vuniroot: Vectorised One Dimensional Root (Zero) Finding

Description

The function vuniroot searches the interval from lower to upper for a root (i.e., zero) of the vectorised 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.

Usage

vuniroot(f, interval, ...,
        lower, upper,
        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,
        n = NULL)

Value

A list with at least three components: root and f.root

give the location of the root and the value of the function evaluated at that point. iter gives the number of iterations used.

Further components may be added in future: component init.it

was added in R 3.1.0.

Arguments

f: the function for which the root is sought.
interval: a matrix with two columns 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 vuniroot 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.
n: integer number; size of input vector to f (only used if lower and upper are of length 1)

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 “upward crossing”) or "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 \(S \cdot f(u) \ge 0\).

vuniroot() uses a C++ subroutine based on "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 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.

References

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

Examples

Run this code

# \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
# }

##--- 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))
tools::assertWarning(
vuniroot(sinc_, cbind(0,5), extendInt="yes", maxiter=4) #-> "just" a warning
 , verbose=TRUE)

## now with  check.conv=TRUE, must signal a convergence error :
tools::assertError(
vuniroot(sinc_, cbind(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 <- 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":
tools::assertError(
vuniroot(froot, u= 0.99, a=a, b=b, interval= cbind(0.1, 10), extendInt="down")
 , verbose=TRUE)

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