Methods for multivariate Quadrature (numerical integration)

Description

This package provides methods to construct multivariate grids, which can be used for multivariate quadrature. This grids can be based on different quadrature rules like Newton-Cotes formulas (trapezoidal-, Simpson-rule, ...) or Gauss-Quadrature (Gauss-Hermite, Gauss-Legendre, ...). For the construction of the multidimensional grid the product-rule or the combination-technique can be applied.

Details

Author(s)

Constantin Weiser (University Mainz / Germany) Maintainer: Constantin Weiser <constantin.weiser@gmail.com>

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

Peter Jaeckel (2005): A note on multivariate Gauss-Hermite quadrature

Examples

myGrid <- createNIGrid(dim=2, type="GLe", level=5)
rescale(myGrid, domain=rbind(c(-1,1),c(-1,1)))
print(myGrid)
plot(myGrid, col="blue")
myFun <- function(x){
   1 - x[,1]^2 * x[,2]^2
}
quadrature(myFun, myGrid)

nodes and weights for 1D - Gauss-Quadrature

Description

This data set stores nodes an weights for Gauss-Quadrature. Syntax:

QuadRules[['type']][['level']]

• type="GLe" Gauss-Legendre; interval [0,1]; max-level 45

• type="nLe" nested-type Gauss-Legendre; interval [0,1]; max-level 25

• type="GKr" Gauss-Kronrod; interval [0,1]; max-level 29

• type="GLa" Gauss-Laguere; interval [0, Inf); max-level 30

• type="GHe" Gauss-Hermite; interval (-Inf, Inf); max-level 45

• type="GHN" Gauss-Hermite (as above, but pre-multiplied weights \hat(w)_i = w_i * \phi(x_i))

• type="nHe" nested-type Gauss-Hermite; interval (-Inf, Inf) max-level 25

• type="nHN" nested-type Gauss-Hermite (as above, but pre-multiplied weights \hat(w)_i = w_i * \phi(x_i))

• type="Leja" Leja-points; interval [0,1]; max-level 141

Format

list of nodes and weights (for organisation see "Syntax" in description section)

Source

- http://keisan.casio.com/exec/system/1329114617 high precission computing (for G..-rules)

- further information in createNIGrid

Examples

nw <- QuadRules[["GHe"]][[2]]

copies an NIGrid-object

Description

copyNIGrid copies an NIGrid-object

Usage

copyNIGrid(object1, object2 = NULL)

Arguments

Value

Returns a NIGrid-object or NULL

Examples

myGrid <- createNIGrid(dim=2, type="GHe", level=5)
myGrid.copy <- copyNIGrid(myGrid)

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

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

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)

get nodes and weights from an NIGrid-object

Description

getNodes and getWeights extract the (potentially rescaled) nodes and weights out of an NIGrid-Object

Usage

getNodes(grid)

getWeights(grid)

Arguments

Value

Returns the nodes or weights of the given grid

See Also

createNIGrid

Examples

myGrid <- createNIGrid(dim=2, type="cNC1", level=3)
getNodes(myGrid)
getWeights(myGrid)

plots an NIGrid-object

Description

Plots the grid points of an NIGrid-object

Usage

## S3 method for class 'NIGrid'
plot(x, plot.dimension = NULL, ...)

Arguments

Examples

myGrid <- createNIGrid(dim=4, type=c("GHe", "cNC1", "GLe", "oNC1"),
                       level=c(3,4,5,6))
plot(myGrid) ## dimension 1-min(3,dim(myGrid)) are plotted
## Free arranged plots
plot(myGrid, plot.dimension=c(4,2,1))
plot(myGrid, plot.dimension=c(1,2))
plot(myGrid, plot.dimension=c(3))

prints characteristic information for an NIGrid-object

Description

Prints characteristic information for an NIGrid-object

Usage

## S3 method for class 'NIGrid'
print(x, ...)

Arguments

Value

Prints the information for an NIGrid-object (i.a. grid size (dimensions, grid points, memory usage), type and support)

Examples

myGrid <- createNIGrid(dim=2, type="GHe", level=5)
print(myGrid)

computes the approximated Integral

Description

quadrature computes the integral for a given function based on an NIGrid-object

Usage

quadrature(f, grid = NULL, ...)

Arguments

Value

The approximated value of the integral

See Also

createNIGrid, rescale

Examples

myGrid <- createNIGrid(dim=2, type="GLe", level=5)
rescale(myGrid, domain=rbind(c(-1,1),c(-1,1)))
plot(myGrid, col="blue")
myFun <- function(x){
   1 - x[,1]^2 * x[,2]^2
}
quadrature(myFun, myGrid)

reads a quadrature-rule from a text file

Description

readRule reads a quadrature-rule from a text file

Usage

readRule(file = NULL)

Arguments

Details

The text file containing the quadrature rule has to be formatted in the following way:

The first line have to declare the domain initial.domain a b, where a and b denotes the lower and upper-bound for the integration domain. This can be either a number or '-Inf'/'Inf' (for example initial.domain 0 1 or initial.domain 0 Inf)

Every following line contains one single node and weight belonging to one level of the rule (format: 'level' 'node' 'weight'). This example shows the use for the "midpoint-rule" (levels: 1 - 3).

> initial.domain 0 1

> 1 0.5 1

> 2 0.25 0.5

> 2 0.75 0.5

> 3 0.166666666666667 0.333333333333333

> 3 0.5 0.333333333333333

> 3 0.833333333333333 0.333333333333333

Value

Returns an object of class 'customRule', which can be used for creating a 'NIGrid' (createNIGrid)

See Also

createNIGrid

Examples

## Not run: myRule <- readRule(file="midpoint_rule.txt")
## Not run: nw <- createNIGrid(d=1, type = myRule.txt, level = 2)

moves, rescales and/or rotates a multidimensional grid.

Description

rescale.NIGrid manipulates a grid for more efficient numerical integration with respect to a given domain (bounded integral) or vector of means and covariance matrix (unbounded integral).

Usage

rescale(object, ...)

## S3 method for class 'NIGrid'
rescale(object, domain = NULL, m = NULL, C = NULL,
  dec.type = 0, ...)

Arguments

Value

This function modifies the "support-attribute" of the grid. The recalculation of the nodes and weights is done when the getNodes or getWeights are used.

References

Peter Jaeckel (2005): A note on multivariate Gauss-Hermite quadrature

See Also

quadrature, createNIGrid

Examples

C = matrix(c(2,0.9,0.9,2),2)
m = c(-.5, .3)
par(mfrow=c(3,1))

myGrid <- createNIGrid(dim=2, type="GHe", level=5)

rescale(myGrid, m=m, C=C, dec.type=0)
plot(myGrid, col="red")

rescale(myGrid, m=m, C=C, dec.type=1)
plot(myGrid, col="green")

rescale(myGrid, m=m, C=C, dec.type=2)
plot(myGrid, col="blue")

returns the size of an NIGrid-object

Description

Returns the size of an NIGrid-object

Usage

size(object, ...)

## S3 method for class 'NIGrid'
size(object, ...)

## S3 method for class 'NIGrid'
dim(x)

Arguments

Value

Returns the grid size in terms of dimensions, number of grid points and used memory

Examples

myGrid <- createNIGrid(dim=2, type="GHe", level=5)
size(myGrid)
dim(myGrid)