numerical_differentiation.sf

#!/usr/bin/ruby

# N-th numerical differentiation of a function `f(x)`, at a given value `x`.

# Formula for first derivative:
#   f'(x) = Limit_{h -> 0} (f(x + h) - f(x))/h

# which can be extended to the n-th derivative as:
#   f^(n)(x) = Limit_{h -> 0} (-1)^(n+1) * 1/h^n * Sum_{k=0..n} (-1)^(k+1) * binomial(n, k) * f(x + h*k)

# See also:
#   https://en.wikipedia.org/wiki/Derivative
#   https://en.wikipedia.org/wiki/Numerical_differentiation

func derivative_nth(f, x, n=1, h=1e-49) {
    (-1)**(n+1) * sum(0..n, {|k|
        (-1)**(k+1) * binomial(n, k) * f(x + h*k)
    }) / h**n
}

var x = 10
var f = {|x| x**3 + x**2 }

say derivative_nth(f, x)          #=> 320
say derivative_nth(f, x, 2)       #=> 62
say derivative_nth(f, x, 3)       #=> 6
say derivative_nth(f, x, 4)       #=> 0