createNIGrid: creates a grid for numerical integration.
In mvQuad: Methods for Multivariate Quadrature

createNIGrid

R Documentation

creates a grid for numerical integration.

Description

createNIGrid Creates a grid for multivariate numerical integration. The Grid can be based on different quadrature- and construction-rules.

Usage

createNIGrid(dim = NULL, type = NULL, level = NULL,
  ndConstruction = "product", level.trans = NULL)

Arguments

`dim`	number of dimensions
`type`	quadrature rule (see Details)
`level`	accuracy level (typically number of grid points for the underlying 1D quadrature rule)
`ndConstruction`	character vector which denotes the construction rule for multidimensional grids. `product` for product rule, returns a "full grid" (default) `sparse` for combination technique, leads to a regular "sparse grid".
`level.trans`	logical variable denotes either to take the levels as number of grid points (FALSE = default) or to transform in that manner that number of grid points = 2^(levels-1) (TRUE). Alternatively `level.trans` can be a function, which takes (n x d)-matrix and returns a matrix with the same dimensions (see the example; this feature is particularly useful for the 'sparse' construction rule, to account for different importance of the dimensions).

Details

The following quadrature rules are supported (build-in).

cNC1, cNC2, ..., cNC6: closed Newton-Cotes Formula of degree 1-6 (1=trapezoidal-rule; 2=Simpson's-rule; ...), initial interval of integration: [0, 1]
oNC0, oNC1, ..., oNC3: open Newton-Cote Formula of degree 0-3 (0=midpoint-rule; ...), initial interval of integration: [0, 1]
GLe, GKr: Gauss-Legendre and Gauss-Kronrod rule for an initial interval of integration: [0, 1]
nLe: nested Gauss-Legendre rule for an initial interval of integration: [0, 1] (Knut Petras (2003). Smolyak cubature of given polynomial degree with few nodes for increasing dimension. Numerische Mathematik 93, 729-753)
GLa: Gauss-Laguerre rule for an initial interval of integration: [0, INF)
GHe: Gauss-Hermite rule for an initial interval of integration: (-INF, INF)
nHe: nested Gauss-Hermite rule for an initial interval of integration: (-INF, INF) (A. Genz and B. D. Keister (1996). Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight." Journal of Computational and Applied Mathematics 71, 299-309)
GHN, nHN: (nested) Gauss-Hermite rule as before but weights are multiplied by the standard normal density (\hat(w)_i = w_i * \phi(x_i)).
Leja: Leja-Points for an initial interval of integration: [0, 1]

The argument type and level can also be vector-value, different for each dimension (the later only for "product rule"; see examples)

Value

Returns an object of class 'NIGrid'. This object is basically an environment containing nodes and weights and a list of features for this special grid. This grid can be used for numerical integration (via quadrature)

References

Philip J. Davis, Philip Rabinowitz (1984): Methods of Numerical Integration

F. Heiss, V. Winschel (2008): Likelihood approximation by numerical integration on sparse grids, Journal of Econometrics

H.-J. Bungartz, M. Griebel (2004): Sparse grids, Acta Numerica

Examples

## 1D-Grid --> closed Newton-Cotes Formula of degree 1 (trapeziodal-rule)
myGrid <- createNIGrid(dim=1, type="cNC1", level=10)
print(myGrid)
## 2D-Grid --> nested Gauss-Legendre rule
myGrid <- createNIGrid(dim=2, type=c("GLe","nLe"), level=c(4, 7))
rescale(myGrid, domain = rbind(c(-1,1),c(-1,1)))
plot(myGrid)
print(myGrid)
myFun <- function(x){
   1-x[,1]^2*x[,2]^2
}
quadrature(f = myFun, grid = myGrid)
## level transformation
levelTrans <- function(x){
  tmp <- as.matrix(x)
  tmp[, 2] <- 2*tmp[ ,2]
  return(tmp)
}
nw <- createNIGrid(dim=2, type="cNC1", level = 3,
   level.trans = levelTrans, ndConstruction = "sparse")
plot(nw)

mvQuad documentation built on Sept. 19, 2023, 5:06 p.m.