Title: Eigensystems

1 Modules and classes

GSL
- Eigen
  - EigenValues < Vector
  - EigenVectors < Matrix
  - Symm (Module)
    - Workspace (Class)
  - Symmv (Module)
    - Workspace (Class)
  - Herm (Module)
    - Workspace (Class)
  - Hermv (Module)
    - Workspace (Class)
    - Vectors < Matrix::Complex

2 Real Symmetric Matrices, GSL::Eigen::Symm module

2.1 Workspace classes

GSL::Eigen::Symm::Workspace::alloc(n)
GSL::Eigen::Symm::Workspace::new(n)
GSL::Eigen::Symmv::Workspace::alloc(n)
GSL::Eigen::Symmv::Workspace::new(n)
GSL::Eigen::Herm::Workspace::alloc(n)
GSL::Eigen::Herm::Workspace::new(n)
GSL::Eigen::Hermv::Workspace::alloc(n)
GSL::Eigen::Hermv::Workspace::new(n)

2.2 Methods to solve eigensystems

GSL::Eigen::symm(A)

GSL::Eigen::symm(A, workspace)

GSL::Matrix#eigen_symm

GSL::Matrix#eigen_symm(workspace)

These methods compute the eigenvalues of the real symmetric matrix. The workspace object workspace can be omitted.

GSL::Eigen::symmv(A)

GSL::Matrix#eigen_symmv

These methods compute the eigenvalues and eigenvectors of the real symmetric matrix, and return an array of two elements: The first is a GSL::Vector object which stores all the eigenvalues. The second is a GSL::Matrix object, whose columns contain eigenvectors.

Singleton method of the GSL::Eigen module, GSL::Eigen::symm

m = GSL::Matrix.new([1.0, 1/2.0, 1/3.0, 1/4.0], [1/2.0, 1/3.0, 1/4.0, 1/5.0],
                   [1/3.0, 1/4.0, 1/5.0, 1/6.0], [1/4.0, 1/5.0, 1/6.0, 1/7.0])
eigval, eigvec = Eigen::symmv(m)

Instance method of GSL::Matrix class
```
eigval, eigvec = m.eigen_symmv
```

3 Complex Hermitian Matrices

GSL::Eigen::herm(A)
GSL::Eigen::herm(A, workspace)
GSL::Matrix::Complex#eigen_herm
GSL::Matrix::Complex#eigen_herm(workspace): These methods compute the eigenvalues of the complex hermitian matrix.
GSL::Eigen::hermv(A)
GSL::Eigen::hermv(A, workspace)
GSL::Matrix::Complex#eigen_hermv
GSL::Matrix::Complex#eigen_hermv(workspace

4 Sorting Eigenvalues and Eigenvectors

GSL::Eigen::symmv_sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC)

GSL::Eigen::Symmv::sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC)

These methods simultaneously sort the eigenvalues stored in the vector eval and the corresponding real eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter type,

GSL::Eigen::SORT_VAL_ASC ascending order in numerical value
GSL::Eigen::SORT_VAL_DESC escending order in numerical value
GSL::Eigen::SORT_ABS_ASC scending order in magnitude
GSL::Eigen::SORT_ABS_DESC descending order in magnitude

The sorting is carried out in-place.

GSL::Eigen::hermv_sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC)

GSL::Eigen::Hermv::sort(eval, evec, type = GSL::Eigen::SORT_VAL_ASC)

These methods simultaneously sort the eigenvalues stored in the vector eval and the corresponding complex eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter type as shown above.

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