multiroot: Solves for n roots of n (nonlinear) equations.
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

Given a vector of n variables, and a set of n (nonlinear) equations in these variables,

estimates the root of the equations, i.e. the variable values where all function values = 0.

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

multiroot(f, start, maxiter = 100, 
          rtol = 1e-6, atol = 1e-8, ctol = 1e-8, 
          useFortran = TRUE, positive = FALSE,
          jacfunc = NULL, jactype = "fullint",
          verbose = FALSE, bandup = 1, banddown = 1, 
          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.
`useFortran`	logical, if `FALSE`, then an R -implementation of the Newton-Raphson method is used - see details.
`positive`	if `TRUE`, the unknowns y are forced to be non-negative numbers.
`jacfunc`	if not `NULL`, a user-supplied R function that estimates the Jacobian of the system of differential equations dydot(i)/dy(j). In some circumstances, supplying `jacfunc` can speed up the computations. The R calling sequence for `jacfunc` is identical to that of `f`. If the Jacobian is a full matrix, `jacfunc` should return a matrix dydot/dy, where the ith row contains the derivative of dy_i/dt with respect to y_j, or a vector containing the matrix elements by columns. If the Jacobian is banded, `jacfunc` should return a matrix containing only the nonzero bands of the jacobian, (dydot/dy), rotated row-wise.
`jactype`	the structure of the Jacobian, one of "fullint", "fullusr", "bandusr", "bandint", or "sparse" - either full or banded and estimated internally or by the user, or arbitrary sparse. If the latter, then the solver will call, `stodes`, else `stode` If the Jacobian is arbitrarily "sparse", then it will be calculated by the solver (i.e. it is not possible to also specify `jacfunc`).
`verbose`	if `TRUE`: full output to the screen, e.g. will output the steady-state settings.
`bandup`	number of non-zero bands above the diagonal, in case the Jacobian is banded.
`banddown`	number of non-zero bands below the diagonal, in case the jacobian is banded.
`parms`	vector or list of parameters used in `f` or `jacfunc`.
`...`	additional arguments passed to function `f`.

start gives the initial guess for each variable; different initial guesses may return different roots.

The input parameters rtol, and atol determine the error control performed by the solver.

The solver will control the vector e of estimated local errors in f, according to an inequality of the form max-norm of ( e/ewt ) <= 1, where ewt is a vector of positive error weights. The values of rtol and atol should all be non-negative.

The form of ewt is:

\bold{rtol} * abs(\bold{x}) + \bold{atol}

where multiplication of two vectors is element-by-element.

In addition, the solver will stop if between two iterations, the maximal change in the values of x is less than ctol.

There is no checking whether the requested precision exceeds the capabilities of the machine.

a list containing:

`root`	the location (x-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`))).

The Fortran implementation of the Newton-Raphson method function (the default) is generally faster than the R implementation. The R implementation has been included for didactic purposes.

multiroot makes use of function stode. Technically, it is just a wrapper around function stode. If the sparsity structure of the Jacobian is known, it may be more efficiently to call stode, stodes, steady, steady.1D, steady.2D, steady.3D.

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  
## 2 simultaneous equations
## =======================================================================

model <- function(x) c(F1 = x[1]^2+ x[2]^2 -1, 
                       F2 = x[1]^2- x[2]^2 +0.5)

(ss <- multiroot(f = model, start = c(1, 1)))

## =======================================================================
## example 2
## 3 equations, two solutions
## =======================================================================

model <- function(x) c(F1 = x[1] + x[2] + x[3]^2 - 12,
                       F2 = x[1]^2 - x[2] + x[3] - 2,
                       F3 = 2 * x[1] - x[2]^2 + x[3] - 1 )

# first solution
(ss <- multiroot(model, c(1, 1, 1), useFortran = FALSE))
(ss <- multiroot(f = model, start = c(1, 1, 1)))

# second solution; use different start values
(ss <- multiroot(model, c(0, 0, 0)))
model(ss$root)

## =======================================================================
## example 2b: same, but with parameters
## 3 equations, two solutions
## =======================================================================

model2 <- function(x, parms) 
      c(F1 = x[1] + x[2] + x[3]^2 - parms[1],
        F2 = x[1]^2 - x[2] + x[3] - parms[2],
        F3 = 2 * x[1] - x[2]^2 + x[3] - parms[3])

# first solution
parms <- c(12, 2, 1)
multiroot(model2, c(1, 1, 1), parms = parms)
multiroot(model2, c(0, 0, 0), parms = parms*2)

## =======================================================================
## example 3: find a matrix
## =======================================================================

f2<-function(x)   {
 X <- matrix(nrow = 5, x)
 X %*% X %*% X -matrix(nrow = 5, data = 1:25, byrow = TRUE)
}
x <- multiroot(f2, start = 1:25 )$root
X <- matrix(nrow = 5, x)

X%*%X%*%X

$root
[1] 0.5000000 0.8660254

$f.root
          F1           F2 
2.323138e-08 2.323308e-08 

$iter
[1] 5

$estim.precis
[1] 2.323223e-08

$root
[1] 1 2 3

$f.root
           F1            F2            F3 
 3.087877e-10  4.794444e-09 -8.678146e-09 

$iter
[1] 6

$estim.precis
[1] 4.593792e-09

$root
[1] 1 2 3

$f.root
           F1            F2            F3 
 3.087877e-10  4.794444e-09 -8.678146e-09 

$iter
[1] 6

$estim.precis
[1] 4.593792e-09

$root
[1] -0.2337207  1.3531901  3.2985649

$f.root
           F1            F2            F3 
 1.092413e-08  1.920978e-07 -4.850423e-08 

$iter
[1] 10

$estim.precis
[1] 8.384205e-08

           F1            F2            F3 
 1.092413e-08  1.920978e-07 -4.850423e-08 
$root
[1] 1 2 3

$f.root
           F1            F2            F3 
 3.087877e-10  4.794444e-09 -8.678146e-09 

$iter
[1] 6

$estim.precis
[1] 4.593792e-09

$root
[1] -0.6406658  1.2471383  4.8366856

$f.root
           F1            F2            F3 
 3.105072e-12  1.661515e-10 -2.408385e-11 

$iter
[1] 11

$estim.precis
[1] 6.444682e-11

     [,1] [,2] [,3] [,4] [,5]
[1,]    1    2    3    4    5
[2,]    6    7    8    9   10
[3,]   11   12   13   14   15
[4,]   16   17   18   19   20
[5,]   21   22   23   24   25