GQdk

0th

Percentile

Sparse Gaussian / Gauss-Hermite Quadrature grid

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.

Aliases
  • GQdk
  • GQN
Examples
# NOT RUN {
GQdk(2,5) # 53 x 3

GQN[[3]][[5]] # a 14 x 4 matrix
# }
Documentation reproduced from package lme4, version 1.1-21, License: GPL (>= 2)

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