multiroot.1D: Solves for n roots of n (nonlinear) equations, created by...
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

multiroot.1D finds the solution to boundary value problems of ordinary differential equations, which are approximated using the method-of-lines approach.

Assumes a banded Jacobian matrix, uses the Newton-Raphson method.

multiroot.1D(f, start, maxiter = 100,
             rtol = 1e-6, atol = 1e-8, ctol = 1e-8, 
             nspec = NULL, dimens = NULL, verbose = FALSE, 
             positive = FALSE, names = NULL, parms = NULL, ...)

`f`	function for which the root is sought; it must return a vector with as many values as the length of `start`. It is called either as `f(x, ...)` if `parms = NULL` or as `f(x, parms, ...)` if `parms` is not `NULL`.
`start`	vector containing initial guesses for the unknown x; if `start` has a name attribute, the names will be used to label the output vector.
`maxiter`	maximal number of iterations allowed.
`rtol`	relative error tolerance, either a scalar or a vector, one value for each element in the unknown x.
`atol`	absolute error tolerance, either a scalar or a vector, one value for each element in x.
`ctol`	a scalar. If between two iterations, the maximal change in the variable values is less than this amount, then it is assumed that the root is found.
`nspec`	the number of species (components) in the model. If `NULL`, then `dimens` should be specified.
`dimens`	the number of boxes in the model. If NULL, then `nspec` should be specified.
`verbose`	if `TRUE`: full output to the screen, e.g. will output the steady-state settings.
`positive`	if `TRUE`, the unknowns y are forced to be non-negative numbers.
`names`	the names of the components; used to label the output, which will be written as a matrix.
`parms`	vector or list of parameters used in `f`.
`...`	additional arguments passed to function `f`.

multiroot.1D is similar to steady.1D, except for the function specification which is simpler in multiroot.1D.

It is to be used to solve (simple) boundary value problems of differential equations.

The following differential equation:

0=f(x,y,y',y'')

with boundary conditions

y(x=a) = ya, at the start and y(x=b)=yb at the end of the integration interval [a,b] is approximated

as follows:

1. First the integration interval x is discretized, for instance:

dx <- 0.01

x <- seq(a,b,by=dx)

where dx should be small enough to keep numerical errors small.

2. Then the first- and second-order derivatives are differenced on this numerical grid. R's diff function is very efficient in taking numerical differences, so it is used to approximate the first-, and second-order derivates as follows.

A first-order derivative y' can be approximated either as:

y'=diff(c(ya,y))/dx: if only the initial condition ya is prescribed,
y'=diff(c(y,yb))/dx: if only the final condition, yb is prescribed,
y'=0.5*(diff(c(ya,y))/dx+diff(c(y,yb))/dx): if initial, ya, and final condition, yb are prescribed.

The latter (centered differences) is to be preferred.

A second-order derivative y” can be approximated by differencing twice.

y”=diff(diff(c(ya,y,yb))/dx)/dx

3. Finally, function multiroot.1D is used to locate the root.

See the examples

a list containing:

`root`	the values of the root.
`f.root`	the value of the function evaluated at the `root`.
`iter`	the number of iterations used.
`estim.precis`	the estimated precision for `root`. It is defined as the mean of the absolute function values (mean(abs(`f.root`))).

multiroot.1D makes use of function steady.1D.

It is NOT guaranteed that the method will converge to the root.

Karline Soetaert <karline.soetaert@nioz.nl>

stode, which uses a different function call.

uniroot.all, to solve for all roots of one (nonlinear) equation

steady, steady.band, steady.1D, steady.2D, steady.3D, steady-state solvers, which find the roots of ODEs or PDEs. The function call differs from multiroot.

jacobian.full, jacobian.band, estimates the Jacobian matrix assuming a full or banded structure.

gradient, hessian, estimates the gradient matrix or the Hessian.

## =======================================================================
## Example 1: simple standard problem
## solve the BVP ODE:
## d2y/dt^2=-3py/(p+t^2)^2
## y(t= -0.1)=-0.1/sqrt(p+0.01)
## y(t=  0.1)= 0.1/sqrt(p+0.01)
## where p = 1e-5
##
## analytical solution y(t) = t/sqrt(p + t^2).
##
## =======================================================================

bvp <- function(y) {
  dy2 <- diff(diff(c(ya, y, yb))/dx)/dx
  return(dy2 + 3*p*y/(p+x^2)^2)
}

dx <- 0.0001
x <- seq(-0.1, 0.1, by = dx)

p  <- 1e-5
ya <- -0.1/sqrt(p+0.01)
yb <-  0.1/sqrt(p+0.01)

print(system.time(
  y <- multiroot.1D(start = runif(length(x)), f = bvp, nspec = 1)
))

plot(x, y$root, ylab = "y", main = "BVP test problem")

# add analytical solution
curve(x/sqrt(p+x*x), add = TRUE, type = "l", col = "red")

## =======================================================================
## Example 2: bvp test problem 28
## solve:
## xi*y'' + y*y' - y=0
## with boundary conditions:
## y0=1
## y1=3/2
## =======================================================================

prob28 <-function(y, xi) {
 dy2 <- diff(diff(c(ya, y, yb))/dx)/dx          # y''
 dy  <- 0.5*(diff(c(ya, y)) +diff(c(y, yb)))/dx # y' - centered differences

 xi*dy2 +dy*y-y
}

ya <- 1
yb <- 3/2
dx <- 0.001
x <- seq(0, 1, by = dx)
N <- length(x)
print(system.time(
  Y1 <- multiroot.1D(f = prob28, start = runif(N), 
                     nspec = 1, xi = 0.1)
))
Y2<- multiroot.1D(f = prob28, start = runif(N), nspec = 1, xi = 0.01)
Y3<- multiroot.1D(f = prob28, start = runif(N), nspec = 1, xi = 0.001)

plot(x, Y3$root, type = "l", col = "green", main = "bvp test problem 28")
lines(x, Y2$root, col = "red")
lines(x, Y1$root, col = "blue")