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  1. #include <stdio.h>
  2. #include <stdlib.h>
  3. #include <stdint.h>
  4. #define _USE_MATH_DEFINES
  5. #include <math.h>
  6. #include <assert.h>
  7. #include <stdarg.h>
  8. #include <vector>
  9. #include <utility>
  10. #include <numeric>
  11.  
  12. // [CONF] 中間結果表示
  13. #define SHOW_ITM
  14.  
  15. #ifdef _WIN32
  16. int fopen_s_wrp(FILE** const p_fp, const char* const fpath, const char* const mode)
  17. {
  18. 	return (int)fopen_s(p_fp, fpath, mode);
  19. }
  20. int sprintf_s_wrp(char buf[], const size_t bufsz, const char* const fmt, ...)
  21. {
  22. 	va_list ap;
  23. 	int nRet;
  24.  
  25. 	va_start(ap, fmt);
  26. 	nRet = vsprintf_s(buf, bufsz, fmt, ap);
  27. 	va_end(ap);
  28.  
  29. 	return nRet;
  30. }
  31. #else
  32. int fopen_s_wrp(FILE** const p_fp, const char* const fpath, const char* const mode)
  33. {
  34. 	*p_fp = fopen(fpath, mode);
  35. 	return (*p_fp == NULL);
  36. }
  37. int sprintf_s_wrp(char buf[], const size_t bufsz, const char* const fmt, ...)
  38. {
  39. 	va_list ap;
  40. 	int nRet;
  41.  
  42. 	va_start(ap, fmt);
  43. 	nRet = vsprintf(buf, fmt, ap);
  44. 	va_end(ap);
  45.  
  46. 	return nRet;
  47. }
  48. #define _countof(arr)	(sizeof(arr) / sizeof(arr[0]))
  49. #endif
  50.  
  51. namespace priroot
  52. {
  53. 	typedef int64_t pow_t;
  54.  
  55. 	/// 1の原始n乗根を表す。
  56. 	/// nの剰余計算を高速化するため、nは2^pに制限する。
  57. 	class nth_root
  58. 	{
  59. 		const pow_t t90;
  60. 		const pow_t m360;
  61. 		std::vector<double> m_sin;
  62.  
  63. 	public:
  64. 		nth_root(const int p)
  65. 			: t90(((pow_t)1 << (p - 2))), m360((t90 << 2) - 1), m_sin((size_t)t90 + 1)
  66. 		{
  67. 			const double k = 0.5 * M_PI;
  68. 			for (pow_t i = 1; i < t90; i++) {
  69. 				m_sin[i] = sin((k * (double)i) / (double)t90);
  70. 			}
  71. 			m_sin[0] = 0.0;
  72. 			m_sin[t90] = 1.0;
  73. 		}
  74.  
  75. 		pow_t n() const { return 4 * t90; }
  76.  
  77. 		/// 1の原始n乗根のk乗を取得する。
  78. 		void pow(const pow_t k_, double& re, double& im) const {
  79. 			pow_t k = (k_ < 0) ? 4 * t90 - k_ : k_;
  80. 			k &= m360;		// nの剰余
  81. 			assert(0 <= k && k < 4 * t90);
  82. 			if (k <= t90) {
  83. 				re = m_sin[t90 - k];
  84. 				im = m_sin[k];
  85. 			}
  86. 			else if (k <= 2 * t90) {
  87. 				re = -m_sin[k - t90];			// = t90 - (2 * t90 - k)
  88. 				im = m_sin[2 * t90 - k];
  89. 			}
  90. 			else if (k <= 3 * t90) {
  91. 				re = -m_sin[3 * t90 - k];		// = t90 - (k - 2 * t90)
  92. 				im = -m_sin[k - 2 * t90];
  93. 			}
  94. 			else {
  95. 				re = m_sin[k - 3 * t90];		// = t90 - (4 * t90 - k)
  96. 				im = -m_sin[4 * t90 - k];
  97. 			}
  98. 		}
  99.  
  100. 	};
  101.  
  102. }	// namespace priroot
  103.  
  104. namespace dftn
  105. {
  106. 	using namespace priroot;
  107.  
  108. 	/// フーリエ変換
  109. 	/// DFTの原理式
  110. 	///   X(k) = Σ[j=0..t360-1]{ x(j) * w^(j*k) }  (k=0..t360-1)
  111. 	/// に従いDFTを計算し、X(k)の実部をfbgn[k..]、虚部をfbgn[(k + t360)..]に格納する。
  112. 	void fouri(
  113. 		std::vector<double>::const_iterator xbgn,
  114. 		std::vector<double>::const_iterator xend,
  115. 		std::vector<double>::iterator fbgn,
  116. 		std::vector<double>::iterator fend,
  117. 		const int p,
  118. 		const nth_root& w)
  119. 	{
  120. 		const pow_t t360 = (pow_t)1 << p;
  121. 		assert(xend - xbgn == (ptrdiff_t)(2 * t360));
  122. 		assert(fend - fbgn == (ptrdiff_t)(2 * t360));
  123. 		assert(w.n() == t360);
  124.  
  125. 		for (pow_t k = 0; k < t360; k++) {
  126. 			double sum_re = 0.0;
  127. 			double sum_im = 0.0;
  128. 			for (pow_t j = 0; j < t360; j++) {
  129. 				const double a = xbgn[j];
  130. 				const double b = xbgn[j + t360];
  131.  
  132. 				double wjk_re, wjk_im;
  133. 				w.pow(j * k, wjk_re, wjk_im);
  134. 				sum_re += a * wjk_re - b * wjk_im;
  135. 				sum_im += a * wjk_im + b * wjk_re;
  136. 			}
  137. 			fbgn[k] = sum_re;
  138. 			fbgn[k + t360] = sum_im;
  139. 		}
  140. 	}
  141.  
  142. 	/// フーリエ逆変換
  143. 	/// IDFTの原理式
  144. 	///   x(j) = Σ[k=0..t360-1]{ X(k) * w^(-j*k) }  (j=0..t360-1)
  145. 	/// に従いIDFTを計算し、x(k)の実部をxbgn[k..]、虚部をxbgn[(k + t360)..]に格納する。
  146. 	void fouriinv(
  147. 		std::vector<double>::const_iterator fbgn,
  148. 		std::vector<double>::const_iterator fend,
  149. 		std::vector<double>::iterator xbgn,
  150. 		std::vector<double>::iterator xend,
  151. 		const int p,
  152. 		const nth_root& w)
  153. 	{
  154. 		const pow_t t360 = (pow_t)1 << p;
  155. 		assert(fend - fbgn == (ptrdiff_t)(2 * t360));
  156. 		assert(xend - xbgn == (ptrdiff_t)(2 * t360));
  157. 		assert(w.n() == t360);
  158.  
  159. 		for (pow_t j = 0; j < t360; j++) {
  160. 			double sum_re = 0.0;
  161. 			double sum_im = 0.0;
  162. 			for (pow_t k = 0; k < t360; k++) {
  163. 				const double a = fbgn[k];
  164. 				const double b = fbgn[k + t360];
  165.  
  166. 				double wjk_re, wjk_im;
  167. 				w.pow(j * k, wjk_re, wjk_im);
  168. 				sum_re += a * wjk_re + b * wjk_im;
  169. 				sum_im += b * wjk_re - a * wjk_im;
  170. 			}
  171. 			xbgn[j] = sum_re / (double)t360;
  172. 			xbgn[j + t360] = sum_im / (double)t360;
  173. 		}
  174. 	}
  175.  
  176. }	// namespace dftn
  177.  
  178. namespace dft4n
  179. {
  180. 	using namespace priroot;
  181.  
  182. 	/// 4n乗根によるフーリエ変換
  183. 	/// x(j) (j=0..(2*t360)-1)から
  184. 	///   x'(j) = (x(j) + i * x(j+t360)) * u^(j/4) (j=0..t360-1)
  185. 	/// を構成し、x'(j)をDFTする。ここで、uは1の原始t360乗根。
  186. 	void fouri4n(
  187. 		std::vector<double>::iterator xbgn,
  188. 		std::vector<double>::iterator xend,
  189. 		std::vector<double>::iterator fbgn,
  190. 		std::vector<double>::iterator fend,
  191. 		const int p,
  192. 		const nth_root& w)
  193. 	{
  194. 		const pow_t t360 = (pow_t)1 << p;
  195. 		assert(xend - xbgn == (ptrdiff_t)(2 * t360));
  196. 		assert(fend - fbgn == (ptrdiff_t)(2 * t360));
  197. 		assert(w.n() == (t360 << 2));
  198.  
  199. 		// { x'(j) }構成
  200. 		// { x(j) + i * x(j+t360) } にu^(j/4)を掛ける。
  201. 		for (pow_t j = 0; j < t360; j++) {
  202. 			const double a = xbgn[j];
  203. 			const double b = xbgn[j + t360];
  204.  
  205. 			// u = w^4としてu^(j/4) = w^(j)
  206. 			double wj_re, wj_im;
  207. 			w.pow(j, wj_re, wj_im);
  208. 			xbgn[j] = a * wj_re - b * wj_im;
  209. 			xbgn[j + t360] = a * wj_im + b * wj_re;
  210. 		}
  211.  
  212. 		// { x'(j) }のフーリエ変換
  213. 		std::vector<double> fr((size_t)t360);
  214. 		for (pow_t k = 0; k < t360; k++) {
  215. 			double sum_re = 0.0;
  216. 			double sum_im = 0.0;
  217. 			for (pow_t j = 0; j < t360; j++) {
  218. 				const double a = xbgn[j];
  219. 				const double b = xbgn[j + t360];
  220.  
  221. 				// u = w^4としてu^(j*k)を掛けて足す
  222. 				// u^(j*k) = w^(4*j*k)
  223. 				double wjk_re, wjk_im;
  224. 				w.pow(4 * j * k, wjk_re, wjk_im);
  225. 				sum_re += a * wjk_re - b * wjk_im;
  226. 				sum_im += a * wjk_im + b * wjk_re;
  227. 			}
  228. 			fbgn[k] = sum_re;
  229. 			fbgn[k + t360] = sum_im;
  230. 		}
  231. 	}
  232.  
  233. 	/// 4n乗根による逆フーリエ変換
  234. 	void fouri4ninv(
  235. 		std::vector<double>::const_iterator fbgn,
  236. 		std::vector<double>::const_iterator fend,
  237. 		std::vector<double>::iterator xbgn,
  238. 		std::vector<double>::iterator xend,
  239. 		const int p,
  240. 		const nth_root& w)
  241. 	{
  242. 		const pow_t t360 = (pow_t)1 << p;
  243. 		assert(fend - fbgn == (ptrdiff_t)(2 * t360));
  244. 		assert(xend - xbgn == (ptrdiff_t)(2 * t360));
  245. 		assert(w.n() == (t360 << 2));
  246.  
  247. 		// { x'(j) }の逆フーリエ変換
  248. 		for (pow_t j = 0; j < t360; j++) {
  249. 			double sum_re = 0.0;
  250. 			double sum_im = 0.0;
  251. 			for (pow_t k = 0; k < t360; k++) {
  252. 				const double a = fbgn[k];
  253. 				const double b = fbgn[k + t360];
  254.  
  255. 				// u = w^4としてu^(j*(-k))を掛けて足す
  256. 				// u^(j*(-k)) = w^(-4*j*k)
  257. 				double wjk_re, wjk_im;
  258. 				w.pow(4 * j * k, wjk_re, wjk_im);
  259. 				sum_re += a * wjk_re + b * wjk_im;
  260. 				sum_im += b * wjk_re - a * wjk_im;
  261. 			}
  262. 			xbgn[j] = sum_re / (double)t360;
  263. 			xbgn[j + t360] = sum_im / (double)t360;
  264. 		}
  265.  
  266. 		// { x(j) }復元
  267. 		for (pow_t j = 0; j < t360; j++) {
  268. 			const double a = xbgn[j];
  269. 			const double b = xbgn[j + t360];
  270.  
  271. 			// u = w^4としてu^(-j/4) = w^(-j)を掛ける
  272. 			double wj_re, wj_im;
  273. 			w.pow(j, wj_re, wj_im);
  274. 			xbgn[j] = a * wj_re + b * wj_im;
  275. 			xbgn[j + t360] = b * wj_re - a * wj_im;
  276. 		}
  277. 	}
  278.  
  279. }	// namespace dftn
  280.  
  281. namespace utest
  282. {
  283. 	using namespace priroot;
  284.  
  285. 	void nth_rootTest()
  286. 	{
  287. 		const int p = 4;
  288.  
  289. 		nth_root w(p);
  290. 		const pow_t t360 = (pow_t)1 << p;
  291. 		for (pow_t k = 0; k <= 2 * t360; k++) {
  292. 			double re, im;
  293. 			w.pow(k, re, im);
  294. 			const double expn = (k % t360 == 0) ? 0.0 : (2 * M_PI) / atan2(im, re);
  295. 			printf("rt.pow(%2lld) = % f + i * % f (w^% 10f=1)\n", k, re, im, expn);
  296. 		}
  297. 	}
  298.  
  299. 	void show_complex_seq(const std::vector<double>& s)
  300. 	{
  301. 		const size_t hsz = s.size() / 2;
  302. 		assert(2 * hsz == s.size());
  303.  
  304. 		printf("Re:");
  305. 		for (size_t i = 0; i < hsz; i++) {
  306. 			printf(" % 12f", s[i]);
  307. 		}
  308. 		printf("\n");
  309. 		printf("Im:");
  310. 		for (size_t i = 0; i < hsz; i++) {
  311. 			printf(" % 12f", s[i + hsz]);
  312. 		}
  313. 		printf("\n");
  314. 	}
  315.  
  316. 	void fouriTest()
  317. 	{
  318. 		const int p = 4;
  319.  
  320. 		const pow_t t360 = (pow_t)1 << p;
  321. 		nth_root w(p);
  322.  
  323. 		// 入力列準備
  324. 		std::vector<double> x((size_t)(2 * t360));
  325. 		std::iota(x.begin(), x.begin() + t360, 0.0);
  326.  
  327. 		printf("input seq:\n");
  328. 		show_complex_seq(x);
  329. 		printf("\n");
  330.  
  331. 		// フーリエ変換
  332. 		std::vector<double> f((size_t)(2 * t360));
  333. 		dftn::fouri(x.begin(), x.end(), f.begin(), f.end(), p, w);
  334. 		x.clear();
  335.  
  336. 		printf("dftn::fouri() result:\n");
  337. 		show_complex_seq(f);
  338. 		printf("\n");
  339.  
  340. 		// 逆フーリエ変換
  341. 		std::vector<double> r((size_t)(2 * t360));
  342. 		dftn::fouriinv(f.begin(), f.end(), r.begin(), r.end(), p, w);
  343. 		f.clear();
  344.  
  345. 		printf("dftn::fouriinv() result:\n");
  346. 		show_complex_seq(r);
  347. 		printf("\n");
  348.  
  349. 		// 照合
  350. 		const size_t sz = r.size();
  351. 		if (sz != (size_t)(2 * t360)) { abort(); }
  352. 		for (size_t i = 0; i < sz / 2; i++) {
  353. 			const double actv = r[i];
  354. 			const double expv = (double)i;
  355. 			const double err = std::abs(actv - expv);
  356. 			if (err >= 0.1) { abort(); }
  357. 		}
  358. 		for (size_t i = sz / 2; i < sz; i++) {
  359. 			const double expv = 0.0;
  360. 			const double actv = r[i];
  361. 			const double err = std::abs(actv - expv);
  362. 			if (err >= 0.1) { abort(); }
  363. 		}
  364. 	}
  365.  
  366. 	void fouri4nTest()
  367. 	{
  368. 		const int p = 4;
  369.  
  370. 		const pow_t t360 = (pow_t)1 << (p - 1);
  371. 		nth_root w(p + 1);
  372.  
  373. 		// 入力列準備
  374. 		std::vector<double> x((size_t)2 * t360);
  375. 		std::iota(x.begin(), x.end(), 0.0);
  376.  
  377. 		printf("input seq:\n");
  378. 		show_complex_seq(x);
  379. 		printf("\n");
  380.  
  381. 		// フーリエ変換
  382. 		std::vector<double> f((size_t)2 * t360);
  383. 		dft4n::fouri4n(x.begin(), x.end(), f.begin(), f.end(), p - 1, w);
  384. 		x.clear();
  385.  
  386. 		printf("dft4n::fouri4n() result:\n");
  387. 		show_complex_seq(f);
  388. 		printf("\n");
  389.  
  390. 		// 逆フーリエ変換
  391. 		std::vector<double> r((size_t)2 * t360);
  392. 		dft4n::fouri4ninv(f.begin(), f.end(), r.begin(), r.end(), p - 1, w);
  393. 		f.clear();
  394.  
  395. 		printf("dft4n::fouri4ninv() result:\n");
  396. 		show_complex_seq(r);
  397. 		printf("\n");
  398.  
  399. 		// 照合
  400. 		const size_t sz = r.size();
  401. 		if (sz != (size_t)(2 * t360)) { abort(); }
  402. 		for (size_t i = 0; i < sz; i++) {
  403. 			const double actv = r[i];
  404. 			const double expv = (double)i;
  405. 			const double err = std::abs(actv - expv);
  406. 			if (err >= 0.1) { abort(); }
  407. 		}
  408. 	}
  409.  
  410. }	// namespace utest
  411.  
  412. int main()
  413. {
  414. #ifdef SHOW_ITM
  415. 	utest::nth_rootTest();
  416. 	utest::fouriTest();
  417. 	utest::fouri4nTest();
  418. #endif
  419. }
  420.  
Success #stdin #stdout 0.01s 5272KB
comments (?)
stdin 
Standard input is empty
stdout 
rt.pow( 0) =  1.000000 + i *  0.000000 (w^  0.000000=1)
rt.pow( 1) =  0.923880 + i *  0.382683 (w^ 16.000000=1)
rt.pow( 2) =  0.707107 + i *  0.707107 (w^  8.000000=1)
rt.pow( 3) =  0.382683 + i *  0.923880 (w^  5.333333=1)
rt.pow( 4) =  0.000000 + i *  1.000000 (w^  4.000000=1)
rt.pow( 5) = -0.382683 + i *  0.923880 (w^  3.200000=1)
rt.pow( 6) = -0.707107 + i *  0.707107 (w^  2.666667=1)
rt.pow( 7) = -0.923880 + i *  0.382683 (w^  2.285714=1)
rt.pow( 8) = -1.000000 + i *  0.000000 (w^  2.000000=1)
rt.pow( 9) = -0.923880 + i * -0.382683 (w^ -2.285714=1)
rt.pow(10) = -0.707107 + i * -0.707107 (w^ -2.666667=1)
rt.pow(11) = -0.382683 + i * -0.923880 (w^ -3.200000=1)
rt.pow(12) = -0.000000 + i * -1.000000 (w^ -4.000000=1)
rt.pow(13) =  0.382683 + i * -0.923880 (w^ -5.333333=1)
rt.pow(14) =  0.707107 + i * -0.707107 (w^ -8.000000=1)
rt.pow(15) =  0.923880 + i * -0.382683 (w^-16.000000=1)
rt.pow(16) =  1.000000 + i *  0.000000 (w^  0.000000=1)
rt.pow(17) =  0.923880 + i *  0.382683 (w^ 16.000000=1)
rt.pow(18) =  0.707107 + i *  0.707107 (w^  8.000000=1)
rt.pow(19) =  0.382683 + i *  0.923880 (w^  5.333333=1)
rt.pow(20) =  0.000000 + i *  1.000000 (w^  4.000000=1)
rt.pow(21) = -0.382683 + i *  0.923880 (w^  3.200000=1)
rt.pow(22) = -0.707107 + i *  0.707107 (w^  2.666667=1)
rt.pow(23) = -0.923880 + i *  0.382683 (w^  2.285714=1)
rt.pow(24) = -1.000000 + i *  0.000000 (w^  2.000000=1)
rt.pow(25) = -0.923880 + i * -0.382683 (w^ -2.285714=1)
rt.pow(26) = -0.707107 + i * -0.707107 (w^ -2.666667=1)
rt.pow(27) = -0.382683 + i * -0.923880 (w^ -3.200000=1)
rt.pow(28) = -0.000000 + i * -1.000000 (w^ -4.000000=1)
rt.pow(29) =  0.382683 + i * -0.923880 (w^ -5.333333=1)
rt.pow(30) =  0.707107 + i * -0.707107 (w^ -8.000000=1)
rt.pow(31) =  0.923880 + i * -0.382683 (w^-16.000000=1)
rt.pow(32) =  1.000000 + i *  0.000000 (w^  0.000000=1)
input seq:
Re:     0.000000     1.000000     2.000000     3.000000     4.000000     5.000000     6.000000     7.000000     8.000000     9.000000    10.000000    11.000000    12.000000    13.000000    14.000000    15.000000
Im:     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000

dftn::fouri() result:
Re:   120.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000    -8.000000
Im:     0.000000   -40.218716   -19.313708   -11.972846    -8.000000    -5.345429    -3.313708    -1.591299     0.000000     1.591299     3.313708     5.345429     8.000000    11.972846    19.313708    40.218716

dftn::fouriinv() result:
Re:    -0.000000     1.000000     2.000000     3.000000     4.000000     5.000000     6.000000     7.000000     8.000000     9.000000    10.000000    11.000000    12.000000    13.000000    14.000000    15.000000
Im:     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000     0.000000    -0.000000     0.000000     0.000000     0.000000     0.000000

input seq:
Re:     0.000000     1.000000     2.000000     3.000000     4.000000     5.000000     6.000000     7.000000
Im:     8.000000     9.000000    10.000000    11.000000    12.000000    13.000000    14.000000    15.000000

dft4n::fouri4n() result:
Re:   -44.043434     5.749926     7.163243     7.453990     7.495150     7.357149     6.757625     2.066352
Im:    81.225363    14.966947     6.565430     2.426773    -0.787931    -4.276089    -9.748028   -26.372466

dft4n::fouri4ninv() result:
Re:     0.000000     1.000000     2.000000     3.000000     4.000000     5.000000     6.000000     7.000000
Im:     8.000000     9.000000    10.000000    11.000000    12.000000    13.000000    14.000000    15.000000
https://ideone.com/uK9v58
language:
C++14 (clang 8.0)
created:
3 years ago
visibility:
public

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