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I want to save the variables m, n, r, X, Y (see code below) in one file e.g file_1 and
repeat for every iteration with a new file name prefarably with
the iteration number e.g., file_2.

In MATLAB I could simply do save(['data_',int2str(i),'.mat'],'variable1', 'variable2' ) so that data(1) will contain m, n, r, X, Y; data(2) will contain m, n, r, X, Y with their new values, and so on

How do I do same in Python?

Test = 5
for i in range(Tests):
 r = 5
 m = 500
 n = 500
 X=np.random.rand(m,n)
 Y=np.random.rand(m,n)

The pickle library is the way to go here.

import pickle
Tests = 5
data = 
for i in range(Tests):
 data['r'] = 5
 data['m'] = 500
 data['n'] = 500
 data['X'] = np.random.rand(data['m'],data['n'])
 data['Y'] = np.random.rand(data['m'],data['n'])
 with open('data.pickle'.format(i), 'wb') as f:
 pickle.dump(data, f)

This allows you to access your dictionary of data later. For example,

with open('data0.pickle', 'rb') as f:
 data = pickle.load(f)
r = data['r']
m = data['m']

and so on.

Here is how you can do the same in python.

import numpy as np
Test = 2
for i in range(Test):
 r = 5
 m = 500
 n = 500
 X=np.random.rand(m,n)
 Y=np.random.rand(m,n)
 file_name = 'test_'+str(i)
 open(file_name,'w').write(str(r) + str(m) + str(n) + str(X) + str(Y))

If you would like to do it without using any libraries here's my solution:

import numpy as np
Test = 5
for i in range(Test):
 r = 5
 m = 500
 n = 500
 X=np.random.rand(m,n)
 Y=np.random.rand(m,n)

 file = open('file_'.format(i+1), 'w')
 file.write('r = ;m = ;n = ;X = ;Y = '.format(r, m, n, X, Y))
 file.close()

Loading saved variables

exec(open('file_1', 'r').read())