How to generate a symbolic multivariate polynomial of a given dimension in SymPy?

[foadsf]

I want to use power series to approximate some PDEs. The first step I need to generate symbolic multivariate polynomials, given a numpy ndarray.

Consider the polynomial below:

I want to take a m dimensional ndarray of D=[d1,...,dm] where djs are non-negative integers, and generate a symbolic multivariate polynomial in the form of symbolic expression. The symbolic expression consists of monomials of the form:

Fo example if D=[2,3] the output should be

For this specific case I could nest two for loops and add the expressions. But I don't know what to do for Ds with arbitrary length. If I could generate the D dimensional ndarrays of A and X without using for loops, then I could use np.sum(np.multiply(A,X)) as Frobenius inner product to get what I need.

I would appreciate if you could help me know how to do this in SymPy. Thanks in advance.

[Kalevi Suominen]

I would first generate the list of monomial indices by using e.g. itertool.product, then create a dictionary containing the indexed coefficients, and finally create a Poly object with given variables from those coefficients. For example, to construct a polynomial of degree 3 or less in 3 variables this could be done:

indices = [i for i in itertools.product(range(4), repeat=3) if sum(i) < 4]
a = IndexBase('a')
coeffs = {i: a[i] for i in indices}
vars = symbols('x:3')
Poly(coeffs, *vars)

Kalevi Suominen

[Kalevi Suominen]

Correction: IndexedBase instead of IndexBase

[Foad Sojoodi Farimani]

Dear Kalevi,

First of thanks a lot for the great instructions. It is a good step forward. I applied your code as:

    from itertools import product

    from sympy import IndexedBase, symbols, Poly

    d=3

    l=4

    indices = [i for i in product(range(l), repeat=d) if sum(i) < l]

    a = IndexedBase('a')

    coeffs = {i: a[i] for i in indices}

    vars = symbols('x:3')

    Poly(coeffs, *vars)

However there are a couple of issues:

• `  if sum(i) < l ` causes the total degree of each monomial to be less than 4, that's not what I want. I want to take a list or ndarray of non-negative integers D=[d1,...,dm] and each monomial should have a degree of  i_j for x_j which is less than d_j , so not the total degree of each monomial but the degree of each variable is important. is there any way to tell `itertools.product` to give the Cartesian product of a list of ranges. something like  itertools.product(range(d1),...,range(dm)). 
• in the second line from bottom how can I change the `3` with a variable?

Thanks for your help again.

Best,

Foad

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[Foad Sojoodi Farimani]

I think I solved the second problem. From here I learned that if I change `symbols('x:3')` to `symbols(f'x1:{d+1}')` the second issue is solved.

[Foad Sojoodi Farimani]

From here I could solve the problem:

    from itertools import product

    from sympy import IndexedBase, symbols, Poly

    D=[2,3]

    d=len(D)

    indices=list(product(*map(range, D)))

    a = IndexedBase('a')

    coeffs = {i: a[i] for i in indices}

    vars = symbols(f'x1:{d+1}')

    Poly(coeffs, *vars)

and the result is:

    Poly(a[1, 2]*x1*x2**2 + a[1, 1]*x1*x2 + a[1, 0]*x1 + a[0, 2]*x2**2 + a[0, 1]*x2 + a[0, 0], x1, x2, domain='EX')

[Foad Sojoodi Farimani]

other solutions have been posted here and here