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.. _nmod-poly-factor:
**nmod_poly_factor.h** -- factorisation of univariate polynomials over integers mod n (word-size n)
===================================================================================================
Types, macros and constants
-------------------------------------------------------------------------------
.. type:: nmod_poly_factor_struct
.. type:: nmod_poly_factor_t
Factorisation
--------------------------------------------------------------------------------
.. function:: void nmod_poly_factor_init(nmod_poly_factor_t fac)
Initialises ``fac`` for use. An ``nmod_poly_factor_t``
represents a polynomial in factorised form as a product of
polynomials with associated exponents.
.. function:: void nmod_poly_factor_clear(nmod_poly_factor_t fac)
Frees all memory associated with ``fac``.
.. function:: void nmod_poly_factor_realloc(nmod_poly_factor_t fac, slong alloc)
Reallocates the factor structure to provide space for
precisely ``alloc`` factors.
.. function:: void nmod_poly_factor_fit_length(nmod_poly_factor_t fac, slong len)
Ensures that the factor structure has space for at
least ``len`` factors. This function takes care
of the case of repeated calls by always at least
doubling the number of factors the structure can hold.
.. function:: void nmod_poly_factor_set(nmod_poly_factor_t res, const nmod_poly_factor_t fac)
Sets ``res`` to the same factorisation as ``fac``.
.. function:: void nmod_poly_factor_print(const nmod_poly_factor_t fac)
Prints the entries of ``fac`` to standard output.
.. function:: void nmod_poly_factor_insert(nmod_poly_factor_t fac, const nmod_poly_t poly, slong exp)
Inserts the factor ``poly`` with multiplicity ``exp`` into
the factorisation ``fac``.
If ``fac`` already contains ``poly``, then ``exp`` simply
gets added to the exponent of the existing entry.
.. function:: void nmod_poly_factor_concat(nmod_poly_factor_t res, const nmod_poly_factor_t fac)
Concatenates two factorisations.
This is equivalent to calling :func:`nmod_poly_factor_insert`
repeatedly with the individual factors of ``fac``.
Does not support aliasing between ``res`` and ``fac``.
.. function:: void nmod_poly_factor_pow(nmod_poly_factor_t fac, slong exp)
Raises ``fac`` to the power ``exp``.
.. function:: int nmod_poly_is_irreducible(const nmod_poly_t f)
Returns 1 if the polynomial ``f`` is irreducible, otherwise returns 0.
.. function:: int nmod_poly_is_irreducible_ddf(const nmod_poly_t f)
Returns 1 if the polynomial ``f`` is irreducible, otherwise returns 0.
Uses fast distinct-degree factorisation.
.. function:: int nmod_poly_is_irreducible_rabin(const nmod_poly_t f)
Returns 1 if the polynomial ``f`` is irreducible, otherwise returns 0.
Uses Rabin irreducibility test.
.. function:: int _nmod_poly_is_squarefree(nn_srcptr f, slong len, nmod_t mod)
Returns 1 if ``(f, len)`` is squarefree, and 0 otherwise. As a
special case, the zero polynomial is not considered squarefree.
There are no restrictions on the length.
.. function:: int nmod_poly_is_squarefree(const nmod_poly_t f)
Returns 1 if ``f`` is squarefree, and 0 otherwise. As a special
case, the zero polynomial is not considered squarefree.
.. function:: void nmod_poly_factor_squarefree(nmod_poly_factor_t res, const nmod_poly_t f)
Sets ``res`` to a square-free factorization of ``f``.
.. function:: int nmod_poly_factor_equal_deg_prob(nmod_poly_t factor, flint_rand_t state, const nmod_poly_t pol, slong d)
Probabilistic equal degree factorisation of ``pol`` into
irreducible factors of degree ``d``. If it passes, a factor is
placed in factor and 1 is returned, otherwise 0 is returned and
the value of factor is undetermined.
Requires that ``pol`` be monic, non-constant and squarefree.
.. function:: void nmod_poly_factor_equal_deg(nmod_poly_factor_t factors, const nmod_poly_t pol, slong d)
Assuming ``pol`` is a product of irreducible factors all of
degree ``d``, finds all those factors and places them in factors.
Requires that ``pol`` be monic, non-constant and squarefree.
.. function:: void nmod_poly_factor_distinct_deg(nmod_poly_factor_t res, const nmod_poly_t poly, slong * const * degs)
Factorises a monic non-constant squarefree polynomial ``poly``
of degree n into factors `f[d]` such that for `1 \leq d \leq n`
`f[d]` is the product of the monic irreducible factors of ``poly``
of degree `d`. Factors `f[d]` are stored in ``res``, and the degree `d`
of the irreducible factors is stored in ``degs`` in the same order
as the factors.
Requires that ``degs`` has enough space for ``(n/2)+1 * sizeof(slong)``.
.. function:: void nmod_poly_factor_distinct_deg_threaded(nmod_poly_factor_t res, const nmod_poly_t poly, slong * const * degs)
Multithreaded version of :func:`nmod_poly_factor_distinct_deg`.
.. function:: void nmod_poly_factor_cantor_zassenhaus(nmod_poly_factor_t res, const nmod_poly_t f)
Factorises a non-constant polynomial ``f`` into monic irreducible
factors using the Cantor-Zassenhaus algorithm.
.. function:: void nmod_poly_factor_berlekamp(nmod_poly_factor_t res, const nmod_poly_t f)
Factorises a non-constant, squarefree polynomial ``f`` into monic
irreducible factors using the Berlekamp algorithm.
.. function:: void nmod_poly_factor_kaltofen_shoup(nmod_poly_factor_t res, const nmod_poly_t poly)
Factorises a non-constant polynomial ``f`` into monic irreducible
factors using the fast version of Cantor-Zassenhaus algorithm proposed by
Kaltofen and Shoup (1998). More precisely this algorithm uses a
“baby step/giant step” strategy for the distinct-degree factorization
step. If :func:`flint_get_num_threads` is greater than one
:func:`nmod_poly_factor_distinct_deg_threaded` is used.
.. function:: ulong nmod_poly_factor_with_berlekamp(nmod_poly_factor_t res, const nmod_poly_t f)
Factorises a general polynomial ``f`` into monic irreducible factors
and returns the leading coefficient of ``f``, or 0 if ``f``
is the zero polynomial.
This function first checks for small special cases, deflates ``f``
if it is of the form `p(x^m)` for some `m > 1`, then performs a
square-free factorisation, and finally runs Berlekamp on all the
individual square-free factors.
.. function:: ulong nmod_poly_factor_with_cantor_zassenhaus(nmod_poly_factor_t res, const nmod_poly_t f)
Factorises a general polynomial ``f`` into monic irreducible factors
and returns the leading coefficient of ``f``, or 0 if ``f``
is the zero polynomial.
This function first checks for small special cases, deflates ``f``
if it is of the form `p(x^m)` for some `m > 1`, then performs a
square-free factorisation, and finally runs Cantor-Zassenhaus on all the
individual square-free factors.
.. function:: ulong nmod_poly_factor_with_kaltofen_shoup(nmod_poly_factor_t res, const nmod_poly_t f)
Factorises a general polynomial ``f`` into monic irreducible factors
and returns the leading coefficient of ``f``, or 0 if ``f``
is the zero polynomial.
This function first checks for small special cases, deflates ``f``
if it is of the form `p(x^m)` for some `m > 1`, then performs a
square-free factorisation, and finally runs Kaltofen-Shoup on all the
individual square-free factors.
.. function:: ulong nmod_poly_factor(nmod_poly_factor_t res, const nmod_poly_t f)
Factorises a general polynomial ``f`` into monic irreducible factors
and returns the leading coefficient of ``f``, or 0 if ``f``
is the zero polynomial.
This function first checks for small special cases, deflates ``f``
if it is of the form `p(x^m)` for some `m > 1`, then performs a
square-free factorisation, and finally runs either Cantor-Zassenhaus
or Berlekamp on all the individual square-free factors.
Currently Cantor-Zassenhaus is used by default unless the modulus is 2, in
which case Berlekamp is used.
.. function:: void _nmod_poly_interval_poly_worker(void * arg_ptr)
Worker function to compute interval polynomials in distinct degree
factorisation. Input/output is stored in
``nmod_poly_interval_poly_arg_t``.
|
