RFC: a "safe" uniroot() function for future R

With main R releases only happening yearly in spring, now is
good time to consider *and* discuss new features for what we
often call "R-devel" and more officially is
  R Under development (unstable) (.....) -- "Unsuffered Consequences"

Here is one such example I hereby expose to public scrutiny:

A few minutes ago, I've committed the following to R-devel
(the 'trunk' in the svn repository for R):

------------------------------------------------------------------------
r62834 | maechler | 2013-05-30 10:01:33 +0200 (Thu, 30 May 2013) | 1 line
Changed paths:
   M doc/NEWS.Rd
   M src/library/stats/NAMESPACE
   M src/library/stats/R/nlm.R
   M src/library/stats/man/uniroot.Rd
   M tests/Examples/stats-Ex.Rout.save

new "safe" uniroot() =: unirootS()  [may change; see R-devel e-mail]
------------------------------------------------------------------------

The help file says

     ?unirootS()? is a ?safe? version of ?uniroot()?, built on
     ?uniroot()?, also useful as drop-in replacement of ?uniroot()? in
     some cases.  ?Safe? means searching for the correct ?interval =
     c(lower,upper)? if ?sign(f(x))? does not satisfy the requirements
     at the interval end points; see the ?Details? section.

We've had this function, called  safeUroot() in our package
copula for a while now, where an earlier not-exported version
has been in my package nor1mix even longer.
When I was tempted to introduce it into yet another CRAN package
I maintain,  I decided that this may be a sign that such a
simple [ utility for / generalization of ] uniroot() should
probably rather be added to R itself.

The function definition, also visible in R's devel.sources, at the bottom of
 https://svn.r-project.org/R/trunk/src/library/stats/R/nlm.R ,
shows that unirootS() is a wrapper for uniroot() and 
is in principle 100% back compatible to uniroot() itself for all
the cases where f(lower) and f(upper) are of differing sign and
hence uniroot() does not give a quick error.
unirootS() just has three new optional arguments, all with their
defaults set such as to remain uniroot() compatible.

So, one option instead of the currently commited one would be to
adopt  unirootS() as "the new uniroot()" and rename current
uniroot to .uniroot() {and still export both}.

The capital "S" in the function name and the 'Sig' name is of
course quite a matter of taste, and this case corresponds to my
taste, but even that is part of the RFC.

unirootS <- function(f, interval, ...,
		     lower = min(interval), upper = max(interval),
		     f.lower = f(lower, ...), f.upper = f(upper, ...),
		     Sig = NULL, check.conv = FALSE,
		     tol = .Machine$double.eps^0.25, maxiter = 1000, trace = 0)
{
    if (   is.null(Sig) && f.lower * f.upper > 0 ||
	is.numeric(Sig) && (Sig*f.lower > 0 || Sig*f.upper < 0)) {
	if(trace)
	    cat(sprintf("search in [%g,%g]%s", lower, upper,
			if(trace >= 2)"\n" else " ... "))
	Delta <- function(u) 0.01* pmax(1e-7, abs(u))
	## Two cases:
	if(is.null(Sig)) {
	    ## case 1)	'Sig' unspecified --> extend (lower, upper) at the same time
	    delta <- Delta(c(lower,upper))
	    while(isTRUE(f.lower*f.upper > 0) && any(iF <- is.finite(c(lower,upper)))) {
		if(iF[1]) f.lower <- f(lower <- lower - delta[1])
		if(iF[2]) f.upper <- f(upper <- upper + delta[2])
		if(trace >= 2)
		    cat(sprintf(" .. modified lower,upper: (%15g,%15g)\n",
				lower,upper))
		delta <- 2 * delta
	    }
	} else {
	    ## case 2) 'Sig' specified --> typically change only *one* of lower, upper
	    ## make sure we have Sig*f(lower) < 0 and Sig*f(upper) > 0:
	    delta <- Delta(lower)
	    while(isTRUE(Sig*f.lower > 0)) {
		f.lower <- f(lower <- lower - delta)
		if(trace) cat(sprintf(" .. modified lower: %g\n", lower))
		delta <- 2 * delta
	    }
	    delta <- Delta(upper)
	    while(isTRUE(Sig*f.upper < 0)) {
		f.upper <- f(upper <- upper + delta)
		if(trace) cat(sprintf(" .. modified upper: %g\n", upper))
		delta <- 2 * delta
	    }
	}
	if(trace && trace < 2)
	    cat(sprintf("extended to [%g, %g]\n", lower, upper))
    }
    if(!isTRUE(f.lower * f.upper <= 0))
	stop("did not succeed extending the interval endpoints for f(lower) * f(upper) <= 0")
    if(check.conv) {
	r <- tryCatch(uniroot(f, ..., lower=lower, upper=upper,
			      f.lower=f.lower, f.upper=f.upper,
			      tol=tol, maxiter=maxiter),
		      warning = function(w)w)
	if(inherits(r, "warning"))
	    stop("convergence problem in zero finding: ", r$message)
	else r
    }
    else
	uniroot(f, ..., lower=lower, upper=upper,
		f.lower=f.lower, f.upper=f.upper,
		tol=tol, maxiter=maxiter)
}

-----------

As said, your comments are very welcome!
Note that I'm less interested in variations which gain 10-20% in
speed benchmarks, rather I'd appreciate proposals for changes
that give a "better" (in your sense) user interface.

Martin Maechler, ETH Zurich