gr_series.h – formal power series over generic rings

We provide two implementations of formal power series:

• gr_series_mod - truncated power series \(R[[x]] / x^n\), with exact reresentation of elements.

• gr_series - infinite power series \(R[[x]]\), with elements represented as finite truncations with an error term \(O(x^n)\) where n is tracked per-element. Polynomials of small degree can be represented exactly without error term.

It should be noted that these are different algebraic structures. For example, \(\mathbb{Q}[[x]]\) is an integral domain, but \(\mathbb{Q}[[x]] / x^n\) is not. Some operations are admissible for gr_series but not for gr_series_mod, and vice versa. Note that for gr_equal(), \(0 = 0\) is true in both \(R[[x]]\) and \(R[[x]] / x^n\), but \(0 + O(x^n) = 0 + O(x^n)\) (or even \(0 + O(x^n) = 0\)) is unknown in \(R[[x]]\) since we do not know whether the big-O term hides some nonzero term. For a computation where either representation may be used, it is recommended to choose gr_series_mod as this will have less overhead.

Context constructors

void gr_series_mod_ctx_init(gr_ctx_t ctx, gr_ctx_t base_ring, slong prec)

Initializes ctx to a ring of truncated power series \(R[[x]] / \langle x^n \rangle\) over the given base_ring. Elements have type gr_poly_t. It is assumed that all inputs are already truncated to length n, and this invariant is enforced for all outputs.

The truncation order n of a gr_series_mod context is fixed. If \(m \le n\), then elements belonging to a context object for \(R[[x]] / x^m\) can be given as input to a context for \(R[[x]] / x^n\), but not vice versa.

void gr_series_ctx_init(gr_ctx_t ctx, gr_ctx_t base_ring, slong prec)

Initializes ctx to a ring of power series \(R[[x]]\) over the given base_ring. Elements have type gr_series_t.

Elements are generally inexact, having an error term \(O(x^n)\) where n may be different for each element. The context parameter prec defines the default precision: tails \(c_n x^n + c_{n+1} x^{n+1} + \ldots\) with \(n \ge prec\) will be replaced by the error term \(O(x^n)\).

The default precision is mutable. Elements can be shared freely between two context objects with the same base ring, even if those context objects have different precision.

Truncated power series

Types and macros

GR_SERIES_MOD_ELEM_CTX(ctx)

Given a gr_series_mod context object, accesses the context object of the base ring.

GR_SERIES_MOD_N(ctx)

Given a gr_series_mod context object, accesses the truncation order n.

Generic methods

Elements of a gr_series_mod ring can be manipulated using the standard gr interface. The following is a list of implemented functions such that gr_series_mod_foo overloads gr_foo in the method table for gr_series_mod; the user can optionally call gr_series_mod_foo directly.

void gr_series_mod_ctx_clear(gr_ctx_t ctx)

int gr_series_mod_ctx_write(gr_stream_t out, gr_ctx_t ctx)

truth_t gr_series_mod_ctx_is_ring(gr_ctx_t ctx)

truth_t gr_series_mod_ctx_is_commutative_ring(gr_ctx_t ctx)

truth_t gr_series_mod_ctx_is_integral_domain(gr_ctx_t ctx)

truth_t gr_series_mod_ctx_is_rational_vector_space(gr_ctx_t ctx)

truth_t gr_series_mod_ctx_is_real_vector_space(gr_ctx_t ctx)

truth_t gr_series_mod_ctx_is_complex_vector_space(gr_ctx_t ctx)

truth_t gr_series_mod_ctx_is_field(gr_ctx_t ctx)

int gr_series_mod_ctx_set_gen_name(gr_ctx_t ctx, const char *s)

int gr_series_mod_ctx_set_gen_names(gr_ctx_t ctx, const char **s)

int gr_series_mod_gens_recursive(gr_vec_t vec, gr_ctx_t ctx)

void gr_series_mod_init(gr_poly_t res, gr_ctx_t ctx)

void gr_series_mod_clear(gr_poly_t res, gr_ctx_t ctx)

void gr_series_mod_swap(gr_poly_t x, gr_poly_t y, gr_ctx_t ctx)

int gr_series_mod_randtest(gr_poly_t res, flint_rand_t state, gr_ctx_t ctx)

int gr_series_mod_write(gr_stream_t out, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_zero(gr_poly_t res, gr_ctx_t ctx)

int gr_series_mod_one(gr_poly_t res, gr_ctx_t ctx)

int gr_series_mod_gen(gr_poly_t res, gr_ctx_t ctx)

int gr_series_mod_set(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_set_other(gr_poly_t res, gr_srcptr x, gr_ctx_t x_ctx, gr_ctx_t ctx)

void gr_series_mod_set_shallow(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_set_si(gr_poly_t res, slong c, gr_ctx_t ctx)

int gr_series_mod_set_ui(gr_poly_t res, ulong c, gr_ctx_t ctx)

int gr_series_mod_set_fmpz(gr_poly_t res, const fmpz_t c, gr_ctx_t ctx)

int gr_series_mod_set_fmpq(gr_poly_t res, const fmpq_t c, gr_ctx_t ctx)

truth_t gr_series_mod_is_zero(const gr_poly_t x, gr_ctx_t ctx)

truth_t gr_series_mod_is_one(const gr_poly_t x, gr_ctx_t ctx)

truth_t gr_series_mod_equal(const gr_poly_t x, const gr_poly_t y, gr_ctx_t ctx)

int gr_series_mod_neg(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_add(gr_poly_t res, const gr_poly_t x, const gr_poly_t y, gr_ctx_t ctx)

int gr_series_mod_sub(gr_poly_t res, const gr_poly_t x, const gr_poly_t y, gr_ctx_t ctx)

int gr_series_mod_mul(gr_poly_t res, const gr_poly_t x, const gr_poly_t y, gr_ctx_t ctx)

int gr_series_mod_inv(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_div(gr_poly_t res, const gr_poly_t x, const gr_poly_t y, gr_ctx_t ctx)

int gr_series_mod_exp(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_log(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_sqrt(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_rsqrt(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_tan(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_asin(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_acos(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_atan(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_asinh(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_acosh(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_mod_atanh(gr_poly_t res, const gr_poly_t x, gr_ctx_t ctx)

Power series

Types and macros

type gr_series_struct

type gr_series_t

A structure containing a gr_poly_t followed by a slong representing the exponent in the error term.

type gr_series_vec_struct

type gr_series_vec_t

Alias for gr_vec_t with gr_series_t elements, provided for convenience.

GR_SERIES_POLY(x)

Macro accessing the polynomial part of a gr_series_t.

GR_SERIES_ERROR(x)

Macro accessing the error of a gr_series_t.

GR_SERIES_ERR_MAX

The maximum allowed n in an error term \(O(x^n)\).

GR_SERIES_ERR_EXACT

A special value of n used to indicate that a series is exact.

GR_SERIES_ELEM_CTX(ctx)

Given a gr_series context object, accesses the context object of the base ring.

GR_SERIES_PREC(ctx)

Given a gr_series context object, accesses the default precision.

Error term manipulation

slong _gr_series_get_error(const gr_series_t f, gr_ctx_t ctx)

Return the exponent \(n\) of the error term \(O(x^n)\) of f. If f is exact, returns GR_SERIES_ERR_EXACT.

truth_t _gr_series_is_exact(const gr_series_t f, gr_ctx_t ctx)

Returns whether f is exact as a power series, i.e. lacks error term \(O(x^n)\). This does not recursively check exactness of the underlying elements.

void _gr_series_set_error(gr_series_t f, slong n, gr_ctx_t ctx)

Add an error term \(O(x^n)\) in-place to f. The exponent n is clamped between 0 and GR_SERIES_ERR_MAX. Terms of order higher than or equal to n will be truncated.

void _gr_series_make_exact(gr_series_t f, gr_ctx_t ctx)

Remove the \(O(x^n)\) error term (if any) from f.

Generic methods

Elements of a gr_series ring can be manipulated using the standard gr interface. The following is a list of implemented functions such that gr_series_foo overloads gr_foo in the method table for gr_series; the user can optionally call gr_series_foo directly.

void gr_series_ctx_clear(gr_ctx_t ctx)

int gr_series_ctx_write(gr_stream_t out, gr_ctx_t ctx)

truth_t gr_series_ctx_is_ring(gr_ctx_t ctx)

truth_t gr_series_ctx_is_commutative_ring(gr_ctx_t ctx)

truth_t gr_series_ctx_is_integral_domain(gr_ctx_t ctx)

truth_t gr_series_ctx_is_rational_vector_space(gr_ctx_t ctx)

truth_t gr_series_ctx_is_real_vector_space(gr_ctx_t ctx)

truth_t gr_series_ctx_is_complex_vector_space(gr_ctx_t ctx)

int gr_series_ctx_set_gen_name(gr_ctx_t ctx, const char *s)

int gr_series_ctx_set_gen_names(gr_ctx_t ctx, const char **s)

void gr_series_init(gr_series_t res, gr_ctx_t ctx)

void gr_series_clear(gr_series_t res, gr_ctx_t ctx)

void gr_series_swap(gr_series_t x, gr_series_t y, gr_ctx_t ctx)

void gr_series_set_shallow(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_randtest(gr_series_t res, flint_rand_t state, gr_ctx_t ctx)

int gr_series_write(gr_stream_t out, const gr_series_t x, gr_ctx_t ctx)

int gr_series_zero(gr_series_t res, gr_ctx_t ctx)

int gr_series_one(gr_series_t res, gr_ctx_t ctx)

int gr_series_set(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_gen(gr_series_t res, gr_ctx_t ctx)

int gr_series_gens_recursive(gr_vec_t vec, gr_ctx_t ctx)

int gr_series_neg(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_set_gr_poly(gr_series_t res, const gr_poly_t x, gr_ctx_t ctx)

int gr_series_set_scalar(gr_series_t res, gr_srcptr x, gr_ctx_t ctx)

int gr_series_set_si(gr_series_t res, slong c, gr_ctx_t ctx)

int gr_series_set_ui(gr_series_t res, ulong c, gr_ctx_t ctx)

int gr_series_set_fmpz(gr_series_t res, const fmpz_t c, gr_ctx_t ctx)

int gr_series_set_fmpq(gr_series_t res, const fmpq_t c, gr_ctx_t ctx)

int gr_series_set_other(gr_series_t res, gr_srcptr x, gr_ctx_t x_ctx, gr_ctx_t ctx)

truth_t gr_series_is_zero(const gr_series_t x, gr_ctx_t ctx)

truth_t gr_series_is_one(const gr_series_t x, gr_ctx_t ctx)

truth_t gr_series_coeff_is_zero(const gr_series_t x, slong i, gr_ctx_t ctx)

truth_t gr_series_equal(const gr_series_t x, const gr_series_t y, gr_ctx_t ctx)

int gr_series_add(gr_series_t res, const gr_series_t x, const gr_series_t y, gr_ctx_t ctx)

int gr_series_sub(gr_series_t res, const gr_series_t x, const gr_series_t y, gr_ctx_t ctx)

int gr_series_mul(gr_series_t res, const gr_series_t x, const gr_series_t y, gr_ctx_t ctx)

int gr_series_inv(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_div(gr_series_t res, const gr_series_t x, const gr_series_t y, gr_ctx_t ctx)

int gr_series_divexact(gr_series_t res, const gr_series_t x, const gr_series_t y, gr_ctx_t ctx)

int gr_series_sqrt(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_rsqrt(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_exp(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_log(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_tan(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_asin(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_acos(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_atan(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_asinh(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_acosh(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_atanh(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_gamma(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_rgamma(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_lgamma(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_digamma(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_erf(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_erfc(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_erfi(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_exp_integral_ei(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_cos_integral(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_cosh_integral(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_sin_integral(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_sinh_integral(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_fresnel(gr_series_t res1, gr_series_t res2, const gr_series_t x, int normalized, gr_ctx_t ctx)

int gr_series_fresnel_s(gr_series_t res, const gr_series_t x, int normalized, gr_ctx_t ctx)

int gr_series_fresnel_c(gr_series_t res, const gr_series_t x, int normalized, gr_ctx_t ctx)

int gr_series_airy(gr_series_t res1, gr_series_t res2, gr_series_t res3, gr_series_t res4, const gr_series_t x, gr_ctx_t ctx)

int gr_series_airy_ai(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_airy_ai_prime(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_airy_bi(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_airy_bi_prime(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_log_integral(gr_series_t res, const gr_series_t x, int offset, gr_ctx_t ctx)

int gr_series_gamma_upper(gr_series_t res, const gr_series_t s, const gr_series_t x, int regularized, gr_ctx_t ctx)

int gr_series_gamma_lower(gr_series_t res, const gr_series_t s, const gr_series_t x, int regularized, gr_ctx_t ctx)

int gr_series_beta_lower(gr_series_t res, const gr_series_t a, const gr_series_t b, const gr_series_t x, int regularized, gr_ctx_t ctx)

int gr_series_polylog(gr_series_t res, const gr_series_t s, const gr_series_t z, gr_ctx_t ctx)

int gr_series_hurwitz_zeta(gr_series_t res, const gr_series_t s, const gr_series_t z, gr_ctx_t ctx)

int gr_series_dirichlet_l(gr_series_t res, const dirichlet_group_t G, const dirichlet_char_t chi, const gr_series_t x, gr_ctx_t ctx)

int gr_series_dirichlet_hardy_theta(gr_series_t res, const dirichlet_group_t G, const dirichlet_char_t chi, const gr_series_t x, gr_ctx_t ctx)

int gr_series_dirichlet_hardy_z(gr_series_t res, const dirichlet_group_t G, const dirichlet_char_t chi, const gr_series_t x, gr_ctx_t ctx)

int gr_series_jacobi_theta(gr_series_t res1, gr_series_t res2, gr_series_t res3, gr_series_t res4, const gr_series_t x, const gr_series_t tau, gr_ctx_t ctx)

int gr_series_jacobi_theta_1(gr_series_t res, const gr_series_t x, const gr_series_t tau, gr_ctx_t ctx)

int gr_series_jacobi_theta_2(gr_series_t res, const gr_series_t x, const gr_series_t tau, gr_ctx_t ctx)

int gr_series_jacobi_theta_3(gr_series_t res, const gr_series_t x, const gr_series_t tau, gr_ctx_t ctx)

int gr_series_jacobi_theta_4(gr_series_t res, const gr_series_t x, const gr_series_t tau, gr_ctx_t ctx)

int gr_series_agm1(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_elliptic_k(gr_series_t res, const gr_series_t x, gr_ctx_t ctx)

int gr_series_weierstrass_p(gr_series_t res, const gr_series_t x, const gr_series_t tau, gr_ctx_t ctx)

int gr_series_hypgeom_pfq(gr_series_t res, const gr_series_vec_t a, const gr_series_vec_t b, const gr_series_t x, int regularized, gr_ctx_t ctx)