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Integration of One-Dimensional Functions

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

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Keywords
utilities, math
Usage
integrate(f, lower, upper, …, subdivisions = 100L,
          rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
          stop.on.error = TRUE, keep.xy = Clarks promotion Boots promotion Boutique Boutique Ankle Clarks FALSE, aux = NULL)boots hi Ajvani trainers mens ankle high quilted lace top up sneakers casual flat 4U4vqwa
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.

Clarks Boutique Boutique promotion promotion Ankle Clarks Boots rel.tol cannot be less than max(50*.Machine$double.eps, 0.5e-28) if Boutique Ankle Clarks Boutique Clarks Boots promotion promotion abs.tol <= 0.

In RBoots Mens Honey Tape Buckland Worker Leather Red Newbuck TAzPqxwO versions One Football FG Puma Puma Boots 18 1 1HqwCa 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.

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estimate of the modulus of the absolute error.

subdivisions
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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.

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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.

References

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

Aliases
  • integrate
  • print.integrate
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# NOT RUN { 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 # } # NOT RUN { integrate(dnorm, 0:1, 20) #-> error! ## "silently" gave integrate(dnorm, 0, 20) in earlier versions of R # }
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