SVD

bool matrix::SVD(

  matrix&  U,                   // 酉矩阵

  matrix&  V,                   // 酉矩阵

  vector&  singular_values      // 奇异值向量

   );

  matrix a= {{0, 1, 2, 3, 4, 5, 6, 7, 8}};

  a=a-4;

  Print("matrix a \n", a);

  a.Reshape(3, 3);

  matrix b=a;

  Print("matrix b \n", b);

//--- 执行 SVD 分解

  matrix U, V;

  vector singular_values;

  b.SVD(U, V, singular_values);

  Print("U \n", U);

  Print("V \n", V);

  Print("singular_values = ", singular_values);



// 检查模块

//--- U * 奇异值对角线 * V = A

  matrix matrix_s;

  matrix_s.Diag(singular_values);

  Print("matrix_s \n", matrix_s);

  matrix matrix_vt=V.Transpose();

  Print("matrix_vt \n", matrix_vt);

  matrix matrix_usvt=(U.MatMul(matrix_s)).MatMul(matrix_vt);

  Print("matrix_usvt \n", matrix_usvt);



  ulong errors=(int)b.Compare(matrix_usvt, 1e-9);

  double res=(errors==0);

  Print("errors=", errors);



//---- 其它检查

  matrix U_Ut=U.MatMul(U.Transpose());

  Print("U_Ut \n", U_Ut);

  Print("Ut_U \n", (U.Transpose()).MatMul(U));



  matrix vt_V=matrix_vt.MatMul(V);

  Print("vt_V \n", vt_V);

  Print("V_vt \n", V.MatMul(matrix_vt));



  /*

  matrix a

  [[-4,-3,-2,-1,0,1,2,3,4]]

  matrix b

  [[-4,-3,-2]

   [-1,0,1]

   [2,3,4]]

  U

  [[-0.7071067811865474,0.5773502691896254,0.408248290463863]

   [-6.827109697437648e-17,0.5773502691896253,-0.8164965809277256]

   [0.7071067811865472,0.5773502691896255,0.4082482904638627]]

  V

  [[0.5773502691896258,-0.7071067811865474,-0.408248290463863]

   [0.5773502691896258,1.779939029415334e-16,0.8164965809277258]

   [0.5773502691896256,0.7071067811865474,-0.408248290463863]]

  singular_values = [7.348469228349533,2.449489742783175,3.277709923350408e-17]



  matrix_s

  [[7.348469228349533,0,0]

   [0,2.449489742783175,0]

   [0,0,3.277709923350408e-17]]

  matrix_vt

  [[0.5773502691896258,0.5773502691896258,0.5773502691896256]

   [-0.7071067811865474,1.779939029415334e-16,0.7071067811865474]

   [-0.408248290463863,0.8164965809277258,-0.408248290463863]]

  matrix_usvt

  [[-3.999999999999997,-2.999999999999999,-2]

   [-0.9999999999999981,-5.977974170712231e-17,0.9999999999999974]

   [2,2.999999999999999,3.999999999999996]]

  errors=0



  U_Ut

  [[0.9999999999999993,-1.665334536937735e-16,-1.665334536937735e-16]

   [-1.665334536937735e-16,0.9999999999999987,-5.551115123125783e-17]

   [-1.665334536937735e-16,-5.551115123125783e-17,0.999999999999999]]

  Ut_U

  [[0.9999999999999993,-5.551115123125783e-17,-1.110223024625157e-16]

   [-5.551115123125783e-17,0.9999999999999987,2.498001805406602e-16]

   [-1.110223024625157e-16,2.498001805406602e-16,0.999999999999999]]

  vt_V

  [[1,-5.551115123125783e-17,0]

   [-5.551115123125783e-17,0.9999999999999996,1.110223024625157e-16]

   [0,1.110223024625157e-16,0.9999999999999996]]

  V_vt

  [[0.9999999999999999,1.110223024625157e-16,1.942890293094024e-16]

   [1.110223024625157e-16,0.9999999999999998,1.665334536937735e-16]

   [1.942890293094024e-16,1.665334536937735e-16,0.9999999999999996]

  */

 }