int {rmutil}R Documentation

Numerical Integration of a Function

Description

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).

Usage

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

Arguments

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.

Author(s)

J.K. Lindsey

Examples

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)


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