线性阿尔法方程Ax = By的最小二乘解 [英] Least Squares Solution of Linear Algerbraic Equation Ax = By in Eigen C++

问题描述

我有一组矩阵形式的线性代数方程， Ax = By 。 A 是 36x20 和 x 的矩阵是大小 20 ， B 为 36x13 和 y 为 13x1 。 Rank（A）= 20 。因为系统是超定的（有比变量更多的方程），所以最小二乘解是可能的，i，e; x =（A ^ TA）^ - 1A ^ TBy 。我想要解决方案，以便剩余误差e = Ax-By应该最小化。

使用 Eigen / Dense C ++的库已经制定了所有的矩阵等。我尝试本页上描述的方法
Eigen Tutorial ！

我猜这个页面描述的方法只适用于方阵。因为当它试图运行这个错误。

  x = A.jacobiSvd（ComputeThinU | ComputeThinV）.solve（B * y）; 
  

错误

  /usr/include/eigen3/Eigen/src/SVD/JacobiSVD.h：在成员函数'const 
中Eigen :: internal :: solve_retval< Eigen :: JacobiSVD< MatrixType，QRPreconditioner>，Rhs& 
 Eigen :: JacobiSVD< MatrixType，QRPreconditioner> :: solve（const 
 Eigen :: MatrixBase< OtherDerived>&）const [with Rhs = 
 Eigen :: GeneralProduct& ; float，36,13>，Eigen :: Matrix< double，-1,1>，4> 
 _MatrixType = Eigen :: Matrix< float，36，20> ;; int QRPreconditioner = 2]'：
 /usr/include/eigen3/Eigen/src/SVD/JacobiSVD.h:658:5：warning：控制到达
结束非空函数[-Wreturn-类型] 
 make [2]：*** [src / CMakeFiles / spacebot_actuationKinematics.dir 
 /ActuationKinematics.cpp.o]错误1 
 make [1]：*** [src / CMakeFiles / spacebot_actuationKinematics.dir / all]错误2 
 make：*** [all]错误2 
  

ComputeThin * 选项只适用于动态大小的矩阵。对于固定大小，您必须使用 ComputeFull * 。不过，在你的情况下，最好使用动态大小矩阵，即 MatrixXf 。固定大小矩阵只对非常小的矩阵有意义。

最后，ColPivHouseholderQR可能是最小二乘解的更好的选择。 SVD有点过分。

I have a set of linear algebraic equations in matrices form, Ax=By. Where A is matrix of 36x20 and x is a vector of size 20, B is 36x13 and y is 13x1. Rank(A)=20. Because system is overdetermined (there are more number of equations than the variables), so least squares solution is possible, i,e; x = (A^TA)^-1A^TBy. I want the solution so that the residual error e = Ax-By should be minimized.

Using Eigen/Dense library of C++ i have formulated all the matrices etc. I tried the method described on this page Eigen Tutorial!

I guess the method described in this page is only for square matrices. Because when it try to run this it gives error.

 x = A.jacobiSvd( ComputeThinU | ComputeThinV ).solve(B*y);

Error

 /usr/include/eigen3/Eigen/src/SVD/JacobiSVD.h: In member function 'const    
 Eigen::internal::solve_retval<Eigen::JacobiSVD<MatrixType, QRPreconditioner>, Rhs> 
 Eigen::JacobiSVD<MatrixType, QRPreconditioner>::solve(const 
 Eigen::MatrixBase<OtherDerived>&) const [with Rhs = 
 Eigen::GeneralProduct<Eigen::Matrix<float, 36, 13>, Eigen::Matrix<double, -1, 1>, 4>; 
 _MatrixType = Eigen::Matrix<float, 36, 20>; int QRPreconditioner = 2]':
 /usr/include/eigen3/Eigen/src/SVD/JacobiSVD.h:658:5: warning: control reaches end of  
 non-void function [-Wreturn-type]
 make[2]: *** [src/CMakeFiles/spacebot_actuationKinematics.dir 
 /ActuationKinematics.cpp.o] Error 1
 make[1]: *** [src/CMakeFiles/spacebot_actuationKinematics.dir/all] Error 2
 make: *** [all] Error 2

解决方案

As explained in the documentation, the ComputeThin* options are only for Dynamic sized matrices. For fixed sizes, you must use ComputeFull*. Nevertheless, in your case it is better to use Dynamic size matrices, i.e., MatrixXf. Fixed size matrices only makes sense for very small ones.

Finally, ColPivHouseholderQR is probably a better choice for least-square solving. SVD is a bit overkill.

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