数学 - Pytyhon - 線型代数学 - 将来はどうなる？ - 人口移動モデル、都市と農村、人口増加の想定、出生、行列、固有値、固有ベクトル、行列式、将来の比率、増加率

対話・おもしろ線形代数
楽天ブックス Yahoo!

学習環境

• Surface
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad
• MyScript Nebo - MyScript(iPad アプリ(iOS))
• 参考書籍
  • Pythonからはじめる数学入門 (楽天ブックス Yahoo!)

対話・おもしろ線形代数 (木村良夫(著)、現代数学社)の第4話(将来はどうなる？)、君も挑戦してみようの解答を求めてみる。あとBASICで書かれたプログラムのコードをPythonで記述してみる。

$A=\left[\begin{array}{cc}0.92& 0.12\\ 0.08& 0.9\end{array}\right]$

$\det \left[\begin{array}{cc}\lambda -0.92& -0.12\\ -0.08& \lambda -0.9\end{array}\right]=0$

${\lambda }^2-1.82\lambda +0.8184=0$

$\lambda =0.91\pm \sqrt{0.0097}$

$\lambda \approx 0.91\pm 0.098=1.008,0.812$

$A\left[\begin{array}{c}x\\ y\end{array}\right]=1.008\left[\begin{array}{c}x\\ y\end{array}\right]$

$\{\begin{array}{l}0.92x+0.12y=1.008x\\ 0.08x+0.9y=1.008y\end{array}$

$\begin{array}{l}0.08x+0.9=1.008\\ x=1.35\end{array}$

よって、 将来の都市と農村の人口の比率は

$1:1.35$

になり、 また、その名人口増加率は1.008% に近づいていく。

コード

#!/usr/bin/env python3
from sympy import Matrix

A = Matrix([[0.92, 0.12],
            [0.08, 0.9]])


X = Matrix([[80.0],
            [20.0]])
PREV = X
print(f'経過年数,都市,農村,比率,都市増加率,農村増加率')
for i in range(100):
    a, b = X
    prev_a, prev_b = PREV
    print(f'{i: 3d},{a: 10.5f},{b: 10.5f},{(a / b):10.5f},{a/prev_a:10.5f},{b/prev_b:10.5f}')
    PREV = X
    X = A * X

入出力結果(Zsh、PowerShell、Terminal、Jupyter(IPython))

% ./sample1.py
経過年数,都市,農村,比率,都市増加率,農村増加率
  0,  80.00000,  20.00000,   4.00000,   1.00000,   1.00000
  1,  76.00000,  24.40000,   3.11475,   0.95000,   1.22000
  2,  72.84800,  28.04000,   2.59800,   0.95853,   1.14918
  3,  70.38496,  31.06384,   2.26582,   0.96619,   1.10784
  4,  68.48182,  33.58825,   2.03886,   0.97296,   1.08127
  5,  67.03387,  35.70797,   1.87728,   0.97886,   1.06311
  6,  65.95612,  37.49989,   1.75884,   0.98392,   1.05018
  7,  65.17961,  39.02639,   1.67014,   0.98823,   1.04071
  8,  64.64841,  40.33812,   1.60266,   0.99185,   1.03361
  9,  64.31711,  41.47618,   1.55070,   0.99488,   1.02821
 10,  64.14888,  42.47393,   1.51031,   0.99738,   1.02406
 11,  64.11384,  43.35845,   1.47869,   0.99945,   1.02082
 12,  64.18775,  44.15171,   1.45380,   1.00115,   1.01830
 13,  64.35094,  44.87156,   1.43411,   1.00254,   1.01630
 14,  64.58745,  45.53248,   1.41849,   1.00368,   1.01473
 15,  64.88435,  46.14623,   1.40606,   1.00460,   1.01348
 16,  65.23115,  46.72235,   1.39614,   1.00534,   1.01248
 17,  65.61934,  47.26861,   1.38822,   1.00595,   1.01169
 18,  66.04202,  47.79129,   1.38188,   1.00644,   1.01106
 19,  66.49362,  48.29553,   1.37681,   1.00684,   1.01055
 20,  66.96959,  48.78546,   1.37274,   1.00716,   1.01014
 21,  67.46628,  49.26448,   1.36947,   1.00742,   1.00982
 22,  67.98072,  49.73534,   1.36685,   1.00763,   1.00956
 23,  68.51050,  50.20026,   1.36474,   1.00779,   1.00935
 24,  69.05369,  50.66108,   1.36305,   1.00793,   1.00918
 25,  69.60872,  51.11926,   1.36169,   1.00804,   1.00904
 26,  70.17434,  51.57603,   1.36060,   1.00813,   1.00894
 27,  70.74952,  52.03238,   1.35972,   1.00820,   1.00885
 28,  71.33344,  52.48910,   1.35901,   1.00825,   1.00878
 29,  71.92546,  52.94687,   1.35845,   1.00830,   1.00872
 30,  72.52504,  53.40622,   1.35799,   1.00834,   1.00868
 31,  73.13179,  53.86760,   1.35762,   1.00837,   1.00864
 32,  73.74536,  54.33138,   1.35733,   1.00839,   1.00861
 33,  74.36549,  54.79787,   1.35709,   1.00841,   1.00859
 34,  74.99200,  55.26732,   1.35690,   1.00842,   1.00857
 35,  75.62472,  55.73995,   1.35674,   1.00844,   1.00855
 36,  76.26353,  56.21593,   1.35662,   1.00845,   1.00854
 37,  76.90836,  56.69542,   1.35652,   1.00846,   1.00853
 38,  77.55914,  57.17855,   1.35644,   1.00846,   1.00852
 39,  78.21584,  57.66543,   1.35637,   1.00847,   1.00852
 40,  78.87842,  58.15615,   1.35632,   1.00847,   1.00851
 41,  79.54689,  58.65081,   1.35628,   1.00847,   1.00851
 42,  80.22123,  59.14948,   1.35625,   1.00848,   1.00850
 43,  80.90147,  59.65223,   1.35622,   1.00848,   1.00850
 44,  81.58762,  60.15913,   1.35620,   1.00848,   1.00850
 45,  82.27971,  60.67022,   1.35618,   1.00848,   1.00850
 46,  82.97776,  61.18558,   1.35617,   1.00848,   1.00849
 47,  83.68181,  61.70524,   1.35615,   1.00848,   1.00849
 48,  84.39189,  62.22926,   1.35614,   1.00849,   1.00849
 49,  85.10805,  62.75769,   1.35614,   1.00849,   1.00849
 50,  85.83033,  63.29056,   1.35613,   1.00849,   1.00849
 51,  86.55877,  63.82793,   1.35613,   1.00849,   1.00849
 52,  87.29342,  64.36984,   1.35612,   1.00849,   1.00849
 53,  88.03433,  64.91633,   1.35612,   1.00849,   1.00849
 54,  88.78154,  65.46744,   1.35612,   1.00849,   1.00849
 55,  89.53511,  66.02322,   1.35612,   1.00849,   1.00849
 56,  90.29509,  66.58371,   1.35611,   1.00849,   1.00849
 57,  91.06153,  67.14894,   1.35611,   1.00849,   1.00849
 58,  91.83448,  67.71897,   1.35611,   1.00849,   1.00849
 59,  92.61400,  68.29383,   1.35611,   1.00849,   1.00849
 60,  93.40014,  68.87357,   1.35611,   1.00849,   1.00849
 61,  94.19295,  69.45822,   1.35611,   1.00849,   1.00849
 62,  94.99250,  70.04784,   1.35611,   1.00849,   1.00849
 63,  95.79884,  70.64245,   1.35611,   1.00849,   1.00849
 64,  96.61203,  71.24212,   1.35611,   1.00849,   1.00849
 65,  97.43212,  71.84687,   1.35611,   1.00849,   1.00849
 66,  98.25918,  72.45675,   1.35611,   1.00849,   1.00849
 67,  99.09325,  73.07181,   1.35611,   1.00849,   1.00849
 68,  99.93441,  73.69209,   1.35611,   1.00849,   1.00849
 69, 100.78271,  74.31763,   1.35611,   1.00849,   1.00849
 70, 101.63821,  74.94849,   1.35611,   1.00849,   1.00849
 71, 102.50097,  75.58469,   1.35611,   1.00849,   1.00849
 72, 103.37105,  76.22630,   1.35611,   1.00849,   1.00849
 73, 104.24853,  76.87336,   1.35611,   1.00849,   1.00849
 74, 105.13345,  77.52590,   1.35611,   1.00849,   1.00849
 75, 106.02588,  78.18399,   1.35611,   1.00849,   1.00849
 76, 106.92589,  78.84766,   1.35611,   1.00849,   1.00849
 77, 107.83354,  79.51696,   1.35611,   1.00849,   1.00849
 78, 108.74889,  80.19195,   1.35611,   1.00849,   1.00849
 79, 109.67201,  80.87267,   1.35611,   1.00849,   1.00849
 80, 110.60297,  81.55916,   1.35611,   1.00849,   1.00849
 81, 111.54183,  82.25148,   1.35611,   1.00849,   1.00849
 82, 112.48866,  82.94968,   1.35611,   1.00849,   1.00849
 83, 113.44353,  83.65381,   1.35611,   1.00849,   1.00849
 84, 114.40651,  84.36391,   1.35611,   1.00849,   1.00849
 85, 115.37765,  85.08004,   1.35611,   1.00849,   1.00849
 86, 116.35705,  85.80225,   1.35611,   1.00849,   1.00849
 87, 117.34475,  86.53059,   1.35611,   1.00849,   1.00849
 88, 118.34084,  87.26511,   1.35611,   1.00849,   1.00849
 89, 119.34539,  88.00586,   1.35611,   1.00849,   1.00849
 90, 120.35846,  88.75291,   1.35611,   1.00849,   1.00849
 91, 121.38013,  89.50629,   1.35611,   1.00849,   1.00849
 92, 122.41048,  90.26608,   1.35611,   1.00849,   1.00849
 93, 123.44957,  91.03231,   1.35611,   1.00849,   1.00849
 94, 124.49748,  91.80504,   1.35611,   1.00849,   1.00849
 95, 125.55429,  92.58434,   1.35611,   1.00849,   1.00849
 96, 126.62006,  93.37024,   1.35611,   1.00849,   1.00849
 97, 127.69489,  94.16283,   1.35611,   1.00849,   1.00849
 98, 128.77884,  94.96213,   1.35611,   1.00849,   1.00849
 99, 129.87199,  95.76823,   1.35611,   1.00849,   1.00849
%