ca.h – exact real and complex numbers¶

A ca_t represents a real or complex number in a form suitable for exact field arithmetic or comparison. Exceptionally, a ca_t may represent a special nonnumerical value, such as an infinity.

Introduction: numbers¶

A Calcium number is a real or complex number represented as an element of a formal field \(K = \mathbb{Q}(a_1, \ldots, a_n)\) where the symbols \(a_k\) denote fixed algebraic or transcendental numbers called extension numbers. For example, \(e^{-2 \pi} - 3 i\) may be represented as \((1 - 3 a_2^2 a_1) / a_2^2\) in the field \(\mathbb{Q}(a_1,a_2)\) with \(a_1 = i, a_2 = e^{\pi}\). Extension numbers and fields are documented in the following separate modules:

The user does not need to construct extension numbers or formal extension fields explicitly: each ca_t contains an internal pointer to its formal field, and operations on Calcium numbers generate and cache fields automatically as needed to express the results.

This representation is not canonical (in general). A given complex number can be represented in different ways depending on the choice of formal field K. Even within a fixed field K, a number can have different representations if there are algebraic relations between the extension numbers. Two numbers x and y can be tested for inequality using numerical evaluation; to test for equality, it may be necessary to eliminate dependencies between extension numbers. One of the central goals of Calcium will be to implement heuristics for such elimination.

Together with each formal field K, Calcium stores a reduction ideal \(I = \{g_1,\ldots,g_m\}\) with \(g_i \in \mathbb{Z}[a_1,\ldots,a_n]\), defining a set of algebraic relations \(g_i(a_1,\ldots,a_n) = 0\). Relations can be absolute, say \(g_i = a_j^2 + 1\), or relative, say \(g_i = 2 a_j - 4 a_k - a_l a_m\). The reduction ideal effectively partitions \(K\) into equivalence classes of complex numbers (e.g. \(i^2 = -1\) or \(2 \log(\pi i) = 4 \log(\sqrt{\pi}) + \pi i\)), enabling simplifications and equality proving.

Extension numbers are always sorted \(a_1 \succ a_2 \succ \ldots \succ a_n\) where \(\succ\) denotes a structural ordering (see ca_cmp_repr()). If the reduction ideal is triangular and the multivariate polynomial arithmetic uses lexicographic ordering, reduction by I eliminates numbers \(a_i\) with higher complexity in the sense of \(\succ\).

The reduction ideal is an imperfect computational crutch: it is not guaranteed to capture all algebraic relations, and reduction is not guaranteed to produce uniquely defined representatives. However, in the specific case of an absolute number field \(K = \mathbb{Q}(a)\) where a is a qqbar_t extension, the reduction ideal (consisting of a single minimal polynomial) is canonical and field elements of K can be chosen canonically.

Introduction: special values¶

In order to provide a closed arithmetic system and express limiting cases of operators and special functions, a ca_t can hold any of the following special values besides ordinary numbers:

Unsigned infinity, a formal object \({\tilde \infty}\) representing the value of \(1 / 0\). More generally, this is the value of meromorphic functions at poles.
Signed infinity, a formal object \(a \cdot \infty\) where the sign \(a\) is a Calcium number with \(|a| = 1\). The most common values are \(+\infty, -\infty, +i \infty, -i \infty\). Signed infinities are used to denote directional limits and logarithmic singularities (for example, \(\log(0) = -\infty\)).
Undefined, a formal object representing the value of indeterminate forms such as \(0 / 0\) and essential singularities such as \(\exp(\tilde \infty)\), where a number or infinity would not make sense as an answer.
Unknown, a meta-value used to signal that the actual desired value could not be computed, either because Calcium does not (yet) have a data structure or algorithm for that case, or because doing so would be unreasonably expensive. This occurs, for example, if Calcium performs a division and is unable to decide whether the result is a number, unsigned infinity or undefined (because testing for zero fails). Wrappers may want to check output variables for Unknown and throw an exception (e.g. NotImplementedError in Python).

The distinction between Calcium numbers (which must represent elements of \(\mathbb{C}\)) and the different kinds of nonnumerical values (infinities, Undefined or Unknown) is essential. Nonnumerical values may not be used as field extension numbers \(a_k\), and the denominator of a formal field element must always represent a nonzero complex number. Accordingly, for any given Calcium value x that is not Unknown, it is exactly known whether x represents A) a number, B) unsigned infinity, C) a signed infinity, or D) Undefined.

Number objects¶

For all types, a type_t is defined as an array of length one of type type_struct, permitting a type_t to be passed by reference.

type ca_struct¶

type ca_t¶: A ca_t contains an index to a field K, and data representing an element x of K. The data is either an inline rational number (fmpq_t), an inline Antic number field element (nf_elem_t) when K is an absolute algebraic number field \(\mathbb{Q}(a)\), or a pointer to a heap-allocated fmpz_mpoly_q_t representing an element of a generic field \(\mathbb{Q}(a_1,\ldots,a_n)\). Special values are encoded using magic bits in the field index.

type ca_ptr¶: Alias for ca_struct *, used for vectors of numbers.

type ca_srcptr¶: Alias for const ca_struct *, used for vectors of numbers when passed as constant input to functions.

Context objects¶

type ca_ctx_struct¶

type ca_ctx_t¶

A ca_ctx_t context object holds a cache of fields K and constituent extension numbers \(a_k\). The field index in an individual ca_t instance represents a shallow reference to the object defining the field K within the context object, so creating many elements of the same field is cheap.

Since context objects are mutable (and may be mutated even when performing read-only operations on ca_t instances), they must not be accessed simultaneously by different threads: in multithreaded environments, the user must use a separate context object for each thread.

void ca_ctx_init(ca_ctx_t ctx)¶: Initializes the context object ctx for use. Any evaluation options stored in the context object are set to default values.

void ca_ctx_clear(ca_ctx_t ctx)¶: Clears the context object ctx, freeing any memory allocated internally. This function should only be called after all ca_t instances referring to this context have been cleared.

void ca_ctx_print(ca_ctx_t ctx)¶: Prints a description of the context ctx to standard output. This will give a complete listing of the cached fields in ctx.

Memory management for numbers¶

void ca_init(ca_t x, ca_ctx_t ctx)¶: Initializes the variable x for use, associating it with the context object ctx. The value of x is set to the rational number 0.

void ca_clear(ca_t x, ca_ctx_t ctx)¶: Clears the variable x.

void ca_swap(ca_t x, ca_t y, ca_ctx_t ctx)¶: Efficiently swaps the variables x and y.

Symbolic expressions¶

void ca_get_fexpr(fexpr_t res, const ca_t x, ulong flags, ca_ctx_t ctx)¶: Sets res to a symbolic expression representing x.

int ca_set_fexpr(ca_t res, const fexpr_t expr, ca_ctx_t ctx)¶: Sets res to the value represented by the symbolic expression expr. Returns 1 on success and 0 on failure. This function essentially just traverses the expression tree using ca arithmetic; it does not provide advanced symbolic evaluation. It is guaranteed to at least be able to parse the output of ca_get_fexpr().

Printing¶

The style of printed output is controlled by ctx->options[CA_OPT_PRINT_FLAGS] (see Context options) which can be set to any combination of the following flags:

CA_PRINT_N¶

Print a decimal approximation of the number. The approximation is guaranteed to be correctly rounded to within one unit in the last place.

If combined with CA_PRINT_REPR, numbers appearing within the symbolic representation will also be printed with decimal approximations.

Warning: printing a decimal approximation requires a computation, which can be expensive. It can also mutate cached data (numerical enclosures of extension numbers), affecting subsequent computations.

CA_PRINT_DIGITS¶: Multiplied by a positive integer, specifies the number of decimal digits to show with CA_PRINT_N. If not given, the default precision is six digits.

CA_PRINT_REPR¶

Print the symbolic representation of the number (including its recursive elements). If used together with CA_PRINT_N, field elements will print as decimal {symbolic} while extension numbers will print as decimal [symbolic].

All extension numbers appearing in the field defining x and in the inner constructions of those extension numbers will be given local labels a, b, etc. for this printing.

CA_PRINT_FIELD¶: For each field element, explicitly print its formal field along with its reduction ideal if present, e.g. QQ or QQ(a,b,c) / <a-b, c^2+1>.

CA_PRINT_DEFAULT¶: The default print style. Equivalent to CA_PRINT_N | CA_PRINT_REPR.

CA_PRINT_DEBUG¶: Verbose print style for debugging. Equivalent to CA_PRINT_N | CA_PRINT_REPR | CA_PRINT_FIELD.

As a special case, small integers are always printed as simple literals.

As illustration, here are the numbers \(-7\), \(2/3\), \((\sqrt{3}+5)/2\) and \(\sqrt{2} (\log(\pi) + \pi i)\) printed in various styles:

# CA_PRINT_DEFAULT
-7
0.666667 {2/3}
3.36603 {(a+5)/2 where a = 1.73205 [a^2-3=0]}
1.61889 + 4.44288*I {a*c+b*c*d where a = 1.14473 [Log(3.14159 {b})], b = 3.14159 [Pi], c = 1.41421 [c^2-2=0], d = I [d^2+1=0]}

# CA_PRINT_N
-7
0.666667
3.36603
1.61889 + 4.44288*I

# CA_PRINT_N | (CA_PRINT_DIGITS * 20)
-7
0.66666666666666666667
3.3660254037844386468
1.6188925298220266685 + 4.4428829381583662470*I

# CA_PRINT_REPR
-7
2/3
(a+5)/2 where a = [a^2-3=0]
a*c+b*c*d where a = Log(b), b = Pi, c = [c^2-2=0], d = [d^2+1=0]

# CA_PRINT_DEBUG
-7
0.666667 {2/3  in  QQ}
3.36603 {(a+5)/2  in  QQ(a)/<a^2-3> where a = 1.73205 [a^2-3=0]}
1.61889 + 4.44288*I {a*c+b*c*d  in  QQ(a,b,c,d)/<c^2-2, d^2+1> where a = 1.14473 [Log(3.14159 {b  in  QQ(b)})], b = 3.14159 [Pi], c = 1.41421 [c^2-2=0], d = I [d^2+1=0]}

void ca_print(const ca_t x, ca_ctx_t ctx)¶: Prints x to standard output.

void ca_fprint(FILE *fp, const ca_t x, ca_ctx_t ctx)¶: Prints x to the file fp.

char *ca_get_str(const ca_t x, ca_ctx_t ctx)¶: Prints x to a string which is returned. The user should free this string by calling flint_free.

void ca_printn(const ca_t x, slong n, ca_ctx_t ctx)¶: Prints an n-digit numerical representation of x to standard output.

Special values¶

void ca_zero(ca_t res, ca_ctx_t ctx)¶
void ca_one(ca_t res, ca_ctx_t ctx)¶
void ca_neg_one(ca_t res, ca_ctx_t ctx)¶: Sets res to the integer 0, 1 or -1. This creates a canonical representation of this number as an element of the trivial field \(\mathbb{Q}\).

void ca_i(ca_t res, ca_ctx_t ctx)¶
void ca_neg_i(ca_t res, ca_ctx_t ctx)¶: Sets res to the imaginary unit \(i = \sqrt{-1}\), or its negation \(-i\). This creates a canonical representation of \(i\) as the generator of the algebraic number field \(\mathbb{Q}(i)\).

void ca_pi(ca_t res, ca_ctx_t ctx)¶: Sets res to the constant \(\pi\). This creates an element of the transcendental number field \(\mathbb{Q}(\pi)\).

void ca_pi_i(ca_t res, ca_ctx_t ctx)¶: Sets res to the constant \(\pi i\). This creates an element of the composite field \(\mathbb{Q}(i,\pi)\) rather than representing \(\pi i\) (or even \(2 \pi i\), which for some purposes would be more elegant) as an atomic quantity.

void ca_euler(ca_t res, ca_ctx_t ctx)¶: Sets res to Euler’s constant \(\gamma\). This creates an element of the (transcendental?) number field \(\mathbb{Q}(\gamma)\).

void ca_unknown(ca_t res, ca_ctx_t ctx)¶: Sets res to the meta-value Unknown.

void ca_undefined(ca_t res, ca_ctx_t ctx)¶: Sets res to Undefined.

void ca_uinf(ca_t res, ca_ctx_t ctx)¶: Sets res to unsigned infinity \({\tilde \infty}\).

void ca_pos_inf(ca_t res, ca_ctx_t ctx)¶
void ca_neg_inf(ca_t res, ca_ctx_t ctx)¶
void ca_pos_i_inf(ca_t res, ca_ctx_t ctx)¶
void ca_neg_i_inf(ca_t res, ca_ctx_t ctx)¶: Sets res to the signed infinity \(+\infty\), \(-\infty\), \(+i \infty\) or \(-i \infty\).

Assignment and conversion¶

void ca_set(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to a copy of x.

void ca_set_si(ca_t res, slong v, ca_ctx_t ctx)¶
void ca_set_ui(ca_t res, ulong v, ca_ctx_t ctx)¶
void ca_set_fmpz(ca_t res, const fmpz_t v, ca_ctx_t ctx)¶
void ca_set_fmpq(ca_t res, const fmpq_t v, ca_ctx_t ctx)¶: Sets res to the integer or rational number v. This creates a canonical representation of this number as an element of the trivial field \(\mathbb{Q}\).

void ca_set_d(ca_t res, double x, ca_ctx_t ctx)¶
void ca_set_d_d(ca_t res, double x, double y, ca_ctx_t ctx)¶: Sets res to the value of x, or the complex value \(x + yi\). NaN is interpreted as Unknown (not Undefined).

void ca_transfer(ca_t res, ca_ctx_t res_ctx, const ca_t src, ca_ctx_t src_ctx)¶

Sets res to src where the corresponding context objects res_ctx and src_ctx may be different.

This operation preserves the mathematical value represented by src, but may result in a different internal representation depending on the settings of the context objects.

Conversion of algebraic numbers¶

void ca_set_qqbar(ca_t res, const qqbar_t x, ca_ctx_t ctx)¶

Sets res to the algebraic number x.

If x is rational, res is set to the canonical representation as an element in the trivial field \(\mathbb{Q}\).

If x is irrational, this function always sets res to an element of a univariate number field \(\mathbb{Q}(a)\). It will not, for example, identify \(\sqrt{2} + \sqrt{3}\) as an element of \(\mathbb{Q}(\sqrt{2}, \sqrt{3})\). However, it may attempt to find a simpler number field than that generated by x itself. For example:

If x is quadratic, it will be expressed as an element of \(\mathbb{Q}(\sqrt{N})\) where N has no small repeated factors (obtained by performing a smooth factorization of the discriminant).
TODO: if possible, coerce x to a low-degree cyclotomic field.

int ca_get_fmpz(fmpz_t res, const ca_t x, ca_ctx_t ctx)¶

int ca_get_fmpq(fmpq_t res, const ca_t x, ca_ctx_t ctx)¶

int ca_get_qqbar(qqbar_t res, const ca_t x, ca_ctx_t ctx)¶

Attempts to evaluate x to an explicit integer, rational or algebraic number. If successful, sets res to this number and returns 1. If unsuccessful, returns 0.

The conversion certainly fails if x does not represent an integer, rational or algebraic number (respectively), but can also fail if x is too expensive to compute under the current evaluation limits. In particular, the evaluation will be aborted if an intermediate algebraic number (or more precisely, the resultant polynomial prior to factorization) exceeds CA_OPT_QQBAR_DEG_LIMIT or the coefficients exceed some multiple of CA_OPT_PREC_LIMIT. Note that evaluation may hit those limits even if the minimal polynomial for x itself is small. The conversion can also fail if no algorithm has been implemented for the functions appearing in the construction of x.

int ca_can_evaluate_qqbar(const ca_t x, ca_ctx_t ctx)¶: Checks if ca_get_qqbar() has a chance to succeed. In effect, this checks if all extension numbers are manifestly algebraic numbers (without doing any evaluation).

Random generation¶

void ca_randtest_rational(ca_t res, flint_rand_t state, slong bits, ca_ctx_t ctx)¶: Sets res to a random rational number with numerator and denominator up to bits bits in size.

void ca_randtest(ca_t res, flint_rand_t state, slong depth, slong bits, ca_ctx_t ctx)¶: Sets res to a random number generated by evaluating a random expression. The algorithm randomly selects between generating a “simple” number (a random rational number or quadratic field element with coefficients up to bits in size, or a random builtin constant), or if depth is nonzero, applying a random arithmetic operation or function to operands produced through recursive calls with depth - 1. The output is guaranteed to be a number, not a special value.

void ca_randtest_special(ca_t res, flint_rand_t state, slong depth, slong bits, ca_ctx_t ctx)¶: Randomly generates either a special value or a number.

void ca_randtest_same_nf(ca_t res, flint_rand_t state, const ca_t x, slong bits, slong den_bits, ca_ctx_t ctx)¶: Sets res to a random element in the same number field as x, with numerator coefficients up to bits in size and denominator up to den_bits in size. This function requires that x is an element of an absolute number field.

Representation properties¶

The following functions deal with the representation of a ca_t and hence can always be decided quickly and unambiguously. The return value for predicates is 0 for false and 1 for true.

int ca_equal_repr(const ca_t x, const ca_t y, ca_ctx_t ctx)¶

Returns whether x and y have identical representation. For field elements, this checks if x and y belong to the same formal field (with generators having identical representation) and are represented by the same rational function within that field.

For special values, this tests equality of the special values, with Unknown handled as if it were a value rather than a meta-value: that is, Unknown = Unknown gives 1, and Unknown = y gives 0 for any other kind of value y. If neither x nor y is Unknown, then representation equality implies that x and y describe to the same mathematical value, but if either operand is Unknown, the result is meaningless for mathematical comparison.

int ca_cmp_repr(const ca_t x, const ca_t y, ca_ctx_t ctx)¶: Compares the representations of x and y in a canonical sort order, returning -1, 0 or 1. This only performs a lexicographic comparison of the representations of x and y; the return value does not say anything meaningful about the numbers represented by x and y.

ulong ca_hash_repr(const ca_t x, ca_ctx_t ctx)¶: Hashes the representation of x.

int ca_is_unknown(const ca_t x, ca_ctx_t ctx)¶: Returns whether x is Unknown.

int ca_is_special(const ca_t x, ca_ctx_t ctx)¶: Returns whether x is a special value or metavalue (not a field element).

int ca_is_qq_elem(const ca_t x, ca_ctx_t ctx)¶: Returns whether x is represented as an element of the rational field \(\mathbb{Q}\).

int ca_is_qq_elem_zero(const ca_t x, ca_ctx_t ctx)¶
int ca_is_qq_elem_one(const ca_t x, ca_ctx_t ctx)¶
int ca_is_qq_elem_integer(const ca_t x, ca_ctx_t ctx)¶: Returns whether x is represented as the element 0, 1 or any integer in the rational field \(\mathbb{Q}\).

int ca_is_nf_elem(const ca_t x, ca_ctx_t ctx)¶: Returns whether x is represented as an element of a univariate algebraic number field \(\mathbb{Q}(a)\).

int ca_is_cyclotomic_nf_elem(slong *p, ulong *q, const ca_t x, ca_ctx_t ctx)¶: Returns whether x is represented as an element of a univariate cyclotomic field, i.e. \(\mathbb{Q}(a)\) where a is a root of unity. If p and q are not NULL and x is represented as an element of a cyclotomic field, this also sets p and q to the minimal integers with \(0 \le p < q\) such that the generating root of unity is \(a = e^{2 \pi i p / q}\). Note that the answer 0 does not prove that x is not a cyclotomic number, and the order q is also not necessarily the generator of the smallest cyclotomic field containing x. For the purposes of this function, only nontrivial cyclotomic fields count; the return value is 0 if x is represented as a rational number.

int ca_is_generic_elem(const ca_t x, ca_ctx_t ctx)¶: Returns whether x is represented as a generic field element; i.e. it is not a special value, not represented as an element of the rational field, and not represented as an element of a univariate algebraic number field.

Value predicates¶

The following predicates check a mathematical property which might not be effectively decidable. The result is a truth_t to allow representing an unknown outcome.

truth_t ca_check_is_number(const ca_t x, ca_ctx_t ctx)¶: Tests if x is a number. The result is T_TRUE is x is a field element (and hence a complex number), T_FALSE if x is an infinity or Undefined, and T_UNKNOWN if x is Unknown.

truth_t ca_check_is_zero(const ca_t x, ca_ctx_t ctx)¶
truth_t ca_check_is_one(const ca_t x, ca_ctx_t ctx)¶
truth_t ca_check_is_neg_one(const ca_t x, ca_ctx_t ctx)¶
truth_t ca_check_is_i(const ca_t x, ca_ctx_t ctx)¶
truth_t ca_check_is_neg_i(const ca_t x, ca_ctx_t ctx)¶: Tests if x is equal to the number \(0\), \(1\), \(-1\), \(i\), or \(-i\).

truth_t ca_check_is_algebraic(const ca_t x, ca_ctx_t ctx)¶
truth_t ca_check_is_rational(const ca_t x, ca_ctx_t ctx)¶
truth_t ca_check_is_integer(const ca_t x, ca_ctx_t ctx)¶: Tests if x is respectively an algebraic number, a rational number, or an integer.

truth_t ca_check_is_real(const ca_t x, ca_ctx_t ctx)¶: Tests if x is a real number. Warning: this returns T_FALSE if x is an infinity with real sign.

truth_t ca_check_is_negative_real(const ca_t x, ca_ctx_t ctx)¶: Tests if x is a negative real number. Warning: this returns T_FALSE if x is negative infinity.

truth_t ca_check_is_imaginary(const ca_t x, ca_ctx_t ctx)¶: Tests if x is an imaginary number. Warning: this returns T_FALSE if x is an infinity with imaginary sign.

truth_t ca_check_is_undefined(const ca_t x, ca_ctx_t ctx)¶: Tests if x is the special value Undefined.

truth_t ca_check_is_infinity(const ca_t x, ca_ctx_t ctx)¶: Tests if x is any infinity (unsigned or signed).

truth_t ca_check_is_uinf(const ca_t x, ca_ctx_t ctx)¶: Tests if x is unsigned infinity \({\tilde \infty}\).

truth_t ca_check_is_signed_inf(const ca_t x, ca_ctx_t ctx)¶: Tests if x is any signed infinity.

truth_t ca_check_is_pos_inf(const ca_t x, ca_ctx_t ctx)¶
truth_t ca_check_is_neg_inf(const ca_t x, ca_ctx_t ctx)¶
truth_t ca_check_is_pos_i_inf(const ca_t x, ca_ctx_t ctx)¶
truth_t ca_check_is_neg_i_inf(const ca_t x, ca_ctx_t ctx)¶: Tests if x is equal to the signed infinity \(+\infty\), \(-\infty\), \(+i \infty\), \(-i \infty\), respectively.

Comparisons¶

truth_t ca_check_equal(const ca_t x, const ca_t y, ca_ctx_t ctx)¶: Tests \(x = y\) as a mathematical equality. The result is T_UNKNOWN if either operand is Unknown. The result may also be T_UNKNOWN if x and y are numerically indistinguishable and cannot be proved equal or unequal by an exact computation.

truth_t ca_check_lt(const ca_t x, const ca_t y, ca_ctx_t ctx)¶
truth_t ca_check_le(const ca_t x, const ca_t y, ca_ctx_t ctx)¶
truth_t ca_check_gt(const ca_t x, const ca_t y, ca_ctx_t ctx)¶
truth_t ca_check_ge(const ca_t x, const ca_t y, ca_ctx_t ctx)¶: Compares x and y, implementing the respective operations \(x < y\), \(x \le y\), \(x > y\), \(x \ge y\). Only real numbers and \(-\infty\) and \(+\infty\) are considered comparable. The result is T_FALSE (not T_UNKNOWN) if either operand is not comparable (being a nonreal complex number, unsigned infinity, or undefined).

Field structure operations¶

void ca_merge_fields(ca_t resx, ca_t resy, const ca_t x, const ca_t y, ca_ctx_t ctx)¶

Sets resx and resy to copies of x and y coerced to a common field. Both x and y must be field elements (not special values).

In the present implementation, this simply merges the lists of generators, avoiding duplication. In the future, it will be able to eliminate generators satisfying algebraic relations.

void ca_condense_field(ca_t res, ca_ctx_t ctx)¶

Attempts to demote the value of res to a trivial subfield of its current field by removing unused generators. In particular, this demotes any obviously rational value to the trivial field \(\mathbb{Q}\).

This function is applied automatically in most operations (arithmetic operations, etc.).

ca_ext_ptr ca_is_gen_as_ext(const ca_t x, ca_ctx_t ctx)¶: If x is a generator of its formal field, \(x = a_k \in \mathbb{Q}(a_1,\ldots,a_n)\), returns a pointer to the extension number defining \(a_k\). If x is not a generator, returns NULL.

Arithmetic¶

void ca_neg(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the negation of x. For numbers, this operation amounts to a direct negation within the formal field. For a signed infinity \(c \infty\), negation gives \((-c) \infty\); all other special values are unchanged.

void ca_add_fmpq(ca_t res, const ca_t x, const fmpq_t y, ca_ctx_t ctx)¶

void ca_add_fmpz(ca_t res, const ca_t x, const fmpz_t y, ca_ctx_t ctx)¶

void ca_add_ui(ca_t res, const ca_t x, ulong y, ca_ctx_t ctx)¶

void ca_add_si(ca_t res, const ca_t x, slong y, ca_ctx_t ctx)¶

void ca_add(ca_t res, const ca_t x, const ca_t y, ca_ctx_t ctx)¶

Sets res to the sum of x and y. For special values, the following rules apply (\(c \infty\) denotes a signed infinity, \(|c| = 1\)):

\(c \infty + d \infty = c \infty\) if \(c = d\)
\(c \infty + d \infty = \text{Undefined}\) if \(c \ne d\)
\(\tilde \infty + c \infty = \tilde \infty + \tilde \infty = \text{Undefined}\)
\(c \infty + z = c \infty\) if \(z \in \mathbb{C}\)
\(\tilde \infty + z = \tilde \infty\) if \(z \in \mathbb{C}\)
\(z + \text{Undefined} = \text{Undefined}\) for any value z (including Unknown)

In any other case involving special values, or if the specific case cannot be distinguished, the result is Unknown.

void ca_sub_fmpq(ca_t res, const ca_t x, const fmpq_t y, ca_ctx_t ctx)¶
void ca_sub_fmpz(ca_t res, const ca_t x, const fmpz_t y, ca_ctx_t ctx)¶
void ca_sub_ui(ca_t res, const ca_t x, ulong y, ca_ctx_t ctx)¶
void ca_sub_si(ca_t res, const ca_t x, slong y, ca_ctx_t ctx)¶
void ca_fmpq_sub(ca_t res, const fmpq_t x, const ca_t y, ca_ctx_t ctx)¶
void ca_fmpz_sub(ca_t res, const fmpz_t x, const ca_t y, ca_ctx_t ctx)¶
void ca_ui_sub(ca_t res, ulong x, const ca_t y, ca_ctx_t ctx)¶
void ca_si_sub(ca_t res, slong x, const ca_t y, ca_ctx_t ctx)¶
void ca_sub(ca_t res, const ca_t x, const ca_t y, ca_ctx_t ctx)¶: Sets res to the difference of x and y. This is equivalent to computing \(x + (-y)\).

void ca_mul_fmpq(ca_t res, const ca_t x, const fmpq_t y, ca_ctx_t ctx)¶

void ca_mul_fmpz(ca_t res, const ca_t x, const fmpz_t y, ca_ctx_t ctx)¶

void ca_mul_ui(ca_t res, const ca_t x, ulong y, ca_ctx_t ctx)¶

void ca_mul_si(ca_t res, const ca_t x, slong y, ca_ctx_t ctx)¶

void ca_mul(ca_t res, const ca_t x, const ca_t y, ca_ctx_t ctx)¶

Sets res to the product of x and y. For special values, the following rules apply (\(c \infty\) denotes a signed infinity, \(|c| = 1\)):

\(c \infty \cdot d \infty = c d \infty\)
\(c \infty \cdot \tilde \infty = \tilde \infty\)
\(\tilde \infty \cdot \tilde \infty = \tilde \infty\)
\(c \infty \cdot z = \operatorname{sgn}(z) c \infty\) if \(z \in \mathbb{C} \setminus \{0\}\)
\(c \infty \cdot 0 = \text{Undefined}\)
\(\tilde \infty \cdot 0 = \text{Undefined}\)
\(z \cdot \text{Undefined} = \text{Undefined}\) for any value z (including Unknown)

In any other case involving special values, or if the specific case cannot be distinguished, the result is Unknown.

void ca_inv(ca_t res, const ca_t x, ca_ctx_t ctx)¶

Sets res to the multiplicative inverse of x. In a univariate algebraic number field, this always produces a rational denominator, but the denominator might not be rationalized in a multivariate field. For special values and zero, the following rules apply:

\(1 / (c \infty) = 1 / \tilde \infty = 0\)
\(1 / 0 = \tilde \infty\)
\(1 / \text{Undefined} = \text{Undefined}\)
\(1 / \text{Unknown} = \text{Unknown}\)

If it cannot be determined whether x is zero or nonzero, the result is Unknown.

void ca_fmpq_div(ca_t res, const fmpq_t x, const ca_t y, ca_ctx_t ctx)¶

void ca_fmpz_div(ca_t res, const fmpz_t x, const ca_t y, ca_ctx_t ctx)¶

void ca_ui_div(ca_t res, ulong x, const ca_t y, ca_ctx_t ctx)¶

void ca_si_div(ca_t res, slong x, const ca_t y, ca_ctx_t ctx)¶

void ca_div_fmpq(ca_t res, const ca_t x, const fmpq_t y, ca_ctx_t ctx)¶

void ca_div_fmpz(ca_t res, const ca_t x, const fmpz_t y, ca_ctx_t ctx)¶

void ca_div_ui(ca_t res, const ca_t x, ulong y, ca_ctx_t ctx)¶

void ca_div_si(ca_t res, const ca_t x, slong y, ca_ctx_t ctx)¶

void ca_div(ca_t res, const ca_t x, const ca_t y, ca_ctx_t ctx)¶

Sets res to the quotient of x and y. This is equivalent to computing \(x \cdot (1 / y)\). For special values and division by zero, this implies the following rules (\(c \infty\) denotes a signed infinity, \(|c| = 1\)):

\((c \infty) / (d \infty) = (c \infty) / \tilde \infty = \tilde \infty / (c \infty) = \tilde \infty / \tilde \infty = \text{Undefined}\)
\(c \infty / z = (c / \operatorname{sgn}(z)) \infty\) if \(z \in \mathbb{C} \setminus \{0\}\)
\(c \infty / 0 = \tilde \infty / 0 = \tilde \infty\)
\(z / (c \infty) = z / \tilde \infty = 0\) if \(z \in \mathbb{C}\)
\(z / 0 = \tilde \infty\) if \(z \in \mathbb{C} \setminus \{0\}\)
\(0 / 0 = \text{Undefined}\)
\(z / \text{Undefined} = \text{Undefined}\) for any value z (including Unknown)
\(\text{Undefined} / z = \text{Undefined}\) for any value z (including Unknown)

In any other case involving special values, or if the specific case cannot be distinguished, the result is Unknown.

void ca_dot(ca_t res, const ca_t initial, int subtract, ca_srcptr x, slong xstep, ca_srcptr y, slong ystep, slong len, ca_ctx_t ctx)¶

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.

void ca_fmpz_poly_evaluate(ca_t res, const fmpz_poly_t poly, const ca_t x, ca_ctx_t ctx)¶
void ca_fmpq_poly_evaluate(ca_t res, const fmpq_poly_t poly, const ca_t x, ca_ctx_t ctx)¶: Sets res to the polynomial poly evaluated at x.

void ca_fmpz_mpoly_evaluate_horner(ca_t res, const fmpz_mpoly_t f, ca_srcptr x, const fmpz_mpoly_ctx_t mctx, ca_ctx_t ctx)¶
void ca_fmpz_mpoly_evaluate_iter(ca_t res, const fmpz_mpoly_t f, ca_srcptr x, const fmpz_mpoly_ctx_t mctx, ca_ctx_t ctx)¶
void ca_fmpz_mpoly_evaluate(ca_t res, const fmpz_mpoly_t f, ca_srcptr x, const fmpz_mpoly_ctx_t mctx, ca_ctx_t ctx)¶: Sets res to the multivariate polynomial f evaluated at the vector of arguments x.

void ca_fmpz_mpoly_q_evaluate(ca_t res, const fmpz_mpoly_q_t f, ca_srcptr x, const fmpz_mpoly_ctx_t mctx, ca_ctx_t ctx)¶: Sets res to the multivariate rational function f evaluated at the vector of arguments x.

void ca_fmpz_mpoly_q_evaluate_no_division_by_zero(ca_t res, const fmpz_mpoly_q_t f, ca_srcptr x, const fmpz_mpoly_ctx_t mctx, ca_ctx_t ctx)¶
void ca_inv_no_division_by_zero(ca_t res, const ca_t x, ca_ctx_t ctx)¶: These functions behave like the normal arithmetic functions, but assume (and do not check) that division by zero cannot occur. Division by zero will result in undefined behavior.

Powers and roots¶

void ca_sqr(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the square of x.

void ca_pow_fmpq(ca_t res, const ca_t x, const fmpq_t y, ca_ctx_t ctx)¶
void ca_pow_fmpz(ca_t res, const ca_t x, const fmpz_t y, ca_ctx_t ctx)¶
void ca_pow_ui(ca_t res, const ca_t x, ulong y, ca_ctx_t ctx)¶
void ca_pow_si(ca_t res, const ca_t x, slong y, ca_ctx_t ctx)¶
void ca_pow(ca_t res, const ca_t x, const ca_t y, ca_ctx_t ctx)¶: Sets res to x raised to the power y. Handling of special values is not yet implemented.

void ca_pow_si_arithmetic(ca_t res, const ca_t x, slong n, ca_ctx_t ctx)¶: Sets res to x raised to the power n. Whereas ca_pow(), ca_pow_si() etc. may create \(x^n\) as an extension number if n is large, this function always perform the exponentiation using field arithmetic.

void ca_sqrt_inert(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_sqrt_nofactor(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_sqrt_factor(ca_t res, const ca_t x, ulong flags, ca_ctx_t ctx)¶

void ca_sqrt(ca_t res, const ca_t x, ca_ctx_t ctx)¶

Sets res to the principal square root of x.

For special values, the following definitions apply:

\(\sqrt{c \infty} = \sqrt{c} \infty\)
\(\sqrt{\tilde \infty} = \tilde \infty\).
Both Undefined and Unknown map to themselves.

The inert version outputs the generator in the formal field \(\mathbb{Q}(\sqrt{x})\) without simplifying.

The factor version writes \(x = A^2 B\) in \(K\) where \(K\) is the field of x, and outputs \(A \sqrt{B}\) or \(-A \sqrt{B}\) (whichever gives the correct sign) as an element of \(K(\sqrt{B})\) or some subfield thereof. This factorization is only a heuristic and is not guaranteed to make \(B\) minimal. Factorization options can be passed through to flags: see ca_factor() for details.

The nofactor version will not perform a general factorization, but may still perform other simplifications. It may in particular attempt to simplify \(\sqrt{x}\) to a single element in \(\overline{\mathbb{Q}}\).

void ca_sqrt_ui(ca_t res, ulong n, ca_ctx_t ctx)¶: Sets res to the principal square root of n.

Complex parts¶

void ca_abs(ca_t res, const ca_t x, ca_ctx_t ctx)¶

Sets res to the absolute value of x.

For special values, the following definitions apply:

\(|c \infty| = |\tilde \infty| = +\infty\).
Both Undefined and Unknown map to themselves.

This function will attempt to simplify its argument through an exact computation. It may in particular attempt to simplify \(|x|\) to a single element in \(\overline{\mathbb{Q}}\).

In the generic case, this function outputs an element of the formal field \(\mathbb{Q}(|x|)\).

void ca_sgn(ca_t res, const ca_t x, ca_ctx_t ctx)¶

Sets res to the sign of x, defined by

\[\begin{split}\operatorname{sgn}(x) = \begin{cases} 0 & x = 0 \\ \frac{x}{|x|} & x \ne 0 \end{cases}\end{split}\]

for numbers. For special values, the following definitions apply:

\(\operatorname{sgn}(c \infty) = c\).
\(\operatorname{sgn}(\tilde \infty) = \operatorname{Undefined}\).
Both Undefined and Unknown map to themselves.

This function will attempt to simplify its argument through an exact computation. It may in particular attempt to simplify \(\operatorname{sgn}(x)\) to a single element in \(\overline{\mathbb{Q}}\).

In the generic case, this function outputs an element of the formal field \(\mathbb{Q}(\operatorname{sgn}(x))\).

void ca_csgn(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res 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. This function gives Undefined for unsigned infinity and \(\operatorname{csgn}(\operatorname{sgn}(c \infty)) = \operatorname{csgn}(c)\) for signed infinities.

void ca_arg(ca_t res, const ca_t x, ca_ctx_t ctx)¶

Sets res to the complex argument (phase) of x, normalized to the range \((-\pi, +\pi]\). The argument of 0 is defined as 0. For special values, the following definitions apply:

\(\operatorname{arg}(c \infty) = \operatorname{arg}(c)\).
\(\operatorname{arg}(\tilde \infty) = \operatorname{Undefined}\).
Both Undefined and Unknown map to themselves.

void ca_re(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the real part of x. The result is Undefined if x is any infinity (including a real infinity).

void ca_im(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the imaginary part of x. The result is Undefined if x is any infinity (including an imaginary infinity).

void ca_conj_deep(ca_t res, const ca_t x, ca_ctx_t ctx)¶
void ca_conj_shallow(ca_t res, const ca_t x, ca_ctx_t ctx)¶
void ca_conj(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the complex conjugate of x. The shallow version creates a new extension element \(\overline{x}\) unless x can be trivially conjugated in-place in the existing field. The deep version recursively conjugates the extension numbers in the field of x.

void ca_floor(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the floor function of x. The result is Undefined if x is any infinity (including a real infinity). For complex numbers, this is presently defined to take the floor of the real part.

void ca_ceil(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the ceiling function of x. The result is Undefined if x is any infinity (including a real infinity). For complex numbers, this is presently defined to take the ceiling of the real part.

Exponentials and logarithms¶

void ca_exp(ca_t res, const ca_t x, ca_ctx_t ctx)¶

Sets res to the exponential function of x.

For special values, the following definitions apply:

\(e^{+\infty} = +\infty\)
\(e^{c \infty} = \tilde \infty\) if \(0 < \operatorname{Re}(c) < 1\).
\(e^{c \infty} = 0\) if \(\operatorname{Re}(c) < 0\).
\(e^{c \infty} = \text{Undefined}\) if \(\operatorname{Re}(c) = 0\).
\(e^{\tilde \infty} = \text{Undefined}\).
Both Undefined and Unknown map to themselves.

The following symbolic simplifications are performed automatically:

\(e^0 = 1\)
\(e^{\log(z)} = z\)
\(e^{(p/q) \log(z)} = z^{p/q}\) (for rational \(p/q\))
\(e^{(p/q) \pi i}\) = algebraic root of unity (for small rational \(p/q\))

In the generic case, this function outputs an element of the formal field \(\mathbb{Q}(e^x)\).

void ca_log(ca_t res, const ca_t x, ca_ctx_t ctx)¶

Sets res to the natural logarithm of x.

For special values and at the origin, the following definitions apply:

For any infinity, \(\log(c\infty) = \log(\tilde \infty) = +\infty\).
\(\log(0) = -\infty\). The result is Unknown if deciding \(x = 0\) fails.
Both Undefined and Unknown map to themselves.

The following symbolic simplifications are performed automatically:

\(\log(1) = 0\)
\(\log\left(e^z\right) = z + 2 \pi i k\)
\(\log\left(\sqrt{z}\right) = \tfrac{1}{2} \log(z) + 2 \pi i k\)
\(\log\left(z^a\right) = a \log(z) + 2 \pi i k\)
\(\log(x) = \log(-x) + \pi i\) for negative real x

In the generic case, this function outputs an element of the formal field \(\mathbb{Q}(\log(x))\).

Trigonometric functions¶

void ca_sin_cos_exponential(ca_t res1, ca_t res2, const ca_t x, ca_ctx_t ctx)¶

void ca_sin_cos_direct(ca_t res1, ca_t res2, const ca_t x, ca_ctx_t ctx)¶

void ca_sin_cos_tangent(ca_t res1, ca_t res2, const ca_t x, ca_ctx_t ctx)¶

void ca_sin_cos(ca_t res1, ca_t res2, const ca_t x, ca_ctx_t ctx)¶

Sets res1 to the sine of x and res2 to the cosine of x. Either res1 or res2 can be NULL to compute only the other function. Various representations are implemented:

The exponential version expresses the sine and cosine in terms of complex exponentials. Simple algebraic values will simplify to rational numbers or elements of cyclotomic fields.
The direct method expresses the sine and cosine in terms of the original functions (perhaps after applying some symmetry transformations, which may interchange sin and cos). Extremely simple algebraic values will automatically simplify to elements of real algebraic number fields.
The tangent version expresses the sine and cosine in terms of \(\tan(x/2)\), perhaps after applying some symmetry transformations. Extremely simple algebraic values will automatically simplify to elements of real algebraic number fields.

By default, the standard function uses the exponential representation as this typically works best for field arithmetic and simplifications, although it has the disadvantage of introducing complex numbers where real numbers would be sufficient. The behavior of the standard function can be changed using the CA_OPT_TRIG_FORM context setting.

For special values, the following definitions apply:

\(\sin(\pm i \infty) = \pm i \infty\)
\(\cos(\pm i \infty) = +\infty\)
All other infinities give \(\operatorname{Undefined}\)

void ca_sin(ca_t res, const ca_t x, ca_ctx_t ctx)¶
void ca_cos(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the sine or cosine of x. These functions are shortcuts for ca_sin_cos().

void ca_tan_sine_cosine(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_tan_exponential(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_tan_direct(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_tan(ca_t res, const ca_t x, ca_ctx_t ctx)¶

Sets res to the tangent of x. The sine_cosine version evaluates the tangent as a quotient of a sine and cosine, the direct version evaluates it directly as a tangent (possibly after transforming the variable), and the exponential version evaluates it in terms of complex exponentials. Simple algebraic values will automatically simplify to elements of trigonometric or cyclotomic number fields.

By default, the standard function uses the exponential representation as this typically works best for field arithmetic and simplifications, although it has the disadvantage of introducing complex numbers where real numbers would be sufficient. The behavior of the standard function can be changed using the CA_OPT_TRIG_FORM context setting.

For special values, the following definitions apply:

At poles, \(\tan((n+\tfrac{1}{2}) \pi) = \tilde \infty\)
\(\tan(e^{i \theta} \infty) = +i, \quad 0 < \theta < \pi\)
\(\tan(e^{i \theta} \infty) = -i, \quad -\pi < \theta < 0\)
\(\tan(\pm \infty) = \tan(\tilde \infty) = \operatorname{Undefined}\)

void ca_cot(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the cotangent x. This is equivalent to computing the reciprocal of the tangent.

void ca_atan_logarithm(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_atan_direct(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_atan(ca_t res, const ca_t x, ca_ctx_t ctx)¶

Sets res to the inverse tangent of x.

The direct version expresses the result as an inverse tangent (possibly after transforming the variable). The logarithm version expresses it in terms of complex logarithms. Simple algebraic inputs will automatically simplify to rational multiples of \(\pi\).

By default, the standard function uses the logarithm representation as this typically works best for field arithmetic and simplifications, although it has the disadvantage of introducing complex numbers where real numbers would be sufficient. The behavior of the standard function can be changed using the CA_OPT_TRIG_FORM context setting (exponential mode results in logarithmic forms).

For special values, the following definitions apply:

\(\operatorname{atan}(\pm i) = \pm i \infty\)
\(\operatorname{atan}(c \infty) = \operatorname{csgn}(c) \pi / 2\)
\(\operatorname{atan}(\tilde \infty) = \operatorname{Undefined}\)

void ca_asin_logarithm(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_acos_logarithm(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_asin_direct(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_acos_direct(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_asin(ca_t res, const ca_t x, ca_ctx_t ctx)¶

void ca_acos(ca_t res, const ca_t x, ca_ctx_t ctx)¶

Sets res to the inverse sine (respectively, cosine) of x.

The direct version expresses the result as an inverse sine or cosine (possibly after transforming the variable). The logarithm version expresses it in terms of complex logarithms. Simple algebraic inputs will automatically simplify to rational multiples of \(\pi\).

By default, the standard function uses the logarithm representation as this typically works best for field arithmetic and simplifications, although it has the disadvantage of introducing complex numbers where real numbers would be sufficient. The behavior of the standard function can be changed using the CA_OPT_TRIG_FORM context setting (exponential mode results in logarithmic forms).

The inverse cosine is presently implemented as \(\operatorname{acos}(x) = \pi/2 - \operatorname{asin}(x)\).

Special functions¶

void ca_gamma(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the gamma function of x.

void ca_erf(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the error function of x.

void ca_erfc(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the complementary error function of x.

void ca_erfi(ca_t res, const ca_t x, ca_ctx_t ctx)¶: Sets res to the imaginary error function of x.

Numerical evaluation¶

void ca_get_acb_raw(acb_t res, const ca_t x, slong prec, ca_ctx_t ctx)¶: Sets res to an enclosure of the numerical value of x. A working precision of prec bits is used internally for the evaluation, without adaptive refinement. If x is any special value, res is set to acb_indeterminate.

void ca_get_acb(acb_t res, const ca_t x, slong prec, ca_ctx_t ctx)¶

void ca_get_acb_accurate_parts(acb_t res, const ca_t x, slong prec, ca_ctx_t ctx)¶

Sets res to an enclosure of the numerical value of x. The working precision is increased adaptively to try to ensure prec accurate bits in the output. The accurate_parts version tries to ensure prec accurate bits for both the real and imaginary part separately.

The refinement is stopped if the working precision exceeds CA_OPT_PREC_LIMIT (or twice the initial precision, if this is larger). The user may call acb_rel_accuracy_bits to check is the calculation was successful.

The output is not rounded down to prec bits (to avoid unnecessary double rounding); the user may call acb_set_round when rounding is desired.

char *ca_get_decimal_str(const ca_t x, slong digits, ulong flags, ca_ctx_t ctx)¶

Returns a decimal approximation of x with precision up to digits. The output is guaranteed to be correct within 1 ulp in the returned digits, but the number of returned digits may be smaller than digits if the numerical evaluation does not succeed.

If flags is set to 1, attempts to achieve full accuracy for both the real and imaginary parts separately.

If x is not finite or a finite enclosure cannot be produced, returns the string “?”.

The user should free the returned string with flint_free.

Rewriting and simplification¶

void ca_rewrite_complex_normal_form(ca_t res, const ca_t x, int deep, ca_ctx_t ctx)¶

Sets res to x rewritten using standardizing transformations over the complex numbers:

Elementary functions are rewritten in terms of (complex) exponentials, roots and logarithms
Complex parts are rewritten using logarithms, square roots, and (deep) complex conjugates
Algebraic numbers are rewritten in terms of cyclotomic fields where applicable

If deep is set, the rewriting is applied recursively to the tower of extension numbers; otherwise, the rewriting is only applied to the top-level extension numbers.

The result is not a normal form in the strong sense (the same number can have many possible representations even after applying this transformation), but in practice this is a powerful heuristic for simplification.

Factorization¶

type ca_factor_struct¶

type ca_factor_t¶: Represents a real or complex number in factored form \(b_1^{e_1} b_2^{e_2} \cdots b_n^{e_n}\) where \(b_i\) and \(e_i\) are ca_t numbers (the exponents need not be integers).

void ca_factor_init(ca_factor_t fac, ca_ctx_t ctx)¶: Initializes fac and sets it to the empty factorization (equivalent to the number 1).

void ca_factor_clear(ca_factor_t fac, ca_ctx_t ctx)¶: Clears the factorization structure fac.

void ca_factor_one(ca_factor_t fac, ca_ctx_t ctx)¶: Sets fac to the empty factorization (equivalent to the number 1).

void ca_factor_print(const ca_factor_t fac, ca_ctx_t ctx)¶: Prints a description of fac to standard output.

void ca_factor_insert(ca_factor_t fac, const ca_t base, const ca_t exp, ca_ctx_t ctx)¶: Inserts \(b^e\) into fac where b is given by base and e is given by exp. If a base element structurally identical to base already exists in fac, the corresponding exponent is incremented by exp; otherwise, this factor is appended.

void ca_factor_get_ca(ca_t res, const ca_factor_t fac, ca_ctx_t ctx)¶: Expands fac back to a single ca_t by evaluating the powers and multiplying out the result.

void ca_factor(ca_factor_t res, const ca_t x, ulong flags, ca_ctx_t ctx)¶: Sets res to a factorization of x of the form \(x = b_1^{e_1} b_2^{e_2} \cdots b_n^{e_n}\). Requires that x is not a special value. The type of factorization is controlled by flags, which can be set to a combination of constants in the following section.

Factorization options¶

The following flags select the structural polynomial factorization to perform over formal fields \(\mathbb{Q}(a_1,\ldots,a_n)\). Each flag in the list strictly encompasses the factorization power of the preceding flag, so it is unnecessary to pass more than one flag.

CA_FACTOR_POLY_NONE¶: No polynomial factorization at all.

CA_FACTOR_POLY_CONTENT¶: Only extract the rational content.

CA_FACTOR_POLY_SQF¶: Perform a squarefree factorization in addition to extracting the rational content.

CA_FACTOR_POLY_FULL¶: Perform a full multivariate polynomial factorization.

The following flags select the factorization to perform over \(\mathbb{Z}\). Integer factorization is applied if x is an element of \(\mathbb{Q}\), and to the extracted rational content of polynomials. Each flag in the list strictly encompasses the factorization power of the preceding flag, so it is unnecessary to pass more than one flag.

CA_FACTOR_ZZ_NONE¶: No integer factorization at all.

CA_FACTOR_ZZ_SMOOTH¶: Perform a smooth factorization to extract small prime factors (heuristically up to CA_OPT_SMOOTH_LIMIT bits) in addition to identifying perfect powers.

CA_FACTOR_ZZ_FULL¶: Perform a complete integer factorization into prime numbers. This is prohibitively slow for general integers exceeding 70-80 digits.

Context options¶

The options member of a ca_ctx_t object is an array of slong values controlling simplification behavior and various other settings. The values of the array at the following indices can be changed by the user (example: ctx->options[CA_OPT_PREC_LIMIT] = 65536).

It is recommended to set options controlling evaluation only at the time when a context object is created. Changing such options later should normally be harmless, but since the update will not apply retroactively to objects that have already been computed and cached, one might not see the expected behavior. Superficial options (printing) can be changed at any time.

CA_OPT_VERBOSE¶: Whether to print debug information. Default value: 0.

CA_OPT_PRINT_FLAGS¶: Printing style. See Printing for details. Default value: CA_PRINT_DEFAULT.

CA_OPT_MPOLY_ORD¶: Monomial ordering to use for multivariate polynomials. Possible values are ORD_LEX, ORD_DEGLEX and ORD_DEGREVLEX. Default value: ORD_LEX. This option must be set before doing any computations.

CA_OPT_PREC_LIMIT¶: Maximum precision to use internally for numerical evaluation with Arb, and in some cases for the magntiude of exact coefficients. This parameter affects the possibility to prove inequalities and find simplifications between related extension numbers. This is not a strict limit; some calculations may use higher precision when there is a good reason to do so. Default value: 4096.

CA_OPT_QQBAR_DEG_LIMIT¶: Maximum degree of qqbar_t elements allowed internally during simplification of algebraic numbers. This limit may be exceeded when the user provides explicit qqbar_t input of higher degree. Default value: 120.

CA_OPT_LOW_PREC¶: Numerical precision to use for fast checks (typically, before attempting more expensive operations). Default value: 64.

CA_OPT_SMOOTH_LIMIT¶: Size in bits for factors in smooth integer factorization. Default value: 32.

CA_OPT_LLL_PREC¶: Precision to use to find integer relations using LLL. Default value: 128.

CA_OPT_POW_LIMIT¶: Largest exponent to expand powers automatically. This only applies in multivariate and transcendental fields: in number fields, CA_OPT_PREC_LIMIT applies instead. Default value: 20.

CA_OPT_USE_GROEBNER¶: Boolean flag for whether to use Gröbner basis computation. This flag and the following limits affect the ability to prove multivariate identities. Default value: 1.

CA_OPT_GROEBNER_LENGTH_LIMIT¶: Maximum length of ideal basis allowed in Buchberger’s algorithm. Default value: 100.

CA_OPT_GROEBNER_POLY_LENGTH_LIMIT¶: Maximum length of polynomials allowed in Buchberger’s algorithm. Default value: 1000.

CA_OPT_GROEBNER_POLY_BITS_LIMIT¶: Maximum coefficient size in bits of polynomials allowed in Buchberger’s algorithm. Default value: 10000.

CA_OPT_VIETA_LIMIT¶: Maximum degree n of algebraic numbers for which to add Vieta’s formulas to the reduction ideal. This must be set relatively low since the number of terms in Vieta’s formulas is \(O(2^n)\) and the resulting Gröbner basis computations can be expensive. Default value: 6.

CA_OPT_TRIG_FORM¶

Default representation of trigonometric functions. The following values are possible:

CA_TRIG_DIRECT¶: Use the direct functions (with some exceptions).

CA_TRIG_EXPONENTIAL¶: Use complex exponentials.

CA_TRIG_SINE_COSINE¶: Use sines and cosines.

CA_TRIG_TANGENT¶: Use tangents.

Default value: CA_TRIG_EXPONENTIAL.

The exponential representation is currently used by default as typically works best for field arithmetic and simplifications, although it has the disadvantage of introducing complex numbers where real numbers would be sufficient. This may change in the future.

Internal representation¶

CA_FMPQ(x)¶

CA_FMPQ_NUMREF(x)¶

CA_FMPQ_DENREF(x)¶: Assuming that x holds an element of the trivial field \(\mathbb{Q}\), this macro returns a pointer which can be used as an fmpq_t, or respectively to the numerator or denominator as an fmpz_t.

CA_MPOLY_Q(x)¶: Assuming that x holds a generic field element as data, this macro returns a pointer which can be used as an fmpz_mpoly_q_t.

CA_NF_ELEM(x)¶: Assuming that x holds an Antic number field element as data, this macro returns a pointer which can be used as an nf_elem_t.

void _ca_make_field_element(ca_t x, ca_field_srcptr new_index, ca_ctx_t ctx)¶: Changes the internal representation of x to that of an element of the field with index new_index in the context object ctx. This may destroy the value of x.

void _ca_make_fmpq(ca_t x, ca_ctx_t ctx)¶: Changes the internal representation of x to that of an element of the trivial field \(\mathbb{Q}\). This may destroy the value of x.

ca.h – exact real and complex numbers¶

Introduction: numbers¶

Introduction: special values¶

Number objects¶

Context objects¶

Memory management for numbers¶

Symbolic expressions¶

Printing¶

Special values¶

Assignment and conversion¶

Conversion of algebraic numbers¶

Random generation¶

Representation properties¶

Value predicates¶

Comparisons¶

Field structure operations¶

Arithmetic¶

Powers and roots¶

Complex parts¶

Exponentials and logarithms¶

Trigonometric functions¶

Special functions¶

Numerical evaluation¶

Rewriting and simplification¶

Factorization¶

Factorization options¶

Context options¶

Internal representation¶

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