# uniroot

##### One Dimensional Root (Zero) Finding

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

- Keywords
- optimize

##### Usage

```
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)
```

##### 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. - check.conv
- logical indicating whether a convergence warning of the
underlying
`uniroot`

should be caught as an error and if non-convergence in`maxiter`

iterations should be an error instead of a warning. - 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.

##### 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 `"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(`

for a numeric value
of `x`, ...)`x`.

The argument passed to `f`

has special semantics and used to be
shared between calls. The function should not copy it.

##### Value

- 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 R3.1.0.

##### source

Based on

##### References

Brent, R. (1973)
*Algorithms for Minimization without Derivatives.*
Englewood Cliffs, NJ: Prentice-Hall.

##### See Also

`polyroot`

for all complex roots of a polynomial;
`optimize`

, `nlm`

.

##### Examples

`library(stats)`

```
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)
```

*Documentation reproduced from package stats, version 3.3, License: Part of R 3.3*