acb_poly.h – polynomials over the complex numbers

An acb_poly_t represents a polynomial over the complex numbers, implemented as an array of coefficients of type acb_struct.

Most functions are provided in two versions: an underscore method which operates directly on pre-allocated arrays of coefficients and generally has some restrictions (such as requiring the lengths to be nonzero and not supporting aliasing of the input and output arrays), and a non-underscore method which performs automatic memory management and handles degenerate cases.

Types, macros and constants

acb_poly_struct

acb_poly_t

Contains a pointer to an array of coefficients (coeffs), the used length (length), and the allocated size of the array (alloc).

An acb_poly_t is defined as an array of length one of type acb_poly_struct, permitting an acb_poly_t to be passed by reference.

Memory management

void acb_poly_init(acb_poly_t poly)

Initializes the polynomial for use, setting it to the zero polynomial.

void acb_poly_clear(acb_poly_t poly)

Clears the polynomial, deallocating all coefficients and the coefficient array.

void acb_poly_fit_length(acb_poly_t poly, slong len)

Makes sure that the coefficient array of the polynomial contains at least len initialized coefficients.

void _acb_poly_set_length(acb_poly_t poly, slong len)

Directly changes the length of the polynomial, without allocating or deallocating coefficients. The value should not exceed the allocation length.

void _acb_poly_normalise(acb_poly_t poly)

Strips any trailing coefficients which are identical to zero.

void acb_poly_swap(acb_poly_t poly1, acb_poly_t poly2)

Swaps poly1 and poly2 efficiently.

Basic properties and manipulation

slong acb_poly_length(const acb_poly_t poly)

Returns the length of poly, i.e. zero if poly is identically zero, and otherwise one more than the index of the highest term that is not identically zero.

slong acb_poly_degree(const acb_poly_t poly)

Returns the degree of poly, defined as one less than its length. Note that if one or several leading coefficients are balls containing zero, this value can be larger than the true degree of the exact polynomial represented by poly, so the return value of this function is effectively an upper bound.

int acb_poly_is_zero(const acb_poly_t poly)

int acb_poly_is_one(const acb_poly_t poly)

int acb_poly_is_x(const acb_poly_t poly)

Returns 1 if poly is exactly the polynomial 0, 1 or x respectively. Returns 0 otherwise.

void acb_poly_zero(acb_poly_t poly)

Sets poly to the zero polynomial.

void acb_poly_one(acb_poly_t poly)

Sets poly to the constant polynomial 1.

void acb_poly_set(acb_poly_t dest, const acb_poly_t src)

Sets dest to a copy of src.

void acb_poly_set_round(acb_poly_t dest, const acb_poly_t src, slong prec)

Sets dest to a copy of src, rounded to prec bits.

void acb_poly_set_coeff_si(acb_poly_t poly, slong n, slong c)

void acb_poly_set_coeff_acb(acb_poly_t poly, slong n, const acb_t c)

Sets the coefficient with index n in poly to the value c. We require that n is nonnegative.

void acb_poly_get_coeff_acb(acb_t v, const acb_poly_t poly, slong n)

Sets v to the value of the coefficient with index n in poly. We require that n is nonnegative.

acb_poly_get_coeff_ptr(poly, n)

Given \(n \ge 0\), returns a pointer to coefficient n of poly, or NULL if n exceeds the length of poly.

void _acb_poly_shift_right(acb_ptr res, acb_srcptr poly, slong len, slong n)

void acb_poly_shift_right(acb_poly_t res, const acb_poly_t poly, slong n)

Sets res to poly divided by \(x^n\), throwing away the lower coefficients. We require that n is nonnegative.

void _acb_poly_shift_left(acb_ptr res, acb_srcptr poly, slong len, slong n)

void acb_poly_shift_left(acb_poly_t res, const acb_poly_t poly, slong n)

Sets res to poly multiplied by \(x^n\). We require that n is nonnegative.

void acb_poly_truncate(acb_poly_t poly, slong n)

Truncates poly to have length at most n, i.e. degree strictly smaller than n.

Input and output

void acb_poly_printd(const acb_poly_t poly, slong digits)

Prints the polynomial as an array of coefficients, printing each coefficient using acb_printd.

void acb_poly_fprintd(FILE * file, const acb_poly_t poly, slong digits)

Prints the polynomial as an array of coefficients to the stream file, printing each coefficient using acb_fprintd.

Random generation

void acb_poly_randtest(acb_poly_t poly, flint_rand_t state, slong len, slong prec, slong mag_bits)

Creates a random polynomial with length at most len.

Comparisons

int acb_poly_equal(const acb_poly_t A, const acb_poly_t B)

Returns nonzero iff A and B are identical as interval polynomials.

int acb_poly_contains(const acb_poly_t poly1, const acb_poly_t poly2)

int acb_poly_contains_fmpz_poly(const acb_poly_t poly1, const fmpz_poly_t poly2)

int acb_poly_contains_fmpq_poly(const acb_poly_t poly1, const fmpq_poly_t poly2)

Returns nonzero iff poly2 is contained in poly1.

int _acb_poly_overlaps(acb_srcptr poly1, slong len1, acb_srcptr poly2, slong len2)

int acb_poly_overlaps(const acb_poly_t poly1, const acb_poly_t poly2)

Returns nonzero iff poly1 overlaps with poly2. The underscore function requires that len1 ist at least as large as len2.

int acb_poly_get_unique_fmpz_poly(fmpz_poly_t z, const acb_poly_t x)

If x contains a unique integer polynomial, sets z to that value and returns nonzero. Otherwise (if x represents no integers or more than one integer), returns zero, possibly partially modifying z.

int acb_poly_is_real(const acb_poly_t poly)

Returns nonzero iff all coefficients in poly have zero imaginary part.

Conversions

void acb_poly_set_fmpz_poly(acb_poly_t poly, const fmpz_poly_t re, slong prec)

void acb_poly_set2_fmpz_poly(acb_poly_t poly, const fmpz_poly_t re, const fmpz_poly_t im, slong prec)

void acb_poly_set_arb_poly(acb_poly_t poly, const arb_poly_t re)

void acb_poly_set2_arb_poly(acb_poly_t poly, const arb_poly_t re, const arb_poly_t im)

void acb_poly_set_fmpq_poly(acb_poly_t poly, const fmpq_poly_t re, slong prec)

void acb_poly_set2_fmpq_poly(acb_poly_t poly, const fmpq_poly_t re, const fmpq_poly_t im, slong prec)

Sets poly to the given real part re plus the imaginary part im, both rounded to prec bits.

void acb_poly_set_acb(acb_poly_t poly, slong src)

void acb_poly_set_si(acb_poly_t poly, slong src)

Sets poly to src.

Bounds

void _acb_poly_majorant(arb_ptr res, acb_srcptr poly, slong len, slong prec)

void acb_poly_majorant(arb_poly_t res, const acb_poly_t poly, slong prec)

Sets res to an exact real polynomial whose coefficients are upper bounds for the absolute values of the coefficients in poly, rounded to prec bits.

Arithmetic

void _acb_poly_add(acb_ptr C, acb_srcptr A, slong lenA, acb_srcptr B, slong lenB, slong prec)

Sets {C, max(lenA, lenB)} to the sum of {A, lenA} and {B, lenB}. Allows aliasing of the input and output operands.

void acb_poly_add(acb_poly_t C, const acb_poly_t A, const acb_poly_t B, slong prec)

void acb_poly_add_si(acb_poly_t C, const acb_poly_t A, slong B, slong prec)

Sets C to the sum of A and B.

void _acb_poly_sub(acb_ptr C, acb_srcptr A, slong lenA, acb_srcptr B, slong lenB, slong prec)

Sets {C, max(lenA, lenB)} to the difference of {A, lenA} and {B, lenB}. Allows aliasing of the input and output operands.

void acb_poly_sub(acb_poly_t C, const acb_poly_t A, const acb_poly_t B, slong prec)

Sets C to the difference of A and B.

void acb_poly_neg(acb_poly_t C, const acb_poly_t A)

Sets C to the negation of A.

void acb_poly_scalar_mul_2exp_si(acb_poly_t C, const acb_poly_t A, slong c)

Sets C to A multiplied by \(2^c\).

void acb_poly_scalar_mul(acb_poly_t C, const acb_poly_t A, const acb_t c, slong prec)

Sets C to A multiplied by c.

void acb_poly_scalar_div(acb_poly_t C, const acb_poly_t A, const acb_t c, slong prec)

Sets C to A divided by c.

void _acb_poly_mullow_classical(acb_ptr C, acb_srcptr A, slong lenA, acb_srcptr B, slong lenB, slong n, slong prec)

void _acb_poly_mullow_transpose(acb_ptr C, acb_srcptr A, slong lenA, acb_srcptr B, slong lenB, slong n, slong prec)

void _acb_poly_mullow_transpose_gauss(acb_ptr C, acb_srcptr A, slong lenA, acb_srcptr B, slong lenB, slong n, slong prec)

void _acb_poly_mullow(acb_ptr C, acb_srcptr A, slong lenA, acb_srcptr B, slong lenB, slong n, slong prec)

Sets {C, n} to the product of {A, lenA} and {B, lenB}, truncated to length n. The output is not allowed to be aliased with either of the inputs. We require \(\mathrm{lenA} \ge \mathrm{lenB} > 0\), \(n > 0\), \(\mathrm{lenA} + \mathrm{lenB} - 1 \ge n\).

The classical version uses a plain loop.

The transpose version evaluates the product using four real polynomial multiplications (via _arb_poly_mullow()).

The transpose_gauss version evaluates the product using three real polynomial multiplications. This is almost always faster than transpose, but has worse numerical stability when the coefficients vary in magnitude.

The default function _acb_poly_mullow() automatically switches been classical and transpose multiplication.

If the input pointers are identical (and the lengths are the same), they are assumed to represent the same polynomial, and its square is computed.

void acb_poly_mullow_classical(acb_poly_t C, const acb_poly_t A, const acb_poly_t B, slong n, slong prec)

void acb_poly_mullow_transpose(acb_poly_t C, const acb_poly_t A, const acb_poly_t B, slong n, slong prec)

void acb_poly_mullow_transpose_gauss(acb_poly_t C, const acb_poly_t A, const acb_poly_t B, slong n, slong prec)

void acb_poly_mullow(acb_poly_t C, const acb_poly_t A, const acb_poly_t B, slong n, slong prec)

Sets C to the product of A and B, truncated to length n. If the same variable is passed for A and B, sets C to the square of A truncated to length n.

void _acb_poly_mul(acb_ptr C, acb_srcptr A, slong lenA, acb_srcptr B, slong lenB, slong prec)

Sets {C, lenA + lenB - 1} to the product of {A, lenA} and {B, lenB}. The output is not allowed to be aliased with either of the inputs. We require \(\mathrm{lenA} \ge \mathrm{lenB} > 0\). This function is implemented as a simple wrapper for _acb_poly_mullow().

If the input pointers are identical (and the lengths are the same), they are assumed to represent the same polynomial, and its square is computed.

void acb_poly_mul(acb_poly_t C, const acb_poly_t A1, const acb_poly_t B2, slong prec)

Sets C to the product of A and B. If the same variable is passed for A and B, sets C to the square of A.

void _acb_poly_inv_series(acb_ptr Qinv, acb_srcptr Q, slong Qlen, slong len, slong prec)

Sets {Qinv, len} to the power series inverse of {Q, Qlen}. Uses Newton iteration.

void acb_poly_inv_series(acb_poly_t Qinv, const acb_poly_t Q, slong n, slong prec)

Sets Qinv to the power series inverse of Q.

void _acb_poly_div_series(acb_ptr Q, acb_srcptr A, slong Alen, acb_srcptr B, slong Blen, slong n, slong prec)

Sets {Q, n} to the power series quotient of {A, Alen} by {B, Blen}. Uses Newton iteration followed by multiplication.

void acb_poly_div_series(acb_poly_t Q, const acb_poly_t A, const acb_poly_t B, slong n, slong prec)

Sets Q to the power series quotient A divided by B, truncated to length n.

void _acb_poly_div(acb_ptr Q, acb_srcptr A, slong lenA, acb_srcptr B, slong lenB, slong prec)

void _acb_poly_rem(acb_ptr R, acb_srcptr A, slong lenA, acb_srcptr B, slong lenB, slong prec)

void _acb_poly_divrem(acb_ptr Q, acb_ptr R, acb_srcptr A, slong lenA, acb_srcptr B, slong lenB, slong prec)

void acb_poly_divrem(acb_poly_t Q, acb_poly_t R, const acb_poly_t A, const acb_poly_t B, slong prec)

Performs polynomial division with remainder, computing a quotient \(Q\) and a remainder \(R\) such that \(A = BQ + R\). The implementation reverses the inputs and performs power series division.

If the leading coefficient of \(B\) contains zero (or if \(B\) is identically zero), returns 0 indicating failure without modifying the outputs. Otherwise returns nonzero.

void _acb_poly_div_root(acb_ptr Q, acb_t R, acb_srcptr A, slong len, const acb_t c, slong prec)

Divides \(A\) by the polynomial \(x - c\), computing the quotient \(Q\) as well as the remainder \(R = f(c)\).

Composition

void _acb_poly_taylor_shift_horner(acb_ptr g, const acb_t c, slong n, slong prec)

void acb_poly_taylor_shift_horner(acb_poly_t g, const acb_poly_t f, const acb_t c, slong prec)

void _acb_poly_taylor_shift_divconquer(acb_ptr g, const acb_t c, slong n, slong prec)

void acb_poly_taylor_shift_divconquer(acb_poly_t g, const acb_poly_t f, const acb_t c, slong prec)

void _acb_poly_taylor_shift_convolution(acb_ptr g, const acb_t c, slong n, slong prec)

void acb_poly_taylor_shift_convolution(acb_poly_t g, const acb_poly_t f, const acb_t c, slong prec)

void _acb_poly_taylor_shift(acb_ptr g, const acb_t c, slong n, slong prec)

void acb_poly_taylor_shift(acb_poly_t g, const acb_poly_t f, const acb_t c, slong prec)

Sets g to the Taylor shift \(f(x+c)\), computed respectively using an optimized form of Horner’s rule, divide-and-conquer, a single convolution, and an automatic choice between the three algorithms.

The underscore methods act in-place on g = f which has length n.

void _acb_poly_compose_horner(acb_ptr res, acb_srcptr poly1, slong len1, acb_srcptr poly2, slong len2, slong prec)

void acb_poly_compose_horner(acb_poly_t res, const acb_poly_t poly1, const acb_poly_t poly2, slong prec)

void _acb_poly_compose_divconquer(acb_ptr res, acb_srcptr poly1, slong len1, acb_srcptr poly2, slong len2, slong prec)

void acb_poly_compose_divconquer(acb_poly_t res, const acb_poly_t poly1, const acb_poly_t poly2, slong prec)

void _acb_poly_compose(acb_ptr res, acb_srcptr poly1, slong len1, acb_srcptr poly2, slong len2, slong prec)

void acb_poly_compose(acb_poly_t res, const acb_poly_t poly1, const acb_poly_t poly2, slong prec)

Sets res to the composition \(h(x) = f(g(x))\) where \(f\) is given by poly1 and \(g\) is given by poly2, respectively using Horner’s rule, divide-and-conquer, and an automatic choice between the two algorithms.

The default algorithm also handles special-form input \(g = ax^n + c\) efficiently by performing a Taylor shift followed by a rescaling.

The underscore methods do not support aliasing of the output with either input polynomial.

void _acb_poly_compose_series_horner(acb_ptr res, acb_srcptr poly1, slong len1, acb_srcptr poly2, slong len2, slong n, slong prec)

void acb_poly_compose_series_horner(acb_poly_t res, const acb_poly_t poly1, const acb_poly_t poly2, slong n, slong prec)

void _acb_poly_compose_series_brent_kung(acb_ptr res, acb_srcptr poly1, slong len1, acb_srcptr poly2, slong len2, slong n, slong prec)

void acb_poly_compose_series_brent_kung(acb_poly_t res, const acb_poly_t poly1, const acb_poly_t poly2, slong n, slong prec)

void _acb_poly_compose_series(acb_ptr res, acb_srcptr poly1, slong len1, acb_srcptr poly2, slong len2, slong n, slong prec)

void acb_poly_compose_series(acb_poly_t res, const acb_poly_t poly1, const acb_poly_t poly2, slong n, slong prec)

Sets res to the power series composition \(h(x) = f(g(x))\) truncated to order \(O(x^n)\) where \(f\) is given by poly1 and \(g\) is given by poly2, respectively using Horner’s rule, the Brent-Kung baby step-giant step algorithm, and an automatic choice between the two algorithms.

The default algorithm also handles special-form input \(g = ax^n\) efficiently.

We require that the constant term in \(g(x)\) is exactly zero. The underscore methods do not support aliasing of the output with either input polynomial.

void _acb_poly_revert_series_lagrange(acb_ptr h, acb_srcptr f, slong flen, slong n, slong prec)

void acb_poly_revert_series_lagrange(acb_poly_t h, const acb_poly_t f, slong n, slong prec)

void _acb_poly_revert_series_newton(acb_ptr h, acb_srcptr f, slong flen, slong n, slong prec)

void acb_poly_revert_series_newton(acb_poly_t h, const acb_poly_t f, slong n, slong prec)

void _acb_poly_revert_series_lagrange_fast(acb_ptr h, acb_srcptr f, slong flen, slong n, slong prec)

void acb_poly_revert_series_lagrange_fast(acb_poly_t h, const acb_poly_t f, slong n, slong prec)

void _acb_poly_revert_series(acb_ptr h, acb_srcptr f, slong flen, slong n, slong prec)

void acb_poly_revert_series(acb_poly_t h, const acb_poly_t f, slong n, slong prec)

Sets \(h\) to the power series reversion of \(f\), i.e. the expansion of the compositional inverse function \(f^{-1}(x)\), truncated to order \(O(x^n)\), using respectively Lagrange inversion, Newton iteration, fast Lagrange inversion, and a default algorithm choice.

We require that the constant term in \(f\) is exactly zero and that the linear term is nonzero. The underscore methods assume that flen is at least 2, and do not support aliasing.

Evaluation

void _acb_poly_evaluate_horner(acb_t y, acb_srcptr f, slong len, const acb_t x, slong prec)

void acb_poly_evaluate_horner(acb_t y, const acb_poly_t f, const acb_t x, slong prec)

void _acb_poly_evaluate_rectangular(acb_t y, acb_srcptr f, slong len, const acb_t x, slong prec)

void acb_poly_evaluate_rectangular(acb_t y, const acb_poly_t f, const acb_t x, slong prec)

void _acb_poly_evaluate(acb_t y, acb_srcptr f, slong len, const acb_t x, slong prec)

void acb_poly_evaluate(acb_t y, const acb_poly_t f, const acb_t x, slong prec)

Sets \(y = f(x)\), evaluated respectively using Horner’s rule, rectangular splitting, and an automatic algorithm choice.

void _acb_poly_evaluate2_horner(acb_t y, acb_t z, acb_srcptr f, slong len, const acb_t x, slong prec)

void acb_poly_evaluate2_horner(acb_t y, acb_t z, const acb_poly_t f, const acb_t x, slong prec)

void _acb_poly_evaluate2_rectangular(acb_t y, acb_t z, acb_srcptr f, slong len, const acb_t x, slong prec)

void acb_poly_evaluate2_rectangular(acb_t y, acb_t z, const acb_poly_t f, const acb_t x, slong prec)

void _acb_poly_evaluate2(acb_t y, acb_t z, acb_srcptr f, slong len, const acb_t x, slong prec)

void acb_poly_evaluate2(acb_t y, acb_t z, const acb_poly_t f, const acb_t x, slong prec)

Sets \(y = f(x), z = f'(x)\), evaluated respectively using Horner’s rule, rectangular splitting, and an automatic algorithm choice.

When Horner’s rule is used, the only advantage of evaluating the function and its derivative simultaneously is that one does not have to generate the derivative polynomial explicitly. With the rectangular splitting algorithm, the powers can be reused, making simultaneous evaluation slightly faster.

Product trees

void _acb_poly_product_roots(acb_ptr poly, acb_srcptr xs, slong n, slong prec)

void acb_poly_product_roots(acb_poly_t poly, acb_srcptr xs, slong n, slong prec)

Generates the polynomial \((x-x_0)(x-x_1)\cdots(x-x_{n-1})\).

acb_ptr * _acb_poly_tree_alloc(slong len)

Returns an initialized data structured capable of representing a remainder tree (product tree) of len roots.

void _acb_poly_tree_free(acb_ptr * tree, slong len)

Deallocates a tree structure as allocated using _acb_poly_tree_alloc.

void _acb_poly_tree_build(acb_ptr * tree, acb_srcptr roots, slong len, slong prec)

Constructs a product tree from a given array of len roots. The tree structure must be pre-allocated to the specified length using _acb_poly_tree_alloc().

Multipoint evaluation

void _acb_poly_evaluate_vec_iter(acb_ptr ys, acb_srcptr poly, slong plen, acb_srcptr xs, slong n, slong prec)

void acb_poly_evaluate_vec_iter(acb_ptr ys, const acb_poly_t poly, acb_srcptr xs, slong n, slong prec)

Evaluates the polynomial simultaneously at n given points, calling _acb_poly_evaluate() repeatedly.

void _acb_poly_evaluate_vec_fast_precomp(acb_ptr vs, acb_srcptr poly, slong plen, acb_ptr * tree, slong len, slong prec)

void _acb_poly_evaluate_vec_fast(acb_ptr ys, acb_srcptr poly, slong plen, acb_srcptr xs, slong n, slong prec)

void acb_poly_evaluate_vec_fast(acb_ptr ys, const acb_poly_t poly, acb_srcptr xs, slong n, slong prec)

Evaluates the polynomial simultaneously at n given points, using fast multipoint evaluation.

Interpolation

void _acb_poly_interpolate_newton(acb_ptr poly, acb_srcptr xs, acb_srcptr ys, slong n, slong prec)

void acb_poly_interpolate_newton(acb_poly_t poly, acb_srcptr xs, acb_srcptr ys, slong n, slong prec)

Recovers the unique polynomial of length at most n that interpolates the given x and y values. This implementation first interpolates in the Newton basis and then converts back to the monomial basis.

void _acb_poly_interpolate_barycentric(acb_ptr poly, acb_srcptr xs, acb_srcptr ys, slong n, slong prec)

void acb_poly_interpolate_barycentric(acb_poly_t poly, acb_srcptr xs, acb_srcptr ys, slong n, slong prec)

Recovers the unique polynomial of length at most n that interpolates the given x and y values. This implementation uses the barycentric form of Lagrange interpolation.

void _acb_poly_interpolation_weights(acb_ptr w, acb_ptr * tree, slong len, slong prec)

void _acb_poly_interpolate_fast_precomp(acb_ptr poly, acb_srcptr ys, acb_ptr * tree, acb_srcptr weights, slong len, slong prec)

void _acb_poly_interpolate_fast(acb_ptr poly, acb_srcptr xs, acb_srcptr ys, slong len, slong prec)

void acb_poly_interpolate_fast(acb_poly_t poly, acb_srcptr xs, acb_srcptr ys, slong n, slong prec)

Recovers the unique polynomial of length at most n that interpolates the given x and y values, using fast Lagrange interpolation. The precomp function takes a precomputed product tree over the x values and a vector of interpolation weights as additional inputs.

Differentiation

void _acb_poly_derivative(acb_ptr res, acb_srcptr poly, slong len, slong prec)

Sets {res, len - 1} to the derivative of {poly, len}. Allows aliasing of the input and output.

void acb_poly_derivative(acb_poly_t res, const acb_poly_t poly, slong prec)

Sets res to the derivative of poly.

void _acb_poly_integral(acb_ptr res, acb_srcptr poly, slong len, slong prec)

Sets {res, len} to the integral of {poly, len - 1}. Allows aliasing of the input and output.

void acb_poly_integral(acb_poly_t res, const acb_poly_t poly, slong prec)

Sets res to the integral of poly.

Elementary functions

void _acb_poly_pow_ui_trunc_binexp(acb_ptr res, acb_srcptr f, slong flen, ulong exp, slong len, slong prec)

Sets {res, len} to {f, flen} raised to the power exp, truncated to length len. Requires that len is no longer than the length of the power as computed without truncation (i.e. no zero-padding is performed). Does not support aliasing of the input and output, and requires that flen and len are positive. Uses binary expontiation.

void acb_poly_pow_ui_trunc_binexp(acb_poly_t res, const acb_poly_t poly, ulong exp, slong len, slong prec)

Sets res to poly raised to the power exp, truncated to length len. Uses binary exponentiation.

void _acb_poly_pow_ui(acb_ptr res, acb_srcptr f, slong flen, ulong exp, slong prec)

Sets res to {f, flen} raised to the power exp. Does not support aliasing of the input and output, and requires that flen is positive.

void acb_poly_pow_ui(acb_poly_t res, const acb_poly_t poly, ulong exp, slong prec)

Sets res to poly raised to the power exp.

void _acb_poly_pow_series(acb_ptr h, acb_srcptr f, slong flen, acb_srcptr g, slong glen, slong len, slong prec)

Sets {h, len} to the power series \(f(x)^{g(x)} = \exp(g(x) \log f(x))\) truncated to length len. This function detects special cases such as g being an exact small integer or \(\pm 1/2\), and computes such powers more efficiently. This function does not support aliasing of the output with either of the input operands. It requires that all lengths are positive, and assumes that flen and glen do not exceed len.

void acb_poly_pow_series(acb_poly_t h, const acb_poly_t f, const acb_poly_t g, slong len, slong prec)

Sets h to the power series \(f(x)^{g(x)} = \exp(g(x) \log f(x))\) truncated to length len. This function detects special cases such as g being an exact small integer or \(\pm 1/2\), and computes such powers more efficiently.

void _acb_poly_pow_acb_series(acb_ptr h, acb_srcptr f, slong flen, const acb_t g, slong len, slong prec)

Sets {h, len} to the power series \(f(x)^g = \exp(g \log f(x))\) truncated to length len. This function detects special cases such as g being an exact small integer or \(\pm 1/2\), and computes such powers more efficiently. This function does not support aliasing of the output with either of the input operands. It requires that all lengths are positive, and assumes that flen does not exceed len.

void acb_poly_pow_acb_series(acb_poly_t h, const acb_poly_t f, const acb_t g, slong len, slong prec)

Sets h to the power series \(f(x)^g = \exp(g \log f(x))\) truncated to length len.

void _acb_poly_sqrt_series(acb_ptr g, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_sqrt_series(acb_poly_t g, const acb_poly_t h, slong n, slong prec)

Sets g to the power series square root of h, truncated to length n. Uses division-free Newton iteration for the reciprocal square root, followed by a multiplication.

The underscore method does not support aliasing of the input and output arrays. It requires that hlen and n are greater than zero.

void _acb_poly_rsqrt_series(acb_ptr g, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_rsqrt_series(acb_poly_t g, const acb_poly_t h, slong n, slong prec)

Sets g to the reciprocal power series square root of h, truncated to length n. Uses division-free Newton iteration.

The underscore method does not support aliasing of the input and output arrays. It requires that hlen and n are greater than zero.

void _acb_poly_log_series(acb_ptr res, acb_srcptr f, slong flen, slong n, slong prec)

void acb_poly_log_series(acb_poly_t res, const acb_poly_t f, slong n, slong prec)

Sets res to the power series logarithm of f, truncated to length n. Uses the formula \(\log(f(x)) = \int f'(x) / f(x) dx\), adding the logarithm of the constant term in f as the constant of integration.

The underscore method supports aliasing of the input and output arrays. It requires that flen and n are greater than zero.

void _acb_poly_atan_series(acb_ptr res, acb_srcptr f, slong flen, slong n, slong prec)

void acb_poly_atan_series(acb_poly_t res, const acb_poly_t f, slong n, slong prec)

Sets res the power series inverse tangent of f, truncated to length n.

Uses the formula

\[\tan^{-1}(f(x)) = \int f'(x) / (1+f(x)^2) dx,\]

adding the function of the constant term in f as the constant of integration.

The underscore method supports aliasing of the input and output arrays. It requires that flen and n are greater than zero.

void _acb_poly_exp_series_basecase(acb_ptr f, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_exp_series_basecase(acb_poly_t f, const acb_poly_t h, slong n, slong prec)

void _acb_poly_exp_series(acb_ptr f, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_exp_series(acb_poly_t f, const acb_poly_t h, slong n, slong prec)

Sets \(f\) to the power series exponential of \(h\), truncated to length \(n\).

The basecase version uses a simple recurrence for the coefficients, requiring \(O(nm)\) operations where \(m\) is the length of \(h\).

The main implementation uses Newton iteration, starting from a small number of terms given by the basecase algorithm. The complexity is \(O(M(n))\). Redundant operations in the Newton iteration are avoided by using the scheme described in [HZ2004].

The underscore methods support aliasing and allow the input to be shorter than the output, but require the lengths to be nonzero.

void _acb_poly_sin_cos_series_basecase(acb_ptr s, acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec, int times_pi)

void acb_poly_sin_cos_series_basecase(acb_poly_t s, acb_poly_t c, const acb_poly_t h, slong n, slong prec, int times_pi)

void _acb_poly_sin_cos_series_tangent(acb_ptr s, acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec, int times_pi)

void acb_poly_sin_cos_series_tangent(acb_poly_t s, acb_poly_t c, const acb_poly_t h, slong n, slong prec, int times_pi)

void _acb_poly_sin_cos_series(acb_ptr s, acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_sin_cos_series(acb_poly_t s, acb_poly_t c, const acb_poly_t h, slong n, slong prec)

Sets s and c to the power series sine and cosine of h, computed simultaneously.

The basecase version uses a simple recurrence for the coefficients, requiring \(O(nm)\) operations where \(m\) is the length of \(h\).

The tangent version uses the tangent half-angle formulas to compute the sine and cosine via _acb_poly_tan_series(). This requires \(O(M(n))\) operations. When \(h = h_0 + h_1\) where the constant term \(h_0\) is nonzero, the evaluation is done as \(\sin(h_0 + h_1) = \cos(h_0) \sin(h_1) + \sin(h_0) \cos(h_1)\), \(\cos(h_0 + h_1) = \cos(h_0) \cos(h_1) - \sin(h_0) \sin(h_1)\), to improve accuracy and avoid dividing by zero at the poles of the tangent function.

The default version automatically selects between the basecase and tangent algorithms depending on the input.

The basecase and tangent versions take a flag times_pi specifying that the input is to be multiplied by \(\pi\).

The underscore methods support aliasing and require the lengths to be nonzero.

void _acb_poly_sin_series(acb_ptr s, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_sin_series(acb_poly_t s, const acb_poly_t h, slong n, slong prec)

void _acb_poly_cos_series(acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_cos_series(acb_poly_t c, const acb_poly_t h, slong n, slong prec)

Respectively evaluates the power series sine or cosine. These functions simply wrap _acb_poly_sin_cos_series(). The underscore methods support aliasing and require the lengths to be nonzero.

void _acb_poly_tan_series(acb_ptr g, acb_srcptr h, slong hlen, slong len, slong prec)

void acb_poly_tan_series(acb_poly_t g, const acb_poly_t h, slong n, slong prec)

Sets g to the power series tangent of h.

For small n takes the quotient of the sine and cosine as computed using the basecase algorithm. For large n, uses Newton iteration to invert the inverse tangent series. The complexity is \(O(M(n))\).

The underscore version does not support aliasing, and requires the lengths to be nonzero.

void _acb_poly_sin_cos_pi_series(acb_ptr s, acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_sin_cos_pi_series(acb_poly_t s, acb_poly_t c, const acb_poly_t h, slong n, slong prec)

void _acb_poly_sin_pi_series(acb_ptr s, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_sin_pi_series(acb_poly_t s, const acb_poly_t h, slong n, slong prec)

void _acb_poly_cos_pi_series(acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_cos_pi_series(acb_poly_t c, const acb_poly_t h, slong n, slong prec)

void _acb_poly_cot_pi_series(acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_cot_pi_series(acb_poly_t c, const acb_poly_t h, slong n, slong prec)

Compute the respective trigonometric functions of the input multiplied by \(\pi\).

void _acb_poly_sinh_cosh_series_basecase(acb_ptr s, acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_sinh_cosh_series_basecase(acb_poly_t s, acb_poly_t c, const acb_poly_t h, slong n, slong prec)

void _acb_poly_sinh_cosh_series_exponential(acb_ptr s, acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_sinh_cosh_series_exponential(acb_poly_t s, acb_poly_t c, const acb_poly_t h, slong n, slong prec)

void _acb_poly_sinh_cosh_series(acb_ptr s, acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_sinh_cosh_series(acb_poly_t s, acb_poly_t c, const acb_poly_t h, slong n, slong prec)

void _acb_poly_sinh_series(acb_ptr s, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_sinh_series(acb_poly_t s, const acb_poly_t h, slong n, slong prec)

void _acb_poly_cosh_series(acb_ptr c, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_cosh_series(acb_poly_t c, const acb_poly_t h, slong n, slong prec)

Sets s and c respectively to the hyperbolic sine and cosine of the power series h, truncated to length n.

The implementations mirror those for sine and cosine, except that the exponential version computes both functions using the exponential function instead of the hyperbolic tangent.

void _acb_poly_sinc_series(acb_ptr s, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_sinc_series(acb_poly_t s, const acb_poly_t h, slong n, slong prec)

Sets s to the sinc function of the power series h, truncated to length n.

Gamma function

void _acb_poly_gamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_gamma_series(acb_poly_t res, const acb_poly_t h, slong n, slong prec)

void _acb_poly_rgamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_rgamma_series(acb_poly_t res, const acb_poly_t h, slong n, slong prec)

void _acb_poly_lgamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_lgamma_series(acb_poly_t res, const acb_poly_t h, slong n, slong prec)

void _acb_poly_digamma_series(acb_ptr res, acb_srcptr h, slong hlen, slong n, slong prec)

void acb_poly_digamma_series(acb_poly_t res, const acb_poly_t h, slong n, slong prec)

Sets res to the series expansion of \(\Gamma(h(x))\), \(1/\Gamma(h(x))\), or \(\log \Gamma(h(x))\), \(\psi(h(x))\), truncated to length n.

These functions first generate the Taylor series at the constant term of h, and then call _acb_poly_compose_series(). The Taylor coefficients are generated using Stirling’s series.

The underscore methods support aliasing of the input and output arrays, and require that hlen and n are greater than zero.

void _acb_poly_rising_ui_series(acb_ptr res, acb_srcptr f, slong flen, ulong r, slong trunc, slong prec)

void acb_poly_rising_ui_series(acb_poly_t res, const acb_poly_t f, ulong r, slong trunc, slong prec)

Sets res to the rising factorial \((f) (f+1) (f+2) \cdots (f+r-1)\), truncated to length trunc. The underscore method assumes that flen, r and trunc are at least 1, and does not support aliasing. Uses binary splitting.

Power sums

void _acb_poly_powsum_series_naive(acb_ptr z, const acb_t s, const acb_t a, const acb_t q, slong n, slong len, slong prec)

void _acb_poly_powsum_series_naive_threaded(acb_ptr z, const acb_t s, const acb_t a, const acb_t q, slong n, slong len, slong prec)

Computes

\[z = S(s,a,n) = \sum_{k=0}^{n-1} \frac{q^k}{(k+a)^{s+t}}\]

as a power series in \(t\) truncated to length len. This function evaluates the sum naively term by term. The threaded version splits the computation over the number of threads returned by flint_get_num_threads().

void _acb_poly_powsum_one_series_sieved(acb_ptr z, const acb_t s, slong n, slong len, slong prec)

Computes

\[z = S(s,1,n) \sum_{k=1}^n \frac{1}{k^{s+t}}\]

as a power series in \(t\) truncated to length len. This function stores a table of powers that have already been calculated, computing \((ij)^r\) as \(i^r j^r\) whenever \(k = ij\) is composite. As a further optimization, it groups all even \(k\) and evaluates the sum as a polynomial in \(2^{-(s+t)}\). This scheme requires about \(n / \log n\) powers, \(n / 2\) multiplications, and temporary storage of \(n / 6\) power series. Due to the extra power series multiplications, it is only faster than the naive algorithm when len is small.

Zeta function

void _acb_poly_zeta_em_choose_param(mag_t bound, ulong * N, ulong * M, const acb_t s, const acb_t a, slong d, slong target, slong prec)

Chooses N and M for Euler-Maclaurin summation of the Hurwitz zeta function, using a default algorithm.

void _acb_poly_zeta_em_bound1(mag_t bound, const acb_t s, const acb_t a, slong N, slong M, slong d, slong wp)

void _acb_poly_zeta_em_bound(arb_ptr vec, const acb_t s, const acb_t a, ulong N, ulong M, slong d, slong wp)

Compute bounds for Euler-Maclaurin evaluation of the Hurwitz zeta function or its power series, using the formulas in [Joh2013].

void _acb_poly_zeta_em_tail_naive(acb_ptr z, const acb_t s, const acb_t Na, acb_srcptr Nasx, slong M, slong len, slong prec)

void _acb_poly_zeta_em_tail_bsplit(acb_ptr z, const acb_t s, const acb_t Na, acb_srcptr Nasx, slong M, slong len, slong prec)

Evaluates the tail in the Euler-Maclaurin sum for the Hurwitz zeta function, respectively using the naive recurrence and binary splitting.

void _acb_poly_zeta_em_sum(acb_ptr z, const acb_t s, const acb_t a, int deflate, ulong N, ulong M, slong d, slong prec)

Evaluates the truncated Euler-Maclaurin sum of order \(N, M\) for the length-d truncated Taylor series of the Hurwitz zeta function \(\zeta(s,a)\) at \(s\), using a working precision of prec bits. With \(a = 1\), this gives the usual Riemann zeta function.

If deflate is nonzero, \(\zeta(s,a) - 1/(s-1)\) is evaluated (which permits series expansion at \(s = 1\)).

void _acb_poly_zeta_cpx_series(acb_ptr z, const acb_t s, const acb_t a, int deflate, slong d, slong prec)

Computes the series expansion of \(\zeta(s+x,a)\) (or \(\zeta(s+x,a) - 1/(s+x-1)\) if deflate is nonzero) to order d.

This function wraps _acb_poly_zeta_em_sum(), automatically choosing default values for \(N, M\) using _acb_poly_zeta_em_choose_param() to target an absolute truncation error of \(2^{-\operatorname{prec}}\).

void _acb_poly_zeta_series(acb_ptr res, acb_srcptr h, slong hlen, const acb_t a, int deflate, slong len, slong prec)

void acb_poly_zeta_series(acb_poly_t res, const acb_poly_t f, const acb_t a, int deflate, slong n, slong prec)

Sets res to the Hurwitz zeta function \(\zeta(s,a)\) where \(s\) a power series and \(a\) is a constant, truncated to length n. To evaluate the usual Riemann zeta function, set \(a = 1\).

If deflate is nonzero, evaluates \(\zeta(s,a) + 1/(1-s)\), which is well-defined as a limit when the constant term of \(s\) is 1. In particular, expanding \(\zeta(s,a) + 1/(1-s)\) with \(s = 1+x\) gives the Stieltjes constants

\[\sum_{k=0}^{n-1} \frac{(-1)^k}{k!} \gamma_k(a) x^k`.\]

If \(a = 1\), this implementation uses the reflection formula if the midpoint of the constant term of \(s\) is negative.

Other special functions

void _acb_poly_polylog_cpx_small(acb_ptr w, const acb_t s, const acb_t z, slong len, slong prec)

void _acb_poly_polylog_cpx_zeta(acb_ptr w, const acb_t s, const acb_t z, slong len, slong prec)

void _acb_poly_polylog_cpx(acb_ptr w, const acb_t s, const acb_t z, slong len, slong prec)

Sets w to the Taylor series with respect to x of the polylogarithm \(\operatorname{Li}_{s+x}(z)\), where s and z are given complex constants. The output is computed to length len which must be positive. Aliasing between w and s or z is not permitted.

The small version uses the standard power series expansion with respect to z, convergent when \(|z| < 1\). The zeta version evaluates the polylogarithm as a sum of two Hurwitz zeta functions. The default version automatically delegates to the small version when z is close to zero, and the zeta version otherwise. For further details, see Algorithms for polylogarithms.

void _acb_poly_polylog_series(acb_ptr w, acb_srcptr s, slong slen, const acb_t z, slong len, slong prec)

void acb_poly_polylog_series(acb_poly_t w, const acb_poly_t s, const acb_t z, slong len, slong prec)

Sets w to the polylogarithm \(\operatorname{Li}_{s}(z)\) where s is a given power series, truncating the output to length len. The underscore method requires all lengths to be positive and supports aliasing between all inputs and outputs.

void _acb_poly_erf_series(acb_ptr res, acb_srcptr z, slong zlen, slong n, slong prec)

void acb_poly_erf_series(acb_poly_t res, const acb_poly_t z, slong n, slong prec)

Sets res to the error function of the power series z, truncated to length n. These methods are provided for backwards compatibility. See acb_hypgeom_erf_series(), acb_hypgeom_erfc_series(), acb_hypgeom_erfi_series().

void _acb_poly_agm1_series(acb_ptr res, acb_srcptr z, slong zlen, slong len, slong prec)

void acb_poly_agm1_series(acb_poly_t res, const acb_poly_t z, slong n, slong prec)

Sets res to the arithmetic-geometric mean of 1 and the power series z, truncated to length n.

void _acb_poly_elliptic_k_series(acb_ptr res, acb_srcptr z, slong zlen, slong len, slong prec)

void acb_poly_elliptic_k_series(acb_poly_t res, const acb_poly_t z, slong n, slong prec)

Sets res to the complete elliptic integral of the first kind of the power series z, truncated to length n.

void _acb_poly_elliptic_p_series(acb_ptr res, acb_srcptr z, slong zlen, const acb_t tau, slong len, slong prec)

void acb_poly_elliptic_p_series(acb_poly_t res, const acb_poly_t z, const acb_t tau, slong n, slong prec)

Sets res to the Weierstrass elliptic function of the power series z, with periods 1 and tau, truncated to length n.

Root-finding

void _acb_poly_root_bound_fujiwara(mag_t bound, acb_srcptr poly, slong len)

void acb_poly_root_bound_fujiwara(mag_t bound, acb_poly_t poly)

Sets bound to an upper bound for the magnitude of all the complex roots of poly. Uses Fujiwara’s bound

\[2 \max \left\{\left|\frac{a_{n-1}}{a_n}\right|, \left|\frac{a_{n-2}}{a_n}\right|^{1/2}, \cdots, \left|\frac{a_1}{a_n}\right|^{1/(n-1)}, \left|\frac{a_0}{2a_n}\right|^{1/n} \right\}\]

where \(a_0, \ldots, a_n\) are the coefficients of poly.

void _acb_poly_root_inclusion(acb_t r, const acb_t m, acb_srcptr poly, acb_srcptr polyder, slong len, slong prec)

Given any complex number \(m\), and a nonconstant polynomial \(f\) and its derivative \(f'\), sets r to a complex interval centered on \(m\) that is guaranteed to contain at least one root of \(f\). Such an interval is obtained by taking a ball of radius \(|f(m)/f'(m)| n\) where \(n\) is the degree of \(f\). Proof: assume that the distance to the nearest root exceeds \(r = |f(m)/f'(m)| n\). Then

\[\left|\frac{f'(m)}{f(m)}\right| = \left|\sum_i \frac{1}{m-\zeta_i}\right| \le \sum_i \frac{1}{|m-\zeta_i|} < \frac{n}{r} = \left|\frac{f'(m)}{f(m)}\right|\]

which is a contradiction (see [Kob2010]).

slong _acb_poly_validate_roots(acb_ptr roots, acb_srcptr poly, slong len, slong prec)

Given a list of approximate roots of the input polynomial, this function sets a rigorous bounding interval for each root, and determines which roots are isolated from all the other roots. It then rearranges the list of roots so that the isolated roots are at the front of the list, and returns the count of isolated roots.

If the return value equals the degree of the polynomial, then all roots have been found. If the return value is smaller, all the remaining output intervals are guaranteed to contain roots, but it is possible that not all of the polynomial’s roots are contained among them.

void _acb_poly_refine_roots_durand_kerner(acb_ptr roots, acb_srcptr poly, slong len, slong prec)

Refines the given roots simultaneously using a single iteration of the Durand-Kerner method. The radius of each root is set to an approximation of the correction, giving a rough estimate of its error (not a rigorous bound).

slong _acb_poly_find_roots(acb_ptr roots, acb_srcptr poly, acb_srcptr initial, slong len, slong maxiter, slong prec)

slong acb_poly_find_roots(acb_ptr roots, const acb_poly_t poly, acb_srcptr initial, slong maxiter, slong prec)

Attempts to compute all the roots of the given nonzero polynomial poly using a working precision of prec bits. If n denotes the degree of poly, the function writes n approximate roots with rigorous error bounds to the preallocated array roots, and returns the number of roots that are isolated.

If the return value equals the degree of the polynomial, then all roots have been found. If the return value is smaller, all the output intervals are guaranteed to contain roots, but it is possible that not all of the polynomial’s roots are contained among them.

The roots are computed numerically by performing several steps with the Durand-Kerner method and terminating if the estimated accuracy of the roots approaches the working precision or if the number of steps exceeds maxiter, which can be set to zero in order to use a default value. Finally, the approximate roots are validated rigorously.

Initial values for the iteration can be provided as the array initial. If initial is set to NULL, default values \((0.4+0.9i)^k\) are used.

The polynomial is assumed to be squarefree. If there are repeated roots, the iteration is likely to find them (with low numerical accuracy), but the error bounds will not converge as the precision increases.

int _acb_poly_validate_real_roots(acb_srcptr roots, acb_srcptr poly, slong len, slong prec)

int acb_poly_validate_real_roots(acb_srcptr roots, const acb_poly_t poly, slong prec)

Given a strictly real polynomial poly (of length len) and isolating intervals for all its complex roots, determines if all the real roots are separated from the non-real roots. If this function returns nonzero, every root enclosure that touches the real axis (as tested by applying arb_contains_zero() to the imaginary part) corresponds to a real root (its imaginary part can be set to zero), and every other root enclosure corresponds to a non-real root (with known sign for the imaginary part).

If this function returns zero, then the signs of the imaginary parts are not known for certain, based on the accuracy of the inputs and the working precision prec.