How Gradient is calcuted

In matlab i have found following answers. let X =

     1     2     3
     3     4     5
     3     2     1
 z=gradient(X)
z =

     1     1     1
     1     1     1
    -1    -1    -1
[z,c]=gradient(X)

z =

     1     1     1
     1     1     1
    -1    -1    -1

c =

     2     2     2
     1     0    -1
     0    -2    -4
But how z and c is calculated,what z and c indicates.

Answer 1

[Sven]

Hi Tinkul,

It is probably clearer if you call your variables z and c different names such as dx and dy. This is because the first and second outputs from gradient is the gradient in each of those directions.

So if you have:

M =
   1     2     3
   3     4     5
   3     2     1

 [dx,dy]=gradient(M)
 dx =
     1     1     1
     1     1     1
    -1    -1    -1

 dy =
     2     2     2
     1     0    -1
     0    -2    -4

You'll notice that dx(1,:) is almost the same as diff(M(1,:)), and dy(:,1) is almost the same as diff(M(:,1)). The main difference is that gradient replicates the last result so that the output has the same size as the input.

Update: this is not quite correct. The two-sided gradient is used in gradient... the one-sided gradient is used in diff.

Does that help? You can also check out the help for gradient which says even more.

[Jan]

No, gradient does not replicate the last element and it is not almost the same as diff. While the values equal the output of diff at the edges, the two-sided difference quotient is used inside.

[Sven]

Thanks Jan, you're right... I should have just pointed to help gradient :)

[Shashank Prasanna]

Quick note, doc gradient is generally more updated than help gradient or even better is the web documentation search. just an fyi.

[Jan]

The web documentation concerns the newest release only. For the locally installed version, the local help is more accurate, when there have been changes.

Answer 2

[Jan]

Beside help gradient you can read the source code also: edit gradient. There you find, that at the edges the single sided difference quotient is calculated - demonstrated on a vector at first:

dx(1) = (x(2) - x(1)) / h;
dx(3) = (x(3) - x(2)) / h;

while h==1 as default. In the inner points the 2nd order two sided difference quotient is created:

dx(2) = (x(3) - x(1)) / (2*h);

For a matrix input the two out puts are calculated a long the rows and the columns respectively.

[Tinkul]

Thanks................