How to define a matrix by vectorization (without for loop) in numpy?

Question

I want to define an NxN matrix A, whose element A(i,j) is sin(i^2 + j).

In MATLAB, I can quickly define this matrix by employing vectorization.

N = 2000;
ii = 1:N;
A = sin(ii.^2 + ii');

How to achieve this in Python? Right now, I am using for loops, which are slow.

import numpy
N = 2000;
A = numpy.empty((N, N));
for i in range(1,N+1): 
    for j in range(1,N+1): 
        A[i-1][j-1] = numpy.sin(j + numpy.power(i, 2.))

Asked By: Archisman Panigrahi

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Source

Answer 1

This can be done with the use of numpy which allows for vectorization of arrays and array-like operations. I’m sure this can be reduced in size, but we can create an array in x and y before making a grid out of these points. We then apply the entire grid to the function A=sin(i**2+j). We then get (I assume ii' is the same as the transpose of ii?)

import numpy as np # Import library
import matplotlib.pyplot as plt # To visualize
N = 2000 # NxN 
x = np.linspace(0,1,N)*np.pi #Define two arrays of length N
y = np.linspace(0,1,N)*np.pi

X, Y = np.meshgrid(x,y) #Create a NxN grid with arrays

A = np.sin(X**2+Y) # Apply grid to function

plt.imshow(A.T, origin='lower', aspect='auto') #Show that it has worked
plt.show()

which gives (updated to match your result in the range [0,pi])

Answered By: t.o.

Answer 2

Many things you can do in MATLAB have exact translations in numpy. Here’s a handy cheat-sheet

N = 2000 # remains unchanged
ii = np.arange(1, N+1)  # In the cheat-sheet section Linear Algebra Equivalents

Now, there’s a slight difference. In MATLAB, everything is treated as a matrix, so a 1-dimensional array is just a 1xN matrix, so transposing it is done just like any other matrix. In numpy, 1-dimensional arrays can’t be transposed, so let’s add a dummy dimension to ii:

ii = ii[None, :]

Now, ii is a 1xN matrix, so we can get A as:

A = np.sin(ii**2 + ii.T);

Numpy takes care of broadcasting the shapes (1, N) and (N, 1) and gives you a result that is (N, N)

^{Note that a' in MATLAB is the conjugate transpose, which would be a.conj().T in numpy, but since these are all real numbers it makes no difference}

Answered By: Pranav Hosangadi

How to define a matrix by vectorization (without for loop) in numpy?

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