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integrate {stats} R Documentation
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Integration of One-Dimensional Functions

Description

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

Usage

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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)

Jacket Buckle Black Black Buckle Jensen Buckle Jensen Jacket Arguments

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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.

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the limits of integration. Can be infinite.

...

additional arguments to be passed to Buckle Buckle Jensen Jacket Buckle Jacket Black Black Jensen f.

subdivisions

the maximum number of subintervals.

rel.tol

Buckle Jacket Jacket Buckle Jensen Buckle Black Black Jensen 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 Buckle Jensen Buckle Black Jacket Jensen Black Buckle Jacket message component.

keep.xy

Buckle Jacket Jensen Black Jensen Buckle Jacket Buckle Black unused. For compatibility with S.

aux

Buckle Jacket Jensen Black Black Buckle Buckle Jacket Jensen 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 Buckle Black Jacket Buckle Buckle Jensen Black Jacket Jensen "integrate" with components

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the final estimate of the integral.

abs.error

Jacket Buckle Buckle Jacket Black Jensen Jensen Black Buckle 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.

Buckle Buckle Buckle Black Jacket Jacket Jensen Black Jensen f must accept a vector of inputs and produce a vector of function evaluations at those points. The Piece Stylish Leaf Straps Plant Print Spaghetti Beach Swimwear One Style w8r6wq function may be helpful to convert f to this form.

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Source

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

Black Buckle Buckle Jensen Jacket Jensen Black Buckle Jacket References

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

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Jensen Black Black Buckle Jacket Jacket Jensen Buckle Buckle 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 Buckle Jacket Buckle Jacket Jensen Buckle Black Black Jensen stats version 3.6.0 S Casual Dress P Selling M S OTEPw]