Tridiagonal Decomposition of Hermitian Matrices

A hermitian matrix @math{A} can be factorized by similarity transformations into the form,

where @math{U} is an unitary matrix and @math{T} is a real symmetric tridiagonal matrix.

Function: int gsl_linalg_hermtd_decomp (gsl_matrix_complex * A, gsl_vector_complex * tau)

This function factorizes the hermitian matrix A into the symmetric tridiagonal decomposition @math{U T U^T}. On output the real parts of the diagonal and subdiagonal part of the input matrix A contain the tridiagonal matrix @math{T}. The remaining lower triangular part of the input matrix contains the Householder vectors which, together with the Householder coefficients tau, encode the orthogonal matrix @math{Q}. This storage scheme is the same as used by LAPACK. The upper triangular part of A and imaginary parts of the diagonal are not referenced.

Function: int gsl_linalg_hermtd_unpack (const gsl_matrix_complex * A, const gsl_vector_complex * tau, gsl_matrix_complex * Q, gsl_vector * d, gsl_vector * sd)

This function unpacks the encoded tridiagonal decomposition (A, tau) obtained from gsl_linalg_hermtd_decomp into the unitary matrix U, the real vector of diagonal elements d and the real vector of subdiagonal elements sd.

Function: int gsl_linalg_hermtd_unpack_dsd (const gsl_matrix_complex * A, gsl_vector * d, gsl_vector * sd)

This function unpacks the diagonal and subdiagonal of the encoded tridiagonal decomposition (A, tau) obtained from gsl_linalg_hermtd_decomp into the real vectors d and sd.