# GQdk

##### 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.

##### 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)*