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.

Note that the comments in the C source code in ‘<R>/src/appl/integrate.c’ give more details, particularly about reasons for failure (internal error code ier >= 1).

In R versions \le 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 4.2.0 Index]