% File src/library/stats/man/nlm.Rd
% Part of the R package, https://www.R-project.org
% Copyright 1995-2018 R Core Team
% Distributed under GPL 2 or later

\name{nlm}
\alias{nlm}
\title{Non-Linear Minimization}
\concept{optimization}
\usage{
nlm(f, p, \dots, 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)
}
\description{
  This function carries out a minimization of the function \code{f}
  using a Newton-type algorithm.  See the references for details.
}
\arguments{
  \item{f}{the function to be minimized, returning a single numeric
    value.  This should be a function with first argument a vector of
    the length of \code{p} followed by any other arguments specified by
    the \code{\dots} argument.

    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 and optionally a
    \code{hessian} attribute.}
  \item{p}{starting parameter values for the minimization.}
  \item{\dots}{additional arguments to be passed to \code{f}.}
  \item{hessian}{if \code{TRUE}, the hessian of \code{f}
    at the minimum is returned.}
  \item{typsize}{an estimate of the size of each parameter
    at the minimum.}
  \item{fscale}{an estimate of the size of \code{f} at the minimum.}
  \item{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{ndigit}{the number of significant digits in the function \code{f}.}
  \item{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{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{steptol}{A positive scalar providing the minimum allowable
    relative step length.}
  \item{iterlim}{a positive integer specifying the maximum number of
    iterations to be performed before the program is terminated.}
  \item{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.}
}
\details{
  Note that arguments after \code{\dots} must be matched exactly.

  If a gradient or hessian is supplied but evaluates to the wrong mode
  or length, it will be ignored if \code{check.analyticals = TRUE} (the
  default) with a warning.  The hessian is not even checked unless the
  gradient is present and passes the sanity checks.

  The C code for the \dQuote{perturbed} Cholesky, \code{choldc()} has
  had a bug in all \R versions before 3.4.1.

  From the three methods available in the original source, we always use
  method \dQuote{1} which is line search.

  The functions supplied should always return finite (including not
  \code{NA} and not \code{NaN}) values: for the function value itself
  non-finite values are replaced by the maximum positive value with a warning.
}
\value{
  A list containing the following components:
  \item{minimum}{the value of the estimated minimum of \code{f}.}
  \item{estimate}{the point at which the minimum value of
    \code{f} is obtained.}
  \item{gradient}{the gradient at the estimated minimum of \code{f}.}
  \item{hessian}{the hessian at the estimated minimum of \code{f} (if
    requested).}
  \item{code}{an integer indicating why the optimization process terminated.
    \describe{
      \item{1:}{relative gradient is close to zero, current iterate is
        probably solution.}
      \item{2:}{successive iterates within tolerance, current iterate
        is probably solution.}
      \item{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.}
      \item{4:}{iteration limit exceeded.}
      \item{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.}
    }
  }
  \item{iterations}{the number of iterations performed.}
}

\source{
  The current code is by Saikat DebRoy and the R Core team, using a C
  translation of Fortran code by Richard H. Jones.
}
\references{
  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 Transactions on Mathematical Software}, \bold{11}, 419--440.
  \doi{10.1145/6187.6192}.
}
\seealso{
  \code{\link{optim}} and \code{\link{nlminb}}.

  \code{\link{constrOptim}} for constrained optimization,
  \code{\link{optimize}} for one-dimensional
  minimization and \code{\link{uniroot}} for root finding.
  \code{\link{deriv}} to calculate analytical derivatives.

  For nonlinear regression, \code{\link{nls}} may be better.
}
\examples{
f <- function(x) sum((x-1:length(x))^2)
nlm(f, c(10,10))
nlm(f, c(10,10), print.level = 2)
utils::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.
\dontrun{demo(nlm)}
}
\keyword{nonlinear}
\keyword{optimize}