如何以编程方式求解多元方程组？

Question

我正在尝试解决多元方程组，这是一些Java代码的结果。在运行时之前，不知道变量的形式和数量。一个例子是

(I) (e-a*d*e-b*d*e+2*b*d*f+2*b*d*e*g)/(-1+a*d+b*d)+f == 0

(II) e*g+((f+e*g)*a*d)/(-1+a*d+b*d)==0

(III) -e*h+((-f-e*g)*d)/(-1+a*d+b*d)==0

(IV) -e*j+((-f-e*g)*c)/(-1+a*d+b*d)==0

我尝试使用只返回输入的Symja和引发错误的SymPy

ZeroDivisionError: polynomial division

变量全部来自区间[0,1]，我需要所有解决方案。 Mathematica能够解决这个问题，但由于它是商业软件，我很遗憾无法在这个项目中使用它。

对于使用哪种软件的建议，我将不胜感激。我真的很想要SymPy工作，我不明白为什么它会抛出这个错误，我很感激。低于MWE到SymPy错误：

from sympy.solvers import solve
from sympy.abc import a,b,c,d,e,f,g,h,j

lst = a,b,c,d,e,f,g,h,j
sys = [(e-a*d*e-b*d*e+2*b*d*f+2*b*d*e*g)/(-1+a*d+b*d)+f,e*g+((f+e*g)*a*d)/(-1+a*d+b*d),-e*h+((-f-e*g)*d)/(-1+a*d+b*d),-e*j+((-f-e*g)*c)/(-1+a*d+b*d)]

solution = solve(sys, lst)
print solution

Mathematica版本是：

eqn = {(e - a*d*e - b*d*e + 2*b*d*f + 2*b*d*e*g)/(-1 + a*d + b*d) + f == 0, e*g + ((f + e*g)*a*d)/(-1 + a*d + b*d) == 0, -e*h + ((-f - e*g)*d)/(-1 + a*d + b*d) == 0, -e*j + ((-f - e*g)*c)/(-1 + a*d + b*d) == 0};
Simplify[Solve[eqn, {a, b, c, d, e, f, g, h, j}]]

输出：

{{e -> 0, f -> 0},
{c -> (1 - 2 a d - 3 b d) j, f -> ((-1 + 2 a d + b d) e)/(-1 + 2 a d + 3 b d), g -> (a d)/(1 - 2 a d - 3 b d), h -> d/(1 - 2 a d - 3 b d)},
{a -> 0, c -> j - 3 b d j, f -> ((-1 + b d) e)/(-1 + 3 b d), g -> 0, h -> d/(1 - 3 b d)},
{a -> (1 - b d)/(2 d), c -> -2 b d j, f -> 0, g -> 1/4 - 1/(4 b d), h -> -(1/(2 b))}}

Answer 1

请注意，您有9个变量和4个方程式。因此，您可以消除四个变量 - 您需要告诉Sympy哪个。

以下代码首先将分母与消除a,b,c,d：

相乘

import sympy as sy
from IPython.display import display  # for pretty printing

# sy.init_printing()  # LaTeX-like pretty printing for IPython

a, b, c, d, e, f, g, h, j = sy.symbols("a, b, c, d, e, f, g, h, j", real=True)
lst = a, b, c, d, e, f, g, h, j
sys0 = sy.Matrix([(e-a*d*e-b*d*e+2*b*d*f+2*b*d*e*g)/(-1+a*d+b*d)+f,
                  e*g+((f+e*g)*a*d)/(-1+a*d+b*d),
                  -e*h+((-f-e*g)*d)/(-1+a*d+b*d),
                  -e*j+((-f-e*g)*c)/(-1+a*d+b*d)])

# Denominator can be factored out:
den = a*d + b*d - 1
sys1 = sy.simplify(sys0*den).expand()
print("Factored out denominator:")
display(sys1)

# Elimnate four variables:
sol1 = sy.solve(sys1, a, b, c, d, dict=True)
print("Solutions:")
display(sol1)
print("Substituting back into the equation gives obviously 0, i.e.:")
display(sy.simplify(sys1.subs(sol1[0])).T)

print("The denominator != 0 results in:")
den1 = sy.simplify(den.subs(sol1[0]))
display(sy.solve(den1))

生成以下输出：

Factored out denominator:
Matrix([
[-a*d*e + a*d*f + 2*b*d*e*g - b*d*e + 3*b*d*f + e - f],
[                   2*a*d*e*g + a*d*f + b*d*e*g - e*g],
[              -a*d*e*h - b*d*e*h - d*e*g - d*f + e*h],
[              -a*d*e*j - b*d*e*j - c*e*g - c*f + e*j]])
Solutions:
[{d: 2*e*h/(4*e*g - e + 3*f),
  c: 2*e*j/(4*e*g - e + 3*f),
  a: g/h,
  b: (-e + f)/(2*e*h)}]
Substituting back into the equation gives obviously 0, i.e.:
Matrix([[0, 0, 0, 0]])
The denominator != 0 results in:
[{e: -f/g}]

因此产生的身份是：

-f/g != e  # denominator - only false, if f,e=0 since f,g,e>=0
a = g/h
b = (-e + f)/(2*e*h)
c = 2*e*j/(4*e*g - e + 3*f)
d = 2*e*h/(4*e*g - e + 3*f)

据我所知，Sympy还不能处理多元不等式（尽管我想证明是错误的）。但结果很简单，可以手工完成：

 0 <= g <= h
 0 <= f-e <= 2*e*h
 0 <= 2*e*j <= 4*e*g - e + 3*f
 0 <= 2*e*h <= 4*e*g - e + 3*f
 0 <= e,f,g,h,j <= 1

案例f,e=0也有效。可以通过检查sys0.subs(e,0).diff(f)是否依赖f来验证它。

Answer 2

如果这个问题被关闭，这里是对Mathematica版本的尝试。然而，它目前缺乏将变量限制在区间[0,1]的条件。

Solve[{
  (e - a d e - b d e + 2 b d f + 2 b d e g)/(-1 + a d + b d) + f == 0,
   e g + ((f + e g) a d)/(-1 + a d + b d) == 0,
  -e h + ((-f - e g) d)/(-1 + a d + b d) == 0,
  -e j + ((-f - e g) c)/(-1 + a d + b d) == 0
  },
 {a, b, c, d, e, f, g, h, j}]

Answer 3

好的，我能够自己使用SAGE解决问题，并提供与Mathematica相同的输出。

sage: a,b,c,d,e,f,g,h,j = var('a,b,c,d,e,f,g,h,j')
sage: qe = [(e-a*d*e-b*d*e+2*b*d*f+2*b*d*e*g)+f*(-1+a*d+b*d),e*g*(-1+a*d+b*d)+((f+e*g)*a*d),-e*h*(-1+a*d+b*d)+((-f-e*g)*d),-e*j*(-1+a*d+b*d)+((-f-e*g)*c)]
sage: print(solve(qe,a,b,c,d,e,f,g,h,j, solution_dict=True))

提供输出

[{g: r5, j: r7, b: r2, d: r4, e: 0, h: r6, c: r3, f: 0, a: r1},
{g: r12, j: r14, b: r8, d: r10, e: r11, h: r13, c: r9, f: -r11*r12, a: -(r10*r8 - 1)/r10},
{g: r15*r18, j: r19, b: -1/6*(4*r15*r16*r18 - r16 + 3*r17)*(4*r15*r16*r18^2/(4*r15*r16*r18 - r16 + 3*r17) + 2*r16*r18/(4*r15*r16*r18 - r16 + 3*r17) - r18)/(r16*r18^2), d: 2*r16*r18/(4*r15*r16*r18 - r16 + 3*r17), e: r16, h: r18, c: 2*r16*r19/(4*r15*r16*r18 - r16 + 3*r17), f: r17, a: r15},
{g: r20*r23, j: 0, b: -1/6*(4*r20*r21*r23 - r21 + 3*r22)*(4*r20*r21*r23^2/(4*r20*r21*r23 - r21 + 3*r22) + 2*r21*r23/(4*r20*r21*r23 - r21 + 3*r22) - r23)/(r21*r23^2), d: 2*r21*r23/(4*r20*r21*r23 - r21 + 3*r22), e: r21, h: r23, c: 0, f: r22, a: r20},
{g: -2*r24*r27 - 1, j: 0, b: r24, d: r25, e: r26, h: r27, c: 0, f: 2*r24*r26*r27 + r26, a: -(r24*r25 - 1)/r25},
{g: 0, j: r30, b: r29, d: 0, e: r31, h: 0, c: r30, f: r31, a: r28},
{g: 0, j: 0, b: r33, d: 0, e: r34, h: 0, c: 0, f: r34, a: r32},
{g: -1, j: 0, b: 0, d: r35, e: r36, h: r37, c: 0, f: r36, a: 1/r35},
{g: r40, j: 0, b: 0, d: r38, e: r39, h: 2*r38*r40 + r38, c: 0, f: r39, a: r40/(2*r38*r40 + r38)},
{g: -1/2, j: 0, b: -1/2*(r41 - r42)/(r41*r43), d: -2/3*r41*r43/(r41 - r42), e: r41, h: r43, c: 0, f: r42, a: -1/2/r43}]