# trapz: Trapezoidal Integration

## Description

Compute the area of a function with values `y`

at the points
`x`

.
## Usage

trapz(x, y) cumtrapz(x, y)

trapzfun(f, a, b, maxit = 25, tol = 1e-07, ...)

## Arguments

x

x-coordinates of points on the x-axis

y

y-coordinates of function values

f

function to be integrated.

a, b

lower and upper border of the integration domain.

maxit

maximum number of steps.

tol

tolerance; stops when improvements are smaller.

...

arguments passed to the function.

## Value

Approximated integral of the function, discretized through the points
`x, y`

, from `min(x)`

to `max(x)`

.
Or a matrix of the same size as `y`

.`trapzfun`

returns a lst with components `value`

the value of
the integral, `iter`

the number of iterations, and `rel.err`

the relative error.

## Details

The points `(x, 0)`

and `(x, y)`

are taken as vertices of a
polygon and the area is computed using `polyarea`

. This approach
matches exactly the approximation for integrating the function using the
trapezoidal rule with basepoints `x`

. `cumtrapz`

computes the cumulative integral of `y`

with respect
to `x`

using trapezoidal integration. `x`

and `y`

must be
vectors of the same length, or `x`

must be a vector and `y`

a
matrix whose first dimension is `length(x)`

.
Inputs `x`

and `y`

can be complex.

`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.

## 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