fmpq.h

Function versions of fmpq_numref and fmpq_denref, for use from languages that cannot expand C macros.

Set n to a copy of the numerator or the denominator of q.

Initialises x and sets it to zero. This does not allocate memory.

Frees the memory used by x. It must be initialised again before reuse.

Put a fraction with nonzero denominator in canonical form: divide numerator and denominator by their GCD and make the denominator positive. A zero numerator gives 0/1. The result is undefined if the denominator is zero. In the underscore version, num and den must not be aliased.

Return nonzero if the fraction is in canonical form, that is, the denominator is positive and coprime to the numerator, and zero is represented as 0/1.

Sets dest to a copy of src. No canonicalisation is done, so this also copies non-canonical fractions.

Swaps op1 and op2 efficiently (only the structs are exchanged).

Set res to 0 or 1.

Sets q to the integer n.

Set the result to the canonical form of p/q. Requires q \ne 0. In the underscore versions rnum and rden must not be aliased.

Sets res to the canonical form of p/q, for any integers p and q \ne 0 (not checked).

Convert from and to GMP rationals; fmpq_get_mpz_frac sets a and b to the numerator and denominator of c. The value is copied as it is, without canonicalisation. These functions are declared only if gmp.h is included before any FLINT header.

Sets dest to the integer or fraction "p" or "p/q" in the string s, written in base base as accepted by mpq_set_str. Returns 0 on success and -1 if the string is not valid, in which case dest is not changed.

Returns f as a double, rounded towards zero (by mpq_get_d). The result is system dependent if |f| is outside the range of double.

Sets dest to src rounded in the direction rnd at the precision of dest, and returns the ternary value of MPFR (the sign of the rounding error). Declared only if mpfr.h is included before any FLINT header.

Write x in base b \in [2, 36] as "p/q", or as "p" if the denominator is 1. If str is NULL, a buffer is allocated with flint_malloc (to be freed with flint_free) and returned; otherwise str must have room for the digits of both components plus three characters, and is returned.

Make the uninitialised mpq_t z a read-only view of f, without copying large components, and release it. f must not change while z is in use, and z must not be modified. Declared only if gmp.h is included before any FLINT header.

Make the uninitialised fmpq_t f a read-only view of the mpq_t z, and release it. z must not change while f is in use, and f must not be modified. fmpq_init_set_readonly is declared only if gmp.h is included before any FLINT header.

Print the fraction to stdout or to file, in decimal, as p/q, or as p if the denominator is 1. Return a positive value on success and a non-positive value on failure.

Set the result to a random canonical fraction whose numerator and denominator have at most bits bits. The distribution favours special values (\pm 1, \pm 2, denominators 1 and 2, values with long runs of bits) that trigger corner cases. For bits equal to 1 the result is an integer. num and den must not be aliased.

As fmpq_randtest, but the result is nonzero. Throws if bits is zero.

Choose a numerator and a nonzero denominator of exactly bits bits each and canonicalise, so the components of the result can have fewer bits. num and den must not be aliased.

Return nonzero if x is 0, 1 or \pm 1 respectively.

Return nonzero if x = y. For canonical inputs this is a comparison of the components.

Returns -1, 0 or 1 according to the sign of x.

Return a negative value, zero or a positive value according as x < y, x = y or x > y.

Compare p/q with r/s, r or c, returning a negative value, zero or a positive value. The denominators must be positive; the fractions need not be in lowest terms.

The height of x = p/q is \max(|p|, q). The first function sets height to it, the second returns its number of bits.

Set dest to -\mathit{src} or |\mathit{src}|.

Set res to \mathit{op1} + \mathit{op2}, \mathit{op1} - \mathit{op2}, \mathit{op1} \cdot \mathit{op2} or \mathit{op1} / \mathit{op2}. fmpq_div throws if op2 is zero.

Set (rnum, rden) to the canonical sum, difference, product or quotient of the canonical fractions (op1num, op1den) and (op2num, op2den). _fmpq_div requires a nonzero divisor (not checked).

Set (rnum, rden) to the canonical sum or product of p_1/q_1 and p_2/q_2, which must be canonical with all four values of absolute value at most COEFF_MAX (that is, small fmpz values). The intermediate results are computed in two words. These are the fast paths of the general functions.

Set res to \mathit{op1} + c or \mathit{op1} - c for an integer c.

Set (rnum, rden) to p/q + r or p/q - r. No GCD is needed: (p \pm qr)/q is canonical when p/q is.

Set the result to the product of a fraction and an integer.

Sets res to \mathit{op}/x. Throws if x is zero.

Set res to x \cdot 2^{\mathit{exp}} or x / 2^{\mathit{exp}}, cancelling powers of two from the denominator or the numerator first.

Set the result to \mathit{res} + \mathit{op1} \cdot \mathit{op2} or \mathit{res} - \mathit{op1} \cdot \mathit{op2}. The product is formed in a temporary, so any aliasing is allowed (except, in the underscore versions, a numerator with a denominator).

Sets dest to 1/\mathit{src} by swapping the components and fixing the sign. src must be nonzero; this is not checked, and zero gives the invalid value 1/0.

Set the result to \mathit{op}^e, with 0^0 = 1. fmpq_pow_si throws if \mathit{op} = 0 and e < 0; the underscore version does not check.

Sets a to b^e and returns 1, or returns 0 (leaving a unchanged) if e does not fit in a slong and b \notin \{0, 1, -1\}, in which case the result could not be stored. Throws if b = 0 and e < 0.

Set the result to the GCD of p/q and r/s, defined as the canonicalisation of \gcd(ps, qr)/(qs). For canonical inputs this is \gcd(p, r)/\operatorname{lcm}(q, s), the largest rational g such that p/q and r/s are both integer multiples of g; it is nonnegative, and zero only if both inputs are zero.

Set g = \gcd(a, b) as in fmpq_gcd and the coprime integer cofactors \bar a = a/g and \bar b = b/g. If a = b = 0, then g = 0 and \bar a = \bar b = 0. Unlike _fmpq_gcd, the inputs must be canonical.

If the denominator d of x = n/d is invertible modulo m = mod, set res to the residue a \in [0, m) with n \equiv ad \pmod m and return 1; otherwise return 0 and leave an unspecified value in res. Requires m > 0.

Given m > 2, 0 \le a < m and N, D > 0 with 2ND < m, find the fraction n/d with |n| \le N, 0 < d \le D, \gcd(n, d) = 1 and n \equiv ad \pmod m, which is unique if it exists. Return 1 and set the result if it exists, otherwise return 0.

Rational reconstruction with the balanced bounds N = D = \lfloor\sqrt{\lfloor (m-1)/2 \rfloor}\rfloor, which satisfy 2ND < m.

Given a nonnegative canonical x, set the result to the next element of the enumeration of the nonnegative rationals by increasing height \max(p, q): for q = 1, 2, 3, \ldots the fractions p/q and q/p for 1 \le p < q coprime to q, in increasing order of p. Starting from zero, every nonnegative rational is generated exactly once: 0, 1, 1/2, 2, 1/3, 3, 2/3, 3/2, 1/4, 4, 3/4, 4/3, 1/5, 5, 2/5, \ldots

As fmpq_next_minimal, with each positive value followed by its negative: 0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, \ldots Starting from zero, this generates every rational exactly once, by increasing height.

Given a nonnegative canonical x, set the result to the next fraction in the breadth-first traversal of the Calkin–Wilf tree [CalkinWilf2000]: 0, 1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, \ldots Starting from zero, every nonnegative rational is generated exactly once.

As fmpq_next_calkin_wilf, with each positive value followed by its negative: 0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, 3/2, \ldots

Set l and r to the fractions immediately below and above x among the fractions with denominator at most Q (the Farey sequence of order Q, extended to all of \mathbb{Q}). Throws if the denominator of x exceeds Q. Any aliasing is allowed.

Set x to the simplest fraction in the closed interval between l and r, where a_1/b_1 is simpler than a_2/b_2 if b_1 < b_2, or b_1 = b_2 and a_1 < a_2. The endpoints need not be canonical, but their denominators must be positive; the result is canonical.

Write the first k \le n terms of the simple continued fraction of x to c and the remainder to rem, and return k. The expansion generated is the shorter of the two. fmpq_get_cfrac uses a half-GCD algorithm and runs in quasi-linear time, fmpq_get_cfrac_naive uses repeated division. See Continued fractions.

Sets x to the value of the continued fraction [c_0; c_1, \ldots, c_{n-1}], for n > 0 and positive c_1, \ldots, c_{n-1}. Uses a balanced product of 2 \times 2 matrices. See Continued fractions.

Returns an upper bound for the number of terms of the continued fraction of x: 1 for an integer, otherwise \lfloor 1.4404\, b \rfloor + 2 where b is the bit length of the denominator. The bound is not sharp.

Set the result to the harmonic number H_n = 1 + 1/2 + \cdots + 1/n, with H_0 = 0. Throws if n \ge 2^{\mathtt{FLINT\_BITS} - 1}.

Set s to s(h, k), for any integers h and k. The naive version evaluates the defining sum with k terms. fmpq_dedekind_sum uses the continued fraction of k/h and costs about as much as an extended GCD.

Initialise, clear, set to the identity, test for the identity, and return the largest bit length of an entry.

Set M to MN, MN, MN^{-1}, ME_q, E_q M or ME_q^{-1}, where E_q = \left(\begin{smallmatrix} q & 1 \\ 1 & 0 \end{smallmatrix}\right) and the inverses use \det N = \pm 1.

Add N^{-1}x to the vector y, or M^{-1}B to the matrix A. The inputs xa, xb and B may be overwritten.

Memory management and appending of terms for _fmpq_cfrac_list_t.

Memory management for _fmpq_ball_t.

Returns nonzero if the endpoints are canonical and the ball lies in (1, \infty).

Given a ball x > 1, append to s continued fraction terms valid for every number in x, multiply M on the right by the corresponding elementary matrices, and replace x by its image under the inverse transformation. With needM zero and an exact ball, M may be left unchanged.

Half-GCD step: for a = x_num > b = x_den > 0, generate the terms valid for every real number in the open interval (a/(b+1), (a+1)/b), append them to s, multiply M on the right by their matrices and replace (a, b) by the reduced pair, in place.