Project 2

454 days ago by vsbhatia10

#ignore - Number 2 completed on separate page. SIZEWEIGHT = [(270,487.7), (170,140), (70,250), (130,60), (420,600), (130,6.4), (20.8,5.6), (5,2.5), (205,275), (83,3), (128,74), (180,120), (175,150), (165, 35), (195,44), (98,62), (240,82), (300,500), (18.4,4.8), (46,22), (55,23), (43,1.8), (48,21), (87,65), (84,60), (114,35), (32,10.1), (40,22), (220,230), (115,76), (65,25), (13.8,1.4)] SIZEWEIGHT_plot=list_plot(SIZEWEIGHT,xmin=0,ymin=0,marker='D',color='red',thickness=2) show(SIZEWEIGHT_plot) a=var('a') b=var('b') x=var('x') model(x)=a*x+b print(find_fit(SIZEWEIGHT,model)) logSIZEWEIGHT = [(log(270), log(487.7)), (log(170),log(140)), (log(70),log(250)), (log(130),log(60)), (log(420),log(600)), (log(130),log(6.4)), (log(20.8),log(5.6)), (log(5),log(2.5)), (log(205),log(275)), (log(83),log(3)), (log(178),log(74)), (log(180),log(120)), (log(175),log(150)), (log(165),log(35)), (log(195),log(44)), (log(98),log(62)), (log(240),log(82)), (log(300),log(500)), (log(18.4),log(4.8)), (log(46),log(22)), (log(55),log(23)), (log(43),log(1.8)), (log(48),log(21)), (log(87),log(65)), (log(84),log(60)), (log(114),log(35)), (log(32),log(10.1)), (log(40),log(22)), (log(220),log(230)), (log(115),log(76)), (log(65),log(25)), (log(13.8),log(1.4))] SIZEWEIGHTlog_plot=list_plot(logSIZEWEIGHT,xmin=0,ymin=0,marker='D',color='red',thickness=2) a=var('a') b=var('b') x=var('x') logreg=plot(1.3675633219929886*x-2.4672658543193333,(x,0,6)) compplot=logreg+SIZEWEIGHTlog_plot show(compplot) model(x)=a*x+b print(find_fit(logSIZEWEIGHT,model)) # The scatter plot shows that, while there seems to be a correlation between the size and weight of various mammals, # there is by no means a perfectly linear relationship present. 
        
verbose 0 (138: primitive.py, options) WARNING: Ignoring option
'marker'=D
verbose 0 (138: primitive.py, options) WARNING: Ignoring option
'thickness'=2
verbose 0 (138: primitive.py, options) 
The allowed options for Point set defined by 32 point(s) are:
    alpha          How transparent the point is.                        

    faceted        If True color the edge of the point.                 

    hue            The color given as a hue.                            

    legend_label   The label for this item in the legend.               

    rgbcolor       The color as an RGB tuple.                           

    size           How big the point is (i.e., area in points^2=(1/72
inch)^2).
    zorder         The layer level in which to draw                     



[a == 1.3629129388504175, b == -59.719147356104074]
verbose 0 (138: primitive.py, options) WARNING: Ignoring option
'marker'=D
verbose 0 (138: primitive.py, options) WARNING: Ignoring option
'thickness'=2
verbose 0 (138: primitive.py, options) 
The allowed options for Point set defined by 32 point(s) are:
    alpha          How transparent the point is.                        

    faceted        If True color the edge of the point.                 

    hue            The color given as a hue.                            

    legend_label   The label for this item in the legend.               

    rgbcolor       The color as an RGB tuple.                           

    size           How big the point is (i.e., area in points^2=(1/72
inch)^2).
    zorder         The layer level in which to draw                     



[a == 1.3675633219929886, b == -2.4672658543193333]
verbose 0 (138: primitive.py, options) WARNING: Ignoring option 'marker'=D
verbose 0 (138: primitive.py, options) WARNING: Ignoring option 'thickness'=2
verbose 0 (138: primitive.py, options) 
The allowed options for Point set defined by 32 point(s) are:
    alpha          How transparent the point is.                               
    faceted        If True color the edge of the point.                        
    hue            The color given as a hue.                                   
    legend_label   The label for this item in the legend.                      
    rgbcolor       The color as an RGB tuple.                                  
    size           How big the point is (i.e., area in points^2=(1/72 inch)^2).
    zorder         The layer level in which to draw                            


[a == 1.3629129388504175, b == -59.719147356104074]
verbose 0 (138: primitive.py, options) WARNING: Ignoring option 'marker'=D
verbose 0 (138: primitive.py, options) WARNING: Ignoring option 'thickness'=2
verbose 0 (138: primitive.py, options) 
The allowed options for Point set defined by 32 point(s) are:
    alpha          How transparent the point is.                               
    faceted        If True color the edge of the point.                        
    hue            The color given as a hue.                                   
    legend_label   The label for this item in the legend.                      
    rgbcolor       The color as an RGB tuple.                                  
    size           How big the point is (i.e., area in points^2=(1/72 inch)^2).
    zorder         The layer level in which to draw                            


[a == 1.3675633219929886, b == -2.4672658543193333]
x=var('x') a=var('a') b=var('b') c=var('c') a=plot(2*x*e^(-(x/1000)^2),(x,-2000,2000)) b=plot(x,(x,-2000,2000)) c=a+b show(c) print(solve(x==2*x*e^(-(x/1000)^2),x)) Stability=vector(RDF,100) Stability[0]=5 for i in range(99): Stability[i+1]=2*Stability[i]*e^(-(Stability[i]/1000)^2) points = [[x,Stability[x]] for x in range(100)] pplot=scatter_plot(points) show(pplot) for i in range(99): print(i,Stability[i]) # The two non-negative equilibrium points are at 0 and 1000 times the square root of the natural log of 2. # The equilibrium point at 0 is unstable and the point at approximately 832.55461 is stable as shown in the second # graph. # In order to find the equilibrium points, we set P(n+1) equal to y and we set P(n) equal to x. After setting r equal # to 2, we solved for x. We then took the derivative of the original equation and found the slope at the equilibrium # points we found by solving for P in the original calculation to determine stability.If the absolute value of the # derivative was less than one at P, we concluded that P was stable.If not, we would concluded that P was unstable # stability and search for oscillation. 
        

[
x == -1000*sqrt(log(2)),
x == 1000*sqrt(log(2)),
x == 0
]

(0, 5.0)
(1, 9.99975000312)
(2, 19.9975002562)
(3, 39.9790097087)
(4, 79.8303228956)
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(6, 309.406586008)
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[
x == -1000*sqrt(log(2)),
x == 1000*sqrt(log(2)),
x == 0
]

(0, 5.0)
(1, 9.99975000312)
(2, 19.9975002562)
(3, 39.9790097087)
(4, 79.8303228956)
(5, 158.646382903)
(6, 309.406586008)
(7, 562.319936191)
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x=var('x') a=var('a') b=var('b') c=var('c') a=plot(2*x*e^(-(x/1000)^2),(x,-2000,2000)) b=plot(x,(x,-2000,2000)) c=a+b show(c) print(solve(x==2*x*e^(-(x/1000)^2),x)) Stability=vector(RDF,100) Stability1=vector(RDF,100) Stability2=vector(RDF,100) Stability3=vector(RDF,100) Stability[0]=5 Stability1[0]=536 Stability2[0]=1000 Stability3[0]=2000 for i in range(99): Stability[i+1]=3*Stability[i]*e^(-(Stability[i]/1000)^2) Stability1[i+1]=3*Stability1[i]*e^(-(Stability1[i]/1000)^2) Stability2[i+1]=3*Stability2[i]*e^(-(Stability2[i]/1000)^2) Stability3[i+1]=3*Stability3[i]*e^(-(Stability3[i]/1000)^2) points = [[x,Stability[x]] for x in range(100)] points1 = [[x,Stability1[x]] for x in range(100)] points2 = [[x,Stability2[x]] for x in range (100)] points3 = [[x,Stability3[x]] for x in range (100)] pplot=scatter_plot(points) pplot1=scatter_plot(points1) pplot2=scatter_plot(points2) pplot3=scatter_plot(points3) show(pplot) show(pplot1) show(pplot2) show(pplot3) for i in range(99): print(i,Stability[i],Stability1[i],Stability2[i],Stability3[i]) # Both equilibrium points are clearly unstable. However, there is some oscillation between the values of 741.151903771 # and 1283.71264271. This means that the equilibrium point essentially only exists in theory and suggesting an # alternative form of stability. # In order to find the equilibrium points, we set P(n+1) equal to y and we set P(n) equal to x. After setting r equal # # to 3, we solved for x. We then took the derivative of the original equation and found the slope at the equilibrium # points we found by solving for P in the original calculation to determine stability.If the absolute value of the # derivative was less than one at P, we concluded that P was stable.If not, we would concluded that P was unstable # stability and search for oscillation. 
        

[
x == -1000*sqrt(log(2)),
x == 1000*sqrt(log(2)),
x == 0
]




(0, 5.0, 536.0, 1000.0, 2000.0)
(1, 14.9996250047, 1206.46569327, 1103.63832351, 109.893833332)
(2, 44.9887519122, 844.296105846, 979.421711502, 325.7239952)
(3, 134.693361942, 1241.75333995, 1125.86589935, 878.808093242)
(4, 396.815242728, 797.068462283, 950.840159549, 1217.89005067)
(5, 1017.00848646, 1266.78152451, 1155.00650144, 829.009228326)
(6, 1084.55686995, 763.651541803, 912.723988508, 1250.86085213)
(7, 1003.52637638, 1278.65011065, 1190.32389729, 784.892442321)
(8, 1099.73283592, 747.867918307, 865.858967209, 1271.69105727)
(9, 984.390365327, 1282.45597377, 1227.36218132, 757.116378753)
(10, 1120.58979508, 742.818111568, 816.328577148, 1280.36958831)
(11, 957.6680629, 1283.42139252, 1257.69616369, 745.585683665)
(12, 1148.23975195, 741.538121582, 775.764917795, 1282.90759456)
(13, 921.629370949, 1283.64642271, 1274.93517643, 742.219284786)
(14, 1182.46290539, 741.239825994, 752.802718045, 1283.52766385)
(15, 876.335068657, 1283.69771718, 1281.39365198, 741.397247645)
(16, 1219.74567439, 741.171834153, 744.227042225, 1283.67070129)
(17, 826.525219873, 1283.70934818, 1283.16447348, 741.207644106)
(18, 1252.25196378, 741.156417184, 741.878715433, 1283.70322518)
(19, 783.033976923, 1283.71198234, 1283.58732392, 741.164533261)
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(24, 1282.96497988, 741.151956064, 741.160450414, 1283.71264271)
(25, 742.143201002, 1283.71274436, 1283.71129333, 741.152050274)
(26, 1283.54104105, 741.151915542, 741.153838881, 1283.71272827)
(27, 741.37951509, 1283.71275128, 1283.71242276, 741.151936872)
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(29, 741.203602314, 1283.71275285, 1283.71267847, 741.151911197)
(30, 1283.70391658, 741.151904292, 741.152002877, 1283.71275202)
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(98, 1283.71275331, 741.151903684, 741.151903684, 1283.71275331)

[
x == -1000*sqrt(log(2)),
x == 1000*sqrt(log(2)),
x == 0
]




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(9, 984.390365327, 1282.45597377, 1227.36218132, 757.116378753)
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(12, 1148.23975195, 741.538121582, 775.764917795, 1282.90759456)
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(87, 741.151903684, 1283.71275331, 1283.71275331, 741.151903684)
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(93, 741.151903684, 1283.71275331, 1283.71275331, 741.151903684)
(94, 1283.71275331, 741.151903684, 741.151903684, 1283.71275331)
(95, 741.151903684, 1283.71275331, 1283.71275331, 741.151903684)
(96, 1283.71275331, 741.151903684, 741.151903684, 1283.71275331)
(97, 741.151903684, 1283.71275331, 1283.71275331, 741.151903684)
(98, 1283.71275331, 741.151903684, 741.151903684, 1283.71275331)
Stability=vector(RDF,100) Stability1=vector(RDF,100) Stability2=vector(RDF,100) Stability3=vector(RDF,100) Stability[0]=5 Stability1[0]=536 Stability2[0]=1000 Stability3[0]=2000 for i in range(99): Stability[i+1]=3.7*Stability[i]*e^(-(Stability[i]/1000)^2) Stability1[i+1]=3.7*Stability1[i]*e^(-(Stability1[i]/1000)^2) Stability2[i+1]=3.7*Stability2[i]*e^(-(Stability2[i]/1000)^2) Stability3[i+1]=3.7*Stability3[i]*e^(-(Stability3[i]/1000)^2) points = [[x,Stability[x]] for x in range(100)] points1 = [[x,Stability1[x]] for x in range(100)] points2 = [[x,Stability2[x]] for x in range (100)] points3 = [[x,Stability3[x]] for x in range (100)] pplot=var('pplot') pplot=scatter_plot(points) pplot1=scatter_plot(points1) pplot2=scatter_plot(points2) pplot3=scatter_plot(points3) show(pplot) show(pplot1) show(pplot2) show(pplot3) for i in range(99): print(i,Stability[i],Stability1[i],Stability2[i],Stability3[i]) # Both equilibrium points are clearly unstable. There is some oscillation but since the points never converge to a # specific value and instead diverge away from the equilibrium points, the equilibrium points essentially only exist # in theory,suggesting an alternative form of stability. # In order to find the equilibrium points, we set P(n+1) equal to y and we set P(n) equal to x. After setting r equal # to 3.7, we solved for x. We then took the derivative of the original equation and found the slope at the equilibrium # points we found by solving for P in the original calculation to determine stability. If the absolute value of the derivative was less than # one at P, we concluded that P was stable.If not, we would concluded that P was unstable stability and search for # oscillation. 
        




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(2, 68.4248675238, 601.505764643, 789.721925748, 492.354096194)
(3, 251.989438521, 1549.92179515, 1566.1145005, 1429.55491842)
(4, 874.99775315, 519.068055378, 498.683278292, 685.277580926)
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Stability=vector(RDF,100) Stability1=vector(RDF,100) Stability2=vector(RDF,100) Stability3=vector(RDF,100) Stability[0]=5 Stability1[0]=536 Stability2[0]=1000 Stability3[0]=2000 for i in range(99): Stability[i+1]=3.8*Stability[i]*e^(-(Stability[i]/1000)^2) Stability1[i+1]=3.8*Stability1[i]*e^(-(Stability1[i]/1000)^2) Stability2[i+1]=3.8*Stability2[i]*e^(-(Stability2[i]/1000)^2) Stability3[i+1]=3.8*Stability3[i]*e^(-(Stability3[i]/1000)^2) points = [[x,Stability[x]] for x in range(100)] points1 = [[x,Stability1[x]] for x in range(100)] points2 = [[x,Stability2[x]] for x in range (100)] points3 = [[x,Stability3[x]] for x in range (100)] pplot=scatter_plot(points) pplot1=scatter_plot(points1) pplot2=scatter_plot(points2) pplot3=scatter_plot(points3) show(pplot) show(pplot1) show(pplot2) show(pplot3) for i in range(99): print(i,Stability[i],Stability1[i],Stability2[i],Stability3[i]) # Both equilibrium points are clearly unstable. There is some oscillation but since the points never converge to a # specific value and instead diverge away from the equilibrium points, the equilibrium points essentially only exist # in theory,suggesting an alternative form of stability. # In order to find the equilibrium points, we set P(n+1) equal to y and we set P(n) equal to x. After setting r equal # to 3.8, we solved for x. We then took the derivative of the original equation and found the slope at the equilibrium # points we found by solving for P in the original calculation to determine stability.If the absolute value of the # derivative was less than one at P, we concluded that P was stable.If not, we would concluded that P was unstable # stability and search for oscillation. 
        




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Stability=vector(RDF,5000) Stability1=vector(RDF,5000) Stability2=vector(RDF,5000) Stability3=vector(RDF,5000) Stability[0]=5 Stability1[0]=536 Stability2[0]=1000 Stability3[0]=2000 for i in range(1000): Stability[i+1]=4.2*Stability[i]*e^(-(Stability[i]/1000)^2) Stability1[i+1]=4.2*Stability1[i]*e^(-(Stability1[i]/1000)^2) Stability2[i+1]=4.2*Stability2[i]*e^(-(Stability2[i]/1000)^2) Stability3[i+1]=4.2*Stability3[i]*e^(-(Stability3[i]/1000)^2) points = [[x,Stability[x]] for x in range(899,1000)] points1 = [[x,Stability1[x]] for x in range(899,1000)] points2 = [[x,Stability2[x]] for x in range (899,1000)] points3 = [[x,Stability3[x]] for x in range (899,1000)] pplot=scatter_plot(points) pplot1=scatter_plot(points1) pplot2=scatter_plot(points2) pplot3=scatter_plot(points3) show(pplot) show(pplot1) show(pplot2) show(pplot3) for i in range(899,1000): print(i,Stability[i],Stability1[i],Stability2[i],Stability3[i]) # Both equilibrium points are clearly unstable. Additionally, there does not appear to be any legitimate case of # oscillation. The points do not seem to have any obvious relation to the equilibrium point. It is difficult to # decipher any alternative form of stability from the graphs. 
        




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