fmpzi.h

Initialise x to zero, respectively clear it.

Swaps x and y in constant time.

Set x to 0, respectively 1.

Sets res to x.

Sets res to a + bi.

Prints x to stdout in the form a+b*I (for example 6+3*I or 30-15*I).

Sets res to a random element whose parts are drawn by fmpz_randtest with up to bits bits.

Returns whether x = y.

Return whether x = 0, respectively x = 1.

Returns whether x is one of the units \pm 1, \pm i.

Returns k \in \{0, 1, 2, 3\} such that i^k x is the canonical associate of x: the one with positive real part a and imaginary part b in (-a, a] (for x = 0, k = 0).

Sets res to the canonical associate i^k x of x.

Returns the number of bits of the larger in absolute value of the real and imaginary parts, 0 for x = 0.

Sets res to the norm N(x) = a^2 + b^2.

Set res to the complex conjugate \bar x, respectively to -x.

Set res to x + y, respectively x - y.

Set res to xy, respectively x^2.

Sets res to x^{e} with x^0 = 1, by left-to-right binary exponentiation, with a direct formula for purely real or purely imaginary x.

Set z to ix, respectively x/i = -ix.

Sets res to i^k z for any integer k.

Sets q to x/y, assuming that the division is exact. Throws if y = 0.

Sets q and r to the quotient and remainder of x by y: x = qy + r, where q is x/y with both coordinates rounded to the nearest integer, ties rounded upwards. Then N(r) \le N(y)/2, and the pair is uniquely determined. r may be NULL. Throws if y = 0.

Sets q and r with x = qy + r and N(r) < N(y), choosing q from a floating-point approximation of x/y when the operands have similar sizes; the choice is implementation-defined. r may be NULL. Throws if y = 0.

Sets res to x / (1 + i)^k for the largest k such that the division is exact, and returns k. For x = 0, sets res to 0 and returns 0.

Set the output to the greatest common divisor of x and y in canonical form (see fmpzi_canonical_unit_i_pow); the GCD of 0 and 0 is 0. The euclidean version is Euclid's algorithm with fmpzi_divrem, the euclidean_improved version uses approximate quotients and finishes in double-precision arithmetic, the binary version is Weilert's (1+i)-ary algorithm [Wei2000] (which falls back to the Euclidean algorithm above 500 bits), and the shortest version finds the GCD as a shortest vector of the lattice generated by x and y. fmpzi_gcd chooses among them.

Return whether n is a Gaussian prime: a unit times 1 + i, a unit times a rational prime p \equiv 3 \pmod 4, or an element whose norm is a rational prime. fmpzi_is_prime proves primality of the relevant rational integer with fmpz_is_prime, fmpzi_is_probabprime uses the probable prime test fmpz_is_probabprime.

Computes the coefficients [\mathtt{nlo}, \mathtt{nhi}) of the product of the Gaussian integer polynomials (poly1, len1) and (poly2, len2), with the coefficient arithmetic done pointwise on fft_small transforms: two transforms per coefficient (real and imaginary parts) and four pointwise operations per complex product, the shorter operand's transforms kept and the longer rolling as in _fmpz_poly_mulmid_classical_fft_small. Squaring shares transforms, pairs symmetric terms and computes diagonal squares with three multiplications. The lengths may be given in either order. Returns 1 on success and 0 when the transformed representation is unavailable or unprofitable in which case nothing is written and the caller should fall back to a different routine.

Sets (res, nhi - nlo) to the coefficients at indices [\mathtt{nlo}, \mathtt{nhi}) of the product of the Gaussian integer polynomials (poly1, len1) and (poly2, len2), choosing an algorithm. Assumes that len1 and len2 are positive and that 0 \le nlo < nhi \le len1 + len2 - 1; the lengths may be given in either order, and res must not alias the inputs. This is the multiplication used by the fmpzi generic ring for polynomial multiplication, low products and middle products.

Sets (res, nhi - nlo) to the coefficients at indices [\mathtt{nlo}, \mathtt{nhi}) of the product of the Gaussian integer polynomials (poly1, len1) and (poly2, len2), using Kronecker substitution. Assumes that len1 and len2 are positive and that 0 \le nlo < nhi \le len1 + len2 - 1; the lengths may be given in either order, and res must not alias the inputs.

Sets (res, nhi - nlo) to the coefficients at indices [\mathtt{nlo}, \mathtt{nhi}) of the product of the Gaussian integer polynomials (poly1, len1) and (poly2, len2), using Schoenhage-Strassen convolution. Assumes that len1 and len2 are positive and that 0 \le nlo < nhi \le len1 + len2 - 1; the lengths may be given in either order, and res must not alias the inputs.