Data structure containing the CMag pointer

mag_init x

Initializes the variable x for use. Its value is set to zero.

mag_clear x

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

mag_swap x y

Swaps x and y efficiently.

_mag_vec_init n

Allocates a vector of length n. All entries are set to zero.

_mag_vec_clear v n

Clears a vector of length n.

mag_allocated_bytes x

Returns the total number of bytes heap-allocated internally by this object. The count excludes the size of the structure itself. Add sizeof(mag_struct) to get the size of the object as a whole.

mag_zero res

Sets res to zero.

mag_one res

Sets res to one.

mag_inf res

Sets res to positive infinity.

mag_is_special x

Returns nonzero iff x is zero or positive infinity.

mag_is_zero x

Returns nonzero iff x is zero.

mag_is_inf x

Returns nonzero iff x is positive infinity.

mag_is_finite x

Returns nonzero iff x is not positive infinity (since there is no NaN value, this function is exactly the logical negation of mag_is_inf).

mag_init_set res x

Initializes res and sets it to the value of x. This operation is always exact.

mag_set res x

Sets res to the value of x. This operation is always exact.

mag_set_fmpz res x

Sets res to an upper bound for \(|x|\). The operation may be inexact even if x is exactly representable.

mag_set_fmpz_lower res x

Sets res to a lower bound for \(|x|\). The operation may be inexact even if x is exactly representable.

mag_set_ui_2exp_si res x y

Sets res to an upper bound for \(|x| \cdot 2^y\).

mag_set_fmpz_2exp_fmpz_lower res x y

Sets res to a lower bound for \(|x| \cdot 2^y\).

mag_get_d x

Returns a double giving an upper bound for x.

mag_get_d_log2_approx x

Returns a double approximating \(\log_2(x)\), suitable for estimating magnitudes (warning: not a rigorous bound). The value is clamped between COEFF_MIN and COEFF_MAX.

mag_get_fmpz_lower res x

Sets res, respectively, to the exact rational number represented by x, the integer exactly representing the ceiling function of x, or the integer exactly representing the floor function of x.

These functions are unsafe: the user must check in advance that x is of reasonable magnitude. If x is infinite or has a bignum exponent, an abort will be raised. If the exponent otherwise is too large or too small, the available memory could be exhausted resulting in undefined behavior.

mag_equal x y

Returns nonzero iff x and y have the same value.

mag_cmp x y

Returns negative, zero, or positive, depending on whether x is smaller, equal, or larger than y.

mag_cmp_2exp_si x y

Returns negative, zero, or positive, depending on whether x is smaller, equal, or larger than \(2^y\).

mag_max res x y

Sets res respectively to the smaller or the larger of x and y.

mag_get_str x Returns a string representation of x. The memory needs to be deallocated with flint_free

mag_print x

Prints x to standard output.

mag_fprint file x

Prints x to the stream file.

mag_dump_str x

Allocates a string and writes a binary representation of x to it that can be read by mag_load_str. The returned string needs to be deallocated with flint_free.

mag_load_str x str

Parses str into x. Returns a nonzero value if str is not formatted correctly.

mag_dump_file stream x

Writes a binary representation of x to stream that can be read by mag_load_file. Returns a nonzero value if the data could not be written.

mag_load_file x stream

Reads x from stream. Returns a nonzero value if the data is not formatted correctly or the read failed. Note that the data is assumed to be delimited by a whitespace or end-of-file, i.e., when writing multiple values with mag_dump_file make sure to insert a whitespace to separate consecutive values.

mag_randtest res state expbits

Sets res to a random finite value, with an exponent up to expbits bits large.

mag_randtest_special res state expbits

Like mag_randtest, but also sometimes sets res to infinity.

mag_add_ui res x y

Sets res to an upper bound for \(x + y\).

mag_add_ui_lower res x y

Sets res to a lower bound for \(x + y\).

mag_add_2exp_fmpz res x e

Sets res to an upper bound for \(x + 2^e\).

mag_add_ui_2exp_si res x y e

Sets res to an upper bound for \(x + y 2^e\).

mag_sub res x y

Sets res to an upper bound for \(\max(x-y, 0)\).

mag_sub_lower res x y

Sets res to a lower bound for \(\max(x-y, 0)\).

mag_mul_2exp_fmpz res x y

Sets res to \(x \cdot 2^y\). This operation is exact.

mag_mul_fmpz res x y

Sets res to an upper bound for \(xy\).

mag_mul_fmpz_lower res x y

Sets res to a lower bound for \(xy\).

mag_addmul z x y

Sets z to an upper bound for \(z + xy\).

mag_div_fmpz res x y

Sets res to an upper bound for \(x / y\).

mag_div_lower res x y

Sets res to a lower bound for \(x / y\).

mag_inv res x

Sets res to an upper bound for \(1 / x\).

mag_inv_lower res x

Sets res to a lower bound for \(1 / x\).

mag_fast_init_set x y

Initialises x and sets it to the value of y.

mag_fast_zero res

Sets res to zero.

mag_fast_is_zero x

Returns nonzero iff x to zero.

mag_fast_mul res x y

Sets res to an upper bound for \(xy\).

mag_fast_addmul z x y

Sets z to an upper bound for \(z + xy\).

mag_fast_add_2exp_si res x e

Sets res to an upper bound for \(x + 2^e\).

mag_fast_mul_2exp_si res x e

Sets res to an upper bound for \(x 2^e\).

mag_pow_fmpz res x e

Sets res to an upper bound for \(x^e\).

mag_pow_fmpz_lower res x e

Sets res to a lower bound for \(x^e\).

mag_sqrt res x

Sets res to an upper bound for \(\sqrt{x}\).

mag_sqrt_lower res x

Sets res to a lower bound for \(\sqrt{x}\).

mag_rsqrt res x

Sets res to an upper bound for \(1/\sqrt{x}\).

mag_rsqrt_lower res x

Sets res to an lower bound for \(1/\sqrt{x}\).

mag_hypot res x y

Sets res to an upper bound for \(\sqrt{x^2 + y^2}\).

mag_root res x n

Sets res to an upper bound for \(x^{1/n}\).

mag_log res x

Sets res to an upper bound for \(\log(\max(1,x))\).

mag_log_lower res x

Sets res to a lower bound for \(\log(\max(1,x))\).

mag_neg_log res x

Sets res to an upper bound for \(-\log(\min(1,x))\), i.e. an upper bound for \(|\log(x)|\) for \(x \le 1\).

mag_neg_log_lower res x

Sets res to a lower bound for \(-\log(\min(1,x))\), i.e. a lower bound for \(|\log(x)|\) for \(x \le 1\).

mag_log_ui res n

Sets res to an upper bound for \(\log(n)\).

mag_log1p res x

Sets res to an upper bound for \(\log(1+x)\). The bound is computed accurately for small x.

mag_exp res x

Sets res to an upper bound for \(\exp(x)\).

mag_exp_lower res x

Sets res to a lower bound for \(\exp(x)\).

mag_expinv res x

Sets res to an upper bound for \(\exp(-x)\).

mag_expinv_lower res x

Sets res to a lower bound for \(\exp(-x)\).

mag_expm1 res x

Sets res to an upper bound for \(\exp(x) - 1\). The bound is computed accurately for small x.

mag_exp_tail res x N

Sets res to an upper bound for \(\sum_{k=N}^{\infty} x^k / k!\).

mag_binpow_uiui res m n

Sets res to an upper bound for \((1 + 1/m)^n\).

mag_geom_series res x N

Sets res to an upper bound for \(\sum_{k=N}^{\infty} x^k\).

mag_const_pi_lower res

Sets res to an upper (respectively lower) bound for \(\pi\).

mag_atan_lower res x

Sets res to an upper (respectively lower) bound for \(\operatorname{atan}(x)\).

mag_sinh_lower res x

Sets res to an upper or lower bound for \(\cosh(x)\) or \(\sinh(x)\).

mag_fac_ui res n

Sets res to an upper bound for \(n!\).

mag_rfac_ui res n

Sets res to an upper bound for \(1/n!\).

mag_bin_uiui res n k

Sets res to an upper bound for the binomial coefficient \({n \choose k}\).

mag_bernoulli_div_fac_ui res n

Sets res to an upper bound for \(|B_n| / n!\) where \(B_n\) denotes a Bernoulli number.

mag_polylog_tail res z s d N

Sets res to an upper bound for

\[`\] \[\sum_{k=N}^{\infty} \frac{z^k \log^d(k)}{k^s}.\]

The bounding strategy is described in algorithms_polylogarithms. Note: in applications where \(s\) in this formula may be real or complex, the user can simply substitute any convenient integer \(s'\) such that \(s' \le \operatorname{Re}(s)\).

mag_hurwitz_zeta_uiui res s a

Sets res to an upper bound for \(\zeta(s,a) = \sum_{k=0}^{\infty} (k+a)^{-s}\). We use the formula

\[`\] \[\zeta(s,a) \le \frac{1}{a^s} + \frac{1}{(s-1) a^{s-1}}\]

which is obtained by estimating the sum by an integral. If \(s \le 1\) or \(a = 0\), the bound is infinite.