lme4 (version 1.1-35.5)

GHrule: Univariate Gauss-Hermite quadrature rule

Description

Create a univariate Gauss-Hermite quadrature rule.

Usage

GHrule(ord, asMatrix = TRUE)

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.

Arguments

ord

scalar integer between 1 and 100 - 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(), then on fastGHQuad's gaussHermiteData(.).

References

Qing Liu and Donald A. Pierce (1994). A Note on Gauss-Hermite Quadrature. Biometrika 81(3), 624--629. tools:::Rd_expr_doi("10.2307/2337136")

See Also

a different interface is available via GQdk().

Examples

(r5  <- GHrule( 5, asMatrix=FALSE))
(r12 <- GHrule(12, asMatrix=FALSE))

## second, fourth, sixth, eighth and tenth central moments of the
## standard Gaussian N(0,1) density:
ps <- seq(2, 10, by = 2)
cbind(p = ps, "E[X^p]" = with(r5,  sapply(ps, function(p) sum(w * z^p)))) # p=10 is wrong for 5-rule
p <- 1:15
GQ12 <- with(r12, sapply(p, function(p) sum(w * z^p)))
cbind(p = p, "E[X^p]" = zapsmall(GQ12))
## standard R numerical integration can do it too:
intL <- lapply(p, function(p) integrate(function(x) x^p * dnorm(x),
                                        -Inf, Inf, rel.tol=1e-11))
integR <- sapply(intL, `[[`, "value")
cbind(p, "E[X^p]" = integR)# no zapsmall() needed here
all.equal(GQ12, integR, tol=0)# => shows small difference
stopifnot(all.equal(GQ12, integR, tol = 1e-10))
(xactMom <- cumprod(seq(1,13, by=2)))
stopifnot(all.equal(xactMom, GQ12[2*(1:7)], tol=1e-14))
## mean relative errors :
mean(abs(GQ12  [2*(1:7)] / xactMom - 1)) # 3.17e-16
mean(abs(integR[2*(1:7)] / xactMom - 1)) # 9.52e-17 {even better}