Ask Your Question

Univariate Polynomial Multiplication

asked 2011-07-21 11:30:28 -0600

anonymous user


What algorithms does sage use for univariate polynomial multiplication?

In particular, I am curious about the algorithm used for multiplication over finite fields and polynomials with coefficients in Z or Q.

More generally, is there an easy way to look under the hood and trace down the exact functions being called in the event I have a question like this again?

edit retag flag offensive close merge delete

3 answers

Sort by ยป oldest newest most voted

answered 2011-07-21 15:24:18 -0600

niles gravatar image

This is a pretty broad question, and I'm certainly not an expert in this area, but maybe I can get you pointed in the right direction. It's easy to look under the hood, but maybe (depending on your experience) not so easy to pin down exactly what's being called. Here's how I proceeded:

sage: R.<t> = PolynomialRing(ZZ) # make the kind of ring I want to check
sage: f = R.random_element(); f  # and an element
3*t^2 + t + 65

See what kind of objects they are:

sage: type(R)
<class 'sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain'>
sage: type(f)
<type 'sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint'>

You can browse these source files in your local sage install at $SAGE_ROOT/devel/sage-main/sage/rings/polynomial, or you can browse them online (linked from the "Source Code" section of the main Sage Development page).

Of course it helps if you know where to look; multiplication is implemented (often, including this case) by the _mul_ method, so you can check it directly with

sage: f._mul_??

This will display the relevant part of the source code. There, you will see that this particular multiplication is implemented by fmpz_poly_mul -- this is a FLINT method, and you'd have to look at the FLINT source if you want to go further in this hunt . . .

A similar procedure will, I think, get you started with finding the implementations of the other multiplication algorithms you mentioned. Good luck!

edit flag offensive delete link more


Oh, I shouldn't have taken so long to write this answer!

niles gravatar imageniles ( 2011-07-21 15:27:48 -0600 )edit

answered 2011-07-21 14:48:18 -0600

Mike Hansen gravatar image

For integer polynomials, FLINT is used to do the multiplication. We look at the _mul_ method since in Sage, __mul__ is used to handle coercion and then dispatches to _mul_.

sage: R.<x> = ZZ[]
sage: x._mul_??
cpdef RingElement _mul_(self, RingElement right):
    Returns self multiplied by right.


        sage: R.<x> = PolynomialRing(ZZ)
        sage: (x - 2)*(x^2 - 8*x + 16)
        x^3 - 10*x^2 + 32*x - 32
    cdef Polynomial_integer_dense_flint x = self._new()
    fmpz_poly_mul(x.__poly, self.__poly,
    return x

You can do a similar check to see that FLINT's fmpq_poly_mul is used for polynomials over the rationals.

The univariate polynomials over finite fields different based on the size of the field. For example, for small primes, FLINT is used:

sage: R.<x> = GF(next_prime(2^5))[]
sage: type(x)
<type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>

But for larger primes, NTL is used:

sage: R.<x> = GF(next_prime(2^64))[]
sage: type(x)
<type 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_p'>

To find out the exact algorithm / heuristics used, you'll need to delve into those packages.

edit flag offensive delete link more

answered 2011-07-21 16:33:10 -0600

mathmajor gravatar image

Thank you for the quick help. This is exactly what I was looking for.

edit flag offensive delete link more

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools


Asked: 2011-07-21 11:30:28 -0600

Seen: 610 times

Last updated: Jul 21 '11