在Mathematica中求解非线性方程组

我在尝试用数学来解非线性方程组。 我试着求解和求解，我也试着定义a{ij}和b{ij}和m33=1数值来简化方程，但Mathematica似乎工作太长，或者我做错了什么。在Mathematica中，我只是试图找到解，但我的代码中也需要一些c/c++库来完成这项工作

“运算符”中的主要等式：

其中“运算符”是透视变换：

u= (m13 + m11*x + m12*y)/(m33 + m31*x + m32*y); 

v= (m23 + m21*x +m22*y)/(m33 + m31*x + m32*y);

我在Mathematica中的输入：

Solve[(b13 + (b11 (m13 + m11 x1 + m12 y1))/(m33 + m31 x1 + 
         m32 y1) + (b12 (m23 + m21 x1 + m22 y1))/(m33 + m31 x1 + 
         m32 y1))/(b33 + (b31 (m13 + m11 x1 + m12 y1))/(m33 + m31 x1 +
          m32 y1) + (b32 (m23 + m21 x1 + m22 y1))/(m33 + m31 x1 + 
         m32 y1)) == (m13 + (m11 (a13 + a11 x1 + a12 y1))/(a33 + 
         a31 x1 + a32 y1) + (m12 (a23 + a21 x1 + a22 y1))/(a33 + 
         a31 x1 + a32 y1))/(m33 + (m31 (a13 + a11 x1 + a12 y1))/(a33 +
          a31 x1 + a32 y1) + (m32 (a23 + a21 x1 + a22 y1))/(a33 + 
         a31 x1 + 
         a32 y1)) && (b23 + (b21 (m13 + m11 x1 + m12 y1))/(m33 + 
         m31 x1 + m32 y1) + (b22 (m23 + m21 x1 + m22 y1))/(m33 + 
         m31 x1 + m32 y1))/(b33 + (b31 (m13 + m11 x1 + m12 y1))/(m33 +
          m31 x1 + m32 y1) + (b32 (m23 + m21 x1 + m22 y1))/(m33 + 
         m31 x1 + 
         m32 y1)) == (m23 + (m21 (a13 + a11 x1 + a12 y1))/(a33 + 
         a31 x1 + a32 y1) + (m22 (a23 + a21 x1 + a22 y1))/(a33 + 
         a31 x1 + a32 y1))/(m33 + (m31 (a13 + a11 x1 + a12 y1))/(a33 +
          a31 x1 + a32 y1) + (m32 (a23 + a21 x1 + a22 y1))/(a33 + 
         a31 x1 + 
         a32 y1)) && (b13 + (b11 (m13 + m11 x2 + m12 y2))/(m33 + 
         m31 x2 + m32 y2) + (b12 (m23 + m21 x2 + m22 y2))/(m33 + 
         m31 x2 + m32 y2))/(b33 + (b31 (m13 + m11 x2 + m12 y2))/(m33 +
          m31 x2 + m32 y2) + (b32 (m23 + m21 x2 + m22 y2))/(m33 + 
         m31 x2 + 
         m32 y2)) == (m13 + (m11 (a13 + a11 x2 + a12 y2))/(a33 + 
         a31 x2 + a32 y2) + (m12 (a23 + a21 x2 + a22 y2))/(a33 + 
         a31 x2 + a32 y2))/(m33 + (m31 (a13 + a11 x2 + a12 y2))/(a33 +
          a31 x2 + a32 y2) + (m32 (a23 + a21 x2 + a22 y2))/(a33 + 
         a31 x2 + 
         a32 y2)) && (b23 + (b21 (m13 + m11 x2 + m12 y2))/(m33 + 
         m31 x2 + m32 y2) + (b22 (m23 + m21 x2 + m22 y2))/(m33 + 
         m31 x2 + m32 y2))/(b33 + (b31 (m13 + m11 x2 + m12 y2))/(m33 +
          m31 x2 + m32 y2) + (b32 (m23 + m21 x2 + m22 y2))/(m33 + 
         m31 x2 + 
         m32 y2)) == (m23 + (m21 (a13 + a11 x2 + a12 y2))/(a33 + 
         a31 x2 + a32 y2) + (m22 (a23 + a21 x2 + a22 y2))/(a33 + 
         a31 x2 + a32 y2))/(m33 + (m31 (a13 + a11 x2 + a12 y2))/(a33 +
          a31 x2 + a32 y2) + (m32 (a23 + a21 x2 + a22 y2))/(a33 + 
         a31 x2 + 
         a32 y2)) && (b13 + (b11 (m13 + m11 x3 + m12 y3))/(m33 + 
         m31 x3 + m32 y3) + (b12 (m23 + m21 x3 + m22 y3))/(m33 + 
         m31 x3 + m32 y3))/(b33 + (b31 (m13 + m11 x3 + m12 y3))/(m33 +
          m31 x3 + m32 y3) + (b32 (m23 + m21 x3 + m22 y3))/(m33 + 
         m31 x3 + 
         m32 y3)) == (m13 + (m11 (a13 + a11 x3 + a12 y3))/(a33 + 
         a31 x3 + a32 y3) + (m12 (a23 + a21 x3 + a22 y3))/(a33 + 
         a31 x3 + a32 y3))/(m33 + (m31 (a13 + a11 x3 + a12 y3))/(a33 +
          a31 x3 + a32 y3) + (m32 (a23 + a21 x3 + a22 y3))/(a33 + 
         a31 x3 + 
         a32 y3)) && (b23 + (b21 (m13 + m11 x3 + m12 y3))/(m33 + 
         m31 x3 + m32 y3) + (b22 (m23 + m21 x3 + m22 y3))/(m33 + 
         m31 x3 + m32 y3))/(b33 + (b31 (m13 + m11 x3 + m12 y3))/(m33 +
          m31 x3 + m32 y3) + (b32 (m23 + m21 x3 + m22 y3))/(m33 + 
         m31 x3 + 
         m32 y3)) == (m23 + (m21 (a13 + a11 x3 + a12 y3))/(a33 + 
         a31 x3 + a32 y3) + (m22 (a23 + a21 x3 + a22 y3))/(a33 + 
         a31 x3 + a32 y3))/(m33 + (m31 (a13 + a11 x3 + a12 y3))/(a33 +
          a31 x3 + a32 y3) + (m32 (a23 + a21 x3 + a22 y3))/(a33 + 
         a31 x3 + 
         a32 y3)) && (b13 + (b11 (m13 + m11 x4 + m12 y4))/(m33 + 
         m31 x4 + m32 y4) + (b12 (m23 + m21 x4 + m22 y4))/(m33 + 
         m31 x4 + m32 y4))/(b33 + (b31 (m13 + m11 x4 + m12 y4))/(m33 +
          m31 x4 + m32 y4) + (b32 (m23 + m21 x4 + m22 y4))/(m33 + 
         m31 x4 + 
         m32 y4)) == (m13 + (m11 (a13 + a11 x4 + a12 y4))/(a33 + 
         a31 x4 + a32 y4) + (m12 (a23 + a21 x4 + a22 y4))/(a33 + 
         a31 x4 + a32 y4))/(m33 + (m31 (a13 + a11 x4 + a12 y4))/(a33 +
          a31 x4 + a32 y4) + (m32 (a23 + a21 x4 + a22 y4))/(a33 + 
         a31 x4 + 
         a32 y4)) && (b23 + (b21 (m13 + m11 x4 + m12 y4))/(m33 + 
         m31 x4 + m32 y4) + (b22 (m23 + m21 x4 + m22 y4))/(m33 + 
         m31 x4 + m32 y4))/(b33 + (b31 (m13 + m11 x4 + m12 y4))/(m33 +
          m31 x4 + m32 y4) + (b32 (m23 + m21 x4 + m22 y4))/(m33 + 
         m31 x4 + 
         m32 y4)) == (m23 + (m21 (a13 + a11 x4 + a12 y4))/(a33 + 
         a31 x4 + a32 y4) + (m22 (a23 + a21 x4 + a22 y4))/(a33 + 
         a31 x4 + a32 y4))/(m33 + (m31 (a13 + a11 x4 + a12 y4))/(a33 +
          a31 x4 + a32 y4) + (m32 (a23 + a21 x4 + a22 y4))/(a33 + 
         a31 x4 + a32 y4)) && m33 == 1, {m11, m12, m13, m21, m22, m23,
   m31, m32}]

---

为了有机会做到这一点，您需要实际的数字来代替参数变量。否则，系统太大，无法处理

为了举例说明，我创建了多项式（忽略分母消失场景），然后对参数进行随机数字替换。我本可以通过替换消除m33，但选择将m33-1==0保留在系统中。这样，任何一个方程都不需要特殊处理。一个可能的效率考虑在其变量是线性的方程子集上进行这样的消除。

In[40]:= exprs = Apply[Subtract, eqns, {1}];
e2 = Together[exprs];
polys = Numerator[e2];

In[62]:= allvars = Variables[polys];
vars = {m11, m12, m13, m21, m22, m23, m31, m32, m33};
params = Complement[allvars, vars]

Out[64]= {a11, a12, a13, a21, a22, a23, a31, a32, a33, b11, b12, b13, \
b21, b22, b23, b31, b32, b33, x1, x2, x3, x4, y1, y2, y3, y4}

In[69]:= SeedRandom[11111];
substitutions = 
  Thread[params -> RandomInteger[{-1000, 1000}, Length[params]]];

In[71]:= numpolys = polys /. substitutions;

我认为系统实际上是欠定的，也就是说，你的方程有冗余（更严格地说，是代数依赖）。因此它与伪随机超平面相交，然后得到一个有限解集

In[73]:= Timing[solns = NSolve[numpolys == 0, vars];]

During evaluation of In[73]:= NSolve::infsolns: Infinite solution set has dimension at least 1. Returning intersection of solutions with (107814 m11)/118505-(177066 m12)/118505-(164294 m13)/118505+(32943 m21)/23701+(186238 m22)/118505-(126102 m23)/118505-(178233 m31)/118505-(185338 m32)/118505+(141088 m33)/118505 == 1.

Out[73]= {357.420000, Null}

以下是本例中的解决方案

In[74]:= solns // N

Out[74]= {{m11 -> -2.22241, m12 -> 0., m13 -> -2.41203, 
  m21 -> -0.539924, m22 -> 2.33146*10^-172, m23 -> -0.585993, 
  m31 -> 0.921382, m32 -> -4.05984*10^-172, 
  m33 -> 1.}, {m11 -> -2.22241, m12 -> 0., m13 -> -2.41203, 
  m21 -> -0.539924, m22 -> 2.33146*10^-172, m23 -> -0.585993, 
  m31 -> 0.921382, m32 -> -4.05984*10^-172, 
  m33 -> 1.}, {m11 -> -2.22241, m12 -> 0., m13 -> -2.41203, 
  m21 -> -0.539924, m22 -> 2.33146*10^-172, m23 -> -0.585993, 
  m31 -> 0.921382, m32 -> -4.05984*10^-172, 
  m33 -> 1.}, {m11 -> -0.029351, m12 -> 0., m13 -> 0.409304, 
  m21 -> 0.0182228, m22 -> -2.19075*10^-169, m23 -> -0.25412, 
  m31 -> -0.0717095, m32 -> 2.05529*10^-169, 
  m33 -> 1.}, {m11 -> -0.029351, m12 -> 0., m13 -> 0.409304, 
  m21 -> 0.0182228, m22 -> -2.19075*10^-169, m23 -> -0.25412, 
  m31 -> -0.0717095, m32 -> 2.05529*10^-169, 
  m33 -> 1.}, {m11 -> -0.029351, m12 -> 0., m13 -> 0.409304, 
  m21 -> 0.0182228, m22 -> -2.19075*10^-169, m23 -> -0.25412, 
  m31 -> -0.0717095, m32 -> 2.05529*10^-169, 
  m33 -> 1.}, {m11 -> 0.541883, m12 -> 0., m13 -> -0.123031, 
  m21 -> -4.58369, m22 -> -5.60174*10^-170, m23 -> 1.0407, 
  m31 -> -4.40445, m32 -> -5.32622*10^-170, 
  m33 -> 1.}, {m11 -> 0.541883, m12 -> 0., m13 -> -0.123031, 
  m21 -> -4.58369, m22 -> -5.60174*10^-170, m23 -> 1.0407, 
  m31 -> -4.40445, m32 -> -5.32622*10^-170, 
  m33 -> 1.}, {m11 -> 0.541883, m12 -> 0., m13 -> -0.123031, 
  m21 -> -4.58369, m22 -> -5.60174*10^-170, m23 -> 1.0407, 
  m31 -> -4.40445, m32 -> -5.32622*10^-170, m33 -> 1.}}

我们用非常小的数值残差检查它们是否满足原始表达式

In[76]:= exprs /. substitutions /. solns

Out[76]= {{-4.44089*10^-16, 1.11022*10^-16, -8.88178*10^-16, 0., 
  0., -1.11022*10^-16, -4.44089*10^-16, 1.11022*10^-16, 
  0.}, {-4.44089*10^-16, 1.11022*10^-16, -8.88178*10^-16, 0., 
  0., -1.11022*10^-16, -4.44089*10^-16, 1.11022*10^-16, 
  0.}, {-4.44089*10^-16, 1.11022*10^-16, -8.88178*10^-16, 0., 
  0., -1.11022*10^-16, -4.44089*10^-16, 1.11022*10^-16, 
  0.}, {-2.22045*10^-16, 1.11022*10^-16, -1.66533*10^-16, 
  1.11022*10^-16, -5.55112*10^-17, 1.11022*10^-16, -1.11022*10^-16, 
  5.55112*10^-17, 0.}, {-2.22045*10^-16, 
  1.11022*10^-16, -1.66533*10^-16, 1.11022*10^-16, -5.55112*10^-17, 
  1.11022*10^-16, -1.11022*10^-16, 5.55112*10^-17, 
  0.}, {-2.22045*10^-16, 1.11022*10^-16, -1.66533*10^-16, 
  1.11022*10^-16, -5.55112*10^-17, 1.11022*10^-16, -1.11022*10^-16, 
  5.55112*10^-17, 0.}, {2.34535*10^-15, -1.11022*10^-15, 
  1.11022*10^-16, 2.22045*10^-16, 1.31839*10^-15, -1.11022*10^-15, 
  1.249*10^-15, -6.66134*10^-16, 
  0.}, {2.34535*10^-15, -1.11022*10^-15, 1.11022*10^-16, 
  2.22045*10^-16, 1.31839*10^-15, -1.11022*10^-15, 
  1.249*10^-15, -6.66134*10^-16, 
  0.}, {2.34535*10^-15, -1.11022*10^-15, 1.11022*10^-16, 
  2.22045*10^-16, 1.31839*10^-15, -1.11022*10^-15, 
  1.249*10^-15, -6.66134*10^-16, 0.}}

---

您正在使用一个程序生成Mathematica输入，对吗？；）我用Mathematica进行替换，得到两个方程，然后复制粘贴4次，得到4个点的8个方程（x1，y1），…，（x4，y4）。@Loud先生，请你澄清一下，

u，v

在哪里进入最上面的方程？@b.gatesucks u，v只是运算的结果，例如（u，v）=A[（x，y）]，然后我们将（u，v）替换成M，得到M[（u，v）]. A、 B，M是相同的算符，只是不同的常数A（ij），B（ij），M（ij）。它似乎是超定的：9个等式，8个未知数。毫不奇怪，a11等的随机替换给出了一个空的解决方案集。对不起，我不明白。请至少告诉我您使用的技术是如何调用的？我还可以给出a（I，j）b（I，j）的实数，所以我们可以测试这个解。这里有人说这个非线性方程可以在三维空间中作为矩阵方程来求解，也许用这种方法来求解它会更简单？例如一个矩阵（0-13001001）B矩阵（-0.4009-1.0787446.14631.61800.8875-159.22720.0003 0.00291）但矩阵B可能存在错误。我基于math.stackexchange响应的建议是尝试将其重新定义为矩阵零空间计算。使用非零公差，您可能能够克服输入参数中的小数值误差。您能更具体一点吗？矩阵零空间如何帮助我？如果你有一个形式为a.M=M.B的矩阵方程，a，B已知，M为未知矩阵，那么a.M-M.B=0，这可以被重新表示为矩阵乘以向量=0，从而进行零空间计算。

In[76]:= exprs /. substitutions /. solns

Out[76]= {{-4.44089*10^-16, 1.11022*10^-16, -8.88178*10^-16, 0., 
  0., -1.11022*10^-16, -4.44089*10^-16, 1.11022*10^-16, 
  0.}, {-4.44089*10^-16, 1.11022*10^-16, -8.88178*10^-16, 0., 
  0., -1.11022*10^-16, -4.44089*10^-16, 1.11022*10^-16, 
  0.}, {-4.44089*10^-16, 1.11022*10^-16, -8.88178*10^-16, 0., 
  0., -1.11022*10^-16, -4.44089*10^-16, 1.11022*10^-16, 
  0.}, {-2.22045*10^-16, 1.11022*10^-16, -1.66533*10^-16, 
  1.11022*10^-16, -5.55112*10^-17, 1.11022*10^-16, -1.11022*10^-16, 
  5.55112*10^-17, 0.}, {-2.22045*10^-16, 
  1.11022*10^-16, -1.66533*10^-16, 1.11022*10^-16, -5.55112*10^-17, 
  1.11022*10^-16, -1.11022*10^-16, 5.55112*10^-17, 
  0.}, {-2.22045*10^-16, 1.11022*10^-16, -1.66533*10^-16, 
  1.11022*10^-16, -5.55112*10^-17, 1.11022*10^-16, -1.11022*10^-16, 
  5.55112*10^-17, 0.}, {2.34535*10^-15, -1.11022*10^-15, 
  1.11022*10^-16, 2.22045*10^-16, 1.31839*10^-15, -1.11022*10^-15, 
  1.249*10^-15, -6.66134*10^-16, 
  0.}, {2.34535*10^-15, -1.11022*10^-15, 1.11022*10^-16, 
  2.22045*10^-16, 1.31839*10^-15, -1.11022*10^-15, 
  1.249*10^-15, -6.66134*10^-16, 
  0.}, {2.34535*10^-15, -1.11022*10^-15, 1.11022*10^-16, 
  2.22045*10^-16, 1.31839*10^-15, -1.11022*10^-15, 
  1.249*10^-15, -6.66134*10^-16, 0.}}