数学 - Python - 代数学 - 連立方程式と高次方程式 - 2元、3元、2次、3次、4次

学習環境

代数への出発 (新装版数学入門シリーズ) (松坂和夫(著)、岩波書店)の第5章(連立方程式と高次方程式)、問4の解答を求めてみる。

1. $\begin{array}{l} {\begin{cases} 3 x^{2} + 6 y^{2} = 3 \\ 3 x^{2} - 4 y^{2} = 43 \end{cases} \\ 10 y^{2} = - 40 \\ y^{2} = - 4 \\ y = \pm 2 i \\ x^{2} \\ = 1 - 2 y^{2} \\ = 1 + 8 \\ = 9 \\ x = \pm 3 \\ x = \pm 3, y = \pm 2 i \end{array}$
  
  （複号任意）
2. $\begin{array}{l} {\begin{cases} 2 x^{2} + 2 x y + 4 y^{2} = 148 \\ 2 x^{2} + 2 x y + y^{2} = 73 \end{cases} \\ 3 y^{2} = 75 \\ y^{2} = 25 \\ y = \pm 5 \\ x^{2} \pm 5 x + 50 = 74 \\ x^{2} \pm 5 x - 24 = 0 \\ (x \pm 8) (x \mp 3) = 0 \\ x = \pm 3, \mp 8 \\ y = \pm 5 \end{array}$
  
  （複号同順）
3. $\begin{array}{l} x^{3} + y^{3} = 18 \\ (x + y) (x^{2} - x y + y^{2}) = 18 \\ x^{2} - x y + y^{2} = 9 \\ x^{2} - x (2 - x) + {(2 - x)}^{2} - 9 = 0 \\ x^{2} - 2 x + x^{2} + 4 - 4 x + x^{2} - 9 = 0 \\ 3 x^{2} - 6 x - 5 = 0 \\ x = \frac{3 \pm \sqrt{9 + 15}}{3} \\ = \frac{3 \pm 2 \sqrt{6}}{3} \\ y = - \frac{3 \pm 2 \sqrt{6}}{3} + 2 \\ = \frac{3 \mp 2 \sqrt{6}}{3} \end{array}$
  
  （複号同順）
  
  $\begin{array}{l} y^{2} = x^{2} - 13 \\ x^{4} - {(x^{2} - 13)}^{2} + 65 = 0 \\ x^{4} - x^{4} + 26 x^{2} - 169 + 65 = 0 \\ 26 x^{2} - 104 = 0 \\ x^{2} = 4 \\ x = \pm 2 \\ y^{2} = 4 - 13 = - 9 \\ y = \pm 3 i \end{array}$
  
  （複号任意）

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, solve, I, sqrt print('4.') x, y, z = symbols('x, y, z') class TestSimultanEousequations(TestCase): def test1(self): self.assertEqual( solve([x ** 2 + 2 * y ** 2 - 1, 3 * x ** 2 - 4 * y ** 2 - 43]), [{x: s1 * 3, y: s2 * 2 * I} for s1 in [-1, 1] for s2 in [-1, 1]] ) def test2(self): self.assertEqual( solve([x ** 2 + x * y + 2 * y ** 2 - 74, 2 * x ** 2 + 2 * x * y + y ** 2 - 73]), [{x: -8, y: 5}, {x: -3, y: -5}, {x: 3, y: 5}, {x: 8, y: -5}] ) def test3(self): self.assertEqual( solve([x + y - 2, x ** 3 + y ** 3 - 18]), [{x: (3 - 2 * sqrt(6)) / 3, y: (3 + 2 * sqrt(6)) / 3}, {x: (3 + 2 * sqrt(6)) / 3, y: (3 - 2 * sqrt(6)) / 3}], ) def test4(self): self.assertEqual( solve([x ** 2 - y ** 2 - 13, x ** 4 - y ** 4 + 65]), [{x: s1 * 2, y: s2 * 3 * I} for s1 in [-1, 1] for s2 in [-1, 1]] ) if __name__ == "__main__": main()

% ./sample4.py -v 4. test1 (__main__.TestSimultanEousequations) ... ok test2 (__main__.TestSimultanEousequations) ... ok test3 (__main__.TestSimultanEousequations) ... ok test4 (__main__.TestSimultanEousequations) ... ok ---------------------------------------------------------------------- Ran 4 tests in 0.382s OK %

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2020年5月7日木曜日

数学 - Python - 代数学 - 連立方程式と高次方程式 - 2元、3元、2次、3次、4次

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