DISCRETE DERIVATIVES

Formulas for discrete approximations to the derivatives of a function can be obtained from the Taylor expansion,

f(x+k d) = f(x) + k d f'(x) + k2 d2 / 2!   f"(x) + k3 d3 / 3!   f"'(x) + ...

The first derivative is

f'(x) = 1/12 { - f(x+2d) + 8 f(x + d) - 8 f(x - d) + f(x - 2d) } + O(d5)

The second derivative is

f"(x) = 1/12 { - f(x+2d) + 16 f(x + d) - 30 f(x) + 16 f(x - d) - f(x - 2d) } + O(d6)

In a similar way one obtains for a function of two variables

fxy = 1/96 { - f(x+2,y+2) + 16 f(x+1,y+1) + 16 f(x-1,y-1) - f(x-2,y-2) + f(x+2,y-2) - 16 f(x+1,y-1) - 16 f(x+1,y-1) + f(x+2,y-2) } + O(d6)

 

Marco Corvi - Page hosted by geocities.com.