\Header{nlm}{Non-Linear Minimization}
\keyword{nonlinear}{nlm}
\keyword{optimize}{nlm}
\begin{Description}\relax
This function carries out a minimization of the function \code{f}
using a Newton-type algorithm.  See the references for details.

This is a preliminary version of this function and it will probably change.\end{Description}
\begin{Usage}
\begin{verbatim}
nlm(f, p, hessian = FALSE, typsize=rep(1, length(p)), fscale=1,
    print.level = 0, ndigit=12, gradtol = 1e-6,
    stepmax = max(1000 * sqrt(sum((p/typsize)^2)), 1000),
    steptol = 1e-6, iterlim = 100, check.analyticals = TRUE, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{f}] the function to be minimized.  If the function value has
an attribute called \code{gradient} or both \code{gradient} and
\code{hessian} attributes, these will be used in the calculation of
updated parameter values.  Otherwise, numerical derivatives are
used. \code{\Link{deriv}} returns a function with suitable
\code{gradient} attribute.  This should be a function a vector of
the length of \code{p} followed by any other arguments specified
in \code{dots}.
\item[\code{p}] starting parameter values for the minimization.
\item[\code{hessian}] if \code{TRUE}, the hessian of \code{f}
at the minimum is returned.
\item[\code{typsize}] an estimate of the size of each parameter
at the minimum.
\item[\code{fscale}] an estimate of the size of \code{f} at the minimum.
\item[\code{print.level}] this argument determines the level of printing
which is done during the minimization process.  The default
value of \code{0} means that no printing occurs, a value of \code{1}
means that initial and final details are printed and a value
of 2 means that full tracing information is printed.
\item[\code{ndigit}] the number of significant digits in the function \code{f}.
\item[\code{gradtol}] a positive scalar giving the tolerance at which the
scaled gradient is considered close enough to zero to
terminate the algorithm.  The scaled gradient is a
measure of the relative change in \code{f} in each direction
\code{p[i]} divided by the relative change in \code{p[i]}.
\item[\code{stepmax}] a positive scalar which gives the maximum allowable
scaled step length.  \code{stepmax} is used to prevent steps which
would cause the optimization function to overflow, to prevent the
algorithm from leaving the area of interest in parameter space, or to
detect divergence in the algorithm. \code{stepmax} would be chosen
small enough to prevent the first two of these occurrences, but should
be larger than any anticipated reasonable step.
\item[\code{steptol}] A positive scalar providing the minimum allowable
relative step length.
\item[\code{iterlim}] a positive integer specifying the maximum number of
iterations to be performed before the program is terminated.
\item[\code{check.analyticals}] a logical scalar specifying whether the
analytic gradients and Hessians, if they are supplied, should be
checked against numerical derivatives at the initial parameter
values. This can help detect incorrectly formulated gradients or
Hessians.
\item[\code{...}] additional arguments to \code{f}.
\end{ldescription}
\end{Arguments}
\begin{Value}
A list containing the following components:
\begin{ldescription}
\item[\code{minimum}] the value of the estimated minimum of \code{f}.
\item[\code{estimate}] the point at which the mininum value of
\code{f} is obtained.
\item[\code{gradient}] the gradient at the estimated minimum of \code{f}.
\item[\code{hessian}] the hessian at the estimated minimum of \code{f} (if requested).
\item[\code{code}] an integer indicating why the optimization process terminated.

1: relative gradient is close to zero, current iterate is
probably solution.

2: successive iterates within tolerance, current iterate
is probably solution.

3: last global step failed to locate a point lower than
\code{estimate}.  Either \code{estimate} is an approximate local
minimum of the function or \code{steptol} is too small.

4: iteration limit exceeded.

5: maximum step size \code{stepmax} exceeded five consecutive
times.  Either the function is unbounded below,
becomes asymptotic to a finite value from above in
some direction or \code{stepmax} is too small.

\end{ldescription}
\end{Value}
\begin{References}\relax
Dennis, J. E. and Schnabel, R. B. (1983) \emph{Numerical Methods for
Unconstrained Optimization and Nonlinear Equations.} Prentice-Hall,
Englewood Cliffs, NJ.

Schnabel, R. B., Koontz, J. E. and Weiss, B. E. (1985) A modular
system of algorithms for unconstrained minimization.
\emph{ACM Trans. Math. Software}, \bold{11}, 419--440.\end{References}
\begin{SeeAlso}\relax
\code{\Link{optimize}} for one-dimensional
minimization and \code{\Link{uniroot}} for root finding.
\code{\Link{deriv}} to calculate analytical derivatives.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
f <- function(x) sum((x-1:length(x))^2)
nlm(f, c(10,10))
nlm(f, c(10,10), print.level = 2)
str(nlm(f, c(5), hessian = TRUE))

f <- function(x, a) sum((x-a)^2)
nlm(f, c(10,10), a=c(3,5))
f <- function(x, a)
{
    res <- sum((x-a)^2)
    attr(res, "gradient") <- 2*(x-a)
    res
}
nlm(f, c(10,10), a=c(3,5))

## more examples, including the use of derivatives.
demo(nlm)
\end{ExampleCode}
\end{Examples}

