### 14.17 Tridiagonal Systems

The functions described in this section efficiently solve symmetric, non-symmetric and cyclic tridiagonal systems with minimal storage. Note that the current implementations of these functions use a variant of Cholesky decomposition, so the tridiagonal matrix must be positive definite. For non-positive definite matrices, the functions return the error code `GSL_ESING`.

Function: int gsl_linalg_solve_tridiag (const gsl_vector * diag, const gsl_vector * e, const gsl_vector * f, const gsl_vector * b, gsl_vector * x)

  This function solves the general N-by-N system A x = b where A is tridiagonal (N >= 2). The super-diagonal and sub-diagonal vectors e and f must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

      A = ( d_0 e_0  0   0  )
          ( f_0 d_1 e_1  0  )
          (  0  f_1 d_2 e_2 )
          (  0   0  f_2 d_3 )

Function: int gsl_linalg_solve_symm_tridiag (const gsl_vector * diag, const gsl_vector * e, const gsl_vector * b, gsl_vector * x)

  This function solves the general N-by-N system A x = b where A is symmetric tridiagonal (N >= 2). The off-diagonal vector e must be one element shorter than the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

      A = ( d_0 e_0  0   0  )
          ( e_0 d_1 e_1  0  )
          (  0  e_1 d_2 e_2 )
          (  0   0  e_2 d_3 )

Function: int gsl_linalg_solve_cyc_tridiag (const gsl_vector * diag, const gsl_vector * e, const gsl_vector * f, const gsl_vector * b, gsl_vector * x)

  This function solves the general N-by-N system A x = b where A is cyclic tridiagonal (N >= 3). The cyclic super-diagonal and sub-diagonal vectors e and f must have the same number of elements as the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

      A = ( d_0 e_0  0  f_3 )
          ( f_0 d_1 e_1  0  )
          (  0  f_1 d_2 e_2 )
          ( e_3  0  f_2 d_3 )

Function: int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector * diag, const gsl_vector * e, const gsl_vector * b, gsl_vector * x)

  This function solves the general N-by-N system A x = b where A is symmetric cyclic tridiagonal (N >= 3). The cyclic off-diagonal vector e must have the same number of elements as the diagonal vector diag. The form of A for the 4-by-4 case is shown below,

      A = ( d_0 e_0  0  e_3 )
          ( e_0 d_1 e_1  0  )
          (  0  e_1 d_2 e_2 )
          ( e_3  0  e_2 d_3 )