pythonarraysnumpy

numpy - evaluate function on a grid of points


What is a good way to produce a numpy array containing the values of a function evaluated on an n-dimensional grid of points?

For example, suppose I want to evaluate the function defined by

def func(x, y):
    return <some function of x and y>

Suppose I want to evaluate it on a two dimensional array of points with the x values going from 0 to 4 in ten steps, and the y values going from -1 to 1 in twenty steps. What's a good way to do this in numpy?

P.S. This has been asked in various forms on StackOverflow many times, but I couldn't find a concisely stated question and answer. I posted this to provide a concise simple solution (below).


Solution

  • shorter, faster and clearer answer, avoiding meshgrid:

    import numpy as np
    
    def func(x, y):
        return np.sin(y * x)
    
    xaxis = np.linspace(0, 4, 10)
    yaxis = np.linspace(-1, 1, 20)
    result = func(xaxis[:,None], yaxis[None,:])
    

    This will be faster in memory if you get something like x^2+y as function, since than x^2 is done on a 1D array (instead of a 2D one), and the increase in dimension only happens when you do the "+". For meshgrid, x^2 will be done on a 2D array, in which essentially every row is the same, causing massive time increases.

    Edit: the "x[:,None]", makes x to a 2D array, but with an empty second dimension. This "None" is the same as using "x[:,numpy.newaxis]". The same thing is done with Y, but with making an empty first dimension.

    Edit: in 3 dimensions:

    def func2(x, y, z):
        return np.sin(y * x)+z
    
    xaxis = np.linspace(0, 4, 10)
    yaxis = np.linspace(-1, 1, 20)
    zaxis = np.linspace(0, 1, 20)
    result2 = func2(xaxis[:,None,None], yaxis[None,:,None],zaxis[None,None,:])
    

    This way you can easily extend to n dimensions if you wish, using as many None or : as you have dimensions. Each : makes a dimension, and each None makes an "empty" dimension. The next example shows a bit more how these empty dimensions work. As you can see, the shape changes if you use None, showing that it is a 3D object in the next example, but the empty dimensions only get filled up whenever you multiply with an object that actually has something in those dimensions (sounds complicated, but the next example shows what i mean)

    In [1]: import numpy
    
    In [2]: a = numpy.linspace(-1,1,20)
    
    In [3]: a.shape
    Out[3]: (20,)
    
    In [4]: a[None,:,None].shape 
    Out[4]: (1, 20, 1)
    
    In [5]: b = a[None,:,None] # this is a 3D array, but with the first and third dimension being "empty"
    In [6]: c = a[:,None,None] # same, but last two dimensions are "empty" here
    
    In [7]: d=b*c 
    
    In [8]: d.shape # only the last dimension is "empty" here
    Out[8]: (20, 20, 1)
    

    edit: without needing to type the None yourself

    def ndm(*args):
        return [x[(None,)*i+(slice(None),)+(None,)*(len(args)-i-1)] for i, x in enumerate(args)]
    
    
    x2,y2,z2  = ndm(xaxis,yaxis,zaxis)
    result3 = func2(x2,y2,z2)
    

    This way, you make the None-slicing to create the extra empty dimensions, by making the first argument you give to ndm as the first full dimension, the second as second full dimension etc- it does the same as the 'hardcoded' None-typed syntax used before.

    Short explanation: doing x2, y2, z2 = ndm(xaxis, yaxis, zaxis) is the same as doing

    x2 = xaxis[:,None,None]
    y2 = yaxis[None,:,None]
    z2 = zaxis[None,None,:]
    

    but the ndm method should also work for more dimensions, without needing to hardcode the None-slices in multiple lines like just shown. This will also work in numpy versions before 1.8, while numpy.meshgrid only works for higher than 2 dimensions if you have numpy 1.8 or higher.