# GHrule: Univariate Gauss-Hermite quadrature rule In lme4: Linear Mixed-Effects Models using 'Eigen' and S4

## Description

Create a univariate Gauss-Hermite quadrature rule.

## Usage

 `1` ``` GHrule(ord, asMatrix = TRUE) ```

## Arguments

 `ord` scalar integer between 1 and 25 - the order, or number of nodes and weights, in the rule. When the function being multiplied by the standard normal density is a polynomial of order 2k-1 the rule of order k integrates the product exactly. `asMatrix` logical scalar - should the result be returned as a matrix. If `FALSE` a data frame is returned. Defaults to `TRUE`.

## Details

This version of Gauss-Hermite quadrature provides the node positions and weights for a scalar integral of a function multiplied by the standard normal density.

Originally based on package SparseGrid's hidden `GQN()`.

## Value

a matrix (or data frame, is `asMatrix` is false) with `ord` rows and three columns which are `z` the node positions, `w` the weights and `ldnorm`, the logarithm of the normal density evaluated at the nodes.

a different interface is available via `GQdk()`.

## Examples

 ```1 2 3 4``` ```(r5 <- GHrule(5, asMatrix=FALSE)) ## second, fourth, sixth, eighth and tenth central moments of the ## standard Gaussian density with(r5, sapply(seq(2, 10, 2), function(p) sum(w * z^p))) ```

### Example output

```Loading required package: Matrix
z          w     ldnorm
1 -2.856970e+00 0.01125741 -5.0000774
2 -1.355626e+00 0.22207592 -1.8377997
3  3.865099e-17 0.53333333 -0.9189385
4  1.355626e+00 0.22207592 -1.8377997
5  2.856970e+00 0.01125741 -5.0000774
   1   3  15 105 825
```

lme4 documentation built on June 22, 2021, 9:07 a.m.