R: Trapezoidal Integration

trapz {pracma} | R Documentation

## 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.

### 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.

### 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.

### See Also

`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

[Package pracma version 2.0.4 Index]