# qqbar.h – algebraic numbers represented by minimal polynomials

A `qqbar_t` represents a real or complex algebraic number (an element of \(\overline{\mathbb{Q}}\)) by its unique reduced minimal polynomial in \(\mathbb{Z}[x]\) and an isolating complex interval. The precision of isolating intervals is maintained automatically to ensure that all operations on `qqbar_t` instances are exact.

This representation is useful for working with individual algebraic numbers of moderate degree (up to 100, say). Arithmetic in this representation is expensive: an arithmetic operation on numbers of degrees m and n involves computing and then factoring an annihilating polynomial of degree mn and potentially also performing numerical root-finding. For doing repeated arithmetic, it is generally more efficient to work with the `ca_t` type in a fixed number field. The `qqbar_t` type is used internally by the `ca_t` type to represent the embedding of number fields in \(\mathbb{R}\) or \(\mathbb{C}\) and to decide predicates for algebraic numbers.

## Types and macros

type qqbar_struct

type qqbar_t

  A qqbar_struct consists of an fmpz_poly_struct and an acb_struct. A qqbar_t is defined as an array of length one of type qqbar_struct, permitting a qqbar_t to be passed by reference.

type qqbar_ptr

  Alias for `qqbar_struct *`, used for qqbar vectors.

type qqbar_srcptr

  Alias for `const qqbar_struct *`, used for qqbar vectors when passed as readonly input to functions.

QQBAR_POLY(x)

  Macro returning a pointer to the minimal polynomial of x which can be used as an fmpz_poly_t.

QQBAR_COEFFS(x)

  Macro returning a pointer to the array of fmpz coefficients of the minimal polynomial of x.

QQBAR_ENCLOSURE(x)

  Macro returning a pointer to the enclosure of x which can be used as an acb_t.

## Memory management

void qqbar_init(qqbar_t res)

  Initializes the variable res for use, and sets its value to zero.

void qqbar_clear(qqbar_t res)

  Clears the variable res, freeing or recycling its allocated memory.

qqbar_ptr _qqbar_vec_init(slong len)

  Returns a pointer to an array of len initialized qqbar_struct:s.

void _qqbar_vec_clear(qqbar_ptr vec, slong len)

  Clears all len entries in the vector vec and frees the vector itself.

## Assignment

void qqbar_swap(qqbar_t x, qqbar_t y)

  Swaps the values of x and y efficiently.

void qqbar_set(qqbar_t res, const qqbar_t x)
void qqbar_set_si(qqbar_t res, slong x)
void qqbar_set_ui(qqbar_t res, ulong x)
void qqbar_set_fmpz(qqbar_t res, const fmpz_t x)
void qqbar_set_fmpq(qqbar_t res, const fmpq_t x)

  Sets res to the value x.

void qqbar_set_re_im(qqbar_t res, const qqbar_t x, const qqbar_t y)

  Sets res to the value \(x + yi\).

int qqbar_set_d(qqbar_t res, double x)
int qqbar_set_re_im_d(qqbar_t res, double x, double y)

  Sets res to the value x or \(x + yi\) respectively. These functions performs error handling: if x and y are finite, the conversion succeeds and the return flag is 1. If x or y is non-finite (infinity or NaN), the conversion fails and the return flag is 0.

## Properties

slong qqbar_degree(const qqbar_t x)

  Returns the degree of x, i.e. the degree of the minimal polynomial.

int qqbar_is_rational(const qqbar_t x)

  Returns whether x is a rational number.

int qqbar_is_integer(const qqbar_t x)

  Returns whether x is an integer (an element of \(\mathbb{Z}\)).

int qqbar_is_algebraic_integer(const qqbar_t x)

  Returns whether x is an algebraic integer, i.e. whether its minimal polynomial has leading coefficient 1.

int qqbar_is_zero(const qqbar_t x)
int qqbar_is_one(const qqbar_t x)
int qqbar_is_neg_one(const qqbar_t x)

  Returns whether x is the number \(0\), \(1\), \(-1\).

int qqbar_is_i(const qqbar_t x)
int qqbar_is_neg_i(const qqbar_t x)

  Returns whether x is the imaginary unit \(i\) (respectively \(-i\)).

int qqbar_is_real(const qqbar_t x)

  Returns whether x is a real number.

void qqbar_height(fmpz_t res, const qqbar_t x)

  Sets res to the height of x (the largest absolute value of the coefficients of the minimal polynomial of x).

slong qqbar_height_bits(const qqbar_t x)

  Returns the height of x (the largest absolute value of the coefficients of the minimal polynomial of x) measured in bits.

int qqbar_within_limits(const qqbar_t x, slong deg_limit, slong bits_limit)

  Checks if x has degree bounded by deg_limit and height bounded by bits_limit bits, returning 0 (false) or 1 (true). If deg_limit is set to 0, the degree check is skipped, and similarly for bits_limit.

int qqbar_binop_within_limits(const qqbar_t x, const qqbar_t y, slong deg_limit, slong bits_limit)

  Checks if \(x + y\), \(x - y\), \(x \cdot y\) and \(x / y\) certainly have degree bounded by deg_limit (by multiplying the degrees for x and y to obtain a trivial bound). For bits_limits, the sum of the bit heights of x and y is checked against the bound (this is only a heuristic). If deg_limit is set to 0, the degree check is skipped, and similarly for bits_limit.

## Conversions

void _qqbar_get_fmpq(fmpz_t num, fmpz_t den, const qqbar_t x)

  Sets num and den to the numerator and denominator of x. Aborts if x is not a rational number.

void qqbar_get_fmpq(fmpq_t res, const qqbar_t x)

  Sets res to x. Aborts if x is not a rational number.

void qqbar_get_fmpz(fmpz_t res, const qqbar_t x)

  Sets res to x. Aborts if x is not an integer.

## Special values

void qqbar_zero(qqbar_t res)

  Sets res to the number 0.

void qqbar_one(qqbar_t res)

  Sets res to the number 1.

void qqbar_i(qqbar_t res)

  Sets res to the imaginary unit \(i\).

void qqbar_phi(qqbar_t res)

  Sets res to the golden ratio \(\varphi = \tfrac{1}{2}(\sqrt{5} + 1)\).

## Input and output

void qqbar_print(const qqbar_t x)

  Prints res to standard output. The output shows the degree and the list of coefficients of the minimal polynomial followed by a decimal representation of the enclosing interval. This function is mainly intended for debugging.

void qqbar_printn(const qqbar_t x, slong n)

  Prints res to standard output. The output shows a decimal approximation to n digits.

void qqbar_printnd(const qqbar_t x, slong n)

  Prints res to standard output. The output shows a decimal approximation to n digits, followed by the degree of the number.

For example, print, printn and printnd with \(n = 6\) give the following output for the numbers 0, 1, \(i\), \(\varphi\), \(\sqrt{2}-\sqrt{3} i\):

    deg 1 [0, 1] 0
    deg 1 [-1, 1] 1.00000
    deg 2 [1, 0, 1] 1.00000*I
    deg 2 [-1, -1, 1] [1.61803398874989484820458683436563811772 +/- 6.00e-39]
    deg 4 [25, 0, 2, 0, 1] [1.4142135623730950488016887242096980786 +/- 8.67e-38] + [-1.732050807568877293527446341505872367 +/- 1.10e-37]*I

    0
    1.00000
    1.00000*I
    1.61803
    1.41421 - 1.73205*I

    0 (deg 1)
    1.00000 (deg 1)
    1.00000*I (deg 2)
    1.61803 (deg 2)
    1.41421 - 1.73205*I (deg 4)

## Random generation

void qqbar_randtest(qqbar_t res, flint_rand_t state, slong deg, slong bits)

  Sets res to a random algebraic number with degree up to deg and with height (measured in bits) up to bits.

void qqbar_randtest_real(qqbar_t res, flint_rand_t state, slong deg, slong bits)

  Sets res to a random real algebraic number with degree up to deg and with height (measured in bits) up to bits.

void qqbar_randtest_nonreal(qqbar_t res, flint_rand_t state, slong deg, slong bits)

  Sets res to a random nonreal algebraic number with degree up to deg and with height (measured in bits) up to bits. Since all algebraic numbers of degree 1 are real, deg must be at least 2.

## Comparisons

int qqbar_equal(const qqbar_t x, const qqbar_t y)

  Returns whether x and y are equal.

int qqbar_equal_fmpq_poly_val(const qqbar_t x, const fmpq_poly_t f, const qqbar_t y)

  Returns whether x is equal to \(f(y)\). This function is more efficient than evaluating \(f(y)\) and comparing the results.

int qqbar_cmp_re(const qqbar_t x, const qqbar_t y)

  Compares the real parts of x and y, returning -1, 0 or +1.

int qqbar_cmp_im(const qqbar_t x, const qqbar_t y)

  Compares the imaginary parts of x and y, returning -1, 0 or +1.

int qqbar_cmpabs_re(const qqbar_t x, const qqbar_t y)

  Compares the absolute values of the real parts of x and y, returning -1, 0 or +1.

int qqbar_cmpabs_im(const qqbar_t x, const qqbar_t y)

  Compares the absolute values of the imaginary parts of x and y, returning -1, 0 or +1.

int qqbar_cmpabs(const qqbar_t x, const qqbar_t y)

  Compares the absolute values of x and y, returning -1, 0 or +1.

int qqbar_cmp_root_order(const qqbar_t x, const qqbar_t y)

  Compares x and y using an arbitrary but convenient ordering defined on the complex numbers. This is useful for sorting the roots of a polynomial in a canonical order.

  We define the root order as follows: real roots come first, in descending order. Nonreal roots are subsequently ordered first by real part in descending order, then in ascending order by the absolute value of the imaginary part, and then in descending order of the sign. This implies that complex conjugate roots are adjacent, with the root in the upper half plane first.

ulong qqbar_hash(const qqbar_t x)

  Returns a hash of x. As currently implemented, this function only hashes the minimal polynomial of x. The user should mix in some bits based on the numerical value if it is critical to distinguish between conjugates of the same minimal polynomial. This function is also likely to produce serial runs of values for lexicographically close minimal polynomials. This is not necessarily a problem for use in hash tables, but if it is important that all bits in the output are random, the user should apply an integer hash function to the output.

## Complex parts

void qqbar_conj(qqbar_t res, const qqbar_t x)

  Sets res to the complex conjugate of x.

void qqbar_re(qqbar_t res, const qqbar_t x)

  Sets res to the real part of x.

void qqbar_im(qqbar_t res, const qqbar_t x)

  Sets res to the imaginary part of x.

void qqbar_re_im(qqbar_t res1, qqbar_t res2, const qqbar_t x)

  Sets res1 to the real part of x and res2 to the imaginary part of x.

void qqbar_abs(qqbar_t res, const qqbar_t x)

  Sets res to the absolute value of x:

void qqbar_abs2(qqbar_t res, const qqbar_t x)

  Sets res to the square of the absolute value of x.

void qqbar_sgn(qqbar_t res, const qqbar_t x)

  Sets res to the complex sign of x, defined as 0 if x is zero and as \(x / |x|\) otherwise.

int qqbar_sgn_re(const qqbar_t x)

  Returns the sign of the real part of x (-1, 0 or +1).

int qqbar_sgn_im(const qqbar_t x)

  Returns the sign of the imaginary part of x (-1, 0 or +1).

int qqbar_csgn(const qqbar_t x)

  Returns the extension of the real sign function taking the value 1 for x strictly in the right half plane, -1 for x strictly in the left half plane, and the sign of the imaginary part when x is on the imaginary axis. Equivalently, \(\operatorname{csgn}(x) = x / \sqrt{x^2}\) except that the value is 0 when x is zero.

## Integer parts

void qqbar_floor(fmpz_t res, const qqbar_t x)

  Sets res to the floor function of x. If x is not real, the value is defined as the floor function of the real part of x.

void qqbar_ceil(fmpz_t res, const qqbar_t x)

  Sets res to the ceiling function of x. If x is not real, the value is defined as the ceiling function of the real part of x.

## Arithmetic

void qqbar_neg(qqbar_t res, const qqbar_t x)

  Sets res to the negation of x.

void qqbar_add(qqbar_t res, const qqbar_t x, const qqbar_t y)
void qqbar_add_fmpq(qqbar_t res, const qqbar_t x, const fmpq_t y)
void qqbar_add_fmpz(qqbar_t res, const qqbar_t x, const fmpz_t y)
void qqbar_add_ui(qqbar_t res, const qqbar_t x, ulong y)
void qqbar_add_si(qqbar_t res, const qqbar_t x, slong y)

  Sets res to the sum of x and y.

void qqbar_sub(qqbar_t res, const qqbar_t x, const qqbar_t y)
void qqbar_sub_fmpq(qqbar_t res, const qqbar_t x, const fmpq_t y)
void qqbar_sub_fmpz(qqbar_t res, const qqbar_t x, const fmpz_t y)
void qqbar_sub_ui(qqbar_t res, const qqbar_t x, ulong y)
void qqbar_sub_si(qqbar_t res, const qqbar_t x, slong y)
void qqbar_fmpq_sub(qqbar_t res, const fmpq_t x, const qqbar_t y)
void qqbar_fmpz_sub(qqbar_t res, const fmpz_t x, const qqbar_t y)
void qqbar_ui_sub(qqbar_t res, ulong x, const qqbar_t y)
void qqbar_si_sub(qqbar_t res, slong x, const qqbar_t y)

  Sets res to the difference of x and y.

void qqbar_mul(qqbar_t res, const qqbar_t x, const qqbar_t y)
void qqbar_mul_fmpq(qqbar_t res, const qqbar_t x, const fmpq_t y)
void qqbar_mul_fmpz(qqbar_t res, const qqbar_t x, const fmpz_t y)
void qqbar_mul_ui(qqbar_t res, const qqbar_t x, ulong y)
void qqbar_mul_si(qqbar_t res, const qqbar_t x, slong y)

  Sets res to the product of x and y.

void qqbar_mul_2exp_si(qqbar_t res, const qqbar_t x, slong e)

  Sets res to x multiplied by \(2^e\).

void qqbar_sqr(qqbar_t res, const qqbar_t x)

  Sets res to the square of x.

void qqbar_inv(qqbar_t res, const qqbar_t x)

  Sets res to the multiplicative inverse of y. Division by zero calls flint_abort.

void qqbar_div(qqbar_t res, const qqbar_t x, const qqbar_t y)
void qqbar_div_fmpq(qqbar_t res, const qqbar_t x, const fmpq_t y)
void qqbar_div_fmpz(qqbar_t res, const qqbar_t x, const fmpz_t y)
void qqbar_div_ui(qqbar_t res, const qqbar_t x, ulong y)
void qqbar_div_si(qqbar_t res, const qqbar_t x, slong y)
void qqbar_fmpq_div(qqbar_t res, const fmpq_t x, const qqbar_t y)
void qqbar_fmpz_div(qqbar_t res, const fmpz_t x, const qqbar_t y)
void qqbar_ui_div(qqbar_t res, ulong x, const qqbar_t y)
void qqbar_si_div(qqbar_t res, slong x, const qqbar_t y)

  Sets res to the quotient of x and y. Division by zero calls flint_abort.

void qqbar_scalar_op(qqbar_t res, const qqbar_t x, const fmpz_t a, const fmpz_t b, const fmpz_t c)

  Sets res to the rational affine transformation \((ax+b)/c\), performed as a single operation. There are no restrictions on a, b and c except that c must be nonzero. Division by zero calls flint_abort.

## Powers and roots

void qqbar_sqrt(qqbar_t res, const qqbar_t x)
void qqbar_sqrt_ui(qqbar_t res, ulong x)

  Sets res to the principal square root of x.

void qqbar_rsqrt(qqbar_t res, const qqbar_t x)

  Sets res to the reciprocal of the principal square root of x. Division by zero calls flint_abort.

void qqbar_pow_ui(qqbar_t res, const qqbar_t x, ulong n)
void qqbar_pow_si(qqbar_t res, const qqbar_t x, slong n)
void qqbar_pow_fmpz(qqbar_t res, const qqbar_t x, const fmpz_t n)
void qqbar_pow_fmpq(qqbar_t res, const qqbar_t x, const fmpq_t n)

  Sets res to x raised to the n-th power. Raising zero to a negative power aborts.

void qqbar_root_ui(qqbar_t res, const qqbar_t x, ulong n)
void qqbar_fmpq_root_ui(qqbar_t res, const fmpq_t x, ulong n)

  Sets res to the principal n-th root of x. The order n must be positive.

void qqbar_fmpq_pow_si_ui(qqbar_t res, const fmpq_t x, slong m, ulong n)

  Sets res to the principal branch of \(x^{m/n}\). The order n must be positive. Division by zero calls flint_abort.

int qqbar_pow(qqbar_t res, const qqbar_t x, const qqbar_t y)

  General exponentiation: if \(x^y\) is an algebraic number, sets res to this value and returns 1. If \(x^y\) is transcendental or undefined, returns 0. Note that this function returns 0 instead of aborting on division zero.

## Numerical enclosures

The following functions guarantee a polished output in which both the real and imaginary parts are accurate to prec bits and exact when exactly representable (that is, when a real or imaginary part is a sufficiently small dyadic number). In some cases, the computations needed to polish the output may be expensive. When polish is unnecessary, `qqbar_enclosure_raw()` may be used instead. Alternatively, `qqbar_cache_enclosure()` can be used to avoid recomputations.

void qqbar_get_acb(acb_t res, const qqbar_t x, slong prec)

  Sets res to an enclosure of x rounded to prec bits.

void qqbar_get_arb(arb_t res, const qqbar_t x, slong prec)

  Sets res to an enclosure of x rounded to prec bits, assuming that x is a real number. If x is not real, res is set to \([\operatorname{NaN} \pm \infty]\).

void qqbar_get_arb_re(arb_t res, const qqbar_t x, slong prec)

  Sets res to an enclosure of the real part of x rounded to prec bits.

void qqbar_get_arb_im(arb_t res, const qqbar_t x, slong prec)

  Sets res to an enclosure of the imaginary part of x rounded to prec bits.

void qqbar_cache_enclosure(qqbar_t res, slong prec)

  Polishes the internal enclosure of res to at least prec bits of precision in-place. Normally, qqbar operations that need high-precision enclosures compute them on the fly without caching the results; if res will be used as an invariant operand for many operations, calling this function as a precomputation step can improve performance.

## Numerator and denominator

void qqbar_denominator(fmpz_t res, const qqbar_t y)

  Sets res to the denominator of y, i.e. the leading coefficient of the minimal polynomial of y.

void qqbar_numerator(qqbar_t res, const qqbar_t y)

  Sets res to the numerator of y, i.e. y multiplied by its denominator.

## Conjugates

void qqbar_conjugates(qqbar_ptr res, const qqbar_t x)

  Sets the entries of the vector res to the d algebraic conjugates of x, including x itself, where d is the degree of x. The output is sorted in a canonical order (as defined by `qqbar_cmp_root_order()`).

## Polynomial evaluation

void _qqbar_evaluate_fmpq_poly(qqbar_t res, const fmpz *poly, const fmpz_t den, slong len, const qqbar_t x)
void qqbar_evaluate_fmpq_poly(qqbar_t res, const fmpq_poly_t poly, const qqbar_t x)
void _qqbar_evaluate_fmpz_poly(qqbar_t res, const fmpz *poly, slong len, const qqbar_t x)
void qqbar_evaluate_fmpz_poly(qqbar_t res, const fmpz_poly_t poly, const qqbar_t x)

  Sets res to the value of the given polynomial poly evaluated at the algebraic number x. These methods detect simple special cases and automatically reduce poly if its degree is greater or equal to that of the minimal polynomial of x. In the generic case, evaluation is done by computing minimal polynomials of representation matrices.

int qqbar_evaluate_fmpz_mpoly_iter(qqbar_t res, const fmpz_mpoly_t poly, qqbar_srcptr x, slong deg_limit, slong bits_limit, const fmpz_mpoly_ctx_t ctx)
int qqbar_evaluate_fmpz_mpoly_horner(qqbar_t res, const fmpz_mpoly_t poly, qqbar_srcptr x, slong deg_limit, slong bits_limit, const fmpz_mpoly_ctx_t ctx)
int qqbar_evaluate_fmpz_mpoly(qqbar_t res, const fmpz_mpoly_t poly, qqbar_srcptr x, slong deg_limit, slong bits_limit, const fmpz_mpoly_ctx_t ctx)

  Sets res to the value of poly evaluated at the algebraic numbers given in the vector x. The number of variables is defined by the context object ctx.

  The parameters deg_limit and bits_limit define evaluation limits: if any temporary result exceeds these limits (not necessarily the final value, in case of cancellation), the evaluation is aborted and 0 (failure) is returned. If evaluation succeeds, 1 is returned.

  The iter version iterates over all terms in succession and computes the powers that appear. The horner version uses a multivariate implementation of the Horner scheme. The default algorithm currently uses the Horner scheme.

## Polynomial roots

void qqbar_roots_fmpz_poly(qqbar_ptr res, const fmpz_poly_t poly, int flags)
void qqbar_roots_fmpq_poly(qqbar_ptr res, const fmpq_poly_t poly, int flags)

  Sets the entries of the vector res to the d roots of the polynomial poly. Roots with multiplicity appear with repetition in the output array. By default, the roots will be sorted in a convenient canonical order (as defined by `qqbar_cmp_root_order()`). Instances of a repeated root always appear consecutively.

  The following flags are supported:

  - QQBAR_ROOTS_IRREDUCIBLE - if set, poly is assumed to be irreducible (it may still have constant content), and no polynomial factorization is performed internally.

  - QQBAR_ROOTS_UNSORTED - if set, the roots will not be guaranteed to be sorted (except for repeated roots being listed consecutively).

void qqbar_eigenvalues_fmpz_mat(qqbar_ptr res, const fmpz_mat_t mat, int flags)
void qqbar_eigenvalues_fmpq_mat(qqbar_ptr res, const fmpq_mat_t mat, int flags)

  Sets the entries of the vector res to the eigenvalues of the square matrix mat. These functions compute the characteristic polynomial of mat and then call `qqbar_roots_fmpz_poly()` with the same flags.

int _qqbar_roots_poly_squarefree(qqbar_ptr roots, qqbar_srcptr coeffs, slong len, slong deg_limit, slong bits_limit)

  Writes to the vector roots the \(d\) roots of the polynomial with algebraic number coefficients coeffs of length len (\(d = len - 1\)).

  Given the polynomial \(f = a_0 + \ldots + a_d x^d\) with coefficients in \(\overline{\mathbb{Q}}\), we construct an annihilating polynomial with coefficients in \(\mathbb{Q}\) as \(g = \prod (\tilde a_0 + \ldots + \tilde a_d x^d)\) where the product is taken over all combinations of algebraic conjugates \(\tilde a_k\) of the input coefficients. The polynomial \(g\) is subsequently factored to find candidate roots.

  The leading coefficient \(a_d\) must be nonzero and the polynomial \(f\) polynomial must be squarefree. To compute roots of a general polynomial which may have repeated roots, it is necessary to perform a squarefree factorization before calling this function. An option is to call `gr_poly_roots()` with a `qqbar` context object, which wraps this function and takes care of the initial squarefree factorization.

  Since the product \(g\) can explode in size very quickly, the deg_limit and bits_limit parameters allow bounding the degree and working precision. The function returns 1 on success and 0 on failure indicating that such a limit has been exceeded. Setting nonpositive values for the limits removes the restrictions; however, the function can still fail and return 0 in that case if \(g\) exceeds machine size.

Note: to compute algebraic number roots of polynomials of various other types, use `gr_poly_roots_other()`.

## Roots of unity and trigonometric functions

The following functions use word-size integers p and q instead of fmpq_t instances to express rational numbers. This is to emphasize that the computations are feasible only with small q in this representation of algebraic numbers since the associated minimal polynomials have degree \(O(q)\). The input p and q do not need to be reduced a priori, but should not be close to the word boundaries (they may be added and subtracted internally).

void qqbar_root_of_unity(qqbar_t res, slong p, ulong q)

  Sets res to the root of unity \(e^{2 \pi i p / q}\).

int qqbar_is_root_of_unity(slong *p, ulong *q, const qqbar_t x)

  If x is not a root of unity, returns 0. If x is a root of unity, returns 1. If p and q are not NULL and x is a root of unity, this also sets p and q to the minimal integers with \(0 \le p < q\) such that \(x = e^{2 \pi i p / q}\).

void qqbar_exp_pi_i(qqbar_t res, slong p, ulong q)

  Sets res to the root of unity \(e^{\pi i p / q}\).

void qqbar_cos_pi(qqbar_t res, slong p, ulong q)
void qqbar_sin_pi(qqbar_t res, slong p, ulong q)
int qqbar_tan_pi(qqbar_t res, slong p, ulong q)
int qqbar_cot_pi(qqbar_t res, slong p, ulong q)
int qqbar_sec_pi(qqbar_t res, slong p, ulong q)
int qqbar_csc_pi(qqbar_t res, slong p, ulong q)

  Sets res to the trigonometric function \(\cos(\pi x)\), \(\sin(\pi x)\), etc., with \(x = \tfrac{p}{q}\). The functions tan, cot, sec and csc return the flag 1 if the value exists, and return 0 if the evaluation point is a pole of the function.

int qqbar_log_pi_i(slong *p, ulong *q, const qqbar_t x)

  If \(y = \operatorname{log}(x) / (\pi i)\) is algebraic, and hence necessarily rational, sets \(y = p / q\) to the reduced such fraction with \(-1 < y \le 1\) and returns 1. If y is not algebraic, returns 0.

int qqbar_atan_pi(slong *p, ulong *q, const qqbar_t x)

  If \(y = \operatorname{atan}(x) / \pi\) is algebraic, and hence necessarily rational, sets \(y = p / q\) to the reduced such fraction with \(|y| < \tfrac{1}{2}\) and returns 1. If y is not algebraic, returns 0.

int qqbar_asin_pi(slong *p, ulong *q, const qqbar_t x)

  If \(y = \operatorname{asin}(x) / \pi\) is algebraic, and hence necessarily rational, sets \(y = p / q\) to the reduced such fraction with \(|y| \le \tfrac{1}{2}\) and returns 1. If y is not algebraic, returns 0.

int qqbar_acos_pi(slong *p, ulong *q, const qqbar_t x)

  If \(y = \operatorname{acos}(x) / \pi\) is algebraic, and hence necessarily rational, sets \(y = p / q\) to the reduced such fraction with \(0 \le y \le 1\) and returns 1. If y is not algebraic, returns 0.

int qqbar_acot_pi(slong *p, ulong *q, const qqbar_t x)

  If \(y = \operatorname{acot}(x) / \pi\) is algebraic, and hence necessarily rational, sets \(y = p / q\) to the reduced such fraction with \(-\tfrac{1}{2} < y \le \tfrac{1}{2}\) and returns 1. If y is not algebraic, returns 0.

int qqbar_asec_pi(slong *p, ulong *q, const qqbar_t x)

  If \(y = \operatorname{asec}(x) / \pi\) is algebraic, and hence necessarily rational, sets \(y = p / q\) to the reduced such fraction with \(0 \le y \le 1\) and returns 1. If y is not algebraic, returns 0.

int qqbar_acsc_pi(slong *p, ulong *q, const qqbar_t x)

  If \(y = \operatorname{acsc}(x) / \pi\) is algebraic, and hence necessarily rational, sets \(y = p / q\) to the reduced such fraction with \(-\tfrac{1}{2} \le y \le \tfrac{1}{2}\) and returns 1. If y is not algebraic, returns 0.

## Guessing and simplification

int qqbar_guess(qqbar_t res, const acb_t z, slong max_deg, slong max_bits, int flags, slong prec)

  Attempts to find an algebraic number res of degree at most max_deg and height at most max_bits bits matching the numerical enclosure z. The return flag indicates success. This is only a heuristic method, and the return flag neither implies a rigorous proof that res is the correct result, nor a rigorous proof that no suitable algebraic number with the given max_deg and max_bits exists. (Proof of nonexistence could in principle be computed, but this is not yet implemented.)

  The working precision prec should normally be the same as the precision used to compute z. It does not make much sense to run this algorithm with precision smaller than O(max_deg · max_bits).

  This function does a single iteration at the target max_deg, max_bits, and prec. For best performance, one should invoke this function repeatedly with successively larger parameters when the size of the intended solution is unknown or may be much smaller than a worst-case bound.

int qqbar_express_in_field(fmpq_poly_t res, const qqbar_t alpha, const qqbar_t x, slong max_bits, int flags, slong prec)

  Attempts to express x in the number field generated by alpha, returning success (0 or 1). On success, res is set to a polynomial f of degree less than the degree of alpha and with height (counting both the numerator and the denominator, when the coefficients of g are put on a common denominator) bounded by max_bits bits, such that \(f(\alpha) = x\).

  (Exception: the max_bits parameter is currently ignored if x is rational, in which case res is just set to the value of x.)

  This function looks for a linear relation heuristically using a working precision of prec bits. If x is expressible in terms of alpha, then this function is guaranteed to succeed when prec is taken large enough. The identity \(f(\alpha) = x\) is checked rigorously, i.e. a return value of 1 implies a proof of correctness. In principle, choosing a sufficiently large prec can be used to prove that x does not lie in the field generated by alpha, but the present implementation does not support doing so automatically.

  This function does a single iteration at the target max_bits and and prec. For best performance, one should invoke this function repeatedly with successively larger parameters when the size of the intended solution is unknown or may be much smaller than a worst-case bound.

## Symbolic expressions and conversion to radicals

void qqbar_get_quadratic(fmpz_t a, fmpz_t b, fmpz_t c, fmpz_t q, const qqbar_t x, int factoring)

  Assuming that x has degree 1 or 2, computes integers a, b, c and q such that

  > \[x = \frac{a + b \sqrt{c}}{q}\]

  and such that c is not a perfect square, q is positive, and q has no content in common with both a and b. In other words, this determines a quadratic field \(\mathbb{Q}(\sqrt{c})\) containing x, and then finds the canonical reduced coefficients a, b and q expressing x in this field. For convenience, this function supports rational x, for which b and c will both be set to zero. The following remarks apply to irrationals.

  The radicand c will not be a perfect square, but will not automatically be squarefree since this would require factoring the discriminant. As a special case, c will be set to \(-1\) if x is a Gaussian rational number. Otherwise, behavior is controlled by the factoring parameter.

  - If factoring is 0, no factorization is performed apart from removing powers of two.

  - If factoring is 1, a complete factorization is performed (c will be minimal). This can be very expensive if the discriminant is large.

  - If factoring is 2, a smooth factorization is performed to remove small factors from c. This is a tradeoff that provides pretty output in most cases while avoiding extreme worst-case slowdown. The smooth factorization guarantees finding all small factors (up to some trial division limit determined internally by FLINT), but large factors are only found heuristically.

int qqbar_set_fexpr(qqbar_t res, const fexpr_t expr)

  Sets res to the algebraic number represented by the symbolic expression expr, returning 1 on success and 0 on failure.

  This function performs a “static” evaluation using qqbar arithmetic, supporting only closed-form expressions with explicitly algebraic subexpressions. It can be used to recover values generated by `qqbar_get_expr_formula()` and variants. For evaluating more complex expressions involving other types of values or requiring symbolic simplifications, the user should preprocess expr so that it is in a form which can be parsed by `qqbar_set_fexpr()`.

  The following expressions are supported:

  - Integer constants

  - Arithmetic operations with algebraic operands

  - Square roots of algebraic numbers

  - Powers with algebraic base and exponent an explicit rational number

  - NumberI, GoldenRatio, RootOfUnity

  - Floor, Ceil, Abs, Sign, Csgn, Conjugate, Re, Im, Max, Min

  - Trigonometric functions with argument an explicit rational number times Pi

  - Exponentials with argument an explicit rational number times Pi * NumberI

  - The Decimal() constructor

  - AlgebraicNumberSerialized() (assuming valid data, which is not checked)

  - PolynomialRootIndexed()

  - PolynomialRootNearest()

  Examples of formulas that are not supported, despite the value being an algebraic number:

  - `Pi - Pi` (general transcendental simplifications are not performed)

  - `1 / Infinity` (only numbers are handled)

  - `Sum(n, For(n, 1, 10))` (only static evaluation is performed)

void qqbar_get_fexpr_repr(fexpr_t res, const qqbar_t x)

  Sets res to a symbolic expression reflecting the exact internal representation of x. The output will have the form `AlgebraicNumberSerialized(List(coeffs), enclosure)`. The output can be converted back to a `qqbar_t` value using `qqbar_set_fexpr()`. This is the recommended format for serializing algebraic numbers as it requires minimal computation, but it has the disadvantage of not being human-readable.

void qqbar_get_fexpr_root_nearest(fexpr_t res, const qqbar_t x)

  Sets res to a symbolic expression unambiguously describing x in the form `PolynomialRootNearest(List(coeffs), point)` where point is an approximation of x guaranteed to be closer to x than any conjugate root. The output can be converted back to a `qqbar_t` value using `qqbar_set_fexpr()`. This is a useful format for human-readable presentation, but serialization and deserialization can be expensive.

void qqbar_get_fexpr_root_indexed(fexpr_t res, const qqbar_t x)

  Sets res to a symbolic expression unambiguously describing x in the form `PolynomialRootIndexed(List(coeffs), index)` where index is the index of x among its conjugate roots in the builtin root sort order. The output can be converted back to a `qqbar_t` value using `qqbar_set_fexpr()`. This is a useful format for human-readable presentation when the numerical value is important, but serialization and deserialization can be expensive.

int qqbar_get_fexpr_formula(fexpr_t res, const qqbar_t x, ulong flags)

  Attempts to express the algebraic number x as a closed-form expression using arithmetic operations, radicals, and possibly exponentials or trigonometric functions, but without using `PolynomialRootNearest` or `PolynomialRootIndexed`. Returns 0 on failure and 1 on success.

  The flags parameter toggles different methods for generating formulas. It can be set to any combination of the following. If flags is 0, only rational numbers will be handled.

  QQBAR_FORMULA_ALL

    Toggles all methods (potentially expensive).

  QQBAR_FORMULA_GAUSSIANS

    Detect Gaussian rational numbers \(a + bi\).

  QQBAR_FORMULA_QUADRATICS

    Solve quadratics in the form \(a + b \sqrt{d}\).

  QQBAR_FORMULA_CYCLOTOMICS

    Detect elements of cyclotomic fields. This works by trying plausible cyclotomic fields (based on the degree of the input), using LLL to find candidate number field elements, and certifying candidates through an exact computation. Detection is heuristic and is not guaranteed to find all cyclotomic numbers.

  QQBAR_FORMULA_CUBICS
  QQBAR_FORMULA_QUARTICS
  QQBAR_FORMULA_QUINTICS

    Solve polynomials of degree 3, 4 and (where applicable) 5 using cubic, quartic and quintic formulas (not yet implemented).

  QQBAR_FORMULA_DEPRESSION

    Use depression to try to generate simpler numbers.

  QQBAR_FORMULA_DEFLATION

    Use deflation to try to generate simpler numbers. This allows handling number of the form \(a^{1/n}\) where a can be represented in closed form.

  QQBAR_FORMULA_SEPARATION

    Try separating real and imaginary parts or sign and magnitude of complex numbers. This allows handling numbers of the form \(a + bi\) or \(m \cdot s\) (with \(m > 0, |s| = 1\)) where a and b or m and s can be represented in closed form. This is only attempted as a fallback after other methods fail: if an explicit Cartesian or magnitude-sign represented is desired, the user should manually separate the number into complex parts before calling `qqbar_get_fexpr_formula()`.

  QQBAR_FORMULA_EXP_FORM
  QQBAR_FORMULA_TRIG_FORM
  QQBAR_FORMULA_RADICAL_FORM
  QQBAR_FORMULA_AUTO_FORM

    Select output form for cyclotomic numbers. The auto form (equivalent to no flags being set) results in radicals for numbers of low degree, trigonometric functions for real numbers, and complex exponentials for nonreal numbers. The other flags (not fully implemented) can be used to force exponential form, trigonometric form, or radical form.

## Internal functions

void qqbar_fmpz_poly_composed_op(fmpz_poly_t res, const fmpz_poly_t A, const fmpz_poly_t B, int op)

  Given nonconstant polynomials A and B, sets res to a polynomial whose roots are \(a+b\), \(a-b\), \(ab\) or \(a/b\) for all roots a of A and all roots b of B. The parameter op selects the arithmetic operation: 0 for addition, 1 for subtraction, 2 for multiplication and 3 for division. If op is 3, B must not have zero as a root.

void qqbar_binary_op(qqbar_t res, const qqbar_t x, const qqbar_t y, int op)

  Performs a binary operation using a generic algorithm. This does not check for special cases.

int _qqbar_validate_uniqueness(acb_t res, const fmpz_poly_t poly, const acb_t z, slong max_prec)

  Given z known to be an enclosure of at least one root of poly, certifies that the enclosure contains a unique root, and in that case sets res to a new (possibly improved) enclosure for the same root, returning 1. Returns 0 if uniqueness cannot be certified.

  The enclosure is validated by performing a single step with the interval Newton method. The working precision is determined from the accuracy of z, but limited by max_prec bits.

  This method slightly inflates the enclosure z to improve the chances that the interval Newton step will succeed. Uniqueness on this larger interval implies uniqueness of the original interval, but not existence; when existence has not been ensured a priori, `_qqbar_validate_existence_uniqueness()` should be used instead.

int _qqbar_validate_existence_uniqueness(acb_t res, const fmpz_poly_t poly, const acb_t z, slong max_prec)

  Given any complex interval z, certifies that the enclosure contains a unique root of poly, and in that case sets res to a new (possibly improved) enclosure for the same root, returning 1. Returns 0 if existence and uniqueness cannot be certified.

  The enclosure is validated by performing a single step with the interval Newton method. The working precision is determined from the accuracy of z, but limited by max_prec bits.

void _qqbar_enclosure_raw(acb_t res, const fmpz_poly_t poly, const acb_t z, slong prec)
void qqbar_enclosure_raw(acb_t res, const qqbar_t x, slong prec)

  Sets res to an enclosure of x accurate to about prec bits (the actual accuracy can be slightly lower, or higher).

  This function uses repeated interval Newton steps to polish the initial enclosure z, doubling the working precision each time. If any step fails to improve the accuracy significantly, the root is recomputed from scratch to higher precision.

  If the initial enclosure is accurate enough, res is set to this value without rounding and without further computation.

int _qqbar_acb_lindep(fmpz *rel, acb_srcptr vec, slong len, int check, slong prec)

  Attempts to find an integer vector rel giving a linear relation between the elements of the real or complex vector vec, using the LLL algorithm.

  The working precision is set to the minimum of prec and the relative accuracy of vec (that is, the difference between the largest magnitude and the largest error magnitude within vec). 95% of the bits within the working precision are used for the LLL matrix, and the remaining 5% bits are used to validate the linear relation by evaluating the linear combination and checking that the resulting interval contains zero. This validation does not prove the existence or nonexistence of a linear relation, but it provides a quick heuristic way to eliminate spurious relations.

  If check is set, the return value indicates whether the validation was successful; otherwise, the return value simply indicates whether the algorithm was executed normally (failure may occur, for example, if the input vector is non-finite).

  In principle, this method can be used to produce a proof that no linear relation exists with coefficients up to a specified bit size, but this has not yet been implemented.