GeographicLib  1.48
EllipticFunction.cpp
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1 /**
2  * \file EllipticFunction.cpp
3  * \brief Implementation for GeographicLib::EllipticFunction class
4  *
5  * Copyright (c) Charles Karney (2008-2016) <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 
17 namespace 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  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  // http://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 
66  Math::real EllipticFunction::RF(real x, real y) {
67  // Carlson, eqs 2.36 - 2.38
68  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 
81  Math::real EllipticFunction::RC(real x, real y) {
82  // Defined only for y != 0 and x >= 0.
83  return ( !(x >= y) ? // x < y and catch nans
84  // http://dlmf.nist.gov/19.2.E18
85  atan(sqrt((y - x) / x)) / sqrt(y - x) :
86  ( x == y ? 1 / sqrt(y) :
87  Math::asinh( y > 0 ?
88  // http://dlmf.nist.gov/19.2.E19
89  // atanh(sqrt((x - y) / x))
90  sqrt((x - y) / y) :
91  // http://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  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  real tolRD = pow(real(0.2) * (numeric_limits<real>::epsilon() * real(0.01)),
130  1/real(8));
131  real
132  A0 = (x + y + z + 2*p)/5,
133  An = A0,
134  delta = (p-x) * (p-y) * (p-z),
135  Q = max(max(abs(A0-x), abs(A0-y)), max(abs(A0-z), abs(A0-p))) / tolRD,
136  x0 = x,
137  y0 = y,
138  z0 = z,
139  p0 = p,
140  mul = 1,
141  mul3 = 1,
142  s = 0;
143  while (Q >= mul * abs(An)) {
144  // Max 7 trips
145  real
146  lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0),
147  d0 = (sqrt(p0)+sqrt(x0)) * (sqrt(p0)+sqrt(y0)) * (sqrt(p0)+sqrt(z0)),
148  e0 = delta/(mul3 * Math::sq(d0));
149  s += RC(1, 1 + e0)/(mul * d0);
150  An = (An + lam)/4;
151  x0 = (x0 + lam)/4;
152  y0 = (y0 + lam)/4;
153  z0 = (z0 + lam)/4;
154  p0 = (p0 + lam)/4;
155  mul *= 4;
156  mul3 *= 64;
157  }
158  real
159  X = (A0 - x) / (mul * An),
160  Y = (A0 - y) / (mul * An),
161  Z = (A0 - z) / (mul * An),
162  P = -(X + Y + Z) / 2,
163  E2 = X*Y + X*Z + Y*Z - 3*P*P,
164  E3 = X*Y*Z + 2*P * (E2 + 2*P*P),
165  E4 = (2*X*Y*Z + P * (E2 + 3*P*P)) * P,
166  E5 = X*Y*Z*P*P;
167  // http://dlmf.nist.gov/19.36.E2
168  // Polynomial is
169  // (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
170  // - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272
171  // - 9*(E3*E4+E2*E5)/68)
172  return ((471240 - 540540 * E2) * E5 +
173  (612612 * E2 - 540540 * E3 - 556920) * E4 +
174  E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
175  E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
176  (4084080 * mul * An * sqrt(An)) + 6 * s;
177  }
178 
179  Math::real EllipticFunction::RD(real x, real y, real z) {
180  // Carlson, eqs 2.28 - 2.34
181  real tolRD = pow(real(0.2) * (numeric_limits<real>::epsilon() * real(0.01)),
182  1/real(8));
183  real
184  A0 = (x + y + 3*z)/5,
185  An = A0,
186  Q = max(max(abs(A0-x), abs(A0-y)), abs(A0-z)) / tolRD,
187  x0 = x,
188  y0 = y,
189  z0 = z,
190  mul = 1,
191  s = 0;
192  while (Q >= mul * abs(An)) {
193  // Max 7 trips
194  real lam = sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0);
195  s += 1/(mul * sqrt(z0) * (z0 + lam));
196  An = (An + lam)/4;
197  x0 = (x0 + lam)/4;
198  y0 = (y0 + lam)/4;
199  z0 = (z0 + lam)/4;
200  mul *= 4;
201  }
202  real
203  X = (A0 - x) / (mul * An),
204  Y = (A0 - y) / (mul * An),
205  Z = -(X + Y) / 3,
206  E2 = X*Y - 6*Z*Z,
207  E3 = (3*X*Y - 8*Z*Z)*Z,
208  E4 = 3 * (X*Y - Z*Z) * Z*Z,
209  E5 = X*Y*Z*Z*Z;
210  // http://dlmf.nist.gov/19.36.E2
211  // Polynomial is
212  // (1 - 3*E2/14 + E3/6 + 9*E2^2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
213  // - E2^3/16 + 3*E3^2/40 + 3*E2*E4/20 + 45*E2^2*E3/272
214  // - 9*(E3*E4+E2*E5)/68)
215  return ((471240 - 540540 * E2) * E5 +
216  (612612 * E2 - 540540 * E3 - 556920) * E4 +
217  E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
218  E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
219  (4084080 * mul * An * sqrt(An)) + 3 * s;
220  }
221 
222  void EllipticFunction::Reset(real k2, real alpha2,
223  real kp2, real alphap2) {
224  // Accept nans here (needed for GeodesicExact)
225  if (k2 > 1)
226  throw GeographicErr("Parameter k2 is not in (-inf, 1]");
227  if (alpha2 > 1)
228  throw GeographicErr("Parameter alpha2 is not in (-inf, 1]");
229  if (kp2 < 0)
230  throw GeographicErr("Parameter kp2 is not in [0, inf)");
231  if (alphap2 < 0)
232  throw GeographicErr("Parameter alphap2 is not in [0, inf)");
233  _k2 = k2;
234  _kp2 = kp2;
235  _alpha2 = alpha2;
236  _alphap2 = alphap2;
237  _eps = _k2/Math::sq(sqrt(_kp2) + 1);
238  // Values of complete elliptic integrals for k = 0,1 and alpha = 0,1
239  // K E D
240  // k = 0: pi/2 pi/2 pi/4
241  // k = 1: inf 1 inf
242  // Pi G H
243  // k = 0, alpha = 0: pi/2 pi/2 pi/4
244  // k = 1, alpha = 0: inf 1 1
245  // k = 0, alpha = 1: inf inf pi/2
246  // k = 1, alpha = 1: inf inf inf
247  //
248  // Pi(0, k) = K(k)
249  // G(0, k) = E(k)
250  // H(0, k) = K(k) - D(k)
251  // Pi(0, k) = K(k)
252  // G(0, k) = E(k)
253  // H(0, k) = K(k) - D(k)
254  // Pi(alpha2, 0) = pi/(2*sqrt(1-alpha2))
255  // G(alpha2, 0) = pi/(2*sqrt(1-alpha2))
256  // H(alpha2, 0) = pi/(2*(1 + sqrt(1-alpha2)))
257  // Pi(alpha2, 1) = inf
258  // H(1, k) = K(k)
259  // G(alpha2, 1) = H(alpha2, 1) = RC(1, alphap2)
260  if (_k2) {
261  // Complete elliptic integral K(k), Carlson eq. 4.1
262  // http://dlmf.nist.gov/19.25.E1
263  _Kc = _kp2 ? RF(_kp2, 1) : Math::infinity();
264  // Complete elliptic integral E(k), Carlson eq. 4.2
265  // http://dlmf.nist.gov/19.25.E1
266  _Ec = _kp2 ? 2 * RG(_kp2, 1) : 1;
267  // D(k) = (K(k) - E(k))/k^2, Carlson eq.4.3
268  // http://dlmf.nist.gov/19.25.E1
269  _Dc = _kp2 ? RD(0, _kp2, 1) / 3 : Math::infinity();
270  } else {
271  _Kc = _Ec = Math::pi()/2; _Dc = _Kc/2;
272  }
273  if (_alpha2) {
274  // http://dlmf.nist.gov/19.25.E2
275  real rj = (_kp2 != 0 && _alphap2 != 0) ? RJ(0, _kp2, 1, _alphap2) :
276  Math::infinity(),
277  // Only use rc if _kp2 = 0.
278  rc = _kp2 ? 0 : (_alphap2 ? RC(1, _alphap2) : Math::infinity());
279  // Pi(alpha^2, k)
280  _Pic = _kp2 != 0 ? _Kc + _alpha2 * rj / 3 : Math::infinity();
281  // G(alpha^2, k)
282  _Gc = _kp2 ? _Kc + (_alpha2 - _k2) * rj / 3 : rc;
283  // H(alpha^2, k)
284  _Hc = _kp2 ? _Kc - (_alphap2 ? _alphap2 * rj : 0) / 3 : rc;
285  } else {
286  _Pic = _Kc; _Gc = _Ec;
287  // Hc = Kc - Dc but this involves large cancellations if k2 is close to
288  // 1. So write (for alpha2 = 0)
289  // Hc = int(cos(phi)^2/sqrt(1-k2*sin(phi)^2),phi,0,pi/2)
290  // = 1/sqrt(1-k2) * int(sin(phi)^2/sqrt(1-k2/kp2*sin(phi)^2,...)
291  // = 1/kp * D(i*k/kp)
292  // and use D(k) = RD(0, kp2, 1) / 3
293  // so Hc = 1/kp * RD(0, 1/kp2, 1) / 3
294  // = kp2 * RD(0, 1, kp2) / 3
295  // using http://dlmf.nist.gov/19.20.E18
296  // Equivalently
297  // RF(x, 1) - RD(0, x, 1)/3 = x * RD(0, 1, x)/3 for x > 0
298  // For k2 = 1 and alpha2 = 0, we have
299  // Hc = int(cos(phi),...) = 1
300  _Hc = _kp2 ? _kp2 * RD(0, 1, _kp2) / 3 : 1;
301  }
302  }
303 
304  /*
305  * Implementation of methods given in
306  *
307  * R. Bulirsch
308  * Numerical Calculation of Elliptic Integrals and Elliptic Functions
309  * Numericshe Mathematik 7, 78-90 (1965)
310  */
311 
312  void EllipticFunction::sncndn(real x, real& sn, real& cn, real& dn)
313  const {
314  // Bulirsch's sncndn routine, p 89.
315  real tolJAC = sqrt(numeric_limits<real>::epsilon() * real(0.01));
316  if (_kp2 != 0) {
317  real mc = _kp2, d = 0;
318  if (_kp2 < 0) {
319  d = 1 - mc;
320  mc /= -d;
321  d = sqrt(d);
322  x *= d;
323  }
324  real c = 0; // To suppress warning about uninitialized variable
325  real m[num_], n[num_];
326  unsigned l = 0;
327  for (real a = 1; l < num_ || GEOGRAPHICLIB_PANIC; ++l) {
328  // This converges quadratically. Max 5 trips
329  m[l] = a;
330  n[l] = mc = sqrt(mc);
331  c = (a + mc) / 2;
332  if (!(abs(a - mc) > tolJAC * a)) {
333  ++l;
334  break;
335  }
336  mc *= a;
337  a = c;
338  }
339  x *= c;
340  sn = sin(x);
341  cn = cos(x);
342  dn = 1;
343  if (sn != 0) {
344  real a = cn / sn;
345  c *= a;
346  while (l--) {
347  real b = m[l];
348  a *= c;
349  c *= dn;
350  dn = (n[l] + a) / (b + a);
351  a = c / b;
352  }
353  a = 1 / sqrt(c*c + 1);
354  sn = sn < 0 ? -a : a;
355  cn = c * sn;
356  if (_kp2 < 0) {
357  swap(cn, dn);
358  sn /= d;
359  }
360  }
361  } else {
362  sn = tanh(x);
363  dn = cn = 1 / cosh(x);
364  }
365  }
366 
367  Math::real EllipticFunction::F(real sn, real cn, real dn) const {
368  // Carlson, eq. 4.5 and
369  // http://dlmf.nist.gov/19.25.E5
370  real cn2 = cn*cn, dn2 = dn*dn,
371  fi = cn2 ? abs(sn) * RF(cn2, dn2, 1) : K();
372  // Enforce usual trig-like symmetries
373  if (cn < 0)
374  fi = 2 * K() - fi;
375  return Math::copysign(fi, sn);
376  }
377 
378  Math::real EllipticFunction::E(real sn, real cn, real dn) const {
379  real
380  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
381  ei = cn2 ? abs(sn) *( _k2 <= 0 ?
382  // Carlson, eq. 4.6 and
383  // http://dlmf.nist.gov/19.25.E9
384  RF(cn2, dn2, 1) - _k2 * sn2 * RD(cn2, dn2, 1) / 3 :
385  ( _kp2 >= 0 ?
386  // http://dlmf.nist.gov/19.25.E10
387  _kp2 * RF(cn2, dn2, 1) +
388  _k2 * _kp2 * sn2 * RD(cn2, 1, dn2) / 3 +
389  _k2 * abs(cn) / dn :
390  // http://dlmf.nist.gov/19.25.E11
391  - _kp2 * sn2 * RD(dn2, 1, cn2) / 3 +
392  dn / abs(cn) ) ) :
393  E();
394  // Enforce usual trig-like symmetries
395  if (cn < 0)
396  ei = 2 * E() - ei;
397  return Math::copysign(ei, sn);
398  }
399 
400  Math::real EllipticFunction::D(real sn, real cn, real dn) const {
401  // Carlson, eq. 4.8 and
402  // http://dlmf.nist.gov/19.25.E13
403  real
404  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
405  di = cn2 ? abs(sn) * sn2 * RD(cn2, dn2, 1) / 3 : D();
406  // Enforce usual trig-like symmetries
407  if (cn < 0)
408  di = 2 * D() - di;
409  return Math::copysign(di, sn);
410  }
411 
412  Math::real EllipticFunction::Pi(real sn, real cn, real dn) const {
413  // Carlson, eq. 4.7 and
414  // http://dlmf.nist.gov/19.25.E14
415  real
416  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
417  pii = cn2 ? abs(sn) * (RF(cn2, dn2, 1) +
418  _alpha2 * sn2 *
419  RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
420  Pi();
421  // Enforce usual trig-like symmetries
422  if (cn < 0)
423  pii = 2 * Pi() - pii;
424  return Math::copysign(pii, sn);
425  }
426 
427  Math::real EllipticFunction::G(real sn, real cn, real dn) const {
428  real
429  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
430  gi = cn2 ? abs(sn) * (RF(cn2, dn2, 1) +
431  (_alpha2 - _k2) * sn2 *
432  RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
433  G();
434  // Enforce usual trig-like symmetries
435  if (cn < 0)
436  gi = 2 * G() - gi;
437  return Math::copysign(gi, sn);
438  }
439 
440  Math::real EllipticFunction::H(real sn, real cn, real dn) const {
441  real
442  cn2 = cn*cn, dn2 = dn*dn, sn2 = sn*sn,
443  // WARNING: large cancellation if k2 = 1, alpha2 = 0, and phi near pi/2
444  hi = cn2 ? abs(sn) * (RF(cn2, dn2, 1) -
445  _alphap2 * sn2 *
446  RJ(cn2, dn2, 1, cn2 + _alphap2 * sn2) / 3) :
447  H();
448  // Enforce usual trig-like symmetries
449  if (cn < 0)
450  hi = 2 * H() - hi;
451  return Math::copysign(hi, sn);
452  }
453 
454  Math::real EllipticFunction::deltaF(real sn, real cn, real dn) const {
455  // Function is periodic with period pi
456  if (cn < 0) { cn = -cn; sn = -sn; }
457  return F(sn, cn, dn) * (Math::pi()/2) / K() - atan2(sn, cn);
458  }
459 
460  Math::real EllipticFunction::deltaE(real sn, real cn, real dn) const {
461  // Function is periodic with period pi
462  if (cn < 0) { cn = -cn; sn = -sn; }
463  return E(sn, cn, dn) * (Math::pi()/2) / E() - atan2(sn, cn);
464  }
465 
466  Math::real EllipticFunction::deltaPi(real sn, real cn, real dn) const {
467  // Function is periodic with period pi
468  if (cn < 0) { cn = -cn; sn = -sn; }
469  return Pi(sn, cn, dn) * (Math::pi()/2) / Pi() - atan2(sn, cn);
470  }
471 
472  Math::real EllipticFunction::deltaD(real sn, real cn, real dn) const {
473  // Function is periodic with period pi
474  if (cn < 0) { cn = -cn; sn = -sn; }
475  return D(sn, cn, dn) * (Math::pi()/2) / D() - atan2(sn, cn);
476  }
477 
478  Math::real EllipticFunction::deltaG(real sn, real cn, real dn) const {
479  // Function is periodic with period pi
480  if (cn < 0) { cn = -cn; sn = -sn; }
481  return G(sn, cn, dn) * (Math::pi()/2) / G() - atan2(sn, cn);
482  }
483 
484  Math::real EllipticFunction::deltaH(real sn, real cn, real dn) const {
485  // Function is periodic with period pi
486  if (cn < 0) { cn = -cn; sn = -sn; }
487  return H(sn, cn, dn) * (Math::pi()/2) / H() - atan2(sn, cn);
488  }
489 
490  Math::real EllipticFunction::F(real phi) const {
491  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
492  return abs(phi) < Math::pi() ? F(sn, cn, dn) :
493  (deltaF(sn, cn, dn) + phi) * K() / (Math::pi()/2);
494  }
495 
496  Math::real EllipticFunction::E(real phi) const {
497  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
498  return abs(phi) < Math::pi() ? E(sn, cn, dn) :
499  (deltaE(sn, cn, dn) + phi) * E() / (Math::pi()/2);
500  }
501 
503  real n = ceil(ang/360 - real(0.5));
504  ang -= 360 * n;
505  real sn, cn;
506  Math::sincosd(ang, sn, cn);
507  return E(sn, cn, Delta(sn, cn)) + 4 * E() * n;
508  }
509 
511  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
512  return abs(phi) < Math::pi() ? Pi(sn, cn, dn) :
513  (deltaPi(sn, cn, dn) + phi) * Pi() / (Math::pi()/2);
514  }
515 
516  Math::real EllipticFunction::D(real phi) const {
517  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
518  return abs(phi) < Math::pi() ? D(sn, cn, dn) :
519  (deltaD(sn, cn, dn) + phi) * D() / (Math::pi()/2);
520  }
521 
522  Math::real EllipticFunction::G(real phi) const {
523  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
524  return abs(phi) < Math::pi() ? G(sn, cn, dn) :
525  (deltaG(sn, cn, dn) + phi) * G() / (Math::pi()/2);
526  }
527 
528  Math::real EllipticFunction::H(real phi) const {
529  real sn = sin(phi), cn = cos(phi), dn = Delta(sn, cn);
530  return abs(phi) < Math::pi() ? H(sn, cn, dn) :
531  (deltaH(sn, cn, dn) + phi) * H() / (Math::pi()/2);
532  }
533 
535  real tolJAC = sqrt(numeric_limits<real>::epsilon() * real(0.01));
536  real n = floor(x / (2 * _Ec) + real(0.5));
537  x -= 2 * _Ec * n; // x now in [-ec, ec)
538  // Linear approximation
539  real phi = Math::pi() * x / (2 * _Ec); // phi in [-pi/2, pi/2)
540  // First order correction
541  phi -= _eps * sin(2 * phi) / 2;
542  // For kp2 close to zero use asin(x/_Ec) or
543  // J. P. Boyd, Applied Math. and Computation 218, 7005-7013 (2012)
544  // https://doi.org/10.1016/j.amc.2011.12.021
545  for (int i = 0; i < num_ || GEOGRAPHICLIB_PANIC; ++i) {
546  real
547  sn = sin(phi),
548  cn = cos(phi),
549  dn = Delta(sn, cn),
550  err = (E(sn, cn, dn) - x)/dn;
551  phi -= err;
552  if (abs(err) < tolJAC)
553  break;
554  }
555  return n * Math::pi() + phi;
556  }
557 
558  Math::real EllipticFunction::deltaEinv(real stau, real ctau) const {
559  // Function is periodic with period pi
560  if (ctau < 0) { ctau = -ctau; stau = -stau; }
561  real tau = atan2(stau, ctau);
562  return Einv( tau * E() / (Math::pi()/2) ) - tau;
563  }
564 
565 } // namespace GeographicLib
static T pi()
Definition: Math.hpp:202
void sncndn(real x, real &sn, real &cn, real &dn) const
void Reset(real k2=0, real alpha2=0)
static T infinity()
Definition: Math.hpp:862
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
Math::real deltaPi(real sn, real cn, real dn) const
Math::real deltaE(real sn, real cn, real dn) const
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.hpp:555
static T asinh(T x)
Definition: Math.hpp:311
static real RG(real x, real y, real z)
static T sq(T x)
Definition: Math.hpp:232
static real RF(real x, real y, real z)
static real RC(real x, real y)
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
Header for GeographicLib::EllipticFunction class.
Math::real F(real phi) const
Math::real deltaH(real sn, real cn, real dn) const
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)
Math::real deltaG(real sn, real cn, real dn) const
Math::real Ed(real ang) const
static T copysign(T x, T y)
Definition: Math.hpp:748
Math::real Einv(real x) const
Exception handling for GeographicLib.
Definition: Constants.hpp:389
Math::real deltaD(real sn, real cn, real dn) const
static real RD(real x, real y, real z)
Math::real deltaEinv(real stau, real ctau) const
static real RJ(real x, real y, real z, real p)
#define GEOGRAPHICLIB_PANIC
Definition: Math.hpp:87
Math::real deltaF(real sn, real cn, real dn) const