| uniroot | R Documentation |
The function 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
‘Details’ section.
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 |
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 |
lower, upper |
the lower and upper end points of the interval to be searched. |
f.lower, f.upper |
the same as |
extendInt |
character string specifying if the interval
|
check.conv |
logical indicating whether a convergence warning of the
underlying |
tol |
the desired accuracy (convergence tolerance). |
maxiter |
the maximum number of iterations. |
trace |
integer number; if positive, tracing information is produced. Higher values giving more 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) * f(u) <= 0.
If it is known how f changes sign at the root
x0, 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:= +/- 1, to
require S = sign(f(x0 +
eps)) at the solution. In that case, the search interval [l,u]
possibly is extended to be such that
S * f(l) <= 0 and S * f(u) >= 0.
uniroot() uses Fortran subroutine ‘"zeroin"’ (from Netlib)
based on 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 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.
A list with at least 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.)
Further components may be added in future: component init.it
was added in R 3.1.0.
Based on ‘zeroin.c’ in https://www.netlib.org/c/brent.shar.
Brent, R. (1973) Algorithms for Minimization without Derivatives. Englewood Cliffs, NJ: Prentice-Hall.
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))
uniroot(sinc, c(0,5), extendInt="yes", maxiter=4) #-> "just" a warning
## now with check.conv=TRUE, must signal a convergence error :
uniroot(sinc, c(0,5), extendInt="yes", maxiter=4, check.conv=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":
uniroot(froot, u= 0.99, a=a, b=b, interval= c(0.1, 10), extendInt="down")
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