numpy.gradient(f, *varargs, **kwargs)
[source]
Return the gradient of an Ndimensional array.
The gradient is computed using second order accurate central differences in the interior points and either first or second order accurate onesides (forward or backwards) differences at the boundaries. The returned gradient hence has the same shape as the input array.
Parameters: 


Returns: 

Assuming that (i.e., has at least 3 continuous derivatives) and let be a nonhomogeneous stepsize, we minimize the “consistency error” between the true gradient and its estimate from a linear combination of the neighboring gridpoints:
By substituting and with their Taylor series expansion, this translates into solving the following the linear system:
The resulting approximation of is the following:
It is worth noting that if (i.e., data are evenly spaced) we find the standard second order approximation:
With a similar procedure the forward/backward approximations used for boundaries can be derived.
[1]  Quarteroni A., Sacco R., Saleri F. (2007) Numerical Mathematics (Texts in Applied Mathematics). New York: Springer. 
[2]  Durran D. R. (1999) Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. New York: Springer. 
[3]  Fornberg B. (1988) Generation of Finite Difference Formulas on Arbitrarily Spaced Grids, Mathematics of Computation 51, no. 184 : 699706. PDF. 
>>> 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. For instance a uniform spacing:
>>> x = np.arange(f.size) >>> np.gradient(f, x) array([1. , 1.5, 2.5, 3.5, 4.5, 5. ])
Or a non uniform one:
>>> x = np.array([0., 1., 1.5, 3.5, 4., 6.], dtype=float) >>> np.gradient(f, x) array([1. , 3. , 3.5, 6.7, 6.9, 2.5])
For two dimensional arrays, the return will be two arrays ordered by axis. In this example the first array stands for the gradient in rows and the second one in columns direction:
>>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float)) [array([[ 2., 2., 1.], [ 2., 2., 1.]]), array([[1. , 2.5, 4. ], [1. , 1. , 1. ]])]
In this example the spacing is also specified: uniform for axis=0 and non uniform for axis=1
>>> dx = 2. >>> y = [1., 1.5, 3.5] >>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float), dx, y) [array([[ 1. , 1. , 0.5], [ 1. , 1. , 0.5]]), array([[2. , 2. , 2. ], [2. , 1.7, 0.5]])]
It is possible to specify how boundaries are treated using edge_order
>>> x = np.array([0, 1, 2, 3, 4]) >>> f = x**2 >>> np.gradient(f, edge_order=1) array([1., 2., 4., 6., 7.]) >>> np.gradient(f, edge_order=2) array([0., 2., 4., 6., 8.])
The axis
keyword can be used to specify a subset of axes of which the gradient is calculated
>>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float), axis=0) array([[ 2., 2., 1.], [ 2., 2., 1.]])
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