acb.h – complex numbers

An acb_t represents a complex number with error bounds. An acb_t consists of a pair of real number balls of type arb_struct, representing the real and imaginary part with separate error bounds.

An acb_t thus represents a rectangle \([m_1-r_1, m_1+r_1] + [m_2-r_2, m_2+r_2] i\) in the complex plane. This is used instead of a disk or square representation (consisting of a complex floating-point midpoint with a single radius), since it allows implementing many operations more conveniently by splitting into ball operations on the real and imaginary parts. It also allows tracking when complex numbers have an exact (for example exactly zero) real part and an inexact imaginary part, or vice versa.

The interface for the acb_t type is slightly less developed than that for the arb_t type. In many cases, the user can easily perform missing operations by directly manipulating the real and imaginary parts.

Types, macros and constants

acb_struct
acb_t

An acb_struct consists of a pair of arb_struct:s. An acb_t is defined as an array of length one of type acb_struct, permitting an acb_t to be passed by reference.

acb_ptr

Alias for acb_struct *, used for vectors of numbers.

acb_srcptr

Alias for const acb_struct *, used for vectors of numbers when passed as constant input to functions.

acb_realref(x)

Macro returning a pointer to the real part of x as an arb_t.

acb_imagref(x)

Macro returning a pointer to the imaginary part of x as an arb_t.

Memory management

void acb_init(acb_t x)

Initializes the variable x for use, and sets its value to zero.

void acb_clear(acb_t x)

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

acb_ptr _acb_vec_init(slong n)

Returns a pointer to an array of n initialized acb_struct:s.

void _acb_vec_clear(acb_ptr v, slong n)

Clears an array of n initialized acb_struct:s.

slong acb_allocated_bytes(const acb_t x)

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

slong _acb_vec_allocated_bytes(acb_srcptr vec, slong len)

Returns the total number of bytes allocated for this vector, i.e. the space taken up by the vector itself plus the sum of the internal heap allocation sizes for all its member elements.

double _acb_vec_estimate_allocated_bytes(slong len, slong prec)

Estimates the number of bytes that need to be allocated for a vector of len elements with prec bits of precision, including the space for internal limb data. See comments for _arb_vec_estimate_allocated_bytes().

Basic manipulation

void acb_zero(acb_t z)
void acb_one(acb_t z)
void acb_onei(acb_t z)

Sets z respectively to 0, 1, \(i = \sqrt{-1}\).

void acb_set(acb_t z, const acb_t x)
void acb_set_ui(acb_t z, slong x)
void acb_set_si(acb_t z, slong x)
void acb_set_d(acb_t z, double x)
void acb_set_fmpz(acb_t z, const fmpz_t x)
void acb_set_arb(acb_t z, const arb_t c)

Sets z to the value of x.

void acb_set_si_si(acb_t z, slong x, slong y)
void acb_set_d_d(acb_t z, double x, double y)
void acb_set_fmpz_fmpz(acb_t z, const fmpz_t x, const fmpz_t y)
void acb_set_arb_arb(acb_t z, const arb_t x, const arb_t y)

Sets the real and imaginary part of z to the values x and y respectively

void acb_set_fmpq(acb_t z, const fmpq_t x, slong prec)
void acb_set_round(acb_t z, const acb_t x, slong prec)
void acb_set_round_fmpz(acb_t z, const fmpz_t x, slong prec)
void acb_set_round_arb(acb_t z, const arb_t x, slong prec)

Sets z to x, rounded to prec bits.

void acb_swap(acb_t z, acb_t x)

Swaps z and x efficiently.

void acb_add_error_mag(acb_t x, const mag_t err)

Adds err to the error bounds of both the real and imaginary parts of x, modifying x in-place.

Input and output

The acb_print… functions print to standard output, while acb_fprint… functions print to the stream file.

void acb_print(const acb_t x)
void acb_fprint(FILE * file, const acb_t x)

Prints the internal representation of x.

void acb_printd(const acb_t x, slong digits)
void acb_fprintd(FILE * file, const acb_t x, slong digits)

Prints x in decimal. The printed value of the radius is not adjusted to compensate for the fact that the binary-to-decimal conversion of both the midpoint and the radius introduces additional error.

void acb_printn(const acb_t x, slong digits, ulong flags)
void acb_fprintn(FILE * file, const acb_t x, slong digits, ulong flags)

Prints a nice decimal representation of x, using the format of arb_get_str() (or the corresponding arb_printn()) for the real and imaginary parts.

By default, the output shows the midpoint of both the real and imaginary parts with a guaranteed error of at most one unit in the last decimal place. In addition, explicit error bounds are printed so that the displayed decimal interval is guaranteed to enclose x.

Any flags understood by arb_get_str() can be passed via flags to control the format of the real and imaginary parts.

Random number generation

void acb_randtest(acb_t z, flint_rand_t state, slong prec, slong mag_bits)

Generates a random complex number by generating separate random real and imaginary parts.

void acb_randtest_special(acb_t z, flint_rand_t state, slong prec, slong mag_bits)

Generates a random complex number by generating separate random real and imaginary parts. Also generates NaNs and infinities.

void acb_randtest_precise(acb_t z, flint_rand_t state, slong prec, slong mag_bits)

Generates a random complex number with precise real and imaginary parts.

void acb_randtest_param(acb_t z, flint_rand_t state, slong prec, slong mag_bits)

Generates a random complex number, with very high probability of generating integers and half-integers.

Precision and comparisons

int acb_is_zero(const acb_t z)

Returns nonzero iff z is zero.

int acb_is_one(const acb_t z)

Returns nonzero iff z is exactly 1.

int acb_is_finite(const acb_t z)

Returns nonzero iff z certainly is finite.

int acb_is_exact(const acb_t z)

Returns nonzero iff z is exact.

int acb_is_int(const acb_t z)

Returns nonzero iff z is an exact integer.

int acb_is_int_2exp_si(const acb_t x, slong e)

Returns nonzero iff z exactly equals \(n 2^e\) for some integer n.

int acb_equal(const acb_t x, const acb_t y)

Returns nonzero iff x and y are identical as sets, i.e. if the real and imaginary parts are equal as balls.

Note that this is not the same thing as testing whether both x and y certainly represent the same complex number, unless either x or y is exact (and neither contains NaN). To test whether both operands might represent the same mathematical quantity, use acb_overlaps() or acb_contains(), depending on the circumstance.

int acb_equal_si(const acb_t x, slong y)

Returns nonzero iff x is equal to the integer y.

int acb_eq(const acb_t x, const acb_t y)

Returns nonzero iff x and y are certainly equal, as determined by testing that arb_eq() holds for both the real and imaginary parts.

int acb_ne(const acb_t x, const acb_t y)

Returns nonzero iff x and y are certainly not equal, as determined by testing that arb_ne() holds for either the real or imaginary parts.

int acb_overlaps(const acb_t x, const acb_t y)

Returns nonzero iff x and y have some point in common.

void acb_union(acb_t z, const acb_t x, const acb_t y, slong prec)

Sets z to a complex interval containing both x and y.

void acb_get_abs_ubound_arf(arf_t u, const acb_t z, slong prec)

Sets u to an upper bound for the absolute value of z, computed using a working precision of prec bits.

void acb_get_abs_lbound_arf(arf_t u, const acb_t z, slong prec)

Sets u to a lower bound for the absolute value of z, computed using a working precision of prec bits.

void acb_get_rad_ubound_arf(arf_t u, const acb_t z, slong prec)

Sets u to an upper bound for the error radius of z (the value is currently not computed tightly).

void acb_get_mag(mag_t u, const acb_t x)

Sets u to an upper bound for the absolute value of x.

void acb_get_mag_lower(mag_t u, const acb_t x)

Sets u to a lower bound for the absolute value of x.

int acb_contains_fmpq(const acb_t x, const fmpq_t y)
int acb_contains_fmpz(const acb_t x, const fmpz_t y)
int acb_contains(const acb_t x, const acb_t y)

Returns nonzero iff y is contained in x.

int acb_contains_zero(const acb_t x)

Returns nonzero iff zero is contained in x.

int acb_contains_int(const acb_t x)

Returns nonzero iff the complex interval represented by x contains an integer.

slong acb_rel_error_bits(const acb_t x)

Returns the effective relative error of x measured in bits. This is computed as if calling arb_rel_error_bits() on the real ball whose midpoint is the larger out of the real and imaginary midpoints of x, and whose radius is the larger out of the real and imaginary radiuses of x.

slong acb_rel_accuracy_bits(const acb_t x)

Returns the effective relative accuracy of x measured in bits, equal to the negative of the return value from acb_rel_error_bits().

slong acb_bits(const acb_t x)

Returns the maximum of arb_bits applied to the real and imaginary parts of x, i.e. the minimum precision sufficient to represent x exactly.

void acb_indeterminate(acb_t x)

Sets x to \([\operatorname{NaN} \pm \infty] + [\operatorname{NaN} \pm \infty]i\), representing an indeterminate result.

void acb_trim(acb_t y, const acb_t x)

Sets y to a a copy of x with both the real and imaginary parts trimmed (see arb_trim()).

int acb_is_real(const acb_t x)

Returns nonzero iff the imaginary part of x is zero. It does not test whether the real part of x also is finite.

int acb_get_unique_fmpz(fmpz_t z, const acb_t x)

If x contains a unique integer, sets z to that value and returns nonzero. Otherwise (if x represents no integers or more than one integer), returns zero.

Complex parts

void acb_get_real(arb_t re, const acb_t z)

Sets re to the real part of z.

void acb_get_imag(arb_t im, const acb_t z)

Sets im to the imaginary part of z.

void acb_arg(arb_t r, const acb_t z, slong prec)

Sets r to a real interval containing the complex argument (phase) of z. We define the complex argument have a discontinuity on \((-\infty,0]\), with the special value \(\operatorname{arg}(0) = 0\), and \(\operatorname{arg}(a+0i) = \pi\) for \(a < 0\). Equivalently, if \(z = a+bi\), the argument is given by \(\operatorname{atan2}(b,a)\) (see arb_atan2()).

void acb_abs(arb_t r, const acb_t z, slong prec)

Sets r to the absolute value of z.

void acb_sgn(acb_t r, const acb_t z, slong prec)

Sets r to the complex sign of z, defined as 0 if z is exactly zero and the projection onto the unit circle \(z / |z| = \exp(i \arg(z))\) otherwise.

void acb_csgn(arb_t r, const acb_t z)

Sets r to the extension of the real sign function taking the value 1 for z strictly in the right half plane, -1 for z strictly in the left half plane, and the sign of the imaginary part when z is on the imaginary axis. Equivalently, \(\operatorname{csgn}(z) = z / \sqrt{z^2}\) except that the value is 0 when z is exactly zero.

Arithmetic

void acb_neg(acb_t z, const acb_t x)

Sets z to the negation of x.

void acb_conj(acb_t z, const acb_t x)

Sets z to the complex conjugate of x.

void acb_add_ui(acb_t z, const acb_t x, ulong y, slong prec)
void acb_add_si(acb_t z, const acb_t x, slong y, slong prec)
void acb_add_fmpz(acb_t z, const acb_t x, const fmpz_t y, slong prec)
void acb_add_arb(acb_t z, const acb_t x, const arb_t y, slong prec)
void acb_add(acb_t z, const acb_t x, const acb_t y, slong prec)

Sets z to the sum of x and y.

void acb_sub_ui(acb_t z, const acb_t x, ulong y, slong prec)
void acb_sub_si(acb_t z, const acb_t x, slong y, slong prec)
void acb_sub_fmpz(acb_t z, const acb_t x, const fmpz_t y, slong prec)
void acb_sub_arb(acb_t z, const acb_t x, const arb_t y, slong prec)
void acb_sub(acb_t z, const acb_t x, const acb_t y, slong prec)

Sets z to the difference of x and y.

void acb_mul_onei(acb_t z, const acb_t x)

Sets z to x multiplied by the imaginary unit.

void acb_div_onei(acb_t z, const acb_t x)

Sets z to x divided by the imaginary unit.

void acb_mul_ui(acb_t z, const acb_t x, ulong y, slong prec)
void acb_mul_si(acb_t z, const acb_t x, slong y, slong prec)
void acb_mul_fmpz(acb_t z, const acb_t x, const fmpz_t y, slong prec)
void acb_mul_arb(acb_t z, const acb_t x, const arb_t y, slong prec)

Sets z to the product of x and y.

void acb_mul(acb_t z, const acb_t x, const acb_t y, slong prec)

Sets z to the product of x and y. If at least one part of x or y is zero, the operations is reduced to two real multiplications. If x and y are the same pointers, they are assumed to represent the same mathematical quantity and the squaring formula is used.

void acb_mul_2exp_si(acb_t z, const acb_t x, slong e)
void acb_mul_2exp_fmpz(acb_t z, const acb_t x, const fmpz_t e)

Sets z to x multiplied by \(2^e\), without rounding.

void acb_sqr(acb_t z, const acb_t x, slong prec)

Sets z to x squared.

void acb_cube(acb_t z, const acb_t x, slong prec)

Sets z to x cubed, computed efficiently using two real squarings, two real multiplications, and scalar operations.

void acb_addmul(acb_t z, const acb_t x, const acb_t y, slong prec)
void acb_addmul_ui(acb_t z, const acb_t x, ulong y, slong prec)
void acb_addmul_si(acb_t z, const acb_t x, slong y, slong prec)
void acb_addmul_fmpz(acb_t z, const acb_t x, const fmpz_t y, slong prec)
void acb_addmul_arb(acb_t z, const acb_t x, const arb_t y, slong prec)

Sets z to z plus the product of x and y.

void acb_submul(acb_t z, const acb_t x, const acb_t y, slong prec)
void acb_submul_ui(acb_t z, const acb_t x, ulong y, slong prec)
void acb_submul_si(acb_t z, const acb_t x, slong y, slong prec)
void acb_submul_fmpz(acb_t z, const acb_t x, const fmpz_t y, slong prec)
void acb_submul_arb(acb_t z, const acb_t x, const arb_t y, slong prec)

Sets z to z minus the product of x and y.

void acb_inv(acb_t z, const acb_t x, slong prec)

Sets z to the multiplicative inverse of x.

void acb_div_ui(acb_t z, const acb_t x, ulong y, slong prec)
void acb_div_si(acb_t z, const acb_t x, slong y, slong prec)
void acb_div_fmpz(acb_t z, const acb_t x, const fmpz_t y, slong prec)
void acb_div_arb(acb_t z, const acb_t x, const arb_t y, slong prec)
void acb_div(acb_t z, const acb_t x, const acb_t y, slong prec)

Sets z to the quotient of x and y.

Mathematical constants

void acb_const_pi(acb_t y, slong prec)

Sets y to the constant \(\pi\).

Powers and roots

void acb_sqrt(acb_t r, const acb_t z, slong prec)

Sets r to the square root of z. If either the real or imaginary part is exactly zero, only a single real square root is needed. Generally, we use the formula \(\sqrt{a+bi} = u/2 + ib/u, u = \sqrt{2(|a+bi|+a)}\), requiring two real square root extractions.

void acb_rsqrt(acb_t r, const acb_t z, slong prec)

Sets r to the reciprocal square root of z. If either the real or imaginary part is exactly zero, only a single real reciprocal square root is needed. Generally, we use the formula \(1/\sqrt{a+bi} = ((a+r) - bi)/v, r = |a+bi|, v = \sqrt{r |a+bi+r|^2}\), requiring one real square root and one real reciprocal square root.

void acb_quadratic_roots_fmpz(acb_t r1, acb_t r2, const fmpz_t a, const fmpz_t b, const fmpz_t c, slong prec)

Sets r1 and r2 to the roots of the quadratic polynomial \(ax^2 + bx + c\). Requires that a is nonzero. This function is implemented so that both roots are computed accurately even when direct use of the quadratic formula would lose accuracy.

void acb_root_ui(acb_t r, const acb_t z, ulong k, slong prec)

Sets r to the principal k-th root of z.

void acb_pow_fmpz(acb_t y, const acb_t b, const fmpz_t e, slong prec)
void acb_pow_ui(acb_t y, const acb_t b, ulong e, slong prec)
void acb_pow_si(acb_t y, const acb_t b, slong e, slong prec)

Sets \(y = b^e\) using binary exponentiation (with an initial division if \(e < 0\)). Note that these functions can get slow if the exponent is extremely large (in such cases acb_pow() may be superior).

void acb_pow_arb(acb_t z, const acb_t x, const arb_t y, slong prec)
void acb_pow(acb_t z, const acb_t x, const acb_t y, slong prec)

Sets \(z = x^y\), computed using binary exponentiation if \(y\) if a small exact integer, as \(z = (x^{1/2})^{2y}\) if \(y\) is a small exact half-integer, and generally as \(z = \exp(y \log x)\).

void acb_unit_root(acb_t res, ulong order, slong prec)

Sets res to \(\exp(\frac{2i\pi}{\mathrm{order}})\) to precision prec.

Exponentials and logarithms

void acb_exp(acb_t y, const acb_t z, slong prec)

Sets y to the exponential function of z, computed as \(\exp(a+bi) = \exp(a) \left( \cos(b) + \sin(b) i \right)\).

void acb_exp_pi_i(acb_t y, const acb_t z, slong prec)

Sets y to \(\exp(\pi i z)\).

void acb_exp_invexp(acb_t s, acb_t t, const acb_t z, slong prec)

Sets \(v = \exp(z)\) and \(w = \exp(-z)\).

void acb_expm1(acb_t res, const acb_t z, slong prec)

Computes \(\exp(z)-1\), using an accurate method when \(z \approx 0\).

void acb_log(acb_t y, const acb_t z, slong prec)

Sets y to the principal branch of the natural logarithm of z, computed as \(\log(a+bi) = \frac{1}{2} \log(a^2 + b^2) + i \operatorname{arg}(a+bi)\).

void acb_log1p(acb_t z, const acb_t x, slong prec)

Sets \(z = \log(1+x)\), computed accurately when \(x \approx 0\).

Trigonometric functions

void acb_sin(acb_t s, const acb_t z, slong prec)
void acb_cos(acb_t c, const acb_t z, slong prec)
void acb_sin_cos(acb_t s, acb_t c, const acb_t z, slong prec)

Sets \(s = \sin(z)\), \(c = \cos(z)\), evaluated as \(\sin(a+bi) = \sin(a)\cosh(b) + i \cos(a)\sinh(b)\), \(\cos(a+bi) = \cos(a)\cosh(b) - i \sin(a)\sinh(b)\).

void acb_tan(acb_t s, const acb_t z, slong prec)

Sets \(s = \tan(z) = \sin(z) / \cos(z)\). For large imaginary parts, the function is evaluated in a numerically stable way as \(\pm i\) plus a decreasing exponential factor.

void acb_cot(acb_t s, const acb_t z, slong prec)

Sets \(s = \cot(z) = \cos(z) / \sin(z)\). For large imaginary parts, the function is evaluated in a numerically stable way as \(\pm i\) plus a decreasing exponential factor.

void acb_sin_pi(acb_t s, const acb_t z, slong prec)
void acb_cos_pi(acb_t s, const acb_t z, slong prec)
void acb_sin_cos_pi(acb_t s, acb_t c, const acb_t z, slong prec)

Sets \(s = \sin(\pi z)\), \(c = \cos(\pi z)\), evaluating the trigonometric factors of the real and imaginary part accurately via arb_sin_cos_pi().

void acb_tan_pi(acb_t s, const acb_t z, slong prec)

Sets \(s = \tan(\pi z)\). Uses the same algorithm as acb_tan(), but evaluates the sine and cosine accurately via arb_sin_cos_pi().

void acb_cot_pi(acb_t s, const acb_t z, slong prec)

Sets \(s = \cot(\pi z)\). Uses the same algorithm as acb_cot(), but evaluates the sine and cosine accurately via arb_sin_cos_pi().

void acb_sinc(acb_t s, const acb_t z, slong prec)

Sets \(s = \operatorname{sinc}(x) = \sin(z) / z\).

void acb_sinc_pi(acb_t s, const acb_t z, slong prec)

Sets \(s = \operatorname{sinc}(\pi x) = \sin(\pi z) / (\pi z)\).

Inverse trigonometric functions

void acb_asin(acb_t res, const acb_t z, slong prec)

Sets res to \(\operatorname{asin}(z) = -i \log(iz + \sqrt{1-z^2})\).

void acb_acos(acb_t res, const acb_t z, slong prec)

Sets res to \(\operatorname{acos}(z) = \tfrac{1}{2} \pi - \operatorname{asin}(z)\).

void acb_atan(acb_t res, const acb_t z, slong prec)

Sets res to \(\operatorname{atan}(z) = \tfrac{1}{2} i (\log(1-iz)-\log(1+iz))\).

Hyperbolic functions

void acb_sinh(acb_t s, const acb_t z, slong prec)
void acb_cosh(acb_t c, const acb_t z, slong prec)
void acb_sinh_cosh(acb_t s, acb_t c, const acb_t z, slong prec)
void acb_tanh(acb_t s, const acb_t z, slong prec)
void acb_coth(acb_t s, const acb_t z, slong prec)

Respectively computes \(\sinh(z) = -i\sin(iz)\), \(\cosh(z) = \cos(iz)\), \(\tanh(z) = -i\tan(iz)\), \(\coth(z) = i\cot(iz)\).

Inverse hyperbolic functions

void acb_asinh(acb_t res, const acb_t z, slong prec)

Sets res to \(\operatorname{asinh}(z) = -i \operatorname{asin}(iz)\).

void acb_acosh(acb_t res, const acb_t z, slong prec)

Sets res to \(\operatorname{acosh}(z) = \log(z + \sqrt{z+1} \sqrt{z-1})\).

void acb_atanh(acb_t res, const acb_t z, slong prec)

Sets res to \(\operatorname{atanh}(z) = -i \operatorname{atan}(iz)\).

Lambert W function

void acb_lambertw_asymp(acb_t res, const acb_t z, const fmpz_t k, slong L, slong M, slong prec)

Sets res to the Lambert W function \(W_k(z)\) computed using L and M terms in the bivariate series giving the asymptotic expansion at zero or infinity. This algorithm is valid everywhere, but the error bound is only finite when \(|\log(z)|\) is sufficiently large.

int acb_lambertw_check_branch(const acb_t w, const fmpz_t k, slong prec)

Tests if w definitely lies in the image of the branch \(W_k(z)\). This function is used internally to verify that a computed approximation of the Lambert W function lies on the intended branch. Note that this will necessarily evaluate to false for points exactly on (or overlapping) the branch cuts, where a different algorithm has to be used.

void acb_lambertw_bound_deriv(mag_t res, const acb_t z, const acb_t ez1, const fmpz_t k)

Sets res to an upper bound for \(|W_k'(z)|\). The input ez1 should contain the precomputed value of \(ez+1\).

Along the real line, the directional derivative of \(W_k(z)\) is understood to be taken. As a result, the user must handle the branch cut discontinuity separately when using this function to bound perturbations in the value of \(W_k(z)\).

void acb_lambertw(acb_t res, const acb_t z, const fmpz_t k, int flags, slong prec)

Sets res to the Lambert W function \(W_k(z)\) where the index k selects the branch (with \(k = 0\) giving the principal branch). The placement of branch cuts follows [CGHJK1996].

If flags is nonzero, nonstandard branch cuts are used.

If flags is set to ACB_LAMBERTW_LEFT, computes \(W_{\mathrm{left}|k}(z)\) which corresponds to \(W_k(z)\) in the upper half plane and \(W_{k+1}(z)\) in the lower half plane, connected continuously to the left of the branch points. In other words, the branch cut on \((-\infty,0)\) is rotated counterclockwise to \((0,+\infty)\). (For \(k = -1\) and \(k = 0\), there is also a branch cut on \((-1/e,0)\), continuous from below instead of from above to maintain counterclockwise continuity.)

If flags is set to ACB_LAMBERTW_MIDDLE, computes \(W_{\mathrm{middle}}(z)\) which corresponds to \(W_{-1}(z)\) in the upper half plane and \(W_{1}(z)\) in the lower half plane, connected continuously through \((-1/e,0)\) with branch cuts on \((-\infty,-1/e)\) and \((0,+\infty)\). \(W_{\mathrm{middle}}(z)\) extends the real analytic function \(W_{-1}(x)\) defined on \((-1/e,0)\) to a complex analytic function, whereas the standard branch \(W_{-1}(z)\) has a branch cut along the real segment.

The algorithm used to compute the Lambert W function is described in [Joh2017b].

Rising factorials

void acb_rising_ui_bs(acb_t z, const acb_t x, ulong n, slong prec)
void acb_rising_ui_rs(acb_t z, const acb_t x, ulong n, ulong step, slong prec)
void acb_rising_ui_rec(acb_t z, const acb_t x, ulong n, slong prec)
void acb_rising_ui(acb_t z, const acb_t x, ulong n, slong prec)
void acb_rising(acb_t z, const acb_t x, const acb_t n, slong prec)

Computes the rising factorial \(z = x (x+1) (x+2) \cdots (x+n-1)\).

The bs version uses binary splitting. The rs version uses rectangular splitting. The rec version uses either bs or rs depending on the input. The default version uses the gamma function unless n is a small integer.

The rs version takes an optional step parameter for tuning purposes (to use the default step length, pass zero).

void acb_rising2_ui_bs(acb_t u, acb_t v, const acb_t x, ulong n, slong prec)
void acb_rising2_ui_rs(acb_t u, acb_t v, const acb_t x, ulong n, ulong step, slong prec)
void acb_rising2_ui(acb_t u, acb_t v, const acb_t x, ulong n, slong prec)

Letting \(u(x) = x (x+1) (x+2) \cdots (x+n-1)\), simultaneously compute \(u(x)\) and \(v(x) = u'(x)\), respectively using binary splitting, rectangular splitting (with optional nonzero step length step to override the default choice), and an automatic algorithm choice.

void acb_rising_ui_get_mag(mag_t bound, const acb_t x, ulong n)

Computes an upper bound for the absolute value of the rising factorial \(z = x (x+1) (x+2) \cdots (x+n-1)\). Not currently optimized for large n.

Gamma function

void acb_gamma(acb_t y, const acb_t x, slong prec)

Computes the gamma function \(y = \Gamma(x)\).

void acb_rgamma(acb_t y, const acb_t x, slong prec)

Computes the reciprocal gamma function \(y = 1/\Gamma(x)\), avoiding division by zero at the poles of the gamma function.

void acb_lgamma(acb_t y, const acb_t x, slong prec)

Computes the logarithmic gamma function \(y = \log \Gamma(x)\).

The branch cut of the logarithmic gamma function is placed on the negative half-axis, which means that \(\log \Gamma(z) + \log z = \log \Gamma(z+1)\) holds for all \(z\), whereas \(\log \Gamma(z) \ne \log(\Gamma(z))\) in general. In the left half plane, the reflection formula with correct branch structure is evaluated via acb_log_sin_pi().

void acb_digamma(acb_t y, const acb_t x, slong prec)

Computes the digamma function \(y = \psi(x) = (\log \Gamma(x))' = \Gamma'(x) / \Gamma(x)\).

void acb_log_sin_pi(acb_t res, const acb_t z, slong prec)

Computes the logarithmic sine function defined by

\[S(z) = \log(\pi) - \log \Gamma(z) + \log \Gamma(1-z)\]

which is equal to

\[S(z) = \int_{1/2}^z \pi \cot(\pi t) dt\]

where the path of integration goes through the upper half plane if \(0 < \arg(z) \le \pi\) and through the lower half plane if \(-\pi < \arg(z) \le 0\). Equivalently,

\[S(z) = \log(\sin(\pi(z-n))) \mp n \pi i, \quad n = \lfloor \operatorname{re}(z) \rfloor\]

where the negative sign is taken if \(0 < \arg(z) \le \pi\) and the positive sign is taken otherwise (if the interval \(\arg(z)\) does not certainly satisfy either condition, the union of both cases is computed). After subtracting n, we have \(0 \le \operatorname{re}(z) < 1\). In this strip, we use use \(S(z) = \log(\sin(\pi(z)))\) if the imaginary part of z is small. Otherwise, we use \(S(z) = i \pi (z-1/2) + \log((1+e^{-2i\pi z})/2)\) in the lower half-plane and the conjugated expression in the upper half-plane to avoid exponent overflow.

The function is evaluated at the midpoint and the propagated error is computed from \(S'(z)\) to get a continuous change when \(z\) is non-real and \(n\) spans more than one possible integer value.

void acb_polygamma(acb_t z, const acb_t s, const acb_t z, slong prec)

Sets res to the value of the generalized polygamma function \(\psi(s,z)\).

If s is a nonnegative order, this is simply the s-order derivative of the digamma function. If \(s = 0\), this function simply calls the digamma function internally. For integers \(s \ge 1\), it calls the Hurwitz zeta function. Note that for small integers \(s \ge 1\), it can be faster to use acb_poly_digamma_series() and read off the coefficients.

The generalization to other values of s is due to Espinosa and Moll [EM2004]:

\[\psi(s,z) = \frac{\zeta'(s+1,z) + (\gamma + \psi(-s)) \zeta(s+1,z)}{\Gamma(-s)}\]
void acb_barnes_g(acb_t res, const acb_t z, slong prec)
void acb_log_barnes_g(acb_t res, const acb_t z, slong prec)

Computes Barnes G-function or the logarithmic Barnes G-function, respectively. The logarithmic version has branch cuts on the negative real axis and is continuous elsewhere in the complex plane, in analogy with the logarithmic gamma function. The functional equation

\[\log G(z+1) = \log \Gamma(z) + \log G(z).\]

holds for all z.

For small integers, we directly use the recurrence relation \(G(z+1) = \Gamma(z) G(z)\) together with the initial value \(G(1) = 1\). For general z, we use the formula

\[\log G(z) = (z-1) \log \Gamma(z) - \zeta'(-1,z) + \zeta'(-1).\]

Zeta function

void acb_zeta(acb_t z, const acb_t s, slong prec)

Sets z to the value of the Riemann zeta function \(\zeta(s)\). Note: for computing derivatives with respect to \(s\), use acb_poly_zeta_series() or related methods.

This is a wrapper of acb_dirichlet_zeta().

void acb_hurwitz_zeta(acb_t z, const acb_t s, const acb_t a, slong prec)

Sets z to the value of the Hurwitz zeta function \(\zeta(s, a)\). Note: for computing derivatives with respect to \(s\), use acb_poly_zeta_series() or related methods.

This is a wrapper of acb_dirichlet_hurwitz().

void acb_bernoulli_poly_ui(acb_t res, ulong n, const acb_t x, slong prec)

Sets res to the value of the Bernoulli polynomial \(B_n(x)\).

Warning: this function is only fast if either n or x is a small integer.

This function reads Bernoulli numbers from the global cache if they are already cached, but does not automatically extend the cache by itself.

Polylogarithms

void acb_polylog(acb_t w, const acb_t s, const acb_t z, slong prec)
void acb_polylog_si(acb_t w, slong s, const acb_t z, slong prec)

Sets w to the polylogarithm \(\operatorname{Li}_s(z)\).

Arithmetic-geometric mean

See Algorithms for the arithmetic-geometric mean for implementation details.

void acb_agm1(acb_t m, const acb_t z, slong prec)

Sets m to the arithmetic-geometric mean \(M(z) = \operatorname{agm}(1,z)\), defined such that the function is continuous in the complex plane except for a branch cut along the negative half axis (where it is continuous from above). This corresponds to always choosing an “optimal” branch for the square root in the arithmetic-geometric mean iteration.

void acb_agm1_cpx(acb_ptr m, const acb_t z, slong len, slong prec)

Sets the coefficients in the array m to the power series expansion of the arithmetic-geometric mean at the point z truncated to length len, i.e. \(M(z+x) \in \mathbb{C}[[x]]\).

Other special functions

void acb_chebyshev_t_ui(acb_t a, ulong n, const acb_t x, slong prec)
void acb_chebyshev_u_ui(acb_t a, ulong n, const acb_t x, slong prec)

Evaluates the Chebyshev polynomial of the first kind \(a = T_n(x)\) or the Chebyshev polynomial of the second kind \(a = U_n(x)\).

void acb_chebyshev_t2_ui(acb_t a, acb_t b, ulong n, const acb_t x, slong prec)
void acb_chebyshev_u2_ui(acb_t a, acb_t b, ulong n, const acb_t x, slong prec)

Simultaneously evaluates \(a = T_n(x), b = T_{n-1}(x)\) or \(a = U_n(x), b = U_{n-1}(x)\). Aliasing between a, b and x is not permitted.

Vector functions

void _acb_vec_zero(acb_ptr A, slong n)

Sets all entries in vec to zero.

int _acb_vec_is_zero(acb_srcptr vec, slong len)

Returns nonzero iff all entries in x are zero.

int _acb_vec_is_real(acb_srcptr v, slong len)

Returns nonzero iff all entries in x have zero imaginary part.

void _acb_vec_set(acb_ptr res, acb_srcptr vec, slong len)

Sets res to a copy of vec.

void _acb_vec_set_round(acb_ptr res, acb_srcptr vec, slong len, slong prec)

Sets res to a copy of vec, rounding each entry to prec bits.

void _acb_vec_neg(acb_ptr res, acb_srcptr vec, slong len)
void _acb_vec_add(acb_ptr res, acb_srcptr vec1, acb_srcptr vec2, slong len, slong prec)
void _acb_vec_sub(acb_ptr res, acb_srcptr vec1, acb_srcptr vec2, slong len, slong prec)
void _acb_vec_scalar_submul(acb_ptr res, acb_srcptr vec, slong len, const acb_t c, slong prec)
void _acb_vec_scalar_addmul(acb_ptr res, acb_srcptr vec, slong len, const acb_t c, slong prec)
void _acb_vec_scalar_mul(acb_ptr res, acb_srcptr vec, slong len, const acb_t c, slong prec)
void _acb_vec_scalar_mul_ui(acb_ptr res, acb_srcptr vec, slong len, ulong c, slong prec)
void _acb_vec_scalar_mul_2exp_si(acb_ptr res, acb_srcptr vec, slong len, slong c)
void _acb_vec_scalar_mul_onei(acb_ptr res, acb_srcptr vec, slong len)
void _acb_vec_scalar_div_ui(acb_ptr res, acb_srcptr vec, slong len, ulong c, slong prec)
void _acb_vec_scalar_div(acb_ptr res, acb_srcptr vec, slong len, const acb_t c, slong prec)
void _acb_vec_scalar_mul_arb(acb_ptr res, acb_srcptr vec, slong len, const arb_t c, slong prec)
void _acb_vec_scalar_div_arb(acb_ptr res, acb_srcptr vec, slong len, const arb_t c, slong prec)
void _acb_vec_scalar_mul_fmpz(acb_ptr res, acb_srcptr vec, slong len, const fmpz_t c, slong prec)
void _acb_vec_scalar_div_fmpz(acb_ptr res, acb_srcptr vec, slong len, const fmpz_t c, slong prec)

Performs the respective scalar operation elementwise.

slong _acb_vec_bits(acb_srcptr vec, slong len)

Returns the maximum of arb_bits() for all entries in vec.

void _acb_vec_set_powers(acb_ptr xs, const acb_t x, slong len, slong prec)

Sets xs to the powers \(1, x, x^2, \ldots, x^{len-1}\).

void _acb_vec_unit_roots(acb_ptr z, slong order, slong prec)

Sets z to the powers \(1,z,z^2,\dots z^{\mathrm{order}-1}\) where \(z=\exp(\frac{2i\pi}{\mathrm{order}})\) to precision prec.

In order to avoid precision loss, this function does not simply compute powers of a primitive root.

void _acb_vec_add_error_arf_vec(acb_ptr res, arf_srcptr err, slong len)
void _acb_vec_add_error_mag_vec(acb_ptr res, mag_srcptr err, slong len)

Adds the magnitude of each entry in err to the radius of the corresponding entry in res.

void _acb_vec_indeterminate(acb_ptr vec, slong len)

Applies acb_indeterminate() elementwise.

void _acb_vec_trim(acb_ptr res, acb_srcptr vec, slong len)

Applies acb_trim() elementwise.

int _acb_vec_get_unique_fmpz_vec(fmpz * res, acb_srcptr vec, slong len)

Calls acb_get_unique_fmpz() elementwise and returns nonzero if all entries can be rounded uniquely to integers. If any entry in vec cannot be rounded uniquely to an integer, returns zero.

void _acb_vec_sort_pretty(acb_ptr vec, slong len)

Sorts the vector of complex numbers based on the real and imaginary parts. This is intended to reveal structure when printing a set of complex numbers, not to apply an order relation in a rigorous way.