GQdk {lme4}

Sparse Gaussian / Gauss-Hermite Quadrature grid

Description

Generate the sparse multidimensional Gaussian quadrature grids.

Currently unused. See GHrule() for the version currently in use in package lme4.

Usage

  GQdk(d = 1L, k = 1L)
  GQN

Arguments

d

integer scalar - the dimension of the function to be integrated with respect to the standard d-dimensional Gaussian density.

k

integer scalar - the order of the grid. A grid of order k provides an exact result for a polynomial of total order of 2k - 1 or less.

Value

GQdk() returns a matrix with d + 1 columns. The first column is the weights and the remaining d columns are the node coordinates.

GQN is a list of lists, containing the non-redundant quadrature nodes and weights for integration of a scalar function of a d-dimensional argument with respect to the density function of the d-dimensional Gaussian density function.
The outer list is indexed by the dimension, d, in the range of 1 to 20. The inner list is indexed by k, the order of the quadrature.

Note

GQN contains only the non-redundant nodes. To regenerate the whole array of nodes, all possible permutations of axes and all possible combinations of \pm 1 must be applied to the axes. This entire array of nodes is exactly what GQdk() reproduces.

The number of nodes gets very large very quickly with increasing d and k. See the charts at http://www.sparse-grids.de.

Examples

GQdk(2,5) # 53 x 3

GQN[[3]][[5]] # a 14 x 4 matrix