The :type:`fmpz_poly_t` data type represents elements of mathbb{Z}[x]. The :type:`fmpz_poly` module provides routines for memory management, basic arithmetic, and conversions from or to other types.
Each coefficient of an :type:`fmpz_poly_t` is an integer of the FLINT :type:`fmpz_t` type. There are two advantages of this model. Firstly, the :type:`fmpz_t` type is memory managed, so the user can manipulate individual coefficients of a polynomial without having to deal with tedious memory management. Secondly, a coefficient of an :type:`fmpz_poly_t` can be changed without changing the size of any of the other coefficients.
Unless otherwise specified, all functions in this section permit aliasing between their input arguments and between their input and output arguments.
The following example computes the square of the polynomial 5x^3 - 1.
#include "fmpz_poly.h"
int main()
{
fmpz_poly_t x, y;
fmpz_poly_init(x);
fmpz_poly_init(y);
fmpz_poly_set_coeff_ui(x, 3, 5);
fmpz_poly_set_coeff_si(x, 0, -1);
fmpz_poly_mul(y, x, x);
fmpz_poly_print(x); flint_printf("\n");
fmpz_poly_print(y); flint_printf("\n");
fmpz_poly_clear(x);
fmpz_poly_clear(y);
}The output is:
4 -1 0 0 5 7 1 0 0 -10 0 0 25
The :type:`fmpz_poly_t` type is a typedef for an array of length 1 of
:type:`fmpz_poly_struct`'s. This permits passing parameters of type
:type:`fmpz_poly_t` by reference in a manner similar to the way GMP
integers of type mpz_t can be passed by reference.
In reality one never deals directly with the struct and simply
deals with objects of type :type:`fmpz_poly_t`. For simplicity we will
think of an :type:`fmpz_poly_t` as a struct, though in practice to
access fields of this struct, one needs to dereference first, e.g.
to access the length field of an :type:`fmpz_poly_t` called
poly1 one writes poly1->length.
An :type:`fmpz_poly_t` is said to be normalised if either length
is zero, or if the leading coefficient of the polynomial is non-zero.
All :type:`fmpz_poly` functions expect their inputs to be normalised,
and unless otherwise specified they produce output that is normalised.
It is recommended that users do not access the fields of an :type:`fmpz_poly_t` or its coefficient data directly, but make use of the functions designed for this purpose, detailed below.
Functions in fmpz_poly do all the memory management for the user.
One does not need to specify the maximum length or number of limbs per
coefficient in advance before using a polynomial object. FLINT
reallocates space automatically as the computation proceeds, if more
space is required. Each coefficient is also managed separately, being
resized as needed, independently of the other coefficients.
.. type:: fmpz_poly_struct
.. type:: fmpz_poly_t
.. function:: void fmpz_poly_init(fmpz_poly_t poly)
Initialises ``poly`` for use, setting its length to zero.
A corresponding call to :func:`fmpz_poly_clear` must be made after
finishing with the ``fmpz_poly_t`` to free the memory used by
the polynomial.
.. function:: void fmpz_poly_init2(fmpz_poly_t poly, slong alloc)
Initialises ``poly`` with space for at least ``alloc`` coefficients
and sets the length to zero. The allocated coefficients are all set to
zero.
.. function:: void fmpz_poly_realloc(fmpz_poly_t poly, slong alloc)
Reallocates the given polynomial to have space for ``alloc``
coefficients. If ``alloc`` is zero the polynomial is cleared
and then reinitialised. If the current length is greater than
``alloc`` the polynomial is first truncated to length ``alloc``.
.. function:: void fmpz_poly_fit_length(fmpz_poly_t poly, slong len)
If ``len`` is greater than the number of coefficients currently
allocated, then the polynomial is reallocated to have space for at
least ``len`` coefficients. No data is lost when calling this
function.
The function efficiently deals with the case where ``fit_length`` is
called many times in small increments by at least doubling the number
of allocated coefficients when length is larger than the number of
coefficients currently allocated.
.. function:: void fmpz_poly_clear(fmpz_poly_t poly)
Clears the given polynomial, releasing any memory used. It must
be reinitialised in order to be used again.
.. function:: void _fmpz_poly_normalise(fmpz_poly_t poly)
Sets the length of ``poly`` so that the top coefficient is non-zero.
If all coefficients are zero, the length is set to zero. This function
is mainly used internally, as all functions guarantee normalisation.
.. function:: void _fmpz_poly_set_length(fmpz_poly_t poly, slong newlen)
Demotes the coefficients of ``poly`` beyond ``newlen`` and sets
the length of ``poly`` to ``newlen``.
.. function:: void fmpz_poly_attach_truncate(fmpz_poly_t trunc, const fmpz_poly_t poly, slong n)
This function sets the uninitialised polynomial ``trunc`` to the low
`n` coefficients of ``poly``, or to ``poly`` if the latter doesn't
have `n` coefficients. The polynomial ``trunc`` not be cleared or used
as the output of any FLINT functions.
.. function:: void fmpz_poly_attach_shift(fmpz_poly_t trunc, const fmpz_poly_t poly, slong n)
This function sets the uninitialised polynomial ``trunc`` to the
high coefficients of ``poly``, i.e. the coefficients not among the low
`n` coefficients of ``poly``. If the latter doesn't have `n`
coefficients ``trunc`` is set to the zero polynomial. The polynomial
``trunc`` not be cleared or used as the output of any FLINT functions.
.. function:: slong fmpz_poly_length(const fmpz_poly_t poly)
Returns the length of ``poly``. The zero polynomial has length zero.
.. function:: slong fmpz_poly_degree(const fmpz_poly_t poly)
Returns the degree of ``poly``, which is one less than its length.
.. function:: void fmpz_poly_set(fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``poly1`` to equal ``poly2``.
.. function:: void fmpz_poly_set_si(fmpz_poly_t poly, slong c)
Sets ``poly`` to the signed integer ``c``.
.. function:: void fmpz_poly_set_ui(fmpz_poly_t poly, ulong c)
Sets ``poly`` to the unsigned integer ``c``.
.. function:: void fmpz_poly_set_fmpz(fmpz_poly_t poly, const fmpz_t c)
Sets ``poly`` to the integer ``c``.
.. function:: int _fmpz_poly_set_str(fmpz * poly, const char * str)
Sets ``poly`` to the polynomial encoded in the null-terminated
string ``str``. Assumes that ``poly`` is allocated as a
sufficiently large array suitable for the number of coefficients
present in ``str``.
Returns `0` if no error occurred. Otherwise, returns a non-zero
value, in which case the resulting value of ``poly`` is undefined.
If ``str`` is not null-terminated, calling this method might result
in a segmentation fault.
.. function:: int fmpz_poly_set_str(fmpz_poly_t poly, const char * str)
Imports a polynomial from a null-terminated string. If the string
``str`` represents a valid polynomial returns `0`, otherwise
returns `1`.
Returns `0` if no error occurred. Otherwise, returns a non-zero value,
in which case the resulting value of ``poly`` is undefined. If
``str`` is not null-terminated, calling this method might result in
a segmentation fault.
.. function:: char * _fmpz_poly_get_str(const fmpz * poly, slong len)
Returns the plain FLINT string representation of the polynomial
``(poly, len)``.
.. function:: char * fmpz_poly_get_str(const fmpz_poly_t poly)
Returns the plain FLINT string representation of the polynomial
``poly``.
.. function:: char * _fmpz_poly_get_str_pretty(const fmpz * poly, slong len, const char * x)
Returns a pretty representation of the polynomial
``(poly, len)`` using the null-terminated string ``x`` as the
variable name.
.. function:: char * fmpz_poly_get_str_pretty(const fmpz_poly_t poly, const char * x)
Returns a pretty representation of the polynomial ``poly`` using the
null-terminated string ``x`` as the variable name.
.. function:: void fmpz_poly_zero(fmpz_poly_t poly)
Sets ``poly`` to the zero polynomial.
.. function:: void fmpz_poly_one(fmpz_poly_t poly)
Sets ``poly`` to the constant polynomial one.
.. function:: void fmpz_poly_zero_coeffs(fmpz_poly_t poly, slong i, slong j)
Sets the coefficients of `x^i, \dotsc, x^{j-1}` to zero.
.. function:: void fmpz_poly_swap(fmpz_poly_t poly1, fmpz_poly_t poly2)
Swaps ``poly1`` and ``poly2``. This is done efficiently without
copying data by swapping pointers, etc.
.. function:: void _fmpz_poly_reverse(fmpz * res, const fmpz * poly, slong len, slong n)
Sets ``(res, n)`` to the reverse of ``(poly, n)``, where
``poly`` is in fact an array of length ``len``. Assumes that
``0 < len <= n``. Supports aliasing of ``res`` and ``poly``,
but the behaviour is undefined in case of partial overlap.
.. function:: void fmpz_poly_reverse(fmpz_poly_t res, const fmpz_poly_t poly, slong n)
This function considers the polynomial ``poly`` to be of length `n`,
notionally truncating and zero padding if required, and reverses
the result. Since the function normalises its result ``res`` may be
of length less than `n`.
.. function:: void fmpz_poly_truncate(fmpz_poly_t poly, slong newlen)
If the current length of ``poly`` is greater than ``newlen``, it
is truncated to have the given length. Discarded coefficients are not
necessarily set to zero.
.. function:: void fmpz_poly_set_trunc(fmpz_poly_t res, const fmpz_poly_t poly, slong n)
Sets ``res`` to a copy of ``poly``, truncated to length ``n``.
.. function:: void fmpz_mod_poly_rand(fmpz_mod_poly_t f, flint_rand_t state, slong len, const fmpz_mod_ctx_t ctx)
Sets `f` to a random polynomial with up to the given length and where
each coefficient has up to the given number of bits. The coefficients
are uniformly generated random numbers in `[0, n)`, where `n` is the modulus given by the context `ctx`.
.. function:: void fmpz_mod_poly_rand_monic(fmpz_mod_poly_t f, flint_rand_t state, slong len, const fmpz_mod_ctx_t ctx)
Sets `f` to a random monic polynomial with up to the given length and where
each coefficient has up to the given number of bits. The coefficients
are uniformly generated random numbers in `[0, n)`, where `n` is the modulus given by the context `ctx`.
.. function:: void fmpz_mod_poly_rand_irreducible(fmpz_mod_poly_t f, flint_rand_t state, slong len, const fmpz_mod_ctx_t ctx)
Sets `f` to a random irreducible polynomial with up to the given length and where each coefficient has up to the given number of bits. The coefficients
are uniformly generated random numbers in `[0, n)`, where `n` is the modulus given by the context `ctx`.
.. function:: void fmpz_poly_randtest(fmpz_poly_t f, flint_rand_t state, slong len, flint_bitcnt_t bits)
Sets `f` to a random polynomial with up to the given length and where
each coefficient has up to the given number of bits. The coefficients
are random numbers in `[1, n)`, where `n` is the modulus given by the context `ctx`.
.. function:: void fmpz_poly_randtest_unsigned(fmpz_poly_t f, flint_rand_t state, slong len, flint_bitcnt_t bits)
Sets `f` to a random polynomial with up to the given length and where
each coefficient has up to the given number of bits.
.. function:: void fmpz_poly_randtest_not_zero(fmpz_poly_t f, flint_rand_t state, slong len, flint_bitcnt_t bits)
As for :func:`fmpz_poly_randtest` except that ``len`` and bits may
not be zero and the polynomial generated is guaranteed not to be the
zero polynomial.
.. function:: void fmpz_poly_randtest_no_real_root(fmpz_poly_t p, flint_rand_t state, slong len, flint_bitcnt_t bits)
Sets ``p`` to a random polynomial without any real root, whose
length is up to ``len`` and where each coefficient has up to the
given number of bits.
.. function:: void fmpz_poly_randtest_irreducible1(fmpz_poly_t pol, flint_rand_t state, slong len, flint_bitcnt_t bits)
void fmpz_poly_randtest_irreducible2(fmpz_poly_t pol, flint_rand_t state, slong len, flint_bitcnt_t bits)
void fmpz_poly_randtest_irreducible(fmpz_poly_t pol, flint_rand_t state, slong len, flint_bitcnt_t bits)
Sets ``p`` to a random irreducible polynomial, whose
length is up to ``len`` and where each coefficient has up to the
given number of bits. There are two algorithms: *irreducible1*
generates an irreducible polynomial modulo a random prime number
and lifts it to the integers; *irreducible2* generates a random
integer polynomial, factors it, and returns a random factor.
The default function chooses randomly between these methods.
.. function:: void fmpz_poly_get_coeff_fmpz(fmpz_t x, const fmpz_poly_t poly, slong n)
Sets `x` to the `n`-th coefficient of ``poly``. Coefficient
numbering is from zero and if `n` is set to a value beyond the end of
the polynomial, zero is returned.
.. function:: slong fmpz_poly_get_coeff_si(const fmpz_poly_t poly, slong n)
Returns coefficient `n` of ``poly`` as a ``slong``. The result is
undefined if the value does not fit into a ``slong``. Coefficient
numbering is from zero and if `n` is set to a value beyond the end of
the polynomial, zero is returned.
.. function:: ulong fmpz_poly_get_coeff_ui(const fmpz_poly_t poly, slong n)
Returns coefficient `n` of ``poly`` as a ``ulong``. The result is
undefined if the value does not fit into a ``ulong``. Coefficient
numbering is from zero and if `n` is set to a value beyond the end of the
polynomial, zero is returned.
.. function:: fmpz * fmpz_poly_get_coeff_ptr(const fmpz_poly_t poly, slong n)
Returns a reference to the coefficient of `x^n` in the polynomial,
as an ``fmpz *``. This function is provided so that individual
coefficients can be accessed and operated on by functions in the
``fmpz`` module. This function does not make a copy of the
data, but returns a reference to the actual coefficient.
Returns ``NULL`` when `n` exceeds the degree of the polynomial.
This function is implemented as a macro.
.. function:: fmpz * fmpz_poly_lead(const fmpz_poly_t poly)
Returns a reference to the leading coefficient of the polynomial,
as an ``fmpz *``. This function is provided so that the leading
coefficient can be easily accessed and operated on by functions in
the ``fmpz`` module. This function does not make a copy of the
data, but returns a reference to the actual coefficient.
Returns ``NULL`` when the polynomial is zero.
This function is implemented as a macro.
.. function:: void fmpz_poly_set_coeff_fmpz(fmpz_poly_t poly, slong n, const fmpz_t x)
Sets coefficient `n` of ``poly`` to the ``fmpz`` value ``x``.
Coefficient numbering starts from zero and if `n` is beyond the current
length of ``poly`` then the polynomial is extended and zero
coefficients inserted if necessary.
.. function:: void fmpz_poly_set_coeff_si(fmpz_poly_t poly, slong n, slong x)
Sets coefficient `n` of ``poly`` to the ``slong`` value ``x``.
Coefficient numbering starts from zero and if `n` is beyond the current
length of ``poly`` then the polynomial is extended and zero
coefficients inserted if necessary.
.. function:: void fmpz_poly_set_coeff_ui(fmpz_poly_t poly, slong n, ulong x)
Sets coefficient `n` of ``poly`` to the ``ulong`` value
``x``. Coefficient numbering starts from zero and if `n` is beyond
the current length of ``poly`` then the polynomial is extended and
zero coefficients inserted if necessary.
.. function:: int fmpz_poly_equal(const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Returns `1` if ``poly1`` is equal to ``poly2``, otherwise
returns `0`. The polynomials are assumed to be normalised.
.. function:: int fmpz_poly_equal_trunc(const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Return `1` if ``poly1`` and ``poly2``, notionally truncated to
length `n` are equal, otherwise return `0`.
.. function:: int fmpz_poly_is_zero(const fmpz_poly_t poly)
Returns `1` if the polynomial is zero and `0` otherwise.
This function is implemented as a macro.
.. function:: int fmpz_poly_is_one(const fmpz_poly_t poly)
Returns `1` if the polynomial is one and `0` otherwise.
.. function:: int fmpz_poly_is_unit(const fmpz_poly_t poly)
Returns `1` if the polynomial is the constant polynomial `\pm 1`,
and `0` otherwise.
.. function:: int fmpz_poly_is_gen(const fmpz_poly_t poly)
Returns `1` if the polynomial is the degree `1` polynomial `x`, and `0`
otherwise.
.. function:: void _fmpz_poly_add(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``res`` to the sum of ``(poly1, len1)`` and
``(poly2, len2)``. It is assumed that ``res`` has
sufficient space for the longer of the two polynomials.
.. function:: void fmpz_poly_add(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the sum of ``poly1`` and ``poly2``.
.. function:: void fmpz_poly_add_series(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Notionally truncate ``poly1`` and ``poly2`` to length `n` and then
set ``res`` to the sum.
.. function:: void _fmpz_poly_sub(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``res`` to ``(poly1, len1)`` minus ``(poly2, len2)``. It
is assumed that ``res`` has sufficient space for the longer of the
two polynomials.
.. function:: void fmpz_poly_sub(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to ``poly1`` minus ``poly2``.
.. function:: void fmpz_poly_sub_series(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Notionally truncate ``poly1`` and ``poly2`` to length `n` and then
set ``res`` to the sum.
.. function:: void fmpz_poly_neg(fmpz_poly_t res, const fmpz_poly_t poly)
Sets ``res`` to ``-poly``.
.. function:: void fmpz_poly_scalar_abs(fmpz_poly_t res, const fmpz_poly_t poly)
Sets ``poly1`` to the polynomial whose coefficients are the absolute
value of those of ``poly2``.
.. function:: void fmpz_poly_scalar_mul_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)
Sets ``poly1`` to ``poly2`` times `x`.
.. function:: void fmpz_poly_scalar_mul_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)
Sets ``poly1`` to ``poly2`` times the signed ``slong x``.
.. function:: void fmpz_poly_scalar_mul_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)
Sets ``poly1`` to ``poly2`` times the ``ulong x``.
.. function:: void fmpz_poly_scalar_mul_2exp(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong exp)
Sets ``poly1`` to ``poly2`` times ``2^exp``.
.. function:: void fmpz_poly_scalar_addmul_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)
.. function:: void fmpz_poly_scalar_addmul_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)
.. function:: void fmpz_poly_scalar_addmul_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)
Sets ``poly1`` to ``poly1 + x * poly2``.
.. function:: void fmpz_poly_scalar_submul_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)
Sets ``poly1`` to ``poly1 - x * poly2``.
.. function:: void fmpz_poly_scalar_fdiv_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)
Sets ``poly1`` to ``poly2`` divided by the ``fmpz_t x``,
rounding coefficients down toward `- \infty`.
.. function:: void fmpz_poly_scalar_fdiv_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)
Sets ``poly1`` to ``poly2`` divided by the ``slong x``,
rounding coefficients down toward `- \infty`.
.. function:: void fmpz_poly_scalar_fdiv_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)
Sets ``poly1`` to ``poly2`` divided by the ``ulong x``,
rounding coefficients down toward `- \infty`.
.. function:: void fmpz_poly_scalar_fdiv_2exp(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)
Sets ``poly1`` to ``poly2`` divided by ``2^x``,
rounding coefficients down toward `- \infty`.
.. function:: void fmpz_poly_scalar_tdiv_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)
Sets ``poly1`` to ``poly2`` divided by the ``fmpz_t x``,
rounding coefficients toward `0`.
.. function:: void fmpz_poly_scalar_tdiv_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)
Sets ``poly1`` to ``poly2`` divided by the ``slong x``,
rounding coefficients toward `0`.
.. function:: void fmpz_poly_scalar_tdiv_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)
Sets ``poly1`` to ``poly2`` divided by the ``ulong x``,
rounding coefficients toward `0`.
.. function:: void fmpz_poly_scalar_tdiv_2exp(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)
Sets ``poly1`` to ``poly2`` divided by ``2^x``,
rounding coefficients toward `0`.
.. function:: void fmpz_poly_scalar_divexact_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)
Sets ``poly1`` to ``poly2`` divided by the ``fmpz_t x``,
assuming the division is exact for every coefficient.
.. function:: void fmpz_poly_scalar_divexact_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)
Sets ``poly1`` to ``poly2`` divided by the ``slong x``,
assuming the coefficient is exact for every coefficient.
.. function:: void fmpz_poly_scalar_divexact_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)
Sets ``poly1`` to ``poly2`` divided by the ``ulong x``,
assuming the coefficient is exact for every coefficient.
.. function:: void fmpz_poly_scalar_mod_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t p)
Sets ``poly1`` to ``poly2``, reducing each coefficient
modulo `p > 0`.
.. function:: void fmpz_poly_scalar_smod_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t p)
Sets ``poly1`` to ``poly2``, symmetrically reducing
each coefficient modulo `p > 0`, that is, choosing the unique
representative in the interval `(-p/2, p/2]`.
.. function:: slong _fmpz_poly_remove_content_2exp(fmpz * pol, slong len)
Remove the 2-content of ``pol`` and return the number `k`
that is the maximal non-negative integer so that `2^k` divides
all coefficients of the polynomial. For the zero polynomial,
`0` is returned.
.. function:: void _fmpz_poly_scale_2exp(fmpz * pol, slong len, slong k)
Scale ``(pol, len)`` to `p(2^k X)` in-place and divide by the
2-content (so that the gcd of coefficients is odd). If ``k``
is negative the polynomial is multiplied by `2^{kd}`.
.. function:: void _fmpz_poly_bit_pack(nn_ptr arr, const fmpz * poly, slong len, flint_bitcnt_t bit_size, int negate)
Packs the coefficients of ``poly`` into bitfields of the given
``bit_size``, negating the coefficients before packing
if ``negate`` is set to `-1`.
.. function:: int _fmpz_poly_bit_unpack(fmpz * poly, slong len, nn_srcptr arr, flint_bitcnt_t bit_size, int negate)
Unpacks the polynomial of given length from the array as packed into
fields of the given ``bit_size``, finally negating the coefficients
if ``negate`` is set to `-1`. Returns borrow, which is nonzero if a
leading term with coefficient `\pm1` should be added at
position ``len`` of ``poly``.
.. function:: void _fmpz_poly_bit_unpack_unsigned(fmpz * poly, slong len, nn_srcptr arr, flint_bitcnt_t bit_size)
Unpacks the polynomial of given length from the array as packed into
fields of the given ``bit_size``. The coefficients are assumed to
be unsigned.
.. function:: void fmpz_poly_bit_pack(fmpz_t f, const fmpz_poly_t poly, flint_bitcnt_t bit_size)
Packs ``poly`` into bitfields of size ``bit_size``, writing the
result to ``f``. The sign of ``f`` will be the same as that of
the leading coefficient of ``poly``.
.. function:: void fmpz_poly_bit_unpack(fmpz_poly_t poly, const fmpz_t f, flint_bitcnt_t bit_size)
Unpacks the polynomial with signed coefficients packed into
fields of size ``bit_size`` as represented by the integer ``f``.
.. function:: void fmpz_poly_bit_unpack_unsigned(fmpz_poly_t poly, const fmpz_t f, flint_bitcnt_t bit_size)
Unpacks the polynomial with unsigned coefficients packed into
fields of size ``bit_size`` as represented by the integer ``f``.
It is required that ``f`` is nonnegative.
.. function:: void _fmpz_poly_mul_classical(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``(res, len1 + len2 - 1)`` to the product of ``(poly1, len1)``
and ``(poly2, len2)``.
Assumes ``len1`` and ``len2`` are positive. Allows zero-padding
of the two input polynomials. No aliasing of inputs with outputs is
allowed.
.. function:: void fmpz_poly_mul_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the product of ``poly1`` and ``poly2``, computed
using the classical or schoolbook method.
.. function:: void _fmpz_poly_mullow_classical(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)
Sets ``(res, n)`` to the first `n` coefficients of ``(poly1, len1)``
multiplied by ``(poly2, len2)``.
Assumes ``0 < n <= len1 + len2 - 1``. Assumes neither ``len1`` nor
``len2`` is zero.
.. function:: void fmpz_poly_mullow_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Sets ``res`` to the first `n` coefficients of ``poly1 * poly2``.
.. function:: void _fmpz_poly_mulhigh_classical(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong start)
Sets the first ``start`` coefficients of ``res`` to zero and the
remainder to the corresponding coefficients of
``(poly1, len1) * (poly2, len2)``.
Assumes ``start <= len1 + len2 - 1``. Assumes neither ``len1`` nor
``len2`` is zero.
.. function:: void fmpz_poly_mulhigh_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong start)
Sets the first ``start`` coefficients of ``res`` to zero and the
remainder to the corresponding coefficients of the product of ``poly1``
and ``poly2``.
.. function:: void _fmpz_poly_mulmid_classical(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``res`` to the middle ``len1 - len2 + 1`` coefficients of
the product of ``(poly1, len1)`` and ``(poly2, len2)``, i.e. the
coefficients from degree ``len2 - 1`` to ``len1 - 1`` inclusive.
Assumes that ``len1 >= len2 > 0``.
.. function:: void fmpz_poly_mulmid_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the middle ``len(poly1) - len(poly2) + 1``
coefficients of ``poly1 * poly2``, i.e. the coefficient from degree
``len2 - 1`` to ``len1 - 1`` inclusive. Assumes that
``len1 >= len2``.
.. function:: void _fmpz_poly_mul_karatsuba(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``(res, len1 + len2 - 1)`` to the product of ``(poly1, len1)``
and ``(poly2, len2)``. Assumes ``len1 >= len2 > 0``. Allows
zero-padding of the two input polynomials. No aliasing of inputs with
outputs is allowed.
.. function:: void fmpz_poly_mul_karatsuba(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the product of ``poly1`` and ``poly2``.
.. function:: void _fmpz_poly_mullow_karatsuba_n(fmpz * res, const fmpz * poly1, const fmpz * poly2, slong n)
Sets ``res`` to the product of ``poly1`` and ``poly2`` and
truncates to the given length. It is assumed that ``poly1`` and
``poly2`` are precisely the given length, possibly zero padded.
Assumes `n` is not zero.
.. function:: void fmpz_poly_mullow_karatsuba_n(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Sets ``res`` to the product of ``poly1`` and ``poly2`` and
truncates to the given length.
.. function:: void _fmpz_poly_mulhigh_karatsuba_n(fmpz * res, const fmpz * poly1, const fmpz * poly2, slong len)
Sets ``res`` to the product of ``poly1`` and ``poly2`` and
truncates at the top to the given length. The first ``len - 1``
coefficients are set to zero. It is assumed that ``poly1`` and
``poly2`` are precisely the given length, possibly zero padded.
Assumes ``len`` is not zero.
.. function:: void fmpz_poly_mulhigh_karatsuba_n(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong len)
Sets the first ``len - 1`` coefficients of the result to zero and the
remaining coefficients to the corresponding coefficients of the product of
``poly1`` and ``poly2``. Assumes ``poly1`` and ``poly2`` are
at most of the given length.
.. function:: void _fmpz_poly_mul_KS(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``(res, len1 + len2 - 1)`` to the product of ``(poly1, len1)``
and ``(poly2, len2)``.
Places no assumptions on ``len1`` and ``len2``. Allows zero-padding
of the two input polynomials. Supports aliasing of inputs and outputs.
.. function:: void fmpz_poly_mul_KS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the product of ``poly1`` and ``poly2``.
.. function:: void _fmpz_poly_mullow_KS(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)
Sets ``(res, n)`` to the lowest `n` coefficients of the product of
``(poly1, len1)`` and ``(poly2, len2)``.
Assumes that ``len1`` and ``len2`` are positive, but does allow
for the polynomials to be zero-padded. The polynomials may be zero,
too. Assumes `n` is positive. Supports aliasing between ``res``,
``poly1`` and ``poly2``.
.. function:: void fmpz_poly_mullow_KS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Sets ``res`` to the lowest `n` coefficients of the product of
``poly1`` and ``poly2``.
.. function:: void _fmpz_poly_mul_SS(fmpz * output, const fmpz * input1, slong length1, const fmpz * input2, slong length2)
Sets ``(output, length1 + length2 - 1)`` to the product of
``(input1, length1)`` and ``(input2, length2)``.
We must have ``len1 > 1`` and ``len2 > 1``. Allows zero-padding
of the two input polynomials. Supports aliasing of inputs and outputs.
.. function:: void fmpz_poly_mul_SS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the product of ``poly1`` and ``poly2``. Uses the
Schönhage-Strassen algorithm.
.. function:: void _fmpz_poly_mullow_SS(fmpz * output, const fmpz * input1, slong length1, const fmpz * input2, slong length2, slong n)
Sets ``(res, n)`` to the lowest `n` coefficients of the product of
``(poly1, len1)`` and ``(poly2, len2)``.
Assumes that ``len1`` and ``len2`` are positive, but does allow
for the polynomials to be zero-padded. We must have ``len1 > 1``
and ``len2 > 1``. Assumes `n` is positive. Supports aliasing between
``res``, ``poly1`` and ``poly2``.
.. function:: void fmpz_poly_mullow_SS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Sets ``res`` to the lowest `n` coefficients of the product of
``poly1`` and ``poly2``.
.. function:: void _fmpz_poly_mul(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``(res, len1 + len2 - 1)`` to the product of ``(poly1, len1)``
and ``(poly2, len2)``. Assumes ``len1 >= len2 > 0``. Allows
zero-padding of the two input polynomials. Does not support aliasing
between the inputs and the output.
.. function:: void fmpz_poly_mul(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the product of ``poly1`` and ``poly2``. Chooses
an optimal algorithm from the choices above.
.. function:: void _fmpz_poly_mullow(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)
Sets ``(res, n)`` to the lowest `n` coefficients of the product of
``(poly1, len1)`` and ``(poly2, len2)``.
Assumes ``len1 >= len2 > 0`` and ``0 < n <= len1 + len2 - 1``.
Allows for zero-padding in the inputs. Does not support aliasing between
the inputs and the output.
.. function:: void fmpz_poly_mullow(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Sets ``res`` to the lowest `n` coefficients of the product of
``poly1`` and ``poly2``.
.. function:: void fmpz_poly_mulhigh_n(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Sets the high `n` coefficients of ``res`` to the high `n` coefficients
of the product of ``poly1`` and ``poly2``, assuming the latter are
precisely `n` coefficients in length, zero padded if necessary. The
remaining `n - 1` coefficients may be arbitrary.
.. function:: void _fmpz_poly_mulhigh(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong start)
Sets all but the low `n` coefficients of `res` to the corresponding
coefficients of the product of `poly1` of length `len1` and `poly2` of
length `len2`, the remaining coefficients being arbitrary. It is assumed
that `len1 >= len2 > 0` and that `0 < n < len1 + len2 - 1`. Aliasing of
inputs is not permitted.
.. function:: void fmpz_poly_mul_SS_precache_init(fmpz_poly_mul_precache_t pre, slong len1, slong bits1, const fmpz_poly_t poly2)
Precompute the FFT of ``poly2`` to enable repeated multiplication of
``poly2`` by polynomials whose length does not exceed ``len1`` and
whose number of bits per coefficient does not exceed ``bits1``.
The value ``bits1`` may be negative, i.e. it may be the result of
calling ``fmpz_poly_max_bits``. The function only considers the
absolute value of ``bits1``.
Suppose ``len2`` is the length of ``poly2`` and
``len = len1 + len2 - 1`` is the maximum output length of a polynomial
multiplication using ``pre``. Then internally ``len`` is rounded up to
a power of two, `2^n` say. The truncated FFT algorithm is used to smooth
performance but note that it can only do this in the range
`(2^{n-1}, 2^n]`. Therefore, it may be more efficient to recompute `pre`
for cases where the output length will fall below `2^{n-1} + 1`. Otherwise
the implementation will zero pad them up to that length.
Note that the Schoenhage-Strassen algorithm is only efficient for
polynomials with relatively large coefficients relative to the length of
the polynomials.
Also note that there are no restrictions on the polynomials. In particular
the polynomial whose FFT is being precached does not have to be either
longer or shorter than the polynomials it is to be multiplied by.
.. function:: void fmpz_poly_mul_precache_clear(fmpz_poly_mul_precache_t pre)
Clear the space allocated by ``fmpz_poly_mul_SS_precache_init``.
.. function:: void _fmpz_poly_mullow_SS_precache(fmpz * output, const fmpz * input1, slong len1, fmpz_poly_mul_precache_t pre, slong trunc)
Write into ``output`` the first ``trunc`` coefficients of
the polynomial ``(input1, len1)`` by the polynomial whose FFT was precached
by ``fmpz_poly_mul_SS_precache_init`` and stored in ``pre``.
For performance reasons it is recommended that all polynomials be truncated
to at most ``trunc`` coefficients if possible.
.. function:: void fmpz_poly_mullow_SS_precache(fmpz_poly_t res, const fmpz_poly_t poly1, fmpz_poly_mul_precache_t pre, slong n)
Set ``res`` to the product of ``poly1`` by the polynomial whose FFT was
precached by ``fmpz_poly_mul_SS_precache_init`` (and stored in pre). The
result is truncated to `n` coefficients (and normalised).
There are no restrictions on the length of ``poly1`` other than those given
in the call to ``fmpz_poly_mul_SS_precache_init``.
.. function:: void fmpz_poly_mul_SS_precache(fmpz_poly_t res, const fmpz_poly_t poly1, fmpz_poly_mul_precache_t pre)
Set ``res`` to the product of ``poly1`` by the polynomial whose FFT was
precached by ``fmpz_poly_mul_SS_precache_init`` (and stored in pre).
There are no restrictions on the length of ``poly1`` other than those given
in the call to ``fmpz_poly_mul_SS_precache_init``.
.. function:: void _fmpz_poly_sqr_KS(fmpz * rop, const fmpz * op, slong len)
Sets ``(rop, 2*len - 1)`` to the square of ``(op, len)``,
assuming that ``len > 0``.
Supports zero-padding in ``(op, len)``. Does not support aliasing.
.. function:: void fmpz_poly_sqr_KS(fmpz_poly_t rop, const fmpz_poly_t op)
Sets ``rop`` to the square of the polynomial ``op`` using
Kronecker segmentation.
.. function:: void _fmpz_poly_sqr_karatsuba(fmpz * rop, const fmpz * op, slong len)
Sets ``(rop, 2*len - 1)`` to the square of ``(op, len)``,
assuming that ``len > 0``.
Supports zero-padding in ``(op, len)``. Does not support aliasing.
.. function:: void fmpz_poly_sqr_karatsuba(fmpz_poly_t rop, const fmpz_poly_t op)
Sets ``rop`` to the square of the polynomial ``op`` using
the Karatsuba multiplication algorithm.
.. function:: void _fmpz_poly_sqr_classical(fmpz * rop, const fmpz * op, slong len)
Sets ``(rop, 2*len - 1)`` to the square of ``(op, len)``,
assuming that ``len > 0``.
Supports zero-padding in ``(op, len)``. Does not support aliasing.
.. function:: void fmpz_poly_sqr_classical(fmpz_poly_t rop, const fmpz_poly_t op)
Sets ``rop`` to the square of the polynomial ``op`` using
the classical or schoolbook method.
.. function:: void _fmpz_poly_sqr(fmpz * rop, const fmpz * op, slong len)
Sets ``(rop, 2*len - 1)`` to the square of ``(op, len)``,
assuming that ``len > 0``.
Supports zero-padding in ``(op, len)``. Does not support aliasing.
.. function:: void fmpz_poly_sqr(fmpz_poly_t rop, const fmpz_poly_t op)
Sets ``rop`` to the square of the polynomial ``op``.
.. function:: void _fmpz_poly_sqrlow_KS(fmpz * res, const fmpz * poly, slong len, slong n)
Sets ``(res, n)`` to the lowest `n` coefficients
of the square of ``(poly, len)``.
Assumes that ``len`` is positive, but does allow for the polynomial
to be zero-padded. The polynomial may be zero, too. Assumes `n` is
positive. Supports aliasing between ``res`` and ``poly``.
.. function:: void fmpz_poly_sqrlow_KS(fmpz_poly_t res, const fmpz_poly_t poly, slong n)
Sets ``res`` to the lowest `n` coefficients
of the square of ``poly``.
.. function:: void _fmpz_poly_sqrlow_karatsuba_n(fmpz * res, const fmpz * poly, slong n)
Sets ``(res, n)`` to the square of ``(poly, n)`` truncated
to length `n`, which is assumed to be positive. Allows for ``poly``
to be zero-padded.
.. function:: void fmpz_poly_sqrlow_karatsuba_n(fmpz_poly_t res, const fmpz_poly_t poly, slong n)
Sets ``res`` to the square of ``poly`` and
truncates to the given length.
.. function:: void _fmpz_poly_sqrlow_classical(fmpz * res, const fmpz * poly, slong len, slong n)
Sets ``(res, n)`` to the first `n` coefficients of the square
of ``(poly, len)``.
Assumes that ``0 < n <= 2 * len - 1``.
.. function:: void fmpz_poly_sqrlow_classical(fmpz_poly_t res, const fmpz_poly_t poly, slong n)
Sets ``res`` to the first `n` coefficients of
the square of ``poly``.
.. function:: void _fmpz_poly_sqrlow(fmpz * res, const fmpz * poly, slong len, slong n)
Sets ``(res, n)`` to the lowest `n` coefficients
of the square of ``(poly, len)``.
Assumes ``len1 >= len2 > 0`` and ``0 < n <= 2 * len - 1``.
Allows for zero-padding in the input. Does not support aliasing
between the input and the output.
.. function:: void fmpz_poly_sqrlow(fmpz_poly_t res, const fmpz_poly_t poly, slong n)
Sets ``res`` to the lowest `n` coefficients
of the square of ``poly``.
.. function:: void _fmpz_poly_pow_multinomial(fmpz * res, const fmpz * poly, slong len, ulong e)
Computes ``res = poly^e``. This uses the J.C.P. Miller pure
recurrence as follows:
If `\ell` is the index of the lowest non-zero coefficient in ``poly``,
as a first step this method zeros out the lowest `e \ell` coefficients of
``res``. The recurrence above is then used to compute the remaining
coefficients.
Assumes ``len > 0``, ``e > 0``. Does not support aliasing.
.. function:: void fmpz_poly_pow_multinomial(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)
Computes ``res = poly^e`` using a generalisation of binomial expansion
called the J.C.P. Miller pure recurrence [1], [2].
If `e` is zero, returns one, so that in particular ``0^0 = 1``.
The formal statement of the recurrence is as follows. Write the input
polynomial as `P(x) = p_0 + p_1 x + \dotsb + p_m x^m` with `p_0 \neq 0`
and let
.. math::
P(x)^n = a(n, 0) + a(n, 1) x + \dotsb + a(n, mn) x^{mn}.
Then `a(n, 0) = p_0^n` and, for all `1 \leq k \leq mn`,
.. math::
a(n, k) =
(k p_0)^{-1} \sum_{i = 1}^m p_i \bigl( (n + 1) i - k \bigr) a(n, k-i).
[1] D. Knuth, The Art of Computer Programming Vol. 2, Seminumerical
Algorithms, Third Edition (Reading, Massachusetts: Addison-Wesley, 1997)
[2] D. Zeilberger, The J.C.P. Miller Recurrence for Exponentiating a
Polynomial, and its q-Analog, Journal of Difference Equations and
Applications, 1995, Vol. 1, pp. 57--60
.. function:: void _fmpz_poly_pow_binomial(fmpz * res, const fmpz * poly, ulong e)
Computes ``res = poly^e`` when poly is of length 2, using binomial
expansion.
Assumes `e > 0`. Does not support aliasing.
.. function:: void fmpz_poly_pow_binomial(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)
Computes ``res = poly^e`` when ``poly`` is of length `2`, using
binomial expansion.
If the length of ``poly`` is not `2`, raises an exception and aborts.
.. function:: void _fmpz_poly_pow_addchains(fmpz * res, const fmpz * poly, slong len, const int * a, int n)
Given a star chain `1 = a_0 < a_1 < \dotsb < a_n = e` computes
``res = poly^e``.
A star chain is an addition chain `1 = a_0 < a_1 < \dotsb < a_n` such
that, for all `i > 0`, `a_i = a_{i-1} + a_j` for some `j < i`.
Assumes that `e > 2`, or equivalently `n > 1`, and ``len > 0``. Does
not support aliasing.
.. function:: void fmpz_poly_pow_addchains(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)
Computes ``res = poly^e`` using addition chains whenever
`0 \leq e \leq 148`.
If `e > 148`, raises an exception and aborts.
.. function:: void _fmpz_poly_pow_binexp(fmpz * res, const fmpz * poly, slong len, ulong e)
Sets ``res = poly^e`` using left-to-right binary exponentiation as
described on p. 461 of [Knu1997]_.
Assumes that ``len > 0``, ``e > 1``. Assumes that ``res`` is
an array of length at least ``e*(len - 1) + 1``. Does not support
aliasing.
.. function:: void fmpz_poly_pow_binexp(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)
Computes ``res = poly^e`` using the binary exponentiation algorithm.
If `e` is zero, returns one, so that in particular ``0^0 = 1``.
.. function:: void _fmpz_poly_pow_small(fmpz * res, const fmpz * poly, slong len, ulong e)
Sets ``res = poly^e`` whenever `0 \leq e \leq 4`.
Assumes that ``len > 0`` and that ``res`` is an array of length
at least ``e*(len - 1) + 1``. Does not support aliasing.
.. function:: void _fmpz_poly_pow(fmpz * res, const fmpz * poly, slong len, ulong e)
Sets ``res = poly^e``, assuming that ``e, len > 0`` and that
``res`` has space for ``e*(len - 1) + 1`` coefficients. Does
not support aliasing.
.. function:: void fmpz_poly_pow(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)
Computes ``res = poly^e``. If `e` is zero, returns one,
so that in particular ``0^0 = 1``.
.. function:: void _fmpz_poly_pow_trunc(fmpz * res, const fmpz * poly, ulong e, slong n)
Sets ``(res, n)`` to ``(poly, n)`` raised to the power `e` and
truncated to length `n`.
Assumes that `e, n > 0`. Allows zero-padding of ``(poly, n)``.
Does not support aliasing of any inputs and outputs.
.. function:: void fmpz_poly_pow_trunc(fmpz_poly_t res, const fmpz_poly_t poly, ulong e, slong n)
Notationally raises ``poly`` to the power `e`, truncates the result
to length `n` and writes the result in ``res``. This is computed
much more efficiently than simply powering the polynomial and truncating.
Thus, if `n = 0` the result is zero. Otherwise, whenever `e = 0` the
result will be the constant polynomial equal to `1`.
This function can be used to raise power series to a power in an
efficient way.
.. function:: void _fmpz_poly_shift_left(fmpz * res, const fmpz * poly, slong len, slong n)
Sets ``(res, len + n)`` to ``(poly, len)`` shifted left by
`n` coefficients.
Inserts zero coefficients at the lower end. Assumes that ``len``
and `n` are positive, and that ``res`` fits ``len + n`` elements.
Supports aliasing between ``res`` and ``poly``.
.. function:: void fmpz_poly_shift_left(fmpz_poly_t res, const fmpz_poly_t poly, slong n)
Sets ``res`` to ``poly`` shifted left by `n` coeffs. Zero
coefficients are inserted.
.. function:: void _fmpz_poly_shift_right(fmpz * res, const fmpz * poly, slong len, slong n)
Sets ``(res, len - n)`` to ``(poly, len)`` shifted right by
`n` coefficients.
Assumes that ``len`` and `n` are positive, that ``len > n``,
and that ``res`` fits ``len - n`` elements. Supports aliasing
between ``res`` and ``poly``, although in this case the top
coefficients of ``poly`` are not set to zero.
.. function:: void fmpz_poly_shift_right(fmpz_poly_t res, const fmpz_poly_t poly, slong n)
Sets ``res`` to ``poly`` shifted right by `n` coefficients. If `n`
is equal to or greater than the current length of ``poly``, ``res``
is set to the zero polynomial.
.. function:: ulong fmpz_poly_max_limbs(const fmpz_poly_t poly)
Returns the maximum number of limbs required to store the absolute value
of coefficients of ``poly``. If ``poly`` is zero, returns `0`.
.. function:: slong fmpz_poly_max_bits(const fmpz_poly_t poly)
Computes the maximum number of bits `b` required to store the absolute
value of coefficients of ``poly``. If all the coefficients of
``poly`` are non-negative, `b` is returned, otherwise `-b` is returned.
.. function:: void fmpz_poly_height(fmpz_t height, const fmpz_poly_t poly)
Computes the height of ``poly``, defined as the largest of the
absolute values of the coefficients of ``poly``. Equivalently, this
gives the infinity norm of the coefficients. If ``poly`` is zero,
the height is `0`.
.. function:: void _fmpz_poly_2norm(fmpz_t res, const fmpz * poly, slong len)
Sets ``res`` to the Euclidean norm of ``(poly, len)``, that is,
the integer square root of the sum of the squares of the coefficients
of ``poly``.
.. function:: void fmpz_poly_2norm(fmpz_t res, const fmpz_poly_t poly)
Sets ``res`` to the Euclidean norm of ``poly``, that is, the
integer square root of the sum of the squares of the coefficients of
``poly``.
.. function:: ulong _fmpz_poly_2norm_normalised_bits(const fmpz * poly, slong len)
Returns an upper bound on the number of bits of the normalised
Euclidean norm of ``(poly, len)``, i.e. the number of bits of
the Euclidean norm divided by the absolute value of the leading
coefficient. The returned value will be no more than 1 bit too
large.
This is used in the computation of the Landau-Mignotte bound.
It is assumed that ``len > 0``. The result only makes sense
if the leading coefficient is nonzero.
.. function:: void _fmpz_poly_gcd_subresultant(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Computes the greatest common divisor ``(res, len2)`` of
``(poly1, len1)`` and ``(poly2, len2)``, assuming
``len1 >= len2 > 0``. The result is normalised to have
positive leading coefficient. Aliasing between ``res``,
``poly1`` and ``poly2`` is supported.
.. function:: void fmpz_poly_gcd_subresultant(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Computes the greatest common divisor ``res`` of ``poly1`` and
``poly2``, normalised to have non-negative leading coefficient.
This function uses the subresultant algorithm as described
in Algorithm 3.3.1 of [Coh1996]_.
.. function:: int _fmpz_poly_gcd_heuristic(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Computes the greatest common divisor ``(res, len2)`` of
``(poly1, len1)`` and ``(poly2, len2)``, assuming
``len1 >= len2 > 0``. The result is normalised to have
positive leading coefficient. Aliasing between ``res``,
``poly1`` and ``poly2`` is not supported. The function
may not always succeed in finding the GCD. If it fails, the
function returns 0, otherwise it returns 1.
.. function:: int fmpz_poly_gcd_heuristic(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Computes the greatest common divisor ``res`` of ``poly1`` and
``poly2``, normalised to have non-negative leading coefficient.
The function may not always succeed in finding the GCD. If it fails,
the function returns 0, otherwise it returns 1.
This function uses the heuristic GCD algorithm (GCDHEU). The basic
strategy is to remove the content of the polynomials, pack them
using Kronecker segmentation (given a bound on the size of the
coefficients of the GCD) and take the integer GCD. Unpack the
result and test divisibility.
.. function:: void _fmpz_poly_gcd_modular(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Computes the greatest common divisor ``(res, len2)`` of
``(poly1, len1)`` and ``(poly2, len2)``, assuming
``len1 >= len2 > 0``. The result is normalised to have
positive leading coefficient. Aliasing between ``res``,
``poly1`` and ``poly2`` is not supported.
.. function:: void fmpz_poly_gcd_modular(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Computes the greatest common divisor ``res`` of ``poly1`` and
``poly2``, normalised to have non-negative leading coefficient.
This function uses the modular GCD algorithm. The basic
strategy is to remove the content of the polynomials, reduce them
modulo sufficiently many primes and do CRT reconstruction until
some bound is reached (or we can prove with trial division that
we have the GCD).
.. function:: void _fmpz_poly_gcd(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Computes the greatest common divisor ``res`` of ``(poly1, len1)``
and ``(poly2, len2)``, assuming ``len1 >= len2 > 0``. The result
is normalised to have positive leading coefficient.
Assumes that ``res`` has space for ``len2`` coefficients.
Aliasing between ``res``, ``poly1`` and ``poly2`` is not
supported.
.. function:: void fmpz_poly_gcd(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Computes the greatest common divisor ``res`` of ``poly1`` and
``poly2``, normalised to have non-negative leading coefficient.
.. function:: void _fmpz_poly_xgcd_modular(fmpz_t r, fmpz * s, fmpz * t, const fmpz * f, slong len1, const fmpz * g, slong len2)
Set `r` to the resultant of ``(f, len1)`` and ``(g, len2)``.
If the resultant is zero, the function returns immediately. Otherwise it
finds polynomials `s` and `t` such that ``s*f + t*g = r``. The length
of `s` will be no greater than ``len2`` and the length of `t` will be
no greater than ``len1`` (both are zero padded if necessary).
It is assumed that ``len1 >= len2 > 0``. No aliasing of inputs and
outputs is permitted.
The function assumes that `f` and `g` are primitive (have Gaussian content
equal to 1). The result is undefined otherwise.
Uses a multimodular algorithm. The resultant is first computed and
extended GCDs modulo various primes `p` are computed and combined using
CRT. When the CRT stabilises the resulting polynomials are simply reduced
modulo further primes until a proven bound is reached.
.. function:: void fmpz_poly_xgcd_modular(fmpz_t r, fmpz_poly_t s, fmpz_poly_t t, const fmpz_poly_t f, const fmpz_poly_t g)
Set `r` to the resultant of `f` and `g`. If the resultant is zero, the
function then returns immediately, otherwise `s` and `t` are found such
that ``s*f + t*g = r``.
The function assumes that `f` and `g` are primitive (have Gaussian content
equal to 1). The result is undefined otherwise.
Uses the multimodular algorithm.
.. function:: void _fmpz_poly_xgcd(fmpz_t r, fmpz * s, fmpz * t, const fmpz * f, slong len1, const fmpz * g, slong len2)
Set `r` to the resultant of ``(f, len1)`` and ``(g, len2)``.
If the resultant is zero, the function returns immediately. Otherwise it
finds polynomials `s` and `t` such that ``s*f + t*g = r``. The length
of `s` will be no greater than ``len2`` and the length of `t` will be
no greater than ``len1`` (both are zero padded if necessary).
The function assumes that `f` and `g` are primitive (have Gaussian content
equal to 1). The result is undefined otherwise.
It is assumed that ``len1 >= len2 > 0``. No aliasing of inputs and
outputs is permitted.
.. function:: void fmpz_poly_xgcd(fmpz_t r, fmpz_poly_t s, fmpz_poly_t t, const fmpz_poly_t f, const fmpz_poly_t g)
Set `r` to the resultant of `f` and `g`. If the resultant is zero, the
function then returns immediately, otherwise `s` and `t` are found such
that ``s*f + t*g = r``.
The function assumes that `f` and `g` are primitive (have Gaussian content
equal to 1). The result is undefined otherwise.
.. function:: void _fmpz_poly_lcm(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``(res, len1 + len2 - 1)`` to the least common multiple
of the two polynomials ``(poly1, len1)`` and ``(poly2, len2)``,
normalised to have non-negative leading coefficient.
Assumes that ``len1 >= len2 > 0``.
Does not support aliasing.
.. function:: void fmpz_poly_lcm(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the least common multiple of the two
polynomials ``poly1`` and ``poly2``, normalised to
have non-negative leading coefficient.
If either of the two polynomials is zero, sets ``res``
to zero.
This ensures that the equality
.. math::
f g = \gcd(f, g) \operatorname{lcm}(f, g)
holds up to sign.
.. function:: void _fmpz_poly_resultant_modular(fmpz_t res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``res`` to the resultant of ``(poly1, len1)`` and
``(poly2, len2)``, assuming that ``len1 >= len2 > 0``.
.. function:: void fmpz_poly_resultant_modular(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Computes the resultant of ``poly1`` and ``poly2``.
For two non-zero polynomials `f(x) = a_m x^m + \dotsb + a_0` and
`g(x) = b_n x^n + \dotsb + b_0` of degrees `m` and `n`, the resultant
is defined to be
.. math::
a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).
For convenience, we define the resultant to be equal to zero if either
of the two polynomials is zero.
This function uses the modular algorithm described
in [Col1971]_.
.. function:: void fmpz_poly_resultant_modular_div(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t div, slong nbits)
Computes the resultant of ``poly1`` and ``poly2`` divided by
``div`` using a slight modification of the above function. It is assumed that
the resultant is exactly divisible by ``div`` and the result ``res``
has at most ``nbits`` bits.
This bypasses the computation of general bounds.
.. function:: void _fmpz_poly_resultant_euclidean(fmpz_t res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``res`` to the resultant of ``(poly1, len1)`` and
``(poly2, len2)``, assuming that ``len1 >= len2 > 0``.
.. function:: void fmpz_poly_resultant_euclidean(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Computes the resultant of ``poly1`` and ``poly2``.
For two non-zero polynomials `f(x) = a_m x^m + \dotsb + a_0` and
`g(x) = b_n x^n + \dotsb + b_0` of degrees `m` and `n`, the resultant
is defined to be
.. math::
a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).
For convenience, we define the resultant to be equal to zero if either
of the two polynomials is zero.
This function uses the algorithm described
in Algorithm 3.3.7 of [Coh1996]_.
.. function:: void _fmpz_poly_resultant(fmpz_t res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``res`` to the resultant of ``(poly1, len1)`` and
``(poly2, len2)``, assuming that ``len1 >= len2 > 0``.
.. function:: void fmpz_poly_resultant(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Computes the resultant of ``poly1`` and ``poly2``.
For two non-zero polynomials `f(x) = a_m x^m + \dotsb + a_0` and
`g(x) = b_n x^n + \dotsb + b_0` of degrees `m` and `n`, the resultant
is defined to be
.. math::
a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).
For convenience, we define the resultant to be equal to zero if either
of the two polynomials is zero.
.. function:: void _fmpz_poly_discriminant(fmpz_t res, const fmpz * poly, slong len)
Set ``res`` to the discriminant of ``(poly, len)``. Assumes
``len > 1``.
.. function:: void fmpz_poly_discriminant(fmpz_t res, const fmpz_poly_t poly)
Set ``res`` to the discriminant of ``poly``. We normalise the
discriminant so that `\operatorname{disc}(f) = (-1)^{(n(n-1)/2)}
\operatorname{res}(f, f')/\operatorname{lc}(f)`, thus
`\operatorname{disc}(f) = \operatorname{lc}(f)^{(2n - 2)} \prod_{i < j} (r_i
- r_j)^2`, where `\operatorname{lc}(f)` is the leading coefficient of `f`,
`n` is the degree of `f` and `r_i` are the roots of `f`.
.. function:: void _fmpz_poly_content(fmpz_t res, const fmpz * poly, slong len)
Sets ``res`` to the non-negative content of ``(poly, len)``.
Aliasing between ``res`` and the coefficients of ``poly`` is
not supported.
.. function:: void fmpz_poly_content(fmpz_t res, const fmpz_poly_t poly)
Sets ``res`` to the non-negative content of ``poly``. The content
of the zero polynomial is defined to be zero. Supports aliasing, that is,
``res`` is allowed to be one of the coefficients of ``poly``.
.. function:: void _fmpz_poly_primitive_part(fmpz * res, const fmpz * poly, slong len)
Sets ``(res, len)`` to ``(poly, len)`` divided by the content
of ``(poly, len)``, and normalises the result to have non-negative
leading coefficient.
Assumes that ``(poly, len)`` is non-zero. Supports aliasing of
``res`` and ``poly``.
.. function:: void fmpz_poly_primitive_part(fmpz_poly_t res, const fmpz_poly_t poly)
Sets ``res`` to ``poly`` divided by the content of ``poly``,
and normalises the result to have non-negative leading coefficient.
If ``poly`` is zero, sets ``res`` to zero.
.. function:: int _fmpz_poly_is_squarefree(const fmpz * poly, slong len)
Returns whether the polynomial ``(poly, len)`` is square-free.
.. function:: int fmpz_poly_is_squarefree(const fmpz_poly_t poly)
Returns whether the polynomial ``poly`` is square-free. A non-zero
polynomial is defined to be square-free if it has no non-unit square
factors. We also define the zero polynomial to be square-free.
Returns `1` if the length of ``poly`` is at most `2`. Returns whether
the discriminant is zero for quadratic polynomials. Otherwise, returns
whether the greatest common divisor of ``poly`` and its derivative has
length `1`.
.. function:: int _fmpz_poly_divrem_basecase(fmpz * Q, fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)
Computes ``(Q, lenA - lenB + 1)``, ``(R, lenA)`` such that
`A = B Q + R` and each coefficient of `R` beyond ``lenB`` is reduced
modulo the leading coefficient of `B`.
If the leading coefficient of `B` is `\pm 1` or the division is exact,
this is the same thing as division over `\mathbb{Q}`.
Assumes that `\operatorname{len}(A), \operatorname{len}(B) > 0`. Allows zero-padding in
``(A, lenA)``. `R` and `A` may be aliased, but apart from this no
aliasing of input and output operands is allowed.
If the flag ``exact`` is `1`, the function stops if an inexact division
is encountered, upon which the function will return `0`. If no inexact
division is encountered, the function returns `1`. Note that this does not
guarantee the remainder of the polynomial division is zero, merely that
its length is less than that of B. This feature is useful for series
division and for divisibility testing (upon testing the remainder).
For ordinary use set the flag ``exact`` to `0`. In this case, no checks
or early aborts occur and the function always returns `1`.
.. function:: void fmpz_poly_divrem_basecase(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)
Computes `Q`, `R` such that `A = B Q + R` and each coefficient of `R`
beyond `\operatorname{len}(B) - 1` is reduced modulo the leading coefficient of `B`.
If the leading coefficient of `B` is `\pm 1` or the division is exact,
this is the same thing as division over `\mathbb{Q}`. An exception is raised
if `B` is zero.
.. function:: int _fmpz_poly_divrem_divconquer_recursive(fmpz * Q, fmpz * BQ, fmpz * W, const fmpz * A, const fmpz * B, slong lenB, int exact)
Computes ``(Q, lenB)``, ``(BQ, 2 lenB - 1)`` such that
`BQ = B \times Q` and `A = B Q + R` where each coefficient of `R` beyond
`\operatorname{len}(B) - 1` is reduced modulo the leading coefficient of `B`. We
assume that `\operatorname{len}(A) = 2 \operatorname{len}(B) - 1`. If the leading coefficient
of `B` is `\pm 1` or the division is exact, this is the same as division
over `\mathbb{Q}`.
Assumes `\operatorname{len}(B) > 0`. Allows zero-padding in ``(A, lenA)``. Requires
a temporary array ``(W, 2 lenB - 1)``. No aliasing of input and output
operands is allowed.
This function does not read the bottom `\operatorname{len}(B) - 1` coefficients from
`A`, which means that they might not even need to exist in allocated
memory.
If the flag ``exact`` is `1`, the function stops if an inexact division
is encountered, upon which the function will return `0`. If no inexact
division is encountered, the function returns `1`. Note that this does not
guarantee the remainder of the polynomial division is zero, merely that
its length is less than that of B. This feature is useful for series
division and for divisibility testing (upon testing the remainder).
For ordinary use set the flag ``exact`` to `0`. In this case, no checks
or early aborts occur and the function always returns `1`.
.. function:: int _fmpz_poly_divrem_divconquer(fmpz * Q, fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)
Computes ``(Q, lenA - lenB + 1)``, ``(R, lenA)`` such that
`A = B Q + R` and each coefficient of `R` beyond `\operatorname{len}(B) - 1` is
reduced modulo the leading coefficient of `B`. If the leading
coefficient of `B` is `\pm 1` or the division is exact, this is
the same as division over `\mathbb{Q}`.
Assumes `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`. Allows zero-padding in
``(A, lenA)``. No aliasing of input and output operands is
allowed.
If the flag ``exact`` is `1`, the function stops if an inexact division
is encountered, upon which the function will return `0`. If no inexact
division is encountered, the function returns `1`. Note that this does not
guarantee the remainder of the polynomial division is zero, merely that
its length is less than that of B. This feature is useful for series
division and for divisibility testing (upon testing the remainder).
For ordinary use set the flag ``exact`` to `0`. In this case, no checks
or early aborts occur and the function always returns `1`.
.. function:: void fmpz_poly_divrem_divconquer(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)
Computes `Q`, `R` such that `A = B Q + R` and each coefficient of `R`
beyond `\operatorname{len}(B) - 1` is reduced modulo the leading coefficient of `B`.
If the leading coefficient of `B` is `\pm 1` or the division is exact,
this is the same as division over `\mathbb{Q}`. An exception is raised if `B`
is zero.
.. function:: int _fmpz_poly_divrem(fmpz * Q, fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)
Computes ``(Q, lenA - lenB + 1)``, ``(R, lenA)`` such that
`A = B Q + R` and each coefficient of `R` beyond `\operatorname{len}(B) - 1` is
reduced modulo the leading coefficient of `B`. If the leading
coefficient of `B` is `\pm 1` or the division is exact, this is
the same thing as division over `\mathbb{Q}`.
Assumes `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`. Allows zero-padding in
``(A, lenA)``. No aliasing of input and output operands is
allowed.
If the flag ``exact`` is `1`, the function stops if an inexact division
is encountered, upon which the function will return `0`. If no inexact
division is encountered, the function returns `1`. Note that this does not
guarantee the remainder of the polynomial division is zero, merely that
its length is less than that of B. This feature is useful for series
division and for divisibility testing (upon testing the remainder).
For ordinary use set the flag ``exact`` to `0`. In this case, no checks
or early aborts occur and the function always returns `1`.
.. function:: void fmpz_poly_divrem(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)
Computes `Q`, `R` such that `A = B Q + R` and each coefficient of `R`
beyond `\operatorname{len}(B) - 1` is reduced modulo the leading coefficient of `B`.
If the leading coefficient of `B` is `\pm 1` or the division is exact,
this is the same as division over `\mathbb{Q}`. An exception is raised if `B`
is zero.
.. function:: int _fmpz_poly_div_basecase(fmpz * Q, fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)
Computes the quotient ``(Q, lenA - lenB + 1)`` of ``(A, lenA)``
divided by ``(B, lenB)``.
Notationally, computes `Q`, `R` such that `A = B Q + R` and each
coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
coefficient of `B`.
If the leading coefficient of `B` is `\pm 1` or the division is exact,
this is the same as division over `\mathbb{Q}`.
Assumes `\operatorname{len}(A), \operatorname{len}(B) > 0`. Allows zero-padding in ``(A, lenA)``.
Requires a temporary array `R` of size at least the (actual) length
of `A`. For convenience, `R` may be ``NULL``. `R` and `A` may be
aliased, but apart from this no aliasing of input and output operands
is allowed.
If the flag ``exact`` is `1`, the function stops if an inexact division
is encountered, upon which the function will return `0`. If no inexact
division is encountered, the function returns `1`. Note that this does not
guarantee the remainder of the polynomial division is zero, merely that
its length is less than that of B. This feature is useful for series
division and for divisibility testing (upon testing the remainder).
For ordinary use set the flag ``exact`` to `0`. In this case, no checks
or early aborts occur and the function always returns `1`.
.. function:: void fmpz_poly_div_basecase(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)
Computes the quotient `Q` of `A` divided by `Q`.
Notationally, computes `Q`, `R` such that `A = B Q + R` and each
coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
coefficient of `B`.
If the leading coefficient of `B` is `\pm 1` or the division is exact,
this is the same as division over `\mathbb{Q}`. An exception is raised if `B`
is zero.
.. function:: int _fmpz_poly_divremlow_divconquer_recursive(fmpz * Q, fmpz * BQ, const fmpz * A, const fmpz * B, slong lenB, int exact)
Divide and conquer division of ``(A, 2 lenB - 1)`` by ``(B, lenB)``,
computing only the bottom `\operatorname{len}(B) - 1` coefficients of `B Q`.
Assumes `\operatorname{len}(B) > 0`. Requires `B Q` to have length at least
`2 \operatorname{len}(B) - 1`, although only the bottom `\operatorname{len}(B) - 1` coefficients will
carry meaningful output. Does not support any aliasing. Allows
zero-padding in `A`, but not in `B`.
If the flag ``exact`` is `1`, the function stops if an inexact division
is encountered, upon which the function will return `0`. If no inexact
division is encountered, the function returns `1`. Note that this does not
guarantee the remainder of the polynomial division is zero, merely that
its length is less than that of B. This feature is useful for series
division and for divisibility testing (upon testing the remainder).
For ordinary use set the flag ``exact`` to `0`. In this case, no checks
or early aborts occur and the function always returns `1`.
.. function:: int _fmpz_poly_div_divconquer_recursive(fmpz * Q, fmpz * temp, const fmpz * A, const fmpz * B, slong lenB, int exact)
Recursive short division in the balanced case.
Computes the quotient ``(Q, lenB)`` of ``(A, 2 lenB - 1)`` upon
division by ``(B, lenB)``. Requires `\operatorname{len}(B) > 0`. Needs a
temporary array ``temp`` of length `2 \operatorname{len}(B) - 1`. Does not support
any aliasing.
For further details, see [Mul2000]_.
If the flag ``exact`` is `1`, the function stops if an inexact division
is encountered, upon which the function will return `0`. If no inexact
division is encountered, the function returns `1`. Note that this does not
guarantee the remainder of the polynomial division is zero, merely that
its length is less than that of B. This feature is useful for series
division and for divisibility testing (upon testing the remainder).
For ordinary use set the flag ``exact`` to `0`. In this case, no checks
or early aborts occur and the function always returns `1`.
.. function:: int _fmpz_poly_div_divconquer(fmpz * Q, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)
Computes the quotient ``(Q, lenA - lenB + 1)`` of ``(A, lenA)``
upon division by ``(B, lenB)``. Assumes that
`\operatorname{len}(A) \geq \operatorname{len}(B) > 0`. Does not support aliasing.
If the flag ``exact`` is `1`, the function stops if an inexact division
is encountered, upon which the function will return `0`. If no inexact
division is encountered, the function returns `1`. Note that this does not
guarantee the remainder of the polynomial division is zero, merely that
its length is less than that of B. This feature is useful for series
division and for divisibility testing (upon testing the remainder).
For ordinary use set the flag ``exact`` to `0`. In this case, no checks
or early aborts occur and the function always returns `1`.
.. function:: void fmpz_poly_div_divconquer(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)
Computes the quotient `Q` of `A` divided by `B`.
Notationally, computes `Q`, `R` such that `A = B Q + R` and each
coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
coefficient of `B`.
If the leading coefficient of `B` is `\pm 1` or the division is exact,
this is the same as division over `\mathbb{Q}`. An exception is raised if `B`
is zero.
.. function:: int _fmpz_poly_div(fmpz * Q, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)
Computes the quotient ``(Q, lenA - lenB + 1)`` of ``(A, lenA)``
divided by ``(B, lenB)``.
Notationally, computes `Q`, `R` such that `A = B Q + R` and each
coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
coefficient of `B`. If the leading coefficient of `B` is `\pm 1` or
the division is exact, this is the same as division over `\mathbb{Q}`.
Assumes `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`. Allows zero-padding in
``(A, lenA)``. Aliasing of input and output operands is not
allowed.
If the flag ``exact`` is `1`, the function stops if an inexact division
is encountered, upon which the function will return `0`. If no inexact
division is encountered, the function returns `1`. Note that this does not
guarantee the remainder of the polynomial division is zero, merely that
its length is less than that of B. This feature is useful for series
division and for divisibility testing (upon testing the remainder).
For ordinary use set the flag ``exact`` to `0`. In this case, no checks
or early aborts occur and the function always returns `1`.
.. function:: void fmpz_poly_div(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)
Computes the quotient `Q` of `A` divided by `B`.
Notationally, computes `Q`, `R` such that `A = B Q + R` and each
coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
coefficient of `B`. If the leading coefficient of `B` is `\pm 1` or
the division is exact, this is the same as division over `Q`. An
exception is raised if `B` is zero.
.. function:: void _fmpz_poly_rem_basecase(fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB)
Computes the remainder ``(R, lenA)`` of ``(A, lenA)`` upon
division by ``(B, lenB)``.
Notationally, computes `Q`, `R` such that `A = B Q + R` and each
coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
coefficient of `B`. If the leading coefficient of `B` is `\pm 1` or
the division is exact, this is the same thing as division over `\mathbb{Q}`.
Assumes that `\operatorname{len}(A), \operatorname{len}(B) > 0`. Allows zero-padding in
``(A, lenA)``. `R` and `A` may be aliased, but apart from this no
aliasing of input and output operands is allowed.
.. function:: void fmpz_poly_rem_basecase(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)
Computes the remainder `R` of `A` upon division by `B`.
Notationally, computes `Q`, `R` such that `A = B Q + R` and each
coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
coefficient of `B`. If the leading coefficient of `B` is `\pm 1` or
the division is exact, this is the same as division over `\mathbb{Q}`. An
exception is raised if `B` is zero.
.. function:: void _fmpz_poly_rem(fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB)
Computes the remainder ``(R, lenA)`` of ``(A, lenA)`` upon division
by ``(B, lenB)``.
Notationally, computes `Q`, `R` such that `A = B Q + R` and each
coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
coefficient of `B`. If the leading coefficient of `B` is `\pm 1` or
the division is exact, this is the same thing as division over `\mathbb{Q}`.
Assumes that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`. Allows zero-padding in
``(A, lenA)``. Aliasing of input and output operands is not allowed.
.. function:: void fmpz_poly_rem(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)
Computes the remainder `R` of `A` upon division by `B`.
Notationally, computes `Q`, `R` such that `A = B Q + R` and each
coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
coefficient of `B`. If the leading coefficient of `B` is `\pm 1` or
the division is exact, this is the same as division over `\mathbb{Q}`. An
exception is raised if `B` is zero.
.. function:: void _fmpz_poly_div_root_fmpz(fmpz * Q, const fmpz * A, slong len, const fmpz_t c)
Computes the quotient ``(Q, len-1)`` of ``(A, len)`` upon
division by `x - c`.
Supports aliasing of ``Q`` and ``A``, but the result is
undefined in case of partial overlap.
.. function:: void fmpz_poly_div_root_fmpz(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_t c)
Computes the quotient ``(Q, len-1)`` of ``(A, len)`` upon
division by `x - c`.
.. function:: void _fmpz_poly_divexact(fmpz * Q, const fmpz * A, slong lenA, const fmpz * B, slong lenB)
void fmpz_poly_divexact(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)
Like :func:`fmpz_poly_div`, but assumes that the division is exact.
.. function:: void _fmpz_poly_divexact_root_fmpq(fmpz * Q, const fmpz * A, slong len, const fmpq_t c)
Computes the quotient ``(Q, len-1)`` of ``(A, len)`` upon
division by `q x - p` where `c = p/q`. Assumes the division is exact.
Supports aliasing of ``Q`` and ``A``, but the result is
undefined in case of partial overlap.
.. function:: void fmpz_poly_divexact_root_fmpq(fmpz_poly_t Q, const fmpz_poly_t A, const fmpq_t c)
Computes the quotient ``(Q, len-1)`` of ``(A, len)`` upon
division by `q x - p` where `c = p/q`. Assumes the division is exact.
.. function:: void _fmpz_poly_preinvert(fmpz * B_inv, const fmpz * B, slong n)
Given a monic polynomial ``B`` of length ``n``, compute a precomputed
inverse ``B_inv`` of length ``n`` for use in the functions below. No
aliasing of ``B`` and ``B_inv`` is permitted. We assume ``n`` is not zero.
.. function:: void fmpz_poly_preinvert(fmpz_poly_t B_inv, const fmpz_poly_t B)
Given a monic polynomial ``B``, compute a precomputed inverse
``B_inv`` for use in the functions below. An exception is raised if
``B`` is zero.
.. function:: void _fmpz_poly_div_preinv(fmpz * Q, const fmpz * A, slong len1, const fmpz * B, const fmpz * B_inv, slong len2)
Given a precomputed inverse ``B_inv`` of the polynomial ``B`` of
length ``len2``, compute the quotient ``Q`` of ``A`` by ``B``.
We assume the length ``len1`` of ``A`` is at least ``len2``. The
polynomial ``Q`` must have space for ``len1 - len2 + 1``
coefficients. No aliasing of operands is permitted.
.. function:: void fmpz_poly_div_preinv(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, const fmpz_poly_t B_inv)
Given a precomputed inverse ``B_inv`` of the polynomial ``B``,
compute the quotient ``Q`` of ``A`` by ``B``. Aliasing of ``B``
and ``B_inv`` is not permitted.
.. function:: void _fmpz_poly_divrem_preinv(fmpz * Q, fmpz * A, slong len1, const fmpz * B, const fmpz * B_inv, slong len2)
Given a precomputed inverse ``B_inv`` of the polynomial ``B`` of
length ``len2``, compute the quotient ``Q`` of ``A`` by ``B``.
The remainder is then placed in ``A``. We assume the length ``len1``
of ``A`` is at least ``len2``. The polynomial ``Q`` must have
space for ``len1 - len2 + 1`` coefficients. No aliasing of operands is
permitted.
.. function:: void fmpz_poly_divrem_preinv(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B, const fmpz_poly_t B_inv)
Given a precomputed inverse ``B_inv`` of the polynomial ``B``,
compute the quotient ``Q`` of ``A`` by ``B`` and the remainder
``R``. Aliasing of ``B`` and ``B_inv`` is not permitted.
.. function:: fmpz ** _fmpz_poly_powers_precompute(const fmpz * B, slong len)
Computes ``2*len - 1`` powers of `x` modulo the polynomial `B` of
the given length. This is used as a kind of precomputed inverse in
the remainder routine below.
.. function:: void fmpz_poly_powers_precompute(fmpz_poly_powers_precomp_t pinv, fmpz_poly_t poly)
Computes ``2*len - 1`` powers of `x` modulo the polynomial `B` of
the given length. This is used as a kind of precomputed inverse in
the remainder routine below.
.. function:: void _fmpz_poly_powers_clear(fmpz ** powers, slong len)
Clean up resources used by precomputed powers which have been computed
by ``_fmpz_poly_powers_precompute``.
.. function:: void fmpz_poly_powers_clear(fmpz_poly_powers_precomp_t pinv)
Clean up resources used by precomputed powers which have been computed
by ``fmpz_poly_powers_precompute``.
.. function:: void _fmpz_poly_rem_powers_precomp(fmpz * A, slong m, const fmpz * B, slong n, fmpz ** const powers)
Set `A` to the remainder of `A` divide `B` given precomputed powers mod `B`
provided by ``_fmpz_poly_powers_precompute``. No aliasing is allowed.
.. function:: void fmpz_poly_rem_powers_precomp(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B, const fmpz_poly_powers_precomp_t B_inv)
Set `R` to the remainder of `A` divide `B` given precomputed powers mod `B`
provided by ``fmpz_poly_powers_precompute``.
.. function:: int _fmpz_poly_divides(fmpz * Q, const fmpz * A, slong lenA, const fmpz * B, slong lenB)
Returns 1 if ``(B, lenB)`` divides ``(A, lenA)`` exactly and
sets `Q` to the quotient, otherwise returns 0.
It is assumed that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0` and that `Q` has space
for `\operatorname{len}(A) - \operatorname{len}(B) + 1` coefficients.
Aliasing of `Q` with either of the inputs is not permitted.
This function is currently unoptimised and provided for convenience
only.
.. function:: int fmpz_poly_divides(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)
Returns 1 if `B` divides `A` exactly and sets `Q` to the quotient,
otherwise returns 0.
This function is currently unoptimised and provided for convenience
only.
.. function:: slong fmpz_poly_remove(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Set ``res`` to ``poly1`` divided by the highest power of ``poly2`` that
divides it and return the power. The divisor ``poly2`` must not be zero or
`\pm 1`, otherwise an exception is raised.
.. function:: void fmpz_poly_divlow_smodp(fmpz * res, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_t p, slong n)
Compute the `n` lowest coefficients of `f` divided by `g`, assuming the
division is exact modulo `p`. The computed coefficients are reduced modulo
`p` using the symmetric remainder system. We require `f` to be at least `n`
in length. The function can handle trailing zeroes, but the low nonzero
coefficient of `g` must be coprime to `p`. This is a bespoke function used
by factoring.
.. function:: void fmpz_poly_divhigh_smodp(fmpz * res, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_t p, slong n)
Compute the `n` highest coefficients of `f` divided by `g`, assuming the
division is exact modulo `p`. The computed coefficients are reduced modulo
`p` using the symmetric remainder system. We require `f` to be as output
by ``fmpz_poly_mulhigh_n`` given polynomials `g` and a polynomial of
length `n` as inputs. The leading coefficient of `g` must be coprime to
`p`. This is a bespoke function used by factoring.
.. function:: void _fmpz_poly_inv_series_basecase(fmpz * Qinv, const fmpz * Q, slong Qlen, slong n)
Computes the first `n` terms of the inverse power series of
``(Q, lenQ)`` using a recurrence.
Assumes that `n \geq 1` and that `Q` has constant term `\pm 1`.
Does not support aliasing.
.. function:: void fmpz_poly_inv_series_basecase(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)
Computes the first `n` terms of the inverse power series of `Q`
using a recurrence, assuming that `Q` has constant term `\pm 1`
and `n \geq 1`.
.. function:: void _fmpz_poly_inv_series_newton(fmpz * Qinv, const fmpz * Q, slong Qlen, slong n)
Computes the first `n` terms of the inverse power series of
``(Q, lenQ)`` using Newton iteration.
Assumes that `n \geq 1` and that `Q` has constant term `\pm 1`.
Does not support aliasing.
.. function:: void fmpz_poly_inv_series_newton(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)
Computes the first `n` terms of the inverse power series of `Q` using
Newton iteration, assuming `Q` has constant term `\pm 1` and `n \geq 1`.
.. function:: void _fmpz_poly_inv_series(fmpz * Qinv, const fmpz * Q, slong Qlen, slong n)
Computes the first `n` terms of the inverse power series of
``(Q, lenQ)``.
Assumes that `n \geq 1` and that `Q` has constant term `\pm 1`.
Does not support aliasing.
.. function:: void fmpz_poly_inv_series(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)
Computes the first `n` terms of the inverse power series of `Q`,
assuming `Q` has constant term `\pm 1` and `n \geq 1`.
.. function:: void _fmpz_poly_div_series_basecase(fmpz * Q, const fmpz * A, slong Alen, const fmpz * B, slong Blen, slong n)
.. function:: void _fmpz_poly_div_series_divconquer(fmpz * Q, const fmpz * A, slong Alen, const fmpz * B, slong Blen, slong n)
.. function:: void _fmpz_poly_div_series(fmpz * Q, const fmpz * A, slong Alen, const fmpz * B, slong Blen, slong n)
Divides ``(A, Alen)`` by ``(B, Blen)`` as power series over `\mathbb{Z}`,
assuming `B` has constant term `\pm 1` and `n \geq 1`.
Aliasing is not supported.
.. function:: void fmpz_poly_div_series_basecase(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, slong n)
.. function:: void fmpz_poly_div_series_divconquer(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, slong n)
.. function:: void fmpz_poly_div_series(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, slong n)
Performs power series division in `\mathbb{Z}[[x]] / (x^n)`. The function
considers the polynomials `A` and `B` as power series of length `n`
starting with the constant terms. The function assumes that `B` has
constant term `\pm 1` and `n \geq 1`.
.. function:: void _fmpz_poly_pseudo_divrem_basecase(fmpz * Q, fmpz * R, ulong * d, const fmpz * A, slong lenA, const fmpz * B, slong lenB, const fmpz_preinvn_t inv)
If `\ell` is the leading coefficient of `B`, then computes `Q`, `R` such
that `\ell^d A = Q B + R`. This function is used for simulating division
over `\mathbb{Q}`.
Assumes that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`. Assumes that `Q` can fit
`\operatorname{len}(A) - \operatorname{len}(B) + 1` coefficients, and that `R` can fit `\operatorname{len}(A)`
coefficients. Supports aliasing of ``(R, lenA)`` and ``(A, lenA)``.
But other than this, no aliasing of the inputs and outputs is supported.
An optional precomputed inverse of the leading coefficient of `B` from
``fmpz_preinvn_init`` can be supplied. Otherwise ``inv`` should be
``NULL``.
Note: ``fmpz.h`` has to be included before ``fmpz_poly.h`` in order for
``fmpz_poly.h`` to declare this function.
.. function:: void fmpz_poly_pseudo_divrem_basecase(fmpz_poly_t Q, fmpz_poly_t R, ulong * d, const fmpz_poly_t A, const fmpz_poly_t B)
If `\ell` is the leading coefficient of `B`, then computes `Q`, `R` such
that `\ell^d A = Q B + R`. This function is used for simulating division
over `\mathbb{Q}`.
.. function:: void _fmpz_poly_pseudo_divrem_divconquer(fmpz * Q, fmpz * R, ulong * d, const fmpz * A, slong lenA, const fmpz * B, slong lenB, const fmpz_preinvn_t inv)
Computes ``(Q, lenA - lenB + 1)``, ``(R, lenA)`` such that
`\ell^d A = B Q + R`, only setting the bottom `\operatorname{len}(B) - 1` coefficients
of `R` to their correct values. The remaining top coefficients of
``(R, lenA)`` may be arbitrary.
Assumes `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`. Allows zero-padding in
``(A, lenA)``. No aliasing of input and output operands is allowed.
An optional precomputed inverse of the leading coefficient of `B` from
``fmpz_preinvn_init`` can be supplied. Otherwise ``inv`` should be
``NULL``.
Note: ``fmpz.h`` has to be included before ``fmpz_poly.h`` in order for
``fmpz_poly.h`` to declare this function.
.. function:: void fmpz_poly_pseudo_divrem_divconquer(fmpz_poly_t Q, fmpz_poly_t R, ulong * d, const fmpz_poly_t A, const fmpz_poly_t B)
Computes `Q`, `R`, and `d` such that `\ell^d A = B Q + R`, where `R` has
length less than the length of `B` and `\ell` is the leading coefficient
of `B`. An exception is raised if `B` is zero.
.. function:: void _fmpz_poly_pseudo_divrem_cohen(fmpz * Q, fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB)
Assumes that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`. Assumes that `Q` can fit
`\operatorname{len}(A) - \operatorname{len}(B) + 1` coefficients, and that `R` can fit `\operatorname{len}(A)`
coefficients. Supports aliasing of ``(R, lenA)`` and ``(A, lenA)``.
But other than this, no aliasing of the inputs and outputs is supported.
.. function:: void fmpz_poly_pseudo_divrem_cohen(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)
This is a variant of ``fmpz_poly_pseudo_divrem`` which computes
polynomials `Q` and `R` such that `\ell^d A = B Q + R`. However, the
value of `d` is fixed at `\max{\{0, \operatorname{len}(A) - \operatorname{len}(B) + 1\}}`.
This function is faster when the remainder is not well behaved, i.e.
where it is not expected to be close to zero. Note that this function
is not asymptotically fast. It is efficient only for short polynomials,
e.g. when `\operatorname{len}(B) < 32`.
.. function:: void _fmpz_poly_pseudo_rem_cohen(fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB)
Assumes that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`. Assumes that `R` can fit
`\operatorname{len}(A)` coefficients. Supports aliasing of ``(R, lenA)`` and
``(A, lenA)``. But other than this, no aliasing of the inputs and
outputs is supported.
.. function:: void fmpz_poly_pseudo_rem_cohen(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)
This is a variant of :func:`fmpz_poly_pseudo_rem` which computes
polynomials `Q` and `R` such that `\ell^d A = B Q + R`, but only
returns `R`. However, the value of `d` is fixed at
`\max{\{0, \operatorname{len}(A) - \operatorname{len}(B) + 1\}}`.
This function is faster when the remainder is not well behaved, i.e.
where it is not expected to be close to zero. Note that this function
is not asymptotically fast. It is efficient only for short polynomials,
e.g. when `\operatorname{len}(B) < 32`.
This function uses the algorithm described
in Algorithm 3.1.2 of [Coh1996]_.
.. function:: void _fmpz_poly_pseudo_divrem(fmpz * Q, fmpz * R, ulong * d, const fmpz * A, slong lenA, const fmpz * B, slong lenB, const fmpz_preinvn_t inv)
If `\ell` is the leading coefficient of `B`, then computes
``(Q, lenA - lenB + 1)``, ``(R, lenB - 1)`` and `d` such that
`\ell^d A = B Q + R`. This function is used for simulating division
over `\mathbb{Q}`.
Assumes that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`. Assumes that `Q` can fit
`\operatorname{len}(A) - \operatorname{len}(B) + 1` coefficients, and that `R` can fit `\operatorname{len}(A)`
coefficients, although on exit only the bottom `\operatorname{len}(B)` coefficients
will carry meaningful data.
Supports aliasing of ``(R, lenA)`` and ``(A, lenA)``. But other
than this, no aliasing of the inputs and outputs is supported.
An optional precomputed inverse of the leading coefficient of `B` from
``fmpz_preinvn_init`` can be supplied. Otherwise ``inv`` should be
``NULL``.
Note: ``fmpz.h`` has to be included before ``fmpz_poly.h`` in order for
``fmpz_poly.h`` to declare this function.
.. function:: void fmpz_poly_pseudo_divrem(fmpz_poly_t Q, fmpz_poly_t R, ulong * d, const fmpz_poly_t A, const fmpz_poly_t B)
Computes `Q`, `R`, and `d` such that `\ell^d A = B Q + R`.
.. function:: void _fmpz_poly_pseudo_div(fmpz * Q, ulong * d, const fmpz * A, slong lenA, const fmpz * B, slong lenB, const fmpz_preinvn_t inv)
Pseudo-division, only returning the quotient.
Note: ``fmpz.h`` has to be included before ``fmpz_poly.h`` in order for
``fmpz_poly.h`` to declare this function.
.. function:: void fmpz_poly_pseudo_div(fmpz_poly_t Q, ulong * d, const fmpz_poly_t A, const fmpz_poly_t B)
Pseudo-division, only returning the quotient.
.. function:: void _fmpz_poly_pseudo_rem(fmpz * R, ulong * d, const fmpz * A, slong lenA, const fmpz * B, slong lenB, const fmpz_preinvn_t inv)
Pseudo-division, only returning the remainder.
Note: ``fmpz.h`` has to be included before ``fmpz_poly.h`` in order for
``fmpz_poly.h`` to declare this function.
.. function:: void fmpz_poly_pseudo_rem(fmpz_poly_t R, ulong * d, const fmpz_poly_t A, const fmpz_poly_t B)
Pseudo-division, only returning the remainder.
.. function:: void _fmpz_poly_derivative(fmpz * rpoly, const fmpz * poly, slong len)
Sets ``(rpoly, len - 1)`` to the derivative of ``(poly, len)``.
Also handles the cases where ``len`` is `0` or `1` correctly.
Supports aliasing of ``rpoly`` and ``poly``.
.. function:: void fmpz_poly_derivative(fmpz_poly_t res, const fmpz_poly_t poly)
Sets ``res`` to the derivative of ``poly``.
.. function:: void _fmpz_poly_nth_derivative(fmpz * rpoly, const fmpz * poly, ulong n, slong len)
Sets ``(rpoly, len - n)`` to the nth derivative of ``(poly, len)``.
Also handles the cases where ``len <= n`` correctly.
Supports aliasing of ``rpoly`` and ``poly``.
.. function:: void fmpz_poly_nth_derivative(fmpz_poly_t res, const fmpz_poly_t poly, ulong n)
Sets ``res`` to the nth derivative of ``poly``.
.. function:: void _fmpz_poly_evaluate_divconquer_fmpz(fmpz_t res, const fmpz * poly, slong len, const fmpz_t a)
Evaluates the polynomial ``(poly, len)`` at the integer `a` using
a divide and conquer approach. Assumes that the length of the polynomial
is at least one. Allows zero padding. Does not allow aliasing between
``res`` and ``x``.
.. function:: void fmpz_poly_evaluate_divconquer_fmpz(fmpz_t res, const fmpz_poly_t poly, const fmpz_t a)
Evaluates the polynomial ``poly`` at the integer `a` using a divide
and conquer approach.
Aliasing between ``res`` and ``a`` is supported, however,
``res`` may not be part of ``poly``.
.. function:: void _fmpz_poly_evaluate_horner_fmpz(fmpz_t res, const fmpz * f, slong len, const fmpz_t a)
Evaluates the polynomial ``(f, len)`` at the integer `a` using
Horner's rule, and sets ``res`` to the result. Aliasing between
``res`` and `a` or any of the coefficients of `f` is not supported.
.. function:: void fmpz_poly_evaluate_horner_fmpz(fmpz_t res, const fmpz_poly_t f, const fmpz_t a)
Evaluates the polynomial `f` at the integer `a` using Horner's rule, and
sets ``res`` to the result.
As expected, aliasing between ``res`` and ``a`` is supported.
However, ``res`` may not be aliased with a coefficient of `f`.
.. function:: void _fmpz_poly_evaluate_fmpz(fmpz_t res, const fmpz * f, slong len, const fmpz_t a)
Evaluates the polynomial ``(f, len)`` at the integer `a` and sets
``res`` to the result. Aliasing between ``res`` and `a` or any
of the coefficients of `f` is not supported.
.. function:: void fmpz_poly_evaluate_fmpz(fmpz_t res, const fmpz_poly_t f, const fmpz_t a)
Evaluates the polynomial `f` at the integer `a` and sets ``res``
to the result.
As expected, aliasing between ``res`` and `a` is supported. However,
``res`` may not be aliased with a coefficient of `f`.
.. function:: void _fmpz_poly_evaluate_divconquer_fmpq(fmpz_t rnum, fmpz_t rden, const fmpz * f, slong len, const fmpz_t anum, const fmpz_t aden)
Evaluates the polynomial ``(f, len)`` at the rational
``(anum, aden)`` using a divide and conquer approach, and sets
``(rnum, rden)`` to the result in lowest terms. Assumes that
the length of the polynomial is at least one.
Aliasing between ``(rnum, rden)`` and ``(anum, aden)`` or any of
the coefficients of `f` is not supported.
.. function:: void fmpz_poly_evaluate_divconquer_fmpq(fmpq_t res, const fmpz_poly_t f, const fmpq_t a)
Evaluates the polynomial `f` at the rational `a` using a divide
and conquer approach, and sets ``res`` to the result.
.. function:: void _fmpz_poly_evaluate_horner_fmpq(fmpz_t rnum, fmpz_t rden, const fmpz * f, slong len, const fmpz_t anum, const fmpz_t aden)
Evaluates the polynomial ``(f, len)`` at the rational
``(anum, aden)`` using Horner's rule, and sets ``(rnum, rden)`` to
the result in lowest terms.
Aliasing between ``(rnum, rden)`` and ``(anum, aden)`` or any of
the coefficients of `f` is not supported.
.. function:: void fmpz_poly_evaluate_horner_fmpq(fmpq_t res, const fmpz_poly_t f, const fmpq_t a)
Evaluates the polynomial `f` at the rational `a` using Horner's rule, and
sets ``res`` to the result.
.. function:: void _fmpz_poly_evaluate_fmpq(fmpz_t rnum, fmpz_t rden, const fmpz * f, slong len, const fmpz_t anum, const fmpz_t aden)
Evaluates the polynomial ``(f, len)`` at the rational
``(anum, aden)`` and sets ``(rnum, rden)`` to the result in lowest
terms.
Aliasing between ``(rnum, rden)`` and ``(anum, aden)`` or any of
the coefficients of `f` is not supported.
.. function:: void fmpz_poly_evaluate_fmpq(fmpq_t res, const fmpz_poly_t f, const fmpq_t a)
Evaluates the polynomial `f` at the rational `a`, and
sets ``res`` to the result.
.. function:: ulong _fmpz_poly_evaluate_mod(const fmpz * poly, slong len, ulong a, ulong n, ulong ninv)
Evaluates ``(poly, len)`` at the value `a` modulo `n` and
returns the result. The last argument ``ninv`` must be set
to the precomputed inverse of `n`, which can be obtained using
the function :func:`n_preinvert_limb`.
.. function:: ulong fmpz_poly_evaluate_mod(const fmpz_poly_t poly, ulong a, ulong n)
Evaluates ``poly`` at the value `a` modulo `n` and returns the result.
.. function:: void fmpz_poly_evaluate_fmpz_vec(fmpz * res, const fmpz_poly_t f, const fmpz * a, slong n)
Evaluates ``f`` at the `n` values given in the vector ``f``,
writing the results to ``res``.
.. function:: double _fmpz_poly_evaluate_horner_d(const fmpz * poly, slong n, double d)
Evaluate ``(poly, n)`` at the double `d`. No attempt is made to do this
efficiently or in a numerically stable way. It is currently only used in
FLINT for quick and dirty evaluations of polynomials with all coefficients
positive.
.. function:: double fmpz_poly_evaluate_horner_d(const fmpz_poly_t poly, double d)
Evaluate ``poly`` at the double `d`. No attempt is made to do this
efficiently or in a numerically stable way. It is currently only used in
FLINT for quick and dirty evaluations of polynomials with all coefficients
positive.
.. function:: double _fmpz_poly_evaluate_horner_d_2exp(slong * exp, const fmpz * poly, slong n, double d)
Evaluate ``(poly, n)`` at the double `d`. Return the result as a double
and an exponent ``exp`` combination. No attempt is made to do this
efficiently or in a numerically stable way. It is currently only used in
FLINT for quick and dirty evaluations of polynomials with all coefficients
positive.
.. function:: double fmpz_poly_evaluate_horner_d_2exp(slong * exp, const fmpz_poly_t poly, double d)
Evaluate ``poly`` at the double `d`. Return the result as a double
and an exponent ``exp`` combination. No attempt is made to do this
efficiently or in a numerically stable way. It is currently only used in
FLINT for quick and dirty evaluations of polynomials with all coefficients
positive.
.. function:: double _fmpz_poly_evaluate_horner_d_2exp2(slong * exp, const fmpz * poly, slong n, double d, slong dexp)
Evaluate ``poly`` at ``d*2^dexp``. Return the result as a double
and an exponent ``exp`` combination. No attempt is made to do this
efficiently or in a numerically stable way. It is currently only used in
FLINT for quick and dirty evaluations of polynomials with all coefficients
positive.
.. function:: void _fmpz_poly_monomial_to_newton(fmpz * poly, const fmpz * roots, slong n)
Converts ``(poly, n)`` in-place from its coefficients given
in the standard monomial basis to the Newton basis
for the roots `r_0, r_1, \ldots, r_{n-2}`.
In other words, this determines output coefficients `c_i` such that
`c_0 + c_1(x-r_0) + c_2(x-r_0)(x-r_1) + \ldots + c_{n-1}(x-r_0)(x-r_1)\cdots(x-r_{n-2})`
is equal to the input polynomial.
Uses repeated polynomial division.
.. function:: void _fmpz_poly_newton_to_monomial(fmpz * poly, const fmpz * roots, slong n)
Converts ``(poly, n)`` in-place from its coefficients given
in the Newton basis for the roots `r_0, r_1, \ldots, r_{n-2}`
to the standard monomial basis. In other words, this evaluates
`c_0 + c_1(x-r_0) + c_2(x-r_0)(x-r_1) + \ldots + c_{n-1}(x-r_0)(x-r_1)\cdots(x-r_{n-2})`
where `c_i` are the input coefficients for ``poly``.
Uses Horner's rule.
.. function:: int _fmpz_poly_interpolate_newton(fmpz * poly, const fmpz * xs, const fmpz * ys, slong n)
int fmpz_poly_interpolate_newton(fmpz_poly_t poly, const fmpz * xs, const fmpz * ys, slong n)
int _fmpz_poly_interpolate_multi_mod(fmpz * poly, const fmpz * xs, const fmpz * ys, slong n)
int fmpz_poly_interpolate_multi_mod(fmpz_poly_t poly, const fmpz * xs, const fmpz * ys, slong n)
int _fmpz_poly_interpolate(fmpz * poly, const fmpz * xs, const fmpz * ys, slong n)
int fmpz_poly_interpolate(fmpz_poly_t poly, const fmpz * xs, const fmpz * ys, slong n)
Sets ``poly`` to the unique interpolating polynomial of
degree at most `n - 1` satisfying `f(x_i) = y_i` for every pair `x_i, y_i` in
``xs`` and ``ys``, assuming that this polynomial has integer
coefficients and that the evaluation points `x_i` are distinct.
If the `x_i` are not distinct or if the interpolating polynomial does
not have integer coefficients, returns 0 indicating failure.
Otherwise returns 1 indicating success.
The *newton* algorithm first interpolates in the Newton basis
and then converts to the monomial basis.
The *multi_mod* algorithm interpolates modulo one or more prime numbers
and combines the results using CRT, using adaptive termination.
The default implementation chooses automatically between the *newton* and
*multi_mod* algorithms.
.. function:: void _fmpz_poly_interpolate_exact_newton(fmpz * poly, const fmpz * xs, const fmpz * ys, slong n)
void fmpz_poly_interpolate_exact_newton(fmpz_poly_t poly, const fmpz * xs, const fmpz * ys, slong n)
void _fmpz_poly_interpolate_exact(fmpz * poly, const fmpz * xs, const fmpz * ys, slong n)
void fmpz_poly_interpolate_exact(fmpz_poly_t poly, const fmpz * xs, const fmpz * ys, slong n)
Like :func:`fmpz_poly_interpolate` etc., but optimized by omitting
error handling. If the `x_i` are not distinct or if the interpolating
polynomial does not have integer coefficients, the behavior is undefined.
The default implementation chooses automatically between the *exact_newton* and
*multi_mod* algorithms.
.. function:: void fmpz_poly_interpolate_fmpz_vec(fmpz_poly_t poly, const fmpz * xs, const fmpz * ys, slong n)
Like :func:`fmpz_poly_interpolate`, but if the `x_i` values are not distinct
or if the interpolating polynomial does not have integer coefficients,
a ``FLINT_INEXACT`` exception is thrown.
.. function:: void _fmpz_poly_compose_horner(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``res`` to the composition of ``(poly1, len1)`` and
``(poly2, len2)``.
Assumes that ``res`` has space for ``(len1-1)*(len2-1) + 1``
coefficients. Assumes that ``poly1`` and ``poly2`` are non-zero
polynomials. Does not support aliasing between any of the inputs and
the output.
.. function:: void fmpz_poly_compose_horner(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the composition of ``poly1`` and ``poly2``.
To be more precise, denoting ``res``, ``poly1``, and ``poly2``
by `f`, `g`, and `h`, sets `f(t) = g(h(t))`.
This implementation uses Horner's method.
.. function:: void _fmpz_poly_compose_divconquer(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Computes the composition of ``(poly1, len1)`` and ``(poly2, len2)``
using a divide and conquer approach and places the result into ``res``,
assuming ``res`` can hold the output of length
``(len1 - 1) * (len2 - 1) + 1``.
Assumes ``len1, len2 > 0``. Does not support aliasing between
``res`` and any of ``(poly1, len1)`` and ``(poly2, len2)``.
.. function:: void fmpz_poly_compose_divconquer(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the composition of ``poly1`` and ``poly2``.
To be precise about the order of composition, denoting ``res``,
``poly1``, and ``poly2`` by `f`, `g`, and `h`, respectively,
sets `f(t) = g(h(t))`.
.. function:: void _fmpz_poly_compose(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)
Sets ``res`` to the composition of ``(poly1, len1)`` and
``(poly2, len2)``.
Assumes that ``res`` has space for ``(len1-1)*(len2-1) + 1``
coefficients. Assumes that ``poly1`` and ``poly2`` are non-zero
polynomials. Does not support aliasing between any of the inputs and
the output.
.. function:: void fmpz_poly_compose(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)
Sets ``res`` to the composition of ``poly1`` and ``poly2``.
To be precise about the order of composition, denoting ``res``,
``poly1``, and ``poly2`` by `f`, `g`, and `h`, respectively,
sets `f(t) = g(h(t))`.
.. function:: void fmpz_poly_inflate(fmpz_poly_t result, const fmpz_poly_t input, ulong inflation)
Sets ``result`` to the inflated polynomial `p(x^n)` where
`p` is given by ``input`` and `n` is given by ``inflation``.
.. function:: void fmpz_poly_deflate(fmpz_poly_t result, const fmpz_poly_t input, ulong deflation)
Sets ``result`` to the deflated polynomial `p(x^{1/n})` where
`p` is given by ``input`` and `n` is given by ``deflation``.
Requires `n > 0`.
.. function:: ulong fmpz_poly_deflation(const fmpz_poly_t input)
Returns the largest integer by which ``input`` can be deflated.
As special cases, returns 0 if ``input`` is the zero polynomial
and 1 if ``input`` is a constant polynomial.
.. function:: void _fmpz_poly_taylor_shift_horner(fmpz * poly, const fmpz_t c, slong n)
Performs the Taylor shift composing ``poly`` by `x+c` in-place.
Uses an efficient version Horner's rule.
.. function:: void fmpz_poly_taylor_shift_horner(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)
Performs the Taylor shift composing ``f`` by `x+c`.
.. function:: void _fmpz_poly_taylor_shift_divconquer(fmpz * poly, const fmpz_t c, slong n)
Performs the Taylor shift composing ``poly`` by `x+c` in-place.
Uses the divide-and-conquer polynomial composition algorithm.
.. function:: void fmpz_poly_taylor_shift_divconquer(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)
Performs the Taylor shift composing ``f`` by `x+c`.
Uses the divide-and-conquer polynomial composition algorithm.
.. function:: void _fmpz_poly_taylor_shift_multi_mod(fmpz * poly, const fmpz_t c, slong n)
Performs the Taylor shift composing ``poly`` by `x+c` in-place.
Uses a multimodular algorithm, distributing the computation
across :func:`flint_get_num_threads` threads.
.. function:: void fmpz_poly_taylor_shift_multi_mod(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)
Performs the Taylor shift composing ``f`` by `x+c`.
Uses a multimodular algorithm, distributing the computation
across :func:`flint_get_num_threads` threads.
.. function:: void _fmpz_poly_taylor_shift(fmpz * poly, const fmpz_t c, slong n)
Performs the Taylor shift composing ``poly`` by `x+c` in-place.
.. function:: void fmpz_poly_taylor_shift(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)
Performs the Taylor shift composing ``f`` by `x+c`.
.. function:: void _fmpz_poly_compose_series_horner(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)
Sets ``res`` to the composition of ``poly1`` and ``poly2``
modulo `x^n`, where the constant term of ``poly2`` is required
to be zero.
Assumes that ``len1, len2, n > 0``, that ``len1, len2 <= n``,
and that ``(len1-1) * (len2-1) + 1 <= n``, and that ``res`` has
space for ``n`` coefficients. Does not support aliasing between any
of the inputs and the output.
This implementation uses the Horner scheme.
.. function:: void fmpz_poly_compose_series_horner(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Sets ``res`` to the composition of ``poly1`` and ``poly2``
modulo `x^n`, where the constant term of ``poly2`` is required
to be zero.
This implementation uses the Horner scheme.
.. function:: void _fmpz_poly_compose_series_brent_kung(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)
Sets ``res`` to the composition of ``poly1`` and ``poly2``
modulo `x^n`, where the constant term of ``poly2`` is required
to be zero.
Assumes that ``len1, len2, n > 0``, that ``len1, len2 <= n``,
and that ``(len1-1) * (len2-1) + 1 <= n``, and that ``res`` has
space for ``n`` coefficients. Does not support aliasing between any
of the inputs and the output.
This implementation uses Brent-Kung algorithm 2.1 [BrentKung1978]_.
.. function:: void fmpz_poly_compose_series_brent_kung(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Sets ``res`` to the composition of ``poly1`` and ``poly2``
modulo `x^n`, where the constant term of ``poly2`` is required
to be zero.
This implementation uses Brent-Kung algorithm 2.1 [BrentKung1978]_.
.. function:: void _fmpz_poly_compose_series(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)
Sets ``res`` to the composition of ``poly1`` and ``poly2``
modulo `x^n`, where the constant term of ``poly2`` is required
to be zero.
Assumes that ``len1, len2, n > 0``, that ``len1, len2 <= n``,
and that ``(len1-1) * (len2-1) + 1 <= n``, and that ``res`` has
space for ``n`` coefficients. Does not support aliasing between any
of the inputs and the output.
This implementation automatically switches between the Horner scheme
and Brent-Kung algorithm 2.1 depending on the size of the inputs.
.. function:: void fmpz_poly_compose_series(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)
Sets ``res`` to the composition of ``poly1`` and ``poly2``
modulo `x^n`, where the constant term of ``poly2`` is required
to be zero.
This implementation automatically switches between the Horner scheme
and Brent-Kung algorithm 2.1 depending on the size of the inputs.
.. function:: void _fmpz_poly_revert_series(fmpz * Qinv, const fmpz * Q, slong Qlen, slong n)
void fmpz_poly_revert_series(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)
Sets ``Qinv`` to the compositional inverse or reversion of ``Q``
as a power series, i.e. computes `Q^{-1}` such that
`Q(Q^{-1}(x)) = Q^{-1}(Q(x)) = x \bmod x^n`.
It is required that `Q_0 = 0` and `Q_1 = \pm 1`.
Wraps :func:`_gr_poly_revert_series` which chooses automatically
between various algorithms.
.. function:: int _fmpz_poly_sqrtrem_classical(fmpz * res, fmpz * r, const fmpz * poly, slong len)
Returns 1 if ``(poly, len)`` can be written in the form `A^2 + R` where
deg(`R`) < deg(``poly``), otherwise returns `0`. If it can be so
written, ``(res, m - 1)`` is set to `A` and ``(res, m)`` is set to
`R`, where `m = \deg(\mathtt{poly})/2 + 1`.
For efficiency reasons, ``r`` must have room for ``len``
coefficients, and may alias ``poly``.
.. function:: int fmpz_poly_sqrtrem_classical(fmpz_poly_t b, fmpz_poly_t r, const fmpz_poly_t a)
If `a` can be written as `b^2 + r` with `\deg(r) < \deg(a)/2`, return
`1` and set `b` and `r` appropriately. Otherwise return `0`.
.. function:: int _fmpz_poly_sqrtrem_divconquer(fmpz * res, fmpz * r, const fmpz * poly, slong len, fmpz * temp)
Returns 1 if ``(poly, len)`` can be written in the form `A^2 + R` where
deg(`R`) < deg(``poly``), otherwise returns `0`. If it can be so
written, ``(res, m - 1)`` is set to `A` and ``(res, m)`` is set to
`R`, where `m = \deg(\mathtt{poly})/2 + 1`.
For efficiency reasons, ``r`` must have room for ``len``
coefficients, and may alias ``poly``. Temporary space of ``len``
coefficients is required.
.. function:: int fmpz_poly_sqrtrem_divconquer(fmpz_poly_t b, fmpz_poly_t r, const fmpz_poly_t a)
If `a` can be written as `b^2 + r` with `\deg(r) < \deg(a)/2`, return
`1` and set `b` and `r` appropriately. Otherwise return `0`.
.. function:: int _fmpz_poly_sqrt_classical(fmpz * res, const fmpz * poly, slong len, int exact)
If ``exact`` is `1` and ``(poly, len)`` is a perfect square, sets
``(res, len / 2 + 1)`` to the square root of ``poly`` with positive
leading coefficient and returns 1. Otherwise returns 0.
If ``exact`` is `0`, allows a remainder after the square root, which is
not computed.
This function first uses various tests to detect nonsquares quickly.
Then, it computes the square root iteratively from top to bottom,
requiring `O(n^2)` coefficient operations.
.. function:: int fmpz_poly_sqrt_classical(fmpz_poly_t b, const fmpz_poly_t a)
If ``a`` is a perfect square, sets ``b`` to the square root of
``a`` with positive leading coefficient and returns 1.
Otherwise returns 0.
.. function:: int _fmpz_poly_sqrt_KS(fmpz * res, const fmpz * poly, slong len)
Heuristic square root. If the return value is `-1`, the function failed,
otherwise it succeeded and the following applies.
If ``(poly, len)`` is a perfect square, sets
``(res, len / 2 + 1)`` to the square root of ``poly`` with positive
leading coefficient and returns 1. Otherwise returns 0.
This function first uses various tests to detect nonsquares quickly.
Then, it computes the square root iteratively from top to bottom.
.. function:: int fmpz_poly_sqrt_KS(fmpz_poly_t b, const fmpz_poly_t a)
Heuristic square root. If the return value is `-1`, the function failed,
otherwise it succeeded and the following applies.
If ``a`` is a perfect square, sets ``b`` to the square root of
``a`` with positive leading coefficient and returns 1.
Otherwise returns 0.
.. function:: int _fmpz_poly_sqrt_divconquer(fmpz * res, const fmpz * poly, slong len, int exact)
If ``exact`` is `1` and ``(poly, len)`` is a perfect square, sets
``(res, len / 2 + 1)`` to the square root of ``poly`` with positive
leading coefficient and returns 1. Otherwise returns 0.
If ``exact`` is `0`, allows a remainder after the square root, which is
not computed.
This function first uses various tests to detect nonsquares quickly.
Then, it computes the square root iteratively from top to bottom.
.. function:: int fmpz_poly_sqrt_divconquer(fmpz_poly_t b, const fmpz_poly_t a)
If ``a`` is a perfect square, sets ``b`` to the square root of
``a`` with positive leading coefficient and returns 1.
Otherwise returns 0.
.. function:: int _fmpz_poly_sqrt(fmpz * res, const fmpz * poly, slong len)
If ``(poly, len)`` is a perfect square, sets ``(res, len / 2 + 1)``
to the square root of ``poly`` with positive leading coefficient
and returns 1. Otherwise returns 0.
.. function:: int fmpz_poly_sqrt(fmpz_poly_t b, const fmpz_poly_t a)
If ``a`` is a perfect square, sets ``b`` to the square root of
``a`` with positive leading coefficient and returns 1.
Otherwise returns 0.
.. function:: int _fmpz_poly_sqrt_series(fmpz * res, const fmpz * poly, slong len, slong n)
Set ``(res, n)`` to the square root of the series ``(poly, n)``, if it
exists, and return `1`, otherwise, return `0`.
If the valuation of ``poly`` is not zero, ``res`` is zero padded
to make up for the fact that the square root may not be known to precision
`n`.
.. function:: int fmpz_poly_sqrt_series(fmpz_poly_t b, const fmpz_poly_t a, slong n)
Set ``b`` to the square root of the series ``a``, where the latter
is taken to be a series of precision `n`. If such a square root exists,
return `1`, otherwise, return `0`.
Note that if the valuation of ``a`` is not zero, ``b`` will
not have precision ``n``. It is given only to the precision to which
the square root can be computed.
.. function:: void _fmpz_poly_power_sums_naive(fmpz * res, const fmpz * poly, slong len, slong n)
Compute the (truncated) power sums series of the monic polynomial
``(poly,len)`` up to length `n` using Newton identities.
.. function:: void fmpz_poly_power_sums_naive(fmpz_poly_t res, const fmpz_poly_t poly, slong n)
Compute the (truncated) power sum series of the monic polynomial
``poly`` up to length `n` using Newton identities.
.. function:: void fmpz_poly_power_sums(fmpz_poly_t res, const fmpz_poly_t poly, slong n)
Compute the (truncated) power sums series of the monic polynomial ``poly``
up to length `n`. That is the power series whose coefficient of degree `i` is
the sum of the `i`-th power of all (complex) roots of the polynomial
``poly``.
.. function:: void _fmpz_poly_power_sums_to_poly(fmpz * res, const fmpz * poly, slong len)
Compute the (monic) polynomial given by its power sums series ``(poly,len)``.
.. function:: void fmpz_poly_power_sums_to_poly(fmpz_poly_t res, const fmpz_poly_t Q)
Compute the (monic) polynomial given its power sums series ``(Q)``.
.. function:: void _fmpz_poly_signature(slong * r1, slong * r2, const fmpz * poly, slong len)
Computes the signature `(r_1, r_2)` of the polynomial
``(poly, len)``. Assumes that the polynomial is squarefree over `\mathbb{Q}`.
.. function:: void fmpz_poly_signature(slong * r1, slong * r2, const fmpz_poly_t poly)
Computes the signature `(r_1, r_2)` of the polynomial ``poly``,
which is assumed to be square-free over `\mathbb{Q}`. The values of `r_1` and
`2 r_2` are the number of real and complex roots of the polynomial,
respectively. For convenience, the zero polynomial is allowed, in which
case the output is `(0, 0)`.
If the polynomial is not square-free, the behaviour is undefined and an
exception may be raised.
This function uses the algorithm described
in Algorithm 4.1.11 of [Coh1996]_.
.. function:: void fmpz_poly_hensel_build_tree(slong * link, fmpz_poly_t * v, fmpz_poly_t * w, const nmod_poly_factor_t fac)
Initialises and builds a Hensel tree consisting of two arrays `v`, `w`
of polynomials and an array of links, called ``link``.
The caller supplies a set of `r` local factors (in the factor structure
``fac``) of some polynomial `F` over `\mathbf{Z}`. They also supply
two arrays of initialised polynomials `v` and `w`, each of length
`2r - 2` and an array ``link``, also of length `2r - 2`.
We will have five arrays: a `v` of ``fmpz_poly_t``'s and a `V` of
``nmod_poly_t``'s and also a `w` and a `W` and ``link``. Here's
the idea: we sort each leaf and node of a factor tree by degree, in
fact choosing to multiply the two smallest factors, then the next two
smallest (factors or products) etc. until a tree is made. The tree
will be stored in the `v`'s. The first two elements of `v` will be the
smallest modular factors, the last two elements of `v` will multiply to
form `F` itself. Since `v` will be rearranging the original factors we
will need to be able to recover the original order. For this we use the
array ``link`` which has nonnegative even numbers and negative numbers.
It is an array of ``slong``\s which aligns with `V` and `v` if
``link`` has a negative number in spot `j` that means `V_j` is an
original modular factor which has been lifted, if ``link[j]`` is a
nonnegative even number then `V_j` stores a product of the two entries
at ``V[link[j]]`` and ``V[link[j]+1]``.
`W` and `w` play the role of the extended GCD, at `V_0`, `V_2`, `V_4`,
etc. we have a new product, `W_0`, `W_2`, `W_4`, etc. are the XGCD
cofactors of the `V`'s. For example,
`V_0 W_0 + V_1 W_1 \equiv 1 \pmod{p^{\ell}}` for some `\ell`. These
will be lifted along with the entries in `V`. It is not enough to just
lift each factor, we have to lift the entire tree and the tree of
XGCD cofactors.
.. function:: void fmpz_poly_hensel_lift(fmpz_poly_t G, fmpz_poly_t H, fmpz_poly_t A, fmpz_poly_t B, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_poly_t h, const fmpz_poly_t a, const fmpz_poly_t b, const fmpz_t p, const fmpz_t p1)
This is the main Hensel lifting routine, which performs a Hensel step
from polynomials mod `p` to polynomials mod `P = p p_1`. One starts with
polynomials `f`, `g`, `h` such that `f = gh \pmod p`. The polynomials
`a`, `b` satisfy `ag + bh = 1 \pmod p`.
The lifting formulae are
.. math::
G = \biggl( \bigl( \frac{f-gh}{p} \bigr) b \bmod g \biggr) p + g
H = \biggl( \bigl( \frac{f-gh}{p} \bigr) a \bmod h \biggr) p + h
B = \biggl( \bigl( \frac{1-aG-bH}{p} \bigr) b \bmod g \biggr) p + b
A = \biggl( \bigl( \frac{1-aG-bH}{p} \bigr) a \bmod h \biggr) p + a
Upon return we have `A G + B H = 1 \pmod P` and `f = G H \pmod P`,
where `G = g \pmod p` etc.
We require that `1 < p_1 \leq p` and that the input polynomials `f, g, h`
have degree at least `1` and that the input polynomials `a` and `b` are
non-zero.
The output arguments `G, H, A, B` may only be aliased with
the input arguments `g, h, a, b`, respectively.
.. function:: void fmpz_poly_hensel_lift_without_inverse(fmpz_poly_t Gout, fmpz_poly_t Hout, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_poly_t h, const fmpz_poly_t a, const fmpz_poly_t b, const fmpz_t p, const fmpz_t p1)
Given polynomials such that `f = gh \pmod p` and `ag + bh = 1 \pmod p`,
lifts only the factors `g` and `h` modulo `P = p p_1`.
See :func:`fmpz_poly_hensel_lift`.
.. function:: void fmpz_poly_hensel_lift_only_inverse(fmpz_poly_t Aout, fmpz_poly_t Bout, const fmpz_poly_t G, const fmpz_poly_t H, const fmpz_poly_t a, const fmpz_poly_t b, const fmpz_t p, const fmpz_t p1)
Given polynomials such that `f = gh \pmod p` and `ag + bh = 1 \pmod p`,
lifts only the cofactors `a` and `b` modulo `P = p p_1`.
See :func:`fmpz_poly_hensel_lift`.
.. function:: void fmpz_poly_hensel_lift_tree_recursive(slong * link, fmpz_poly_t * v, fmpz_poly_t * w, fmpz_poly_t f, slong j, slong inv, const fmpz_t p0, const fmpz_t p1)
Takes a current Hensel tree ``(link, v, w)`` and a pair `(j,j+1)`
of entries in the tree and lifts the tree from mod `p_0` to
mod `P = p_0 p_1`, where `1 < p_1 \leq p_0`.
Set ``inv`` to `-1` if restarting Hensel lifting, `0` if stopping
and `1` otherwise.
Here `f = g h` is the polynomial whose factors we are trying to lift.
We will have that ``v[j]`` is the product of ``v[link[j]]`` and
``v[link[j] + 1]`` as described above.
Does support aliasing of `f` with one of the polynomials in
the lists `v` and `w`. But the polynomials in these two lists
are not allowed to be aliases of each other.
.. function:: void fmpz_poly_hensel_lift_tree(slong * link, fmpz_poly_t * v, fmpz_poly_t * w, fmpz_poly_t f, slong r, const fmpz_t p, slong e0, slong e1, slong inv)
Computes `p_0 = p^{e_0}` and `p_1 = p^{e_1 - e_0}` for a small prime `p`
and `P = p^{e_1}`.
If we aim to lift to `p^b` then `f` is the polynomial whose factors we
wish to lift, made monic mod `p^b`. As usual, ``(link, v, w)`` is an
initialised tree.
This starts the recursion on lifting the *product tree* for lifting
from `p^{e_0}` to `p^{e_1}`. The value of ``inv`` corresponds to that
given for the function :func:`fmpz_poly_hensel_lift_tree_recursive`. We
set `r` to the number of local factors of `f`.
In terms of the notation, above `P = p^{e_1}`, `p_0 = p^{e_0}` and
`p_1 = p^{e_1-e_0}`.
Assumes that `f` is monic.
Assumes that `1 < p_1 \leq p_0`, that is, `0 < e_1 \leq e_0`.
.. function:: slong _fmpz_poly_hensel_start_lift(fmpz_poly_factor_t lifted_fac, slong * link, fmpz_poly_t * v, fmpz_poly_t * w, const fmpz_poly_t f, const nmod_poly_factor_t local_fac, slong N)
This function takes the local factors in ``local_fac``
and Hensel lifts them until they are known mod `p^N`, where
`N \geq 1`.
These lifted factors will be stored (in the same ordering) in
``lifted_fac``. It is assumed that ``link``, ``v``, and
``w`` are initialized arrays of ``fmpz_poly_t``'s with at least
`2*r - 2` entries and that `r \geq 2`. This is done outside of
this function so that you can keep them for restarting Hensel lifting
later. The product of local factors must be squarefree.
The return value is an exponent which must be passed to the function
:func:`_fmpz_poly_hensel_continue_lift` as ``prev_exp`` if the
Hensel lifting is to be resumed.
Currently, supports the case when `N = 1` for convenience,
although it is preferable in this case to simply iterate
over the local factors and convert them to polynomials over
`\mathbf{Z}`.
.. function:: slong _fmpz_poly_hensel_continue_lift(fmpz_poly_factor_t lifted_fac, slong * link, fmpz_poly_t * v, fmpz_poly_t * w, const fmpz_poly_t f, slong prev, slong curr, slong N, const fmpz_t p)
This function restarts a stopped Hensel lift.
It lifts from ``curr`` to `N`. It also requires ``prev``
(to lift the cofactors) given as the return value of the function
:func:`_fmpz_poly_hensel_start_lift` or the function
:func:`_fmpz_poly_hensel_continue_lift`. The current lifted factors
are supplied in ``lifted_fac`` and upon return are updated
there. As usual ``link``, ``v``, and ``w`` describe the
current Hensel tree, `r` is the number of local factors and `p` is
the small prime modulo whose power we are lifting to. It is required
that ``curr`` be at least `1` and that ``N > curr``.
Currently, supports the case when ``prev`` and ``curr``
are equal.
.. function:: void fmpz_poly_hensel_lift_once(fmpz_poly_factor_t lifted_fac, const fmpz_poly_t f, const nmod_poly_factor_t local_fac, slong N)
This function does a Hensel lift.
It lifts local factors stored in ``local_fac`` of `f` to `p^N`,
where `N \geq 2`. The lifted factors will be stored in ``lifted_fac``.
This lift cannot be restarted. This function is a convenience function
intended for end users. The product of local factors must be squarefree.
The functions in this section are not intended to be particularly fast. They are intended mainly as a debugging aid.
For the string output functions there are two variants. The first uses a
simple string representation of polynomials which prints only the length
of the polynomial and the integer coefficients, whilst the latter variant,
appended with _pretty, uses a more traditional string
representation of polynomials which prints a variable name as part of the
representation.
The first string representation is given by a sequence of integers, in
decimal notation, separated by white space. The first integer gives the
length of the polynomial; the remaining integers are the coefficients.
For example 5x^3 - x + 1 is represented by the string
"4 1 -1 0 5", and the zero polynomial is represented by "0".
The coefficients may be signed and arbitrary precision.
The string representation of the functions appended by _pretty
includes only the non-zero terms of the polynomial, starting with the
one of highest degree. Each term starts with a coefficient, prepended
with a sign, followed by the character *, followed by a variable
name, which must be passed as a string parameter to the function,
followed by a caret ^ followed by a non-negative exponent.
If the sign of the leading coefficient is positive, it is omitted. Also
the exponents of the degree 1 and 0 terms are omitted, as is the
variable and the * character in the case of the degree 0
coefficient. If the coefficient is plus or minus one, the coefficient
is omitted, except for the sign.
Some examples of the _pretty representation are:
5*x^3+7*x-4 x^2+3 -x^4+2*x-1 x+1 5
.. function:: int _fmpz_poly_print(const fmpz * poly, slong len)
Prints the polynomial ``(poly, len)`` to ``stdout``.
In case of success, returns a positive value. In case of failure,
returns a non-positive value.
.. function:: int fmpz_poly_print(const fmpz_poly_t poly)
Prints the polynomial to ``stdout``.
In case of success, returns a positive value. In case of failure,
returns a non-positive value.
.. function:: int _fmpz_poly_print_pretty(const fmpz * poly, slong len, const char * x)
Prints the pretty representation of ``(poly, len)`` to ``stdout``,
using the string ``x`` to represent the indeterminate.
In case of success, returns a positive value. In case of failure,
returns a non-positive value.
.. function:: int fmpz_poly_print_pretty(const fmpz_poly_t poly, const char * x)
Prints the pretty representation of ``poly`` to ``stdout``,
using the string ``x`` to represent the indeterminate.
In case of success, returns a positive value. In case of failure,
returns a non-positive value.
.. function:: int _fmpz_poly_fprint(FILE * file, const fmpz * poly, slong len)
Prints the polynomial ``(poly, len)`` to the stream ``file``.
In case of success, returns a positive value. In case of failure,
returns a non-positive value.
.. function:: int fmpz_poly_fprint(FILE * file, const fmpz_poly_t poly)
Prints the polynomial to the stream ``file``.
In case of success, returns a positive value. In case of failure,
returns a non-positive value.
.. function:: int _fmpz_poly_fprint_pretty(FILE * file, const fmpz * poly, slong len, const char * x)
Prints the pretty representation of ``(poly, len)`` to the stream
``file``, using the string ``x`` to represent the indeterminate.
In case of success, returns a positive value. In case of failure,
returns a non-positive value.
.. function:: int fmpz_poly_fprint_pretty(FILE * file, const fmpz_poly_t poly, const char * x)
Prints the pretty representation of ``poly`` to the stream ``file``,
using the string ``x`` to represent the indeterminate.
In case of success, returns a positive value. In case of failure,
returns a non-positive value.
.. function:: int fmpz_poly_read(fmpz_poly_t poly)
Reads a polynomial from ``stdin``, storing the result in ``poly``.
In case of success, returns a positive number. In case of failure,
returns a non-positive value.
.. function:: int fmpz_poly_read_pretty(fmpz_poly_t poly, char **x)
Reads a polynomial in pretty format from ``stdin``.
For further details, see the documentation for the function
:func:`fmpz_poly_fread_pretty`.
.. function:: int fmpz_poly_fread(FILE * file, fmpz_poly_t poly)
Reads a polynomial from the stream ``file``, storing the result
in ``poly``.
In case of success, returns a positive number. In case of failure,
returns a non-positive value.
.. function:: int fmpz_poly_fread_pretty(FILE * file, fmpz_poly_t poly, char **x)
Reads a polynomial from the file ``file`` and sets ``poly``
to this polynomial. The string ``*x`` is set to the variable
name that is used in the input.
Returns a positive value, equal to the number of characters read from
the file, in case of success. Returns a non-positive value in case of
failure, which could either be a read error or the indicator of a
malformed input.
.. function:: void fmpz_poly_get_nmod_poly(nmod_poly_t Amod, const fmpz_poly_t A)
Sets the coefficients of ``Amod`` to the coefficients in ``A``,
reduced by the modulus of ``Amod``.
.. function:: void fmpz_poly_set_nmod_poly(fmpz_poly_t A, const nmod_poly_t Amod)
Sets the coefficients of ``A`` to the residues in ``Amod``,
normalised to the interval `-m/2 \le r < m/2` where `m` is the modulus.
.. function:: void fmpz_poly_set_nmod_poly_unsigned(fmpz_poly_t A, const nmod_poly_t Amod)
Sets the coefficients of ``A`` to the residues in ``Amod``,
normalised to the interval `0 \le r < m` where `m` is the modulus.
.. function:: void _fmpz_poly_CRT_ui_precomp(fmpz * res, const fmpz * poly1, slong len1, const fmpz_t m1, nn_srcptr poly2, slong len2, ulong m2, ulong m2inv, fmpz_t m1m2, ulong c, int sign)
Sets the coefficients in ``res`` to the CRT reconstruction modulo
`m_1m_2` of the residues ``(poly1, len1)`` and ``(poly2, len2)``
which are images modulo `m_1` and `m_2` respectively.
The caller must supply the precomputed product of the input moduli as
`m_1m_2`, the inverse of `m_1` modulo `m_2` as `c`, and
the precomputed inverse of `m_2` (in the form computed by
``n_preinvert_limb``) as ``m2inv``.
If ``sign`` = 0, residues `0 \le r < m_1 m_2` are computed, while
if ``sign`` = 1, residues `-m_1 m_2/2 \le r < m_1 m_2/2` are computed.
Coefficients of ``res`` are written up to the maximum of
``len1`` and ``len2``.
.. function:: void _fmpz_poly_CRT_ui(fmpz * res, const fmpz * poly1, slong len1, const fmpz_t m1, nn_srcptr poly2, slong len2, ulong m2, ulong m2inv, int sign)
This function is identical to ``_fmpz_poly_CRT_ui_precomp``,
apart from automatically computing `m_1m_2` and `c`. It also
aborts if `c` cannot be computed.
.. function:: void fmpz_poly_CRT_ui(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_t m, const nmod_poly_t poly2, int sign)
Given ``poly1`` with coefficients modulo ``m`` and ``poly2``
with modulus `n`, sets ``res`` to the CRT reconstruction modulo `mn`
with coefficients satisfying `-mn/2 \le c < mn/2` (if sign = 1)
or `0 \le c < mn` (if sign = 0).
.. function:: void _fmpz_poly_product_roots_fmpz_vec(fmpz * poly, const fmpz * xs, slong n)
Sets ``(poly, n + 1)`` to the monic polynomial which is the product
of `(x - x_0)(x - x_1) \cdots (x - x_{n-1})`, the roots `x_i` being
given by ``xs``.
Aliasing of the input and output is not allowed.
.. function:: void fmpz_poly_product_roots_fmpz_vec(fmpz_poly_t poly, const fmpz * xs, slong n)
Sets ``poly`` to the monic polynomial which is the product
of `(x - x_0)(x - x_1) \cdots (x - x_{n-1})`, the roots `x_i` being
given by ``xs``.
.. function:: void _fmpz_poly_product_roots_fmpq_vec(fmpz * poly, const fmpq * xs, slong n)
Sets ``(poly, n + 1)`` to the product of
`(q_0 x - p_0)(q_1 x - p_1) \cdots (q_{n-1} x - p_{n-1})`, the roots
`p_i/q_i` being given by ``xs``.
.. function:: void fmpz_poly_product_roots_fmpq_vec(fmpz_poly_t poly, const fmpq * xs, slong n)
Sets ``poly`` to the polynomial which is the product
of `(q_0 x - p_0)(q_1 x - p_1) \cdots (q_{n-1} x - p_{n-1})`, the roots
`p_i/q_i` being given by ``xs``.
.. function:: fmpz ** _fmpz_poly_tree_alloc(slong len)
Allocates space for a subproduct tree of the given length, having
linear factors at the lowest level.
Entry `i` in the tree is a pointer to a single array of limbs,
capable of storing `\lfloor n / 2^i \rfloor` subproducts of
degree `2^i` adjacently, plus a trailing entry if `n / 2^i` is
not an integer.
For example, a tree of length 7 built from monic linear factors has
the following structure, where spaces have been inserted
for illustrative purposes::
X1 X1 X1 X1 X1 X1 X1
XX1 XX1 XX1 X1
XXXX1 XX1 X1
XXXXXXX1
.. function:: void _fmpz_poly_tree_free(fmpz ** tree, slong len)
Free the allocated space for the subproduct.
.. function:: void _fmpz_poly_tree_build_fmpq_vec(fmpz ** tree, const fmpq * roots, slong len)
Builds a subproduct tree in the preallocated space from
the ``len`` monic linear factors `(q_i x-p_i)` the roots `p_i/q_i`
being given by ``xs``. The top level product is not computed.
.. function:: void _fmpz_poly_bound_roots(fmpz_t bound, const fmpz * poly, slong len)
void fmpz_poly_bound_roots(fmpz_t bound, const fmpz_poly_t poly)
Computes a nonnegative integer ``bound`` that bounds the absolute
value of all complex roots of ``poly``. Uses Fujiwara's bound
.. math::
2 \max \left(
\left|\frac{a_{n-1}}{a_n}\right|,
\left|\frac{a_{n-2}}{a_n}\right|^{\frac{1}{2}}, \dotsc,
\left|\frac{a_1}{a_n}\right|^{\frac{1}{n-1}},
\left|\frac{a_0}{2a_n}\right|^{\frac{1}{n}}
\right)
where the coefficients of the polynomial are `a_0, \ldots, a_n`.
.. function:: void _fmpz_poly_num_real_roots_sturm(slong * n_neg, slong * n_pos, const fmpz * pol, slong len)
Sets ``n_neg`` and ``n_pos`` to the number of negative and
positive roots of the polynomial ``(pol, len)`` using Sturm
sequence. The Sturm sequence is computed via subresultant
remainders obtained by repeated call to the function
``_fmpz_poly_pseudo_rem_cohen``.
The polynomial is assumed to be squarefree, of degree larger than 1
and with non-zero constant coefficient.
.. function:: slong fmpz_poly_num_real_roots_sturm(const fmpz_poly_t pol)
Returns the number of real roots of the squarefree polynomial ``pol``
using Sturm sequence.
The polynomial is assumed to be squarefree.
.. function:: slong _fmpz_poly_num_real_roots(const fmpz * pol, slong len)
Returns the number of real roots of the squarefree polynomial
``(pol, len)``.
The polynomial is assumed to be squarefree.
.. function:: slong fmpz_poly_num_real_roots(const fmpz_poly_t pol)
Returns the number of real roots of the squarefree polynomial ``pol``.
The polynomial is assumed to be squarefree.
.. function:: void _fmpz_poly_cyclotomic(fmpz * a, ulong n, nn_ptr factors, slong num_factors, ulong phi)
Sets ``a`` to the lower half of the cyclotomic polynomial `\Phi_n(x)`,
given `n \ge 3` which must be squarefree.
A precomputed array containing the prime factors of `n` must be provided,
as well as the value of the Euler totient function `\phi(n)` as ``phi``.
If `n` is even, 2 must be the first factor in the list.
The degree of `\Phi_n(x)` is exactly `\phi(n)`. Only the low
`(\phi(n) + 1) / 2` coefficients are written; the high coefficients
can be obtained afterwards by copying the low coefficients
in reverse order, since `\Phi_n(x)` is a palindrome for `n \ne 1`.
We use the sparse power series algorithm described as Algorithm 4
[ArnoldMonagan2011]_. The algorithm is based on the identity
.. math::
\Phi_n(x) = \prod_{d|n} (x^d - 1)^{\mu(n/d)}.
Treating the polynomial as a power series, the multiplications and
divisions can be done very cheaply using repeated additions and
subtractions. The complexity is `O(2^k \phi(n))` where `k` is the
number of prime factors in `n`.
To improve efficiency for small `n`, we treat the ``fmpz``
coefficients as machine integers when there is no risk of overflow.
The following bounds are given in Table 6 of [ArnoldMonagan2011]_:
For `n < 10163195`, the largest coefficient in any `\Phi_n(x)`
has 27 bits, so machine arithmetic is safe on 32 bits.
For `n < 169828113`, the largest coefficient in any `\Phi_n(x)`
has 60 bits, so machine arithmetic is safe on 64 bits.
Further, the coefficients are always `\pm 1` or 0 if there are
exactly two prime factors, so in this case machine arithmetic can be
used as well.
Finally, we handle two special cases: if there is exactly one prime
factor `n = p`, then `\Phi_n(x) = 1 + x + x^2 + \ldots + x^{n-1}`,
and if `n = 2m`, we use `\Phi_n(x) = \Phi_m(-x)` to fall back
to the case when `n` is odd.
.. function:: void fmpz_poly_cyclotomic(fmpz_poly_t poly, ulong n)
Sets ``poly`` to the `n`-th cyclotomic polynomial, defined as
`\Phi_n(x) = \prod_{\omega} (x-\omega)`
where `\omega` runs over all the `n`-th primitive roots of unity.
We factor `n` into `n = qs` where `q` is squarefree,
and compute `\Phi_q(x)`. Then `\Phi_n(x) = \Phi_q(x^s)`.
.. function:: ulong _fmpz_poly_is_cyclotomic(const fmpz * poly, slong len)
ulong fmpz_poly_is_cyclotomic(const fmpz_poly_t poly)
If ``poly`` is a cyclotomic polynomial, returns the index `n` of this
cyclotomic polynomial. If ``poly`` is not a cyclotomic polynomial,
returns 0.
.. function:: void _fmpz_poly_cos_minpoly(fmpz * coeffs, ulong n)
void fmpz_poly_cos_minpoly(fmpz_poly_t poly, ulong n)
Sets ``poly`` to the minimal polynomial of `2 \cos(2 \pi / n)`.
For suitable choice of `n`, this gives the minimal polynomial
of `2 \cos(a \pi)` or `2 \sin(a \pi)` for any rational `a`.
The cosine is multiplied by a factor two since this gives
a monic polynomial with integer coefficients. One can obtain
the minimal polynomial for `\cos(2 \pi / n)` by making
the substitution `x \to x / 2`.
For `n > 2`, the degree of the polynomial is `\varphi(n) / 2`.
For `n = 1, 2`, the degree is 1. For `n = 0`, we define
the output to be the constant polynomial 1.
See [WaktinsZeitlin1993]_.
.. function:: void _fmpz_poly_swinnerton_dyer(fmpz * coeffs, ulong n)
void fmpz_poly_swinnerton_dyer(fmpz_poly_t poly, ulong n)
Sets ``poly`` to the Swinnerton-Dyer polynomial `S_n`, defined as
the integer polynomial
`S_n = \prod (x \pm \sqrt{2} \pm \sqrt{3} \pm \sqrt{5} \pm \ldots \pm \sqrt{p_n})`
where `p_n` denotes the `n`-th prime number and all combinations
of signs are taken. This polynomial has degree `2^n` and is
irreducible over the integers (it is the minimal polynomial
of `\sqrt{2} + \ldots + \sqrt{p_n}`).
.. function:: void _fmpz_poly_chebyshev_t(fmpz * coeffs, ulong n)
void fmpz_poly_chebyshev_t(fmpz_poly_t poly, ulong n)
Sets ``poly`` to the Chebyshev polynomial of the first kind `T_n(x)`,
defined by `T_n(x) = \cos(n \cos^{-1}(x))`, for `n\ge0`. The coefficients are
calculated using a hypergeometric recurrence.
.. function:: void _fmpz_poly_chebyshev_u(fmpz * coeffs, ulong n)
void fmpz_poly_chebyshev_u(fmpz_poly_t poly, ulong n)
Sets ``poly`` to the Chebyshev polynomial of the first kind `U_n(x)`,
defined by `(n+1) U_n(x) = T'_{n+1}(x)`, for `n\ge0`.
The coefficients are calculated using a hypergeometric recurrence.
.. function:: void _fmpz_poly_legendre_pt(fmpz * coeffs, ulong n)
Sets ``coeffs`` to the coefficient array of the shifted Legendre
polynomial `\tilde{P_n}(x)`, defined by `\tilde{P_n}(x) = P_n(2x-1)`, for `n\ge0`.
The coefficients are calculated using a hypergeometric recurrence.
The length of the array will be ``n+1``.
See ``fmpq_poly`` for the Legendre polynomials.
.. function:: void fmpz_poly_legendre_pt(fmpz_poly_t poly, ulong n)
Sets ``poly`` to the shifted Legendre polynomial `\tilde{P_n}(x)`,
defined by `\tilde{P_n}(x) = P_n(2x-1)`, for `n\ge0`. The coefficients are
calculated using a hypergeometric recurrence. See ``fmpq_poly``
for the Legendre polynomials.
.. function:: void _fmpz_poly_hermite_h(fmpz * coeffs, ulong n)
Sets ``coeffs`` to the coefficient array of the Hermite
polynomial `H_n(x)`, defined by `H'_n(x) = 2nH_{n-1}(x)`, for `n\ge0`.
The coefficients are calculated using a hypergeometric recurrence.
The length of the array will be ``n+1``.
.. function:: void fmpz_poly_hermite_h(fmpz_poly_t poly, ulong n)
Sets ``poly`` to the Hermite polynomial `H_n(x)`,
defined by `H'_n(x) = 2nH_{n-1}(x)`, for `n\ge0`. The coefficients are
calculated using a hypergeometric recurrence.
.. function:: void _fmpz_poly_hermite_he(fmpz * coeffs, ulong n)
Sets ``coeffs`` to the coefficient array of the Hermite
polynomial `He_n(x)`, defined by
`He_n(x) = 2^{-\tfrac{n}{2}}H_n\left(\frac{x}{\sqrt2}\right)`, for `n\ge0`.
The coefficients are calculated using a hypergeometric recurrence.
The length of the array will be ``n+1``.
.. function:: void fmpz_poly_hermite_he(fmpz_poly_t poly, ulong n)
Sets ``poly`` to the Hermite polynomial `He_n(x)`,
defined by `He_n(x) = 2^{-\tfrac{n}{2}}H_n\left(\frac{x}{\sqrt2}\right)`, for `n\ge0`.
The coefficients are calculated using a hypergeometric recurrence.
.. function:: void _fmpz_poly_fibonacci(fmpz * coeffs, ulong n)
Sets ``coeffs`` to the coefficient array of the `n`-th Fibonacci polynomial.
The coefficients are calculated using a hypergeometric recurrence.
.. function:: void fmpz_poly_fibonacci(fmpz_poly_t poly, ulong n)
Sets ``poly`` to the `n`-th Fibonacci polynomial.
The coefficients are calculated using a hypergeometric recurrence.
Eulerian numbers are the coefficients to the Eulerian polynomials
A_n(x) = \sum_{m = 0}^{n} A(n, m) x^m,
where the Eulerian polynomials are defined by the exponential generating function
\frac{x - 1}{x - e^{(x - 1) t}}
= \sum_{n = 0}^{\infty} A_n(x) \frac{t^n}{n!}.
The Eulerian numbers can be expressed explicitly via the formula
A(n, m) = \sum_{k = 0}^{m + 1} (-1)^k \binom{n + 1}{k} (m + 1 - k)^n.
Note: Not to be confused with Euler numbers and polynomials.
.. function:: void fmpz_eulerian_polynomial(fmpz_poly_t res, ulong n)
Sets ``res`` to the Eulerian polynomial `A_n(x)`, where we define
`A_0(x) = 1`. The polynomial is calculated via a recursive relation.
.. function:: void _fmpz_poly_eta_qexp(fmpz * f, slong r, slong len)
void fmpz_poly_eta_qexp(fmpz_poly_t f, slong r, slong n)
Sets `f` to the `q`-expansion to length `n` of the
Dedekind eta function (without the leading factor
`q^{1/24}`) raised to the power `r`, i.e.
`(q^{-1/24} \eta(q))^r = \prod_{k=1}^{\infty} (1 - q^k)^r`.
In particular, `r = -1` gives the generating function
of the partition function `p(k)`, and `r = 24` gives,
after multiplication by `q`,
the modular discriminant `\Delta(q)` which generates
the Ramanujan tau function `\tau(k)`.
This function uses sparse formulas for `r = 1, 2, 3, 4, 6`
and otherwise reduces to one of those cases using power series arithmetic.
.. function:: void _fmpz_poly_theta_qexp(fmpz * f, slong r, slong len)
void fmpz_poly_theta_qexp(fmpz_poly_t f, slong r, slong n)
Sets `f` to the `q`-expansion to length `n` of the
Jacobi theta function raised to the power `r`, i.e. `\vartheta(q)^r`
where `\vartheta(q) = 1 + 2 \sum_{k=1}^{\infty} q^{k^2}`.
This function uses sparse formulas for `r = 1, 2`
and otherwise reduces to those cases using power series arithmetic.
.. function:: void fmpz_poly_CLD_bound(fmpz_t res, const fmpz_poly_t f, slong n)
Compute a bound on the `n` coefficient of `fg'/g` where `g` is any
factor of `f`.