I have some problems with the derivative in Matlab

Question

In MATLAB:

Using the X and Y values below, write a MATLAB function SECOND_DERIV in MATLAB. The output of the function should be the approximate value for the second derivative of the data at x, the input variable of the function. Use the forward difference method and interpolate to get your final answer;

X=[1,1.2,1.44,1.73,2.07,2.49,2.99,3.58,4.3,5.16,6.19,7.43,8.92,10.7,12.84,15.41,18.49];

Y=[18.89,19.25,19.83,20.71,21.96,23.6,25.56,27.52,28.67,27.2,19.38,-2.05,-50.9,-152.82,-354.73,-741.48,-1465.11];

This is my coding:

function output = SECOND_DERIV(R)

X=[1,1.2,1.44,1.73,2.07,2.49,2.99,3.58,4.3,5.16,6.19,7.43,8.92,10.7,12.84,15.41,18.49];  
Y=[18.89,19.25,19.83,20.71,21.96,23.6,25.56,27.52,28.67,27.2,19.38,-2.05,-50.9,-152.82,-354.73,-741.48,-1465.11];  

%forward difference method first time.   
XX=X(1:end-1)

%first derivative.    
dydx=diff(Y)./diff(X)

%second derivative.    
dydx2=diff(dydx)

%forward difference method second time.   
XXX=XX(1:end-1)

%get the second derivative from input x.   
output= interp1(XXX,dydx2,x,'linear','extrap')

end

I do not know what wrong with it.

This is the result I got from my course's web

Is it significant that the X array appears to be a geometric series, with ratio of 1.2? If so, this would make your problem much easier because you'd be able to use the standard second-difference formula for regularly spaced samples, provided you work with log(x) rather than x directly. — rwp

oliver oliver · Accepted Answer · 2018-03-25T08:09:37

First, there is no "the" approximate value but rather only "an" approximate value among an infinite set of approximation schemes. In that sense your excercise is ill-defined (but, to be fair, there is probably something you had in the lessons, that completes information).

Using forward differences twice is almost as bad an approximation as it can get. With each forward difference you are displacing the abscissa of the preferred (central difference) approximation by half a sample distance towards the "past".

For the first difference this can be justified by the fact that you might want to stick with the original X-samples. But in the second step you introduce a second displacement by half a sample distance. In order to keep approximation error at least reasonably low, the least you can do is to correct the displacement afterwards by one sample distance towards the "future". This doesn't bring you exactly back to central differences because of non-equidistance, but it's the minimal correction that should be done for the sake of accuracy.

Hence I would replace

XXX=XX(1:end-1)

by

XXX=XX(2:end)

But again, like so many school excercises, the problem is ill-defined and it is difficult to tell from the distance, if this is what is expected from you.

I have some problems with the derivative in Matlab

1 Answers