GeographicLib 1.52
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EllipticFunction.cpp
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1/**
2 * \file EllipticFunction.cpp
3 * \brief Implementation for GeographicLib::EllipticFunction class
4 *
5 * Copyright (c) Charles Karney (2008-2020) <charles@karney.com> and licensed
6 * under the MIT/X11 License. For more information, see
7 * https://geographiclib.sourceforge.io/
8 **********************************************************************/
9
11
12#if defined(_MSC_VER)
13// Squelch warnings about constant conditional expressions
14# pragma warning (disable: 4127)
15#endif
16
17namespace GeographicLib {
18
19 using namespace std;
20
21 /*
22 * Implementation of methods given in
23 *
24 * B. C. Carlson
25 * Computation of elliptic integrals
26 * Numerical Algorithms 10, 13-26 (1995)
27 */
28
29 Math::real EllipticFunction::RF(real x, real y, real z) {
30 // Carlson, eqs 2.2 - 2.7
31 static const real tolRF =
32 pow(3 * numeric_limits<real>::epsilon() * real(0.01), 1/real(8));
33 real
34 A0 = (x + y + z)/3,
35 An = A0,
36 Q = max(max(abs(A0-x), abs(A0-y)), abs(A0-z)) / tolRF,
37 x0 = x,
38 y0 = y,
39 z0 = z,
40 mul = 1;
41 while (Q >= mul * abs(An)) {
42 // Max 6 trips
43 real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0);
44 An = (An + lam)/4;
45 x0 = (x0 + lam)/4;
46 y0 = (y0 + lam)/4;
47 z0 = (z0 + lam)/4;
48 mul *= 4;
49 }
50 real
51 X = (A0 - x) / (mul * An),
52 Y = (A0 - y) / (mul * An),
53 Z = - (X + Y),
54 E2 = X*Y - Z*Z,
55 E3 = X*Y*Z;
56 // https://dlmf.nist.gov/19.36.E1
57 // Polynomial is
58 // (1 - E2/10 + E3/14 + E2^2/24 - 3*E2*E3/44
59 // - 5*E2^3/208 + 3*E3^2/104 + E2^2*E3/16)
60 // convert to Horner form...
61 return (E3 * (6930 * E3 + E2 * (15015 * E2 - 16380) + 17160) +
62 E2 * ((10010 - 5775 * E2) * E2 - 24024) + 240240) /
63 (240240 * sqrt(An));
64 }
65
67 // Carlson, eqs 2.36 - 2.38
68 static const real tolRG0 =
69 real(2.7) * sqrt((numeric_limits<real>::epsilon() * real(0.01)));
70 real xn = sqrt(x), yn = sqrt(y);
71 if (xn < yn) swap(xn, yn);
72 while (abs(xn-yn) > tolRG0 * xn) {
73 // Max 4 trips
74 real t = (xn + yn) /2;
75 yn = sqrt(xn * yn);
76 xn = t;
77 }
78 return Math::pi() / (xn + yn);
79 }
80
82 // Defined only for y != 0 and x >= 0.
83 return ( !(x >= y) ? // x < y and catch nans
84 // https://dlmf.nist.gov/19.2.E18
85 atan(sqrt((y - x) / x)) / sqrt(y - x) :
86 ( x == y ? 1 / sqrt(y) :
87 asinh( y > 0 ?
88 // https://dlmf.nist.gov/19.2.E19
89 // atanh(sqrt((x - y) / x))
90 sqrt((x - y) / y) :
91 // https://dlmf.nist.gov/19.2.E20
92 // atanh(sqrt(x / (x - y)))
93 sqrt(-x / y) ) / sqrt(x - y) ) );
94 }
95
96 Math::real EllipticFunction::RG(real x, real y, real z) {
97 if (z == 0)
98 swap(y, z);
99 // Carlson, eq 1.7
100 return (z * RF(x, y, z) - (x-z) * (y-z) * RD(x, y, z) / 3
101 + sqrt(x * y / z)) / 2;
102 }
103
105 // Carlson, eqs 2.36 - 2.39
106 static const real tolRG0 =
107 real(2.7) * sqrt((numeric_limits<real>::epsilon() * real(0.01)));
108 real
109 x0 = sqrt(max(x, y)),
110 y0 = sqrt(min(x, y)),
111 xn = x0,
112 yn = y0,
113 s = 0,
114 mul = real(0.25);
115 while (abs(xn-yn) > tolRG0 * xn) {
116 // Max 4 trips
117 real t = (xn + yn) /2;
118 yn = sqrt(xn * yn);
119 xn = t;
120 mul *= 2;
121 t = xn - yn;
122 s += mul * t * t;
123 }
124 return (Math::sq( (x0 + y0)/2 ) - s) * Math::pi() / (2 * (xn + yn));
125 }
126
127 Math::real EllipticFunction::RJ(real x, real y, real z, real p) {
128 // Carlson, eqs 2.17 - 2.25
129 static const real
130 tolRD = pow(real(0.2) * (numeric_limits<real>::epsilon() * real(0.01)),
131 1/real(8));
132 real
133 A0 = (x + y + z + 2*p)/5,
134 An = A0,
135 delta = (p-x) * (p-y) * (p-z),
136 Q = max(max(abs(A0-x), abs(A0-y)), max(abs(A0-z), abs(A0-p))) / tolRD,
137 x0 = x,
138 y0 = y,
139 z0 = z,
140 p0 = p,
141 mul = 1,
142 mul3 = 1,
143 s = 0;
144 while (Q >= mul * abs(An)) {
145 // Max 7 trips
146 real
147 lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0),
148 d0 = (sqrt(p0)+sqrt(x0)) * (sqrt(p0)+sqrt(y0)) * (sqrt(p0)+sqrt(z0)),
149 e0 = delta/(mul3 * Math::sq(d0));
150 s += RC(1, 1 + e0)/(mul * d0);
151 An = (An + lam)/4;
152 x0 = (x0 + lam)/4;
153 y0 = (y0 + lam)/4;
154 z0 = (z0 + lam)/4;
155 p0 = (p0 + lam)/4;
156 mul *= 4;
157 mul3 *= 64;
158 }
159 real
160 X = (A0 - x) / (mul * An),
161 Y = (A0 - y) / (mul * An),
162 Z = (A0 - z) / (mul * An),
163 P = -(X + Y + Z) / 2,
164 E2 = X*Y + X*Z + Y*Z - 3*P*P,
165 E3 = X*Y*Z + 2*P * (E2 + 2*P*P),
166 E4 = (2*X*Y*Z + P * (E2 + 3*P*P)) * P,
167 E5 = X*Y*Z*P*P;
168 // https://dlmf.nist.gov/19.36.E2
169 // Polynomial is
170 // (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
171 // - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272
172 // - 9*(E3*E4+E2*E5)/68)
173 return ((471240 - 540540 * E2) * E5 +
174 (612612 * E2 - 540540 * E3 - 556920) * E4 +
175 E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
176 E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
177 (4084080 * mul * An * sqrt(An)) + 6 * s;
178 }
179
180 Math::real EllipticFunction::RD(real x, real y, real z) {
181 // Carlson, eqs 2.28 - 2.34
182 static const real
183 tolRD = pow(real(0.2) * (numeric_limits<real>::epsilon() * real(0.01)),
184 1/real(8));
185 real
186 A0 = (x + y + 3*z)/5,
187 An = A0,
188 Q = max(max(abs(A0-x), abs(A0-y)), abs(A0-z)) / tolRD,
189 x0 = x,
190 y0 = y,
191 z0 = z,
192 mul = 1,
193 s = 0;
194 while (Q >= mul * abs(An)) {
195 // Max 7 trips
196 real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0);
197 s += 1/(mul * sqrt(z0) * (z0 + lam));
198 An = (An + lam)/4;
199 x0 = (x0 + lam)/4;
200 y0 = (y0 + lam)/4;
201 z0 = (z0 + lam)/4;
202 mul *= 4;
203 }
204 real
205 X = (A0 - x) / (mul * An),
206 Y = (A0 - y) / (mul * An),
207 Z = -(X + Y) / 3,
208 E2 = X*Y - 6*Z*Z,
209 E3 = (3*X*Y - 8*Z*Z)*Z,
210 E4 = 3 * (X*Y - Z*Z) * Z*Z,
211 E5 = X*Y*Z*Z*Z;
212 // https://dlmf.nist.gov/19.36.E2
213 // Polynomial is
214 // (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
215 // - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272
216 // - 9*(E3*E4+E2*E5)/68)
217 return ((471240 - 540540 * E2) * E5 +
218 (612612 * E2 - 540540 * E3 - 556920) * E4 +
219 E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
220 E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
221 (4084080 * mul * An * sqrt(An)) + 3 * s;
222 }
223
224 void EllipticFunction::Reset(real k2, real alpha2,
225 real kp2, real alphap2) {
226 // Accept nans here (needed for GeodesicExact)
227 if (k2 > 1)
228 throw GeographicErr("Parameter k2 is not in (-inf, 1]");
229 if (alpha2 > 1)
230 throw GeographicErr("Parameter alpha2 is not in (-inf, 1]");
231 if (kp2 < 0)
232 throw GeographicErr("Parameter kp2 is not in [0, inf)");
233 if (alphap2 < 0)
234 throw GeographicErr("Parameter alphap2 is not in [0, inf)");
235 _k2 = k2;
236 _kp2 = kp2;
237 _alpha2 = alpha2;
238 _alphap2 = alphap2;
239 _eps = _k2/Math::sq(sqrt(_kp2) + 1);
240 // Values of complete elliptic integrals for k = 0,1 and alpha = 0,1
241 // K E D
242 // k = 0: pi/2 pi/2 pi/4
243 // k = 1: inf 1 inf
244 // Pi G H
245 // k = 0, alpha = 0: pi/2 pi/2 pi/4
246 // k = 1, alpha = 0: inf 1 1
247 // k = 0, alpha = 1: inf inf pi/2
248 // k = 1, alpha = 1: inf inf inf
249 //
250 // Pi(0, k) = K(k)
251 // G(0, k) = E(k)
252 // H(0, k) = K(k) - D(k)
253 // Pi(0, k) = K(k)
254 // G(0, k) = E(k)
255 // H(0, k) = K(k) - D(k)
256 // Pi(alpha2, 0) = pi/(2*sqrt(1-alpha2))
257 // G(alpha2, 0) = pi/(2*sqrt(1-alpha2))
258 // H(alpha2, 0) = pi/(2*(1 + sqrt(1-alpha2)))
259 // Pi(alpha2, 1) = inf
260 // H(1, k) = K(k)
261 // G(alpha2, 1) = H(alpha2, 1) = RC(1, alphap2)
262 if (_k2 != 0) {
263 // Complete elliptic integral K(k), Carlson eq. 4.1
264 // https://dlmf.nist.gov/19.25.E1
265 _Kc = _kp2 != 0 ? RF(_kp2, 1) : Math::infinity();
266 // Complete elliptic integral E(k), Carlson eq. 4.2
267 // https://dlmf.nist.gov/19.25.E1
268 _Ec = _kp2 != 0 ? 2 * RG(_kp2, 1) : 1;
269 // D(k) = (K(k) - E(k))/k^2, Carlson eq.4.3
270 // https://dlmf.nist.gov/19.25.E1
271 _Dc = _kp2 != 0 ? RD(0, _kp2, 1) / 3 : Math::infinity();
272 } else {
273 _Kc = _Ec = Math::pi()/2; _Dc = _Kc/2;
274 }
275 if (_alpha2 != 0) {
276 // https://dlmf.nist.gov/19.25.E2
277 real rj = (_kp2 != 0 && _alphap2 != 0) ? RJ(0, _kp2, 1, _alphap2) :
279 // Only use rc if _kp2 = 0.
280 rc = _kp2 != 0 ? 0 :
281 (_alphap2 != 0 ? RC(1, _alphap2) : Math::infinity());
282 // Pi(alpha^2, k)
283 _Pic = _kp2 != 0 ? _Kc + _alpha2 * rj / 3 : Math::infinity();
284 // G(alpha^2, k)
285 _Gc = _kp2 != 0 ? _Kc + (_alpha2 - _k2) * rj / 3 : rc;
286 // H(alpha^2, k)
287 _Hc = _kp2 != 0 ? _Kc - (_alphap2 != 0 ? _alphap2 * rj : 0) / 3 : rc;
288 } else {
289 _Pic = _Kc; _Gc = _Ec;
290 // Hc = Kc - Dc but this involves large cancellations if k2 is close to
291 // 1. So write (for alpha2 = 0)
292 // Hc = int(cos(phi)^2/sqrt(1-k2*sin(phi)^2),phi,0,pi/2)
293 // = 1/sqrt(1-k2) * int(sin(phi)^2/sqrt(1-k2/kp2*sin(phi)^2,...)
294 // = 1/kp * D(i*k/kp)
295 // and use D(k) = RD(0, kp2, 1) / 3
296 // so Hc = 1/kp * RD(0, 1/kp2, 1) / 3
297 // = kp2 * RD(0, 1, kp2) / 3
298 // using https://dlmf.nist.gov/19.20.E18
299 // Equivalently
300 // RF(x, 1) - RD(0, x, 1)/3 = x * RD(0, 1, x)/3 for x > 0
301 // For k2 = 1 and alpha2 = 0, we have
302 // Hc = int(cos(phi),...) = 1
303 _Hc = _kp2 != 0 ? _kp2 * RD(0, 1, _kp2) / 3 : 1;
304 }
305 }
306
307 /*
308 * Implementation of methods given in
309 *
310 * R. Bulirsch
311 * Numerical Calculation of Elliptic Integrals and Elliptic Functions
312 * Numericshe Mathematik 7, 78-90 (1965)
313 */
314
315 void EllipticFunction::sncndn(real x, real& sn, real& cn, real& dn) const {
316 // Bulirsch's sncndn routine, p 89.
317 static const real tolJAC =
318 sqrt(numeric_limits<real>::epsilon() * real(0.01));
319 if (_kp2 != 0) {
320 real mc = _kp2, d = 0;
321 if (_kp2 < 0) {
322 d = 1 - mc;
323 mc /= -d;
324 d = sqrt(d);
325 x *= d;
326 }
327 real c = 0; // To suppress warning about uninitialized variable
328 real m[num_], n[num_];
329 unsigned l = 0;
330 for (real a = 1; l < num_ || GEOGRAPHICLIB_PANIC; ++l) {
331 // This converges quadratically. Max 5 trips
332 m[l] = a;
333 n[l] = mc = sqrt(mc);
334 c = (a + mc) / 2;
335 if (!(abs(a - mc) > tolJAC * a)) {
336 ++l;
337 break;
338 }
339 mc *= a;
340 a = c;
341 }
342 x *= c;
343 sn = sin(x);
344 cn = cos(x);
345 dn = 1;
346 if (sn != 0) {
347 real a = cn / sn;
348 c *= a;
349 while (l--) {
350 real b = m[l];
351 a *= c;
352 c *= dn;
353 dn = (n[l] + a) / (b + a);
354 a = c / b;
355 }
356 a = 1 / sqrt(c*c + 1);
357 sn = sn < 0 ? -a : a;
358 cn = c * sn;
359 if (_kp2 < 0) {
360 swap(cn, dn);
361 sn /= d;
362 }
363 }
364 } else {
365 sn = tanh(x);
366 dn = cn = 1 / cosh(x);
367 }
368 }
369
370 Math::real EllipticFunction::F(real sn, real cn, real dn) const {
371 // Carlson, eq. 4.5 and
372 // https://dlmf.nist.gov/19.25.E5
373 real cn2 = cn*cn, dn2 = dn*dn,
374 fi = cn2 != 0 ? abs(sn) * RF(cn2, dn2, 1) : K();
375 // Enforce usual trig-like symmetries
376 if (cn < 0)
377 fi = 2 * K() - fi;
378 return copysign(fi, sn);
379 }
380
381 Math::real EllipticFunction::E(real sn, real cn, real dn) const {
382 real
383 cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
384 ei = cn2 != 0 ?
385 abs(sn) * ( _k2 <= 0 ?
386 // Carlson, eq. 4.6 and
387 // https://dlmf.nist.gov/19.25.E9
388 RF(cn2, dn2, 1) - _k2 * sn2 * RD(cn2, dn2, 1) / 3 :
389 ( _kp2 >= 0 ?
390 // https://dlmf.nist.gov/19.25.E10
391 _kp2 * RF(cn2, dn2, 1) +
392 _k2 * _kp2 * sn2 * RD(cn2, 1, dn2) / 3 +
393 _k2 * abs(cn) / dn :
394 // https://dlmf.nist.gov/19.25.E11
395 - _kp2 * sn2 * RD(dn2, 1, cn2) / 3 +
396 dn / abs(cn) ) ) :
397 E();
398 // Enforce usual trig-like symmetries
399 if (cn < 0)
400 ei = 2 * E() - ei;
401 return copysign(ei, sn);
402 }
403
404 Math::real EllipticFunction::D(real sn, real cn, real dn) const {
405 // Carlson, eq. 4.8 and
406 // https://dlmf.nist.gov/19.25.E13
407 real
408 cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
409 di = cn2 != 0 ? abs(sn) * sn2 * RD(cn2, dn2, 1) / 3 : D();
410 // Enforce usual trig-like symmetries
411 if (cn < 0)
412 di = 2 * D() - di;
413 return copysign(di, sn);
414 }
415
416 Math::real EllipticFunction::Pi(real sn, real cn, real dn) const {
417 // Carlson, eq. 4.7 and
418 // https://dlmf.nist.gov/19.25.E14
419 real
420 cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
421 pii = cn2 != 0 ? abs(sn) * (RF(cn2, dn2, 1) +
422 _alpha2 * sn2 *
423 RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
424 Pi();
425 // Enforce usual trig-like symmetries
426 if (cn < 0)
427 pii = 2 * Pi() - pii;
428 return copysign(pii, sn);
429 }
430
431 Math::real EllipticFunction::G(real sn, real cn, real dn) const {
432 real
433 cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
434 gi = cn2 != 0 ? abs(sn) * (RF(cn2, dn2, 1) +
435 (_alpha2 - _k2) * sn2 *
436 RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
437 G();
438 // Enforce usual trig-like symmetries
439 if (cn < 0)
440 gi = 2 * G() - gi;
441 return copysign(gi, sn);
442 }
443
444 Math::real EllipticFunction::H(real sn, real cn, real dn) const {
445 real
446 cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
447 // WARNING: large cancellation if k2 = 1, alpha2 = 0, and phi near pi/2
448 hi = cn2 != 0 ? abs(sn) * (RF(cn2, dn2, 1) -
449 _alphap2 * sn2 *
450 RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
451 H();
452 // Enforce usual trig-like symmetries
453 if (cn < 0)
454 hi = 2 * H() - hi;
455 return copysign(hi, sn);
456 }
457
458 Math::real EllipticFunction::deltaF(real sn, real cn, real dn) const {
459 // Function is periodic with period pi
460 if (cn < 0) { cn = -cn; sn = -sn; }
461 return F(sn, cn, dn) * (Math::pi()/2) / K() - atan2(sn, cn);
462 }
463
464 Math::real EllipticFunction::deltaE(real sn, real cn, real dn) const {
465 // Function is periodic with period pi
466 if (cn < 0) { cn = -cn; sn = -sn; }
467 return E(sn, cn, dn) * (Math::pi()/2) / E() - atan2(sn, cn);
468 }
469
470 Math::real EllipticFunction::deltaPi(real sn, real cn, real dn) const {
471 // Function is periodic with period pi
472 if (cn < 0) { cn = -cn; sn = -sn; }
473 return Pi(sn, cn, dn) * (Math::pi()/2) / Pi() - atan2(sn, cn);
474 }
475
476 Math::real EllipticFunction::deltaD(real sn, real cn, real dn) const {
477 // Function is periodic with period pi
478 if (cn < 0) { cn = -cn; sn = -sn; }
479 return D(sn, cn, dn) * (Math::pi()/2) / D() - atan2(sn, cn);
480 }
481
482 Math::real EllipticFunction::deltaG(real sn, real cn, real dn) const {
483 // Function is periodic with period pi
484 if (cn < 0) { cn = -cn; sn = -sn; }
485 return G(sn, cn, dn) * (Math::pi()/2) / G() - atan2(sn, cn);
486 }
487
488 Math::real EllipticFunction::deltaH(real sn, real cn, real dn) const {
489 // Function is periodic with period pi
490 if (cn < 0) { cn = -cn; sn = -sn; }
491 return H(sn, cn, dn) * (Math::pi()/2) / H() - atan2(sn, cn);
492 }
493
495 real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
496 return abs(phi) < Math::pi() ? F(sn, cn, dn) :
497 (deltaF(sn, cn, dn) + phi) * K() / (Math::pi()/2);
498 }
499
501 real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
502 return abs(phi) < Math::pi() ? E(sn, cn, dn) :
503 (deltaE(sn, cn, dn) + phi) * E() / (Math::pi()/2);
504 }
505
507 real n = ceil(ang/360 - real(0.5));
508 ang -= 360 * n;
509 real sn, cn;
510 Math::sincosd(ang, sn, cn);
511 return E(sn, cn, Delta(sn, cn)) + 4 * E() * n;
512 }
513
515 real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
516 return abs(phi) < Math::pi() ? Pi(sn, cn, dn) :
517 (deltaPi(sn, cn, dn) + phi) * Pi() / (Math::pi()/2);
518 }
519
521 real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
522 return abs(phi) < Math::pi() ? D(sn, cn, dn) :
523 (deltaD(sn, cn, dn) + phi) * D() / (Math::pi()/2);
524 }
525
527 real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
528 return abs(phi) < Math::pi() ? G(sn, cn, dn) :
529 (deltaG(sn, cn, dn) + phi) * G() / (Math::pi()/2);
530 }
531
533 real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
534 return abs(phi) < Math::pi() ? H(sn, cn, dn) :
535 (deltaH(sn, cn, dn) + phi) * H() / (Math::pi()/2);
536 }
537
539 static const real tolJAC =
540 sqrt(numeric_limits<real>::epsilon() * real(0.01));
541 real n = floor(x / (2 * _Ec) + real(0.5));
542 x -= 2 * _Ec * n; // x now in [-ec, ec)
543 // Linear approximation
544 real phi = Math::pi() * x / (2 * _Ec); // phi in [-pi/2, pi/2)
545 // First order correction
546 phi -= _eps * sin(2 * phi) / 2;
547 // For kp2 close to zero use asin(x/_Ec) or
548 // J. P. Boyd, Applied Math. and Computation 218, 7005-7013 (2012)
549 // https://doi.org/10.1016/j.amc.2011.12.021
550 for (int i = 0; i < num_ || GEOGRAPHICLIB_PANIC; ++i) {
551 real
552 sn = sin(phi),
553 cn = cos(phi),
554 dn = Delta(sn, cn),
555 err = (E(sn, cn, dn) - x)/dn;
556 phi -= err;
557 if (!(abs(err) > tolJAC))
558 break;
559 }
560 return n * Math::pi() + phi;
561 }
562
563 Math::real EllipticFunction::deltaEinv(real stau, real ctau) const {
564 // Function is periodic with period pi
565 if (ctau < 0) { ctau = -ctau; stau = -stau; }
566 real tau = atan2(stau, ctau);
567 return Einv( tau * E() / (Math::pi()/2) ) - tau;
568 }
569
570} // namespace GeographicLib
Header for GeographicLib::EllipticFunction class.
#define GEOGRAPHICLIB_PANIC
Definition: Math.hpp:61
void sncndn(real x, real &sn, real &cn, real &dn) const
static real RJ(real x, real y, real z, real p)
Math::real deltaG(real sn, real cn, real dn) const
static real RG(real x, real y, real z)
Math::real deltaE(real sn, real cn, real dn) const
Math::real F(real phi) const
static real RC(real x, real y)
Math::real Einv(real x) const
static real RD(real x, real y, real z)
void Reset(real k2=0, real alpha2=0)
Math::real Delta(real sn, real cn) const
Math::real deltaD(real sn, real cn, real dn) const
Math::real Ed(real ang) const
Math::real deltaH(real sn, real cn, real dn) const
Math::real deltaF(real sn, real cn, real dn) const
static real RF(real x, real y, real z)
Math::real deltaPi(real sn, real cn, real dn) const
Math::real deltaEinv(real stau, real ctau) const
Exception handling for GeographicLib.
Definition: Constants.hpp:315
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:126
static T sq(T x)
Definition: Math.hpp:171
static T infinity()
Definition: Math.cpp:276
static T pi()
Definition: Math.hpp:149
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)