how to optimize iteration on exponential in numpy

Question

Say I have the following numpy array

n = 50
a = np.array(range(1, 1000)) / 1000.

I would like to execute this line of code

%timeit v = [a ** k for k in range(0, n)]
1000 loops, best of 3: 2.01 ms per loop

However, this line of code will ultimately be executed in a loop, therefore I have performance issues.

Is there a way to optimize the loop? For example, the result of a specific calculation i in the list comprehension is simply the result of the previous calculation result in the loop, multiplied by a again.

I don't mind storing the results in a 2d-array instead of arrays in a list. That would probably be cleaner. By the way, I also tried the following, but it yields similar performance results:

    k = np.array(range(0, n))
    ones = np.ones(n)
    temp = np.outer(a, ones)

And then performed the following calculation

%timeit temp ** k
1000 loops, best of 3: 1.96 ms per loop

or

%timeit np.power(temp, k)
1000 loops, best of 3: 1.92 ms per loop

But both yields similar results to the list comprehension above. By the way, n will always be an integer in my case.

Answer 1

In quick tests cumprod seems to be faster.

In [225]: timeit v = np.array([a ** k for k in range(0, n)])
2.76 ms ± 1.62 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

In [228]: %%timeit 
     ...: A=np.broadcast_to(a[:,None],(len(a),50))
     ...: v1=np.cumprod(A,axis=1)
     ...: 
208 µs ± 42.3 ns per loop (mean ± std. dev. of 7 runs, 1000 loops each)

To compare values I have to tweak ranges, since v includes a 0 power, while v1 starts with a 1 power:

In [224]: np.allclose(np.array(v)[1:], v1.T[:-1])
Out[224]: True

But the timings suggest that cumprod is worth refining.

The proposed duplicate was Efficient way to compute the Vandermonde matrix. That still has good ideas.