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### 15.1 Real Symmetric Matrices

For real symmetric matrices, the library uses the symmetric bidiagonalization and QR reduction method. This is described in Golub & van Loan, section 8.3. The computed eigenvalues are accurate to an absolute accuracy of \epsilon ||A||_2, where \epsilon is the machine precision.

— Function: gsl_eigen_symm_workspace * gsl_eigen_symm_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues of n-by-n real symmetric matrices. The size of the workspace is O(2n).

— Function: void gsl_eigen_symm_free (gsl_eigen_symm_workspace * w)

This function frees the memory associated with the workspace w.

— Function: int gsl_eigen_symm (gsl_matrix * A, gsl_vector * eval, gsl_eigen_symm_workspace * w)

This function computes the eigenvalues of the real symmetric matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The eigenvalues are stored in the vector eval and are unordered.

— Function: gsl_eigen_symmv_workspace * gsl_eigen_symmv_alloc (const size_t n)

This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real symmetric matrices. The size of the workspace is O(4n).

— Function: void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace * w)

This function frees the memory associated with the workspace w.

— Function: int gsl_eigen_symmv (gsl_matrix * A, gsl_vector * eval, gsl_matrix * evec, gsl_eigen_symmv_workspace * w)

This function computes the eigenvalues and eigenvectors of the real symmetric matrix A. Additional workspace of the appropriate size must be provided in w. The diagonal and lower triangular part of A are destroyed during the computation, but the strict upper triangular part is not referenced. The eigenvalues are stored in the vector eval and are unordered. The corresponding eigenvectors are stored in the columns of the matrix evec. For example, the eigenvector in the first column corresponds to the first eigenvalue. The eigenvectors are guaranteed to be mutually orthogonal and normalised to unit magnitude.

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