Alok Ranjan - 8 months ago 58

C Question

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?

Answer

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)
...
```