

fractions(MASS)                              R Documentation

_R_a_t_i_o_n_a_l _A_p_p_r_o_x_i_m_a_t_i_o_n

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

     Find rational approximations to the components of a
     real numeric object using a standard continued fraction
     method.

_U_s_a_g_e_:

     rational(x, cycles=10, max.denominator=2000)

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

       x: Any object of mode numeric.

  cycles: The maximum number of steps to be used in the con-
          tinued fraction approximation process.

max.denominator: An early termination criterion.  If any
          partial denominator exceeds `max.denominator' the
          continued fraction stops at that point.

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

     Each component is first expanded in a continued frac-
     tion of the form

     `x = floor(x) + 1/(p1 + 1/(p2 + ...{})))'

     where `p1', `p2', ...{} are positive integers, termi-
     nating either at `cycles' terms or when a `pj >
     max.denominator'.  The continued fraction is then re-
     arranged to retrieve the numerator and denominator as
     integers.

     The numerators and denominators are then combined into
     a character vector that becomes the `"fracs"' attribute
     and used in printed representations.

     Arithmetic operations on `"fractions"' objects have
     full floating point accuracy, but the character repre-
     sentation printed out may not.

_V_a_l_u_e_:

     An object of class `"fractions"'.  A structure with
     `.Data' component the same as the input numeric `x',
     but with the rational approximations held as a charac-
     ter vector attribute, `"fracs"'.  Arithmetic operations
     on `"fractions"' objects is possible.

_S_e_e _A_l_s_o_:

     `rational'

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

     X <- matrix(runif(25), 5, 5)
     solve(X, X/5)

                 [,1]        [,2]       [,3]        [,4]        [,5]
     [1,]  2.0000e-01  3.7199e-17 1.2214e-16  5.7887e-17 -8.7841e-17
     [2,] -1.1473e-16  2.0000e-01 7.0955e-17  2.0300e-17 -1.0566e-16
     [3,]  2.7975e-16  1.3653e-17 2.0000e-01 -1.3397e-16  1.5577e-16
     [4,] -2.9196e-16  2.0412e-17 1.5618e-16  2.0000e-01 -2.1921e-16
     [5,] -3.6476e-17 -3.6430e-17 3.6432e-17  4.7690e-17  2.0000e-01

     fractions(solve(X, X/5))

          [,1] [,2] [,3] [,4] [,5]
     [1,] 1/5    0    0    0    0
     [2,]   0  1/5    0    0    0
     [3,]   0    0  1/5    0    0
     [4,]   0    0    0  1/5    0
     [5,]   0    0    0    0  1/5

     fractions(solve(X, X/5)) + 1

          [,1] [,2] [,3] [,4] [,5]
     [1,] 6/5    1    1    1    1
     [2,]   1  6/5    1    1    1
     [3,]   1    1  6/5    1    1
     [4,]   1    1    1  6/5    1
     [5,]   1    1    1    1  6/5

