arb.h – real numbers

An arb_t represents a ball over the real numbers, that is, an interval \([m \pm r] \equiv [m-r, m+r]\) where the midpoint \(m\) and the radius \(r\) are (extended) real numbers and \(r\) is nonnegative (possibly infinite). The result of an (approximate) operation done on arb_t variables is a ball which contains the result of the (mathematically exact) operation applied to any choice of points in the input balls. In general, the output ball is not the smallest possible.

The precision parameter passed to each function roughly indicates the precision to which calculations on the midpoint are carried out (operations on the radius are always done using a fixed, small precision.)

For arithmetic operations, the precision parameter currently simply specifies the precision of the corresponding arf_t operation. In the future, the arithmetic might be made faster by incorporating sloppy rounding (typically equivalent to a loss of 1-2 bits of effective working precision) when the result is known to be inexact (while still propagating errors rigorously, of course). Arithmetic operations done on exact input with exactly representable output are always guaranteed to produce exact output.

For more complex operations, the precision parameter indicates a minimum working precision (algorithms might allocate extra internal precision to attempt to produce an output accurate to the requested number of bits, especially when the required precision can be estimated easily, but this is not generally required).

If the precision is increased and the inputs either are exact or are computed with increased accuracy as well, the output should converge proportionally, absent any bugs. The general intended strategy for using ball arithmetic is to add a few guard bits, and then repeat the calculation as necessary with an exponentially increasing number of guard bits (Ziv’s strategy) until the result is exact enough for one’s purposes (typically the first attempt will be successful).

The following balls with an infinite or NaN component are permitted, and may be returned as output from functions.

  • The ball \([+\infty \pm c]\), where \(c\) is finite, represents the point at positive infinity. Such a ball can always be replaced by \([+\infty \pm 0]\) while preserving mathematical correctness (this is currently not done automatically by the library).
  • The ball \([-\infty \pm c]\), where \(c\) is finite, represents the point at negative infinity. Such a ball can always be replaced by \([-\infty \pm 0]\) while preserving mathematical correctness (this is currently not done automatically by the library).
  • The ball \([c \pm \infty]\), where \(c\) is finite or infinite, represents the whole extended real line \([-\infty,+\infty]\). Such a ball can always be replaced by \([0 \pm \infty]\) while preserving mathematical correctness (this is currently not done automatically by the library). Note that there is no way to represent a half-infinite interval such as \([0,\infty]\).
  • The ball \([\operatorname{NaN} \pm c]\), where \(c\) is finite or infinite, represents an indeterminate value (the value could be any extended real number, or it could represent a function being evaluated outside its domain of definition, for example where the result would be complex). Such an indeterminate ball can always be replaced by \([\operatorname{NaN} \pm \infty]\) while preserving mathematical correctness (this is currently not done automatically by the library).

Types, macros and constants

arb_struct
arb_t

An arb_struct consists of an arf_struct (the midpoint) and a mag_struct (the radius). An arb_t is defined as an array of length one of type arb_struct, permitting an arb_t to be passed by reference.

arb_ptr

Alias for arb_struct *, used for vectors of numbers.

arb_srcptr

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

arb_midref(x)

Macro returning a pointer to the midpoint of x as an arf_t.

arb_radref(x)

Macro returning a pointer to the radius of x as a mag_t.

Memory management

void arb_init(arb_t x)

Initializes the variable x for use. Its midpoint and radius are both set to zero.

void arb_clear(arb_t x)

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

arb_ptr _arb_vec_init(slong n)

Returns a pointer to an array of n initialized arb_struct entries.

void _arb_vec_clear(arb_ptr v, slong n)

Clears an array of n initialized arb_struct entries.

void arb_swap(arb_t x, arb_t y)

Swaps x and y efficiently.

slong arb_allocated_bytes(const arb_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(arb_struct) to get the size of the object as a whole.

slong _arb_vec_allocated_bytes(arb_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 _arb_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. This function returns a double to avoid overflow issues when both len and prec are large.

This is only an approximation of the physical memory that will be used by an actual vector. In practice, the space varies with the content of the numbers; for example, zeros and small integers require no internal heap allocation even if the precision is huge. The estimate assumes that exponents will not be bignums. The actual amount may also be higher or lower due to overhead in the memory allocator or overcommitment by the operating system.

Assignment and rounding

void arb_set(arb_t y, const arb_t x)
void arb_set_arf(arb_t y, const arf_t x)
void arb_set_si(arb_t y, slong x)
void arb_set_ui(arb_t y, ulong x)
void arb_set_d(arb_t y, double x)
void arb_set_fmpz(arb_t y, const fmpz_t x)

Sets y to the value of x without rounding.

void arb_set_fmpz_2exp(arb_t y, const fmpz_t x, const fmpz_t e)

Sets y to \(x \cdot 2^e\).

void arb_set_round(arb_t y, const arb_t x, slong prec)
void arb_set_round_fmpz(arb_t y, const fmpz_t x, slong prec)

Sets y to the value of x, rounded to prec bits.

void arb_set_round_fmpz_2exp(arb_t y, const fmpz_t x, const fmpz_t e, slong prec)

Sets y to \(x \cdot 2^e\), rounded to prec bits.

void arb_set_fmpq(arb_t y, const fmpq_t x, slong prec)

Sets y to the rational number x, rounded to prec bits.

int arb_set_str(arb_t res, const char * inp, slong prec)

Sets res to the value specified by the human-readable string inp. The input may be a decimal floating-point literal, such as “25”, “0.001”, “7e+141” or “-31.4159e-1”, and may also consist of two such literals separated by the symbol “+/-” and optionally enclosed in brackets, e.g. “[3.25 +/- 0.0001]”, or simply “[+/- 10]” with an implicit zero midpoint. The output is rounded to prec bits, and if the binary-to-decimal conversion is inexact, the resulting error is added to the radius.

The symbols “inf” and “nan” are recognized (a nan midpoint results in an indeterminate interval, with infinite radius).

Returns 0 if successful and nonzero if unsuccessful. If unsuccessful, the result is set to an indeterminate interval.

char * arb_get_str(const arb_t x, slong n, ulong flags)

Returns a nice human-readable representation of x, with at most n digits of the midpoint printed.

With default flags, the output can be parsed back with arb_set_str(), and this is guaranteed to produce an interval containing the original interval x.

By default, the output is rounded so that the value given for the midpoint is correct up to 1 ulp (unit in the last decimal place).

If ARB_STR_MORE is added to flags, more (possibly incorrect) digits may be printed.

If ARB_STR_NO_RADIUS is added to flags, the radius is not included in the output if at least 1 digit of the midpoint can be printed.

By adding a multiple m of ARB_STR_CONDENSE to flags, strings of more than three times m consecutive digits are condensed, only printing the leading and trailing m digits along with brackets indicating the number of digits omitted (useful when computing values to extremely high precision).

Assignment of special values

void arb_zero(arb_t x)

Sets x to zero.

void arb_one(arb_t f)

Sets x to the exact integer 1.

void arb_pos_inf(arb_t x)

Sets x to positive infinity, with a zero radius.

void arb_neg_inf(arb_t x)

Sets x to negative infinity, with a zero radius.

void arb_zero_pm_inf(arb_t x)

Sets x to \([0 \pm \infty]\), representing the whole extended real line.

void arb_indeterminate(arb_t x)

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

Input and output

The arb_print… functions print to standard output, while arb_fprint… functions print to the stream file.

void arb_print(const arb_t x)
void arb_fprint(FILE * file, const arb_t x)

Prints the internal representation of x.

void arb_printd(const arb_t x, slong digits)
void arb_fprintd(FILE * file, const arb_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 arb_printn(const arb_t x, slong digits, ulong flags)
void arb_fprintn(FILE * file, const arb_t x, slong digits, ulong flags)

Prints a nice decimal representation of x. By default, the output shows the midpoint with a guaranteed error of at most one unit in the last decimal place. In addition, an explicit error bound is printed so that the displayed decimal interval is guaranteed to enclose x. See arb_get_str() for details.

Random number generation

void arb_randtest(arb_t x, flint_rand_t state, slong prec, slong mag_bits)

Generates a random ball. The midpoint and radius will both be finite.

void arb_randtest_exact(arb_t x, flint_rand_t state, slong prec, slong mag_bits)

Generates a random number with zero radius.

void arb_randtest_precise(arb_t x, flint_rand_t state, slong prec, slong mag_bits)

Generates a random number with radius around \(2^{-\text{prec}}\) the magnitude of the midpoint.

void arb_randtest_wide(arb_t x, flint_rand_t state, slong prec, slong mag_bits)

Generates a random number with midpoint and radius chosen independently, possibly giving a very large interval.

void arb_randtest_special(arb_t x, flint_rand_t state, slong prec, slong mag_bits)

Generates a random interval, possibly having NaN or an infinity as the midpoint and possibly having an infinite radius.

void arb_get_rand_fmpq(fmpq_t q, flint_rand_t state, const arb_t x, slong bits)

Sets q to a random rational number from the interval represented by x. A denominator is chosen by multiplying the binary denominator of x by a random integer up to bits bits.

The outcome is undefined if the midpoint or radius of x is non-finite, or if the exponent of the midpoint or radius is so large or small that representing the endpoints as exact rational numbers would cause overflows.

Radius and interval operations

void arb_get_mid_arb(arb_t m, const arb_t x)

Sets m to the midpoint of x.

void arb_get_rad_arb(arb_t r, const arb_t x)

Sets r to the radius of x.

void arb_add_error_arf(arb_t x, const arf_t err)
void arb_add_error_mag(arb_t x, const mag_t err)
void arb_add_error(arb_t x, const arb_t err)

Adds the absolute value of err to the radius of x (the operation is done in-place).

void arb_add_error_2exp_si(arb_t x, slong e)
void arb_add_error_2exp_fmpz(arb_t x, const fmpz_t e)

Adds \(2^e\) to the radius of x.

void arb_union(arb_t z, const arb_t x, const arb_t y, slong prec)

Sets z to a ball containing both x and y.

int arb_intersection(arb_t z, const arb_t x, const arb_t y, slong prec)

If x and y overlap according to arb_overlaps(), then z is set to a ball containing the intersection of x and y and a nonzero value is returned. Otherwise zero is returned and the value of z is undefined. If x or y contains NaN, the result is NaN.

void arb_nonnegative_part(arb_t res, const arb_t x)

Sets res to the intersection of x with \([0,\infty]\). If x is nonnegative, an exact copy is made. If x is finite and contains negative numbers, an interval of the form \([r/2 \pm r/2]\) is produced, which certainly contains no negative points. In the special case when x is strictly negative, res is set to zero.

void arb_get_abs_ubound_arf(arf_t u, const arb_t x, slong prec)

Sets u to the upper bound for the absolute value of x, rounded up to prec bits. If x contains NaN, the result is NaN.

void arb_get_abs_lbound_arf(arf_t u, const arb_t x, slong prec)

Sets u to the lower bound for the absolute value of x, rounded down to prec bits. If x contains NaN, the result is NaN.

void arb_get_ubound_arf(arf_t u, const arb_t x, long prec)

Sets u to the upper bound for the value of x, rounded up to prec bits. If x contains NaN, the result is NaN.

void arb_get_lbound_arf(arf_t u, const arb_t x, long prec)

Sets u to the lower bound for the value of x, rounded down to prec bits. If x contains NaN, the result is NaN.

void arb_get_mag(mag_t z, const arb_t x)

Sets z to an upper bound for the absolute value of x. If x contains NaN, the result is positive infinity.

void arb_get_mag_lower(mag_t z, const arb_t x)

Sets z to a lower bound for the absolute value of x. If x contains NaN, the result is zero.

void arb_get_mag_lower_nonnegative(mag_t z, const arb_t x)

Sets z to a lower bound for the signed value of x, or zero if x overlaps with the negative half-axis. If x contains NaN, the result is zero.

void arb_get_interval_fmpz_2exp(fmpz_t a, fmpz_t b, fmpz_t exp, const arb_t x)

Computes the exact interval represented by x, in the form of an integer interval multiplied by a power of two, i.e. \(x = [a, b] \times 2^{\text{exp}}\). The result is normalized by removing common trailing zeros from a and b.

This method aborts if x is infinite or NaN, or if the difference between the exponents of the midpoint and the radius is so large that allocating memory for the result fails.

Warning: this method will allocate a huge amount of memory to store the result if the exponent difference is huge. Memory allocation could succeed even if the required space is far larger than the physical memory available on the machine, resulting in swapping. It is recommended to check that the midpoint and radius of x both are within a reasonable range before calling this method.

void arb_set_interval_arf(arb_t x, const arf_t a, const arf_t b, slong prec)
void arb_set_interval_mpfr(arb_t x, const mpfr_t a, const mpfr_t b, slong prec)

Sets x to a ball containing the interval \([a, b]\). We require that \(a \le b\).

void arb_get_interval_arf(arf_t a, arf_t b, const arb_t x, slong prec)
void arb_get_interval_mpfr(mpfr_t a, mpfr_t b, const arb_t x)

Constructs an interval \([a, b]\) containing the ball x. The MPFR version uses the precision of the output variables.

slong arb_rel_error_bits(const arb_t x)

Returns the effective relative error of x measured in bits, defined as the difference between the position of the top bit in the radius and the top bit in the midpoint, plus one. The result is clamped between plus/minus ARF_PREC_EXACT.

slong arb_rel_accuracy_bits(const arb_t x)

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

slong arb_bits(const arb_t x)

Returns the number of bits needed to represent the absolute value of the mantissa of the midpoint of x, i.e. the minimum precision sufficient to represent x exactly. Returns 0 if the midpoint of x is a special value.

void arb_trim(arb_t y, const arb_t x)

Sets y to a trimmed copy of x: rounds x to a number of bits equal to the accuracy of x (as indicated by its radius), plus a few guard bits. The resulting ball is guaranteed to contain x, but is more economical if x has less than full accuracy.

int arb_get_unique_fmpz(fmpz_t z, const arb_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.

This method aborts if there is a unique integer but that integer is so large that allocating memory for the result fails.

Warning: this method will allocate a huge amount of memory to store the result if there is a unique integer and that integer is huge. Memory allocation could succeed even if the required space is far larger than the physical memory available on the machine, resulting in swapping. It is recommended to check that the midpoint of x is within a reasonable range before calling this method.

void arb_floor(arb_t y, const arb_t x, slong prec)
void arb_ceil(arb_t y, const arb_t x, slong prec)

Sets y to a ball containing \(\lfloor x \rfloor\) and \(\lceil x \rceil\) respectively, with the midpoint of y rounded to at most prec bits.

void arb_get_fmpz_mid_rad_10exp(fmpz_t mid, fmpz_t rad, fmpz_t exp, const arb_t x, slong n)

Assuming that x is finite and not exactly zero, computes integers mid, rad, exp such that \(x \in [m-r, m+r] \times 10^e\) and such that the larger out of mid and rad has at least n digits plus a few guard digits. If x is infinite or exactly zero, the outputs are all set to zero.

int arb_can_round_arf(const arb_t x, slong prec, arf_rnd_t rnd)
int arb_can_round_mpfr(const arb_t x, slong prec, mpfr_rnd_t rnd)

Returns nonzero if rounding the midpoint of x to prec bits in the direction rnd is guaranteed to give the unique correctly rounded floating-point approximation for the real number represented by x.

In other words, if this function returns nonzero, applying arf_set_round(), or arf_get_mpfr(), or arf_get_d() to the midpoint of x is guaranteed to return a correctly rounded arf_t, mpfr_t (provided that prec is the precision of the output variable), or double (provided that prec is 53). Moreover, arf_get_mpfr() is guaranteed to return the correct ternary value according to MPFR semantics.

Note that the mpfr version of this function takes an MPFR rounding mode symbol as input, while the arf version takes an arf rounding mode symbol. Otherwise, the functions are identical.

This function may perform a fast, inexact test; that is, it may return zero in some cases even when correct rounding actually is possible.

To be conservative, zero is returned when x is non-finite, even if it is an “exact” infinity.

Comparisons

int arb_is_zero(const arb_t x)

Returns nonzero iff the midpoint and radius of x are both zero.

int arb_is_nonzero(const arb_t x)

Returns nonzero iff zero is not contained in the interval represented by x.

int arb_is_one(const arb_t f)

Returns nonzero iff x is exactly 1.

int arb_is_finite(const arb_t x)

Returns nonzero iff the midpoint and radius of x are both finite floating-point numbers, i.e. not infinities or NaN.

int arb_is_exact(const arb_t x)

Returns nonzero iff the radius of x is zero.

int arb_is_int(const arb_t x)

Returns nonzero iff x is an exact integer.

int arb_is_int_2exp_si(const arb_t x, slong e)

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

int arb_equal(const arb_t x, const arb_t y)

Returns nonzero iff x and y are equal as balls, i.e. have both the same midpoint and radius.

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

int arb_equal_si(const arb_t x, slong y)

Returns nonzero iff x is equal to the integer y.

int arb_is_positive(const arb_t x)
int arb_is_nonnegative(const arb_t x)
int arb_is_negative(const arb_t x)
int arb_is_nonpositive(const arb_t x)

Returns nonzero iff all points p in the interval represented by x satisfy, respectively, \(p > 0\), \(p \ge 0\), \(p < 0\), \(p \le 0\). If x contains NaN, returns zero.

int arb_overlaps(const arb_t x, const arb_t y)

Returns nonzero iff x and y have some point in common. If either x or y contains NaN, this function always returns nonzero (as a NaN could be anything, it could in particular contain any number that is included in the other operand).

int arb_contains_arf(const arb_t x, const arf_t y)
int arb_contains_fmpq(const arb_t x, const fmpq_t y)
int arb_contains_fmpz(const arb_t x, const fmpz_t y)
int arb_contains_si(const arb_t x, slong y)
int arb_contains_mpfr(const arb_t x, const mpfr_t y)
int arb_contains(const arb_t x, const arb_t y)

Returns nonzero iff the given number (or ball) y is contained in the interval represented by x.

If x is contains NaN, this function always returns nonzero (as it could represent anything, and in particular could represent all the points included in y). If y contains NaN and x does not, it always returns zero.

int arb_contains_int(const arb_t x)

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

int arb_contains_zero(const arb_t x)
int arb_contains_negative(const arb_t x)
int arb_contains_nonpositive(const arb_t x)
int arb_contains_positive(const arb_t x)
int arb_contains_nonnegative(const arb_t x)

Returns nonzero iff there is any point p in the interval represented by x satisfying, respectively, \(p = 0\), \(p < 0\), \(p \le 0\), \(p > 0\), \(p \ge 0\). If x contains NaN, returns nonzero.

int arb_eq(const arb_t x, const arb_t y)
int arb_ne(const arb_t x, const arb_t y)
int arb_lt(const arb_t x, const arb_t y)
int arb_le(const arb_t x, const arb_t y)
int arb_gt(const arb_t x, const arb_t y)
int arb_ge(const arb_t x, const arb_t y)

Respectively performs the comparison \(x = y\), \(x \ne y\), \(x < y\), \(x \le y\), \(x > y\), \(x \ge y\) in a mathematically meaningful way. If the comparison \(t \, (\operatorname{op}) \, u\) holds for all \(t \in x\) and all \(u \in y\), returns 1. Otherwise, returns 0.

The balls x and y are viewed as subintervals of the extended real line. Note that balls that are formally different can compare as equal under this definition: for example, \([-\infty \pm 3] = [-\infty \pm 0]\). Also \([-\infty] \le [\infty \pm \infty]\).

The output is always 0 if either input has NaN as midpoint.

Arithmetic

void arb_neg(arb_t y, const arb_t x)
void arb_neg_round(arb_t y, const arb_t x, slong prec)

Sets y to the negation of x.

void arb_abs(arb_t y, const arb_t x)

Sets y to the absolute value of x. No attempt is made to improve the interval represented by x if it contains zero.

void arb_sgn(arb_t y, const arb_t x)

Sets y to the sign function of x. The result is \([0 \pm 1]\) if x contains both zero and nonzero numbers.

void arb_min(arb_t z, const arb_t x, const arb_t y, slong prec)
void arb_max(arb_t z, const arb_t x, const arb_t y, slong prec)

Sets z respectively to the minimum and the maximum of x and y.

void arb_add(arb_t z, const arb_t x, const arb_t y, slong prec)
void arb_add_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
void arb_add_ui(arb_t z, const arb_t x, ulong y, slong prec)
void arb_add_si(arb_t z, const arb_t x, slong y, slong prec)
void arb_add_fmpz(arb_t z, const arb_t x, const fmpz_t y, slong prec)

Sets \(z = x + y\), rounded to prec bits. The precision can be ARF_PREC_EXACT provided that the result fits in memory.

void arb_add_fmpz_2exp(arb_t z, const arb_t x, const fmpz_t m, const fmpz_t e, slong prec)

Sets \(z = x + m \cdot 2^e\), rounded to prec bits. The precision can be ARF_PREC_EXACT provided that the result fits in memory.

void arb_sub(arb_t z, const arb_t x, const arb_t y, slong prec)
void arb_sub_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
void arb_sub_ui(arb_t z, const arb_t x, ulong y, slong prec)
void arb_sub_si(arb_t z, const arb_t x, slong y, slong prec)
void arb_sub_fmpz(arb_t z, const arb_t x, const fmpz_t y, slong prec)

Sets \(z = x - y\), rounded to prec bits. The precision can be ARF_PREC_EXACT provided that the result fits in memory.

void arb_mul(arb_t z, const arb_t x, const arb_t y, slong prec)
void arb_mul_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
void arb_mul_si(arb_t z, const arb_t x, slong y, slong prec)
void arb_mul_ui(arb_t z, const arb_t x, ulong y, slong prec)
void arb_mul_fmpz(arb_t z, const arb_t x, const fmpz_t y, slong prec)

Sets \(z = x \cdot y\), rounded to prec bits. The precision can be ARF_PREC_EXACT provided that the result fits in memory.

void arb_mul_2exp_si(arb_t y, const arb_t x, slong e)
void arb_mul_2exp_fmpz(arb_t y, const arb_t x, const fmpz_t e)

Sets y to x multiplied by \(2^e\).

void arb_addmul(arb_t z, const arb_t x, const arb_t y, slong prec)
void arb_addmul_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
void arb_addmul_si(arb_t z, const arb_t x, slong y, slong prec)
void arb_addmul_ui(arb_t z, const arb_t x, ulong y, slong prec)
void arb_addmul_fmpz(arb_t z, const arb_t x, const fmpz_t y, slong prec)

Sets \(z = z + x \cdot y\), rounded to prec bits. The precision can be ARF_PREC_EXACT provided that the result fits in memory.

void arb_submul(arb_t z, const arb_t x, const arb_t y, slong prec)
void arb_submul_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
void arb_submul_si(arb_t z, const arb_t x, slong y, slong prec)
void arb_submul_ui(arb_t z, const arb_t x, ulong y, slong prec)
void arb_submul_fmpz(arb_t z, const arb_t x, const fmpz_t y, slong prec)

Sets \(z = z - x \cdot y\), rounded to prec bits. The precision can be ARF_PREC_EXACT provided that the result fits in memory.

void arb_inv(arb_t y, const arb_t x, slong prec)

Sets z to \(1 / x\).

void arb_div(arb_t z, const arb_t x, const arb_t y, slong prec)
void arb_div_arf(arb_t z, const arb_t x, const arf_t y, slong prec)
void arb_div_si(arb_t z, const arb_t x, slong y, slong prec)
void arb_div_ui(arb_t z, const arb_t x, ulong y, slong prec)
void arb_div_fmpz(arb_t z, const arb_t x, const fmpz_t y, slong prec)
void arb_fmpz_div_fmpz(arb_t z, const fmpz_t x, const fmpz_t y, slong prec)
void arb_ui_div(arb_t z, ulong x, const arb_t y, slong prec)

Sets \(z = x / y\), rounded to prec bits. If y contains zero, z is set to \(0 \pm \infty\). Otherwise, error propagation uses the rule

\[\left| \frac{x}{y} - \frac{x+\xi_1 a}{y+\xi_2 b} \right| = \left|\frac{x \xi_2 b - y \xi_1 a}{y (y+\xi_2 b)}\right| \le \frac{|xb|+|ya|}{|y| (|y|-b)}\]

where \(-1 \le \xi_1, \xi_2 \le 1\), and where the triangle inequality has been applied to the numerator and the reverse triangle inequality has been applied to the denominator.

void arb_div_2expm1_ui(arb_t z, const arb_t x, ulong n, slong prec)

Sets \(z = x / (2^n - 1)\), rounded to prec bits.

Powers and roots

void arb_sqrt(arb_t z, const arb_t x, slong prec)
void arb_sqrt_arf(arb_t z, const arf_t x, slong prec)
void arb_sqrt_fmpz(arb_t z, const fmpz_t x, slong prec)
void arb_sqrt_ui(arb_t z, ulong x, slong prec)

Sets z to the square root of x, rounded to prec bits.

If \(x = m \pm x\) where \(m \ge r \ge 0\), the propagated error is bounded by \(\sqrt{m} - \sqrt{m-r} = \sqrt{m} (1 - \sqrt{1 - r/m}) \le \sqrt{m} (r/m + (r/m)^2)/2\).

void arb_sqrtpos(arb_t z, const arb_t x, slong prec)

Sets z to the square root of x, assuming that x represents a nonnegative number (i.e. discarding any negative numbers in the input interval).

void arb_hypot(arb_t z, const arb_t x, const arb_t y, slong prec)

Sets z to \(\sqrt{x^2 + y^2}\).

void arb_rsqrt(arb_t z, const arb_t x, slong prec)
void arb_rsqrt_ui(arb_t z, ulong x, slong prec)

Sets z to the reciprocal square root of x, rounded to prec bits. At high precision, this is faster than computing a square root.

void arb_sqrt1pm1(arb_t z, const arb_t x, slong prec)

Sets \(z = \sqrt{1+x}-1\), computed accurately when \(x \approx 0\).

void arb_root_ui(arb_t z, const arb_t x, ulong k, slong prec)

Sets z to the k-th root of x, rounded to prec bits. This function selects between different algorithms. For large k, it evaluates \(\exp(\log(x)/k)\). For small k, it uses arf_root() at the midpoint and computes a propagated error bound as follows: if input interval is \([m-r, m+r]\) with \(r \le m\), the error is largest at \(m-r\) where it satisfies

\[ \begin{align}\begin{aligned}m^{1/k} - (m-r)^{1/k} = m^{1/k} [1 - (1-r/m)^{1/k}]\\= m^{1/k} [1 - \exp(\log(1-r/m)/k)]\\\le m^{1/k} \min(1, -\log(1-r/m)/k)\\= m^{1/k} \min(1, \log(1+r/(m-r))/k).\end{aligned}\end{align} \]

This is evaluated using mag_log1p().

void arb_root(arb_t z, const arb_t x, ulong k, slong prec)

Alias for arb_root_ui(), provided for backwards compatibility.

void arb_sqr(arb_t y, const arb_t x, slong prec)

Sets y to be the square of x.

void arb_pow_fmpz_binexp(arb_t y, const arb_t b, const fmpz_t e, slong prec)
void arb_pow_fmpz(arb_t y, const arb_t b, const fmpz_t e, slong prec)
void arb_pow_ui(arb_t y, const arb_t b, ulong e, slong prec)
void arb_ui_pow_ui(arb_t y, ulong b, ulong e, slong prec)
void arb_si_pow_ui(arb_t y, slong b, ulong e, slong prec)

Sets \(y = b^e\) using binary exponentiation (with an initial division if \(e < 0\)). Provided that b and e are small enough and the exponent is positive, the exact power can be computed by setting the precision to ARF_PREC_EXACT.

Note that these functions can get slow if the exponent is extremely large (in such cases arb_pow() may be superior).

void arb_pow_fmpq(arb_t y, const arb_t x, const fmpq_t a, slong prec)

Sets \(y = b^e\), computed as \(y = (b^{1/q})^p\) if the denominator of \(e = p/q\) is small, and generally as \(y = \exp(e \log b)\).

Note that this function can get slow if the exponent is extremely large (in such cases arb_pow() may be superior).

void arb_pow(arb_t z, const arb_t x, const arb_t y, slong prec)

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

Exponentials and logarithms

void arb_log_ui(arb_t z, ulong x, slong prec)
void arb_log_fmpz(arb_t z, const fmpz_t x, slong prec)
void arb_log_arf(arb_t z, const arf_t x, slong prec)
void arb_log(arb_t z, const arb_t x, slong prec)

Sets \(z = \log(x)\).

At low to medium precision (up to about 4096 bits), arb_log_arf() uses table-based argument reduction and fast Taylor series evaluation via _arb_atan_taylor_rs(). At high precision, it falls back to MPFR. The function arb_log() simply calls arb_log_arf() with the midpoint as input, and separately adds the propagated error.

void arb_log_ui_from_prev(arb_t log_k1, ulong k1, arb_t log_k0, ulong k0, slong prec)

Computes \(\log(k_1)\), given \(\log(k_0)\) where \(k_0 < k_1\). At high precision, this function uses the formula \(\log(k_1) = \log(k_0) + 2 \operatorname{atanh}((k_1-k_0)/(k_1+k_0))\), evaluating the inverse hyperbolic tangent using binary splitting (for best efficiency, \(k_0\) should be large and \(k_1 - k_0\) should be small). Otherwise, it ignores \(\log(k_0)\) and evaluates the logarithm the usual way.

void arb_log1p(arb_t z, const arb_t x, slong prec)

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

void arb_log_base_ui(arb_t res, const arb_t x, ulong b, slong prec)

Sets res to \(\log_b(x)\). The result is computed exactly when possible.

void arb_exp(arb_t z, const arb_t x, slong prec)

Sets \(z = \exp(x)\). Error propagation is done using the following rule: assuming \(x = m \pm r\), the error is largest at \(m + r\), and we have \(\exp(m+r) - \exp(m) = \exp(m) (\exp(r)-1) \le r \exp(m+r)\).

void arb_expm1(arb_t z, const arb_t x, slong prec)

Sets \(z = \exp(x)-1\), computed accurately when \(x \approx 0\).

void arb_exp_invexp(arb_t z, arb_t w, const arb_t x, slong prec)

Sets \(z = \exp(x)\) and \(w = \exp(-x)\). The second exponential is computed from the first using a division, but propagated error bounds are computed separately.

Trigonometric functions

void arb_sin(arb_t s, const arb_t x, slong prec)
void arb_cos(arb_t c, const arb_t x, slong prec)
void arb_sin_cos(arb_t s, arb_t c, const arb_t x, slong prec)

Sets \(s = \sin(x)\), \(c = \cos(x)\). Error propagation uses the rule \(|\sin(m \pm r) - \sin(m)| \le \min(r,2)\).

void arb_sin_pi(arb_t s, const arb_t x, slong prec)
void arb_cos_pi(arb_t c, const arb_t x, slong prec)
void arb_sin_cos_pi(arb_t s, arb_t c, const arb_t x, slong prec)

Sets \(s = \sin(\pi x)\), \(c = \cos(\pi x)\).

void arb_tan(arb_t y, const arb_t x, slong prec)

Sets \(y = \tan(x) = \sin(x) / \cos(y)\).

void arb_cot(arb_t y, const arb_t x, slong prec)

Sets \(y = \cot(x) = \cos(x) / \sin(y)\).

void arb_sin_cos_pi_fmpq(arb_t s, arb_t c, const fmpq_t x, slong prec)
void arb_sin_pi_fmpq(arb_t s, const fmpq_t x, slong prec)
void arb_cos_pi_fmpq(arb_t c, const fmpq_t x, slong prec)

Sets \(s = \sin(\pi x)\), \(c = \cos(\pi x)\) where \(x\) is a rational number (whose numerator and denominator are assumed to be reduced). We first use trigonometric symmetries to reduce the argument to the octant \([0, 1/4]\). Then we either multiply by a numerical approximation of \(\pi\) and evaluate the trigonometric function the usual way, or we use algebraic methods, depending on which is estimated to be faster. Since the argument has been reduced to the first octant, the first of these two methods gives full accuracy even if the original argument is close to some root other the origin.

void arb_tan_pi(arb_t y, const arb_t x, slong prec)

Sets \(y = \tan(\pi x)\).

void arb_cot_pi(arb_t y, const arb_t x, slong prec)

Sets \(y = \cot(\pi x)\).

void arb_sec(arb_t res, const arb_t x, slong prec)

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

void arb_csc(arb_t res, const arb_t x, slong prec)

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

void arb_sinc(arb_t z, const arb_t x, slong prec)

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

void arb_sinc_pi(arb_t z, const arb_t x, slong prec)

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

Inverse trigonometric functions

void arb_atan_arf(arb_t z, const arf_t x, slong prec)
void arb_atan(arb_t z, const arb_t x, slong prec)

Sets \(z = \operatorname{atan}(x)\).

At low to medium precision (up to about 4096 bits), arb_atan_arf() uses table-based argument reduction and fast Taylor series evaluation via _arb_atan_taylor_rs(). At high precision, it falls back to MPFR. The function arb_atan() simply calls arb_atan_arf() with the midpoint as input, and separately adds the propagated error.

The function arb_atan_arf() uses lookup tables if possible, and otherwise falls back to arb_atan_arf_bb().

void arb_atan2(arb_t z, const arb_t b, const arb_t a, slong prec)

Sets r to an the argument (phase) of the complex number \(a + bi\), with the branch cut discontinuity on \((-\infty,0]\). We define \(\operatorname{atan2}(0,0) = 0\), and for \(a < 0\), \(\operatorname{atan2}(0,a) = \pi\).

void arb_asin(arb_t z, const arb_t x, slong prec)

Sets \(z = \operatorname{asin}(x) = \operatorname{atan}(x / \sqrt{1-x^2})\). If \(x\) is not contained in the domain \([-1,1]\), the result is an indeterminate interval.

void arb_acos(arb_t z, const arb_t x, slong prec)

Sets \(z = \operatorname{acos}(x) = \pi/2 - \operatorname{asin}(x)\). If \(x\) is not contained in the domain \([-1,1]\), the result is an indeterminate interval.

Hyperbolic functions

void arb_sinh(arb_t s, const arb_t x, slong prec)
void arb_cosh(arb_t c, const arb_t x, slong prec)
void arb_sinh_cosh(arb_t s, arb_t c, const arb_t x, slong prec)

Sets \(s = \sinh(x)\), \(c = \cosh(x)\). If the midpoint of \(x\) is close to zero and the hyperbolic sine is to be computed, evaluates \((e^{2x}\pm1) / (2e^x)\) via arb_expm1() to avoid loss of accuracy. Otherwise evaluates \((e^x \pm e^{-x}) / 2\).

void arb_tanh(arb_t y, const arb_t x, slong prec)

Sets \(y = \tanh(x) = \sinh(x) / \cosh(x)\), evaluated via arb_expm1() as \(\tanh(x) = (e^{2x} - 1) / (e^{2x} + 1)\) if \(|x|\) is small, and as \(\tanh(\pm x) = 1 - 2 e^{\mp 2x} / (1 + e^{\mp 2x})\) if \(|x|\) is large.

void arb_coth(arb_t y, const arb_t x, slong prec)

Sets \(y = \coth(x) = \cosh(x) / \sinh(x)\), evaluated using the same strategy as arb_tanh().

void arb_sech(arb_t res, const arb_t x, slong prec)

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

void arb_csch(arb_t res, const arb_t x, slong prec)

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

Inverse hyperbolic functions

void arb_atanh(arb_t z, const arb_t x, slong prec)

Sets \(z = \operatorname{atanh}(x)\).

void arb_asinh(arb_t z, const arb_t x, slong prec)

Sets \(z = \operatorname{asinh}(x)\).

void arb_acosh(arb_t z, const arb_t x, slong prec)

Sets \(z = \operatorname{acosh}(x)\). If \(x < 1\), the result is an indeterminate interval.

Constants

The following functions cache the computed values to speed up repeated calls at the same or lower precision. For further implementation details, see Algorithms for mathematical constants.

void arb_const_pi(arb_t z, slong prec)

Computes \(\pi\).

void arb_const_sqrt_pi(arb_t z, slong prec)

Computes \(\sqrt{\pi}\).

void arb_const_log_sqrt2pi(arb_t z, slong prec)

Computes \(\log \sqrt{2 \pi}\).

void arb_const_log2(arb_t z, slong prec)

Computes \(\log(2)\).

void arb_const_log10(arb_t z, slong prec)

Computes \(\log(10)\).

void arb_const_euler(arb_t z, slong prec)

Computes Euler’s constant \(\gamma = \lim_{k \rightarrow \infty} (H_k - \log k)\) where \(H_k = 1 + 1/2 + \ldots + 1/k\).

void arb_const_catalan(arb_t z, slong prec)

Computes Catalan’s constant \(C = \sum_{n=0}^{\infty} (-1)^n / (2n+1)^2\).

void arb_const_e(arb_t z, slong prec)

Computes \(e = \exp(1)\).

void arb_const_khinchin(arb_t z, slong prec)

Computes Khinchin’s constant \(K_0\).

void arb_const_glaisher(arb_t z, slong prec)

Computes the Glaisher-Kinkelin constant \(A = \exp(1/12 - \zeta'(-1))\).

void arb_const_apery(arb_t z, slong prec)

Computes Apery’s constant \(\zeta(3)\).

Lambert W function

void arb_lambertw(arb_t res, const arb_t x, int flags, slong prec)

Computes the Lambert W function, which solves the equation \(w e^w = x\).

The Lambert W function has infinitely many complex branches \(W_k(x)\), two of which are real on a part of the real line. The principal branch \(W_0(x)\) is selected by setting flags to 0, and the \(W_{-1}\) branch is selected by setting flags to 1. The principal branch is real-valued for \(x \ge -1/e\) (taking values in \([-1,+\infty)\)) and the \(W_{-1}\) branch is real-valued for \(-1/e \le x < 0\) and takes values in \((-\infty,-1]\). Elsewhere, the Lambert W function is complex and acb_lambertw() should be used.

The implementation first computes a floating-point approximation heuristically and then computes a rigorously certified enclosure around this approximation. Some asymptotic cases are handled specially. The algorithm used to compute the Lambert W function is described in [Joh2017b], which follows the main ideas in [CGHJK1996].

Gamma function and factorials

void arb_rising_ui_bs(arb_t z, const arb_t x, ulong n, slong prec)
void arb_rising_ui_rs(arb_t z, const arb_t x, ulong n, ulong step, slong prec)
void arb_rising_ui_rec(arb_t z, const arb_t x, ulong n, slong prec)
void arb_rising_ui(arb_t z, const arb_t x, ulong n, slong prec)
void arb_rising(arb_t z, const arb_t x, const arb_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 arb_rising_fmpq_ui(arb_t z, const fmpq_t x, ulong n, slong prec)

Computes the rising factorial \(z = x (x+1) (x+2) \cdots (x+n-1)\) using binary splitting. If the denominator or numerator of x is large compared to prec, it is more efficient to convert x to an approximation and use arb_rising_ui().

void arb_rising2_ui_bs(arb_t u, arb_t v, const arb_t x, ulong n, slong prec)
void arb_rising2_ui_rs(arb_t u, arb_t v, const arb_t x, ulong n, ulong step, slong prec)
void arb_rising2_ui(arb_t u, arb_t v, const arb_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 arb_fac_ui(arb_t z, ulong n, slong prec)

Computes the factorial \(z = n!\) via the gamma function.

void arb_doublefac_ui(arb_t z, ulong n, slong prec)

Computes the double factorial \(z = n!!\) via the gamma function.

void arb_bin_ui(arb_t z, const arb_t n, ulong k, slong prec)
void arb_bin_uiui(arb_t z, ulong n, ulong k, slong prec)

Computes the binomial coefficient \(z = {n \choose k}\), via the rising factorial as \({n \choose k} = (n-k+1)_k / k!\).

void arb_gamma(arb_t z, const arb_t x, slong prec)
void arb_gamma_fmpq(arb_t z, const fmpq_t x, slong prec)
void arb_gamma_fmpz(arb_t z, const fmpz_t x, slong prec)

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

void arb_lgamma(arb_t z, const arb_t x, slong prec)

Computes the logarithmic gamma function \(z = \log \Gamma(x)\). The complex branch structure is assumed, so if \(x \le 0\), the result is an indeterminate interval.

void arb_rgamma(arb_t z, const arb_t x, slong prec)

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

void arb_digamma(arb_t y, const arb_t x, slong prec)

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

Zeta function

void arb_zeta_ui_vec_borwein(arb_ptr z, ulong start, slong num, ulong step, slong prec)

Evaluates \(\zeta(s)\) at \(\mathrm{num}\) consecutive integers s beginning with start and proceeding in increments of step. Uses Borwein’s formula ([Bor2000], [GS2003]), implemented to support fast multi-evaluation (but also works well for a single s).

Requires \(\mathrm{start} \ge 2\). For efficiency, the largest s should be at most about as large as prec. Arguments approaching LONG_MAX will cause overflows. One should therefore only use this function for s up to about prec, and then switch to the Euler product.

The algorithm for single s is basically identical to the one used in MPFR (see [MPFR2012] for a detailed description). In particular, we evaluate the sum backwards to avoid storing more than one \(d_k\) coefficient, and use integer arithmetic throughout since it is convenient and the terms turn out to be slightly larger than \(2^\mathrm{prec}\). The only numerical error in the main loop comes from the division by \(k^s\), which adds less than 1 unit of error per term. For fast multi-evaluation, we repeatedly divide by \(k^{\mathrm{step}}\). Each division reduces the input error and adds at most 1 unit of additional rounding error, so by induction, the error per term is always smaller than 2 units.

void arb_zeta_ui_asymp(arb_t x, ulong s, slong prec)
void arb_zeta_ui_euler_product(arb_t z, ulong s, slong prec)

Computes \(\zeta(s)\) using the Euler product. This is fast only if s is large compared to the precision. Both methods are trivial wrappers for _acb_dirichlet_euler_product_real_ui().

void arb_zeta_ui_bernoulli(arb_t x, ulong s, slong prec)

Computes \(\zeta(s)\) for even s via the corresponding Bernoulli number.

void arb_zeta_ui_borwein_bsplit(arb_t x, ulong s, slong prec)

Computes \(\zeta(s)\) for arbitrary \(s \ge 2\) using a binary splitting implementation of Borwein’s algorithm. This has quasilinear complexity with respect to the precision (assuming that \(s\) is fixed).

void arb_zeta_ui_vec(arb_ptr x, ulong start, slong num, slong prec)
void arb_zeta_ui_vec_even(arb_ptr x, ulong start, slong num, slong prec)
void arb_zeta_ui_vec_odd(arb_ptr x, ulong start, slong num, slong prec)

Computes \(\zeta(s)\) at num consecutive integers (respectively num even or num odd integers) beginning with \(s = \mathrm{start} \ge 2\), automatically choosing an appropriate algorithm.

void arb_zeta_ui(arb_t x, ulong s, slong prec)

Computes \(\zeta(s)\) for nonnegative integer \(s \ne 1\), automatically choosing an appropriate algorithm. This function is intended for numerical evaluation of isolated zeta values; for multi-evaluation, the vector versions are more efficient.

void arb_zeta(arb_t z, const arb_t s, slong prec)

Sets z to the value of the Riemann zeta function \(\zeta(s)\).

For computing derivatives with respect to \(s\), use arb_poly_zeta_series().

void arb_hurwitz_zeta(arb_t z, const arb_t s, const arb_t a, slong prec)

Sets z to the value of the Hurwitz zeta function \(\zeta(s,a)\).

For computing derivatives with respect to \(s\), use arb_poly_zeta_series().

Bernoulli numbers and polynomials

void arb_bernoulli_ui(arb_t b, ulong n, slong prec)
void arb_bernoulli_fmpz(arb_t b, const fmpz_t n, slong prec)

Sets \(b\) to the numerical value of the Bernoulli number \(B_n\) approximated to prec bits.

The internal precision is increased automatically to give an accurate result. Note that, with huge fmpz input, the output will have a huge exponent and evaluation will accordingly be slower.

A single division from the exact fraction of \(B_n\) is used if this value is in the global cache or the exact numerator roughly is larger than prec bits. Otherwise, the Riemann zeta function is used (see arb_bernoulli_ui_zeta()).

This function reads \(B_n\) from the global cache if the number is already cached, but does not automatically extend the cache by itself.

void arb_bernoulli_ui_zeta(arb_t b, ulong n, slong prec)

Sets \(b\) to the numerical value of \(B_n\) accurate to prec bits, computed using the formula \(B_{2n} = (-1)^{n+1} 2 (2n)! \zeta(2n) / (2 \pi)^n\).

To avoid potential infinite recursion, we explicitly call the Euler product implementation of the zeta function. This method will only give high accuracy if the precision is small enough compared to \(n\) for the Euler product to converge rapidly.

void arb_bernoulli_poly_ui(arb_t res, ulong n, const arb_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.

void arb_power_sum_vec(arb_ptr res, const arb_t a, const arb_t b, slong len, slong prec)

For n from 0 to len - 1, sets entry n in the output vector res to

\[S_n(a,b) = \frac{1}{n+1}\left(B_{n+1}(b) - B_{n+1}(a)\right)\]

where \(B_n(x)\) is a Bernoulli polynomial. If a and b are integers and \(b \ge a\), this is equivalent to

\[S_n(a,b) = \sum_{k=a}^{b-1} k^n.\]

The computation uses the generating function for Bernoulli polynomials.

Polylogarithms

void arb_polylog(arb_t w, const arb_t s, const arb_t z, slong prec)
void arb_polylog_si(arb_t w, slong s, const arb_t z, slong prec)

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

Other special functions

void arb_fib_fmpz(arb_t z, const fmpz_t n, slong prec)
void arb_fib_ui(arb_t z, ulong n, slong prec)

Computes the Fibonacci number \(F_n\). Uses the binary squaring algorithm described in [Tak2000]. Provided that n is small enough, an exact Fibonacci number can be computed by setting the precision to ARF_PREC_EXACT.

void arb_agm(arb_t z, const arb_t x, const arb_t y, slong prec)

Sets z to the arithmetic-geometric mean of x and y.

void arb_chebyshev_t_ui(arb_t a, ulong n, const arb_t x, slong prec)
void arb_chebyshev_u_ui(arb_t a, ulong n, const arb_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 arb_chebyshev_t2_ui(arb_t a, arb_t b, ulong n, const arb_t x, slong prec)
void arb_chebyshev_u2_ui(arb_t a, arb_t b, ulong n, const arb_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.

void arb_bell_sum_bsplit(arb_t res, const fmpz_t n, const fmpz_t a, const fmpz_t b, const fmpz_t mmag, slong prec)
void arb_bell_sum_taylor(arb_t res, const fmpz_t n, const fmpz_t a, const fmpz_t b, const fmpz_t mmag, slong prec)

Helper functions for Bell numbers, evaluating the sum \(\sum_{k=a}^{b-1} k^n / k!\). If mmag is non-NULL, it may be used to indicate that the target error tolerance should be \(2^{mmag - prec}\).

void arb_bell_fmpz(arb_t res, const fmpz_t n, slong prec)
void arb_bell_ui(arb_t res, ulong n, slong prec)

Sets res to the Bell number \(B_n\). If the number is too large to fit exactly in prec bits, a numerical approximation is computed efficiently.

The algorithm to compute Bell numbers, including error analysis, is described in detail in [Joh2015].

void arb_euler_number_fmpz(arb_t res, const fmpz_t n, slong prec)
void arb_euler_number_ui(arb_t res, ulong n, slong prec)

Sets res to the Euler number \(E_n\), which is defined by having the exponential generating function \(1 / \cosh(x)\).

The Euler product for the Dirichlet beta function (_acb_dirichlet_euler_product_real_ui()) is used to compute a numerical approximation. If prec is more than enough to represent the result exactly, the exact value is automatically determined from a lower-precision approximation.

void arb_partitions_fmpz(arb_t res, const fmpz_t n, slong prec)
void arb_partitions_ui(arb_t res, ulong n, slong prec)

Sets res to the partition function \(p(n)\). When n is large and \(\log_2 p(n)\) is more than twice prec, the leading term in the Hardy-Ramanujan asymptotic series is used together with an error bound. Otherwise, the exact value is computed and rounded.

Internals for computing elementary functions

void _arb_atan_taylor_naive(mp_ptr y, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N, int alternating)
void _arb_atan_taylor_rs(mp_ptr y, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N, int alternating)

Computes an approximation of \(y = \sum_{k=0}^{N-1} x^{2k+1} / (2k+1)\) (if alternating is 0) or \(y = \sum_{k=0}^{N-1} (-1)^k x^{2k+1} / (2k+1)\) (if alternating is 1). Used internally for computing arctangents and logarithms. The naive version uses the forward recurrence, and the rs version uses a division-avoiding rectangular splitting scheme.

Requires \(N \le 255\), \(0 \le x \le 1/16\), and xn positive. The input x and output y are fixed-point numbers with xn fractional limbs. A bound for the ulp error is written to error.

void _arb_exp_taylor_naive(mp_ptr y, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N)
void _arb_exp_taylor_rs(mp_ptr y, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N)

Computes an approximation of \(y = \sum_{k=0}^{N-1} x^k / k!\). Used internally for computing exponentials. The naive version uses the forward recurrence, and the rs version uses a division-avoiding rectangular splitting scheme.

Requires \(N \le 287\), \(0 \le x \le 1/16\), and xn positive. The input x is a fixed-point number with xn fractional limbs, and the output y is a fixed-point number with xn fractional limbs plus one extra limb for the integer part of the result.

A bound for the ulp error is written to error.

void _arb_sin_cos_taylor_naive(mp_ptr ysin, mp_ptr ycos, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N)
void _arb_sin_cos_taylor_rs(mp_ptr ysin, mp_ptr ycos, mp_limb_t * error, mp_srcptr x, mp_size_t xn, ulong N, int sinonly, int alternating)

Computes approximations of \(y_s = \sum_{k=0}^{N-1} (-1)^k x^{2k+1} / (2k+1)!\) and \(y_c = \sum_{k=0}^{N-1} (-1)^k x^{2k} / (2k)!\). Used internally for computing sines and cosines. The naive version uses the forward recurrence, and the rs version uses a division-avoiding rectangular splitting scheme.

Requires \(N \le 143\), \(0 \le x \le 1/16\), and xn positive. The input x and outputs ysin, ycos are fixed-point numbers with xn fractional limbs. A bound for the ulp error is written to error.

If sinonly is 1, only the sine is computed; if sinonly is 0 both the sine and cosine are computed. To compute sin and cos, alternating should be 1. If alternating is 0, the hyperbolic sine is computed (this is currently only intended to be used together with sinonly).

int _arb_get_mpn_fixed_mod_log2(mp_ptr w, fmpz_t q, mp_limb_t * error, const arf_t x, mp_size_t wn)

Attempts to write \(w = x - q \log(2)\) with \(0 \le w < \log(2)\), where w is a fixed-point number with wn limbs and ulp error error. Returns success.

int _arb_get_mpn_fixed_mod_pi4(mp_ptr w, fmpz_t q, int * octant, mp_limb_t * error, const arf_t x, mp_size_t wn)

Attempts to write \(w = |x| - q \pi/4\) with \(0 \le w < \pi/4\), where w is a fixed-point number with wn limbs and ulp error error. Returns success.

The value of q mod 8 is written to octant. The output variable q can be NULL, in which case the full value of q is not stored.

slong _arb_exp_taylor_bound(slong mag, slong prec)

Returns n such that \(\left|\sum_{k=n}^{\infty} x^k / k!\right| \le 2^{-\mathrm{prec}}\), assuming \(|x| \le 2^{\mathrm{mag}} \le 1/4\).

void arb_exp_arf_bb(arb_t z, const arf_t x, slong prec, int m1)

Computes the exponential function using the bit-burst algorithm. If m1 is nonzero, the exponential function minus one is computed accurately.

Aborts if x is extremely small or large (where another algorithm should be used).

For large x, repeated halving is used. In fact, we always do argument reduction until \(|x|\) is smaller than about \(2^{-d}\) where \(d \approx 16\) to speed up convergence. If \(|x| \approx 2^m\), we thus need about \(m+d\) squarings.

Computing \(\log(2)\) costs roughly 100-200 multiplications, so is not usually worth the effort at very high precision. However, this function could be improved by using \(\log(2)\) based reduction at precision low enough that the value can be assumed to be cached.

void _arb_exp_sum_bs_simple(fmpz_t T, fmpz_t Q, mp_bitcnt_t * Qexp, const fmpz_t x, mp_bitcnt_t r, slong N)
void _arb_exp_sum_bs_powtab(fmpz_t T, fmpz_t Q, mp_bitcnt_t * Qexp, const fmpz_t x, mp_bitcnt_t r, slong N)

Computes T, Q and Qexp such that \(T / (Q 2^{\text{Qexp}}) = \sum_{k=1}^N (x/2^r)^k/k!\) using binary splitting. Note that the sum is taken to N inclusive and omits the constant term.

The powtab version precomputes a table of powers of x, resulting in slightly higher memory usage but better speed. For best efficiency, N should have many trailing zero bits.

void arb_exp_arf_rs_generic(arb_t res, const arf_t x, slong prec, int minus_one)

Computes the exponential function using a generic version of the rectangular splitting strategy, intended for intermediate precision.

void _arb_atan_sum_bs_simple(fmpz_t T, fmpz_t Q, mp_bitcnt_t * Qexp, const fmpz_t x, mp_bitcnt_t r, slong N)
void _arb_atan_sum_bs_powtab(fmpz_t T, fmpz_t Q, mp_bitcnt_t * Qexp, const fmpz_t x, mp_bitcnt_t r, slong N)

Computes T, Q and Qexp such that \(T / (Q 2^{\text{Qexp}}) = \sum_{k=1}^N (-1)^k (x/2^r)^{2k} / (2k+1)\) using binary splitting. Note that the sum is taken to N inclusive, omits the linear term, and requires a final multiplication by \((x/2^r)\) to give the true series for atan.

The powtab version precomputes a table of powers of x, resulting in slightly higher memory usage but better speed. For best efficiency, N should have many trailing zero bits.

void arb_atan_arf_bb(arb_t z, const arf_t x, slong prec)

Computes the arctangent of x. Initially, the argument-halving formula

\[\operatorname{atan}(x) = 2 \operatorname{atan}\left(\frac{x}{1+\sqrt{1+x^2}}\right)\]

is applied up to 8 times to get a small argument. Then a version of the bit-burst algorithm is used. The functional equation

\[\operatorname{atan}(x) = \operatorname{atan}(p/q) + \operatorname{atan}(w), \quad w = \frac{qx-p}{px+q}, \quad p = \lfloor qx \rfloor\]

is applied repeatedly instead of integrating a differential equation for the arctangent, as this appears to be more efficient.

void arb_sin_cos_arf_generic(arb_t s, arb_t c, const arf_t x, slong prec)

Computes the sine and cosine of x using a generic strategy. This function gets called internally by the main sin and cos functions when the precision for argument reduction or series evaluation based on lookup tables is exhausted.

This function first performs a cheap test to see if \(|x| < \pi / 2 - \varepsilon\). If the test fails, it uses \(\pi\) to reduce the argument to the first octant, and then evaluates the sin and cos functions recursively (this call cannot result in infinite recursion).

If no argument reduction is needed, this function uses a generic version of the rectangular splitting algorithm if the precision is not too high, and otherwise invokes the asymptotically fast bit-burst algorithm.

void arb_sin_cos_arf_bb(arb_t s, arb_t c, const arf_t x, slong prec)

Computes the sine and cosine of x using the bit-burst algorithm. It is required that \(|x| < \pi / 2\) (this is not checked).

Vector functions

void _arb_vec_zero(arb_ptr vec, slong n)

Sets all entries in vec to zero.

int _arb_vec_is_zero(arb_srcptr vec, slong len)

Returns nonzero iff all entries in x are zero.

int _arb_vec_is_finite(arb_srcptr x, slong len)

Returns nonzero iff all entries in x certainly are finite.

void _arb_vec_set(arb_ptr res, arb_srcptr vec, slong len)

Sets res to a copy of vec.

void _arb_vec_set_round(arb_ptr res, arb_srcptr vec, slong len, slong prec)

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

void _arb_vec_swap(arb_ptr vec1, arb_ptr vec2, slong len)

Swaps the entries of vec1 and vec2.

void _arb_vec_neg(arb_ptr B, arb_srcptr A, slong n)
void _arb_vec_sub(arb_ptr C, arb_srcptr A, arb_srcptr B, slong n, slong prec)
void _arb_vec_add(arb_ptr C, arb_srcptr A, arb_srcptr B, slong n, slong prec)
void _arb_vec_scalar_mul(arb_ptr res, arb_srcptr vec, slong len, const arb_t c, slong prec)
void _arb_vec_scalar_div(arb_ptr res, arb_srcptr vec, slong len, const arb_t c, slong prec)
void _arb_vec_scalar_mul_fmpz(arb_ptr res, arb_srcptr vec, slong len, const fmpz_t c, slong prec)
void _arb_vec_scalar_mul_2exp_si(arb_ptr res, arb_srcptr src, slong len, slong c)
void _arb_vec_scalar_addmul(arb_ptr res, arb_srcptr vec, slong len, const arb_t c, slong prec)

Performs the respective scalar operation elementwise.

void _arb_vec_dot(arb_t res, arb_srcptr vec1, arb_srcptr vec2, slong len2, slong prec)

Sets res to the dot product of vec1 and vec2.

void _arb_vec_get_mag(mag_t bound, arb_srcptr vec, slong len, slong prec)

Sets bound to an upper bound for the entries in vec.

slong _arb_vec_bits(arb_srcptr x, slong len)

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

void _arb_vec_set_powers(arb_ptr xs, const arb_t x, slong len, slong prec)

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

void _arb_vec_add_error_arf_vec(arb_ptr res, arf_srcptr err, slong len)
void _arb_vec_add_error_mag_vec(arb_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 _arb_vec_indeterminate(arb_ptr vec, slong len)

Applies arb_indeterminate() elementwise.

void _arb_vec_trim(arb_ptr res, arb_srcptr vec, slong len)

Applies arb_trim() elementwise.

int _arb_vec_get_unique_fmpz_vec(fmpz * res, arb_srcptr vec, slong len)

Calls arb_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.