# Finding the second derivative using central difference formula

• Kanashii
In summary: I'm not the OPhis (or her) function as object of study is ##\ \pi(16x^2-x^4)\ ##. The second derivative at ##x=2## is ##-16\pi##
Kanashii
Thread moved from the technical forums, so no Homework Help Template is shown.

## Homework Statement

Develop aprogram that will determine the second derivative of pi(16 x^2 - y^4) at y=2 with step sizes of 0.1, 0.01, 0.001…. until the absolute error (numerical-analytical) converges to 0.00001. Use the 2nd order Central Difference Formula.
User Input: y, tolerance
Output: h, second derivative, error

## Homework Equations

[f(x+h) - 2f(x) + f(x-h)]/h^2

## The Attempt at a Solution

Code:
do
{
n[0]= h;
n[i+1]=n/10;
f= function (y,n);
error[0]= error_function(true_value,f);
error[i+1]= error_function(f,f[i-1]);
printf("%lf        %lf       %lf\n",n,f,error);
i++;
}
while (error > tolerance || error != tolerance);

When I input 0.00001 (tolerance) and 2 (y) into the program I created, the program crashes but it got the values of h, f right and also some values for the error. I do know what to change.
Thank you.

Output:

h ----------------------------f----------------------------error
0.100000--------------- -50.328314 --------------- 0.062832
0.010000-------------- -50.266111 --------------- 50.328314
0.001000--------------- -50.265489 --------------- 0.062204
0.000100--------------- -50.265481--------------- 0.000622
0.000010--------------- -50.265498--------------- 0.000007
0.000001--------------- -50.249582--------------- 0.000017
0.000000--------------- -54.001248--------------- 0.015916
0.000000--------------- -284.217094--------------- 3.751666
0.000000--------------- 0.000000--------------- 230.215846

This table would also go on and the values do not make any sense.

Last edited:
Hi Kanashii,

Does your program work in double precision ? Excel does, and it has no problem
Fortran:
       0.1 -50.3283143105095 -0.06283185
0.01 -50.2661107763487 -0.00062832
0.001 -50.2654887593678 -0.00000630
0.0001 -50.2654776874780  0.00000477
0.00001 -50.2654866164213 -0.00000416
0.000001 -50.2699510880988 -0.00446863
1E-07 -53.5736601294126 -3.30817767
But, as you see, it runs out of steam only a little bit later after hitting 4e-5 error. Phew...

Perhaps I'm being a bit slow today, but can you explain how "pi(16 x^2 - y^4)" defines a function?

Pick 16x^2-x^4

BvU said:
Pick 16x^2-x^4
Thanks. Still not ringing a bell. I've not come across this before (at least not in this form). Do you happen to have a reference I can have a look at?

Bit slow all right ...
• I'm not the OP
• his (or her) function as object of study is ##\ \pi(16x^2-x^4)\ ##. The second derivative at ##x=2## is ##-16\pi##

BvU said:
Bit slow all right ...
• I'm not the OP
• his (or her) function as object of study is ##\ \pi(16x^2-x^4)\ ##. The second derivative at ##x=2## is ##-16\pi##
Ah So there was a typo in the original (no y in the expression) and the function was meant to be ##f(x) = \pi(16x^2-x^4)##. Thanks for the clarification. My mind was going places like implicit differentiation...

How is Kanashii doing ?

## 1. What is the central difference formula for finding the second derivative?

The central difference formula is a method for approximating the second derivative of a function at a given point by using the function values at two neighboring points. It is given by (f(x+h) - 2f(x) + f(x-h)) / h^2, where h is the step size.

## 2. How accurate is the central difference formula for finding the second derivative?

The central difference formula is a second-order accurate method, meaning that the error in the approximation is on the order of h^2. This means that as the step size decreases, the approximation becomes more accurate.

## 3. Can the central difference formula be used for any type of function?

Yes, the central difference formula can be used for any type of function, as long as the function is differentiable at the point of interest. It is commonly used in numerical methods for solving differential equations.

## 4. How does the central difference formula compare to other methods for finding the second derivative?

The central difference formula is a simple and straightforward method for approximating the second derivative. However, it may not be the most accurate method, as there are other techniques that can provide more accurate results, such as using higher-order finite difference formulas or using analytic derivatives if available.

## 5. Are there any limitations or drawbacks to using the central difference formula?

One limitation of the central difference formula is that it requires the function values at two neighboring points, which may not always be available or easy to obtain. In addition, the accuracy of the approximation can be affected by the choice of the step size, with smaller step sizes generally resulting in more accurate results.