gr_series.h

Initialises ctx as the ring R[[x]] / (x^n) of truncated power series over base_ring. Elements have type gr_poly_t. All inputs are assumed to have length at most n, and all outputs are truncated to length n. The truncation order is fixed; elements of a context with order m \le n can be used as inputs in a context with order n, but not the other way round. For n = 0 this is the zero ring.

Initialises ctx as the ring R[[x]] of power series over base_ring. Elements have type gr_series_t. Each element carries its own error term O(x^m); the default precision prec is the order from which tails are replaced by an error term in the results of operations. The default precision can be changed (GR_SERIES_PREC), and elements can be shared between contexts with the same base ring and different precisions.

Context methods: clearing, printing a description, naming the generator, and listing the generators of the tower of rings (the generator x after those of the base ring).

Structural predicates. The ring inherits the properties of the base ring, except that it is an integral domain or a field only for n = 1 (when it is isomorphic to R), and that for n = 0 it is the zero ring.

Element methods: memory management, a random element for testing, and output in the form f (mod x^n).

Assignment: 0, 1, the generator x (which is 0 for n \le 1), a copy, and conversions into the ring (with truncation to length n).

Predicates, decided coefficientwise in the base ring.

Arithmetic modulo x^n: products by gr_poly_mullow, inverses and quotients by gr_poly_inv_series and gr_poly_div_series, which require an invertible constant term of the divisor (GR_DOMAIN if it is zero or not invertible).

Elementary functions modulo x^n, computed with the corresponding gr_poly series functions (Elementary functions of power series). The value at the constant term must exist in the base ring, and the coefficients involve divisions by small integers, so these functions fail in small positive characteristic.

Returns the exponent m of the error term O(x^m) of x, or GR_SERIES_ERR_EXACT if x is exact.

Returns whether x has no error term. This does not check whether the coefficients themselves are exact.

Adds an error term O(x^m) to x in place, discarding the coefficients of order m and higher. The exponent is clamped to 0 \le m \le GR_SERIES_ERR_MAX.

Removes the error term of x, keeping its coefficients.

Context methods: clearing, printing a description, naming the generator, and listing the generators of the tower of rings.

Structural predicates, inherited from the base ring: R[[x]] is a ring, a commutative ring or an integral domain exactly when R is.

Element methods: memory management (an initialised series is the exact zero), a random element for testing, and output in the form f + O(x^m).

Assignment of exact values (a polynomial longer than the default precision is truncated and given an error term), of the generator, and of copies. gr_series_set_other converts elements of the base ring, polynomials and power series over other rings in a variable of the same name (keeping their error terms), and anything else that converts into the base ring, as a constant.

Sets res to the series whose coefficients are the enclosures [m_i \pm r_i] (gr_set_interval_mid_rad in the base ring) and whose error term is the smaller of those of m and r.

Predicates. A known coefficient that differs decides T_FALSE, but a series with an error term can never be shown to be zero, one or equal to another series: the answer is then T_UNKNOWN. A coefficient of order beyond the error term is T_UNKNOWN.

Ring operations. The product of an exact zero with anything is the exact zero; otherwise the error term of a product is the smaller of the error terms of the factors (and of the default precision, if the exact product is longer).

Inverse and quotients in R[[x]]. The inverse needs an invertible constant term: it returns GR_DOMAIN for the exact zero and GR_UNABLE for a series whose constant term is hidden in the error term. Division first removes the common power of x from numerator and denominator (a quotient such as x / x = 1 exists in R[[x]]), returning GR_DOMAIN if the denominator has larger valuation than the numerator and GR_UNABLE if the valuations cannot be decided.

Elementary functions of power series, computed by the gr_poly series functions to the precision allowed by the error term of the input and the default precision. If the constant term is unknown (error term O(x^0)), GR_UNABLE is returned.

Special functions of power series. These are currently implemented only over the base rings arb and acb, through the series functions of arb_hypgeom, acb_hypgeom, acb_dirichlet, acb_elliptic and acb_modular (for example arb_hypgeom_gamma_series), and return GR_UNABLE for other base rings. The flags normalized, offset and regularized have the meanings of the corresponding arb_hypgeom functions. For the functions of two or more series arguments, all arguments except one must currently be constant series.