\Header{splinefun}{Interpolating Splines}
\alias{spline}{splinefun}
\keyword{math}{splinefun}
\keyword{dplot}{splinefun}
\begin{Description}\relax
Perform cubic spline interpolation of given data points, returning
either a list of points obtained by the interpolation or a function
performing the interpolation.
\end{Description}
\begin{Usage}
\begin{verbatim}
splinefun(x, y = NULL, method = "fmm")

spline(x, y = NULL, n = 3*length(x), method = "fmm",
       xmin = min(x), xmax = max(x))
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x,y}] vectors giving the coordinates of the points to be
interpolated.  Alternatively a single plotting structure can be
specified: see \code{\Link{xy.coords}.}
\item[\code{method}] specifies the type of spline to be used.  Possible
values are \code{"fmm"}, \code{"natural"} and \code{"periodic"}.
\item[\code{n}] interpolation takes place at \code{n} equally spaced points
spanning the interval [\code{xmin}, \code{xmax}].
\item[\code{xmin}] left-hand endpoint of the interpolation interval.
\item[\code{xmax}] right-hand endpoint of the interpolation interval.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
If \code{method = "fmm"}, the spline used is that of Forsythe, Malcolm
and Moler (an exact cubic is fitted through the four points at each
end of the data, and this is used to determine the end conditions).
Natural splines are used when \code{method = "natural"}, and periodic
splines when \code{method = "periodic"}.

These interpolation splines can also be used for extrapolation, that is
prediction at points outside the range of \code{x}.  Extrapolation
makes little sense for \code{method = "fmm"}; for natural splines it
is linear using the slope of the interpolating curve at the nearest
data point.
\end{Details}
\begin{Value}
\code{spline} returns a list containing components \code{x} and
\code{y} which give the ordinates where interpolation took place and
the interpolated values.

\code{splinefun} returns a function which will perform cubic spline
interpolation of the given data points.  This is often more useful
than \code{spline}.
\end{Value}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Forsythe, G. E., Malcolm, M. A. and Moler, C. B. (1977)
\emph{Computer Methods for Mathematical Computations}.
\end{References}
\begin{SeeAlso}\relax
\code{\Link{approx}} and \code{\Link{approxfun}} for constant and
linear interpolation.

Package \pkg{splines}, especially \code{\Link{interpSpline}}
and \code{\Link{periodicSpline}} for interpolation splines.
That package also generates spline bases that can be used for
regression splines.

\code{\Link{smooth.spline}} in package \pkg{modreg} for
smoothing splines.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
op <- par(mfrow = c(2,1), mgp = c(2,.8,0), mar = .1+c(3,3,3,1))
n <- 9
x <- 1:n
y <- rnorm(n)
plot(x, y, main = paste("spline[fun](.) through", n, "points"))
lines(spline(x, y))
lines(spline(x, y, n = 201), col = 2)

y <- (x-6)^2
plot(x, y, main = "spline(.) -- 3 methods")
lines(spline(x, y, n = 201), col = 2)
lines(spline(x, y, n = 201, method = "natural"), col = 3)
lines(spline(x, y, n = 201, method = "periodic"), col = 4)
legend(6,25, c("fmm","natural","periodic"), col=2:4, lty=1)

f <- splinefun(x, y)
ls(envir = environment(f))
splinecoef <- eval(expression(z), envir = environment(f))
curve(f(x), 1, 10, col = "green", lwd = 1.5)
points(splinecoef, col = "purple", cex = 2)
par(op)
\end{ExampleCode}
\end{Examples}

