stats (version 3.3)

integrate: Integration of One-Dimensional Functions

Description

Adaptive quadrature of functions of one variable over a finite or infinite interval.

Usage

integrate(f, lower, upper, ..., subdivisions = 100L,
          rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
          stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)

Arguments

f

an Rfunction taking a numeric first argument and returning a numeric vector of the same length. Returning a non-finite element will generate an error.

lower, upper

the limits of integration. Can be infinite.

...

additional arguments to be passed to f.

subdivisions

the maximum number of subintervals.

rel.tol

relative accuracy requested.

abs.tol

absolute accuracy requested.

stop.on.error

logical. If true (the default) an error stops the function. If false some errors will give a result with a warning in the message component.

keep.xy

unused. For compatibility with S.

aux

unused. For compatibility with S.

Value

• A list of class "integrate" with components
• value
  the final estimate of the integral.
• abs.error
  estimate of the modulus of the absolute error.
• subdivisions
  the number of subintervals produced in the subdivision process.
• message
  "OK" or a character string giving the error message.
• call
  the matched call.

source

Based on QUADPACK routines dqags and dqagi by R. Piessens and E. deDoncker--Kapenga, available from Netlib.

Details

Note that arguments after ... must be matched exactly.

If one or both limits are infinite, the infinite range is mapped onto a finite interval.

For a finite interval, globally adaptive interval subdivision is used in connection with extrapolation by Wynn's Epsilon algorithm, with the basic step being Gauss--Kronrod quadrature.

rel.tol cannot be less than max(50*.Machine$double.eps, 0.5e-28) if abs.tol <= 0<="" code="">.

References

R. Piessens, E. deDoncker--Kapenga, C. Uberhuber, D. Kahaner (1983) Quadpack: a Subroutine Package for Automatic Integration; Springer Verlag.

Examples

integrate(dnorm, -1.96, 1.96)
integrate(dnorm, -Inf, Inf)

## a slowly-convergent integral
integrand <- function(x) {1/((x+1)*sqrt(x))}
integrate(integrand, lower = 0, upper = Inf)

## don't do this if you really want the integral from 0 to Inf
integrate(integrand, lower = 0, upper = 10)
integrate(integrand, lower = 0, upper = 100000)
integrate(integrand, lower = 0, upper = 1000000, stop.on.error = FALSE)

## some functions do not handle vector input properly
f <- function(x) 2.0
try(integrate(f, 0, 1))
integrate(Vectorize(f), 0, 1)  ## correct
integrate(function(x) rep(2.0, length(x)), 0, 1)  ## correct

## integrate can fail if misused
integrate(dnorm, 0, 2)
integrate(dnorm, 0, 20)
integrate(dnorm, 0, 200)
integrate(dnorm, 0, 2000)
integrate(dnorm, 0, 20000) ## fails on many systems
integrate(dnorm, 0, Inf)   ## works