deriv.ci.func <- function(deriv.matrix, conf = 0.95)
{
# deriv.ci.func: 
# Given a matrix of bootstrapped 2nd derivative values (of the form 
# deriv.matrix = sp.bootderiv.Nrep.df), this function extracts 
# the lower and upper (100*conf)% confidence limits for the second
# derivatives.
#
# conf should be chosen so that (Nrep+1)*(1-conf)/2 is an integer:
# otherwise it will not be possible to find integers l and u such that the
# l'th and u'th ordered values give us exactly a conf% confidence interval. 
# 
# syntax:  sp.deriv.ci <- deriv.ci.func(sp.bootderiv.Nrep.df, 0.95)
# example:  cb.deriv.ci <- deriv.ci.func(cb.bootderiv.399.10, 0.95)
#
# However, this function is rarely called directly: almost always called
# from within the function sp.plot. 
# 
	Nyears <- ncol(deriv.matrix)
	Nrep <- nrow(deriv.matrix)	#
# Nrep is the number of bootstrap replicates in the 2nd derivative 
# matrix. 
# Find alp: alp=(1-conf)/2:
	alp <- (1 - conf)/2	# 
# Interpolation is not ideal, so discard some rows of the bootstrap
# matrix if (Nrep+1)*alp is not an integer:
	if(abs((Nrep + 1) * alp - round((Nrep + 1) * alp)) > 1e-05)
		stop("need to discard rows of deriv.matrix, or change\n\tconf, so that (Nrep+1)*(1-conf)/2 is an integer."
			)
	assign("deriv.matrix", deriv.matrix, frame = 1)
	lowerpt <- (Nrep + 1) * alp
	upperpt <- (Nrep + 1) * (1 - alp)
	assign("upperpt", upperpt, frame = 1)
	assign("lowerpt", lowerpt, frame = 1)	#
# The confidence interval goes from the (Nrep+1)*alp 'th ordered
# bootstrap value (low) to the (Nrep+1)*(1-alp) 'th ordered bootstrap
# value (high). 
	inner.func <- function(yr)
	{
		sort(deriv.matrix[, yr])[c(lowerpt, upperpt)]
	}
	deriv.ci <- sapply(seq(1, Nyears), inner.func)
	dimnames(deriv.ci) <- list(c("lower", "upper"), NULL)
	deriv.ci
}