\Header{approxfun}{Interpolation Functions}
\alias{approx}{approxfun}
\keyword{arith}{approxfun}
\keyword{dplot}{approxfun}
\begin{Description}\relax
Return a list of points which linearly interpolate given data points,
or a function performing the linear (or constant) interpolation.\end{Description}
\begin{Usage}
\begin{verbatim}
approx   (x, y, xout, method="linear", n=50,
          yleft, yright, rule = 1, f=0, ties = mean)
approxfun(x, y,       method="linear",
          yleft, yright, rule = 1, f=0, ties = mean)
\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{xout}] an optional set of values specifying where interpolation
is to take place.
\item[\code{method}] specifies the interpolation method to be used.  Choices
are \code{"linear"} or \code{"constant"}.
\item[\code{n}] If \code{xout} is not specified, interpolation takes place at
\code{n} equally spaced points spanning the interval [\code{min(x)},
\code{max(x)}].
\item[\code{yleft}] the value to be returned when input \code{x} values
less than \code{min(x)}. The default is defined by the value
of \code{rule} given below.
\item[\code{yright}] the value to be returned when input \code{x} values
greater than \code{max(x)}. The default is defined by the value
of \code{rule} given below.
\item[\code{rule}] an integer describing how interpolation is to take place
outside the interval [\code{min(x)}, \code{max(x)}].
If \code{rule} is \code{1} then \code{NA}s are returned for such
points and if it is \code{2}, the value at the closest data extreme
is used.
\item[\code{f}] For \code{method="constant"} a number between 0 and 1
inclusive, indicating a compromise between left- and
right-continuous step functions. If \code{y0} and \code{y1} are the
values to the left and right of the point then the value is
\code{y0*(1-f)+y1*f} so that \code{f=0} is right-continuous and
\code{f=1} is left-continuous.
\item[\code{ties}] Handling of tied \code{x} values. Either a function
with a single vector argument returning a single number result or
the string \code{"ordered"}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The inputs can contain missing values which are deleted, so at least
two complete \code{(x, y)} pairs are required.  If there are
duplicated (tied) \code{x} values and \code{ties} is a function it is
applied to the \code{y} values for each distinct \code{x} value.
Useful functions in this context include \code{\Link{mean}},
\code{\Link{min}}, and \code{\Link{max}}.  If \code{ties="ordered"}
the \code{x} values are assumed to be already ordered.  The first
\code{y} value will be used for interpolation to the left and the last
one for interpolation to the right.\end{Details}
\begin{Value}
\code{approx} returns a list with components \code{x} and \code{y},
containing \code{n} coordinates which interpolate the given data
points according to the \code{method} (and \code{rule}) desired.

The function \code{approxfun} returns a function performing (linear or
constant) interpolation of the given data points.  For a given set of
\code{x} values, this function will return the corresponding
interpolated values.  This is often more useful than \code{approx}.\end{Value}
\begin{SeeAlso}\relax
\code{\Link{spline}} and \code{\Link{splinefun}} for spline
interpolation.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
x <- 1:10
y <- rnorm(10)
par(mfrow = c(2,1))
plot(x, y, main = "approx(.) and approxfun(.)")
points(approx(x, y), col = 2, pch = "*")
points(approx(x, y, method = "constant"), col = 4, pch = "*")

f <- approxfun(x, y)
curve(f(x), 0, 10, col = "green")
points(x, y)
is.function(fc <- approxfun(x, y, method = "const")) # TRUE
curve(fc(x), 0, 10, col = "darkblue", add = TRUE)

## Show treatment of `ties' :

x <- c(2,2:4,4,4,5,5,7,7,7)
y <- c(1:6, 5:4, 3:1)
approx(x,y, xout=x)$y # warning
(ay <- approx(x,y, xout=x, ties = "ordered")$y)
stopifnot(ay == c(2,2,3,6,6,6,4,4,1,1,1))
approx(x,y, xout=x, ties = min)$y
approx(x,y, xout=x, ties = max)$y

\end{ExampleCode}
\end{Examples}

