What does numpy.gradient do?

Question

So I know what the gradient of a (mathematical) function is, so I feel like I should know what numpy.gradient does. But I don\'t. The documentation is not reall

Answer 1

Here is what is going on. The Taylor series expansion guides us on how to approximate the derivative, given the value at close points. The simplest comes from the first order Taylor series expansion for a C^2 function (two continuous derivatives)...

• f(x+h) = f(x) + f'(x)h+f''(xi)h^2/2.

One can solve for f'(x)...

• f'(x) = [f(x+h) - f(x)]/h + O(h).

Can we do better? Yes indeed. If we assume C^3, then the Taylor expansion is

• f(x+h) = f(x) + f'(x)h + f''(x)h^2/2 + f'''(xi) h^3/6, and
• f(x-h) = f(x) - f'(x)h + f''(x)h^2/2 - f'''(xi) h^3/6.

Subtracting these (both the h^0 and h^2 terms drop out!) and solve for f'(x):

• f'(x) = [f(x+h) - f(x-h)]/(2h) + O(h^2).

So, if we have a discretized function defined on equal distant partitions: x = x_0,x_0+h(=x_1),....,x_n=x_0+h*n, then numpy gradient will yield a "derivative" array using the first order estimate on the ends and the better estimates in the middle.

Example 1. If you don't specify any spacing, the interval is assumed to be 1. so if you call

f = np.array([5, 7, 4, 8])

what you are saying is that f(0) = 5, f(1) = 7, f(2) = 4, and f(3) = 8. Then

np.gradient(f) 

will be: f'(0) = (7 - 5)/1 = 2, f'(1) = (4 - 5)/(2*1) = -0.5, f'(2) = (8 - 7)/(2*1) = 0.5, f'(3) = (8 - 4)/1 = 4.

Example 2. If you specify a single spacing, the spacing is uniform but not 1.

For example, if you call

np.gradient(f, 0.5)

this is saying that h = 0.5, not 1, i.e., the function is really f(0) = 5, f(0.5) = 7, f(1.0) = 4, f(1.5) = 8. The net effect is to replace h = 1 with h = 0.5 and all the results will be doubled.

Example 3. Suppose the discretized function f(x) is not defined on uniformly spaced intervals, for instance f(0) = 5, f(1) = 7, f(3) = 4, f(3.5) = 8, then there is a messier discretized differentiation function that the numpy gradient function uses and you will get the discretized derivatives by calling

np.gradient(f, np.array([0,1,3,3.5]))

Lastly, if your input is a 2d array, then you are thinking of a function f of x, y defined on a grid. The numpy gradient will output the arrays of "discretized" partial derivatives in x and y.