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characters.h
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#ifndef __CHARACTERS_H__
#define __CHARACTERS_H__
#include "slint.h"
#include <iostream>
#include <complex>
#include <cmath>
#include <vector>
#include <set>
#include <fftw3.h>
#ifdef USE_MPFI
#ifdef USE_ARB
#error "Can only use one of MPFI and ARB."
#endif
#include "mpfi_fft.h"
#endif
#ifdef USE_ARB
#include "acb.h"
#endif
inline std::complex<double> e(double z) {
std::complex<double>twopii(0,2*M_PI);
return std::exp(twopii * z);
}
class DirichletGroup;
class DirichletCharacter {
public:
DirichletGroup * parent;
long m; // the label
DirichletCharacter() {}
DirichletCharacter(DirichletGroup * parent_, long m_);
~DirichletCharacter() {}
long exponent(long n);
std::complex<double> max(long* index);
std::complex<double> sum(long end);
std::complex<double> value(long n);
#ifdef USE_ARB
void value(acb_t out, long n);
#endif
bool is_primitive();
bool is_primitive_at_two();
bool is_primitive_at_odd_part();
bool is_even();
void primitive_part_at_two(long * index, long * conductor);
void primitive_part_at_known_p(long * index, long * conductor, long j);
long conductor(long * index = NULL);
std::complex<double> gauss_sum();
void values_mod_p(long &p, long * chi_values);
std::set<long> galois_orbit();
std::complex<double> L_one();
};
class DirichletGroup {
//
//
public:
long q; // the modulus
long q_odd; // the odd part of the modulus
long q_even;
int k; // the number of odd prime factors of q
std::vector<long> * primes;
std::vector<int> * exponents;
std::vector<long> * generators;
long ** A; // exponent vectors:
// for each m coprime to q we store an array
// with the property that
//
// m == g[j]**A[m][k] mod p[j]**e[j]
//
// where g[j] is a primitive root mod p[j]**e[j],
// and p[j] is the j-th prime factor of q.
// (We don't actually store g[k], p[j], or e[j] anywhere.)
long * B;
long * PHI; // PHI[j] = phi(q/phi(p[j]**e[j])). This will make it easier
// to compute the characters.
long phi_q_odd;
long phi_q; // phi(q)
std::complex<double> * zeta_powers_odd; // zeta_powers[n] == e(n/phi(q))
std::complex<double> * zeta_powers_even; // zeta_powers[n] == e(n/phi(q))
long index_from_primitive_character(long q0, long m);
int prec; // working precision when using ARB
#ifdef USE_MPFI
mpfi_c_t * zeta_powers_odd_mpfi;
mpfi_c_t * zeta_powers_even_mpfi;
mpfi_c_t twopii_on_qodd;
mpfi_c_t twopii_on_qeven;
void DFTsum(mpfi_c_t * out, mpfi_c_t * in);
#endif
#ifdef USE_ARB
acb_t * zeta_powers_odd_acb;
acb_t * zeta_powers_even_acb;
acb_t twopii_on_qodd;
acb_t twopii_on_qeven;
#endif
void DFTsum(std::complex<double> * out, std::complex<double> * in);
bool is_coprime_to_q(long n) {
if(q_even > 1 && n % 2 == 0)
return false;
else if(q_odd == 1)
return true;
else
return A[n % q_odd][0] != -1;
}
DirichletGroup() {
q = 0;
}
DirichletGroup(long q_, int prec_ = 0) : q(q_), prec(prec_) {
if(q == 0) return;
q_even = 1;
q_odd = q;
while(q_odd % 2 == 0) {
q_odd /= 2;
q_even *= 2;
}
k = 0;
phi_q = euler_phi(q);
dft_translation_built = false;
if(q_odd > 1) {
primes = new std::vector<long>();
exponents = new std::vector<int>();
generators = new std::vector<long>();
factors(q_odd, primes, exponents);
k = primes->size();
phi_q_odd = euler_phi(q_odd);
PHI = new long[k];
A = new long*[q_odd];
for(int n = 0; n < q_odd; n++) {
A[n] = new long[k];
}
zeta_powers_odd = new std::complex<double>[phi_q_odd];
#ifdef USE_MPFI
zeta_powers_odd_mpfi = new mpfi_c_t[phi_q_odd];
#endif
#ifdef USE_ARB
if(prec > 0) {
zeta_powers_odd_acb = new acb_t[phi_q_odd];
}
#endif
for(long j = 0; j < k; j++) {
long x = pow(primes->at(j), exponents->at(j));
long g = primitive_root(x);
generators->push_back(g);
long phi = (pow(primes->at(j), exponents->at(j) - 1) * (primes->at(j) - 1));
PHI[j] = phi_q_odd/phi;
long a = 1;
for(long l = 0; l < phi; l++) {
for(long m = a; m < q_odd; m += x) {
A[m][j] = l;
}
a = (a * g) % x;
}
}
//
// for each m, 0 <= m < q, if (m,q) > 1, store
// this as a flag in A[m][0]
//
for(long m = 0; m < q_odd; m++) {
if(GCD(m,q_odd) > 1) {
A[m][0] = -1;
}
}
#ifdef USE_MPFI
mpfi_c_init(twopii_on_qodd);
mpfi_c_zero(twopii_on_qodd);
mpfi_const_pi(twopii_on_qodd[0].im);
mpfi_mul_ui(twopii_on_qodd[0].im, twopii_on_qodd[0].im, 2ul);
mpfi_div_ui(twopii_on_qodd[0].im, twopii_on_qodd[0].im, phi_q_odd);
#endif
#ifdef USE_ARB
if(prec > 0) {
acb_init(twopii_on_qodd);
acb_zero(twopii_on_qodd);
arb_const_pi(acb_imagref(twopii_on_qodd), prec);
acb_mul_ui(twopii_on_qodd, twopii_on_qodd, 2ul, prec);
acb_div_ui(twopii_on_qodd, twopii_on_qodd, phi_q_odd, prec);
}
#endif
for(unsigned long n = 0; n < phi_q_odd; n++) {
zeta_powers_odd[n] = e(n/(double)phi_q_odd);
#ifdef USE_MPFI
mpfi_c_init(zeta_powers_odd_mpfi[n]);
mpfi_c_mul_ui(zeta_powers_odd_mpfi[n], twopii_on_qodd, n);
mpfi_c_exp(zeta_powers_odd_mpfi[n], zeta_powers_odd_mpfi[n]);
#endif
#ifdef USE_ARB
if(prec > 0) {
acb_init(zeta_powers_odd_acb[n]);
acb_mul_ui(zeta_powers_odd_acb[n], twopii_on_qodd, n, prec);
acb_exp(zeta_powers_odd_acb[n], zeta_powers_odd_acb[n], prec);
}
#endif
}
} // end of initialization of everything having to do
// with q_odd
else {
phi_q_odd = 1;
}
if(q_even > 4) {
B = new long[q_even];
zeta_powers_even = new std::complex<double>[q_even/4];
#ifdef USE_MPFI
zeta_powers_even_mpfi = new mpfi_c_t[q_even/4];
mpfi_c_init(twopii_on_qeven);
mpfi_c_zero(twopii_on_qeven);
mpfi_const_pi(twopii_on_qeven[0].im);
mpfi_mul_ui(twopii_on_qeven[0].im, twopii_on_qeven[0].im, 2ul);
mpfi_div_ui(twopii_on_qeven[0].im, twopii_on_qeven[0].im, q_even/4);
#endif
#ifdef USE_ARB
if(prec > 0) {
zeta_powers_even_acb = new acb_t[q_even/4];
acb_init(twopii_on_qeven);
acb_zero(twopii_on_qeven);
arb_const_pi(acb_imagref(twopii_on_qeven), prec);
acb_mul_ui(twopii_on_qeven, twopii_on_qeven, 2ul, prec);
acb_div_ui(twopii_on_qeven, twopii_on_qeven, q_even/4, prec);
}
#endif
for(unsigned long n = 0; n < q_even/4; n++) {
zeta_powers_even[n] = e(4*n/double(q_even));
#ifdef USE_MPFI
mpfi_c_init(zeta_powers_even_mpfi[n]);
mpfi_c_mul_ui(zeta_powers_even_mpfi[n], twopii_on_qeven, n);
mpfi_c_exp(zeta_powers_even_mpfi[n], zeta_powers_even_mpfi[n]);
#endif
#ifdef USE_ARB
if(prec > 0) {
acb_init(zeta_powers_even_acb[n]);
acb_mul_ui(zeta_powers_even_acb[n], twopii_on_qeven, n, prec);
acb_exp(zeta_powers_even_acb[n], zeta_powers_even_acb[n], prec);
}
#endif
}
long pow_five = 1;
for(long e = 0; e < q_even/4; e++) {
B[pow_five] = e;
B[pow_five - 1] = 1;
B[q_even - pow_five] = e;
B[q_even - pow_five - 1] = -1;
pow_five *= 5;
pow_five %= q_even;
}
}
}
~DirichletGroup() {
if(q == 0) return;
if(q_odd > 1) {
delete [] zeta_powers_odd;
for(int n = 0; n < q_odd; n++) {
delete [] A[n];
}
#ifdef USE_MPFI
for(int n = 0; n < phi_q_odd; n++) {
mpfi_c_clear(zeta_powers_odd_mpfi[n]);
}
delete [] zeta_powers_odd_mpfi;
mpfi_c_clear(twopii_on_qodd);
#endif
#ifdef USE_ARB
if(prec > 0) {
for(int n = 0; n < phi_q_odd; n++) {
acb_clear(zeta_powers_odd_acb[n]);
}
delete [] zeta_powers_odd_acb;
acb_clear(twopii_on_qodd);
}
#endif
delete [] A;
delete [] PHI;
delete primes;
delete exponents;
delete generators;
}
if(q_even > 4) {
delete [] B;
delete [] zeta_powers_even;
#ifdef USE_MPFI
for(int n = 0; n < q_even/4; n++) {
mpfi_c_clear(zeta_powers_even_mpfi[n]);
}
delete [] zeta_powers_even_mpfi;
mpfi_c_clear(twopii_on_qeven);
#endif
#ifdef USE_ARB
if(prec > 0) {
for(int n = 0; n < q_even/4; n++) {
acb_clear(zeta_powers_even_acb[n]);
}
delete [] zeta_powers_even_acb;
acb_clear(twopii_on_qeven);
}
#endif
}
if(dft_translation_built) {
delete [] dft_translation;
delete [] idft_translation;
delete [] dft_lengths;
}
}
long chi_odd_exponent(long m, long n) {
long x = 0;
for(int j = 0; j < k; j++) {
x += A[m][j]*A[n][j]*PHI[j];
//x = x % phi_q_odd;
if(x > 4294967296)
x = x % phi_q_odd;
}
if(x >= phi_q_odd)
x = x % phi_q_odd;
return x;
}
long chi_even_exponent(long m, long n) {
long x = B[m]*B[n];
if(B[m-1] == -1 && B[n-1] == -1)
x += q_even/8;
return x % (q_even/4);
}
std::complex<double> chi(long m, long n) {
std::complex<double> even_part = 1.0;
std::complex<double> odd_part = 1.0;
if(q_even > 1) {
if(m % 2 == 0 || n % 2 == 0) {
return 0;
}
else if(q_even == 2) even_part = 1.0;
else if(q_even == 4) {
if(m % 4 == 3 && n % 4 == 3) even_part = -1.0;
else even_part = 1.0;
}
else {
even_part = zeta_powers_even[chi_even_exponent(m % q_even, n % q_even)];
}
}
if(m >= q_odd)
m %= q_odd;
if(n >= q_odd)
n %= q_odd;
if(q_odd > 1) {
if(A[m][0] == -1 || A[n][0] == -1)
return 0;
else
odd_part = zeta_powers_odd[chi_odd_exponent(m, n)];
if(q_even == 1)
return odd_part;
}
return odd_part * even_part;
}
long exponent(long m, long n) {
// return the a such that chi(m,n) == e(a/phi(q))
long exponent;
long odd_exponent = 0;
if(q_odd > 1) {
odd_exponent = chi_odd_exponent(m % q_odd, n % q_odd);
}
long even_exponent = 0;
if(q_even > 4) {
even_exponent = chi_even_exponent(m % q_even, n % q_even);
even_exponent *= 2;
// the function just above computes the exponent of
// e(1/ (q_even/4) ), but we want the exponent of
// e(1/phi(q_even)) = e(1/(q_even/2))
}
else if(q_even == 4) {
if(m % q_even == 3 && n % q_even == 3) {
even_exponent = 1;
}
else {
even_exponent = 0;
}
}
if(q_even == 1) { // special case because phi(1) != 1/2.
exponent = odd_exponent;
}
else {
exponent = odd_exponent * q_even/2 + even_exponent * phi_q_odd;
}
// we now have the value of chi(m) as e(exponent/phi(q))
// it could be equal to phi(q), though, and in that case we
// want it to be zero...
if(exponent == phi_q) {
exponent -= phi_q;
}
return exponent;
}
#ifdef USE_MPFI
void chi(mpfi_c_t out, long m, long n) {
if(q_even > 1) {
if(m % 2 == 0 || n % 2 == 0) {
mpfi_c_zero(out);
return;
}
else if(q_even == 2) {
mpfi_c_set_ui(out, 1, 0);
}
else if(q_even == 4) {
if(m % 4 == 3 && n % 4 == 3) {
mpfi_set_si(out[0].re, -1);
mpfi_set_si(out[0].im, 0);
}
else {
mpfi_c_set_ui(out, 1, 0);
}
}
else {
mpfi_c_set(out, zeta_powers_even_mpfi[chi_even_exponent(m % q_even, n % q_even)]);
}
}
else mpfi_c_set_ui(out, 1, 0);
if(m >= q_odd)
m %= q_odd;
if(n >= q_odd);
n %= q_odd;
if(q_odd > 1) {
if(A[m][0] == -1 || A[n][0] == -1)
mpfi_c_zero(out);
else
mpfi_c_mul(out, out, zeta_powers_odd_mpfi[chi_odd_exponent(m, n)]);
}
}
#endif
#ifdef USE_ARB
void chi(acb_t out, long m, long n) {
if(q_even > 1) {
if(m % 2 == 0 || n % 2 == 0) {
acb_zero(out);
return;
}
else if(q_even == 2) {
acb_set_ui(out, 1);
}
else if(q_even == 4) {
if(m % 4 == 3 && n % 4 == 3) {
acb_set_si(out, -1);
//arb_set_si(acb_realref(out), -1, prec);
//arb_set_si(acb_imagref(out), 0, prec);
//mpfi_set_si(out[0].im, 0);
}
else {
acb_set_ui(out, 1);
//mpfi_c_set_ui(out, 1, 0);
}
}
else {
acb_set(out, zeta_powers_even_acb[chi_even_exponent(m % q_even, n % q_even)]);
//mpfi_c_set(out, zeta_powers_even_mpfi[chi_even_exponent(m % q_even, n % q_even)]);
}
}
//else mpfi_c_set_ui(out, 1, 0);
else acb_set_ui(out, 1);
if(m >= q_odd)
m %= q_odd;
if(n >= q_odd)
n %= q_odd;
if(q_odd > 1) {
if(A[m][0] == -1 || A[n][0] == -1)
acb_zero(out);
else
//mpfi_c_mul(out, out, zeta_powers_odd_mpfi[chi_odd_exponent(m, n)]);
acb_mul(out, out, zeta_powers_odd_acb[chi_odd_exponent(m, n)], prec);
}
}
#endif
DirichletCharacter character(long m) {
return DirichletCharacter(this, m);
}
void DFTsum_direct (std::complex<double> * out, std::complex<double> * in) {
//
// Set out[n] to the sum
//
// sum_{k=1}^{q-1} in[k] chi(n,k)
//
// (computed naively, for testing purposes, or because
// something better hasn't been implemented yet.)
for(int n = 0; n < q; n++) {
std::complex<double> S = 0.0;
for(int k = 0; k < q; k++) {
S += in[k] * chi(n,k);
}
out[n] = S;
}
}
#ifdef USE_MPFI
void DFTsum_direct (mpfi_c_t * out, unsigned long * in) {
mpfi_c_t x, y;
mpfi_c_init(x);
mpfi_c_init(y);
for(int n = 0; n < q; n++) {
mpfi_c_zero(out[n]);
if(!is_coprime_to_q(n)) continue;
for(int k = 0; k < q; k++) {
if(!is_coprime_to_q(k)) continue;
chi(y, n, k);
mpfi_c_mul_ui(x, y, in[k]);
mpfi_c_add(out[n], out[n], x);
}
}
mpfi_c_clear(x);
mpfi_c_clear(y);
}
#endif
void all_sums(std::complex<double> * out, long end) {
std::complex<double> * dft_in = new std::complex<double>[q]();
for(long k = 0; k <= end; k++) {
dft_in[k] = 1.0;
}
DFTsum(out, dft_in);
delete [] dft_in;
}
#ifdef USE_MPFI
void all_sums(mpfi_c_t * out, long end) {
mpfi_c_t * dft_in = new mpfi_c_t[q];
for(long k = 0; k < q; k++) {
mpfi_c_init(dft_in[k]);
if(k <= end) {
mpfi_c_set_ui(dft_in[k], 1ul, 0ul);
}
else {
mpfi_c_zero(dft_in[k]);
}
}
DFTsum(out, dft_in);
for(int k = 0; k < q; k++) {
mpfi_c_clear(dft_in[k]);
}
delete [] dft_in;
}
#endif
int * dft_translation;
int * idft_translation;
int * dft_lengths;
int dft_dimension;
bool dft_translation_built;
void build_dft_translation(bool force=false) {
if(dft_translation_built && !force) return;
if(dft_translation_built && force) {
delete [] dft_translation;
delete [] idft_translation;
delete [] dft_lengths;
}
int even_dimension;
if(q_even <= 2) even_dimension = 0;
else if(q_even == 4) even_dimension = 1;
else even_dimension = 2;
dft_dimension = k + even_dimension;
dft_lengths = new int[dft_dimension];
if(even_dimension >= 1)
dft_lengths[0] = 2;
if(even_dimension == 2)
dft_lengths[1] = q_even/4;
for(int n = 0; n < k; n++) {
long p = primes->at(n);
long e = exponents->at(n);
dft_lengths[n+even_dimension] = (p - 1)*pow(p, e-1);
}
dft_translation = new int[q];
idft_translation = new int[q];
int dlog;
// m == g[j]**A[m][k] mod p[j]**e[j]
for(int n = 0; n < q; n++) {
if((q_even > 1 && n % 2 == 0) || (q_odd > 1 && A[n % q_odd][0] == -1)) {
dft_translation[n] = -1;
idft_translation[n] = -1;
}
else {
int dft_index = 0;
int idft_index = 0;
if(even_dimension >= 1) {
idft_index += (n % 4 == 3);
dft_index += (n % 4 == 3);
if(dft_dimension > 1) {
idft_index *= dft_lengths[1];
dft_index *= dft_lengths[1];
}
}
if(even_dimension == 2) {
dlog = B[n % q_even];
idft_index += dlog;
if(dlog > 0)
dft_index += (q_even/4 - B[n % q_even]);
if(dft_dimension > 2) {
idft_index *= dft_lengths[2];
dft_index *= dft_lengths[2];
}
}
for(int j = 0; j < k; j++) {
dlog = A[n % q_odd][j];
idft_index += dlog;
if(dlog > 0)
dft_index += (dft_lengths[j + even_dimension] - A[n % q_odd][j]);
if(j < k-1) {
idft_index *= dft_lengths[j + even_dimension + 1];
dft_index *= dft_lengths[j + even_dimension + 1];
}
}
dft_translation[n] = dft_index;
idft_translation[n] = idft_index;
}
}
dft_translation_built = true;
}
void print_dft_translation() {
build_dft_translation();
for(int n = 0; n < q; n++) {
if(idft_translation[n] != -1) {
std::cout << n << " " << idft_translation[n] << std::endl;
}
}
}
std::set< std::set<long> > galois_orbits() {
std::set< std::set<long> > orbits;
std::set<long> not_in_orbit_yet;
for(long n = 1; n < q; n++) {
if(GCD(n, q) == 1) not_in_orbit_yet.insert(n);
}
if(q == 1) not_in_orbit_yet.insert(1);
while(not_in_orbit_yet.size() > 0) {
long next = *(not_in_orbit_yet.begin());
auto o = character(next).galois_orbit();
orbits.insert(o);
for(long n : o) {
not_in_orbit_yet.erase(n);
}
}
return orbits;
}
std::vector< std::set<long> > ordered_galois_orbits() {
std::vector< std::set<long> > orbits;
std::vector< long > orders;
std::set<long> not_in_orbit_yet;
for(long n = 1; n < q; n++) {
if(GCD(n, q) == 1) not_in_orbit_yet.insert(n);
}
if(q == 1) not_in_orbit_yet.insert(1);
while(not_in_orbit_yet.size() > 0) {
long next = *(not_in_orbit_yet.begin());
orders.push_back(order_mod(next, q));
auto o = character(next).galois_orbit();
orbits.push_back(o);
for(long n : o) {
not_in_orbit_yet.erase(n);
}
}
// now we order the orbits.
for(int k1 = 0; k1 < orbits.size(); k1++) {
for(int k2 = k1 + 1; k2 < orbits.size(); k2++) {
if(orders[k1] > orders[k2]) {
long z = orders[k1];
orders[k1] = orders[k2];
orders[k2] = z;
orbits[k1].swap(orbits[k2]);
}
else if(orders[k1] == orders[k2]) {
long t1 = 0; long t2 = 0;
long k = 2;
while(t1 == t2 && (k < q)) {
std::complex<double> z1 = 0;
for(long n : orbits[k1]) {
z1 = z1 + chi(n,k);
}
std::complex<double> z2 = 0;
for(long n : orbits[k2]) {
z2 = z2 + chi(n,k);
}
t1 = (long)std::round(z1.real());
t1 = (t1 * q)/(long)orbits[k1].size();
t2 = (long)std::round(z2.real());
t2 = (t2 * q)/(long)orbits[k2].size();
k++;
while(GCD(k,q) != 1) k++;
}
if(t1 == t2) {
std::cout << "Warning. Can't distinguish character orbits. Didn't expect this to happen" << std::endl;
std::cerr << "Warning. Can't distinguish character orbits. Didn't expect this to happen" << std::endl;
}
if(t2 < t1) {
orbits[k1].swap(orbits[k2]);
}
}
}
}
return orbits;
}
};
inline long DirichletGroup::index_from_primitive_character(long q0, long primitive_index) {
//
// Given a conductor q0 dividing q and an index of a character
// mod q0, return the index of the character mod q which is the character
// induced by chi_q0(m, -)
//
if(q0 == 1) return 1;
if(primitive_index == 1) return 1;
if(q % q0 != 0) return -1;
std::vector<long> moduli; // We're going to do this for each prime power
std::vector<long> local_indices; // dividing the modulus, and then CRT the results
// together.
for(int j = 0; j < k; j++) {
long p = (*primes)[j];
long modulus_at_p = ipow(p, (*exponents)[j]);
moduli.push_back(modulus_at_p);
long conductor_at_p = 1;
long qq = q0;
while(qq % p == 0) {
conductor_at_p *= p;
qq = qq/p;
}
long primitive_index_at_p = primitive_index % conductor_at_p;
long index_at_p;
if(conductor_at_p == 1) index_at_p = 1;
else {
index_at_p = PowerMod(primitive_index_at_p, modulus_at_p/conductor_at_p, modulus_at_p);
}
local_indices.push_back(index_at_p);
}
// now we need to deal with two
// At 2, we can do almost the same thing, except that we also need to deal
// with +- 1.
long conductor_at_two = 1;
long modulus_at_two = q_even;
long qq = q0;
while(qq % 2 == 0) {
conductor_at_two *= 2;
qq = qq/2;
}
long primitive_index_at_two = primitive_index % conductor_at_two;
long index_at_two;
if(conductor_at_two == 1) index_at_two = 1;
else {
index_at_two = PowerMod(primitive_index_at_two, modulus_at_two/conductor_at_two, modulus_at_two);
// This is right up to a sign of +- 1. We need to adjust to make sure
// that the indices are the same mod 4
if(index_at_two % 4 != primitive_index_at_two % 4) {
index_at_two = modulus_at_two - index_at_two;
}
}
moduli.push_back(modulus_at_two);
local_indices.push_back(index_at_two);
return CRT(local_indices, moduli);
}
inline void DirichletGroup::DFTsum (std::complex<double> * out, std::complex<double> * in) {
if(q < 4) {
// just don't want to deal with problems with 2 right now.
DFTsum_direct(out, in);
return;
}
build_dft_translation();
std::complex<double> *a, *X;
a = new std::complex<double>[phi_q];
X = new std::complex<double>[phi_q];
for(int n = 0; n < q; n++) {
if(dft_translation[n] == -1) continue;
else a[dft_translation[n]] = in[n];
}
fftw_plan plan = fftw_plan_dft(dft_dimension, dft_lengths,
(fftw_complex *)a,
(fftw_complex *)X,
FFTW_FORWARD, FFTW_ESTIMATE);
fftw_execute(plan);
for(int n = 0; n < q; n++) {
if(idft_translation[n] == -1) out[n] = 0.0;
else out[n] = X[idft_translation[n]];
}
fftw_destroy_plan(plan);
delete [] a;
delete [] X;
}
#ifdef USE_MPFI
inline void DirichletGroup::DFTsum (mpfi_c_t * out, mpfi_c_t * in) {
//if(q < 4) {
// // just don't want to deal with problems with 2 right now.
// DFTsum_direct(out, in);
// return;
//}
build_dft_translation();
mpfi_c_t *a;//, *X;
a = new mpfi_c_t[phi_q];
//X = new mpfi_c_t[phi_q];
for(int n = 0; n < phi_q; n++) {
mpfi_c_init(a[n]);
//mpfi_c_init(X[n]);
}
for(int n = 0; n < q; n++) {
// This might be inefficient.
// Since the mpfi_c array is really just an array of pointers,
// we might be able to just copy the pointers without copying
// the data. The fft wouldn't be operating on continuous memory, then,
// but it still might be better than this copy.
if(dft_translation[n] == -1) continue;
else mpfi_c_set(a[dft_translation[n]], in[n]);
}
unsigned long * lengths = new unsigned long[dft_dimension];
for(int n = 0; n < dft_dimension; n++) lengths[n] = dft_lengths[n];
ndft(a, phi_q, dft_dimension, lengths);
//fftw_plan plan = fftw_plan_dft(dft_dimension, dft_lengths,
// (fftw_complex *)a,
// (fftw_complex *)X,
// FFTW_BACKWARD, FFTW_ESTIMATE);
for(int n = 0; n < q; n++) {
if(idft_translation[n] == -1) mpfi_c_zero(out[n]);
else mpfi_c_set(out[n], a[idft_translation[n]]);
}
for(int n = 0; n < phi_q; n++) {
mpfi_c_clear(a[n]);
}
delete [] a;
}
#endif
inline DirichletCharacter::DirichletCharacter(DirichletGroup * parent_, long m_) : parent(parent_), m(m_) {
//
// This doesn't actually do anything. Computing characters
// using DirichletCharacter is not going to be any faster
// than using DirichletGroup::chi(), at least for now.
//
}
inline std::complex<double> DirichletCharacter::gauss_sum() {
std::complex<double> S = 0;
long q = parent->q;
std::complex<double> z = e(1.0/q);
std::complex<double> x = z;
for(long n = 1; n < q; n++) {
S = S + value(n) * x;
x = x * z;
}
return S;
}
inline std::complex<double> DirichletCharacter::max(long * index) {
// return the max of the partial sums of the character,
// and set *index to the spot where the max occurs,
// unless index = 0
//
// note that *index is always <= (q-1)/2
//if(parent->A[m][0] == -1) {
// if(index != 0){
// *index = -1;
// }
// return 0;
//}
std::complex<double> S(0,0);
double absmax = 0.0;
std::complex<double> current_max = 0.0;
long max_location = 0;
for(long n = 0; n <= (parent->q-1)/2; n++) {
S = S + value(n);
if(abs(S) > absmax) {
absmax = abs(S);
current_max = S;
max_location = n;
}
}
if(index != 0) {
*index = max_location;
}
return current_max;
}
inline std::complex<double> DirichletCharacter::sum(long end) {
std::complex<double> S(0,0);
for(long n = 0; n <= end; n++) {
S = S + value(n);
}
return S;
}
// Is this right...?
//
//long DirichletCharacter::exponent(long n) {
// long x = 0;