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.
GQdk(d = 1L, k = 1L) GQN
d |
integer scalar - the dimension of the function
to be integrated with respect to the standard
|
k |
integer scalar - the order of the grid. A grid
of order |
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.
GQN contains only the non-redundant nodes. To regenerate
the whole array of nodes, all possible permutations of
axes and all possible combinations of +/- 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.
GQdk(2,5) # 53 x 3 GQN[[3]][[5]] # a 14 x 4 matrix
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