vuniroot.Rd
\name{vuniroot}
\title{Vectorised One Dimensional Root (Zero) Finding}
\alias{vuniroot}
\usage{
vuniroot(f, interval, \dots,
lower, upper,
f.lower = f(lower, \dots), f.upper = f(upper, \dots),
extendInt = c("no", "yes", "downX", "upX"), check.conv = FALSE,
tol = .Machine$double.eps^0.25, maxiter = 1000, trace = 0,
n = NULL)
}
\arguments{
\item{f}{the function for which the root is sought.}
\item{interval}{a matrix with two columns containing the end-points of the interval
to be searched for the root.}
\item{\dots}{additional named or unnamed arguments to be passed
to \code{f}}
\item{lower, upper}{the lower and upper end points of the interval to
be searched.}
\item{f.lower, f.upper}{the same as \code{f(upper)} and
\code{f(lower)}, respectively. Passing these values from the caller
where they are often known is more economical as soon as \code{f()}
contains non-trivial computations.}
\item{extendInt}{character string specifying if the interval
\code{c(lower,upper)} should be extended or directly produce an error
when \code{f()} does not have differing signs at the endpoints. The
default, \code{"no"}, keeps the search interval and hence produces
an error. Can be abbreviated.}
\item{check.conv}{logical indicating whether a convergence warning of the
underlying \code{\link{vuniroot}} should be caught as an error and if
non-convergence in \code{maxiter} iterations should be an error
instead of a warning.}
\item{tol}{the desired accuracy (convergence tolerance).}
\item{maxiter}{the maximum number of iterations.}
\item{trace}{integer number; if positive, tracing information is
produced. Higher values giving more details.}
\item{n}{integer number; size of input vector to \code{f}
(only used if \code{lower} and \code{upper} are of length 1)}
}
\description{
The function \code{vuniroot} searches the interval from \code{lower}
to \code{upper} for a root (i.e., zero) of the vectorised function \code{f} with
respect to its first argument.
Setting \code{extendInt} to a non-\code{"no"} string, means searching
for the correct \code{interval = c(lower,upper)} if \code{sign(f(x))}
does not satisfy the requirements at the interval end points; see the
\sQuote{Details} section.
}
\details{
Note that arguments after \code{\dots} must be matched exactly.
Either \code{interval} or both \code{lower} and \code{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 \code{extendInt="no"}, the default. Otherwise, if
\code{extendInt="yes"}, the interval is extended on both sides, in
search of a sign change, i.e., until the search interval \eqn{[l,u]}
satisfies \eqn{f(l) \cdot f(u) \le 0}{f(l) * f(u) <= 0}.
If it is \emph{known how} \eqn{f} changes sign at the root
\eqn{x_0}{x0}, that is, if the function is increasing or decreasing there,
\code{extendInt} can (and typically should) be specified as
\code{"upX"} (for \dQuote{upward crossing}) or \code{"downX"},
respectively. Equivalently, define \eqn{S := \pm 1}{S:= +/- 1}, to
require \eqn{S = \mathrm{sign}(f(x_0 + \epsilon))}{S = sign(f(x0 +
eps))} at the solution. In that case, the search interval \eqn{[l,u]}
possibly is extended to be such that \eqn{S\cdot f(l)\le 0}{%
S * f(l) <= 0} and \eqn{S \cdot f(u) \ge 0}{S * f(u) >= 0}.
\code{vuniroot()} uses a C++ subroutine based on \file{"zeroin"} (from Netlib)
and algorithms given in the reference below. They assume a
continuous function (which then is known to have at least one root in
the interval).
Convergence is declared either if \code{f(x) == 0} or the change in
\code{x} for one step of the algorithm is less than \code{tol} (plus an
allowance for representation error in \code{x}).
If the algorithm does not converge in \code{maxiter} steps, a warning
is printed and the current approximation is returned.
\code{f} will be called as \code{f(\var{x}, ...)} for a numeric value
of \var{x}.
The argument passed to \code{f} has special semantics and used to be
shared between calls. The function should not copy it.
}
\value{
A list with at least three components: \code{root} and \code{f.root}
give the location of the root and the value of the function evaluated
at that point. \code{iter} gives the number of
iterations used.
Further components may be added in future: component \code{init.it}
was added in \R 3.1.0.
}
\source{
Based on \file{zeroin.c} in \url{https://netlib.org/c/brent.shar}.
}
\references{
Brent, R. (1973)
\emph{Algorithms for Minimization without Derivatives.}
Englewood Cliffs, NJ: Prentice-Hall.
}
\seealso{
\code{\link{uniroot}} for the standard single root solver
\code{\link{polyroot}} for all complex roots of a polynomial;
\code{\link{optimize}}, \code{\link{nlm}}.
}
\examples{\donttest{
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 <- vuniroot(f, lower=c(0, 0), upper=c(1,1), tol = 0.0001, a = c(1/3,2/3)))
## same example with scalars for lower and upper -- using the n argument
str(xmin <- vuniroot(f, lower=0, upper=1, tol = 0.0001, n=2, a = c(1/3,2/3)))
## handheld calculator example: fixed point of cos(.):
vuniroot(function(x) cos(x) - x, lower = -pi, upper = pi, tol = 1e-9)$root
str(vuniroot(function(x) x*(x^2-1) + .5, lower = -2, upper = 2,
tol = 0.0001))
str(vuniroot(function(x) x*(x^2-1) + .5, lower = -2, upper = 2,
tol = 1e-10))
## Find the smallest value x for which exp(x) > 0 (numerically):
r <- vuniroot(function(x) 1e80*exp(x) - 1e-300, cbind(-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
}% donttest because printed output is so much platform dependent
##--- vuniroot() with new interval extension + checking features: --------------
f1 <- function(x) (121 - x^2)/(x^2+1)
f2 <- function(x) exp(-x)*(x - 12)
tools::assertCondition(vuniroot(f1, cbind(0,10)),
"error", verbose=TRUE)
tools::assertCondition(vuniroot(f2, cbind(0, 2)),
"error", verbose=TRUE)
##--> error: f() .. end points not of opposite sign
## where as 'extendInt="yes"' simply first enlarges the search interval:
u1 <- vuniroot(f1, cbind(0,10),extendInt="yes", trace=1)
u2 <- vuniroot(f2, cbind(0,2), extendInt="yes", trace=2)
stopifnot(all.equal(u1$root, 11, tolerance = 1e-5),
all.equal(u2$root, 12, tolerance = 6e-6))
## The *danger* of interval extension:
## No way to find a zero of a positive function, but
## numerically, f(-|M|) becomes zero :
tools::assertCondition(u3 <- vuniroot(exp, cbind(0,2), extendInt="yes", trace=TRUE),
"error", verbose=TRUE)
## Nonsense example (must give an error):
tools::assertCondition( vuniroot(function(x) 1, cbind(0,1), extendInt="yes"),
"error", verbose=TRUE)
## Convergence checking :
sinc_ <- function(x) ifelse(x == 0, 1, sin(x)/x)
curve(sinc_, -6,18); abline(h=0,v=0, lty=3, col=adjustcolor("gray", 0.8))
\dontshow{tools::assertWarning(}
vuniroot(sinc_, cbind(0,5), extendInt="yes", maxiter=4) #-> "just" a warning
\dontshow{ , verbose=TRUE)}
## now with check.conv=TRUE, must signal a convergence error :
\dontshow{tools::assertError(}
vuniroot(sinc_, cbind(0,5), extendInt="yes", maxiter=4, check.conv=TRUE)
\dontshow{ , verbose=TRUE)}
### Weibull cumulative hazard (example origin, Ravi Varadhan):
cumhaz <- function(t, a, b) b * (t/b)^a
froot <- function(x, u, a, b) cumhaz(x, a, b) - u
n <- 10
u <- -log(runif(n))
a <- 1/2
b <- 1
## Find failure times
ru <- vuniroot(froot, u=u, a=a, b=b, interval= cbind(rep(1.e-14,n), rep(1e4,n)),
extendInt="yes")$root
ru2 <- vuniroot(froot, u=u, a=a, b=b, interval= cbind(rep(0.01,n), rep(10,n)),
extendInt="yes")$root
stopifnot(all.equal(ru, ru2, tolerance = 6e-6))
r1 <- vuniroot(froot, u= 0.99, a=a, b=b, interval= cbind(0.01, 10),
extendInt="up")
stopifnot(all.equal(0.99, cumhaz(r1$root, a=a, b=b)))
## An error if 'extendInt' assumes "wrong zero-crossing direction":
\dontshow{tools::assertError(}
vuniroot(froot, u= 0.99, a=a, b=b, interval= cbind(0.1, 10), extendInt="down")
\dontshow{ , verbose=TRUE)}
}
\keyword{optimize}
