1 /** 2 * Elliptic integrals. 3 * The functions are named similarly to the names used in Mathematica. 4 * 5 * License: BSD style: $(LICENSE) 6 * Copyright: Based on the CEPHES math library, which is 7 * Copyright (C) 1994 Stephen L. Moshier (moshier@world.std.com). 8 * Authors: Stephen L. Moshier (original C code). Conversion to D by Don Clugston 9 * 10 * References: 11 * $(LINK http://en.wikipedia.org/wiki/Elliptic_integral) 12 * 13 * Eric W. Weisstein. "Elliptic Integral of the First Kind." From MathWorld--A Wolfram Web Resource. $(LINK http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html) 14 * 15 * $(LINK http://www.netlib.org/cephes/ldoubdoc.html) 16 * 17 * Macros: 18 * TABLE_SV = <table border=1 cellpadding=4 cellspacing=0> 19 * <caption>Special Values</caption> 20 * $0</table> 21 * SVH = $(TR $(TH $1) $(TH $2)) 22 * SV = $(TR $(TD $1) $(TD $2)) 23 * GAMMA = Γ 24 * INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>) 25 * POWER = $1<sup>$2</sup> 26 * NAN = $(RED NAN) 27 */ 28 /** 29 * Macros: 30 * TABLE_SV = <table border=1 cellpadding=4 cellspacing=0> 31 * <caption>Special Values</caption> 32 * $0</table> 33 * SVH = $(TR $(TH $1) $(TH $2)) 34 * SV = $(TR $(TD $1) $(TD $2)) 35 * 36 * NAN = $(RED NAN) 37 * SUP = <span style="vertical-align:super;font-size:smaller">$0</span> 38 * GAMMA = Γ 39 * INTEGRAL = ∫ 40 * INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>) 41 * POWER = $1<sup>$2</sup> 42 * BIGSUM = $(BIG Σ <sup>$2</sup><sub>$(SMALL $1)</sub>) 43 * CHOOSE = $(BIG () <sup>$(SMALL $1)</sup><sub>$(SMALL $2)</sub> $(BIG )) 44 */ 45 46 module tango.math.Elliptic; 47 48 import tango.math.Math; 49 import tango.math.IEEE; 50 51 /* These functions are based on code from: 52 Cephes Math Library, Release 2.3: October, 1995 53 Copyright 1984, 1987, 1995 by Stephen L. Moshier 54 */ 55 56 57 /** 58 * Incomplete elliptic integral of the first kind 59 * 60 * Approximates the integral 61 * F(phi | m) = $(INTEGRATE 0, phi) dt/ (sqrt( 1- m $(POWER sin, 2) t)) 62 * 63 * of amplitude phi and modulus m, using the arithmetic - 64 * geometric mean algorithm. 65 */ 66 67 real ellipticF(real phi, real m ) 68 { 69 real a, b, c, e, temp, t, K; 70 int d, mod, sign, npio2; 71 72 if( m == 0.0L ) 73 return phi; 74 a = 1.0L - m; 75 if( a == 0.0L ) { 76 if ( fabs(phi) >= PI_2 ) return real.infinity; 77 return log( tan( 0.5L*(PI_2 + phi) ) ); 78 } 79 npio2 = cast(int)floor( phi/PI_2 ); 80 if ( npio2 & 1 ) 81 npio2 += 1; 82 if ( npio2 ) { 83 K = ellipticKComplete( a ); 84 phi = phi - npio2 * PI_2; 85 } else 86 K = 0.0L; 87 if( phi < 0.0L ){ 88 phi = -phi; 89 sign = -1; 90 } else sign = 0; 91 b = sqrt(a); 92 t = tan( phi ); 93 if( fabs(t) > 10.0L ) { 94 /* Transform the amplitude */ 95 e = 1.0L/(b*t); 96 /* ... but avoid multiple recursions. */ 97 if( fabs(e) < 10.0L ){ 98 e = atan(e); 99 if( npio2 == 0 ) 100 K = ellipticKComplete( a ); 101 temp = K - ellipticF( e, m ); 102 goto done; 103 } 104 } 105 a = 1.0L; 106 c = sqrt(m); 107 d = 1; 108 mod = 0; 109 110 while( fabs(c/a) > real.epsilon ) { 111 temp = b/a; 112 phi = phi + atan(t*temp) + mod * PI; 113 mod = cast(int)((phi + PI_2)/PI); 114 t = t * ( 1.0L + temp )/( 1.0L - temp * t * t ); 115 c = 0.5L * ( a - b ); 116 temp = sqrt( a * b ); 117 a = 0.5L * ( a + b ); 118 b = temp; 119 d += d; 120 } 121 temp = (atan(t) + mod * PI)/(d * a); 122 123 done: 124 if ( sign < 0 ) 125 temp = -temp; 126 temp += npio2 * K; 127 return temp; 128 } 129 130 131 /** 132 * Incomplete elliptic integral of the second kind 133 * 134 * Approximates the integral 135 * 136 * E(phi | m) = $(INTEGRATE 0, phi) sqrt( 1- m $(POWER sin, 2) t) dt 137 * 138 * of amplitude phi and modulus m, using the arithmetic - 139 * geometric mean algorithm. 140 */ 141 142 real ellipticE(real phi, real m) 143 { 144 real a, b, c, e, temp, t, E; 145 int d, mod, npio2, sign; 146 147 if ( m == 0.0L ) return phi; 148 real lphi = phi; 149 npio2 = cast(int)floor( lphi/PI_2 ); 150 if( npio2 & 1 ) 151 npio2 += 1; 152 lphi = lphi - npio2 * PI_2; 153 if( lphi < 0.0L ){ 154 lphi = -lphi; 155 sign = -1; 156 } else { 157 sign = 1; 158 } 159 a = 1.0L - m; 160 E = ellipticEComplete( a ); 161 if( a == 0.0L ) { 162 temp = sin( lphi ); 163 goto done; 164 } 165 t = tan( lphi ); 166 b = sqrt(a); 167 if ( fabs(t) > 10.0L ) { 168 /* Transform the amplitude */ 169 e = 1.0L/(b*t); 170 /* ... but avoid multiple recursions. */ 171 if( fabs(e) < 10.0L ){ 172 e = atan(e); 173 temp = E + m * sin( lphi ) * sin( e ) - ellipticE( e, m ); 174 goto done; 175 } 176 } 177 c = sqrt(m); 178 a = 1.0L; 179 d = 1; 180 e = 0.0L; 181 mod = 0; 182 183 while( fabs(c/a) > real.epsilon ) { 184 temp = b/a; 185 lphi = lphi + atan(t*temp) + mod * PI; 186 mod = cast(int)((lphi + PI_2)/PI); 187 t = t * ( 1.0L + temp )/( 1.0L - temp * t * t ); 188 c = 0.5L*( a - b ); 189 temp = sqrt( a * b ); 190 a = 0.5L*( a + b ); 191 b = temp; 192 d += d; 193 e += c * sin(lphi); 194 } 195 196 temp = E / ellipticKComplete( 1.0L - m ); 197 temp *= (atan(t) + mod * PI)/(d * a); 198 temp += e; 199 200 done: 201 if( sign < 0 ) 202 temp = -temp; 203 temp += npio2 * E; 204 return temp; 205 } 206 207 208 /** 209 * Complete elliptic integral of the first kind 210 * 211 * Approximates the integral 212 * 213 * K(m) = $(INTEGRATE 0, π/2) dt/ (sqrt( 1- m $(POWER sin, 2) t)) 214 * 215 * where m = 1 - x, using the approximation 216 * 217 * P(x) - log x Q(x). 218 * 219 * The argument x is used rather than m so that the logarithmic 220 * singularity at x = 1 will be shifted to the origin; this 221 * preserves maximum accuracy. 222 * 223 * x must be in the range 224 * 0 <= x <= 1 225 * 226 * This is equivalent to ellipticF(PI_2, 1-x). 227 * 228 * K(0) = π/2. 229 */ 230 231 real ellipticKComplete(real x) 232 in { 233 // assert(x>=0.0L && x<=1.0L); 234 } 235 body{ 236 237 __gshared immutable real [] P = [ 238 0x1.62e42fefa39ef35ap+0, // 1.3862943611198906189 239 0x1.8b90bfbe8ed811fcp-4, // 0.096573590279993142323 240 0x1.fa05af797624c586p-6, // 0.030885144578720423267 241 0x1.e979cdfac7249746p-7, // 0.01493761594388688915 242 0x1.1f4cc8890cff803cp-7, // 0.0087676982094322259125 243 0x1.7befb3bb1fa978acp-8, // 0.0057973684116620276454 244 0x1.2c2566aa1d5fe6b8p-8, // 0.0045798659940508010431 245 0x1.7333514e7fe57c98p-8, // 0.0056640695097481470287 246 0x1.09292d1c8621348cp-7, // 0.0080920667906392630755 247 0x1.b89ab5fe793a6062p-8, // 0.0067230886765842542487 248 0x1.28e9c44dc5e26e66p-9, // 0.002265267575136470585 249 0x1.c2c43245d445addap-13, // 0.00021494216542320112406 250 0x1.4ee247035a03e13p-20 // 1.2475397291548388385e-06 251 ]; 252 253 __gshared immutable real [] Q = [ 254 0x1p-1, // 0.5 255 0x1.fffffffffff635eap-4, // 0.12499999999999782631 256 0x1.1fffffff8a2bea1p-4, // 0.070312499993302227507 257 0x1.8ffffe6f40ec2078p-5, // 0.04882812208418620146 258 0x1.323f4dbf7f4d0c2ap-5, // 0.037383701182969303058 259 0x1.efe8a028541b50bp-6, // 0.030267864612427881354 260 0x1.9d58c49718d6617cp-6, // 0.025228683455123323041 261 0x1.4d1a8d2292ff6e2ep-6, // 0.020331037356569904872 262 0x1.b637687027d664aap-7, // 0.013373304362459048444 263 0x1.687a640ae5c71332p-8, // 0.0055004591221382442135 264 0x1.0f9c30a94a1dcb4ep-10, // 0.001036110372590318803 265 0x1.d321746708e92d48p-15 // 5.568631677757315399e-05 266 ]; 267 268 enum real LOG4 = 0x1.62e42fefa39ef358p+0; // log(4) 269 270 if( x > real.epsilon ) 271 return poly(x,P) - log(x) * poly(x,Q); 272 if ( x == 0.0L ) 273 return real.infinity; 274 return LOG4 - 0.5L * log(x); 275 } 276 277 /** 278 * Complete elliptic integral of the second kind 279 * 280 * Approximates the integral 281 * 282 * E(m) = $(INTEGRATE 0, π/2) sqrt( 1- m $(POWER sin, 2) t) dt 283 * 284 * where m = 1 - x, using the approximation 285 * 286 * P(x) - x log x Q(x). 287 * 288 * Though there are no singularities, the argument m1 is used 289 * rather than m for compatibility with ellipticKComplete(). 290 * 291 * E(1) = 1; E(0) = π/2. 292 * m must be in the range 0 <= m <= 1. 293 */ 294 295 real ellipticEComplete(real x) 296 in { 297 assert(x>=0 && x<=1.0); 298 } 299 body { 300 __gshared immutable real [] P = [ 301 0x1.c5c85fdf473f78f2p-2, // 0.44314718055994670505 302 0x1.d1591f9e9a66477p-5, // 0.056805192715569305834 303 0x1.65af6a7a61f587cp-6, // 0.021831373198011179718 304 0x1.7a4d48ed00d5745ap-7, // 0.011544857605264509506 305 0x1.d4f5fe4f93b60688p-8, // 0.0071557756305783152481 306 0x1.4cb71c73bac8656ap-8, // 0.0050768322432573952962 307 0x1.4a9167859a1d0312p-8, // 0.0050440671671840438539 308 0x1.dd296daa7b1f5b7ap-8, // 0.0072809117068399675418 309 0x1.04f2c29224ba99b6p-7, // 0.0079635095646944542686 310 0x1.0f5820e2d80194d8p-8, // 0.0041403847015715420009 311 0x1.95ee634752ca69b6p-11, // 0.00077425232385887751162 312 0x1.0c58aa9ab404f4fp-15 // 3.1989378120323412946e-05 313 ]; 314 315 __gshared immutable real [] Q = [ 316 0x1.ffffffffffffb1cep-3, // 0.24999999999999986434 317 0x1.7ffffffff29eaa0cp-4, // 0.093749999999239422678 318 0x1.dfffffbd51eb098p-5, // 0.058593749514839092674 319 0x1.5dffd791cb834c92p-5, // 0.04272453406734691973 320 0x1.1397b63c2f09a8ep-5, // 0.033641677787700181541 321 0x1.c567cde5931e75bcp-6, // 0.02767367465121309044 322 0x1.75e0cae852be9ddcp-6, // 0.022819708015315777007 323 0x1.12bb968236d4e434p-6, // 0.016768357258894633433 324 0x1.1f6572c1c402d07cp-7, // 0.0087706384979640787504 325 0x1.452c6909f88b8306p-9, // 0.0024808767529843311337 326 0x1.1f7504e72d664054p-12, // 0.00027414045912208516032 327 0x1.ad17054dc46913e2p-18 // 6.3939381343012054851e-06 328 ]; 329 if (x==0) 330 return 1.0L; 331 return 1.0L + x * poly(x,P) - log(x) * (x * poly(x,Q) ); 332 } 333 334 debug (UnitTest) 335 { 336 unittest { 337 assert( ellipticF(1, 0)==1); 338 assert(ellipticEComplete(0)==1); 339 assert(ellipticEComplete(1)==PI_2); 340 assert(feqrel(ellipticKComplete(1),PI_2)>= real.mant_dig-1); 341 assert(ellipticKComplete(0)==real.infinity); 342 // assert(ellipticKComplete(1)==0); //-real.infinity); 343 344 real x=0.5653L; 345 assert(ellipticKComplete(1-x) == ellipticF(PI_2, x) ); 346 assert(ellipticEComplete(1-x) == ellipticE(PI_2, x) ); 347 } 348 } 349 350 351 /** 352 * Incomplete elliptic integral of the third kind 353 * 354 * Approximates the integral 355 * 356 * PI(n; phi | m) = $(INTEGRATE t=0, phi) dt/((1 - n $(POWER sin,2)t) * sqrt( 1- m $(POWER sin, 2) t)) 357 * 358 * of amplitude phi, modulus m, and characteristic n using Gauss-Legendre 359 * quadrature. 360 * 361 * Note that ellipticPi(PI_2, m, 1) is infinite for any m. 362 */ 363 real ellipticPi(real phi, real m, real n) 364 { 365 // BUGS: This implementation suffers from poor precision. 366 __gshared immutable double [] t = [ 367 0.9931285991850949, 0.9639719272779138, 368 0.9122344282513259, 0.8391169718222188, 369 0.7463319064601508, 0.6360536807265150, 370 0.5108670019508271, 0.3737060887154195, 371 0.2277858511416451, 0.7652652113349734e-1 372 ]; 373 __gshared immutable double [] w =[ 374 0.1761400713915212e-1, 0.4060142980038694e-1, 375 0.6267204833410907e-1, 0.8327674157670475e-1, 376 0.1019301198172404, 0.1181945319615184, 377 0.1316886384491766, 0.1420961093183820, 378 0.1491729864726037, 0.1527533871307258 379 ]; 380 bool b1 = (m==1) && abs(phi-90)<=1e-8; 381 bool b2 = (n==1) && abs(phi-90)<=1e-8; 382 if (b1 || b2) return real.infinity; 383 real c1 = 0.87266462599716e-2 * phi; 384 real c2 = c1; 385 double x = 0; 386 for (int i=0; i< t.length; ++i) { 387 real c0 = c2 * t[i]; 388 real t1 = c1 + c0; 389 real t2 = c1 - c0; 390 real s1 = sin(t1); // sin(c1 * (1 + t[i])) 391 real s2 = sin(t2); // sin(c1 * (1 - t[i])) 392 real f1 = 1.0 / ((1.0 - n * s1 * s1) * sqrt(1.0 - m * s1 * s1)); 393 real f2 = 1.0 / ((1.0 - n * s2 * s2) * sqrt(1.0 - m * s2 * s2)); 394 x+= w[i]*(f1+f2); 395 } 396 return c1 * x; 397 } 398 399 /** 400 * Complete elliptic integral of the third kind 401 */ 402 real ellipticPiComplete(real m, real n) 403 in { 404 assert(m>=-1.0 && m<=1.0); 405 } 406 body { 407 return ellipticPi(PI_2, m, n); 408 }