### 14.13 Bidiagonalization

A general matrix A can be factorized by similarity transformations into the form,

    A = U B V^T

where U and V are orthogonal matrices and B is a N-by-N bidiagonal matrix with non-zero entries only on the diagonal and superdiagonal. The size of U is M-by-N and the size of V is N-by-N.

Function: int gsl_linalg_bidiag_decomp (gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V)

  This function factorizes the M-by-N matrix A into bidiagonal form U B V^T. The diagonal and superdiagonal of the matrix B are stored in the diagonal and superdiagonal of A. The orthogonal matrices U and V are stored as compressed Householder vectors in the remaining elements of A. The Householder coefficients are stored in the vectors tau_U and tau_V. The length of tau_U must equal the number of elements in the diagonal of A and the length of tau_V should be one element shorter.

Function: int gsl_linalg_bidiag_unpack (const gsl_matrix * A, const gsl_vector * tau_U, gsl_matrix * U, const gsl_vector * tau_V, gsl_matrix * V, gsl_vector * diag, gsl_vector * superdiag)

  This function unpacks the bidiagonal decomposition of A produced by `gsl_linalg_bidiag_decomp`, (A, tau_U, tau_V) into the separate orthogonal matrices U, V and the diagonal vector diag and superdiagonal superdiag. Note that U is stored as a compact M-by-N orthogonal matrix satisfying U^T U = I for efficiency.

Function: int gsl_linalg_bidiag_unpack2 (gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V, gsl_matrix * V)

  This function unpacks the bidiagonal decomposition of A produced by `gsl_linalg_bidiag_decomp`, (A, tau_U, tau_V) into the separate orthogonal matrices U, V and the diagonal vector diag and superdiagonal superdiag. The matrix U is stored in-place in A.

Function: int gsl_linalg_bidiag_unpack_B (const gsl_matrix * A, gsl_vector * diag, gsl_vector * superdiag)

  This function unpacks the diagonal and superdiagonal of the bidiagonal decomposition of A from `gsl_linalg_bidiag_decomp`, into the diagonal vector diag and superdiagonal vector superdiag.