Rmpfr (version 0.7-1)

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, ...),
        verbose = FALSE,
        tol = .Machine$double.eps^0.25, maxiter = 1000,
        epsC = NULL)

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.

verbose

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

tol

the desired accuracy (convergence tolerance).

maxiter

the maximum number of 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 sense and produces a warning.

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.)

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).

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

polyroot for all complex roots of a polynomial; optimize, nlm.

Examples

# NOT RUN {
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( <angle> )
(rtan <- unirootR(tan., interval = mpfr(c(80,100), 200), tol = 1e-40))
## finds 90 {"ok"}, and now gives a warning
# }
# NOT RUN {
## 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

# }