# [python] How do you get the magnitude of a vector in Numpy?

In keeping with the "There's only one obvious way to do it", how do you get the magnitude of a vector (1D array) in Numpy?

``````def mag(x):
return math.sqrt(sum(i**2 for i in x))
``````

The above works, but I cannot believe that I must specify such a trivial and core function myself.

This question is related to `python` `numpy`

## The answer is

Yet another alternative is to use the `einsum` function in numpy for either arrays:

``````In [1]: import numpy as np

In [2]: a = np.arange(1200.0).reshape((-1,3))

In [3]: %timeit [np.linalg.norm(x) for x in a]
100 loops, best of 3: 3.86 ms per loop

In [4]: %timeit np.sqrt((a*a).sum(axis=1))
100000 loops, best of 3: 15.6 µs per loop

In [5]: %timeit np.sqrt(np.einsum('ij,ij->i',a,a))
100000 loops, best of 3: 8.71 µs per loop
``````

or vectors:

``````In [5]: a = np.arange(100000)

In [6]: %timeit np.sqrt(a.dot(a))
10000 loops, best of 3: 80.8 µs per loop

In [7]: %timeit np.sqrt(np.einsum('i,i', a, a))
10000 loops, best of 3: 60.6 µs per loop
``````

There does, however, seem to be some overhead associated with calling it that may make it slower with small inputs:

``````In [2]: a = np.arange(100)

In [3]: %timeit np.sqrt(a.dot(a))
100000 loops, best of 3: 3.73 µs per loop

In [4]: %timeit np.sqrt(np.einsum('i,i', a, a))
100000 loops, best of 3: 4.68 µs per loop
``````

Fastest way I found is via inner1d. Here's how it compares to other numpy methods:

``````import numpy as np
from numpy.core.umath_tests import inner1d

V = np.random.random_sample((10**6,3,)) # 1 million vectors
A = np.sqrt(np.einsum('...i,...i', V, V))
B = np.linalg.norm(V,axis=1)
C = np.sqrt((V ** 2).sum(-1))
D = np.sqrt((V*V).sum(axis=1))
E = np.sqrt(inner1d(V,V))

print [np.allclose(E,x) for x in [A,B,C,D]] # [True, True, True, True]

import cProfile
cProfile.run("np.sqrt(np.einsum('...i,...i', V, V))") # 3 function calls in 0.013 seconds
cProfile.run('np.linalg.norm(V,axis=1)')              # 9 function calls in 0.029 seconds
cProfile.run('np.sqrt((V ** 2).sum(-1))')             # 5 function calls in 0.028 seconds
cProfile.run('np.sqrt((V*V).sum(axis=1))')            # 5 function calls in 0.027 seconds
cProfile.run('np.sqrt(inner1d(V,V))')                 # 2 function calls in 0.009 seconds
``````

inner1d is ~3x faster than linalg.norm and a hair faster than einsum

If you are worried at all about speed, you should instead use:

``````mag = np.sqrt(x.dot(x))
``````

Here are some benchmarks:

``````>>> import timeit
>>> timeit.timeit('np.linalg.norm(x)', setup='import numpy as np; x = np.arange(100)', number=1000)
0.0450878
>>> timeit.timeit('np.sqrt(x.dot(x))', setup='import numpy as np; x = np.arange(100)', number=1000)
0.0181372
``````

EDIT: The real speed improvement comes when you have to take the norm of many vectors. Using pure numpy functions doesn't require any for loops. For example:

``````In [1]: import numpy as np

In [2]: a = np.arange(1200.0).reshape((-1,3))

In [3]: %timeit [np.linalg.norm(x) for x in a]
100 loops, best of 3: 4.23 ms per loop

In [4]: %timeit np.sqrt((a*a).sum(axis=1))
100000 loops, best of 3: 18.9 us per loop

In [5]: np.allclose([np.linalg.norm(x) for x in a],np.sqrt((a*a).sum(axis=1)))
Out[5]: True
``````

You can do this concisely using the toolbelt vg. It's a light layer on top of numpy and it supports single values and stacked vectors.

``````import numpy as np
import vg

x = np.array([1, 2, 3, 4, 5])
mag1 = np.linalg.norm(x)
mag2 = vg.magnitude(x)
print mag1 == mag2
# True
``````

I created the library at my last startup, where it was motivated by uses like this: simple ideas which are far too verbose in NumPy.

use the function norm in scipy.linalg (or numpy.linalg)

``````>>> from scipy import linalg as LA
>>> a = 10*NP.random.randn(6)
>>> a
array([  9.62141594,   1.29279592,   4.80091404,  -2.93714318,
17.06608678, -11.34617065])
>>> LA.norm(a)
23.36461979210312

>>> # compare with OP's function:
>>> import math
>>> mag = lambda x : math.sqrt(sum(i**2 for i in x))
>>> mag(a)
23.36461979210312
``````