Data structure containing the CArb pointer

Createst a new CArbPoly structure encapsulated in ArbPoly.

Use ArbPoly in f.

Use new ArbPoly ptr in f.

arb_poly_init poly

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

arb_poly_clear poly

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

arb_poly_fit_length poly len

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

_arb_poly_set_length poly len

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

_arb_poly_normalise poly

Strips any trailing coefficients which are identical to zero.

arb_poly_allocated_bytes x

Returns the total number of bytes heap-allocated internally by this object. The count excludes the size of the structure itself. Add sizeof(arb_poly_struct) to get the size of the object as a whole.

arb_poly_length 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.

arb_poly_degree 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.

arb_poly_is_zero poly

arb_poly_is_one poly

arb_poly_is_x poly

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

arb_poly_zero poly

arb_poly_one poly

Sets poly to the constant 0 respectively 1.

arb_poly_set dest src

Sets dest to a copy of src.

arb_poly_set_round dest src prec

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

arb_poly_set_trunc dest src n

arb_poly_set_trunc_round dest src n prec

Sets dest to a copy of src, truncated to length n and rounded to prec bits.

arb_poly_set_coeff_si poly n c

arb_poly_set_coeff_arb poly n c

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

arb_poly_get_coeff_arb v poly n

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

_arb_poly_shift_right res poly len n

arb_poly_shift_right res poly n

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

_arb_poly_shift_left res poly len n

arb_poly_shift_left res poly n

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

arb_poly_truncate poly n

Truncates poly to have length at most n, i.e. degree strictly smaller than n. We require that n is nonnegative.

arb_poly_valuation poly

Returns the degree of the lowest term that is not exactly zero in poly. Returns -1 if poly is the zero polynomial.

arb_poly_set_fmpz_poly poly src prec

arb_poly_set_fmpq_poly poly src prec

arb_poly_set_si poly src

Sets poly to src, rounding the coefficients to prec bits.

arb_poly_printd poly digits

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

arb_poly_fprintd file poly digits

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

arb_poly_randtest poly state len prec mag_bits

Creates a random polynomial with length at most len.

arb_poly_contains poly1 poly2

arb_poly_contains_fmpz_poly poly1 poly2

arb_poly_contains_fmpq_poly poly1 poly2

Returns nonzero iff poly1 contains poly2.

arb_poly_equal A B

Returns nonzero iff A and B are equal as polynomial balls, i.e. all coefficients have equal midpoint and radius.

_arb_poly_overlaps poly1 len1 poly2 len2

arb_poly_overlaps poly1 poly2

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

arb_poly_get_unique_fmpz_poly z 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.

_arb_poly_majorant res poly len prec

arb_poly_majorant res poly 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.

_arb_poly_add C A lenA B lenB prec

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

arb_poly_add C A B prec

arb_poly_add_si C A B prec

Sets C to the sum of A and B.

_arb_poly_sub C A lenA B lenB prec

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

arb_poly_sub C A B prec

Sets C to the difference of A and B.

arb_poly_add_series C A B len prec

Sets C to the sum of A and B, truncated to length len.

arb_poly_sub_series C A B len prec

Sets C to the difference of A and B, truncated to length len.

arb_poly_neg C A

Sets C to the negation of A.

arb_poly_scalar_mul_2exp_si C A c

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

arb_poly_scalar_mul C A c prec

Sets C to A multiplied by c.

arb_poly_scalar_div C A c prec

Sets C to A divided by c.

_arb_poly_mullow_classical C A lenA B lenB n prec

_arb_poly_mullow_block C A lenA B lenB n prec

_arb_poly_mullow C A lenA B lenB n 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. This has good numerical stability but gets slow for large n.

The block version decomposes the product into several subproducts which are computed exactly over the integers.

It first attempts to find an integer \(c\) such that \(A(2^c x)\) and \(B(2^c x)\) have slowly varying coefficients, to reduce the number of blocks.

The scaling factor \(c\) is chosen in a quick, heuristic way by picking the first and last nonzero terms in each polynomial. If the indices in \(A\) are \(a_2, a_1\) and the log-2 magnitudes are \(e_2, e_1\), and the indices in \(B\) are \(b_2, b_1\) with corresponding magnitudes \(f_2, f_1\), then we compute \(c\) as the weighted arithmetic mean of the slopes, rounded to the nearest integer:

\[`\] \[c = \left\lfloor \frac{(e_2 - e_1) + (f_2 + f_1)}{(a_2 - a_1) + (b_2 - b_1)} + \frac{1}{2} \right \rfloor.\]

This strategy is used because it is simple. It is not optimal in all cases, but will typically give good performance when multiplying two power series with a similar decay rate.

The default algorithm chooses the classical algorithm for short polynomials and the block algorithm for long polynomials.

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.

arb_poly_mullow_classical C A B n prec

arb_poly_mullow_block C A B n prec

arb_poly_mullow C A B n 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.

_arb_poly_mul C A lenA B lenB 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 _arb_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.

arb_poly_mul C A B 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.

_arb_poly_inv_series Q A Alen len prec

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

arb_poly_inv_series Q A n prec

Sets Q to the power series inverse of A, truncated to length n.

_arb_poly_div_series Q A Alen B Blen n prec

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

arb_poly_div_series Q A B n prec

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

_arb_poly_div Q A lenA B lenB prec

_arb_poly_rem R A lenA B lenB prec

_arb_poly_divrem Q R A lenA B lenB prec

arb_poly_divrem Q R A B 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.

_arb_poly_div_root Q R A len c prec

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

_arb_poly_taylor_shift g c n prec

arb_poly_taylor_shift g f c prec

Sets g to the Taylor shift \(f(x+c)\). The underscore methods act in-place on g = f which has length n.

_arb_poly_compose res poly1 len1 poly2 len2 prec

arb_poly_compose res poly1 poly2 prec

Sets res to the composition \(h(x) = f(g(x))\) where \(f\) is given by poly1 and \(g\) is given by poly2. The underscore method does not support aliasing of the output with either input polynomial.

_arb_poly_compose_series res poly1 len1 poly2 len2 n prec

arb_poly_compose_series res poly1 poly2 n 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. Wraps _gr_poly_compose_series which chooses automatically between various algorithms.

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

_arb_poly_revert_series_lagrange h f flen n prec

arb_poly_revert_series_lagrange h f n prec

_arb_poly_revert_series_newton h f flen n prec

arb_poly_revert_series_newton h f n prec

_arb_poly_revert_series_lagrange_fast h f flen n prec

arb_poly_revert_series_lagrange_fast h f n prec

_arb_poly_revert_series h f flen n prec

arb_poly_revert_series h f n 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.

_arb_poly_evaluate_horner y f len x prec

arb_poly_evaluate_horner y f x prec

_arb_poly_evaluate_rectangular y f len x prec

arb_poly_evaluate_rectangular y f x prec

_arb_poly_evaluate y f len x prec

arb_poly_evaluate y f x prec

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

_arb_poly_evaluate_acb_horner y f len x prec

arb_poly_evaluate_acb_horner y f x prec

_arb_poly_evaluate_acb_rectangular y f len x prec

arb_poly_evaluate_acb_rectangular y f x prec

_arb_poly_evaluate_acb y f len x prec

arb_poly_evaluate_acb y f x prec

Sets \(y = f(x)\) where \(x\) is a complex number, evaluating the polynomial respectively using Horner's rule, rectangular splitting, and an automatic algorithm choice.

_arb_poly_evaluate2_horner y z f len x prec

arb_poly_evaluate2_horner y z f x prec

_arb_poly_evaluate2_rectangular y z f len x prec

arb_poly_evaluate2_rectangular y z f x prec

_arb_poly_evaluate2 y z f len x prec

arb_poly_evaluate2 y z f x 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.

_arb_poly_evaluate2_acb_horner y z f len x prec

arb_poly_evaluate2_acb_horner y z f x prec

_arb_poly_evaluate2_acb_rectangular y z f len x prec

arb_poly_evaluate2_acb_rectangular y z f x prec

_arb_poly_evaluate2_acb y z f len x prec

arb_poly_evaluate2_acb y z f x prec

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

_arb_poly_product_roots poly xs n prec

arb_poly_product_roots poly xs n prec

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

_arb_poly_product_roots_complex poly r rn c cn prec

arb_poly_product_roots_complex poly r rn c cn prec

Generates the polynomial

\[`\] \[\left(\prod_{i=0}^{rn-1} (x-r_i)\right) \left(\prod_{i=0}^{cn-1} (x-c_i)(x-\bar{c_i})\right)\]

having rn real roots given by the array r and having \(2cn\) complex roots in conjugate pairs given by the length-cn array c. Either rn or cn or both may be zero.

Note that only one representative from each complex conjugate pair is supplied (unless a pair is supposed to be repeated with higher multiplicity). To construct a polynomial from complex roots where the conjugate pairs have not been distinguished, use acb_poly_product_roots instead.

_arb_poly_tree_alloc len

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

_arb_poly_tree_free tree len

Deallocates a tree structure as allocated using _arb_poly_tree_alloc.

_arb_poly_tree_build tree roots len prec

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

_arb_poly_evaluate_vec_iter ys poly plen xs n prec

arb_poly_evaluate_vec_iter ys poly xs n prec

Evaluates the polynomial simultaneously at n given points, calling _arb_poly_evaluate repeatedly.

_arb_poly_evaluate_vec_fast_precomp vs poly plen tree len prec

_arb_poly_evaluate_vec_fast ys poly plen xs n prec

arb_poly_evaluate_vec_fast ys poly xs n prec

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

_arb_poly_interpolate_newton poly xs ys n prec

arb_poly_interpolate_newton poly xs ys n 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.

_arb_poly_interpolate_barycentric poly xs ys n prec

arb_poly_interpolate_barycentric poly xs ys n 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.

_arb_poly_interpolation_weights w tree len prec

_arb_poly_interpolate_fast_precomp poly ys tree weights len prec

_arb_poly_interpolate_fast poly xs ys len prec

arb_poly_interpolate_fast poly xs ys n 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.

_arb_poly_derivative res poly len prec

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

arb_poly_derivative res poly prec

Sets res to the derivative of poly.

_arb_poly_nth_derivative res poly n len prec

Sets {res, len - n} to the nth derivative of {poly, len}. Does nothing if len <= n. Allows aliasing of the input and output.

arb_poly_nth_derivative res poly prec

Sets res to the nth derivative of poly.

_arb_poly_integral res poly len prec

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

arb_poly_integral res poly prec

Sets res to the integral of poly.

_arb_poly_borel_transform res poly len prec

arb_poly_borel_transform res poly prec

Computes the Borel transform of the input polynomial, mapping \(\sum_k a_k x^k\) to \(\sum_k (a_k / k!) x^k\). The underscore method allows aliasing.

_arb_poly_inv_borel_transform res poly len prec

arb_poly_inv_borel_transform res poly prec

Computes the inverse Borel transform of the input polynomial, mapping \(\sum_k a_k x^k\) to \(\sum_k a_k k! x^k\). The underscore method allows aliasing.

_arb_poly_binomial_transform_basecase b a alen len prec

arb_poly_binomial_transform_basecase b a len prec

_arb_poly_binomial_transform_convolution b a alen len prec

arb_poly_binomial_transform_convolution b a len prec

_arb_poly_binomial_transform b a alen len prec

arb_poly_binomial_transform b a len prec

Computes the binomial transform of the input polynomial, truncating the output to length len. The binomial transform maps the coefficients \(a_k\) in the input polynomial to the coefficients \(b_k\) in the output polynomial via \(b_n = \sum_{k=0}^n (-1)^k {n \choose k} a_k\). The binomial transform is equivalent to the power series composition \(f(x) \to (1-x)^{-1} f(x/(x-1))\), and is its own inverse.

The basecase version evaluates coefficients one by one from the definition, generating the binomial coefficients by a recurrence relation.

The convolution version uses the identity \(T(f(x)) = B^{-1}(e^x B(f(-x)))\) where \(T\) denotes the binomial transform operator and \(B\) denotes the Borel transform operator. This only costs a single polynomial multiplication, plus some scalar operations.

The default version automatically chooses an algorithm.

The underscore methods do not support aliasing, and assume that the lengths are nonzero.

_arb_poly_graeffe_transform b a len prec

arb_poly_graeffe_transform b a prec

Computes the Graeffe transform of input polynomial.

The Graeffe transform \(G\) of a polynomial \(P\) is defined through the equation \(G(x^2) = \pm P(x)P(-x)\). The sign is given by \((-1)^d\), where \(d = deg(P)\). The Graeffe transform has the property that its roots are exactly the squares of the roots of P.

The underscore method assumes that a and b are initialized, a is of length len, and b is of length at least len. Both methods allow aliasing.

_arb_poly_pow_ui_trunc_binexp res f flen exp len 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 exponentiation.

arb_poly_pow_ui_trunc_binexp res poly exp len prec

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

_arb_poly_pow_ui res f flen exp 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.

arb_poly_pow_ui res poly exp prec

Sets res to poly raised to the power exp.

_arb_poly_pow_series h f flen g glen len 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.

arb_poly_pow_series h f g len 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.

_arb_poly_pow_arb_series h f flen g len 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.

arb_poly_pow_arb_series h f g len prec

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

_arb_poly_sqrt_series g h hlen n prec

arb_poly_sqrt_series g h n 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.

_arb_poly_rsqrt_series g h hlen n prec

arb_poly_rsqrt_series g h n 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.

_arb_poly_log_series res f flen n prec

arb_poly_log_series res f n 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.

_arb_poly_log1p_series res f flen n prec

arb_poly_log1p_series res f n prec

Computes the power series \(\log(1+f)\), with better accuracy when the constant term of f is small.

_arb_poly_atan_series res f flen n prec

arb_poly_atan_series res f n prec

_arb_poly_asin_series res f flen n prec

arb_poly_asin_series res f n prec

_arb_poly_acos_series res f flen n prec

arb_poly_acos_series res f n prec

Sets res respectively to the power series inverse tangent, inverse sine and inverse cosine of f, truncated to length n.

Uses the formulas

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

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

The underscore methods supports aliasing of the input and output arrays. They require that flen and n are greater than zero.

_arb_poly_exp_series_basecase f h hlen n prec

arb_poly_exp_series_basecase f h n prec

_arb_poly_exp_series f h hlen n prec

arb_poly_exp_series f h n 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.

_arb_poly_sin_cos_series s c h hlen n prec

arb_poly_sin_cos_series s c h n prec

Sets s and c to the power series sine and cosine of h, computed simultaneously. The underscore method supports aliasing and requires the lengths to be nonzero.

_arb_poly_sin_series s h hlen n prec

arb_poly_sin_series s h n prec

_arb_poly_cos_series c h hlen n prec

arb_poly_cos_series c h n prec

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

_arb_poly_tan_series g h hlen len prec

arb_poly_tan_series g h n 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.

_arb_poly_sin_cos_pi_series s c h hlen n prec

arb_poly_sin_cos_pi_series s c h n prec

_arb_poly_sin_pi_series s h hlen n prec

arb_poly_sin_pi_series s h n prec

_arb_poly_cos_pi_series c h hlen n prec

arb_poly_cos_pi_series c h n prec

_arb_poly_cot_pi_series c h hlen n prec

arb_poly_cot_pi_series c h n prec

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

_arb_poly_sinh_cosh_series_basecase s c h hlen n prec

arb_poly_sinh_cosh_series_basecase s c h n prec

_arb_poly_sinh_cosh_series_exponential s c h hlen n prec

arb_poly_sinh_cosh_series_exponential s c h n prec

_arb_poly_sinh_cosh_series s c h hlen n prec

arb_poly_sinh_cosh_series s c h n prec

_arb_poly_sinh_series s h hlen n prec

arb_poly_sinh_series s h n prec

_arb_poly_cosh_series c h hlen n prec

arb_poly_cosh_series c h n 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.

_arb_poly_sinc_series s h hlen n prec

arb_poly_sinc_series s h n prec

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

_arb_poly_sinc_pi_series s h hlen n prec

arb_poly_sinc_pi_series s h n prec

Compute the sinc function of the input multiplied by \(\pi\).

_arb_poly_lambertw_series res z zlen flags len prec

arb_poly_lambertw_series res z flags len prec

Sets res to the Lambert W function of the power series z. If flags is 0, the principal branch is computed; if flags is 1, the second real branch \(W_{-1}(z)\) is computed. The underscore method allows aliasing, but assumes that the lengths are nonzero.

_arb_poly_gamma_series res h hlen n prec

arb_poly_gamma_series res h n prec

_arb_poly_rgamma_series res h hlen n prec

arb_poly_rgamma_series res h n prec

_arb_poly_lgamma_series res h hlen n prec

arb_poly_lgamma_series res h n prec

_arb_poly_digamma_series res h hlen n prec

arb_poly_digamma_series res h n 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 _arb_poly_compose_series. The Taylor coefficients are generated using the Riemann zeta function if the constant term of h is a small integer, and with Stirling's series otherwise.

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

_arb_poly_rising_ui_series res f flen r trunc prec

arb_poly_rising_ui_series res f r trunc 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.

_arb_poly_zeta_series res h hlen a deflate len prec

arb_poly_zeta_series res s a deflate n 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.

_arb_poly_riemann_siegel_theta_series res h hlen n prec

arb_poly_riemann_siegel_theta_series res h n prec

Sets res to the series expansion of the Riemann-Siegel theta function

\[`\] \[\theta(h) = \arg \left(\Gamma\left(\frac{2ih+1}{4}\right)\right) - \frac{\log \pi}{2} h\]

where the argument of the gamma function is chosen continuously as the imaginary part of the log gamma function.

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

_arb_poly_riemann_siegel_z_series res h hlen n prec

arb_poly_riemann_siegel_z_series res h n prec

Sets res to the series expansion of the Riemann-Siegel Z-function

\[`\] \[Z(h) = e^{i\theta(h)} \zeta(1/2+ih).\]

The zeros of the Z-function on the real line precisely correspond to the imaginary parts of the zeros of the Riemann zeta function on the critical line.

The underscore method supports aliasing of the input and output arrays, and requires that the lengths are greater than zero.

_arb_poly_root_bound_fujiwara bound poly len

arb_poly_root_bound_fujiwara bound 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.

_arb_poly_newton_convergence_factor convergence_factor poly len convergence_interval prec

Given an interval \(I\) specified by convergence_interval, evaluates a bound for \(C = \sup_{t,u \in I} \frac{1}{2} |f''(t)| / |f'(u)|\), where \(f\) is the polynomial defined by the coefficients {poly, len}. The bound is obtained by evaluating \(f'(I)\) and \(f''(I)\) directly. If \(f\) has large coefficients, \(I\) must be extremely precise in order to get a finite factor.

_arb_poly_newton_step xnew poly len x convergence_interval convergence_factor prec

Performs a single step with Newton's method.

The input consists of the polynomial \(f\) specified by the coefficients {poly, len}, an interval \(x = [m-r, m+r]\) known to contain a single root of \(f\), an interval \(I\) (convergence_interval) containing \(x\) with an associated bound (convergence_factor) for \(C = \sup_{t,u \in I} \frac{1}{2} |f''(t)| / |f'(u)|\), and a working precision prec.

The Newton update consists of setting \(x' = [m'-r', m'+r']\) where \(m' = m - f(m) / f'(m)\) and \(r' = C r^2\). The expression \(m - f(m) / f'(m)\) is evaluated using ball arithmetic at a working precision of prec bits, and the rounding error during this evaluation is accounted for in the output. We now check that \(x' \in I\) and \(m' < m\). If both conditions are satisfied, we set xnew to \(x'\) and return nonzero. If either condition fails, we set xnew to \(x\) and return zero, indicating that no progress was made.

_arb_poly_newton_refine_root r poly len start convergence_interval convergence_factor eval_extra_prec prec

Refines a precise estimate of a polynomial root to high precision by performing several Newton steps, using nearly optimally chosen doubling precision steps.

The inputs are defined as for _arb_poly_newton_step, except for the precision parameters: prec is the target accuracy and eval_extra_prec is the estimated number of guard bits that need to be added to evaluate the polynomial accurately close to the root (typically, if the polynomial has large coefficients of alternating signs, this needs to be approximately the bit size of the coefficients).

_arb_poly_swinnerton_dyer_ui poly n trunc prec

arb_poly_swinnerton_dyer_ui poly n prec

Computes the Swinnerton-Dyer polynomial \(S_n\), which has degree \(2^n\) and is the rational minimal polynomial of the sum of the square roots of the first n prime numbers.

If prec is set to zero, a precision is chosen automatically such that arb_poly_get_unique_fmpz_poly should be successful. Otherwise a working precision of prec bits is used.

The underscore version accepts an additional trunc parameter. Even when computing a truncated polynomial, the array poly must have room for \(2^n + 1\) coefficients, used as temporary space.