uniroot.all: Finds many (all) roots of one equation within an interval
In rootSolve: Nonlinear Root Finding, Equilibrium and Steady-State Analysis of Ordinary Differential Equations

Description Usage Arguments Details Value Note Author(s) See Also Examples

The function uniroot.all searches the interval from lower to upper for several roots (i.e., zero's) of a function f with respect to its first argument.

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uniroot.all(f, interval, lower = min(interval), upper = max(interval), 
            tol = .Machine$double.eps^0.2, maxiter = 1000, 
            trace = 0, n = 100, ...)

`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.
`lower`	the lower end point of the interval to be searched.
`upper`	the upper end point of the interval to be searched.
`tol`	the desired accuracy (convergence tolerance). Passed to function uniroot
`maxiter`	the maximum number of iterations. Passed to function uniroot
`trace`	integer number; if positive, tracing information is produced. Higher values giving more details. Passed to function uniroot
`n`	number of subintervals in which the root is sought.
`...`	additional named or unnamed arguments to be passed to `f` (but beware of partial matching to other arguments).

f will be called as f(x, ...) for a numeric value of x.

Run demo(Jacobandroots) for an example of the use of uniroot.all for steady-state analysis.

See also second example of gradient This example is discussed in the book by Soetaert and Herman (2009).

a vector with the roots found in the interval

The function calls uniroot, the basic R-function.

It is not guaranteed that all roots will be recovered.

This will depend on n, the number of subintervals in which the interval is divided.

If the function "touches" the X-axis (i.e. the root is a saddle point), then this root will generally not be retrieved. (but chances of this are pretty small).

Whereas unitroot passes values one at a time to the function, uniroot.all passes a vector of values to the function. Therefore f should be written such that it can handle a vector of values. See last example.

Karline Soetaert <karline.soetaert@nioz.nl>

uniroot for more information about input.

## =======================================================================
##   Mathematical examples  
## =======================================================================

# a well-behaved case...
fun <- function (x) cos(2*x)^3

curve(fun(x), 0, 10,main = "uniroot.all")

All <- uniroot.all(fun, c(0, 10))
points(All, y = rep(0, length(All)), pch = 16, cex = 2)

# a difficult case...
f <- function (x) 1/cos(1+x^2)
AA <- uniroot.all(f, c(-5, 5))
curve(f(x), -5, 5, n = 500, main = "uniroot.all")
points(AA, rep(0, length(AA)), col = "red", pch = 16)

f(AA)  # !!!

## =======================================================================
## Ecological modelling example  
## =======================================================================

# Example from the book of Soetaert and Herman(2009)
# A practical guide to ecological modelling -
# using R as a simulation platform. Springer

r   <- 0.05
K   <- 10
bet <- 0.1
alf <- 1

# the model : density-dependent growth and sigmoid-type mortality rate
rate <- function(x, r = 0.05) r*x*(1-x/K) - bet*x^2/(x^2+alf^2)

# find all roots within the interval [0,10]
Eq   <- uniroot.all(rate, c(0, 10))

# jacobian evaluated at all roots: 
# This is just one value - and therefore jacobian = eigenvalue
# the sign of eigenvalue: stability of the root: neg=stable, 0=saddle, pos=unstable

eig <- vector()
for (i in 1:length(Eq)) 
  eig[i] <- sign (gradient(rate, Eq[i]))

curve(rate(x), ylab = "dx/dt", from = 0, to = 10,
      main = "Budworm model, roots", 
      sub = "Example from Soetaert and Herman, 2009")
abline(h = 0)
points(x = Eq, y = rep(0, length(Eq)), pch = 21, cex = 2,
       bg = c("grey", "black", "white")[eig+2] )
legend("topleft", pch = 22, pt.cex = 2,
       c("stable", "saddle", "unstable"),
       col = c("grey", "black", "white"), 
       pt.bg = c("grey", "black", "white"))
       
## =======================================================================
## Vectorisation:
## =======================================================================
# from R-help Digest, Vol 130, Issue 27
#https://stat.ethz.ch/pipermail/r-help/2013-December/364799.html

integrand1 <- function(x) 1/x*dnorm(x) 
integrand2 <- function(x) 1/(2*x-50)*dnorm(x) 
integrand3 <- function(x, C) 1/(x+C)

res <- function(C) {
  integrate(integrand1, lower = 1, upper = 50)$value + 
  integrate(integrand2, lower = 50, upper = 100)$value - 
  integrate(integrand3, C = C, lower = 1, upper = 100)$value
}

# uniroot passes one value at a time to the function, so res can be used as such
uniroot(res, c(1, 1000))

# Need to vectorise the function to use uniroot.all:
res <- Vectorize(res)
uniroot.all(res, c(1,1000))

 [1]  2208.227  1811.285  1741.080 -2735.492 -2273.341  3422.914  2635.566
 [8] 11794.351 11794.351  2635.566  3422.914 -2273.341 -2735.492  1741.080
[15]  1811.285  2208.227
$root
[1] 837.0516

$f.root
[1] 1.622351e-11

$iter
[1] 9

$init.it
[1] NA

$estim.prec
[1] 6.103516e-05

[1] 837.0516