize) for a in srange (0, num_iter): new_N = initial_N[-1]+(Nprime(initial_N[-1])*stepsize) initial_N.append(new_N) initial_N[-1]
3.79795164396582
#3 Nprime(N) = 0.2*N*((1-N)/100) initial_N = [15,68,450] stepsize = 0.1 final_t = 100 num_iter = ((final_t)/stepsize) for a in srange (0, num_iter): new_N = initial_N[-1]+(Nprime(initial_N[-1])*stepsize) initial_N.append(new_N) initial_N[-1]
5.43969334166354
#3 [ctd.] Nprime(N) = 0.2*N*((1-N)/100) initial_N = [200] stepsize = 0.1 final_t = 100 num_iter = ((final_t)/stepsize) for a in srange (0, num_iter): new_N = initial_N[-1]+(Nprime(initial_N[-1])*stepsize) initial_N.append(new_N) initial_N
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6.36483722720870, 6.35800796406837, 6.35119471260695, 6.34439741669398, 6.33761602046113, 6.33085046830066, 6.32410070486393, 6.31736667505985, 6.31064832405343, 6.30394559726426, 6.29725844036506, 6.29058679928019, 6.28393062018419, 6.27728984950037, 6.27066443389934, 6.26405432029760, 6.25745945585614, 6.25087978797897, 6.24431526431181, 6.23776583274065, 6.23123144139038, 6.22471203862342, 6.21820757303839, 6.21171799346872, 6.20524324898134, 6.19878328887533, 6.19233806268061, 6.18590752015665, 6.17949161129109, 6.17309028629855, 6.16670349561925, 6.16033118991779, 6.15397332008188, 6.14762983722104, 6.14130069266539, 6.13498583796438, 6.12868522488556, 6.12239880541340, 6.11612653174797, 6.10986835630385, 6.10362423170884, 6.09739411080280, 6.09117794663647, 6.08497569247028, 6.07878730177318, 6.07261272822150, 6.06645192569775, 6.06030484828953, 6.05417145028835, 6.04805168618851, 6.04194551068599, 6.03585287867730, 6.02977374525843, 6.02370806572368, 6.01765579556461, 6.01161689046895, 6.00559130631949, 5.99957899919304, 5.99357992535937, 5.98759404128011, 5.98162130360773, 5.97566166918450, 5.96971509504142, 5.96378153839724, 5.95786095665737, 5.95195330741293, 5.94605854843969, 5.94017663769708, 5.93430753332721, 5.92845119365386, 5.92260757718148, 5.91677664259426, 5.91095834875511, 5.90515265470472, 5.89935951966058, 5.89357890301607, 5.88781076433946, 5.88205506337299, 5.87631176003196, 5.87058081440375, 5.86486218674694, 5.85915583749038, 5.85346172723228, 5.84777981673929, 5.84211006694563, 5.83645243895215, 5.83080689402551, 5.82517339359724, 5.81955189926286, 5.81394237278107, 5.80834477607282, 5.80275907122049, 5.79718522046701, 5.79162318621502, 5.78607293102605, 5.78053441761962, 5.77500760887249, 5.76949246781775, 5.76398895764408, 5.75849704169484, 5.75301668346733, 5.74754784661198, 5.74209049493148, 5.73664459238007, 5.73121010306269, 5.72578699123421, 5.72037522129866, 5.71497475780843, 5.70958556546352, 5.70420760911074, 5.69884085374299, 5.69348526449848, 5.68814080665997, 5.68280744565403, 5.67748514705028, 5.67217387656069, 5.66687360003882, 5.66158428347906, 5.65630589301597, 5.65103839492350, 5.64578175561431, 5.64053594163902, 5.63530091968557, 5.63007665657842, 5.62486311927795, 5.61966027487968, 5.61446809061364, 5.60928653384366, 5.60411557206668, 5.59895517291208, 5.59380530414100, 5.58866593364571, 5.58353702944886, 5.57841855970290, 5.57331049268939, 5.56821279681835, 5.56312544062758, 5.55804839278208, 5.55298162207333, 5.54792509741873, 5.54287878786090, 5.53784266256709, 5.53281669082853, 5.52780084205983, 5.52279508579835, 5.51779939170357, 5.51281372955649, 5.50783806925904, 5.50287238083346, 5.49791663442168, 5.49297080028475, 5.48803484880225, 5.48310875047168, 5.47819247590787, 5.47328599584244, 5.46838928112315, 5.46350230271339, 5.45862503169158, 5.45375743925060, 5.44889949669722, 5.44405117545153, 5.43921244704644, 5.43438328312703, 5.42956365545007, 5.42475353588344, 5.41995289640560, 5.41516170910503, 5.41037994617970, 5.40560757993653, 5.40084458279086, 5.39609092726593, 5.39134658599232, 5.38661153170746, 5.38188573725510, 5.37716917558477]
#4 list_plot(zip(srange(0,100,0.1), initial_N), axes_labels = ["t","N"])
#4 [ctd.] list_plot((srange(0,100,0.1), initial_N), axes_labels = ["t","N"])
3D rendering not yet implemented
# When "zip" is omitted, the result is a cube in 3D space.
#5 a(x)=0.2*x*(1-(x/100)) def eulers_method(initial_val,num_step,step,n): val=[initial_val] counter=srange(0,num_step*step,step) for i in counter: final=val[-1] next_value=final+(n(final)*step) val.append(next_value) return val eulers_method(10,100,.1,n)
[10, 11.0000000000000, 12.1000000000000, 13.3100000000000, 14.6410000000000, 16.1051000000000, 17.7156100000000, 19.4871710000000, 21.4358881000000, 23.5794769100000, 25.9374246010000, 28.5311670611000, 31.3842837672100, 34.5227121439310, 37.9749833583241, 41.7724816941565, 45.9497298635722, 50.5447028499294, 55.5991731349223, 61.1590904484145, 67.2749994932560, 74.0024994425816, 81.4027493868398, 89.5430243255237, 98.4973267580761, 108.347059433884, 119.181765377272, 131.099941914999, 144.209936106499, 158.630929717149, 174.494022688864, 191.943424957750, 211.137767453526, 232.251544198878, 255.476698618766, 281.024368480643, 309.126805328707, 340.039485861577, 374.043434447735, 411.447777892509, 452.592555681760, 497.851811249936, 547.636992374929, 602.400691612422, 662.640760773664, 728.904836851031, 801.795320536134, 881.974852589747, 970.172337848722, 1067.18957163359, 1173.90852879695, 1291.29938167665, 1420.42931984431, 1562.47225182874, 1718.71947701162, 1890.59142471278, 2079.65056718406, 2287.61562390246, 2516.37718629271, 2768.01490492198, 3044.81639541418, 3349.29803495560, 3684.22783845116, 4052.65062229628, 4457.91568452590, 4903.70725297849, 5394.07797827634, 5933.48577610398, 6526.83435371438, 7179.51778908581, 7897.46956799439, 8687.21652479383, 9555.93817727322, 10511.5319950005, 11562.6851945006, 12718.9537139507, 13990.8490853457, 15389.9339938803, 16928.9273932683, 18621.8201325951, 20484.0021458547, 22532.4023604401, 24785.6425964841, 27264.2068561326, 29990.6275417458, 32989.6902959204, 36288.6593255124, 39917.5252580637, 43909.2777838700, 48300.2055622570, 53130.2261184827, 58443.2487303310, 64287.5736033641, 70716.3309637005, 77787.9640600706, 85566.7604660776, 94123.4365126854, 103535.780163954, 113889.358180349, 125278.293998384, 137806.123398223]
#6 a(x)=0.2*x*(1-(x/100)) def eulers_method(initial_val,time_final,step,function): val=[initial_val] time=[0] counter=srange(0,time_final,step) for i in counter: new_time=time[-1]+step time.append(new_time) final=val[-1] next_value=final+(function(final)*step) val.append(next_value) list_one=zip(time,val) return list_one eulers_method(10,10,.1,n)
[(0, 10), (0.100000000000000, 11.0000000000000), (0.200000000000000, 12.1000000000000), (0.300000000000000, 13.3100000000000), (0.400000000000000, 14.6410000000000), (0.500000000000000, 16.1051000000000), (0.600000000000000, 17.7156100000000), (0.700000000000000, 19.4871710000000), (0.800000000000000, 21.4358881000000), (0.900000000000000, 23.5794769100000), (1.00000000000000, 25.9374246010000), (1.10000000000000, 28.5311670611000), (1.20000000000000, 31.3842837672100), (1.30000000000000, 34.5227121439310), (1.40000000000000, 37.9749833583241), (1.50000000000000, 41.7724816941565), (1.60000000000000, 45.9497298635722), (1.70000000000000, 50.5447028499294), (1.80000000000000, 55.5991731349223), (1.90000000000000, 61.1590904484145), (2.00000000000000, 67.2749994932560), (2.10000000000000, 74.0024994425816), (2.20000000000000, 81.4027493868398), (2.30000000000000, 89.5430243255237), (2.40000000000000, 98.4973267580761), (2.50000000000000, 108.347059433884), (2.60000000000000, 119.181765377272), (2.70000000000000, 131.099941914999), (2.80000000000000, 144.209936106499), (2.90000000000000, 158.630929717149), (3.00000000000000, 174.494022688864), (3.10000000000000, 191.943424957750), (3.20000000000000, 211.137767453526), (3.30000000000000, 232.251544198878), (3.40000000000000, 255.476698618766), (3.50000000000000, 281.024368480643), (3.60000000000000, 309.126805328707), (3.70000000000000, 340.039485861577), (3.80000000000000, 374.043434447735), (3.90000000000000, 411.447777892509), (4.00000000000000, 452.592555681760), (4.10000000000000, 497.851811249936), (4.20000000000000, 547.636992374929), (4.30000000000000, 602.400691612422), (4.40000000000000, 662.640760773664), (4.50000000000000, 728.904836851031), (4.60000000000000, 801.795320536134), (4.70000000000000, 881.974852589747), (4.80000000000000, 970.172337848722), (4.90000000000000, 1067.18957163359), (5.00000000000000, 1173.90852879695), (5.10000000000000, 1291.29938167665), (5.20000000000000, 1420.42931984431), (5.30000000000000, 1562.47225182874), (5.40000000000000, 1718.71947701162), (5.50000000000000, 1890.59142471278), (5.60000000000000, 2079.65056718406), (5.70000000000000, 2287.61562390246), (5.80000000000000, 2516.37718629271), (5.90000000000000, 2768.01490492198), (5.99999999999999, 3044.81639541418), (6.09999999999999, 3349.29803495560), (6.19999999999999, 3684.22783845116), (6.29999999999999, 4052.65062229628), (6.39999999999999, 4457.91568452590), (6.49999999999999, 4903.70725297849), (6.59999999999999, 5394.07797827634), (6.69999999999999, 5933.48577610398), (6.79999999999999, 6526.83435371438), (6.89999999999999, 7179.51778908581), (6.99999999999999, 7897.46956799439), (7.09999999999999, 8687.21652479383), (7.19999999999999, 9555.93817727322), (7.29999999999999, 10511.5319950005), (7.39999999999999, 11562.6851945006), (7.49999999999999, 12718.9537139507), (7.59999999999999, 13990.8490853457), (7.69999999999999, 15389.9339938803), (7.79999999999999, 16928.9273932683), (7.89999999999999, 18621.8201325951), (7.99999999999999, 20484.0021458547), (8.09999999999999, 22532.4023604401), (8.19999999999999, 24785.6425964841), (8.29999999999999, 27264.2068561326), (8.39999999999999, 29990.6275417458), (8.49999999999999, 32989.6902959204), (8.59999999999999, 36288.6593255124), (8.69999999999999, 39917.5252580637), (8.79999999999998, 43909.2777838700), (8.89999999999998, 48300.2055622570), (8.99999999999998, 53130.2261184827), (9.09999999999998, 58443.2487303310), (9.19999999999998, 64287.5736033641), (9.29999999999998, 70716.3309637005), (9.39999999999998, 77787.9640600706), (9.49999999999998, 85566.7604660776), (9.59999999999998, 94123.4365126854), (9.69999999999998, 103535.780163954), (9.79999999999998, 113889.358180349), (9.89999999999998, 125278.293998384), (9.99999999999998, 137806.123398223)]
#7 a(x)=0.2*x*(1-(x/100)) def eulers_method(initial_val,time_final,step,function): val=[initial_val] counter=srange(0,time_final,step) for i in counter: final=val[-1] next_value=final+(function(final)*step) val.append(next_value) list_one=zip(counter,val) return list_one eulers_method(10,10,.1,n)
[(0.000000000000000, 10), (0.100000000000000, 11.0000000000000), (0.200000000000000, 12.1000000000000), (0.300000000000000, 13.3100000000000), (0.400000000000000, 14.6410000000000), (0.500000000000000, 16.1051000000000), (0.600000000000000, 17.7156100000000), (0.700000000000000, 19.4871710000000), (0.800000000000000, 21.4358881000000), (0.900000000000000, 23.5794769100000), (1.00000000000000, 25.9374246010000), (1.10000000000000, 28.5311670611000), (1.20000000000000, 31.3842837672100), (1.30000000000000, 34.5227121439310), (1.40000000000000, 37.9749833583241), (1.50000000000000, 41.7724816941565), (1.60000000000000, 45.9497298635722), (1.70000000000000, 50.5447028499294), (1.80000000000000, 55.5991731349223), (1.90000000000000, 61.1590904484145), (2.00000000000000, 67.2749994932560), (2.10000000000000, 74.0024994425816), (2.20000000000000, 81.4027493868398), (2.30000000000000, 89.5430243255237), (2.40000000000000, 98.4973267580761), (2.50000000000000, 108.347059433884), (2.60000000000000, 119.181765377272), (2.70000000000000, 131.099941914999), (2.80000000000000, 144.209936106499), (2.90000000000000, 158.630929717149), (3.00000000000000, 174.494022688864), (3.10000000000000, 191.943424957750), (3.20000000000000, 211.137767453526), (3.30000000000000, 232.251544198878), (3.40000000000000, 255.476698618766), (3.50000000000000, 281.024368480643), (3.60000000000000, 309.126805328707), (3.70000000000000, 340.039485861577), (3.80000000000000, 374.043434447735), (3.90000000000000, 411.447777892509), (4.00000000000000, 452.592555681760), (4.10000000000000, 497.851811249936), (4.20000000000000, 547.636992374929), (4.30000000000000, 602.400691612422), (4.40000000000000, 662.640760773664), (4.50000000000000, 728.904836851031), (4.60000000000000, 801.795320536134), (4.70000000000000, 881.974852589747), (4.80000000000000, 970.172337848722), (4.90000000000000, 1067.18957163359), (5.00000000000000, 1173.90852879695), (5.10000000000000, 1291.29938167665), (5.20000000000000, 1420.42931984431), (5.30000000000000, 1562.47225182874), (5.40000000000000, 1718.71947701162), (5.50000000000000, 1890.59142471278), (5.60000000000000, 2079.65056718406), (5.70000000000000, 2287.61562390246), (5.80000000000000, 2516.37718629271), (5.90000000000000, 2768.01490492198), (5.99999999999999, 3044.81639541418), (6.09999999999999, 3349.29803495560), (6.19999999999999, 3684.22783845116), (6.29999999999999, 4052.65062229628), (6.39999999999999, 4457.91568452590), (6.49999999999999, 4903.70725297849), (6.59999999999999, 5394.07797827634), (6.69999999999999, 5933.48577610398), (6.79999999999999, 6526.83435371438), (6.89999999999999, 7179.51778908581), (6.99999999999999, 7897.46956799439), (7.09999999999999, 8687.21652479383), (7.19999999999999, 9555.93817727322), (7.29999999999999, 10511.5319950005), (7.39999999999999, 11562.6851945006), (7.49999999999999, 12718.9537139507), (7.59999999999999, 13990.8490853457), (7.69999999999999, 15389.9339938803), (7.79999999999999, 16928.9273932683), (7.89999999999999, 18621.8201325951), (7.99999999999999, 20484.0021458547), (8.09999999999999, 22532.4023604401), (8.19999999999999, 24785.6425964841), (8.29999999999999, 27264.2068561326), (8.39999999999999, 29990.6275417458), (8.49999999999999, 32989.6902959204), (8.59999999999999, 36288.6593255124), (8.69999999999999, 39917.5252580637), (8.79999999999998, 43909.2777838700), (8.89999999999998, 48300.2055622570), (8.99999999999998, 53130.2261184827), (9.09999999999998, 58443.2487303310), (9.19999999999998, 64287.5736033641), (9.29999999999998, 70716.3309637005), (9.39999999999998, 77787.9640600706), (9.49999999999998, 85566.7604660776), (9.59999999999998, 94123.4365126854), (9.69999999999998, 103535.780163954), (9.79999999999998, 113889.358180349), (9.89999999999998, 125278.293998384)]
#8 & 9 f(x)=-2.1*x def eulers_method(initial_val,time_final,step,function): val=[initial_val] counter=srange(0,time_final,step) for i in counter: final= val[-1] next_value=final+(function(final)*step) val.append(next_value) list_one=zip(counter,val) return list_one result1 = eulers_method(5,10,1,f(x)) result2 = eulers_method(5,10,0.5,f(x)) result3 = eulers_method(5,10,0.1,f(x)) list_plot(result1,plotjoined=True) list_plot(result2,plotjoined=True) list_plot(result3,plotjoined=True)