Eig – fxbook.net

Eig

计算方阵的本征值和右本征向量。

bool matrix::Eig(
  matrix&  eigen_vectors,         // 本征向量矩阵
  vector&  eigen_values           // 本征向量向量
   );

特征值和特征向量的复解

bool matrix::Eig(
  matrix<complex>& eigen_vectors, // 特征向量矩阵
  vector<complex>& eigen_values   // 特征值向量
   );

参数

eigen_vectors: [输出] 垂直本征向量矩阵。
eigen_values: [输出] 本征值向量。

返回值

成功时返回 true，否则返回 false。

注意

如果计算特征值时遇到复解，则停止计算，并将错误代码设置为4019 (ERR_MATH_OVERFLOW)。使用Eig方法的复数重载在复数空间中获得完整的解决方案

如果一个复特征值的虚部等于零，那么它就是一个实特征值。这可以在下面的示例中看到。

示例：

void OnStart()
  {
   matrix matrix_a =
     {
        {-3.474589, 1.106384, -9.091977,-3.925227 },
        {-5.522139, 2.366887,-15.162351,-6.357512 },
        { 8.394926,-2.960067, 22.292115, 9.524129 },
        { 7.803242,-2.080287, 19.217706, 8.186645 }
     };

   matrix eigen_vectors;
   vector eigen_values;

   bool res=matrix_a.Eig(eigen_vectors, eigen_values);
   Print("res=",res,"  error=",GetLastError());
   Print("Eigen vectors:\n",eigen_vectors, "\nEigen Values:\n",eigen_values);

//--- 检查正确性 A * v = lambda * v
   int vectors=0;
   for(ulong n=0; n<eigen_values.Size(); n++)
     {
      vector eigen_vector=eigen_vectors.Col(n);
      vector vector_d1   =eigen_vector*eigen_values[n];
      vector vector_d2   =matrix_a.MatMul(eigen_vector);

      ulong errors=vector_d1.Compare(vector_d2,1e-13);
      if(errors==0)
         vectors++;
     }
   Print("vectors=",vectors);

//--- 复解
   matrix<complex> eigen_vectors_c;
   vector<complex> eigen_values_c;
   ResetLastError();
   res=matrix_a.Eig(eigen_vectors_c,eigen_values_c);
   Print("res=",res,"  error=",GetLastError());
   Print("Eigen vectors:\n",eigen_vectors_c, "\nEigen Values:\n",eigen_values_c);

//--- 检查正确性 A * v = lambda * v
   matrixc matrix_c;
   matrix_c.Assign(matrix_a);
   vectors=0;
   for(ulong n=0; n<eigen_values_c.Size(); n++)
     {
      vectorc eigen_vector_c=eigen_vectors_c.Col(n);
      vectorc vector_c1     =eigen_vector_c*eigen_values_c[n];
      vectorc vector_c2     =matrix_c.MatMul(eigen_vector_c);

      ulong errors=vector_c1.Compare(vector_c2,1e-13);
      if(errors==0)
         vectors++;
     }
   Print("vectors=",vectors);
  }
/* Result
   res=true  error=4019
   Eigen vectors:
   [[0.2649667608713664]
    [0.4488818803991876]
    [-0.6527335897527492]
    [-0.5497604331807768]]
   Eigen Values:
   [28.94158645962942]
   vectors=1
   res=true  error=0
   Eigen vectors:
   [[(0.2649667608713664,0),(0.2227392219204762,0.3745470492013296),(0.403285439089771,-0.1345146135716524),(-0.2050778513598178,0)]
    [(0.4488818803991876,0),(-0.2613452530342438,-0.3685707727819327),(-0.3968506126372395,0.5257002255533059),(0.8014703373356256,0)]
    [(-0.6527335897527492,0),(-0.3418479634708521,-0.3299830378162041),(-0.3553021031180292,-0.03685418759727989),(-0.05618703726964112,0)]
    [(-0.5497604331807768,0),(0.523227993452828,0.3262508584080381),(0.3512835596143666,0.3666300350184092),(0.558955624442244,0)]]
   Eigen Values:
   [(28.94158645962942,0),(0.01022501104810897,0.02954980822331488),(0.01022501104810897,-0.02954980822331488),(0.4090215182743634,0)]
   vectors=3
*/

Cholesky EigVals