https://github.com/cran/pracma
Tip revision: f0ba8a5d88ce30d08c8312fd9b882f2ff051d91b authored by Hans W. Borchers on 25 August 2018, 21:00:11 UTC
version 2.1.5
version 2.1.5
Tip revision: f0ba8a5
trapz.Rd
\name{trapz}
\alias{trapz}
\alias{cumtrapz}
\alias{trapzfun}
\title{Trapezoidal Integration}
\description{
Compute the area of a function with values \code{y} at the points
\code{x}.
}
\usage{
trapz(x, y)
cumtrapz(x, y)
trapzfun(f, a, b, maxit = 25, tol = 1e-07, ...)
}
\arguments{
\item{x}{x-coordinates of points on the x-axis}
\item{y}{y-coordinates of function values}
\item{f}{function to be integrated.}
\item{a, b}{lower and upper border of the integration domain.}
\item{maxit}{maximum number of steps.}
\item{tol}{tolerance; stops when improvements are smaller.}
\item{...}{arguments passed to the function.}
}
\details{
The points \code{(x, 0)} and \code{(x, y)} are taken as vertices of a
polygon and the area is computed using \code{polyarea}. This approach
matches exactly the approximation for integrating the function using the
trapezoidal rule with basepoints \code{x}.
\code{cumtrapz} computes the cumulative integral of \code{y} with respect
to \code{x} using trapezoidal integration. \code{x} and \code{y} must be
vectors of the same length, or \code{x} must be a vector and \code{y} a
matrix whose first dimension is \code{length(x)}.
Inputs \code{x} and \code{y} can be complex.
\code{trapzfun} realizes trapezoidal integration and stops when the
differencefrom one step to the next is smaller than tolerance (or the
of iterations get too big). The function will only be evaluated once
on each node.
}
\value{
Approximated integral of the function, discretized through the points
\code{x, y}, from \code{min(x)} to \code{max(x)}.
Or a matrix of the same size as \code{y}.
\code{trapzfun} returns a lst with components \code{value} the value of
the integral, \code{iter} the number of iterations, and \code{rel.err}
the relative error.
}
\seealso{
\code{\link{polyarea}}
}
\examples{
# Calculate the area under the sine curve from 0 to pi:
n <- 101
x <- seq(0, pi, len = n)
y <- sin(x)
trapz(x, y) #=> 1.999835504
# Use a correction term at the boundary: -h^2/12*(f'(b)-f'(a))
h <- x[2] - x[1]
ca <- (y[2]-y[1]) / h
cb <- (y[n]-y[n-1]) / h
trapz(x, y) - h^2/12 * (cb - ca) #=> 1.999999969
# Use two complex inputs
z <- exp(1i*pi*(0:100)/100)
ct <- cumtrapz(z, 1/z)
ct[101] #=> 0+3.14107591i
f <- function(x) x^(3/2) #
trapzfun(f, 0, 1) #=> 0.4 with 11 iterations
}
\keyword{ math }