MatLab - gradient command

Question

I am trying to learn various functions and commands in MatLab. I have a question regarding the gradient command.

Say I define the following:

x = 0:1:10;

f = @(x) x.^2 + 2*x -1;

h = gradient(f(x))

This then gives me the following vector:

h =  3     4     6     8    10    12    14    16    18    20    21

I see that the values are correct when x is between 1 and 9, but this is incorrect for x = 0 and x = 10. When x = 0, the gradient should be 2, and when x = 10, the gradient should be 22. So why does MatLab give erroneous answers for these two input values?

If anyone could explain this to me I would greatly appreciate it!

Solution 1

Actually the result is correct. When

>> x0 = 0
>> f(x0) 
    -1

such that the gradient is indeed 3. Similarly for x=10, as f(10) = 119 and f(9)=98, so the gradient is indeed = 21.

The discrepancy between these results and the analytical result is because the gradient is a numerical approximation to the derivative with associated boundary issues.

Consider further what would have happened if you had given less data points, say only two points - the algorithm would give you the gradient as the difference between the points divided by the interval. This is what is happening at the boundary.

Solution 2

I think you're looking at a boundary problem. Expand x and you'll get the right answer. Remember that you are performing a numerical calculation