R: Integration of One-Dimensional Functions

integrate {stats} | R Documentation

## 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 R function 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.

### 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`.

In R versions <= 3.2.x, the first entries of `lower` and `upper` were used whereas an error is signalled now if they are not of length one.

### 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.

### Note

Like all numerical integration routines, these evaluate the function on a finite set of points. If the function is approximately constant (in particular, zero) over nearly all its range it is possible that the result and error estimate may be seriously wrong.

When integrating over infinite intervals do so explicitly, rather than just using a large number as the endpoint. This increases the chance of a correct answer – any function whose integral over an infinite interval is finite must be near zero for most of that interval.

For values at a finite set of points to be a fair reflection of the behaviour of the function elsewhere, the function needs to be well-behaved, for example differentiable except perhaps for a small number of jumps or integrable singularities.

`f` must accept a vector of inputs and produce a vector of function evaluations at those points. The `Vectorize` function may be helpful to convert `f` to this form.

### Source

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

### 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

    integrate(dnorm, 0:1, 20) #-> error!
    ## "silently" gave  integrate(dnorm, 0, 20)  in earlier versions of R

[Package stats version 3.5.1 Index]