mvQuad (version 1.0-8)

createNIGrid: 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)

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)

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)

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

See Also

rescale, quadrature, print, plot and size

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)