docopulae (version 0.2.1)

nint_integrate: Integrate

Description

nint_integrate performs summation and integration of some scalar-valued function over some space.

Usage

nint_integrate(f, space, ...)

Arguments

f

the scalar-valued function (integrand) to be integrated.

space

some space.

...

other arguments passed to f.

Value

• nint_integrate returns a single numeric.

Details

nint_integrate uses nint_integrateNCube and nint_integrateNFunc to handle interval and function dimensions. See their help pages on how to deploy different solutions.

The order of dimensions is optimized for efficiency. Therefore interchangeability (except for function dimensions) is assumed.

See Also

nint_space, nint_transform, nint_integrateNCube, nint_integrateNFunc, fisherI

Examples

## discrete
## a) scatter
s = nint_space(nint_scatDim(1:3),
               nint_scatDim(c(0, 2, 5)))
s
## (1, 0), (2, 2), (3, 5)
nint_integrate(function(x) abs(x[1] - x[2]), s) # 1 + 0 + 2 == 3

## b) grid
s = nint_space(nint_gridDim(1:3),
               nint_gridDim(c(0, 2, 5)))
s
## (1, 0), (1, 2), (1, 5), (2, 0), ..., (3, 2), (3, 5)
nint_integrate(function(x) ifelse(sum(x) < 5, 1, 0), s) # 5


## continous
## c)
s = nint_space(nint_intvDim(1, 3),
               nint_intvDim(1, Inf))
s
nint_integrate(function(x) 1/x[2]**2, s) # 2

## d) infinite, no transform
s = nint_space(nint_intvDim(-Inf, Inf))
nint_integrate(sin, s) # 0

## e) infinite, transform
s = nint_space(nint_intvDim(-Inf, Inf),
               nint_intvDim(-Inf, Inf))
## probability integral transform
tt = nint_transform(function(x) prod(dnorm(x)), s, 1:2, list(
    g=function(x) pnorm(x),
    gij=function(y) { t1 = qnorm(y); cbind(t1, 1/dnorm(t1)) }))
tt$space
nint_integrate(tt$f, tt$space) # 1


## functionally dependent
## f) area of triangle
s = nint_space(nint_intvDim(0, 1),
               nint_funcDim(function(x) nint_intvDim(x[1]/2, 1 - x[1]/2)) )
s
nint_integrate(function(x) 1, s) # 0.5

## g) area of circle
s = nint_space(
    nint_intvDim(-1, 1),
    nint_funcDim(function(x) nint_intvDim( c(-1, 1) * sin(acos(x[1])) ))
)
s
nint_integrate(function(x) 1, s) # pi

## h) volume of sphere
s = nint_space(s[[1]],
               s[[2]],
               nint_funcDim(function(x) {
                   r = sin(acos(x[1]))
                   nint_intvDim(c(-1, 1) * r*cos(asin(x[2] / r)))
               }) )
s
nint_integrate(function(x) 1, s) # 4*pi/3