Rmpfr (version 0.9-5)

unirootR: One Dimensional Root (Zero) Finding -- in pure R

Description

The function unirootR searches the interval from lower to upper for a root (i.e., zero) of the function f with respect to its first argument.

unirootR() is “clone” of uniroot(), written entirely in R, in a way that it works with mpfr-numbers as well.

Usage

unirootR(f, interval, ...,
        lower = min(interval), upper = max(interval),
        f.lower = f(lower, ...), f.upper = f(upper, ...),
        extendInt = c("no", "yes", "downX", "upX"),
        trace = 0, verbose = as.logical(trace),
        verbDigits = max(3, min(20, -log10(tol)/2)),
        tol = .Machine$double.eps^0.25, maxiter = 1000L,
        check.conv = FALSE,
        warn.no.convergence = !check.conv,
        epsC = NULL)

Value

A list with four components: root and f.root give the location of the root and the value of the function evaluated at that point. iter and estim.prec give the number of iterations used and an approximate estimated precision for root. (If the root occurs at one of the endpoints, the estimated precision is

NA.)

Arguments

f

the function for which the root is sought.

interval

a vector containing the end-points of the interval to be searched for the root.

...

additional named or unnamed arguments to be passed to f

lower, upper

the lower and upper end points of the interval to be searched.

f.lower, f.upper

the same as f(upper) and f(lower), respectively. Passing these values from the caller where they are often known is more economical as soon as f() contains non-trivial computations.

extendInt

character string specifying if the interval c(lower,upper) should be extended or directly produce an error when f() does not have differing signs at the endpoints. The default, "no", keeps the search interval and hence produces an error. Can be abbreviated.

trace

integer number; if positive, tracing information is produced. Higher values giving more details.

verbose

logical (or integer) indicating if (and how much) verbose output should be produced during the iterations.

verbDigits

used only if verbose is true, indicates the number of digits numbers should be printed with, using format(., digits=verbDigits).

tol

the desired accuracy (convergence tolerance).

maxiter

the maximum number of iterations.

check.conv

logical indicating whether non convergence should be caught as an error, notably non-convergence in maxiter iterations should be an error instead of a warning.

warn.no.convergence

if set to FALSE there's no warning about non-convergence. Useful to just run a few iterations.

epsC

positive number or NULL in which case a smart default is sought. This should specify the “achievable machine precision” for the given numbers and their arithmetic.

The default will set this to .Machine$double.eps for double precision numbers, and will basically use 2 ^ - min(getPrec(f.lower), getPrec(f.upper)) when that works (as, e.g., for mpfr-numbers) otherwise.

This is factually a lower bound for the achievable lower bound, and hence, setting tol smaller than epsC is typically non-sensical and produces a warning.

Details

Note that arguments after ... must be matched exactly.

Either interval or both lower and upper must be specified: the upper endpoint must be strictly larger than the lower endpoint. The function values at the endpoints must be of opposite signs (or zero), for extendInt="no", the default. Otherwise, if extendInt="yes", the interval is extended on both sides, in search of a sign change, i.e., until the search interval \([l,u]\) satisfies \(f(l) \cdot f(u) \le 0\).

If it is known how \(f\) changes sign at the root \(x_0\), that is, if the function is increasing or decreasing there, extendInt can (and typically should) be specified as "upX" (for “upward crossing”) or "downX", respectively. Equivalently, define \(S := \pm 1\), to require \(S = \mathrm{sign}(f(x_0 + \epsilon))\) at the solution. In that case, the search interval \([l,u]\) possibly is extended to be such that \(S\cdot f(l)\le 0\) and \(S \cdot f(u) \ge 0\).

The function only uses R code with basic arithmetic, such that it should also work with “generalized” numbers (such as mpfr-numbers) as long the necessary Ops methods are defined for those.

The underlying algorithm assumes a continuous function (which then is known to have at least one root in the interval).

Convergence is declared either if f(x) == 0 or the change in x for one step of the algorithm is less than tol (plus an allowance for representation error in x).

If the algorithm does not converge in maxiter steps, a warning is printed and the current approximation is returned.

f will be called as f(x, ...) for a (generalized) numeric value of x.

References

Brent, R. (1973), see uniroot.

See Also

R's own (stats package) uniroot. polyroot for all complex roots of a polynomial; optimize, nlm.

Examples

require(utils) # for str

## some platforms hit zero exactly on the first step:
## if so the estimated precision is 2/3.
f <- function (x,a) x - a
str(xmin <- unirootR(f, c(0, 1), tol = 0.0001, a = 1/3))

## handheld calculator example: fixpoint of cos(.):
rc <- unirootR(function(x) cos(x) - x, lower=-pi, upper=pi, tol = 1e-9)
rc$root

## the same with much higher precision:
rcM <- unirootR(function(x) cos(x) - x,
                 interval= mpfr(c(-3,3), 300), tol = 1e-40)
rcM
x0 <- rcM$root
stopifnot(all.equal(cos(x0), x0,
                    tol = 1e-40))## 40 digits accurate!

str(unirootR(function(x) x*(x^2-1) + .5, lower = -2, upper = 2,
            tol = 0.0001), digits.d = 10)
str(unirootR(function(x) x*(x^2-1) + .5, lower = -2, upper = 2,
            tol = 1e-10 ), digits.d = 10)

## A sign change of f(.), but not a zero but rather a "pole":
tan. <- function(x) tan(x * (Const("pi",200)/180))# == tan(  )
(rtan <- unirootR(tan., interval = mpfr(c(80,100), 200), tol = 1e-40))
## finds 90 {"ok"}, and now gives a warning
stopifnot(all.equal(rtan$root, 90, tolerance = 1e-38))

## Find the smallest value x for which exp(x) > 0 (numerically):
r <- unirootR(function(x) 1e80*exp(x)-1e-300, c(-1000,0), tol = 1e-15)
str(r, digits.d = 15) ##> around -745, depending on the platform.

exp(r$root)     # = 0, but not for r$root * 0.999...
minexp <- r$root * (1 - 10*.Machine$double.eps)
exp(minexp)     # typically denormalized

## --- using mpfr-numbers :

## Find the smallest value x for which exp(x) > 0 ("numerically");
## Note that mpfr-numbers underflow *MUCH* later than doubles:
## one of the smallest mpfr-numbers {see also ?mpfr-class } :
(ep.M <- mpfr(2, 55) ^ - ((2^30 + 1) * (1 - 1e-15)))
r <- unirootR(function(x) 1e99* exp(x) - ep.M, mpfr(c(-1e20, 0), 200))
r # 97 iterations; f.root is very similar to ep.M

## interval extension 'extendInt'  --------------

f1 <- function(x) (121 - x^2)/(x^2+1)
f2 <- function(x) exp(-x)*(x - 12)
tools::assertError(unirootR(f1, c(0,10)), verbose=TRUE)
##--> error: f() .. end points not of opposite sign

## where as  'extendInt="yes"'  simply first enlarges the search interval:
u1 <- unirootR(f1, c(0,10),extendInt="yes", trace=1)
u2 <- unirootR(f2, mpfr(c(0,2), 128), extendInt="yes", trace=2, verbose=FALSE, tol = 1e-25)
stopifnot(all.equal(u1$root, 11, tolerance = 1e-5),
          all.equal(u2$root, 12, tolerance = 1e-23))

## The *danger* of interval extension:
## No way to find a zero of a positive function, but
## numerically, f(-|M|) becomes zero :
u3 <- unirootR(exp, c(0,2), extendInt="yes", trace=TRUE)

## Nonsense example (must give an error):
tools::assertCondition( unirootR(function(x) 1, 0:1, extendInt="yes"),
                       "error", verbose=TRUE)