Trapezoidal Integration
Compute the area of a function with values y at the points
x.
trapz(x, y) cumtrapz(x, y) trapzfun(f, a, b, maxit = 25, tol = 1e-07, ...)
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. |
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.
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.
# 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
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