Line data Source code
1 : /* Copyright (C) 2021 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 :
15 : #include "pari.h"
16 : #include "paripriv.h"
17 :
18 : /***********************************************************************/
19 : /** LAMBERT's W_K FUNCTIONS **/
20 : /***********************************************************************/
21 : /* roughly follows Veberic, https://arxiv.org/abs/1003.1628 */
22 :
23 : static double
24 18 : dblL1L2(double L1)
25 : {
26 18 : double L2 = log(-L1), LI = 1 / L1, N2, N3, N4, N5;
27 18 : N2 = (L2-2.)/2.; N3 = (6.+L2*(-9.+2.*L2))/6.;
28 18 : N4 = (-12.+L2*(36.+L2*(-22.+3*L2)))/12.;
29 18 : N5 = (60.+L2*(-300.+L2*(350.+L2*(-125.+12*L2))))/60.;
30 18 : return L1-L2+L2*LI*(1+LI*(N2+LI*(N3+LI*(N4+LI*N5))));
31 : }
32 :
33 : /* rough approximation to W0(a > -1/e), < 1% relative error */
34 : double
35 385946 : dbllambertW0(double a)
36 : {
37 385946 : if (a < -0.2583)
38 : {
39 0 : const double c2 = -1./3, c3 = 11./72, c4 = -43./540, c5 = 769./17280;
40 0 : double p = sqrt(2 * (M_E * a + 1));
41 0 : if (a < -0.3243) return -1+p*(1+p*(c2+p*c3));
42 0 : return -1+p*(1+p*(c2+p*(c3+p*(c4+p*c5))));
43 : }
44 : else
45 : {
46 385946 : double Wd = log(1.+a);
47 385946 : Wd *= (1.-log(Wd/a))/(1.+Wd);
48 385946 : if (a < 0.6482 && a > -0.1838) return Wd;
49 341159 : return Wd*(1.-log(Wd/a))/(1.+Wd);
50 : }
51 : }
52 : /* uniform approximation to W0, at least 15 bits. */
53 : static double
54 438 : dbllambertW0init(double a)
55 : {
56 438 : if (a < -0.323581)
57 : {
58 30 : const double c2 = 1./3., c3 = 11./72., c4 = 43./540., c5 = 769./17280.;
59 30 : const double c6 = 221./8505., c7 = 680863./43545600.;
60 30 : const double c8 = 1963./204120., c9 = 226287557./37623398400.;
61 30 : double p = M_E * a + 1;
62 30 : if (p <= 0) return -1;
63 20 : p = -sqrt(2 * p);
64 20 : return -(1.+p*(1.+p*(c2+p*(c3+p*(c4+p*(c5+p*(c6+p*(c7+p*(c8+p*c9)))))))));
65 : }
66 408 : if (a < 0.145469)
67 : {
68 96 : const double a1 = 5.931375, a2 = 11.392205, a3 = 7.338883, a4 = 0.653449;
69 96 : const double b1 = 6.931373, b2 = 16.823494, b3 = 16.430723, b4 = 5.115235;
70 96 : double n = 1.+a*(a1+a*(a2+a*(a3+a*a4)));
71 96 : double d = 1.+a*(b1+a*(b2+a*(b3+a*b4)));
72 96 : return a * n / d;
73 : }
74 312 : if (a < 8.706658)
75 : {
76 306 : const double a1 = 2.445053, a2 = 1.343664, a3 = 0.148440, a4 = 0.000804;
77 306 : const double b1 = 3.444708, b2 = 3.292489, b3 = 0.916460, b4 = 0.053068;
78 306 : double n = 1.+a*(a1+a*(a2+a*(a3+a*a4)));
79 306 : double d = 1.+a*(b1+a*(b2+a*(b3+a*b4)));
80 306 : return a * n / d;
81 : }
82 : else
83 : {
84 6 : double w = log(1.+a);
85 6 : w *= (1.-log(w/a)) / (1.+w);
86 6 : return w * (1.-log(w/a)) / (1.+w);
87 : }
88 : }
89 :
90 : /* rough approximation to W_{-1}(0 > a > -1/e), < 1% relative error */
91 : double
92 71247 : dbllambertW_1(double a)
93 : {
94 71247 : if (a < -0.2464)
95 : {
96 240 : const double c2 = -1./3, c3 = 11./72, c4 = -43./540, c5 = 769./17280;
97 240 : double p = -sqrt(2 * (M_E * a + 1));
98 240 : if (a < -0.3243) return -1+p*(1+p*(c2+p*c3));
99 150 : return -1+p*(1+p*(c2+p*(c3+p*(c4+p*c5))));
100 : }
101 : else
102 : {
103 : double Wd;
104 71007 : a = -a; Wd = -log(a);
105 71007 : Wd *= (1.-log(Wd/a))/(1.-Wd);
106 71007 : if (a < 0.0056) return -Wd;
107 4944 : return -Wd*(1.-log(Wd/a))/(1.-Wd);
108 : }
109 : }
110 : /* uniform approximation to W_{-1}, at least 15 bits. */
111 : static double
112 96 : dbllambertW_1init(double a)
113 : {
114 96 : if (a < -0.302985)
115 : {
116 18 : const double c2 = 1./3., c3 = 11./72., c4 = 43./540., c5 = 769./17280.;
117 18 : const double c6 = 221./8505., c7 = 680863./43545600.;
118 18 : const double c8 = 1963./204120., c9 = 226287557./37623398400.;
119 18 : double p = M_E * a + 1;
120 18 : if (p <= 0) return -1;
121 18 : p = sqrt(2 * p);
122 18 : return -(1.+p*(1.+p*(c2+p*(c3+p*(c4+p*(c5+p*(c6+p*(c7+p*(c8+p*c9)))))))));
123 : }
124 78 : if (a <= -0.051012)
125 : {
126 66 : const double a0 = -7.814176, a1 = 253.888101, a2 = 657.949317;
127 66 : const double b1 = -60.439587, b2 = 99.985670, b3 = 682.607399;
128 66 : const double b4 = 962.178439, b5 = 1477.934128;
129 66 : double n = a0+a*(a1+a*a2);
130 66 : double d = 1+a*(b1+a*(b2+a*(b3+a*(b4+a*b5))));
131 66 : return n / d;
132 : }
133 12 : return dblL1L2(log(-a));
134 : }
135 :
136 : /* uniform approximation to more than 46 bits, 50 bits away from -1/e;
137 : * branch = -1 or 0 */
138 : static double
139 540 : dbllambertWfritsch(GEN ga, int branch)
140 : {
141 : double a, z, w1, q, w;
142 540 : if (expo(ga) >= 0x3fe)
143 : { /* branch = 0 */
144 6 : double w = dbllog2(ga) * M_LN2; /* ~ log(1+a) ~ log a */
145 6 : return w * (1.+w-log(w)) / (1.+w);
146 : }
147 534 : a = rtodbl(ga);
148 534 : w = branch? dbllambertW_1init(a): dbllambertW0init(a);
149 534 : if (w == -1.|| w == 0.) return w;
150 524 : z = log(a / w) - w; w1 = 1. + w;
151 524 : q = 2. * w1 * (w1 + (2./3.) * z);
152 524 : return w * (1 + (z / w1) * (q - z) / (q - 2 * z));
153 : }
154 :
155 : static double
156 6 : dbllambertWhalleyspec(double loga)
157 : {
158 6 : double w = dblL1L2(loga);
159 : for(;;)
160 0 : {
161 6 : double n = w + log(-w) - loga, d = 1 - w, r = n / (d + n / d);
162 6 : w *= 1 - r; if (r < 2.e-15) return w;
163 : }
164 : }
165 : /* k = 0 or -1. */
166 : static GEN
167 546 : lambertW(GEN z, long k, long prec)
168 : {
169 546 : pari_sp av = avma;
170 546 : long bit = prec2nbits(prec), L = -(bit / 3 + 10), ct = 0, p;
171 : double wd;
172 : GEN w, vp;
173 :
174 546 : if (gequal0(z) && !k) return real_0(prec);
175 546 : z = gtofp(z, prec);
176 546 : if (k == -1)
177 : {
178 102 : long e = expo(z);
179 102 : if (signe(z) >= 0) pari_err_DOMAIN("lambertw", "z", ">", gen_0, z);
180 102 : wd = e < -512? dbllambertWhalleyspec(dbllog2(z) * M_LN2)
181 102 : : dbllambertWfritsch(z, -1);
182 : }
183 : else
184 444 : wd = dbllambertWfritsch(z, 0);
185 546 : if (fabs(wd + 1) < 1e-5)
186 : {
187 12 : long prec2 = prec + EXTRAPREC64;
188 12 : GEN Z = rtor(z, prec2);
189 12 : GEN t = addrs(mulrr(Z, gexp(gen_1, prec2)), 1);
190 12 : if (signe(t) <= 0) { set_avma(av); return real_m1(prec); }
191 0 : if (realprec(t) < prec)
192 : {
193 0 : prec2 += prec - realprec(t);
194 0 : Z = rtor(z, prec2);
195 0 : t = addrs(mulrr(Z, gexp(gen_1, prec2)), 1);
196 : }
197 0 : t = sqrtr(shiftr(t, 1));
198 0 : w = gprec_w(k == -1? subsr(-1, t) : subrs(t, 1), prec);
199 0 : p = prec; vp = NULL;
200 : }
201 : else
202 : { /* away from -1/e: can reduce accuracy and self-correct */
203 : long pb;
204 534 : w = wd == 0.? z: dbltor(wd);
205 534 : vp = cgetg(30, t_VECSMALL); pb = bit;
206 1196 : while (pb > BITS_IN_LONG * 3/4)
207 662 : { vp[++ct] = nbits2prec(pb); pb = (pb + 2) / 3; }
208 534 : p = vp[ct]; w = gprec_w(w, p);
209 : }
210 534 : if ((k == -1 && (bit < 192 || bit > 640)) || (k == 0 && bit > 1024))
211 : {
212 : for(;;)
213 13 : {
214 : GEN t, ew, n, d;
215 91 : ew = mplog(divrr(w, z)); n = addrr(w, ew); d = addrs(w, 1);
216 91 : t = divrr(n, shiftr(d, 1));
217 91 : w = mulrr(w, subsr(1, divrr(n, addrr(d, t))));
218 91 : if (p >= prec && expo(n) - expo(d) - expo(w) <= L) break;
219 13 : if (vp) { if (--ct) p = vp[ct]; w = gprec_w(w, ct? p: prec); }
220 : }
221 : }
222 : else
223 : {
224 : for(;;)
225 167 : {
226 : GEN t, ew, wew, n, d;
227 623 : ew = mpexp(w); wew = mulrr(w, ew); n = subrr(wew, z); d = addrr(ew, wew);
228 623 : t = divrr(mulrr(addrs(w, 2), n), shiftr(addrs(w, 1), 1));
229 623 : w = subrr(w, divrr(n, subrr(d, t)));
230 623 : if (p >= prec && expo(n) - expo(d) - expo(w) <= L) break;
231 167 : if (vp) { if (--ct) p = vp[ct]; w = gprec_w(w, ct? p: prec); }
232 : }
233 : }
234 534 : return gc_upto(av, w);
235 : }
236 :
237 : /*********************************************************************/
238 : /* Complex branches */
239 : /*********************************************************************/
240 :
241 : /* x *= (1 - (x + log(x) - L) / (x + 1)); L = log(z) + 2IPi * k */
242 : static GEN
243 542366 : lamaux(GEN x, GEN L, long *pe, long prec)
244 : {
245 542366 : GEN n = gsub(gadd(x, glog(x, prec)), L);
246 542366 : if (pe) *pe = maxss(4, -gexpo(n));
247 542366 : if (gequal0(imag_i(n))) n = real_i(n);
248 542366 : return gmul(x, gsubsg(1, gdiv(n, gaddsg(1, x))));
249 : }
250 :
251 : /* Complex branches, experimental */
252 : static GEN
253 67224 : lambertWC(GEN z, long branch, long prec)
254 : {
255 67224 : pari_sp av = avma;
256 : GEN w, pii2k, zl, lzl, L, Lz;
257 67224 : long bit0, si, j, fl = 0, lim = 6, lp = DEFAULTPREC, bit = prec2nbits(prec);
258 :
259 67224 : si = gsigne(imag_i(z)); if (!si) z = real_i(z);
260 67224 : pii2k = gmulsg(branch, PiI2(lp));
261 67224 : zl = gtofp(z, lp); lzl = glog(zl, lp);
262 : /* From here */
263 67224 : if (branch == 0 || branch * si < 0
264 22782 : || (si == 0 && gsigne(z) < 0 && branch == -1))
265 : {
266 44562 : GEN lnzl1 = gaddsg(1, glog(gneg(zl), lp));
267 44562 : if (si == 0) si = gsigne(lnzl1);
268 44562 : if ((branch == 0 || branch * si < 0) && gexpo(lnzl1) < -1)
269 : { /* close to -1/e */
270 2064 : w = gaddsg(1, gmul(z, gexp(gen_1, prec)));
271 2064 : w = gprec_wtrunc(w, lp);
272 2064 : w = gsqrt(gmul2n(w, 1), lp);
273 2064 : w = branch * si < 0? gsubsg(-1, w): gaddsg(-1, w);
274 2064 : lim = 10; fl = 1;
275 : }
276 44562 : if (branch == 0 && !fl && gexpo(lzl) < 0) { w = zl; fl = 1; }
277 : }
278 67224 : if (!fl)
279 : {
280 58416 : if (branch)
281 : {
282 43896 : GEN lr = glog(pii2k, lp);
283 43896 : w = gadd(gsub(gadd(pii2k, lzl), lr), gdiv(gsub(lr, lzl), pii2k));
284 : }
285 : else
286 : {
287 14520 : GEN p = gaddsg(1, gmul(z, gexp(gen_1, lp)));
288 14520 : w = gexpo(p) > 0? lzl: gaddgs(gsqrt(p, lp), -1);
289 : }
290 : }
291 : /* to here: heuristic */
292 67224 : L = gadd(lzl, pii2k);
293 411600 : for (j = 1; j < lim; j++) w = lamaux(w, L, NULL, lp);
294 67224 : Lz = NULL;
295 67224 : if (branch == 0 || branch == -1)
296 : {
297 44742 : Lz = glog(z, prec);
298 44742 : if (branch == -1)
299 : {
300 22260 : long flag = 1;
301 22260 : if (!si && signe(z) <= 0 && signe(addrs(Lz, 1))) flag = 0;
302 22260 : if (flag) Lz = gsub(Lz, PiI2(prec));
303 : }
304 : }
305 67224 : w = lamaux(w, L, &bit0, lp);
306 197990 : while (bit0 < bit || (Lz && gexpo(gsub(gadd(w, glog(w, prec)), Lz)) > 16-bit))
307 : {
308 130766 : long p = nbits2prec(bit0 <<= 1);
309 130766 : L = gadd(gmulsg(branch, PiI2(p)), glog(gprec_w(z, p), p));
310 130766 : w = lamaux(gprec_w(w, p), L, NULL, p);
311 : }
312 67224 : return gc_GEN(av, gprec_w(w, nbits2prec(bit)));
313 : }
314 :
315 : /* exp(t (1 + O(t^n))), n >= 0 */
316 : static GEN
317 132 : serexp0(long v, long n)
318 : {
319 132 : GEN y = cgetg(n+3, t_SER), t;
320 : long i;
321 132 : y[1] = evalsigne(1) | evalvarn(v) | evalvalser(0);
322 132 : gel(y,2) = gen_1; if (!n) return y;
323 126 : gel(y,3) = gen_1; if (n == 1) return y;
324 1110 : for (i=2, t = gen_2; i < n; i++, t = muliu(t,i)) gel(y,i+2) = mkfrac(gen_1,t);
325 102 : gel(y,i+2) = mkfrac(gen_1,t); return y;
326 : }
327 :
328 : /* series expansion of W at -1/e */
329 : static GEN
330 6 : Wbra(long N)
331 : {
332 6 : GEN v = cgetg(N + 2, t_VEC);
333 : long n;
334 6 : gel(v, 1) = gen_m1;
335 6 : gel(v, 2) = gen_1;
336 48 : for (n = 2; n <= N; n++)
337 : {
338 42 : GEN t = gel(v,n), a = gen_0;
339 42 : long k, K = (n - 1) >> 1;
340 114 : for (k = 1; k <= K; k++) t = gadd(t, gmul2n(gel(v,n-2*k), -k));
341 168 : for (k = 2; k < n; k++) a = gadd(a, gmul(gel(v,k+1), gel(v,n+2-k)));
342 42 : gel(v,n+1) = gsub(gdivgs(t, -n-1), gmul2n(a, -1));
343 : }
344 6 : return RgV_to_RgX(v, 0);
345 : }
346 :
347 : static GEN
348 132 : reverse(GEN y)
349 : {
350 132 : GEN z = ser2rfrac_i(y);
351 132 : long l = lg(z);
352 132 : return RgX_to_ser(RgXn_reverse(z, l-2), l-1);
353 : }
354 : static GEN
355 156 : serlambertW(GEN y, long branch, long prec)
356 : {
357 : long n, vy, val, v;
358 156 : GEN t = NULL;
359 :
360 156 : if (!signe(y)) return gcopy(y);
361 156 : v = valser(y);
362 156 : if (v < 0) pari_err_DOMAIN("lambertw","valuation", "<", gen_0, y);
363 150 : if (v > 0 && branch)
364 0 : pari_err_DOMAIN("lambertw [k != 0]", "x", "~", gen_0, y);
365 150 : vy = varn(y); n = lg(y)-3;
366 414 : for (val = 1; val < n; val++)
367 372 : if (!gequal0(polcoef_i(y, val, vy))) break;
368 150 : if (v)
369 : {
370 60 : t = serexp0(vy, n / val);
371 60 : setvalser(t, 1); t = reverse(t); /* rev(x exp(x)) */
372 : }
373 : else
374 : {
375 90 : GEN y0 = gel(y,2), x = glambertW(y0, branch, prec);
376 90 : if (val > n) return scalarser(x, vy, n+1);
377 84 : y = serchop0(y);
378 84 : if (gequalm1(x))
379 : { /* y0 ~ -1/e, branch = 0 or -1 */
380 12 : GEN p = gmul(shiftr(gexp(gen_1,prec), 1), y);
381 12 : if (odd(val)) pari_err(e_MISC, "odd valuation at branch point");
382 6 : p = gsqrt(p, prec); if (odd(branch)) p = gneg(p);
383 6 : n -= val >> 1;
384 6 : t = RgXn_eval(Wbra(n), ser2rfrac_i(p), n);
385 6 : return gtoser(t, varn(t), lg(p));
386 : }
387 72 : t = serexp0(vy, n / val);
388 : /* (x + t) exp(x + t) = (y0 + t y0/x) * exp(t) */
389 72 : t = gmul(deg1pol_shallow(gdiv(y0,x), y0, vy), t);
390 72 : t = gadd(x, reverse(serchop0(t)));
391 : }
392 132 : return normalizeser(gsubst(t, vy, y));
393 : }
394 :
395 : static GEN
396 24 : lambertp(GEN x)
397 : {
398 24 : pari_sp av = avma;
399 : long k, minv;
400 : GEN y;
401 :
402 24 : if (gequal0(x)) return gcopy(x);
403 24 : minv = equaliu(padic_p(x), 2)? 2: 1;
404 24 : if (valp(x) < minv) { x = leafcopy(x); setvalp(x, minv); }
405 24 : k = Qp_exp_prec(x);
406 24 : if (k < 0) return NULL;
407 24 : y = gpowgs(cvstop2(k, x), k - 1);
408 342 : for (k--; k; k--)
409 318 : y = gsub(gpowgs(cvstop2(k, x), k - 1), gdivgu(gmul(x, y), k + 1));
410 24 : return gc_upto(av, gmul(x, y));
411 : }
412 :
413 : /* y a t_REAL */
414 : static int
415 1884 : useC(GEN y, long k)
416 : {
417 1884 : if (signe(y) > 0 || (k && k != -1)) return k ? 1: 0;
418 840 : return gsigne(addsr(1, logr_abs(y))) > 0;
419 : }
420 : static GEN
421 69240 : glambertW_i(void *E, GEN y, long prec)
422 : {
423 : pari_sp av;
424 69240 : long k = (long)E, p;
425 : GEN z;
426 69240 : if (gequal0(y))
427 : {
428 18 : if (k) pari_err_DOMAIN("glambertW","argument","",gen_0,y);
429 12 : return gcopy(y);
430 : }
431 69222 : switch(typ(y))
432 : {
433 1884 : case t_REAL:
434 1884 : p = minss(prec, realprec(y));
435 1884 : return useC(y, k)? lambertWC(y, k, p): lambertW(y, k, p);
436 24 : case t_PADIC: z = lambertp(y);
437 24 : if (!z) pari_err_DOMAIN("glambertW(t_PADIC)","argument","",gen_0,y);
438 24 : return z;
439 65886 : case t_COMPLEX:
440 65886 : p = precision(y);
441 65886 : return lambertWC(y, k, p? p: prec);
442 1428 : default:
443 1428 : av = avma; if (!(z = toser_i(y))) break;
444 156 : return gc_upto(av, serlambertW(z, k, prec));
445 : }
446 1272 : return trans_evalgen("lambert", E, glambertW_i, y, prec);
447 : }
448 :
449 : GEN
450 67968 : glambertW(GEN y, long k, long prec) { return glambertW_i((void*)k, y, prec); }
451 : GEN
452 0 : mplambertW(GEN y, long prec) { return lambertW(y, 0, prec); }
453 :
454 : /*********************************************************************/
455 : /* Application */
456 : /*********************************************************************/
457 : /* Solve x - a * log(x) = b with a > 0 and b >= a * (1 - log(a)). */
458 : GEN
459 0 : mplambertx_logx(GEN a, GEN b, long bit)
460 : {
461 0 : pari_sp av = avma;
462 0 : GEN e = gexp(gneg(gdiv(b, a)), nbits2prec(bit));
463 0 : return gc_upto(av, gmul(gneg(a), lambertW(gneg(gdiv(e, a)), -1, bit)));
464 : }
465 : /* Special case a = 1, b = log(y): solve e^x / x = y with y >= exp(1). */
466 : GEN
467 0 : mplambertX(GEN y, long bit)
468 : {
469 0 : pari_sp av = avma;
470 0 : return gc_upto(av, gneg(lambertW(gneg(ginv(y)), -1, bit)));
471 : }
472 :
473 : /* Solve x * log(x) - a * x = b; if b < 0, assume a >= 1 + log |b|. */
474 : GEN
475 0 : mplambertxlogx_x(GEN a, GEN b, long bit)
476 : {
477 0 : pari_sp av = avma;
478 0 : long s = gsigne(b);
479 : GEN e;
480 0 : if (!s) return gen_0;
481 0 : e = gexp(gneg(a), nbits2prec(bit));
482 0 : return gc_upto(av, gdiv(b, lambertW(gmul(b, e), s > 0? 0: -1, bit)));
483 : }
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