Numerical Differentiation

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Description

Numerical function differentiation for orders n=1..4 using finite difference approximations.

Usage

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fderiv(f, x, n = 1, h = 0,
        method = c("central", "forward", "backward"), ...)

Arguments

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 n=0 function values will be returned.

h

step size: if h=0 step size will be set automatically.

method

one of “central”, “forward”, or “backward”.

...

more variables to be passed to function f.

Details

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”.

Value

Vector of the same length as x.

Note

Numerical differentiation suffers from the conflict between round-off and truncation errors.

References

Kiusalaas, J. (2005). Numerical Methods in Engineering with Matlab. Cambridge University Press.

See Also

numderiv, taylor

Examples

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## 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)

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