Fraction fields

abs(a::fmpq)

Return the absolute value of $a$.

height(a::fmpq)

Return the height of the fraction $a$, namely the largest of the absolute values of the numerator and denominator.

height_bits(a::fmpq)

Return the number of bits of the height of the fraction $a$.

<<(a::fmpq, b::Int)

Return $2^b\times a$.

>>(a::fmpq, b::Int)

Return $2^b/a$.

isless(a::fmpq, b::fmpq)

Return true if $a < b$, otherwise return false.

isless(a::Integer, b::fmpq)

Return true if $a < b$, otherwise return false.

isless(a::fmpq, b::Integer)

Return true if $a < b$, otherwise return false.

isless(a::fmpq, b::fmpz)

Return true if $a < b$, otherwise return false.

isless(a::fmpz, b::fmpq)

Return true if $a < b$, otherwise return false.

mod(a::fmpq, b::fmpz)

Return $a \pmod{b}$ where $b$ is an integer coprime to the denominator of $a$.

mod(a::fmpq, b::Integer)

Return $a \pmod{b}$ where $b$ is an integer coprime to the denominator of $a$.

reconstruct(a::fmpz, b::fmpz)

Attempt to find a rational number $n/d$ such that $0 \leq |n| \leq \lfloor\sqrt{m/2}\rfloor$ and $0 < d \leq \lfloor\sqrt{m/2}\rfloor$ such that gcd$(n, d) = 1$ and $a \equiv nd^{-1} \pmod{m}$. If no solution exists, an exception is thrown.

reconstruct(a::fmpz, b::Integer)

Attempt to find a rational number $n/d$ such that $0 \leq |n| \leq \lfloor\sqrt{m/2}\rfloor$ and $0 < d \leq \lfloor\sqrt{m/2}\rfloor$ such that gcd$(n, d) = 1$ and $a \equiv nd^{-1} \pmod{m}$. If no solution exists, an exception is thrown.

reconstruct(a::Integer, b::fmpz)

Attempt to find a rational number $n/d$ such that $0 \leq |n| \leq \lfloor\sqrt{m/2}\rfloor$ and $0 < d \leq \lfloor\sqrt{m/2}\rfloor$ such that gcd$(n, d) = 1$ and $a \equiv nd^{-1} \pmod{m}$. If no solution exists, an exception is thrown.

reconstruct(a::Integer, b::Integer)

Attempt to find a rational number $n/d$ such that $0 \leq |n| \leq \lfloor\sqrt{m/2}\rfloor$ and $0 < d \leq \lfloor\sqrt{m/2}\rfloor$ such that gcd$(n, d) = 1$ and $a \equiv nd^{-1} \pmod{m}$. If no solution exists, an exception is thrown.

next_minimal(a::fmpq)

Given $a$, returns the next rational number in the sequence obtained by enumerating all positive denominators $q$, and for each $q$ enumerating the numerators $1 \le p < q$ in order and generating both $p/q$ and $q/p$, but skipping all gcd$(p,q) \neq 1$. Starting with zero, this generates every nonnegative rational number once and only once, with the first few entries being $0, 1, 1/2, 2, 1/3, 3, 2/3, 3/2, 1/4, 4, 3/4, 4/3, \ldots$. 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. If $a < 0$ we throw a DomainError().

next_signed_minimal(a::fmpq)

Given a signed rational number $a$ assumed to be in canonical form, returns the next element in the minimal-height sequence generated by next_minimal but with negative numbers interleaved. The sequence begins $0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, \ldots$. Starting with zero, this generates every rational number once and only once, in order of minimal height.

next_calkin_wilf(a::fmpq)

Given $a$ return 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.

next_signed_calkin_wilf(a::fmpq)

Given a signed rational number $a$ returns the next element in the Calkin-Wilf sequence with negative numbers interleaved. The sequence begins $0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, \ldots$. Starting with zero, this generates every rational number once and only once, but not in order of minimal height.

harmonic(n::Int)

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

bernoulli(n::Int)

Computes the Bernoulli number $B_n$ for nonnegative $n$.

bernoulli_cache(n::Int)

Precomputes and caches all the Bernoulli numbers up to $B_n$. This is much faster than repeatedly calling bernoulli(k). Once cached, subsequent calls to bernoulli(k) for any $k \le n$ will read from the cache, making them virtually free.

dedekind_sum(h::fmpz, k::fmpz)

dedekind_sum(h::fmpz, k::Integer)

Computes the Dedekind sum $s(h,k)$ for arbitrary $h$ and $k$.

dedekind_sum(h::Integer, k::fmpz)

Computes the Dedekind sum $s(h,k)$ for arbitrary $h$ and $k$.

dedekind_sum(h::Integer, k::Integer)

Computes the Dedekind sum $s(h,k)$ for arbitrary $h$ and $k$.