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Plotting life history vectors in sage

asked 2018-04-15 13:08:56 -0500

maeve.mccarthy gravatar image

updated 2018-04-16 01:35:56 -0500

slelievre gravatar image

I'm teaching a math bio class where I need to plot a life history graph. The matrix equation is new=A*old, where A is a square matrix. I want to create a matrix of the population at each iteration (for say 10 iterations) and plot each column separately. I'm having trouble figuring it out and would appreciate some help.

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answered 2018-04-15 13:32:46 -0500

slelievre gravatar image

updated 2018-04-15 16:13:39 -0500

One way is to define the matrix a and the list p of vectors of populations, initially containing just the vector of initial populations:

sage: a = matrix(RR, 2, [1.02, -0.01, 0.03, 1.03])
sage: p = [vector(RR, 2, [300, 200])]

then extend the list by applying a number of times the matrix to the last vector:

sage: for k in range(20):
....:     p.append(a * p[-1])

then observe the evolution of populations:

sage: p
[(300.000000000000, 200.000000000000),
 (304.000000000000, 215.000000000000),
 (307.930000000000, 230.570000000000),
 (311.782900000000, 246.725000000000),
 (315.551308000000, 263.480237000000),
 (319.227531790000, 280.851183350000),
 (322.803570592300, 298.853544804200),
 (326.271106556104, 317.503258266095),
 (329.621496104565, 336.816489210761),
 (332.845761134549, 356.809628770221),
 (335.934580069538, 377.499290467364),
 (338.878278766255, 398.902306583471),
 (341.666821275745, 421.035724143963),
 (344.289800459820, 443.916800506554),
 (346.736428463951, 467.562998535545),
 (348.995527047875, 491.991981345530),
 (351.055517775377, 517.221606597332),
 (352.904412064911, 543.269920328514),
 (354.529801102924, 570.155150300316),
 (355.918845621980, 597.895698842414),
 (357.058265545995, 626.510135176345)]

or plot it as a set of points:

sage: point2d(p)
Launched png viewer for Graphics object consisting of 1 graphics primitive

or one could color the first point in red:

sage: point2d(p[0], color='red') + point2d(p[1:])
Launched png viewer for Graphics object consisting of 2 graphics primitives

or one could use line2d instead of point2d, or one could use animate to animate the evolution.

If you want to plot the evolution of each population:

sage: x, y = zip(*p)
sage: px = list_plot(x, color='green')
sage: py = list_plot(y, color='purple')
sage: px
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: py
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: px + py
Launched png viewer for Graphics object consisting of 2 graphics primitives
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Comments

This helps a lot. Thank you!

maeve.mccarthy gravatar imagemaeve.mccarthy ( 2018-04-15 14:45:13 -0500 )edit

How do I plot each column separately? This gives the first column vs the second column. Suppose I want to plot the column entries from the first column on the y-axis and the index on the x-axis.

maeve.mccarthy gravatar imagemaeve.mccarthy ( 2018-04-15 14:58:30 -0500 )edit

Edited answer to address your follow-up question.

slelievre gravatar imageslelievre ( 2018-04-15 16:14:06 -0500 )edit

Thanks so much! You've really rescued me!

maeve.mccarthy gravatar imagemaeve.mccarthy ( 2018-04-15 17:03:13 -0500 )edit
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answered 2018-04-15 13:29:47 -0500

tmonteil gravatar image

updated 2018-04-15 13:30:16 -0500

We need much more input to help seriously, like the size of the matrix, a concrete example of A, an example of picture you want to obtain.

Waiting for more details, here is a sample of what you can easily get:

sage: A = matrix([[2,1],[1,1]])
sage: A
[2 1]
[1 1]
sage: C = matrix([[2],[3]])
sage: C
[2]
[3]
sage: A * C
[7]
[5]
sage: L = []
sage: L = [C]
sage: for i in range(10):
....:     C = A*C
....:     L.append(C)
sage: L
[
[2]  [7]  [19]  [50]  [131]  [343]  [898]  [2351]  [6155]  [16114]
[3], [5], [12], [31], [ 81], [212], [555], [1453], [3804], [ 9959],

[42187]
[26073]
]
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Comments

Thank you!

maeve.mccarthy gravatar imagemaeve.mccarthy ( 2018-04-15 14:45:18 -0500 )edit

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Asked: 2018-04-15 13:08:56 -0500

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Last updated: Apr 16