Also in the documentation1:
>>> y = np.array([1, 2, 4, 7, 11, 16], dtype=np.float)
>>> j = np.gradient(y)
>>> j
array([ 1. , 1.5, 2.5, 3.5, 4.5, 5. ])
Gradient is defined as (change in
y)/(change inx).x, here, is the list index, so the difference between adjacent values is 1.At the boundaries, the first difference is calculated. This means that at each end of the array, the gradient given is simply, the difference between the end two values (divided by 1)
Away from the boundaries the gradient for a particular index is given by taking the difference between the the values either side and dividing by 2.
So, the gradient of y, above, is calculated thus:
j[0] = (y[1]-y[0])/1 = (2-1)/1 = 1
j[1] = (y[2]-y[0])/2 = (4-1)/2 = 1.5
j[2] = (y[3]-y[1])/2 = (7-2)/2 = 2.5
j[3] = (y[4]-y[2])/2 = (11-4)/2 = 3.5
j[4] = (y[5]-y[3])/2 = (16-7)/2 = 4.5
j[5] = (y[5]-y[4])/1 = (16-11)/1 = 5
You could find the minima of all the absolute values in the resulting array to find the turning points of a curve, for example.
1The array is actually called x in the example in the docs, I've changed it to y to avoid confusion.
Top answer 1 of 4
196
Also in the documentation1:
>>> y = np.array([1, 2, 4, 7, 11, 16], dtype=np.float)
>>> j = np.gradient(y)
>>> j
array([ 1. , 1.5, 2.5, 3.5, 4.5, 5. ])
Gradient is defined as (change in
y)/(change inx).x, here, is the list index, so the difference between adjacent values is 1.At the boundaries, the first difference is calculated. This means that at each end of the array, the gradient given is simply, the difference between the end two values (divided by 1)
Away from the boundaries the gradient for a particular index is given by taking the difference between the the values either side and dividing by 2.
So, the gradient of y, above, is calculated thus:
j[0] = (y[1]-y[0])/1 = (2-1)/1 = 1
j[1] = (y[2]-y[0])/2 = (4-1)/2 = 1.5
j[2] = (y[3]-y[1])/2 = (7-2)/2 = 2.5
j[3] = (y[4]-y[2])/2 = (11-4)/2 = 3.5
j[4] = (y[5]-y[3])/2 = (16-7)/2 = 4.5
j[5] = (y[5]-y[4])/1 = (16-11)/1 = 5
You could find the minima of all the absolute values in the resulting array to find the turning points of a curve, for example.
1The array is actually called x in the example in the docs, I've changed it to y to avoid confusion.
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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.
NumPy
numpy.org › doc › stable › reference › generated › numpy.gradient.html
numpy.gradient — NumPy v2.4 Manual
>>> dx = 2 >>> dy = 3 >>> x = np.arange(0, 6, dx) >>> y = np.arange(0, 9, dy) >>> xs, ys = np.meshgrid(x, y) >>> zs = xs + 2 * ys >>> np.gradient(zs, dy, dx) # Passing two scalars (array([[2., 2., 2.], [2., 2., 2.], [2., 2., 2.]]), array([[1., 1., 1.], [1., 1., 1.], [1., 1., 1.]]))
Videos
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np.gradient() — A Simple Illustrated Guide - YouTube
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Coding Gradient Descent with Numpy | Step-by-Step Python Guide ...
32:33
np.diff() - A Simple Guide - YouTube
14:27
Numpy gradient - YouTube
Reddit
reddit.com › r/learnpython › need help in understanding np.gradient for calculating derivatives
r/learnpython on Reddit: Need help in understanding np.gradient for calculating derivatives
June 30, 2023 -
Hi, I'm trying to expand my knowledge in Machine Learning, I came across the np.gradient function, I wanted to understand how it relates to Taylor's Series for estimating values. The documentation seemed a bit confusing for novice.
Top answer 1 of 2
7
One definition of the derivative is f'(x) = (f(x+h)-f(x))/h where h goes to 0. Computers cannot store infinitely small numbers, so they might set h=1e-6 (that is 0.000001). It's a tradeoff because while we want h to be as small as possible, at some point the errors due to computer precision begin to dominate. Given any function that the computer can calculate, it can approximate the derivative. def f(x): return np.sin(x) x = np.arange(-2,2,0.01) y = f(x) dfdx = (f(x+h)-f(x))/h plt.plot(x,y) plt.plot(x,dfdx) plt.show() Assuming that the function is reasonably smooth (i.e. the derivative above exists), another definition of the derivative is f'(x) = (f(x+h)-f(x-h))/(2h) where h goes to 0. Going from x-h to x+h means 2 steps, that's the reason for 2h. Which works just as well. These methods are named finite difference to contrast from the normal derivative definition where h is infinitely small. The first one is the forward difference and the second one is called central difference. The backward difference is (f(x)-f(x-h))/2. Let's assume we want to write a derivative function. It takes a function f and values of x, and gives back f'(x). def f(x): return np.sin(x) def d(fun, x): return (fun(x+h)-fun(x))/h x = np.arange(-2,2,0.01) y = f(x) dfdx = d(f,x) plt.plot(x,y) plt.plot(x,dfdx) plt.show() By passing the function into the function, the derivative function can just call fun wherever it wants/needs to get the derivative. Now things become a bit more inconvenient. For some reason we do not know f. We only know y, i.e. f(x) for some values of x. Let's say that x is evenly spaced as usual. Then our best guess for h is not really tiny but identical to the spacing between neighboring x values. With the forward difference we need to take care at the rightmost value because we cannot just add +h to get a value even further out. Instead we use the backward difference. For values in the middle we decide to use the central difference instead of the forward difference. def f(x): return np.sin(x) def d(y, h=1): dfdx = [(y[1]-y[0])/h] for i in range(1,len(y)-1): dfdx.append((y[i+1]-y[i-1])/2/h) dfdx.append((y[i]-y[i-1])/h) return dfdx h = 0.01 x = np.arange(-2,2,h) y = f(x) dfdx = d(y,h) plt.plot(x,y) plt.plot(x,dfdx) plt.show() The implementation above corresponds to np.gradient in the one-dimensional case where varargs is set to case 1 or 2. The case where varargs is set to 3 or 4 would use x directly in d instead of h. However at that point the formula is more complicated as they mention in the documentation. Effectively any point has a hd (the forward step size) and a hs (the backward step size) and the formula is not just (f(x+hd)-f(x-hs))/(hd+hs) but instead that bigger expression given in the documentation, where the values of hd,hs act as some kind of weights. np.gradient is basically backwards, central and forward difference combined. When you have values like f(1),f(2),f(2+h) and want the derivative at 2, the code notices that 2 and 2+h are very close together and puts greater weight on that (and mostly ignores f(1)). The important part so far is that np.gradient when given a vector with N elements calculates N one-dimensional derivatives, which is not the typical idea of a gradient. np.gradient does support more dimensions which might make things clearer. So in the 1D case, we essentially go through all values from left to right and then consider that value and its direct left and right neighbor to quantify the uptrend or downtrend. In the 2D case, np.gradient still does this, but additionally also walks from top to bottom and does the same. So in 2D it returns 2 arrays, one for left-right and one for top-bottom. The actual definition of the gradient by finite differences is [(f(x+h,y)-f(x,y))/h, (f(x,y+h)-f(x,y))/h] in 2D. These values are indeed returned by np.gradient, the left part is in the first array and the right part in the second array. Say we are in 2D and want the gradient at x=3 and y=0, then we can plug it into np.gradient like this: hx = 1e-6 hy = 1e-3 x = [3,3+hx] y = [0,0+hy] xx,yy = np.meshgrid(x,y) def f(x,y): return x**2-2*x*np.sin(y) + 1/x grad = np.gradient(f(xx,yy), y,x) # Note the order. print(grad[1][0,0], grad[0][0,0]) # Note the order. This is dfdx, dfdy. but if the function f can be calculated by a computer, it makes more sense to just use automatic differentiation instead of finite differences. Automatic differentiation has no h that needs to be chosen carefully. It's always as accurate is possible. import torch x = torch.tensor([3.],requires_grad=True) y = torch.tensor([0.],requires_grad=True) z = x**2-2*x*torch.sin(y) + 1/x z.backward() print(x.grad, y.grad) So what's the deal with the Taylor series? It's just a minor piece in the derivation of that more general expression used by np.gradient. We just start by claiming that we can express the gradient by adding together function values in the direct neighborhood. f'(x) = a f(x) + b f(x+hd) + c f(x-hs) Given that finite differences do work out, this approach should work as well and generalize the idea. Expand f(x+hd) and f(x-hs) with their series: f(x+hd) = f(x) + hd f'(x) + hd^2 f''(x)/2 + ... f(x-hs) = f(x) - hs f'(x) + hs^2 f''(x)/2 + ... Then plug it in and reshape: f'(x) = a f(x) + b f(x) + b hd f'(x) + b hd^2 f''(x)/2 + c f(x) - c hs f'(x) + c hs^2 f''(x)/2 = (a+b+c) f(x) + (b hd - c hs) f'(x) + (b hd^2 + c hs^2 )/2 f''(x) 0 = (a+b+c) f(x) + (b hd - c hs - 1) f'(x) + (b hd^2 + c hs^2 )/2 f''(x) The = in the middle is actually more of an approximately equal sign. We won't be able to reach 0 for all f(x) as claimed on the left hand size, but we can get pretty close. We do NOT want to minimize the right-hand-side. We want it to reach 0 (it can go below 0 right now). To turn this into a minimization problem, we square it. This way we get a positive number always and it really becomes a matter of minimization. We COULD also take the absolute value instead of squaring, but it's pain to work this through and the end result are exactly the same parameters anyway. To minimize: E2 with E = (a+b+c) f(x) + (b hd - c hs - 1) f'(x) + (b hd2 + c hs2 )/2 f''(x) One requirement for an optimum is that the gradient is 0. In this case we take the derivatives with respect to a,b,c because we want to find the optimal a,b,c. First a reminder of the chain rule: dE2 /dt = 2E dE/dt for whatever t is. It's optional to do this but a bit less messy than working it through individually. In particular we have dE^2/da = 2E dE/da = 2E f(x) dE^2/db = 2E dE/db = 2E (f(x) + hd f'(x) + hd^2 f''(x)/2) dE^2/dc = 2E dE/dc = 2E (f(x) - hs f'(x) + hs^2 f''(x)/2) We want ALL three of them to be 0 at the same time. This can only happen if E is 0. 0 := (a+b+c) f(x) + (b hd - c hs - 1) f'(x) + (b hd2 + c hs2 )/2 f''(x) and we want this to be 0 for any f, f', f'' for any value of x. The only way for this to happen is if each coefficient is 0, i.e. a+b+c = 0 b hd - c hs = 1 b hd^2 + c hs^2 = 0 We would need to check the second derivative to make sure that this is a minimum, not a maximum, but given the problem it is fairly clear. So why did we stop exactly after f'' in the Taylor series? It's because this way we get exactly 3 unknowns and 3 equations, which is the most convenient to solve. Multiply the second equation by hd then subtract the third from it. (b hd^2 - c hs hd) - (b hd^2 + c hs^2) = hd -c hs^2 - c hs hd = hd c hs (hs + hd) = -hd c = -hd/hs/(hs+hd) = -hd^2 / (hs hd (hs+hd)) where the last step is just so it looks exactly like in np.gradient. Insert c into the second equation. b hd + hd/hs/(hs+hd) hs = 1 b hd + hd/(hs+hd) = 1 b + 1/(hs+hd) = 1/hd b = 1/hd - 1/(hs+hd) b = (hs(hs+hd) - hs hd) / [hs hd (hs+hd)] b = hs^2 / [hs hd (hs+hd)] From the first equation we know that a = -b-c = (hd2 - hs2 )/(hs hd (hs+hd)). So here's your summary: If you have a function that can be calculated by a computer, use torch or tensorflow or any other framework for automatic differentiation. If you have a function that can be calculated by a computer but such a framework is not available, np.gradient is still a bad idea because it is inefficient. Note for the 2D gradient we needed three values, f(x,y), f(x+dx,y), f(x,y+dy). But with np.gradient we would first need to set up arrays where it is almost natural to also include f(x+dx,y+dy) which is not needed for gradient calculations. It's more natural to set up some loop that increments x once, then y once, then z once, and so on. Many solvers in scipy.optimize work with finite differences. If you have a function that cannot be calculated by a computer, np.gradient may be useful. In practice this means that you have data from some experiment. Even there, the concept of a Taylor series plays no role here UNLESS the data was taken on an unevenly spaced grid.
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You might enjoy this stackoverflow post on the same question
Scaler
scaler.com › home › topics › what is the numpy.gradient() method in numpy?
What is the numpy.gradient() method in Numpy? - Scaler Topics
May 4, 2023 - The gradient is calculated using ... function by utilizing either the first or second-order correct one-sides (in either direction) differences at the boundaries and second-order accurate central differences in the interior locations...
Educative
educative.io › answers › how-to-use-the-numpygradient-function-for-a-2d-array-in-python
How to use the numpy.gradient function for a 2D array in Python
We can use the numpy.gradient() function to find the gradient of an N-dimensional array.
SciPy
docs.scipy.org › doc › numpy-1.10.1 › reference › generated › numpy.gradient.html
numpy.gradient — NumPy v1.10 Manual
The gradient is computed using second order accurate central differences in the interior and either first differences or second order accurate one-sides (forward or backwards) differences at the boundaries. The returned gradient hence has the same shape as the input array.
NumPy
numpy.org › doc › 2.1 › reference › generated › numpy.gradient.html
numpy.gradient — NumPy v2.1 Manual
>>> dx = 2 >>> dy = 3 >>> x = np.arange(0, 6, dx) >>> y = np.arange(0, 9, dy) >>> xs, ys = np.meshgrid(x, y) >>> zs = xs + 2 * ys >>> np.gradient(zs, dy, dx) # Passing two scalars (array([[2., 2., 2.], [2., 2., 2.], [2., 2., 2.]]), array([[1., 1., 1.], [1., 1., 1.], [1., 1., 1.]]))
SciPy
docs.scipy.org › doc › numpy-1.9.3 › reference › generated › numpy.gradient.html
numpy.gradient — NumPy v1.9 Manual
The gradient is computed using second order accurate central differences in the interior and either first differences or second order accurate one-sides (forward or backwards) differences at the boundaries. The returned gradient hence has the same shape as the input array.
NumPy
numpy.org › doc › 2.0 › reference › generated › numpy.gradient.html
numpy.gradient — NumPy v2.0 Manual
>>> f = np.array([1, 2, 4, 7, 11, 16], dtype=float) >>> np.gradient(f) array([1. , 1.5, 2.5, 3.5, 4.5, 5. ]) >>> np.gradient(f, 2) array([0.5 , 0.75, 1.25, 1.75, 2.25, 2.5 ]) Spacing can be also specified with an array that represents the coordinates of the values F along the dimensions.
SciPy
docs.scipy.org › doc › numpy-1.8.1 › reference › generated › numpy.gradient.html
numpy.gradient — NumPy v1.8 Manual
The gradient is computed using central differences in the interior and first differences at the boundaries. The returned gradient hence has the same shape as the input array. Examples · >>> x = np.array([1, 2, 4, 7, 11, 16], dtype=np.float) >>> np.gradient(x) array([ 1.
Medium
medium.com › @whyamit404 › understanding-derivatives-with-numpy-e54d65fcbc52
Understanding Derivatives with NumPy | by whyamit404 | Medium
February 8, 2025 - In NumPy, we don’t have a dedicated function for derivatives. Instead, we use np.gradient(). This function calculates the derivative using numerical differentiation.
AskPython
askpython.com › home › numpy gradient: returning the gradient of n-dimensional array
Numpy Gradient: Returning the Gradient of N-dimensional Array - AskPython
December 29, 2022 - Let us try calculating the gradient of a two-dimensional array with a uniform spacing of ‘2’, as shown below. ar =([[1.2, 3.4, 5.6], [7.8, 9.0, 0.1]]) np.gradient(ar,2)
Finxter
blog.finxter.com › home › learn python blog › np.gradient() — a simple illustrated guide
np.gradient() - A Simple Illustrated Guide - Be on the Right Side of Change
June 24, 2022 - In Python, the numpy.gradient() function approximates the gradient of an N-dimensional array. It uses the second-order accurate central differences in the interior points and either first or second-order accurate one-sided differences at the ...
Kodeclik
kodeclik.com › numpy-gradient
Python numpy.gradient()
October 16, 2024 - numpy.gradient() computes the gradient of a function represented in an n-dimensional array using finite differences.
IncludeHelp
includehelp.com › python › what-does-numpy-gradient-do.aspx
Python - What does numpy.gradient() do?
The numpy.gradient() method is used to find the gradient of an N-dimensional array. The gradient is computed using second-order accurate central differences in the interior points and either first or second-order accurate one-sides (forward or backward) differences at the boundaries.
Naukri
naukri.com › code360 › library › numpy-gradient-method-in-numpy
numpy.gradient() Method in Numpy - Naukri Code 360
February 5, 2025 - Almost there... just a few more seconds
GeeksforGeeks
geeksforgeeks.org › machine learning › numpy-gradient-descent-optimizer-of-neural-networks
Numpy Gradient - Descent Optimizer of Neural Networks - GeeksforGeeks
March 29, 2023 - import numpy as np def GD(f, start, lr, n_iter=50, tol=1e-05): res = start for _ in range(n_iter): # gradient is calculated using the np.gradient function. new_val = -lr * np.gradient(f) if np.all(np.abs(new_val) <= tol): break res += new_val # we return a vector as the gradient can be multivariable function. # if the function has 1 dependent variable then it returns a scalar value. return res f = np.array([2, 4], dtype=float) # low learning rate doesn't allow to converge at global minima print(f'The vector notation of global minima: {GD(f,10,0.001)}') Output: [9.9 9.9] The value returned by the algorithm is not even close to 0.
Beautiful Soup
tedboy.github.io › numpy › generated › numpy.gradient.html
1.6.121. numpy.gradient — Numpy API
The gradient is computed using second order accurate central differences in the interior and either first differences or second order accurate one-sides (forward or backwards) differences at the boundaries. The returned gradient hence has the same shape as the input array.
NumPy
numpy.org › doc › 2.3 › reference › generated › numpy.gradient.html
numpy.gradient — NumPy v2.3 Manual
>>> dx = 2 >>> dy = 3 >>> x = np.arange(0, 6, dx) >>> y = np.arange(0, 9, dy) >>> xs, ys = np.meshgrid(x, y) >>> zs = xs + 2 * ys >>> np.gradient(zs, dy, dx) # Passing two scalars (array([[2., 2., 2.], [2., 2., 2.], [2., 2., 2.]]), array([[1., 1., 1.], [1., 1., 1.], [1., 1., 1.]]))