[Rd] unstable corner of parameter space for qbeta?

[Ben Bolker]

  I've discovered an infelicity (I guess) in qbeta(): it's not a bug,
since there's a clear warning about lack of convergence of the numerical
algorithm ("full precision may not have been achieved").  I can work
around this, but I'm curious why it happens and whether there's a better
workaround -- it doesn't seem to be in a particularly extreme corner of
parameter space. It happens, e.g., for 	these parameters:

phi <- 1.1
i <- 0.01
t <- 0.001
shape1 = i/phi  ##  0.009090909
shape2 = (1-i)/phi  ## 0.9
qbeta(t,shape1,shape2)  ##  5.562685e-309
##  brute-force uniroot() version, see below
Qbeta0(t,shape1,shape2)  ## 0.9262824

  The qbeta code is pretty scary to read: the warning "full precision
may not have been achieved" is triggered here:

https://github.com/wch/r-source/blob/f8d4d7d48051860cc695b99db9be9cf439aee743/src/nmath/qbeta.c#L530

  Any thoughts?  Should I report this on the bug list?


A more general illustration:
http://www.math.mcmaster.ca/bolker/misc/qbeta.png

===
fun <- function(phi,i=0.01,t=0.001, f=qbeta) {
  f(t,shape1=i/phi,shape2=(1-i)/phi, lower.tail=FALSE)
}
## brute-force beta quantile function
Qbeta0 <- function(t,shape1,shape2,lower.tail=FALSE) {
  fn <- function(x) {pbeta(x,shape1,shape2,lower.tail=lower.tail)-t}
  uniroot(fn,interval=c(0,1))$root
}
Qbeta <- Vectorize(Qbeta0,c("t","shape1","shape2"))
curve(fun,from=1,to=4)
curve(fun(x,f=Qbeta),add=TRUE,col=2)