Finds many (all) roots of one equation within an interval

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Description

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.

Usage

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

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.

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).

maxiter

the maximum number of iterations.

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).

Details

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).

Value

a vector with the roots found in the interval

Note

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.

Author(s)

Karline Soetaert <karline.soetaert@nioz.nl>

See Also

uniroot for more information about input.

Examples

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## =======================================================================
##   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))