hilbert              package:fUtilities              R Documentation

_H_i_l_b_e_r_t _M_a_t_r_i_x

_D_e_s_c_r_i_p_t_i_o_n:

     Creates a Hilbert matrix.

_U_s_a_g_e:

     hilbert(n)

_A_r_g_u_m_e_n_t_s:

       n: an integer value, the dimension of the square matrix. 

_D_e_t_a_i_l_s:

     In linear algebra, a Hilbert matrix is a matrix with the unit 
     fraction elements.

     The Hilbert matrices are canonical examples of ill-conditioned 
     matrices, making them notoriously difficult to use in numerical 
     computation. For example, the 2-norm condition number of a 5x5
     Hilbert matrix above is about 4.8e5.

     The Hilbert matrix is symmetric and positive definite.

_V_a_l_u_e:

     'hilbert' generates a Hilbert matrix of order 'n'.

_R_e_f_e_r_e_n_c_e_s:

     Hilbert D.,  _Collected papers_, vol. II, article 21.

     Beckermann B, (2000); _The condition number of real Vandermonde,
     Krylov and positive  definite Hankel matrices_,  Numerische
     Mathematik 85, 553-577, 2000.

     Choi, M.D., (1983); _Tricks or Treats with the Hilbert Matrix_,
     American Mathematical Monthly 90, 301-312, 1983.

     Todd, J., (1954); _The Condition Number of the Finite Segment of
     the Hilbert Matrix_, National Bureau of Standards, Applied
     Mathematics Series 39, 109-116. 

     Wilf, H.S., (1970); _Finite Sections of Some Classical
     Inequalities_,  Heidelberg, Springer.

_E_x_a_m_p_l_e_s:

     ## Create a Hilbert Matrix:
        H = hilbert(5)
        H                              

