
uniroot
searches the interval from lower
to upper
for a root (i.e., zero) of the function f
with
respect to its first argument. Setting extendInt
to a non-"no"
string, 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
uniroot(f, interval, ...,
lower = min(interval), upper = max(interval),
f.lower = f(lower, ...), f.upper = f(upper, ...),
extendInt = c("no", "yes", "downX", "upX"), check.conv = FALSE,
tol = .Machine$double.eps^0.25, maxiter = 1000, trace = 0)
f
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.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.uniroot
should be caught as an error and if
non-convergence in maxiter
iterations should be an error
instead of a warning.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
.) Further components may be added in future: component init.it
was added in R3.1.0.
...
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 "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 $S \cdot f(u) \ge 0$.
uniroot()
uses Fortran subroutine
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 numeric value
of x.
The argument passed to f
has special semantics and used to be
shared between calls. The function should not copy it.
polyroot
for all complex roots of a polynomial;
optimize
, nlm
.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 <- uniroot(f, c(0, 1), tol = 0.0001, a = 1/3))
## handheld calculator example: fixed point of cos(.):
uniroot(function(x) cos(x) - x, lower = -pi, upper = pi, tol = 1e-9)$root
str(uniroot(function(x) x*(x^2-1) + .5, lower = -2, upper = 2,
tol = 0.0001))
str(uniroot(function(x) x*(x^2-1) + .5, lower = -2, upper = 2,
tol = 1e-10))
## Find the smallest value x for which exp(x) > 0 (numerically):
r <- uniroot(function(x) 1e80*exp(x) - 1e-300, c(-1000, 0), tol = 1e-15)
str(r, digits.d = 15) # around -745, depending on the platform.
exp(r$root) # = 0, but not for r$root * 0.999...
minexp <- r$root * (1 - 10*.Machine$double.eps)
exp(minexp) # typically denormalized
##--- uniroot() with new interval extension + checking features: --------------
f1 <- function(x) (121 - x^2)/(x^2+1)
f2 <- function(x) exp(-x)*(x - 12)
try(uniroot(f1, c(0,10)))
try(uniroot(f2, c(0, 2)))
##--> error: f() .. end points not of opposite sign
## where as 'extendInt="yes"' simply first enlarges the search interval:
u1 <- uniroot(f1, c(0,10),extendInt="yes", trace=1)
u2 <- uniroot(f2, c(0,2), extendInt="yes", trace=2)
stopifnot(all.equal(u1$root, 11, tolerance = 1e-5),
all.equal(u2$root, 12, tolerance = 6e-6))
## The *danger* of interval extension:
## No way to find a zero of a positive function, but
## numerically, f(-|M|) becomes zero :
u3 <- uniroot(exp, c(0,2), extendInt="yes", trace=TRUE)
## Nonsense example (must give an error):
tools::assertCondition( uniroot(function(x) 1, 0:1, extendInt="yes"),
"error", verbose=TRUE)
## Convergence checking :
sinc <- function(x) ifelse(x == 0, 1, sin(x)/x)
curve(sinc, -6,18); abline(h=0,v=0, lty=3, col=adjustcolor("gray", 0.8))
tools::assertWarning(
uniroot(sinc, c(0,5), extendInt="yes", maxiter=4) #-> "just" a warning
, verbose=TRUE)
## now with check.conv=TRUE, must signal a convergence error :
tools::assertError(
uniroot(sinc, c(0,5), extendInt="yes", maxiter=4, check.conv=TRUE)
, verbose=TRUE)
### Weibull cumulative hazard (example origin, Ravi Varadhan):
cumhaz <- function(t, a, b) b * (t/b)^a
froot <- function(x, u, a, b) cumhaz(x, a, b) - u
n <- 1000
u <- -log(runif(n))
a <- 1/2
b <- 1
## Find failure times
ru <- sapply(u, function(x)
uniroot(froot, u=x, a=a, b=b, interval= c(1.e-14, 1e04),
extendInt="yes")$root)
ru2 <- sapply(u, function(x)
uniroot(froot, u=x, a=a, b=b, interval= c(0.01, 10),
extendInt="yes")$root)
stopifnot(all.equal(ru, ru2, tolerance = 6e-6))
r1 <- uniroot(froot, u= 0.99, a=a, b=b, interval= c(0.01, 10),
extendInt="up")
stopifnot(all.equal(0.99, cumhaz(r1$root, a=a, b=b)))
## An error if 'extendInt' assumes "wrong zero-crossing direction":
tools::assertError(
uniroot(froot, u= 0.99, a=a, b=b, interval= c(0.1, 10), extendInt="down")
, verbose=TRUE)
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