Numerical Differentiation
Numerical function differentiation for orders n=1..4 using
finite difference approximations.
fderiv(f, x, n = 1, h = 0,
method = c("central", "forward", "backward"), ...)f |
function to be differentiated. |
x |
point(s) where differentiation will take place. |
n |
order of derivative, should only be between 1 and 8;
for |
h |
step size: if |
method |
one of “central”, “forward”, or “backward”. |
... |
more variables to be passed to function |
Derivatives are computed applying central difference formulas that stem from the Taylor series approximation. These formulas have a convergence rate of O(h^2).
Use the ‘forward’ (right side) or ‘backward’ (left side) method if the function can only be computed or is only defined on one side. Otherwise, always use the central difference formulas.
Optimal step sizes depend on the accuracy the function can be computed with.
Assuming internal functions with an accuracy 2.2e-16, appropriate step
sizes might be 5e-6, 1e-4, 5e-4, 2.5e-3 for n=1,...,4 and
precisions of about 10^-10, 10^-8, 5*10^-7, 5*10^-6 (at best).
For n>4 a recursion (or finite difference) formula will be applied,
cd. the Wikipedia article on “finite difference”.
Vector of the same length as x.
Numerical differentiation suffers from the conflict between round-off and truncation errors.
Kiusalaas, J. (2005). Numerical Methods in Engineering with Matlab. Cambridge University Press.
## Not run:
f <- sin
xs <- seq(-pi, pi, length.out = 100)
ys <- f(xs)
y1 <- fderiv(f, xs, n = 1, method = "backward")
y2 <- fderiv(f, xs, n = 2, method = "backward")
y3 <- fderiv(f, xs, n = 3, method = "backward")
y4 <- fderiv(f, xs, n = 4, method = "backward")
plot(xs, ys, type = "l", col = "gray", lwd = 2,
xlab = "", ylab = "", main = "Sinus and its Derivatives")
lines(xs, y1, col=1, lty=2)
lines(xs, y2, col=2, lty=3)
lines(xs, y3, col=3, lty=4)
lines(xs, y4, col=4, lty=5)
grid()
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.