An CArb consists of a pair of Arb:s.

Create new Acb

Apply function f to Acb

Apply function f to new Acb

Apply function f to real part of Acb

Apply function f to imaginary part of Acb

acb_realref z

acb_realref returns a CArb pointer to the real part of z.

acb_imagref z

acb_imagref returns a CArb pointer to the imaginary part of z.

acb_init x

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

acb_clear x

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

_acb_vec_init n

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

_acb_vec_clear v n

Clears an array of n initialized acb_struct:s.

acb_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(acb_struct) to get the size of the object as a whole.

_acb_vec_allocated_bytes vec 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.

_acb_vec_estimate_allocated_bytes len 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.

acb_onei z

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

acb_set_arb z c

Sets z to the value of x.

acb_set_arb_arb z x y

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

acb_set_round_arb z x prec

Sets z to x, rounded to prec bits.

acb_swap z x

Swaps z and x efficiently.

acb_add_error_arb x err

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

acb_get_mid m x

Sets m to the midpoint of x.

acb_fprint file x

Prints the internal representation of x.

acb_printd file x digits

Prints x in decimal to stdout. 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.

acb_fprintd file x digits

Prints x in decimal to stream file. 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.

acb_fprintn file x digits 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.

acb_randtest z state prec mag_bits

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

acb_randtest_special z state prec mag_bits

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

acb_randtest_precise z state prec mag_bits

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

acb_randtest_param z state prec mag_bits

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

acb_is_zero z

Returns nonzero iff z is zero.

acb_is_one z

Returns nonzero iff z is exactly 1.

acb_is_finite z

Returns nonzero iff z certainly is finite.

acb_is_exact z

Returns nonzero iff z is exact.

acb_is_int z

Returns nonzero iff z is an exact integer.

acb_is_int_2exp_si x e

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

acb_equal x 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.

acb_equal_si x y

Returns nonzero iff x is equal to the integer y.

acb_eq x 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.

acb_ne x 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.

acb_overlaps x y

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

acb_union z x y prec

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

acb_get_abs_ubound_arf u z prec

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

acb_get_abs_lbound_arf u z prec

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

acb_get_rad_ubound_arf u z prec

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

acb_get_mag u x

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

acb_get_mag_lower u x

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

acb_contains x y

Returns nonzero iff y is contained in x.

acb_contains_zero x

Returns nonzero iff zero is contained in x.

acb_contains_int x

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

acb_contains_interior x y

Tests if y is contained in the interior of x. This predicate always evaluates to false if x and y are both real-valued, since an imaginary part of 0 is not considered contained in the interior of the point interval 0. More generally, the same problem occurs for intervals with an exact real or imaginary part. Such intervals must be handled specially by the user where a different interpretation is intended.

acb_rel_error_bits 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.

acb_rel_accuracy_bits x

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

acb_rel_one_accuracy_bits x

Given a ball with midpoint m and radius r, returns an approximation of the relative accuracy of \([\max(1,|m|) \pm r]\) measured in bits.

acb_bits 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.

acb_indeterminate x

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

acb_trim y x

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

acb_is_real x

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

acb_get_unique_fmpz z 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.

acb_get_real re z

Sets re to the real part of z.

acb_get_imag im z

Sets im to the imaginary part of z.

acb_arg r z 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).

acb_abs r z prec

Sets r to the absolute value of z.

acb_sgn r z 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.

acb_csgn r 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.

acb_neg_round z x prec

Sets z to the negation of x.

acb_conj z x

Sets z to the complex conjugate of x.

acb_add z x y prec

Sets z to the sum of x and y.

acb_sub z x y prec

Sets z to the difference of x and y.

acb_mul_onei z x

Sets z to x multiplied by the imaginary unit.

acb_div_onei z x

Sets z to x divided by the imaginary unit.

acb_mul_arb z x y prec

Sets z to the product of x and y.

acb_mul z x y 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.

acb_mul_2exp_fmpz z x e

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

acb_sqr z x prec

Sets z to x squared.

acb_cube z x prec

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

acb_addmul_arb z x y prec

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

acb_submul_arb z x y prec

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

acb_inv z x prec

Sets z to the multiplicative inverse of x.

acb_div z x y prec

Sets z to the quotient of x and y.

acb_dot_precise res s subtract x xstep y ystep len prec

Computes the dot product of the vectors x and y, setting res to \(s + (-1)^{subtract} \sum_{i=0}^{len-1} x_i y_i\).

The initial term s is optional and can be omitted by passing NULL (equivalently, \(s = 0\)). The parameter subtract must be 0 or 1. The length len is allowed to be negative, which is equivalent to a length of zero. The parameters xstep or ystep specify a step length for traversing subsequences of the vectors x and y; either can be negative to step in the reverse direction starting from the initial pointer. Aliasing is allowed between res and s but not between res and the entries of x and y.

The default version determines the optimal precision for each term and performs all internal calculations using mpn arithmetic with minimal overhead. This is the preferred way to compute a dot product; it is generally much faster and more precise than a simple loop.

The simple version performs fused multiply-add operations in a simple loop. This can be used for testing purposes and is also used as a fallback by the default version when the exponents are out of range for the optimized code.

The precise version computes the dot product exactly up to the final rounding. This can be extremely slow and is only intended for testing.

acb_approx_dot res s subtract x xstep y ystep len prec

Computes an approximate dot product without error bounds. The radii of the inputs are ignored (only the midpoints are read) and only the midpoint of the output is written.

acb_dot_ui res initial subtract x xstep y ystep len prec

Equivalent to acb_dot, but with integers in the array y. The uiui and siui versions take an array of double-limb integers as input; the siui version assumes that these represent signed integers in two's complement form.

acb_const_pi y prec

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

acb_sqrt r z 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.

acb_sqrt_analytic r z analytic prec

Computes the square root. If analytic is set, gives a NaN-containing result if z touches the branch cut.

acb_rsqrt r z 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.

acb_rsqrt_analytic r z analytic prec

Computes the reciprocal square root. If analytic is set, gives a NaN-containing result if z touches the branch cut.

acb_quadratic_roots_fmpz r1 r2 a b c 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.

acb_root_ui r z k prec

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

acb_pow_si y b e 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).

acb_pow z x y 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 \(z = \exp(y \log x)\).

acb_pow_analytic r x y analytic prec

Computes the power \(x^y\). If analytic is set, gives a NaN-containing result if x touches the branch cut (unless y is an integer).

acb_unit_root res order prec

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

acb_exp y z prec

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

acb_exp_pi_i y z prec

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

acb_exp_invexp s t z prec

Sets \(s = \exp(z)\) and \(t = \exp(-z)\).

acb_expm1 res z prec

Sets res to \(\exp(z)-1\), using a more accurate method when \(z \approx 0\).

acb_log y z prec

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

acb_log_analytic r z analytic prec

Computes the natural logarithm. If analytic is set, gives a NaN-containing result if z touches the branch cut.

acb_log1p z x prec

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

acb_sin_cos s c z prec

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

acb_tan s z 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.

acb_cot s z 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.

acb_sin_cos_pi s c z 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.

acb_tan_pi s z 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.

acb_cot_pi s z 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.

acb_sec res z prec

Computes \(\sec(z) = 1 / \cos(z)\).

acb_csc res z prec

Computes \(\csc(x) = 1 / \sin(z)\).

acb_csc_pi res z prec

Computes \(\csc(\pi x) = 1 / \sin(\pi z)\). Evaluates the sine accurately via acb_sin_pi.

acb_sinc s z prec

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

acb_sinc_pi s z prec

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

acb_asin res z prec

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

acb_acos res z prec

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

acb_atan res z prec

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

acb_coth s z prec

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

acb_sech res z prec

Computes \(\operatorname{sech}(z) = 1 / \cosh(z)\).

acb_csch res z prec

Computes \(\operatorname{csch}(z) = 1 / \sinh(z)\).

acb_asinh res z prec

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

acb_acosh res z prec

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

acb_atanh res z prec

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

acb_lambertw_asymp res z k L M 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.

acb_lambertw_check_branch w k 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.

acb_lambertw_bound_deriv res z ez1 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 \(W_k(z)\).

acb_lambertw res z k flags 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].

acb_rising_ui z x n prec

Computes the rising factorial \(z = x (x+1) (x+2) \cdots (x+n-1)\). These functions are aliases for acb_hypgeom_rising_ui and acb_hypgeom_rising.

acb_gamma y x prec

Computes the gamma function \(y = \Gamma(x)\). This is an alias for acb_hypgeom_gamma.

acb_rgamma y x prec

Computes the reciprocal gamma function \(y = 1/\Gamma(x)\), avoiding division by zero at the poles of the gamma function. This is an alias for acb_hypgeom_rgamma.

acb_lgamma y x prec

Computes the logarithmic gamma function \(y = \log \Gamma(x)\). This is an alias for acb_hypgeom_lgamma.

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.

acb_digamma y x prec

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

acb_log_sin_pi res z 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.

acb_polygamma res s z 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]:

\[`\] \[\psi(s,z) = \frac{\zeta'(s+1,z) + (\gamma + \psi(-s)) \zeta(s+1,z)}{\Gamma(-s)}\]

acb_log_barnes_g res z 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

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

acb_zeta z s 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.

acb_hurwitz_zeta z s a 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.

acb_bernoulli_poly_ui res n x 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.

acb_polylog_si w s z prec

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

acb_agm1 m z 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.

acb_agm1_cpx m z len 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. \(M(z+x) \in \mathbb{C}[[x]]\).

acb_agm m x y prec

Sets m to the arithmetic-geometric mean of x and y. The square roots in the AGM iteration are chosen so as to form the "optimal" AGM sequence. This gives a well-defined function of x and y except when \(x / y\) is a negative real number, in which case there are two optimal AGM sequences. In that case, an arbitrary but consistent choice is made (if a decision cannot be made due to inexact arithmetic, the union of both choices is returned).

acb_chebyshev_u_ui a n x prec

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

acb_chebyshev_u2_ui a b n x 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.

acb_real_abs res z analytic prec

The absolute value is extended to \(+z\) in the right half plane and \(-z\) in the left half plane, with a discontinuity on the vertical line \(\operatorname{Re}(z) = 0\).

acb_real_sgn res z analytic prec

The sign function is extended to \(+1\) in the right half plane and \(-1\) in the left half plane, with a discontinuity on the vertical line \(\operatorname{Re}(z) = 0\). If analytic is not set, this is effectively the same function as acb_csgn.

acb_real_heaviside res z analytic prec

The Heaviside step function (or unit step function) is extended to \(+1\) in the right half plane and \(0\) in the left half plane, with a discontinuity on the vertical line \(\operatorname{Re}(z) = 0\).

acb_real_floor res z analytic prec

The floor function is extended to a piecewise constant function equal to \(n\) in the strips with real part \((n,n+1)\), with discontinuities on the vertical lines \(\operatorname{Re}(z) = n\).

acb_real_ceil res z analytic prec

The ceiling function is extended to a piecewise constant function equal to \(n+1\) in the strips with real part \((n,n+1)\), with discontinuities on the vertical lines \(\operatorname{Re}(z) = n\).

acb_real_max res x y analytic prec

The real function \(\max(x,y)\) is extended to a piecewise analytic function of two variables by returning \(x\) when \(\operatorname{Re}(x) \ge \operatorname{Re}(y)\) and returning \(y\) when \(\operatorname{Re}(x) < \operatorname{Re}(y)\), with discontinuities where \(\operatorname{Re}(x) = \operatorname{Re}(y)\).

acb_real_min res x y analytic prec

The real function \(\min(x,y)\) is extended to a piecewise analytic function of two variables by returning \(x\) when \(\operatorname{Re}(x) \le \operatorname{Re}(y)\) and returning \(y\) when \(\operatorname{Re}(x) > \operatorname{Re}(y)\), with discontinuities where \(\operatorname{Re}(x) = \operatorname{Re}(y)\).

acb_real_sqrtpos res z analytic prec

Extends the real square root function on \([0,+\infty)\) to the usual complex square root on the cut plane. Like arb_sqrtpos, only the nonnegative part of z is considered if z is purely real and analytic is not set. This is useful for integrating \(\sqrt{f(x)}\) where it is known that \(f(x) \ge 0\): unlike acb_sqrt_analytic, no spurious imaginary terms \([\pm \varepsilon] i\) are created when the balls computed for \(f(x)\) straddle zero.

_acb_vec_zero A n

Sets all entries in vec to zero.

_acb_vec_is_zero vec len

Returns nonzero iff all entries in x are zero.

_acb_vec_is_real v len

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

_acb_vec_set res vec len

Sets res to a copy of vec.

_acb_vec_set_round res vec len prec

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

_acb_vec_swap vec1 vec2 len

Swaps the entries of vec1 and vec2.

_acb_vec_scalar_div_fmpz res vec len c prec

Performs the respective scalar operation elementwise.

_acb_vec_bits vec len

Returns the maximum of arb_bits for all entries in vec.

_acb_vec_set_powers xs x len prec

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

_acb_vec_unit_roots z order len prec

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

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

_acb_vec_add_error_mag_vec res err len

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

_acb_vec_indeterminate vec len

Applies acb_indeterminate elementwise.

_acb_vec_trim res vec len

Applies acb_trim elementwise.

_acb_vec_get_unique_fmpz_vec res vec 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.

_acb_vec_sort_pretty vec 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.