4

Mathemticaで非線形連立方程式を解こうとしています。SolveとNSolveを試しました。また、方程式を簡略化するためにa_{ij}とb_{ij}とm33= 1の数値を定義しようとしましたが、Mathematicaが長すぎるか、何か間違ったことをしているようです。Mathematicaでは解決策を見つけようとしています。 、しかし、コードでこれを行うには、c / c++libも必要です。

「演算子」の主な方程式：

M[A[(x,y)]]=B[M[(x,y)]]

ここで、「演算子」は遠近法変換です。

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

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

Mathematicaでの私の入力：

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}]
4

1 に答える 1

5

これを行うには、パラメータ変数の代わりに実際の数値が必要です。そうしないと、システムが大きすぎて処理できなくなります。

説明するために、多項式を作成し (分母が消失するシナリオを無視して)、パラメーターにランダムな数値置換を行いました。置き換えによって m33 を排除することもできましたが、システムに m33-1==0 を残すことにしました。そうすれば、どの方程式に対しても特別な処理は必要ありませんでした。効率化のために、変数が線形である方程式のサブセットに対してそのような消去を行うことを検討するかもしれません。

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;

NSolve は、システムが実際には劣決定であると判断しました。つまり、方程式には冗長性があります (より専門的に言えば、代数依存性)。したがって、疑似乱数超平面と交差し、有限の解セットが得られました。

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}

この場合の解決策は次のとおりです。

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.}}

非常に小さな数値残差で元の式を満たすことを確認します。

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.}}
于 2012-08-02T17:54:14.003 に答える