321 lines
11 KiB
Plaintext
321 lines
11 KiB
Plaintext
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/*=============================================================================
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This file is part of FLINT.
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FLINT is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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FLINT is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with FLINT; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2009, 2010, 2011 Sebastian Pancratz
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******************************************************************************/
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*******************************************************************************
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Memory management
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We represent a rational function over $\mathbf{Q}$ as the quotient
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of two coprime integer polynomials of type \code{fmpz_poly_t},
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enforcing that the leading coefficient of the denominator is
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positive. The zero function is represented as $0/1$.
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*******************************************************************************
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void fmpz_poly_q_init(fmpz_poly_q_t rop)
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Initialises \code{rop}.
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void fmpz_poly_q_clear(fmpz_poly_q_t rop)
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Clears the object \code{rop}.
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fmpz_poly_struct * fmpz_poly_q_numref(const fmpz_poly_q_t op)
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Returns a reference to the numerator of \code{op}.
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fmpz_poly_struct * fmpz_poly_q_denref(const fmpz_poly_q_t op)
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Returns a reference to the denominator of \code{op}.
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void fmpz_poly_q_canonicalise(fmpz_poly_q_t rop)
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Brings \code{rop} into canonical form, only assuming that
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the denominator is non-zero.
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int fmpz_poly_q_is_canonical(const fmpz_poly_q_t op)
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Checks whether the rational function \code{op} is in
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canonical form.
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*******************************************************************************
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Randomisation
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*******************************************************************************
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void fmpz_poly_q_randtest(fmpz_poly_q_t poly, flint_rand_t state,
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slong len1, mp_bitcnt_t bits1,
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slong len2, mp_bitcnt_t bits2)
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Sets \code{poly} to a random rational function.
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void fmpz_poly_q_randtest_not_zero(fmpz_poly_q_t poly, flint_rand_t state,
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slong len1, mp_bitcnt_t bits1,
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slong len2, mp_bitcnt_t bits2)
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Sets \code{poly} to a random non-zero rational function.
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*******************************************************************************
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Assignment
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*******************************************************************************
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void fmpz_poly_q_set(fmpz_poly_q_t rop, const fmpz_poly_q_t op)
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Sets the element \code{rop} to the same value as the element \code{op}.
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void fmpz_poly_q_set_si(fmpz_poly_q_t rop, slong op)
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Sets the element \code{rop} to the value given by the \code{slong}
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\code{op}.
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void fmpz_poly_q_swap(fmpz_poly_q_t op1, fmpz_poly_q_t op2)
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Swaps the elements \code{op1} and \code{op2}.
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This is done efficiently by swapping pointers.
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void fmpz_poly_q_zero(fmpz_poly_q_t rop)
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Sets \code{rop} to zero.
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void fmpz_poly_q_one(fmpz_poly_q_t rop)
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Sets \code{rop} to one.
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void fmpz_poly_q_neg(fmpz_poly_q_t rop, const fmpz_poly_q_t op)
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Sets the element \code{rop} to the additive inverse of \code{op}.
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void fmpz_poly_q_inv(fmpz_poly_q_t rop, const fmpz_poly_q_t op)
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Sets the element \code{rop} to the multiplicative inverse of \code{op}.
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Assumes that the element \code{op} is non-zero.
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*******************************************************************************
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Comparison
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*******************************************************************************
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int fmpz_poly_q_is_zero(const fmpz_poly_q_t op)
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Returns whether the element \code{op} is zero.
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int fmpz_poly_q_is_one(const fmpz_poly_q_t op)
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Returns whether the element \code{rop} is equal to the constant
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polynomial $1$.
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int fmpz_poly_q_equal(const fmpz_poly_q_t op1, const fmpz_poly_q_t op2)
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Returns whether the two elements \code{op1} and \code{op2} are equal.
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*******************************************************************************
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Addition and subtraction
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*******************************************************************************
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void fmpz_poly_q_add(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op1, const fmpz_poly_q_t op2)
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Sets \code{rop} to the sum of \code{op1} and \code{op2}.
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void fmpz_poly_q_sub(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op1, const fmpz_poly_q_t op2)
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Sets \code{rop} to the difference of \code{op1} and \code{op2}.
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void fmpz_poly_q_addmul(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op1, const fmpz_poly_q_t op2)
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Adds the product of \code{op1} and \code{op2} to \code{rop}.
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void fmpz_poly_q_submul(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op1, const fmpz_poly_q_t op2)
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Subtracts the product of \code{op1} and \code{op2} from \code{rop}.
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*******************************************************************************
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Scalar multiplication and division
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*******************************************************************************
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void fmpz_poly_q_scalar_mul_si(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op, slong x)
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Sets \code{rop} to the product of the rational function \code{op}
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and the \code{slong} integer $x$.
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void fmpz_poly_q_scalar_mul_mpz(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op, const mpz_t x)
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Sets \code{rop} to the product of the rational function \code{op}
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and the \code{mpz_t} integer $x$.
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void fmpz_poly_q_scalar_mul_mpq(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op, const mpq_t x)
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Sets \code{rop} to the product of the rational function \code{op}
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and the \code{mpq_t} rational $x$.
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void fmpz_poly_q_scalar_div_si(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op, slong x)
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Sets \code{rop} to the quotient of the rational function \code{op}
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and the \code{slong} integer $x$.
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void fmpz_poly_q_scalar_div_mpz(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op, const mpz_t x)
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Sets \code{rop} to the quotient of the rational function \code{op}
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and the \code{mpz_t} integer $x$.
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void fmpz_poly_q_scalar_div_mpq(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op, const mpq_t x)
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Sets \code{rop} to the quotient of the rational function \code{op}
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and the \code{mpq_t} rational $x$.
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*******************************************************************************
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Multiplication and division
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*******************************************************************************
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void fmpz_poly_q_mul(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op1, const fmpz_poly_q_t op2)
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Sets \code{rop} to the product of \code{op1} and \code{op2}.
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void fmpz_poly_q_div(fmpz_poly_q_t rop,
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const fmpz_poly_q_t op1, const fmpz_poly_q_t op2)
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Sets \code{rop} to the quotient of \code{op1} and \code{op2}.
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*******************************************************************************
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Powering
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*******************************************************************************
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void fmpz_poly_q_pow(fmpz_poly_q_t rop, const fmpz_poly_q_t op, ulong exp)
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Sets \code{rop} to the \code{exp}-th power of \code{op}.
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The corner case of \code{exp == 0} is handled by setting \code{rop} to
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the constant function $1$. Note that this includes the case $0^0 = 1$.
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*******************************************************************************
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Derivative
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*******************************************************************************
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void fmpz_poly_q_derivative(fmpz_poly_q_t rop, const fmpz_poly_q_t op)
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Sets \code{rop} to the derivative of \code{op}.
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*******************************************************************************
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Evaluation
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*******************************************************************************
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int fmpz_poly_q_evaluate(mpq_t rop, const fmpz_poly_q_t f, const mpq_t a)
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Sets \code{rop} to $f$ evaluated at the rational $a$.
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If the denominator evaluates to zero at $a$, returns non-zero and
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does not modify any of the variables. Otherwise, returns $0$ and
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sets \code{rop} to the rational $f(a)$.
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*******************************************************************************
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Input and output
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The following three methods enable users to construct elements of type\\
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\code{fmpz_poly_q_t} from strings or to obtain string representations of
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such elements.
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The format used is based on the FLINT format for integer polynomials of
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type \code{fmpz_poly_t}, which we recall first:
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A non-zero polynomial $a_0 + a_1 X + \dotsb + a_n X^n$ of length
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$n + 1$ is represented by the string \code{"n+1 a_0 a_1 ... a_n"},
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where there are two space characters following the length and single
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space characters separating the individual coefficients. There is no
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leading or trailing white-space. The zero polynomial is simply
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represented by \code{"0"}.
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We adapt this notation for rational functions as follows. We denote the
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zero function by \code{"0"}. Given a non-zero function with numerator
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and denominator string representations \code{num} and \code{den},
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respectively, we use the string \code{num/den} to represent the rational
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function, unless the denominator is equal to one, in which case we simply
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use \code{num}.
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There is also a \code{_pretty} variant available, which bases the string
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parts for the numerator and denominator on the output of the function
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\code{fmpz_poly_get_str_pretty} and introduces parentheses where
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necessary.
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Note that currently these functions are not optimised for performance and
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are intended to be used only for debugging purposes or one-off input and
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output, rather than as a low-level parser.
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*******************************************************************************
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int fmpz_poly_q_set_str(fmpz_poly_q_t rop, const char *s)
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Sets \code{rop} to the rational function given
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by the string \code{s}.
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char * fmpz_poly_q_get_str(const fmpz_poly_q_t op)
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Returns the string representation of
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the rational function \code{op}.
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char * fmpz_poly_q_get_str_pretty(const fmpz_poly_q_t op, const char *x)
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Returns the pretty string representation of
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the rational function \code{op}.
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int fmpz_poly_q_print(const fmpz_poly_q_t op)
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Prints the representation of the rational
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function \code{op} to \code{stdout}.
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int fmpz_poly_q_print_pretty(const fmpz_poly_q_t op, const char *x)
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Prints the pretty representation of the rational
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function \code{op} to \code{stdout}.
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