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DST.cpp
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/**
* \file DST.cpp
* \brief Implementation for GeographicLib::DST class
*
* Copyright (c) Charles Karney (2022) <karney@alum.mit.edu> and licensed under
* the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#include <GeographicLib/DST.hpp>
#include <vector>
#include "kissfft.hh"
namespace GeographicLib {
using namespace std;
DST::DST(int N)
: _nN(N < 0 ? 0 : N)
, _fft(make_shared<fft_t>(fft_t(2 * _nN, false)))
{}
void DST::reset(int N) {
N = N < 0 ? 0 : N;
if (N == _nN) return;
_nN = N;
_fft->assign(2 * _nN, false);
}
void DST::fft_transform(real data[], real F[], bool centerp) const {
// Implement DST-III (centerp = false) or DST-IV (centerp = true).
// Elements (0,N], resp. [0,N), of data should be set on input for centerp
// = false, resp. true. F must have a size of at least N and on output
// elements [0,N) of F contain the transform.
if (_nN == 0) return;
if (centerp) {
for (int i = 0; i < _nN; ++i) {
data[_nN+i] = data[_nN-1-i];
data[2*_nN+i] = -data[i];
data[3*_nN+i] = -data[_nN-1-i];
}
} else {
data[0] = 0; // set [0]
for (int i = 1; i < _nN; ++i)
data[_nN+i] = data[_nN-i]; // set [N+1,2*N-1]
for (int i = 0; i < 2*_nN; ++i)
data[2*_nN+i] = -data[i]; // [2*N, 4*N-1]
}
vector<complex<real>> ctemp(2*_nN);
_fft->transform_real(data, ctemp.data());
if (centerp) {
real d = -Math::pi()/(4*_nN);
for (int i = 0, j = 1; i < _nN; ++i, j+=2)
ctemp[j] *= exp(complex<real>(0, j*d));
}
for (int i = 0, j = 1; i < _nN; ++i, j+=2) {
F[i] = -ctemp[j].imag() / (2*_nN);
}
}
void DST::fft_transform2(real data[], real F[]) const {
// Elements [0,N), of data should be set to the N grid center values and F
// should have size of at least 2*N. On input elements [0,N) of F contain
// the size N transform; on output elements [0,2*N) of F contain the size
// 2*N transform.
fft_transform(data, F+_nN, true);
// Copy DST-IV order N tx to [0,N) elements of data
for (int i = 0; i < _nN; ++i) data[i] = F[i+_nN];
for (int i = _nN; i < 2*_nN; ++i)
// (DST-IV order N - DST-III order N) / 2
F[i] = (data[2*_nN-1-i] - F[2*_nN-1-i])/2;
for (int i = 0; i < _nN; ++i)
// (DST-IV order N + DST-III order N) / 2
F[i] = (data[i] + F[i])/2;
}
void DST::transform(function<real(real)> f, real F[]) const {
vector<real> data(4 * _nN);
real d = Math::pi()/(2 * _nN);
for (int i = 1; i <= _nN; ++i)
data[i] = f( i * d );
fft_transform(data.data(), F, false);
}
void DST::refine(function<real(real)> f, real F[]) const {
vector<real> data(4 * _nN);
real d = Math::pi()/(4 * _nN);
for (int i = 0; i < _nN; ++i)
data[i] = f( (2*i + 1) * d );
fft_transform2(data.data(), F);
}
Math::real DST::eval(real sinx, real cosx, const real F[], int N) {
// Evaluate
// y = sum(F[i] * sin((2*i+1) * x), i, 0, N-1)
// using Clenshaw summation.
// Approx operation count = (N + 5) mult and (2 * N + 2) add
real
ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
y0 = N & 1 ? F[--N] : 0, y1 = 0; // accumulators for sum
// Now N is even
while (N > 0) {
// Unroll loop x 2, so accumulators return to their original role
y1 = ar * y0 - y1 + F[--N];
y0 = ar * y1 - y0 + F[--N];
}
return sinx * (y0 + y1); // sin(x) * (y0 + y1)
}
Math::real DST::integral(real sinx, real cosx, const real F[], int N) {
// Evaluate
// y = -sum(F[i]/(2*i+1) * cos((2*i+1) * x), i, 0, N-1)
// using Clenshaw summation.
// Approx operation count = (N + 5) mult and (2 * N + 2) add
real
ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
y0 = 0, y1 = 0; // accumulators for sum
for (--N; N >= 0; --N) {
real t = ar * y0 - y1 + F[N]/(2*N+1);
y1 = y0; y0 = t;
}
return cosx * (y1 - y0); // cos(x) * (y1 - y0)
}
Math::real DST::integral(real sinx, real cosx, real siny, real cosy,
const real F[], int N) {
// return integral(siny, cosy, F, N) - integral(sinx, cosx, F, N);
real
// 2*cos(y-x)*cos(y+x) -> 2 * cos(2 * x)
ac = +2 * (cosy * cosx + siny * sinx) * (cosy * cosx - siny * sinx),
// -2*sin(y-x)*sin(y+x) -> 0
as = -2 * (siny * cosx - cosy * sinx) * (siny * cosx + cosy * sinx),
y0 = 0, y1 = 0, z0 = 0, z1 = 0; // accumulators for sum
for (--N; N >= 0; --N) {
real
ty = ac * y0 + as * z0 - y1 + F[N]/(2*N+1),
tz = as * y0 + ac * z0 - z1;
y1 = y0; y0 = ty;
z1 = z0; z0 = tz;
}
// B[0] - B[1] = [y0-y1, z0-z1]
// F[0] = [cosy + cosx, cosy - cosx] -> [2 * cosx, 0]
// (B[0] - B[1]) . F[0]
// = [(y0 - y1) * (cosy + cosx) + (z0 - z1) * (cosy - cosx),
// (y0 - y1) * (cosy - cosx) + (z0 - z1) * (cosy + cosx),
// return -(2nd element)
return (y1 - y0) * (cosy - cosx) + (z1 - z0) * (cosy + cosx);
}
} // namespace GeographicLib
DST.hpp
Header for GeographicLib::DST class.
real
GeographicLib::Math::real real
Definition GeodSolve.cpp:28
GeographicLib::DST::reset
void reset(int N)
Definition DST.cpp:24
GeographicLib::DST::transform
void transform(std::function< real(real)> f, real F[]) const
Definition DST.cpp:79
GeographicLib::DST::DST
DST(int N=0)
Definition DST.cpp:19
GeographicLib::DST::eval
static real eval(real sinx, real cosx, const real F[], int N)
Definition DST.cpp:95
GeographicLib::DST::N
int N() const
Definition DST.hpp:93
GeographicLib::DST::refine
void refine(std::function< real(real)> f, real F[]) const
Definition DST.cpp:87
GeographicLib::DST::integral
static real integral(real sinx, real cosx, const real F[], int N)
Definition DST.cpp:112
GeographicLib::Math::pi
static T pi()
Definition Math.hpp:199
GeographicLib::Math::real
double real
Definition Math.hpp:110
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12
std
Definition NearestNeighbor.hpp:806