int                  package:rmutil                  R Documentation

_N_u_m_e_r_i_c_a_l _I_n_t_e_g_r_a_t_i_o_n _o_f _a _F_u_n_c_t_i_o_n

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

     `int' performs numerical integration of a given function using
     either Romberg integration or algorithm 614 of the collected
     algorithms from ACM. Only the former is vectorized. The latter
     appeared in ACM-Trans. Math. Software, Vol.10, No. 2, Jun., 1984,
     p. 152-160 and uses formulae optimal in certain Hardy spaces
     h(p,d); see Sikorski,K., Optimal quadrature algorithms in HP
     spaces, Num. Math., 39, 405-410 (1982).

     Functions may have singularities at one or both end-points of the
     interval (a,b).

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

     int(f, a="-infty", b="infty", type="Romberg", eps=1.0e-6, max, d, p=0)

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

       f: The function (of one variable) to integrate, returning either
          a scalar or a vector.

       a: A scalar or vector giving the lower bound. If non-numeric,
          taken to be -infty, in which case it must be the same for the
          whole vector.

       b: A scalar or vector giving the upper bound. If non-numeric,
          taken to be infty, in which case it must be the same for the
          whole vector.

    type: The algorithm to be used, by default Romberg integration.
          Otherwise, it uses the TOMS614 algorithm.

     eps: Precision.

     max: For Romberg, the maximum number of steps, by default set to
          16. For TOMS614, the maximum number of function evaluations,
          by default set to 100.

       d: For Romberg, the number of extrapolation points so that 2k is
          the order of integration, by default set to 5; d=2 is
          Simpson's rule. For TOMS614, heuristic termination = any real
          number; deterministic termination = a number in the range 0 <
          d < pi/2 by default, set to 1.

       p: For TOMS614, p = 0: heuristic termination, p = 1:
          deterministic termination with the infinity norm, p > 1:
          deterministic termination with the p-th norm.

_A_u_t_h_o_r(_s):

     J.K. Lindsey

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

     f <- function(x) sin(x)+cos(x)-x^2
     int(f, a=0, b=2)
     #
     f <- function(x) exp(-(x-2)^2/2)/sqrt(2*pi)
     int(f, a=0:3)
     1-pnorm(0:3, 2)

