Rational numbers (opaque pointer)

newFmpq

Construct a Fmpq.

withFmpq x f

Execute f with x.

withFmpqNum x f

Execute f on the numerator of x.

withFmpqDen x f

Execute f on the denominator of x.

withNewFmpq f

Execture f with a new Fmpq.

fmpq_init x

Initialises the fmpq_t variable x for use. Its value is set to 0.

fmpq_clear x

Clears the fmpq_t variable x. To use the variable again, it must be re-initialised with fmpq_init.

fmpq_canonicalise res

Puts res in canonical form: the numerator and denominator are reduced to lowest terms, and the denominator is made positive. If the numerator is zero, the denominator is set to one.

If the denominator is zero, the outcome of calling this function is undefined, regardless of the value of the numerator.

_fmpq_canonicalise num den

Does the same thing as fmpq_canonicalise, but for numerator and denominator given explicitly as fmpz_t variables. Aliasing of num and den is not allowed.

fmpq_is_canonical x

Returns nonzero if fmpq_t x is in canonical form (as produced by fmpq_canonicalise), and zero otherwise.

_fmpq_is_canonical num den

Does the same thing as fmpq_is_canonical, but for numerator and denominator given explicitly as fmpz_t variables.

fmpq_set dest src

Sets dest to a copy of src. No canonicalisation is performed.

fmpq_swap op1 op2

Swaps the two rational numbers op1 and op2.

fmpq_neg dest src

Sets dest to the additive inverse of src.

fmpq_abs dest src

Sets dest to the absolute value of src.

fmpq_zero res

Sets the value of res to 0.

fmpq_one res

Sets the value of res to \(1\).

fmpq_is_zero res

Returns nonzero if res has value 0, and returns zero otherwise.

fmpq_is_one res

Returns nonzero if res has value \(1\), and returns zero otherwise.

fmpq_is_pm1 res

Returns nonzero if res has value \(\pm{1}\) and zero otherwise.

fmpq_equal x y

Returns nonzero if x and y are equal, and zero otherwise. Assumes that x and y are both in canonical form.

fmpq_sgn x

Returns the sign of the rational number \(x\).

fmpq_cmp x y

Returns negative if \(x < y\), zero if \(x = y\), and positive if \(x > y\).

fmpq_cmp_si x y

Returns negative if \(x < y\), zero if \(x = y\), and positive if \(x > y\).

fmpq_equal_ui x y

Returns \(1\) if \(x = y\), otherwise returns \(0\).

fmpq_equal_si x y

Returns \(1\) if \(x = y\), otherwise returns \(0\).

fmpq_height height x

Sets height to the height of \(x\), defined as the larger of the absolute values of the numerator and denominator of \(x\).

fmpq_height_bits x

Returns the number of bits in the height of \(x\).

fmpq_set_fmpz_frac res p q

Sets res to the canonical form of the fraction p / q. This is equivalent to assigning the numerator and denominator separately and calling fmpq_canonicalise.

fmpq_get_mpz_frac a b c

Sets a, b to the numerator and denominator of c respectively.

fmpq_set_si res p q

Sets res to the canonical form of the fraction p / q.

_fmpq_set_si rnum rden p q

Sets (rnum, rden) to the canonical form of the fraction p / q. rnum and rden may not be aliased.

fmpq_set_ui res p q

Sets res to the canonical form of the fraction p / q.

_fmpq_set_ui rnum rden p q

Sets (rnum, rden) to the canonical form of the fraction p / q. rnum and rden may not be aliased.

fmpq_set_mpq dest src

Sets the value of dest to that of the mpq_t variable src.

fmpq_set_str dest s base

Sets the value of dest to the value represented in the string s in base base.

Returns 0 if no error occurs. Otherwise returns -1 and dest is set to zero.

fmpq_init_set_mpz_frac_readonly z p q

Assuming z is an fmpz_t which will not be cleaned up, this temporarily copies p and q into the numerator and denominator of z for read only operations only. The user must not run fmpq_clear on z.

fmpq_get_d f

Returns \(f\) as a double, rounding towards zero if f cannot be represented exactly. The return is system dependent if f is too large or too small to fit in a double.

fmpq_get_mpq dest src

Sets the value of dest

fmpq_get_mpfr dest src rnd

Sets the MPFR variable dest to the value of src, rounded to the nearest representable binary floating-point value in direction rnd. Returns the sign of the rounding, according to MPFR conventions.

_fmpq_get_str str b num den

Prints the string representation of \(x\) in base \(b \in [2, 36]\) to a suitable buffer.

If str is not NULL, this is used as the buffer and also the return value. If str is NULL, allocates sufficient space and returns a pointer to the string.

flint_mpq_init_set_readonly z f

Sets the uninitialised mpq_t \(z\) to the value of the readonly fmpq_t \(f\).

Note that it is assumed that \(f\) does not change during the lifetime of \(z\).

The rational \(z\) has to be cleared by a call to flint_mpq_clear_readonly.

The suggested use of the two functions is as follows:

fmpq_t f;
...
{
    mpq_t z;

    flint_mpq_init_set_readonly(z, f);
    foo(..., z);
    flint_mpq_clear_readonly(z);
}

This provides a convenient function for user code, only requiring to work with the types fmpq_t and mpq_t.

flint_mpq_clear_readonly z

Clears the readonly mpq_t \(z\).

fmpq_init_set_readonly f z

Sets the uninitialised fmpq_t \(f\) to a readonly version of the rational \(z\).

Note that the value of \(z\) is assumed to remain constant throughout the lifetime of \(f\).

The fmpq_t \(f\) has to be cleared by calling the function fmpq_clear_readonly.

The suggested use of the two functions is as follows:

mpq_t z;
...
{
    fmpq_t f;

    fmpq_init_set_readonly(f, z);
    foo(..., f);
    fmpq_clear_readonly(f);
}

fmpq_clear_readonly f

Clears the readonly fmpq_t \(f\).

fmpq_fprint file x

Prints x as a fraction to the stream file. The numerator and denominator are printed verbatim as integers, with a forward slash (/) printed in between.

In case of success, returns a positive number. In case of failure, returns a non-positive number.

_fmpq_fprint file num den

Does the same thing as fmpq_fprint, but for numerator and denominator given explicitly as fmpz_t variables.

In case of success, returns a positive number. In case of failure, returns a non-positive number.

fmpq_print x

Prints x as a fraction. The numerator and denominator are printed verbatim as integers, with a forward slash (/) printed in between.

In case of success, returns a positive number. In case of failure, returns a non-positive number.

_fmpq_print num den

Does the same thing as fmpq_print, but for numerator and denominator given explicitly as fmpz_t variables.

In case of success, returns a positive number. In case of failure, returns a non-positive number.

fmpq_randtest res state bits

Sets res to a random value, with numerator and denominator having up to bits bits. The fraction will be in canonical form. This function has an increased probability of generating special values which are likely to trigger corner cases.

_fmpq_randtest num den state bits

Does the same thing as fmpq_randtest, but for numerator and denominator given explicitly as fmpz_t variables. Aliasing of num and den is not allowed.

fmpq_randtest_not_zero res state bits

As per fmpq_randtest, but the result will not be \(0\). If bits is set to \(0\), an exception will result.

fmpq_randbits res state bits

Sets res to a random value, with numerator and denominator both having exactly bits bits before canonicalisation, and then puts res in canonical form. Note that as a result of the canonicalisation, the resulting numerator and denominator can be slightly smaller than bits bits.

_fmpq_randbits num den state bits

Does the same thing as fmpq_randbits, but for numerator and denominator given explicitly as fmpz_t variables. Aliasing of num and den is not allowed.

fmpq_add res op1 op2

Sets res respectively to op1 + op2, op1 - op2, op1 * op2, or op1 / op2. Assumes that the inputs are in canonical form, and produces output in canonical form. Division by zero results in an error. Aliasing between any combination of the variables is allowed.

_fmpq_add rnum rden op1num op1den op2num op2den

Sets (rnum, rden) to the canonical form of the sum, difference, product or quotient respectively of the fractions represented by (op1num, op1den) and (op2num, op2den). Aliasing between any combination of the variables is allowed, whilst no numerator is aliased with a denominator.

_fmpq_add_si rnum rden p q r

Sets (rnum, rden) to the canonical form of the sum or difference respectively of the fractions represented by (p, q) and (r, 1). Numerators may not be aliased with denominators.

_fmpq_mul_si rnum rden p q r

Sets (rnum, rden) to the product of (p, q) and the integer \(r\).

fmpq_add_si res op1 c

Sets res to the sum or difference respectively, of the fraction op1 and the integer \(c\).

_fmpq_mul_ui rnum rden p q r

Sets (rnum, rden) to the product of (p, q) and the integer \(r\).

fmpq_mul_ui res op1 c

Sets res to the product of op1 and the integer \(c\).

fmpq_addmul res op1 op2

Sets res to res + op1 * op2 or res - op1 * op2 respectively, placing the result in canonical form. Aliasing between any combination of the variables is allowed.

_fmpq_addmul rnum rden op1num op1den op2num op2den

Sets (rnum, rden) to the canonical form of the fraction (rnum, rden) + (op1num, op1den) * (op2num, op2den) or (rnum, rden) - (op1num, op1den) * (op2num, op2den) respectively. Aliasing between any combination of the variables is allowed, whilst no numerator is aliased with a denominator.

fmpq_inv dest src

Sets dest to 1 / src. The result is placed in canonical form, assuming that src is already in canonical form.

_fmpq_pow_si rnum rden opnum opden e

Sets res to op raised to the power~`e`, where~`e` is a slong. If \(e\) is \(0\) and op is \(0\), then res will be set to \(1\).

fmpq_pow_fmpz a b e

Set res to op raised to the power~`e`. Return \(1\) for success and \(0\) for failure.

fmpq_mul_fmpz res op x

Sets res to the product of the rational number op and the integer x.

fmpq_div_fmpz res op x

Sets res to the quotient of the rational number op and the integer x.

fmpq_mul_2exp res x exp

Sets res to x multiplied by 2^exp.

fmpq_div_2exp res x exp

Sets res to x divided by 2^exp.

_fmpq_gcd rnum rden p q r s

Set (rnum, rden) to the gcd of (p, q) and (r, s) which we define to be the canonicalisation of gcd`(ps, qr)/(qs). (This is apparently Euclid's original definition and is stable under scaling of numerator and denominator. It also agrees with the gcd on the integers. Note that it does not agree with gcd as defined in fmpq_poly`.) This definition agrees with the result as output by Sage and Pari/GP.

fmpq_gcd res op1 op2

Set res to the gcd of op1 and op2. See the low level function _fmpq_gcd for our definition of gcd.

_fmpq_gcd_cofactors gnum gden abar bbar anum aden bnum bden

Set \(g\) to \(\operatorname{gcd}(a,b)\) as per fmpq_gcd and also compute \(\overline{a} = a/g\) and \(\overline{b} = b/g\). Unlike fmpq_gcd, this function requires canonical inputs.

_fmpq_add_small rnum rden p1 q1 p2 q2

Sets (rnum, rden) to the sum of (p1, q1) and (p2, q2). Assumes that (p1, q1) and (p2, q2) are in canonical form and that all inputs are between COEFF_MIN and COEFF_MAX.

_fmpq_mul_small rnum rden p1 q1 p2 q2

Sets (rnum, rden) to the product of (p1, q1) and (p2, q2). Assumes that (p1, q1) and (p2, q2) are in canonical form and that all inputs are between COEFF_MIN and COEFF_MAX.

_fmpq_mod_fmpz res num den mod

Sets the integer res to the residue \(a\) of \(x = n/d\) = (num, den) modulo the positive integer \(m\) = mod, defined as the \(0 \le a < m\) satisfying \(n \equiv a d \pmod m\). If such an \(a\) exists, 1 will be returned, otherwise 0 will be returned.

_fmpq_reconstruct_fmpz_2_naive n d a m N D

Reconstructs a rational number from its residue \(a\) modulo \(m\).

Given a modulus \(m > 2\), a residue \(0 \le a < m\), and positive \(N, D\) satisfying \(2ND < m\), this function attempts to find a fraction \(n/d\) with \(0 \le |n| \le N\) and \(0 < d \le D\) such that \(\gcd(n,d) = 1\) and \(n \equiv ad \pmod m\). If a solution exists, then it is also unique. The function returns 1 if successful, and 0 to indicate that no solution exists.

_fmpq_reconstruct_fmpz n d a m

Reconstructs a rational number from its residue \(a\) modulo \(m\), returning 1 if successful and 0 if no solution exists. Uses the balanced bounds \(N = D = \lfloor\sqrt{\frac{m-1}{2}}\rfloor\).

_fmpq_next_minimal rnum rden num den

Given \(x\) which is assumed to be nonnegative and in canonical form, sets res to the next rational number in the sequence obtained by enumerating all positive denominators \(q\), for each \(q\) enumerating the numerators \(1 \le p < q\) in order and generating both \(p/q\) and \(q/p\), but skipping all \(\gcd(p,q) \ne 1\). Starting with zero, this generates every nonnegative rational number once and only once, with the first few entries being:

\(0, 1, 1/2, 2, 1/3, 3, 2/3, 3/2, 1/4, 4, 3/4, 4/3, 1/5, 5, 2/5, \ldots.\)

This enumeration produces the rational numbers in order of minimal height. It has the disadvantage of being somewhat slower to compute than the Calkin-Wilf enumeration.

_fmpq_next_signed_minimal rnum rden num den

Given a signed rational number \(x\) assumed to be in canonical form, sets res to the next element in the minimal-height sequence generated by fmpq_next_minimal but with negative numbers interleaved:

\(0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, \ldots.\)

Starting with zero, this generates every rational number once and only once, in order of minimal height.

_fmpq_next_calkin_wilf rnum rden num den

Given \(x\) which is assumed to be nonnegative and in canonical form, sets res to the next number in the breadth-first traversal of the Calkin-Wilf tree. Starting with zero, this generates every nonnegative rational number once and only once, with the first few entries being:

\(0, 1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, \ldots.\)

Despite the appearance of the initial entries, the Calkin-Wilf enumeration does not produce the rational numbers in order of height: some small fractions will appear late in the sequence. This order has the advantage of being faster to produce than the minimal-height order.

_fmpq_next_signed_calkin_wilf rnum rden num den

Given a signed rational number \(x\) assumed to be in canonical form, sets res to the next element in the Calkin-Wilf sequence with negative numbers interleaved:

\(0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, \ldots.\)

Starting with zero, this generates every rational number once and only once, but not in order of minimal height.

fmpq_farey_neighbors l r x Q

Set \(l\) and \(r\) to the fractions directly below and above \(x\) in the Farey sequence of order \(Q\). This function will throw if \(x\) is not canonical or \(Q\) is less than the denominator of \(x\).

fmpq_farey_mediant x l r

Set \(x\) to the mediant of \(l\) and \(r\)

fmpq_simplest_between x l r

Set \(x\) to the simplest fraction in the closed interval \([l, r]\). The underscore version makes the additional assumption that \(l \le r\). The endpoints \(l\) and \(r\) do not need to be reduced, but their denominators do need to be positive. \(x\) will be always be returned in canonical form. A canonical fraction \(a_1/b_1\) is defined to be simpler than \(a_2/b_2\) iff \(b_1<b_2\) or \(b_1=b_2\) and \(a_1<a_2\).

fmpq_get_cfrac c rem x n

Generates up to \(n\) terms of the (simple) continued fraction expansion of \(x\), writing the coefficients to the vector \(c\) and the remainder \(r\) to the rem variable. The return value is the number \(k\) of generated terms. The output satisfies

\[x = c_0 + \cfrac{1}{c_1 + \cfrac{1}{c_2 + \cfrac{1}{ \ddots + \cfrac{1}{c_{k-1} + r }}}}\] ]

If \(r\) is zero, the continued fraction expansion is complete. If \(r\) is nonzero, \(1/r\) can be passed back as input to generate \(c_k, c_{k+1}, \ldots\). Calls to fmpq_get_cfrac can therefore be chained to generate the continued fraction incrementally, extracting any desired number of coefficients at a time.

In general, a rational number has exactly two continued fraction expansions. By convention, we generate the shorter one. The longer expansion can be obtained by replacing the last coefficient \(a_{k-1}\) by the pair of coefficients \(a_{k-1} - 1, 1\).

The behaviour of this function in corner cases is as follows:

- if \(x\) is infinite (anything over 0), rem will be zero and the return is \(k=0\) regardless of \(n\).

• [else (if \(x\) is finite),]
• if \(n <= 0\), rem will be \(1/x\) (allowing for infinite in the case \(x=0\)) and the return is \(k=0\)
• else (if \(n > 0\)), rem will finite and the return is \(0 < k \le n\).

Essentially, if this function is called with canonical \(x\) and \(n > 0\), then rem will be canonical. Therefore, applications relying on canonical fmpq_t's should not call this function with \(n <= 0\).

fmpq_set_cfrac x c n

Sets \(x\) to the value of the continued fraction

\[`\] \[x = c_0 + \cfrac{1}{c_1 + \cfrac{1}{c_2 + \cfrac{1}{ \ddots + \cfrac{1}{c_{n-1}}}}}\]

where all \(c_i\) except \(c_0\) should be nonnegative. It is assumed that \(n > 0\).

For large \(n\), this function implements a subquadratic algorithm. The convergents are given by a chain product of 2 by 2 matrices. This product is split in half recursively to balance the size of the coefficients.

fmpq_cfrac_bound x

Returns an upper bound for the number of terms in the continued fraction expansion of \(x\). The computed bound is not necessarily sharp.

We use the fact that the smallest denominator that can give a continued fraction of length \(n\) is the Fibonacci number \(F_{n+1}\).

fmpq_get_cfrac_st c rem x n

Generates up to \(n\) terms of the continued fraction expansion of \(x\), writing the coefficients to the vector \(c\) and the remainder \(r\) to the rem variable. The return value is the number \(k\) of generated terms. The output satisfies

\[ x = c_0 - \cfrac{1}{c_1 - \cfrac{1}{c_2 - \cfrac{1}{ \ddots - \cfrac{1}{c_{k-1} - r }}}} \]

This expansion is closely related to the modular group. In terms of the generators of the modular group \(S\) and \(T\) the expandion of \(x=h/k\) satisfies:

\[ T^{c_0} S T^{c_1} \cdots T^{c_{k-1}} S \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} h \\ k \end{pmatrix} \]

fmpq_set_cfrac_st x c n

Returns the value \(x\) corresponding to the continued fraction \(c\).

_fmpq_harmonic_ui num den n

Computes the harmonic number \(H_n = 1 + 1/2 + 1/3 + \dotsb + 1/n\). Table lookup is used for \(H_n\) whose numerator and denominator fit in single limb. For larger \(n\), a divide and conquer strategy is used.

fmpq_dedekind_sum s h k

Computes \(s(h,k)\) for arbitrary \(h\) and \(k\). The naive version uses a straightforward implementation of the defining sum using fmpz arithmetic and is slow for large \(k\).