fmpq.h – rational numbers

The fmpq_t data type represents rational numbers as fractions of multiprecision integers.

An fmpq_t is an array of length 1 of type fmpq, with fmpq being implemented as a pair of fmpz’s representing numerator and denominator.

This format is designed to allow rational numbers with small numerators or denominators to be stored and manipulated efficiently. When components no longer fit in single machine words, the cost of fmpq_t arithmetic is roughly the same as that of mpq_t arithmetic, plus a small amount of overhead.

A fraction is said to be in canonical form if the numerator and denominator have no common factor and the denominator is positive. Except where otherwise noted, all functions in the fmpq module assume that inputs are in canonical form, and produce outputs in canonical form. The user can manipulate the numerator and denominator of an fmpq_t as arbitrary integers, but then becomes responsible for canonicalising the number (for example by calling fmpq_canonicalise) before passing it to any library function.

For most operations, both a function operating on fmpq_t’s and an underscore version operating on fmpz_t components are provided. The underscore functions may perform less error checking, and may impose limitations on aliasing between the input and output variables, but generally assume that the components are in canonical form just like the non-underscore functions.

Types, macros and constants

type fmpq

An fmpq is implemented as a struct containing two fmpz’s, one for the numerator, and one for the denominator.

type fmpq_t

An array of length 1 of fmpq’s. This is used to pass fmpq’s around by reference without fuss, similar to the way mpq_t’s work.

fmpz *fmpq_numref(const fmpq_t x)

fmpz *fmpq_denref(const fmpq_t x)

Returns respectively a pointer to the numerator and denominator of x.

Memory management

void fmpq_init(fmpq_t x)

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

void fmpq_clear(fmpq_t x)

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

Canonicalisation

void fmpq_canonicalise(fmpq_t 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.

void _fmpq_canonicalise(fmpz_t num, fmpz_t 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.

int fmpq_is_canonical(const fmpq_t x)

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

int _fmpq_is_canonical(const fmpz_t num, const fmpz_t den)

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

Basic assignment

void fmpq_set(fmpq_t dest, const fmpq_t src)

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

void fmpq_swap(fmpq_t op1, fmpq_t op2)

Swaps the two rational numbers op1 and op2.

void fmpq_neg(fmpq_t dest, const fmpq_t src)

Sets dest to the additive inverse of src.

void fmpq_abs(fmpq_t dest, const fmpq_t src)

Sets dest to the absolute value of src.

void fmpq_zero(fmpq_t res)

Sets the value of res to 0.

void fmpq_one(fmpq_t res)

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

Comparison

int fmpq_is_zero(const fmpq_t res)

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

int fmpq_is_one(const fmpq_t res)

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

int fmpq_is_pm1(const fmpq_t res)

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

int fmpq_equal(const fmpq_t x, const fmpq_t y)

int fmpq_equal_fmpz(const fmpq_t x, const fmpz_t y)

int fmpq_equal_si(fmpq_t x, slong y)

int fmpq_equal_ui(fmpq_t x, ulong y)

Returns nonzero if x and y are equal, and zero otherwise.

int fmpq_sgn(const fmpq_t x)

Returns the sign of the rational number \(x\). That is, returns \(-1\) if \(x < 0\), \(1\) if \(x > 0\) and \(0\) if \(x = 0\).

int fmpq_cmp(const fmpq_t x, const fmpq_t y)

int fmpq_cmp_fmpz(const fmpq_t x, const fmpz_t y)

int fmpq_cmp_si(const fmpq_t x, slong y)

int fmpq_cmp_ui(const fmpq_t x, ulong y)

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

void fmpq_height(fmpz_t height, const fmpq_t x)

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

flint_bitcnt_t fmpq_height_bits(const fmpq_t x)

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

Conversion

void fmpq_set_fmpz_frac(fmpq_t res, const fmpz_t p, const fmpz_t 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.

void fmpq_get_mpz_frac(mpz_t a, mpz_t b, fmpq_t c)

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

Note: Requires that gmp.h has been included before any FLINT header is included.

void fmpq_set_si(fmpq_t res, slong p, ulong q)

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

void _fmpq_set_si(fmpz_t rnum, fmpz_t rden, slong p, ulong q)

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

void fmpq_set_ui(fmpq_t res, ulong p, ulong q)

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

void _fmpq_set_ui(fmpz_t rnum, fmpz_t rden, ulong p, ulong q)

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

void fmpq_set_mpq(fmpq_t dest, const mpq_t src)

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

Note: Requires that gmp.h has been included before any FLINT header is included.

int fmpq_set_str(fmpq_t dest, const char *s, int 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.

double fmpq_get_d(const fmpq_t 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.

void fmpq_get_mpq(mpq_t dest, const fmpq_t src)

Sets the value of dest

Note: Requires that gmp.h has been included before any FLINT header is included.

int fmpq_get_mpfr(mpfr_t dest, const fmpq_t src, mpfr_rnd_t 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.

Note: Requires that mpfr.h has been included before any FLINT header is included.

char *_fmpq_get_str(char *str, int b, const fmpz_t num, const fmpz_t den)

char *fmpq_get_str(char *str, int b, const fmpq_t x)

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.

void flint_mpq_init_set_readonly(mpq_t z, const fmpq_t 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.

Note: Requires that gmp.h has been included before any FLINT header is included.

void flint_mpq_clear_readonly(mpq_t z)

Clears the readonly mpq_t \(z\).

Note: Requires that gmp.h has been included before any FLINT header is included.

void fmpq_init_set_readonly(fmpq_t f, const mpq_t 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);
}

void fmpq_clear_readonly(fmpq_t f)

Clears the readonly fmpq_t \(f\).

Input and output

int fmpq_fprint(FILE *file, const fmpq_t 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.

int _fmpq_fprint(FILE *file, const fmpz_t num, const fmpz_t 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.

int fmpq_print(const fmpq_t 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.

int _fmpq_print(const fmpz_t num, const fmpz_t 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.

Random number generation

void fmpq_randtest(fmpq_t res, flint_rand_t state, flint_bitcnt_t bits)

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

void _fmpq_randtest(fmpz_t num, fmpz_t den, flint_rand_t state, flint_bitcnt_t 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.

void fmpq_randtest_not_zero(fmpq_t res, flint_rand_t state, flint_bitcnt_t bits)

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

void fmpq_randbits(fmpq_t res, flint_rand_t state, flint_bitcnt_t 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.

void _fmpq_randbits(fmpz_t num, fmpz_t den, flint_rand_t state, flint_bitcnt_t 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.

Arithmetic

void fmpq_add(fmpq_t res, const fmpq_t op1, const fmpq_t op2)

void fmpq_sub(fmpq_t res, const fmpq_t op1, const fmpq_t op2)

void fmpq_mul(fmpq_t res, const fmpq_t op1, const fmpq_t op2)

void fmpq_div(fmpq_t res, const fmpq_t op1, const fmpq_t op2)

Sets res respectively to op1 + op2, op1 - op2, op1 * op2, or op1 / op2. Division by zero results in an error. Aliasing between any combination of the variables is allowed.

void _fmpq_add(fmpz_t rnum, fmpz_t rden, const fmpz_t op1num, const fmpz_t op1den, const fmpz_t op2num, const fmpz_t op2den)

void _fmpq_sub(fmpz_t rnum, fmpz_t rden, const fmpz_t op1num, const fmpz_t op1den, const fmpz_t op2num, const fmpz_t op2den)

void _fmpq_mul(fmpz_t rnum, fmpz_t rden, const fmpz_t op1num, const fmpz_t op1den, const fmpz_t op2num, const fmpz_t op2den)

void _fmpq_div(fmpz_t rnum, fmpz_t rden, const fmpz_t op1num, const fmpz_t op1den, const fmpz_t op2num, const fmpz_t 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.

void _fmpq_add_si(fmpz_t rnum, fmpz_t rden, const fmpz_t p, const fmpz_t q, slong r)

void _fmpq_sub_si(fmpz_t rnum, fmpz_t rden, const fmpz_t p, const fmpz_t q, slong r)

void _fmpq_add_ui(fmpz_t rnum, fmpz_t rden, const fmpz_t p, const fmpz_t q, ulong r)

void _fmpq_sub_ui(fmpz_t rnum, fmpz_t rden, const fmpz_t p, const fmpz_t q, ulong r)

void _fmpq_add_fmpz(fmpz_t rnum, fmpz_t rden, const fmpz_t p, const fmpz_t q, const fmpz_t r)

void _fmpq_sub_fmpz(fmpz_t rnum, fmpz_t rden, const fmpz_t p, const fmpz_t q, const fmpz_t 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.

void fmpq_add_si(fmpq_t res, const fmpq_t op1, slong c)

void fmpq_sub_si(fmpq_t res, const fmpq_t op1, slong c)

void fmpq_add_ui(fmpq_t res, const fmpq_t op1, ulong c)

void fmpq_sub_ui(fmpq_t res, const fmpq_t op1, ulong c)

void fmpq_add_fmpz(fmpq_t res, const fmpq_t op1, const fmpz_t c)

void fmpq_sub_fmpz(fmpq_t res, const fmpq_t op1, const fmpz_t c)

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

void _fmpq_mul_si(fmpz_t rnum, fmpz_t rden, const fmpz_t p, const fmpz_t q, slong r)

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

void fmpq_mul_si(fmpq_t res, const fmpq_t op1, slong c)

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

void _fmpq_mul_ui(fmpz_t rnum, fmpz_t rden, const fmpz_t p, const fmpz_t q, ulong r)

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

void fmpq_mul_ui(fmpq_t res, const fmpq_t op1, ulong c)

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

void fmpq_addmul(fmpq_t res, const fmpq_t op1, const fmpq_t op2)

void fmpq_submul(fmpq_t res, const fmpq_t op1, const fmpq_t op2)

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

void _fmpq_addmul(fmpz_t rnum, fmpz_t rden, const fmpz_t op1num, const fmpz_t op1den, const fmpz_t op2num, const fmpz_t op2den)

void _fmpq_submul(fmpz_t rnum, fmpz_t rden, const fmpz_t op1num, const fmpz_t op1den, const fmpz_t op2num, const fmpz_t 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.

void fmpq_inv(fmpq_t dest, const fmpq_t src)

Sets dest to 1 / src.

void _fmpq_pow_si(fmpz_t rnum, fmpz_t rden, const fmpz_t opnum, const fmpz_t opden, slong e)

void fmpq_pow_si(fmpq_t res, const fmpq_t op, slong 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 \(1\).

int fmpq_pow_fmpz(fmpq_t a, const fmpq_t b, const fmpz_t e)

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

void fmpq_mul_fmpz(fmpq_t res, const fmpq_t op, const fmpz_t x)

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

void fmpq_div_fmpz(fmpq_t res, const fmpq_t op, const fmpz_t x)

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

void fmpq_mul_2exp(fmpq_t res, const fmpq_t x, flint_bitcnt_t exp)

Sets res to x multiplied by 2^exp.

void fmpq_div_2exp(fmpq_t res, const fmpq_t x, flint_bitcnt_t exp)

Sets res to x divided by 2^exp.

void _fmpq_gcd(fmpz_t rnum, fmpz_t rden, const fmpz_t p, const fmpz_t q, const fmpz_t r, const fmpz_t s)

Set (rnum, rden) to the gcd of (p, q) and (r, s) which we define to be the canonicalisation of \(\operatorname{gcd}(ps, qr)/(qs)\). Does not assume that (rnum, rden), (p, q) or (r, s) are canonical. (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.

void fmpq_gcd(fmpq_t res, const fmpq_t op1, const fmpq_t op2)

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

void _fmpq_gcd_cofactors(fmpz_t gnum, fmpz_t gden, fmpz_t abar, fmpz_t bbar, const fmpz_t anum, const fmpz_t aden, const fmpz_t bnum, const fmpz_t bden)

void fmpq_gcd_cofactors(fmpq_t g, fmpz_t abar, fmpz_t bbar, const fmpq_t a, const fmpq_t b)

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(), _fmpq_gcd_cofactors() requires canonical inputs.

void _fmpq_add_small(fmpz_t rnum, fmpz_t rden, slong p1, ulong q1, slong p2, ulong 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.

void _fmpq_mul_small(fmpz_t rnum, fmpz_t rden, slong p1, ulong q1, slong p2, ulong 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.

Modular reduction and rational reconstruction

int _fmpq_mod_fmpz(fmpz_t res, const fmpz_t num, const fmpz_t den, const fmpz_t mod)

int fmpq_mod_fmpz(fmpz_t res, const fmpq_t x, const fmpz_t 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.

int _fmpq_reconstruct_fmpz_2_naive(fmpz_t n, fmpz_t d, const fmpz_t a, const fmpz_t m, const fmpz_t N, const fmpz_t D)

int _fmpq_reconstruct_fmpz_2(fmpz_t n, fmpz_t d, const fmpz_t a, const fmpz_t m, const fmpz_t N, const fmpz_t D)

int fmpq_reconstruct_fmpz_2(fmpq_t res, const fmpz_t a, const fmpz_t m, const fmpz_t N, const fmpz_t 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.

int _fmpq_reconstruct_fmpz(fmpz_t n, fmpz_t d, const fmpz_t a, const fmpz_t m)

int fmpq_reconstruct_fmpz(fmpq_t res, const fmpz_t a, const fmpz_t 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 \(N = D = \lfloor\sqrt{\frac{m-1}{2}}\rfloor\).

Rational enumeration

void _fmpq_next_minimal(fmpz_t rnum, fmpz_t rden, const fmpz_t num, const fmpz_t den)

void fmpq_next_minimal(fmpq_t res, const fmpq_t x)

Given \(x = \mathtt{num} / \mathtt{den}\), 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.

void _fmpq_next_signed_minimal(fmpz_t rnum, fmpz_t rden, const fmpz_t num, const fmpz_t den)

void fmpq_next_signed_minimal(fmpq_t res, const fmpq_t x)

Given a signed rational number \(x = \mathtt{num} / \mathtt{den}\), 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.

void _fmpq_next_calkin_wilf(fmpz_t rnum, fmpz_t rden, const fmpz_t num, const fmpz_t den)

void fmpq_next_calkin_wilf(fmpq_t res, const fmpq_t x)

Given \(x = \mathtt{num} / \mathtt{den}\), 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.

void _fmpq_next_signed_calkin_wilf(fmpz_t rnum, fmpz_t rden, const fmpz_t num, const fmpz_t den)

void fmpq_next_signed_calkin_wilf(fmpq_t res, const fmpq_t x)

Given a signed rational number \(x = \mathtt{num} / \mathtt{den}\), 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.

void fmpq_farey_neighbors(fmpq_t l, fmpq_t r, const fmpq_t x, const fmpz_t 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 \(Q\) is less than the denominator of \(x\).

void _fmpq_simplest_between(fmpz_t x_num, fmpz_t x_den, const fmpz_t l_num, const fmpz_t l_den, const fmpz_t r_num, const fmpz_t r_den)

void fmpq_simplest_between(fmpq_t x, const fmpq_t l, const fmpq_t 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 canonical, but their denominators do need to be positive. \(x\) will 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 \(a_1<a_2\).

Continued fractions

slong fmpq_get_cfrac(fmpz *c, fmpq_t rem, const fmpq_t x, slong n)

slong fmpq_get_cfrac_naive(fmpz *c, fmpq_t rem, const fmpq_t x, slong 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 behavior 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 \(n \le 0\), 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 \(n \le 0\).

void fmpq_set_cfrac(fmpq_t x, const fmpz *c, slong 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.

slong fmpq_cfrac_bound(const fmpq_t 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 \(F_{n+1}\).

Special functions

void _fmpq_harmonic_ui(fmpz_t num, fmpz_t den, ulong n)

void fmpq_harmonic_ui(fmpq_t x, ulong 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.

Dedekind sums

Most of the definitions and relations used in the following section are given by Apostol [Apostol1997]. The Dedekind sum \(s(h,k)\) is defined for all integers \(h\) and \(k\) as

\[s(h,k) = \sum_{i=1}^{k-1} \left(\left(\frac{i}{k}\right)\right) \left(\left(\frac{hi}{k}\right)\right)\]

where

\[\begin{split}((x))=\begin{cases} x-\lfloor x\rfloor-1/2 &\mbox{if } x\in\mathbf{Q}\setminus\mathbf{Z}\\ 0 &\mbox{if }x\in\mathbf{Z}. \end{cases}\end{split}\]

If \(0 < h < k\) and \((h,k) = 1\), this reduces to

\[s(h,k) = \sum_{i=1}^{k-1} \frac{i}{k} \left(\frac{hi}{k}-\left\lfloor\frac{hi}{k}\right\rfloor -\frac{1}{2}\right).\]

The main formula for evaluating the series above is the following. Letting \(r_0 = k\), \(r_1 = h\), \(r_2, r_3, \ldots, r_n, r_{n+1} = 1\) be the remainder sequence in the Euclidean algorithm for computing GCD of \(h\) and \(k\),

\[s(h,k) = \frac{1-(-1)^n}{8} - \frac{1}{12} \sum_{i=1}^{n+1} (-1)^i \left(\frac{1+r_i^2+r_{i-1}^2}{r_i r_{i-1}}\right).\]

Writing \(s(h,k) = p/q\), some useful properties employed are \(|s| < k / 12\), \(q \mid 6k\) and \(2|p| < k^2\).

void fmpq_dedekind_sum(fmpq_t s, const fmpz_t h, const fmpz_t k)

void fmpq_dedekind_sum_naive(fmpq_t s, const fmpz_t h, const fmpz_t 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 \(k\).