GeographicLib 2.3

DST.cpp
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1/**
2 * \file DST.cpp
3 * \brief Implementation for GeographicLib::DST class
4 *
5 * Copyright (c) Charles Karney (2022) <karney@alum.mit.edu> and licensed under
6 * the MIT/X11 License. For more information, see
7 * https://geographiclib.sourceforge.io/
8 **********************************************************************/
9
10#include <GeographicLib/DST.hpp>
11#include <vector>
12
13#include "kissfft.hh"
14
15 namespace GeographicLib {
16
17 using namespace std;
18
19 DST::DST(int N)
20 : _N(N < 0 ? 0 : N)
21 , _fft(make_shared<fft_t>(fft_t(2 * _N, false)))
22 {}
23
24 void DST::reset(int N) {
25 N = N < 0 ? 0 : N;
26 if (N == _N) return;
27 _N = N;
28 _fft->assign(2 * _N, false);
29 }
30
31 void DST::fft_transform(real data[], real F[], bool centerp) const {
32 // Implement DST-III (centerp = false) or DST-IV (centerp = true).
33
34 // Elements (0,N], resp. [0,N), of data should be set on input for centerp
35 // = false, resp. true. F must have a size of at least N and on output
36 // elements [0,N) of F contain the transform.
37 if (_N == 0) return;
38 if (centerp) {
39 for (int i = 0; i < _N; ++i) {
40 data[_N+i] = data[_N-1-i];
41 data[2*_N+i] = -data[i];
42 data[3*_N+i] = -data[_N-1-i];
43 }
44 } else {
45 data[0] = 0; // set [0]
46 for (int i = 1; i < _N; ++i) data[_N+i] = data[_N-i]; // set [N+1,2*N-1]
47 for (int i = 0; i < 2*_N; ++i) data[2*_N+i] = -data[i]; // [2*N, 4*N-1]
48 }
49 vector<complex<real>> ctemp(2*_N);
50 _fft->transform_real(data, ctemp.data());
51 if (centerp) {
52 real d = -Math::pi()/(4*_N);
53 for (int i = 0, j = 1; i < _N; ++i, j+=2)
54 ctemp[j] *= exp(complex<real>(0, j*d));
55 }
56 for (int i = 0, j = 1; i < _N; ++i, j+=2) {
57 F[i] = -ctemp[j].imag() / (2*_N);
58 }
59 }
60
61 void DST::fft_transform2(real data[], real F[]) const {
62 // Elements [0,N), of data should be set to the N grid center values and F
63 // should have size of at least 2*N. On input elements [0,N) of F contain
64 // the size N transform; on output elements [0,2*N) of F contain the size
65 // 2*N transform.
66 fft_transform(data, F+_N, true);
67 // Copy DST-IV order N tx to [0,N) elements of data
68 for (int i = 0; i < _N; ++i) data[i] = F[i+_N];
69 for (int i = _N; i < 2*_N; ++i)
70 // (DST-IV order N - DST-III order N) / 2
71 F[i] = (data[2*_N-1-i] - F[2*_N-1-i])/2;
72 for (int i = 0; i < _N; ++i)
73 // (DST-IV order N + DST-III order N) / 2
74 F[i] = (data[i] + F[i])/2;
75 }
76
77 void DST::transform(function<real(real)> f, real F[]) const {
78 vector<real> data(4 * _N);
79 real d = Math::pi()/(2 * _N);
80 for (int i = 1; i <= _N; ++i)
81 data[i] = f( i * d );
82 fft_transform(data.data(), F, false);
83 }
84
85 void DST::refine(function<real(real)> f, real F[]) const {
86 vector<real> data(4 * _N);
87 real d = Math::pi()/(4 * _N);
88 for (int i = 0; i < _N; ++i)
89 data[i] = f( (2*i + 1) * d );
90 fft_transform2(data.data(), F);
91 }
92
93 Math::real DST::eval(real sinx, real cosx, const real F[], int N) {
94 // Evaluate
95 // y = sum(F[i] * sin((2*i+1) * x), i, 0, N-1)
96 // using Clenshaw summation.
97 // Approx operation count = (N + 5) mult and (2 * N + 2) add
98 real
99 ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
100 y0 = N & 1 ? F[--N] : 0, y1 = 0; // accumulators for sum
101 // Now N is even
102 while (N > 0) {
103 // Unroll loop x 2, so accumulators return to their original role
104 y1 = ar * y0 - y1 + F[--N];
105 y0 = ar * y1 - y0 + F[--N];
106 }
107 return sinx * (y0 + y1); // sin(x) * (y0 + y1)
108 }
109
110 Math::real DST::integral(real sinx, real cosx, const real F[], int N) {
111 // Evaluate
112 // y = -sum(F[i]/(2*i+1) * cos((2*i+1) * x), i, 0, N-1)
113 // using Clenshaw summation.
114 // Approx operation count = (N + 5) mult and (2 * N + 2) add
115 real
116 ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
117 y0 = 0, y1 = 0; // accumulators for sum
118 for (--N; N >= 0; --N) {
119 real t = ar * y0 - y1 + F[N]/(2*N+1);
120 y1 = y0; y0 = t;
121 }
122 return cosx * (y1 - y0); // cos(x) * (y1 - y0)
123 }
124
125 Math::real DST::integral(real sinx, real cosx, real siny, real cosy,
126 const real F[], int N) {
127 // return integral(siny, cosy, F, N) - integral(sinx, cosx, F, N);
128 real
129 // 2*cos(y-x)*cos(y+x) -> 2 * cos(2 * x)
130 ac = +2 * (cosy * cosx + siny * sinx) * (cosy * cosx - siny * sinx),
131 // -2*sin(y-x)*sin(y+x) -> 0
132 as = -2 * (siny * cosx - cosy * sinx) * (siny * cosx + cosy * sinx),
133 y0 = 0, y1 = 0, z0 = 0, z1 = 0; // accumulators for sum
134 for (--N; N >= 0; --N) {
135 real
136 ty = ac * y0 + as * z0 - y1 + F[N]/(2*N+1),
137 tz = as * y0 + ac * z0 - z1;
138 y1 = y0; y0 = ty;
139 z1 = z0; z0 = tz;
140 }
141 // B[0] - B[1] = [y0-y1, z0-z1]
142 // F[0] = [cosy + cosx, cosy - cosx] -> [2 * cosx, 0]
143 // (B[0] - B[1]) . F[0]
144 // = [(y0 - y1) * (cosy + cosx) + (z0 - z1) * (cosy - cosx),
145 // (y0 - y1) * (cosy - cosx) + (z0 - z1) * (cosy + cosx),
146 // return -(2nd element)
147 return (y1 - y0) * (cosy - cosx) + (z1 - z0) * (cosy + cosx);
148 }
149
150} // namespace GeographicLib
DST.hpp
Header for GeographicLib::DST class.
real
GeographicLib::Math::real real
Definition: GeodSolve.cpp:29
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:77
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:93
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:85
GeographicLib::DST::integral
static real integral(real sinx, real cosx, const real F[], int N)
Definition: DST.cpp:110
GeographicLib::Math::pi
static T pi()
Definition: Math.hpp:183
GeographicLib::Math::real
double real
Definition: Math.hpp:100
GeographicLib
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
std
Definition: NearestNeighbor.hpp:812