harmonics.c File Reference

Calculates the harmonic polynomials. More...

#include "harmonics.h"
#include "hpdyad.h"
#include "stdbool.h"
#include "tools.h"
#include <math.h>

Include dependency graph for harmonics.c:

Functions
double	coeffs_c_inner (long long n, long long i, long long k, long long dim)
	Computes cₙ,ᵢ,ₖ＝ ∏_{j=i+1}^{⌊n/2⌋-k} (2n＋d−2−4k−2j).
double	coeffs_c_outer (long long n, long long k, long long dim)
	Computes cₙ,ₖ＝2⁻ᵏ/ cₙ,₀,ₖ ∏ⱼ₌₁ᵏ(2n＋d−2j)∕((2n＋d＋2−4j)(2n＋d−4j)).
void	coeffs_c_inner_hpdyad (unsigned int n, unsigned int i, unsigned int k, unsigned int dim, hpdyad_t *out)
	Computes cₙ,ᵢ,ₖ＝ ∏_{j=i+1}^{⌊n/2⌋-k} (2n＋d−2−4k−2j) as hpdyad.
void	harmonic_h_inner_term_scalar_hpdyad (unsigned int n, unsigned int i, unsigned int k, unsigned int dim, hpdyad_t *out)
	Computes (-2)**(-i) · c_{n,i,k} · binom(i+k,k) as hpdyad.
void	harmonic_h_inner_term_multi_hpdyad (unsigned int dim, const unsigned int *alpha, const unsigned int *beta, const unsigned int *theta1, const unsigned int *theta2, hpdyad_t *out)
	Computes binom(i,β) × binom(α,θ₁) × θ₂!/(θ₂−θ₁)! as hpdyad.
double	harmonic_h_inner_sum (unsigned int k, unsigned int dim, const unsigned int *alpha, const unsigned int *gamma, unsigned int alphaAbs)
	Computes the inner sum h_inner(α,γ,k) using exact hpdyad arithmetic.
unsigned long long	precompute_harmonic_h_inner_chunk_size (unsigned int alphaAbs, unsigned int kMax, unsigned int dim, const unsigned int *alpha, unsigned long long *chunk_offset, unsigned long long *valid_count)
	Computes chunk offsets and valid entry counts for precomputed inner harmonic sums corresponding to k = 0, ..., floor(|alpha|/2). For each k, chunk_offset[k] stores the starting offset in the coeffs array where values corresponding to |gamma| = |alpha| - k are stored. valid_count[k] stores the number of valid gamma for that k.
void	precompute_harmonic_h_inner_sum (unsigned int alphaAbs, unsigned int dim, const unsigned int *alpha, const unsigned long long *chunk_offset, double *coeffs, unsigned int *exponents)
	Precomputes and stores inner harmonic sums h_inner(α,γ,k) and exponents (2γ-α) for all k = 0, ..., floor(|alpha|/2) and all gamma with |gamma| = |alpha| - k satisfying 2γ[i] >= α[i]. Coefficients are stored in coeffs starting at offsets given by chunk_offset[k]. Exponents are stored in exponents with stride dim, i.e., exponent i for entry n of chunk k is at exponents[(chunk_offset[k] + n) * dim + i]. Gamma is iterated in graded lexicographic order with total degree |alpha| - k.
double	harmonic_h (unsigned int k, unsigned int dim, const double *z, unsigned int alphaAbs, const unsigned long long *chunk_offset, const unsigned long long *valid_count, const double *coeffs, const unsigned int *exponents)
	Calculates the homogeneous harmonic polynomial h₍α,k₎ of degree |α|−2k such that y^α = ∑ₖ (y·y)^k h₍α,k₎(y); explicitly, h₍α,k₎(y)=c_{|α|,k} ∑{|γ|=|α|−k} y^{2γ−α} h_inner(α,γ,k). Uses precomputed coefficients and exponents to avoid multi-index iteration.

Detailed Description

Calculates the harmonic polynomials.

Function Documentation

coeffs_c_inner()

double coeffs_c_inner	(	long long	n,
		long long	i,
		long long	k,
		long long	dim )

Computes cₙ,ᵢ,ₖ＝ ∏_{j=i+1}^{⌊n/2⌋-k} (2n＋d−2−4k−2j).

Parameters

[in]	n	index (total of alpha).
[in]	i	lower bound of product (exclusive).
[in]	k	index (specifies degree |alpha| - 2k).
[in]	dim	dimension of the zeta function inputs.

Returns

cₙ,ᵢ,ₖ.

coeffs_c_inner_hpdyad()

void coeffs_c_inner_hpdyad	(	unsigned int	n,
		unsigned int	i,
		unsigned int	k,
		unsigned int	dim,
		hpdyad_t *	out )

Computes cₙ,ᵢ,ₖ＝ ∏_{j=i+1}^{⌊n/2⌋-k} (2n＋d−2−4k−2j) as hpdyad.

Parameters

[in]	n	index (total of alpha).
[in]	i	lower bound of product (exclusive).
[in]	k	index (specifies degree |alpha| - 2k).
[in]	dim	dimension.
[out]	out	result hpdyad.

coeffs_c_outer()

double coeffs_c_outer	(	long long	n,
		long long	k,
		long long	dim )

Computes cₙ,ₖ＝2⁻ᵏ/ cₙ,₀,ₖ ∏ⱼ₌₁ᵏ(2n＋d−2j)∕((2n＋d＋2−4j)(2n＋d−4j)).

Parameters

[in]	n	index (total of alpha).
[in]	k	index (specifies degree |alpha| - 2k).
[in]	dim	dimension of the zeta function inputs.

Returns

cₙ,ₖ.

harmonic_h()

double harmonic_h	(	unsigned int	k,
		unsigned int	dim,
		const double *	z,
		unsigned int	alphaAbs,
		const unsigned long long *	chunk_offset,
		const unsigned long long *	valid_count,
		const double *	coeffs,
		const unsigned int *	exponents )

Calculates the homogeneous harmonic polynomial h₍α,k₎ of degree |α|−2k such that y^α = ∑ₖ (y·y)^k h₍α,k₎(y); explicitly, h₍α,k₎(y)=c_{|α|,k} ∑{|γ|=|α|−k} y^{2γ−α} h_inner(α,γ,k). Uses precomputed coefficients and exponents to avoid multi-index iteration.

Parameters

[in]	k	specifies degree |alpha| - 2k.
[in]	dim	dimension of alpha, gamma and y.
[in]	z	vector of the polynomial.
[in]	alphaAbs	total of alpha.
[in]	chunk_offset	starting offsets for each k.
[in]	valid_count	number of valid entries for each k.
[in]	coeffs	array storing precomputed inner harmonic sums.
[in]	exponents	array storing precomputed exponents (2γ-α).

Returns

h₍α,k₎(z).

harmonic_h_inner_sum()

double harmonic_h_inner_sum	(	unsigned int	k,
		unsigned int	dim,
		const unsigned int *	alpha,
		const unsigned int *	gamma,
		unsigned int	alphaAbs )

Computes the inner sum h_inner(α,γ,k) using exact hpdyad arithmetic.

Precondition

k ≤ ⌊alphaAbs/2⌋

Parameters

[in]	k	specifies degree |alpha| - 2k.
[in]	dim	dimension of alpha, beta and gamma.
[in]	alpha	upper multi-index.
[in]	gamma	fixed multi-index gamma.
[in]	alphaAbs	total of alpha.

Returns

h_inner(α,γ,k).

harmonic_h_inner_term_multi_hpdyad()

void harmonic_h_inner_term_multi_hpdyad	(	unsigned int	dim,
		const unsigned int *	alpha,
		const unsigned int *	beta,
		const unsigned int *	theta1,
		const unsigned int *	theta2,
		hpdyad_t *	out )

Computes binom(i,β) × binom(α,θ₁) × θ₂!/(θ₂−θ₁)! as hpdyad.

Parameters

[in]	dim	dimension of the multi-indices.
[in]	alpha	upper multi-index α.
[in]	beta	lower multi-index β.
[in]	theta1	multi-index α+β−γ.
[in]	theta2	multi-index γ−β.
[out]	out	result hpdyad (always positive).

harmonic_h_inner_term_scalar_hpdyad()

void harmonic_h_inner_term_scalar_hpdyad	(	unsigned int	n,
		unsigned int	i,
		unsigned int	k,
		unsigned int	dim,
		hpdyad_t *	out )

Computes (-2)**(-i) · c_{n,i,k} · binom(i+k,k) as hpdyad.

Parameters

[in]	n	index (total of alpha).
[in]	i	index (total of beta).
[in]	k	index (specifies degree |alpha| - 2k).
[in]	dim	dimension.
[out]	out	result hpdyad.

precompute_harmonic_h_inner_chunk_size()

unsigned long long precompute_harmonic_h_inner_chunk_size	(	unsigned int	alphaAbs,
		unsigned int	kMax,
		unsigned int	dim,
		const unsigned int *	alpha,
		unsigned long long *	chunk_offset,
		unsigned long long *	valid_count )

Computes chunk offsets and valid entry counts for precomputed inner harmonic sums corresponding to k = 0, ..., floor(|alpha|/2). For each k, chunk_offset[k] stores the starting offset in the coeffs array where values corresponding to |gamma| = |alpha| - k are stored. valid_count[k] stores the number of valid gamma for that k.

Parameters

[in]	alphaAbs	total of alpha.
[in]	kMax	floor(|alpha|/2).
[in]	dim	dimension of alpha and gamma.
[in]	alpha	upper multi-index.
[out]	chunk_offset	array of length kMax+1 storing offsets.
[out]	valid_count	array of length kMax+1 storing valid entry counts.

Returns

total length required for coeffs array.

precompute_harmonic_h_inner_sum()

void precompute_harmonic_h_inner_sum	(	unsigned int	alphaAbs,
		unsigned int	dim,
		const unsigned int *	alpha,
		const unsigned long long *	chunk_offset,
		double *	coeffs,
		unsigned int *	exponents )

Precomputes and stores inner harmonic sums h_inner(α,γ,k) and exponents (2γ-α) for all k = 0, ..., floor(|alpha|/2) and all gamma with |gamma| = |alpha| - k satisfying 2γ[i] >= α[i]. Coefficients are stored in coeffs starting at offsets given by chunk_offset[k]. Exponents are stored in exponents with stride dim, i.e., exponent i for entry n of chunk k is at exponents[(chunk_offset[k] + n) * dim + i]. Gamma is iterated in graded lexicographic order with total degree |alpha| - k.

Precomputes and stores inner harmonic sums h_inner(α,γ,k) and exponents (2γ-α) for all k = 0, ..., floor(|alpha|/2) and all gamma with |gamma| = |alpha| - k satisfying 2γ[i] >= α[i]. Coefficients are stored in coeffs starting at offsets given by chunk_offset[k]. Exponents are stored in exponents with stride dim, i.e., exponent i for entry n of chunk k is at exponents[(chunk_offset[k] + n) * dim + i]. Gamma is ordered identically to harmonic_h_update_outer_index.

Parameters

[in]	alphaAbs	total of alpha.
[in]	dim	dimension of alpha and gamma.
[in]	alpha	upper multi-index.
[in]	chunk_offset	starting offsets for each k.
[out]	coeffs	array storing precomputed inner harmonic sums.
[out]	exponents	array storing precomputed exponents (2γ-α), size = totalSize * dim.