CODEWITHSUNDEEP
cnumerical-methods
Alok Ranjan
Alok Ranjan

Reputation: 31

Is there any better way to find nth derivative of a function in C?

I want to write a subroutine that calculates the nth derivative of a function given by the subroutine of the form:

double func(double x)
{
    // a mathematical expression representing a smooth curve 
    //returns the value of the function at x      
}

i wrote the following subroutine:

double nthderive(double (*f)(double),double x0,int n)
{   
   const double delta=1.0e-8;
   double x1=x0 - delta;
   double x2=x0 + delta;
   if(n==0) {
       return f(x0);
   }
   else {
       return (nthderive(f,x2,n-1)-nthderive(f,x1,n-1))/(2*delta);
   }
}

Can someone suggest a better algorithm for finding the nth derivative?

Upvotes: 0

Views: 1680

Answers (2)

StefanM
StefanM

Reputation: 781

The nth derivative can be computed using Cauchy's integral lemma.

Transform the integral over the path into a "standard integral" (see Line integral).

Then you can integrate that expression (e.g. with trapezoidal rule).

If you want to calculate high order derivatives this method will most likely be stable.

Upvotes: 2

user1196549
user1196549

Reputation:

Not counting the numerical instability issue, that makes adjustment of Delta difficult and precludes the use for high order derivatives (say n > 3 !), the recursive solution is pretty inefficient, as it takes 2^n function evaluations, where n+1 are enough.

Indeed, you compute

f' = f(+1) - f(-1)
f'' = f'(+1) - f'(-1) = f(+2) - f(0) - f(0) + f(-2)
f''' = f''(+1) - f''(-1) = f(+3) - f(+1) - f(+1) + f(-1) - f(+1) + f(-1) + f(-1) - f(-3)
...

when the Binomial formula tells you

f' = f(+1) - f(-1)
f'' = f(+2) - 2f(0) + f(-2)
f''' = f(+3) - 3f(+1) + 3f(-1) - f(-3)
...

Upvotes: 0

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