Generator optimization for multidimensional polynomial indicators

Question

HI, I am trying to find a general expression for obtaining the exponents of a multidimensional polynomial of order orderand with n_variables, like the one that is presented in this reference in equation (3).

Here is my current code that uses a generator itertools.product.

def generalized_taylor_expansion_exponents( order, n_variables ):
    """
    Find the exponents of a multivariate polynomial expression of order
    `order` and `n_variable` number of variables. 
    """
    exps = (p for p in itertools.product(range(order+1), repeat=n_variables) if sum(p) <= order)
    # discard the first element, which is all zeros..
    exps.next()
    return exps

Desired Result:

for i in generalized_taylor_expansion_exponents(order=3, n_variables=3): 
    print i

(0, 0, 1)
(0, 0, 2)
(0, 0, 3)
(0, 1, 0)
(0, 1, 1)
(0, 1, 2)
(0, 2, 0)
(0, 2, 1)
(0, 3, 0)
(1, 0, 0)
(1, 0, 1)
(1, 0, 2)
(1, 1, 0)
(1, 1, 1)
(1, 2, 0)
(2, 0, 0)
(2, 0, 1)
(2, 1, 0)
(3, 0, 0)

In fact, this code runs quickly because the generator object is created only. If I want to populate the list with values ​​from this generator, execution really starts to slow down, mainly due to the large number of calls to sum. Typical values ​​for orderand n_variablesare 5 and 10, respectively.

How can I significantly increase the speed of execution?

Thanks for any help.

Davide Lasagna

Answer 1

, , , . .

def generalized_taylor_expansion_exponents( order, n_variables ):
    """
    Find the exponents of a multivariate polynomial expression of order
    `order` and `n_variable` number of variables. 
    """
    pattern = [0] * n_variables
    for current_sum in range(1, order+1):
        pattern[0] = current_sum
        yield tuple(pattern)
        while pattern[-1] < current_sum:
            for i in range(2, n_variables + 1):
                if 0 < pattern[n_variables - i]:
                    pattern[n_variables - i] -= 1
                    if 2 < i:
                        pattern[n_variables - i + 1] = 1 + pattern[-1]
                        pattern[-1] = 0
                    else:
                        pattern[-1] += 1
                    break
            yield tuple(pattern)
        pattern[-1] = 0

Answer 2

, :

def _gtee_helper(order, n_variables):
    if n_variables == 0:
        yield ()
        return
    for i in range(order + 1):
        for result in _gtee_helper(order - i, n_variables - 1):
            yield (i,) + result


def generalized_taylor_expansion_exponents(order, n_variables):
    """
    Find the exponents of a multivariate polynomial expression of order
    `order` and `n_variable` number of variables. 
    """
    result = _gtee_helper(order, n_variables)
    result.next() # discard the first element, which is all zeros
    return result