I'm trying to obtain the following S2 from a given S1,

$$ S2(i, j) = \sum _{k=0} ^i \sum _{l=0} ^j S1(k, i-k, j, l) e^{ -\sqrt{-1} (i-k)^2 (j-l) } $$

and I've written the following code using four loops. However, when I increase the parameter N, the computation becomes too slow.

import numpy as np

S2=np.zeros((N,N),dtype ='complex_')

for i in range(N):
     for j in range(N):
          for k in range(i+1):
               for l in range(j+1):

Are there more fast methods without using the loops, such as numpy's trace? For the k summation, by inverse-ordering 0 axis or 1 axis for S1, trace seemed to be available with i-N as the offset with weight of the exponential, but a tuple cannot be the offset.

  • \$\begingroup\$ How is this "without using loops"? I see four for loops in the code. I think your summary title is incorrect. \$\endgroup\$ Feb 22 at 20:37

1 Answer 1


This can be vectorised, and when that's done it does run faster, though the time complexity does not improve and is still O(n^4). The proposed implementation has a speedup of roughly 35x.

Your current algorithm - which is a reasonable first step prior to optimisation - uses nested loops and direct indices. Whereas it computes fewer elements when compared to a fully-vectorised multiplication with pre- and post-triangularisation, the latter is more efficient at scale.

Recognise that the following three expressions:

  • for k in range(i+1)
  • for l in range(j+1)
  • S1[k,i-k,j,l]

all imply triangular matrices. Triangularising S1 can be done with a partial assignment to an initially-zero matrix using modified triu_indices. Triangularising the k and l assignments can be done after your calculation with a tril call.

A quick profile shows that the heart of the calculation - the Euler-form exp(i*theta) - is the bottleneck. Euler-form polar space calculations in Numpy are slow. Convert this to a rectangular expression and calculate the cos and sin in two halves of a contiguous array.

Further optimisations may be possible where you calculate only the non-zero segment and then reshape it. I have not shown this.


from timeit import timeit

import numpy as np
import pandas as pd
import seaborn as sns
import cProfile

from matplotlib import pyplot as plt

def old_integrate(S1: np.ndarray) -> np.ndarray:
    N = S1.shape[0]
    S2 = np.zeros((N, N), dtype='complex_')

    for i in range(N):
        for j in range(N):
            for k in range(i + 1):
                for l in range(j + 1):
                    S2[i, j] += S1[k, i - k, j, l] * np.exp(-1j * (i - k) ** 2 * (j - l))

    return S2

def slide_s1(S1: np.ndarray, N: int) -> np.ndarray:
    # Achieve the effect of S1[k, i-k, :, :]
    S1_slid = np.zeros_like(S1)
    k, i = np.triu_indices(N)
    S1_slid[i, k, :, :] = S1[k, i-k, :, :]
    return S1_slid

def addend_expr(
    S1_slid: np.ndarray,
    i_minus_k: np.ndarray,
    j_minus_l: np.ndarray,
) -> np.ndarray:
    # return S1_slid * np.exp(-1j * (i_minus_k**2 * j_minus_l)) #  820 ms

    '''theta = i_minus_k**2 * j_minus_l  # 437 ms
    result = S1_slid * (
        np.cos(theta) - 1j * np.sin(theta)
    return result'''

    # 268 ms
    theta = -i_minus_k**2 * j_minus_l
    rect = np.empty((*S1_slid.shape, 2), dtype=np.float64)
    np.cos(theta, out=rect[..., 0])
    np.sin(theta, out=rect[..., 1])
    as_complex = rect.view(dtype=np.complex128)[..., 0]
    as_complex *= S1_slid
    return as_complex

def make_addends(S1_slid: np.ndarray, N: int):
    # Numeric indices over any of the dimensions
    idx = np.arange(N)

    # Broadcast index difference to two-dimensional matrix
    idx_diff = idx[:, np.newaxis] - idx

    # Four-dimensional term: (j - l)
    j_minus_l = idx_diff[np.newaxis, np.newaxis, :, :]

    # Four-dimensional term: (i - k)
    i_minus_k = idx_diff[:, :, np.newaxis, np.newaxis]

    # Full addends before triangularisation
    addends = addend_expr(S1_slid, i_minus_k, j_minus_l)

    # Triangularise
    return np.tril(addends)

def integrate(S1: np.ndarray) -> np.ndarray:
    N = S1.shape[0]
    S1_slid = slide_s1(S1, N)
    addends = make_addends(S1_slid, N)

    # sum ("integrate").
    S2 = np.sum(addends, axis=(1, 3))
    return S2

def synthetic(N: int) -> np.ndarray:
    return np.arange(N**4).reshape((N, N, N, N))

def test() -> None:
    # Regression test.

    # Original input data
    N = 3
    S1 = synthetic(N)

    s2_expected = np.array((
        (  0,   7.00000000 +  0.00000000j,  21.00000000 +  0.00000000j),
        ( 36,  80.48362767 - 10.09765182j, 121.40263435 - 27.10299716j),
        (108, 184.34527389 - 16.92451601j, 238.92174068 - 79.19827981j),

    for method in (old_integrate, integrate):
        S2 = method(S1)
        assert np.allclose(S2, s2_expected, rtol=1e-10, atol=0)

def profile() -> None:
    N = 60
            'S1': synthetic(N),

def compare() -> None:
    times = []

    for n in np.round(10**np.linspace(0.5, 1.6, 5)):
        S1 = synthetic(int(n))
        for method in (old_integrate, integrate):
            def run():
                return method(S1)
            t = timeit(run, number=1)
            times.append((n, method.__name__, t))

    df = pd.DataFrame(times, columns=('n', 'method', 't'))
    ax = sns.lineplot(data=df, x='n', hue='method', y='t')
    ax.set(xscale='log', yscale='log')

if __name__ == '__main__':


         99 function calls (96 primitive calls) in 0.357 seconds

   Ordered by: internal time

   ncalls  tottime  percall  cumtime  percall filename:lineno(function)
        1    0.264    0.264    0.264    0.264 274360.py:32(addend_expr)
      8/5    0.049    0.006    0.061    0.012 {built-in method numpy.core._multiarray_umath.implement_array_function}
        1    0.012    0.012    0.012    0.012 {method 'reduce' of 'numpy.ufunc' objects}
        1    0.012    0.012    0.357    0.357 <string>:1(<module>)

time complexity

  • \$\begingroup\$ For the expressions S1[k,i-k,j,l] and for k in range(i+1), aren't the two corresponding triangularizings are equivalent? Then, further triangularizing for for k in range(i+1) is not necessary. \$\endgroup\$
    – user16308
    Feb 23 at 16:07
  • \$\begingroup\$ @user16308 You're right! Thanks; edited. \$\endgroup\$
    – Reinderien
    Feb 23 at 16:28
  • \$\begingroup\$ You're welcome! Thanks for the answer \$\endgroup\$
    – user16308
    Feb 23 at 16:31
  • \$\begingroup\$ When the N is increased a lot to ~200, which is the case I am studying, the memory problem may pop up, only for the vectorisation, not for the explicit for loops. I may need to figure out how the vectorisations can avoid the memory problem. \$\endgroup\$
    – user16308
    Feb 24 at 16:00

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