Line data Source code
1 : /* Copyright (C) 2015 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 : #define DEBUGLEVEL DEBUGLEVEL_gammamellininv
19 :
20 : /*******************************************************************/
21 : /* Computation of inverse Mellin */
22 : /* transforms of gamma products. */
23 : /*******************************************************************/
24 : /* Handle complex Vga whose sum is real */
25 : static GEN
26 26012 : sumVga(GEN Vga) { return real_i(vecsum(Vga)); }
27 :
28 : /* ac != 0 */
29 : static double
30 617311 : lemma526_i(double ac, double c, double t, double B)
31 : {
32 617311 : double D = -B/ac; /* sgn(t) = sgn(a) = - sgn(D) */
33 617311 : if (D <= 0)
34 : {
35 519081 : if (D > -100)
36 : {
37 84210 : D = -exp(D) / t;
38 84210 : if (D < - 1/M_E) return 0;
39 83881 : D = dbllambertW_1(D);
40 : }
41 : else
42 : { /* avoid underflow, use asymptotic expansion */
43 434871 : double U = D - log(t);
44 434871 : D = U - log(-U);
45 : }
46 518752 : return pow(maxdd(t, -t * D), c);
47 : }
48 : else
49 : {
50 98230 : if (D < 100)
51 9870 : D = dbllambertW0(-exp(D) / t);
52 : else
53 : { /* avoid overflow, use asymptotic expansion */
54 88360 : double U = D - log(-t);
55 88360 : D = U - log(U);
56 : }
57 98230 : return pow(-t * D, c);
58 : }
59 : }
60 : /* b > 0, c > 0; solve x^a exp(-b x^(1/c)) < e^(-B) for x >= 0 */
61 : double
62 14 : dbllemma526(double a, double b, double c, double B)
63 : {
64 : double ac;
65 14 : if (!a) return B <= 0? 0: pow(B/b, c);
66 14 : ac = a*c; if (B < 0) B = 1e-9;
67 14 : return lemma526_i(ac, c, ac/b, B);
68 : }
69 : /* Same, special case b/c = 2Pi, the only one needed: for c = d/2 */
70 : double
71 2560107 : dblcoro526(double a, double c, double B)
72 : {
73 2560107 : if (!a) return B <= 0? 0: pow(B/(2*M_PI*c), c);
74 617297 : if (B < 0) B = 1e-9;
75 617297 : return lemma526_i(a*c, c, a/(2*M_PI), B);
76 : }
77 :
78 : static const double MELLININV_CUTOFF = 121.; /* C*C */
79 :
80 : /* x real */
81 : static GEN
82 11158 : RMOD2(GEN x) { return gsub(x, gmul2n(gdiventgs(x,2), 1)); }
83 : /* x real or complex, return canonical representative for x mod 2Z */
84 : static GEN
85 11158 : MOD2(GEN x)
86 11158 : { return typ(x) == t_COMPLEX? mkcomplex(RMOD2(gel(x,1)), gel(x,2)): RMOD2(x); }
87 : static GEN
88 3388 : RgV_MOD2(GEN x)
89 14546 : { pari_APPLY_same(MOD2(gel(x,i))); }
90 :
91 : /* classes of poles of the gamma factor mod 2Z, sorted by increasing
92 : * Re(s) mod 2 (in [0,2[).*/
93 : static GEN
94 3388 : gammapoles(GEN Vga, long *pdV, long bit)
95 : {
96 3388 : long i, m, emax, l = lg(Vga);
97 3388 : GEN P, B = RgV_MOD2(Vga), V = cgetg(l, t_VEC);
98 3388 : P = gen_indexsort(B, (void*)lexcmp, cmp_nodata);
99 9464 : for (i = m = 1; i < l;)
100 : {
101 6076 : GEN u = gel(B, P[i]);
102 : long k;
103 11158 : for(k = i+1; k < l; k++)
104 : {
105 7770 : GEN v = gsub(u, gel(B, P[k]));
106 7770 : if (!gequal0(v) && (!isinexactreal(v) || gexpo(v) > -bit)) break;
107 : }
108 6076 : gel(V, m++) = vecslice(P,i,k-1);
109 6076 : i = k;
110 : }
111 3388 : setlg(V, m); emax = 0;
112 9464 : for (i = 1; i < m; i++)
113 : {
114 6076 : long j, e = 0, li = lg(gel(V,i))-1;
115 6076 : GEN b = gel(B, gel(V,i)[1]);
116 18466 : for (j = 1; j < m; j++)
117 12390 : if (j != i) e -= gexpo(gsub(gel(B, gel(V,j)[1]), b));
118 6076 : emax = maxss(emax, e * li);
119 : }
120 9464 : for (i = 1; i < m; i++) gel(V,i) = vecpermute(Vga, gel(V,i));
121 3388 : *pdV = emax; return V;
122 : }
123 :
124 : static GEN
125 553987 : sercoeff(GEN x, long n, long prec)
126 : {
127 553987 : long N = n - valser(x);
128 553987 : return (N < 0)? gen_0: gprec_wtrunc(gel(x, N+2), prec);
129 : }
130 :
131 : /* prod_i Gamma(s/2 + (m+LA[i])/2), set t *= prod_i (s/2 + (m+LA[i])/2) */
132 : static GEN
133 12390 : get_gamma(GEN *pt, GEN LA, GEN m, int round, long precdl, long prec)
134 : {
135 12390 : long i, l = lg(LA);
136 12390 : GEN pr = NULL, t = *pt;
137 33187 : for (i = 1; i < l; i++)
138 : {
139 20797 : GEN u, g, a = gmul2n(gadd(m, gel(LA,i)), -1);
140 20797 : if (round) a = ground(a);
141 20797 : u = deg1pol_shallow(ghalf, a, 0);
142 20797 : g = ggamma(RgX_to_ser(u, precdl), prec);
143 20797 : pr = pr? gmul(pr, g): g;
144 20797 : t = t? gmul(t, u): u;
145 : }
146 12390 : *pt = t; return pr;
147 : }
148 : /* generalized power series expansion of inverse Mellin around x = 0;
149 : * m-th derivative */
150 : static GEN
151 3388 : Kderivsmallinit(GEN ldata, GEN Vga, long m, long bit)
152 : {
153 3388 : const double C2 = MELLININV_CUTOFF;
154 3388 : long prec2, N, j, l, dLA, limn, d = lg(Vga)-1;
155 : GEN piA, LA, L, M, mat;
156 :
157 3388 : LA = gammapoles(Vga, &dLA, bit); N = lg(LA)-1;
158 3388 : prec2 = nbits2prec(dLA + bit * (1 + M_PI*d/C2));
159 : #if BITS_IN_LONG == 32
160 484 : if (odd(prec2lg(prec2))) prec2 += BITS_IN_LONG;
161 : #endif
162 3388 : if (ldata) Vga = ldata_get_gammavec(ldata_newprec(ldata, prec2));
163 3388 : L = cgetg(N+1, t_VECSMALL);
164 3388 : M = cgetg(N+1, t_VEC);
165 3388 : mat = cgetg(N+1, t_VEC);
166 3388 : limn = ceil(2*M_LN2*bit / (d * dbllambertW0(C2/(M_PI*M_E*d))));
167 3388 : l = limn + 2;
168 9464 : for (j = 1; j <= N; j++)
169 : {
170 6076 : GEN S = gel(LA,j);
171 6076 : GEN C, c, mj, G = NULL, t = NULL, tj = NULL;
172 6076 : long i, k, n, jj, lj = L[j] = lg(S)-1, precdl = lj+3;
173 :
174 6076 : gel(M,j) = mj = gsubsg(2, gel(S, vecindexmin(real_i(S))));
175 18466 : for (jj = 1; jj <= N; jj++)
176 : { /* if jj = j, poles come from this class only */
177 12390 : GEN g = get_gamma((jj==j)? &tj: &t, gel(LA,jj), mj, jj==j, precdl, prec2);
178 12390 : G = G? gmul(G, g): g;
179 : }
180 6076 : c = cgetg(limn+2,t_COL); gel(c,1) = G;
181 321398 : for (n=1; n <= limn; n++)
182 : {
183 315322 : GEN A = utoineg(2*n), T = RgX_Rg_translate(tj, A);
184 : /* T = exact polynomial, may vanish at 0 (=> pole in c[n+1]) */
185 315322 : if (t) T = RgX_mul(T, RgX_Rg_translate(t, A)); /* no pole here */
186 315322 : gel(c,n+1) = gdiv(gel(c,n), T);
187 : }
188 6076 : gel(mat, j) = C = cgetg(lj+1, t_COL);
189 17234 : for (k = 1; k <= lj; k++)
190 : {
191 11158 : GEN L = cgetg(l, t_POL);
192 565145 : for (n = 2; n < l; n++) gel(L,n) = sercoeff(gel(c,n), -k, prec2);
193 11158 : L[1] = evalsigne(1)|evalvarn(0); gel(C,k) = L;
194 : }
195 : /* C[k] = \sum_n c_{j,k} t^n =: C_k(t) in Dokchitser's Algo 3.3
196 : * m-th derivative of t^(-M+2) sum_k (-ln t)^k/k! C_k(t^2) */
197 6076 : if (m)
198 : {
199 119 : mj = gsubgs(mj, 2);
200 238 : for (i = 1; i <= m; i++, mj = gaddgs(mj,1))
201 322 : for (k = 1; k <= lj; k++)
202 : {
203 203 : GEN c = gel(C,k), d = RgX_shift_shallow(gmul2n(RgX_deriv(c),1), 1);
204 203 : c = RgX_Rg_mul(c, mj);
205 203 : if (k < lj) c = RgX_add(c, gel(C,k+1));
206 203 : gel(C,k) = RgX_sub(d, c);
207 : }
208 119 : gel(M,j) = gaddgs(mj,2);
209 : }
210 17234 : for (k = 1; k <= lj; k++)
211 : {
212 11158 : GEN c = gel(C,k);
213 11158 : if (k > 2) c = RgX_Rg_div(c, mpfact(k-1));
214 11158 : gel(C,k) = RgX_to_RgC(c, lgpol(c));
215 : }
216 : }
217 : /* Algo 3.3: * \phi^(m)(t) = sum_j t^m_j sum_k (-ln t)^k mat[j,k](t^2) */
218 3388 : piA = gsubsg(m*d, sumVga(Vga));
219 3388 : if (!gequal0(piA)) piA = powPis(gmul2n(piA,-1), prec2);
220 3388 : return mkvec5(L, RgV_neg(M), mat, mkvecsmall(prec2), piA);
221 : }
222 :
223 : /* Evaluate a vector considered as a polynomial using Horner. */
224 : static GEN
225 316562 : evalvec(GEN vec, long N, GEN u)
226 : {
227 316562 : GEN S = gen_0;
228 : long n;
229 316562 : N = minss(N, lg(vec)-1);
230 11043089 : for (n = N; n >= 1; n--) S = gmul(u, gadd(gel(vec,n), S));
231 316562 : return S;
232 : }
233 :
234 : /* gammamellininvinit accessors */
235 : static double
236 5817683 : get_tmax(long bitprec)
237 5817683 : { return (M_LN2 / MELLININV_CUTOFF) * bitprec ; }
238 : static GEN
239 5726 : GMi_get_Vga(GEN K) { return gel(K,2); }
240 : static long
241 14023567 : GMi_get_degree(GEN K) { return lg(gel(K,2))-1; }
242 : static long
243 6964204 : GMi_get_m(GEN K) { return itos( gel(K,3) ); }
244 : static GEN /* [lj,mj,mat,prec2], Kderivsmall only */
245 7142874 : GMi_get_VS(GEN K) { return gel(K,4); }
246 : /* K[5] = [Ms,cd,A2], Kderivlarge only */
247 : static long/*Kderivlarge*/
248 6964204 : GMi_get_status(GEN K) { return itos(gmael3(K,5,1,2)); }
249 : static GEN/*Kderivlarge*/
250 6964204 : GMi_get_M(GEN K) { return gmael3(K,5,1,1); }
251 : static GEN/*Kderivlarge*/
252 6964204 : GMi_get_cd(GEN K) { return gmael(K,5,2); }
253 : static GEN/*Kderivlarge*/
254 13928408 : GMi_get_A2(GEN K) { return gmael(K,5,3); }
255 :
256 : static double
257 7053539 : GMi_get_tmax(GEN K, long bitprec)
258 7053539 : { return (typ(GMi_get_VS(K)) == t_INT)? -1.0 : get_tmax(bitprec); }
259 :
260 : /* Compute m-th derivative of inverse Mellin at x by generalized power series
261 : * around x = 0; x2d = x^(2/d), x is possibly NULL (don't bother about
262 : * complex branches). Assume |x|^(2/d) <= tmax = M_LN2*bitprec/MELLININV_CUTOFF*/
263 : static GEN
264 89335 : Kderivsmall(GEN K, GEN x, GEN x2d, long bitprec)
265 : {
266 89335 : GEN VS = GMi_get_VS(K), L = gel(VS,1), M = gel(VS,2), mat = gel(VS,3);
267 89335 : GEN d2, Lx, x2, S, pi, piA = gel(VS,5);
268 89335 : long j, k, prec = gel(VS,4)[1], d = GMi_get_degree(K), limn, N = lg(L)-1;
269 89335 : double xd, Wd, Ed = M_LN2*bitprec / d;
270 :
271 89335 : xd = maxdd(M_PI*dblmodulus(x2d), 1E-13); /* pi |x|^2/d unless x tiny */
272 89335 : if (xd > Ed) pari_err_BUG("Kderivsmall (x2d too large)");
273 : /* Lemma 5.2.6 (2), a = 1 + log(Pi x^(2/d)) = log(e / xd),
274 : * B = log(2)*bitprec / d = Ed */
275 89335 : Wd = dbllambertW0( Ed / (M_E*xd) ); /* solution of w exp(w) = B exp(-a)*/
276 89335 : limn = (long) ceil(2*Ed/Wd);
277 89335 : pi = mppi(prec);
278 89335 : d2 = gdivsg(d,gen_2);
279 89335 : if (x)
280 1150 : x = gmul(gtofp(x,prec), gpow(pi,d2,prec));
281 : else
282 88185 : x = gpow(gmul(gtofp(x2d,prec),pi), d2, prec);
283 : /* at this stage, x has been replaced by pi^(d/2) x */
284 89335 : x2 = gsqr(x);
285 89335 : Lx = gpowers(gneg(glog(x,prec)), vecsmall_max(L));
286 89335 : S = gen_0;
287 247467 : for (j = 1; j <= N; ++j)
288 : {
289 158132 : long lj = L[j];
290 158132 : GEN s = gen_0;
291 474694 : for (k = 1; k <= lj; k++)
292 316562 : s = gadd(s, gmul(gel(Lx,k), evalvec(gmael(mat,j,k), limn, x2)));
293 158132 : S = gadd(S, gmul(gpow(x, gel(M,j), prec), s));
294 : }
295 89335 : if (!gequal0(piA)) S = gmul(S, piA);
296 89335 : return S;
297 : }
298 :
299 : /* In Klarge, we conpute K(t) as (asymptotic) * F(z), where F ~ 1 is given by
300 : * a continued fraction and z = Pi t^(2/d). If we take 2n terms in F (n terms
301 : * in Euler form), F_n(z) - F(z) is experimentally in exp(- C sqrt(n*z))
302 : * where C ~ 8 for d > 2 [HEURISTIC] and C = 4 (theorem) for d = 1 or d = 2
303 : * and vga = [0,1]. For e^(-E) absolute error, we want
304 : * exp(-C sqrt(nz)) < e^-(E+a), where a ~ ln(asymptotic)
305 : * i.e. 2n > (E+a)^2 / t^(2/d) * 2/(C^2 Pi); C^2*Pi/2 ~ 100.5 ~ 101
306 : *
307 : * In fact, this model becomes wrong for z large: we use instead
308 : *
309 : * exp(- sqrt(D * nz/log(z+1))) < e^-(E+a),
310 : * i.e. 2n > (E+a)^2 * log(1 + Pi t^(2/d))/ t^(2/d) * 2/(D Pi); */
311 : static double
312 6986562 : get_D(long d) { return d <= 2 ? 157. : 180.; }
313 : /* if (abs), absolute error rather than relative */
314 : static void
315 6964204 : Kderivlarge_optim(GEN K, int abs, GEN t2d, double cd, long *pbitprec, long *pnlim)
316 : {
317 6964204 : GEN A2 = GMi_get_A2(K);
318 6964204 : long bitprec = *pbitprec, d = GMi_get_degree(K);
319 6964204 : const double D = get_D(d), td = dblmodulus(t2d);
320 6964204 : double a, rtd, E = M_LN2*bitprec;
321 :
322 : /* t = 0 can happen with finite continued fraction or easyvga */
323 6964204 : if (!td) { *pnlim = 0; return; }
324 6964197 : rtd = (typ(t2d) == t_COMPLEX)? gtodouble(gel(t2d,1)): td;
325 : /* A2/2 = A, log(td) = (2/d)*log t */
326 6964197 : a = d*gtodouble(A2)*log2(td)/2 - (M_PI/M_LN2)*d*rtd + log2(cd);/*log2 K(t)~a*/
327 : /* if bitprec <= 0, caller should return K(t) ~ 0 */
328 6964197 : bitprec += 64;
329 6964197 : if (abs)
330 : {
331 6959530 : bitprec += ceil(a);
332 6959530 : if (a <= -65) E = M_LN2*bitprec; /* guarantees E <= initial E */
333 : }
334 6964197 : *pbitprec = bitprec;
335 6964197 : *pnlim = ceil(E*E * log2(1+M_PI*td) / (D*td));
336 : }
337 :
338 : /* Compute m-th derivative of inverse Mellin at t by continued fraction of
339 : * asymptotic expansion; t2d = t^(2/d). If t is NULL, "lfun" mode: don't
340 : * bother about complex branches + use absolute (rather than relative)
341 : * accuracy */
342 : static GEN
343 6964204 : Kderivlarge(GEN K, GEN t, GEN t2d, long bitprec0)
344 : {
345 : GEN tdA, P, S, pi, z;
346 6964204 : const long d = GMi_get_degree(K);
347 6964204 : GEN M = GMi_get_M(K), cd = GMi_get_cd(K), A2 = GMi_get_A2(K);
348 6964204 : long prec, nlim, status = GMi_get_status(K), m = GMi_get_m(K), bitprec = bitprec0;
349 :
350 6964204 : Kderivlarge_optim(K, !t, t2d, gtodouble(cd), &bitprec, &nlim);
351 6964204 : if (bitprec <= 0) return gen_0;
352 6879070 : prec = nbits2prec(bitprec);
353 6879070 : t2d = gtofp(t2d, prec);
354 6879070 : if (t)
355 4674 : tdA = gpow(t, gdivgu(A2,d), prec);
356 : else
357 6874396 : tdA = gpow(t2d, gdivgu(A2,2), prec);
358 6879070 : pi = mppi(prec); z = gmul(pi, t2d);
359 6879070 : P = gmul(gmul(cd, tdA), gexp(gmulsg(-d, z), prec));
360 6879070 : if (m) P = gmul(P, gpowgs(mulsr(-2, pi), m));
361 6879070 : if (status == 2) /* finite continued fraction */
362 1256023 : S = (lg(M) == 2)? gel(M,1): poleval(M, ginv(z));
363 : else
364 : {
365 5623047 : S = contfraceval_inv(M, z, nlim/2);
366 5623047 : if (DEBUGLEVEL>3)
367 : {
368 0 : GEN S0 = contfraceval_inv(M, z, minss(nlim/2 + 1, minss(lg(gel(M, 1)) - 1, lg(gel(M, 2)))));
369 0 : long e = gexpo(gmul(P, gsub(S,S0)));
370 0 : if (-e < bitprec0)
371 0 : err_printf("Kderivlarge: e = %ld, bit = %ld\n",e,bitprec0);
372 : }
373 5623047 : if (status == 1) S = gmul(S, gsubsg(1, ginv(gmul(z, pi))));
374 : }
375 6879070 : return gmul(P, S);
376 : }
377 :
378 : /* Dokchitser's coefficients used for asymptotic expansion of inverse Mellin
379 : * 2 <= p <= min(n+1, d), c = 2n-p+1; sh = (sh(x)/x)^(d-p) */
380 : static GEN
381 1065349 : vp(long p, long c, GEN SMd, GEN sh)
382 : {
383 1065349 : GEN s, ve = cgetg(p+2, t_VEC);
384 : long m, j, k;
385 :
386 1065349 : gel(ve,1) = gen_1; gel(ve,2) = utoipos(c);
387 3107450 : for (j = 2; j <= p; j++) gel(ve,j+1) = gdivgs(gmulgu(gel(ve,j), c), j);
388 1065349 : s = gel(SMd, 1);
389 4172799 : for (m = 1; m <= p; m++)
390 : {
391 3107450 : GEN t, c = gel(SMd, m+1);
392 3107450 : if (gequal0(c)) continue;
393 1639055 : t = gel(ve, m+1);
394 3283885 : for (k = 1; k <= m/2; k++)
395 1644830 : t = gadd(t, gmul(gel(ve, m-2*k+1), RgX_coeff(sh, k)));
396 1639055 : s = gadd(s, gmul(c, t));
397 : }
398 1065349 : return s;
399 : }
400 :
401 : static GEN
402 6783 : get_SM(GEN Vga)
403 : {
404 6783 : long k, m, d = lg(Vga)-1;
405 6783 : GEN pol, nS1, SM, C, t = vecsum(Vga);
406 :
407 6783 : pol = roots_to_pol(gmulgs(Vga, -d), 0); /* deg(pol) = d */
408 6783 : SM = cgetg(d+2, t_VEC); gel(SM,1) = gen_1;
409 6783 : if (gequal0(t))
410 : { /* shortcut */
411 9688 : for (m = 1; m <= d; m++) gel(SM,m+1) = gel(pol,d+2-m);
412 2436 : return SM;
413 : }
414 4347 : nS1 = gpowers(gneg(t), d); C = matpascal(d);
415 19012 : for (m = 1; m <= d; m++)
416 : {
417 14665 : pari_sp av = avma;
418 14665 : GEN s = gmul(gel(nS1, m+1), gcoeff(C, d+1, m+1));
419 49210 : for (k = 1; k <= m; k++)
420 : {
421 34545 : GEN e = gmul(gel(nS1, m-k+1), gcoeff(C, d-k+1, m-k+1));
422 34545 : s = gadd(s, gmul(e, RgX_coeff(pol, d-k)));
423 : }
424 14665 : gel(SM, m+1) = gc_upto(av, s);
425 : }
426 4347 : return SM;
427 : }
428 :
429 : static GEN
430 6783 : get_SMd(GEN Vga)
431 : {
432 6783 : GEN M, SM = get_SM(Vga);
433 6783 : long p, m, d = lg(Vga)-1;
434 :
435 6783 : M = cgetg(d, t_VEC);
436 21917 : for (p = 2; p <= d; p++)
437 : {
438 15134 : GEN a = gen_1, c;
439 15134 : long D = d - p;
440 15134 : gel(M, p-1) = c = cgetg(p+2, t_COL);
441 15134 : gel(c, 1) = gel(SM, p+1);
442 58954 : for (m = 1; m <= p; m++)
443 : {
444 43820 : a = muliu(a, D + m);
445 43820 : gel(c, m+1) = gmul(gel(SM, p-m+1), a);
446 : }
447 : }
448 6783 : return M;
449 : }
450 :
451 : /* Asymptotic expansion of inverse Mellin, to length nlimmax. Set status = 0
452 : * (regular), 1 (one Hankel determinant vanishes => contfracinit will fail)
453 : * or 2 (same as 1, but asymptotic expansion is finite!)
454 : *
455 : * If status = 2, the asymptotic expansion is finite so return only
456 : * the necessary number of terms nlim <= nlimmax + d. */
457 : static GEN
458 25641 : Klargeinit(GEN Vga, long nlimmax, long *status, long prec)
459 : {
460 25641 : long d = lg(Vga) - 1, p, n, cnt;
461 : GEN M, SMd, se, vsinh, vd;
462 :
463 25641 : if (Vgaeasytheta(Vga)) { *status = 2; return mkvec(gen_1); }
464 : /* d >= 2 */
465 6783 : *status = 0;
466 6783 : if (prec) prec += nbits2extraprec((prec >> 1) + BITS_IN_LONG);
467 6783 : SMd = get_SMd(Vga);
468 6783 : se = gsinh(RgX_to_ser(pol_x(0), d+2), 0); setvalser(se,0);
469 6783 : se = gdeflate(se, 0, 2); /* se(x^2) = sinh(x)/x */
470 6783 : vsinh = gpowers(se, d);
471 6783 : vd = gpowers(utoipos(2*d), d);
472 6783 : M = cgetg(nlimmax + d + 1, t_VEC); gel(M,1) = gen_1;
473 476369 : for (n = 2, cnt = 0; n <= nlimmax || cnt; n++)
474 : {
475 476369 : pari_sp av = avma;
476 476369 : long ld = minss(d, n);
477 476369 : GEN s = gen_0;
478 1541718 : for (p = 2; p <= ld; p++)
479 : {
480 1065349 : GEN z = vp(p, 2*n-1-p, gel(SMd, p-1), gel(vsinh, d-p+1));
481 1065349 : s = gadd(s, gmul(gdiv(z, gel(vd, p+1)), gel(M, n+1-p)));
482 : }
483 476369 : if (prec && !isinexact(s)) s = gtofp(s, prec);
484 476369 : gel(M,n) = s = gc_upto(av, gdivgs(s, 1-n));
485 476369 : if (gequal0(s))
486 : {
487 154 : cnt++; *status = 1;
488 154 : if (cnt >= d-1) { *status = 2; n -= d-2; break; }
489 : }
490 : else
491 : {
492 476215 : if (n >= nlimmax) { n++; break; }
493 469558 : cnt = 0;
494 : }
495 : }
496 6783 : setlg(M, n); return M;
497 : }
498 :
499 : /* remove trailing zeros from vector. */
500 : static void
501 266 : stripzeros(GEN M)
502 : {
503 : long i;
504 469 : for(i = lg(M)-1; i >= 1; --i)
505 469 : if (!gequal0(gel(M, i))) break;
506 266 : setlg(M, i+1);
507 266 : }
508 :
509 : /* Asymptotic expansion of the m-th derivative of inverse Mellin, to length
510 : * nlimmax. If status = 2, the asymptotic expansion is finite so return only
511 : * the necessary number of terms nlim <= nlimmax + d. */
512 : static GEN
513 22379 : gammamellininvasymp_i(GEN Vga, long nlimmax, long m, long *status, long prec)
514 : {
515 : GEN M, A, Aadd;
516 : long d, i, nlim, n;
517 :
518 22379 : M = Klargeinit(Vga, nlimmax, status, prec);
519 22379 : if (!m) return M;
520 266 : d = lg(Vga)-1;
521 : /* half the exponent of t in asymptotic expansion. */
522 266 : A = gdivgu(gaddsg(1-d, sumVga(Vga)), 2*d);
523 266 : if (*status == 2) M = shallowconcat(M, zerovec(m));
524 266 : nlim = lg(M)-1;
525 266 : Aadd = sstoQ(2-d, 2*d); /* (1/d) - (1/2) */
526 560 : for (i = 1; i <= m; i++, A = gadd(A,Aadd))
527 8050 : for (n = nlim-1; n >= 1; --n)
528 15512 : gel(M, n+1) = gsub(gel(M, n+1),
529 15512 : gmul(gel(M, n), gsub(A, uutoQ(n-1, d))));
530 266 : stripzeros(M); return M;
531 : }
532 :
533 : INLINE int
534 22365 : RgV_is_CV(GEN x)
535 : {
536 : long i;
537 85120 : for (i = lg(x)-1; i > 0; i--)
538 : {
539 84532 : long t = typ(gel(x,i));
540 84532 : if (!is_real_t(t) && t!= t_COMPLEX) return 0;
541 : }
542 588 : return 1;
543 : }
544 :
545 : static GEN
546 22393 : get_Vga(GEN x, GEN *ldata)
547 : {
548 22393 : if (typ(x)==t_VEC && RgV_is_CV(x)) { *ldata = NULL; return x; }
549 21805 : *ldata = lfunmisc_to_ldata_shallow_i(x);
550 21805 : if (*ldata) x = ldata_get_gammavec(*ldata);
551 21805 : return x;
552 : }
553 : GEN
554 28 : gammamellininvasymp(GEN Vga, long nlim, long m)
555 : {
556 28 : pari_sp av = avma;
557 : long status;
558 : GEN ldata;
559 28 : Vga = get_Vga(Vga, &ldata);
560 28 : if (!is_vec_t(typ(Vga)) || lg(Vga) == 1)
561 7 : pari_err_TYPE("gammamellininvasymp",Vga);
562 21 : return gc_GEN(av, gammamellininvasymp_i(Vga, nlim, m, &status, 0));
563 : }
564 :
565 : /* Does the continued fraction of the asymptotic expansion M at oo of inverse
566 : * Mellin transform attached to Vga have zero Hankel determinants ? */
567 : static long
568 3381 : ishankelspec(GEN Vga)
569 : {
570 3381 : long status, i, d = lg(Vga)-1;
571 : GEN M;
572 :
573 3381 : if (d == 5 || d == 7)
574 21 : { /* known bad cases: a x 5 or 7 */
575 133 : GEN v1 = gel(Vga, 1);
576 588 : for (i = 2; i <= d; ++i)
577 476 : if (!gequal(gel(Vga,i), v1)) break;
578 133 : if (i > d) return 1;
579 : }
580 3248 : else if (d==10 || d==14)
581 : { /* [ a x 5 or 7, (a+1) x 5 or 7] */
582 7 : long d2 = d>>1;
583 7 : long s0 = 1, s1 = 0, sm1 = 0;
584 7 : GEN v1 = gel(Vga, 1);
585 70 : for (i = 2; i <= d; i++)
586 : {
587 63 : GEN s = gsub(gel(Vga,i),v1);
588 63 : if (gequal0(s)) s0++;
589 35 : else if (gequal1(s)) s1++;
590 0 : else if (gequalm1(s)) sm1++;
591 : }
592 7 : if (s0==d2 && (s1==d2 || sm1==d2)) return 1;
593 : }
594 : /* Heuristic: if 6 first terms in contfracinit don't fail, assume OK */
595 3262 : M = Klargeinit(Vga, 7, &status, 0);
596 3262 : return !contfracinit_i(M, 6);
597 : }
598 :
599 : /* Initialize data for computing m-th derivative of inverse Mellin */
600 : GEN
601 22372 : gammamellininvinit(GEN Vga, long m, long bitprec)
602 : {
603 22372 : const double C2 = MELLININV_CUTOFF;
604 22372 : pari_sp ltop = avma;
605 : GEN A2, M, VS, VL, cd, ldata;
606 22372 : long nlimmax, status, d, prec = nbits2prec((4*bitprec)/3);
607 22372 : double E = M_LN2*bitprec, tmax = get_tmax(bitprec); /* = E/C2 */
608 :
609 22372 : if (m < 0)
610 7 : pari_err_DOMAIN("gammamellininvinit", "derivation order", "<", gen_0, stoi(m));
611 22365 : Vga = get_Vga(Vga, &ldata); d = lg(Vga)-1;
612 22365 : if (!is_vec_t(typ(Vga)) || !d) pari_err_TYPE("gammamellininvinit",Vga);
613 22358 : nlimmax = ceil(E * log2(1+M_PI*tmax) * C2 / get_D(d));
614 22358 : A2 = gaddsg(m*(2-d) + 1-d, sumVga(Vga));
615 22358 : cd = (d <= 2)? gen_2: gsqrt(gdivgu(int2n(d+1), d), nbits2prec(bitprec));
616 : /* if in Klarge, we have |t| > tmax = E/C2, thus nlim < E*C2/D. */
617 22358 : M = gammamellininvasymp_i(Vga, nlimmax, m, &status, prec);
618 22358 : if (status == 2)
619 : {
620 18970 : tmax = -1.; /* only use Klarge */
621 18970 : VS = gen_0;
622 : }
623 : else
624 : {
625 3388 : VS = Kderivsmallinit(ldata, Vga, m, bitprec);
626 3388 : if (status == 0 && ishankelspec(Vga)) status = 1;
627 3388 : if (status == 1)
628 : { /* a Hankel determinant vanishes => contfracinit is undefined.
629 : So compute K(t) / (1 - 1/(pi^2*t)) instead of K(t)*/
630 133 : GEN t = ginv(mppi(prec));
631 : long i;
632 20902 : for (i = 2; i < lg(M); ++i)
633 20769 : gel(M, i) = gadd(gel(M, i), gmul(gel(M, i-1), t));
634 : }
635 : else
636 3255 : M = RgC_gtofp(M, prec); /* convert from rationals to t_REAL: faster */
637 3388 : M = contfracinit(M, lg(M)-2);
638 : }
639 22358 : VL = mkvec3(mkvec2(M, stoi(status)), cd, A2);
640 22358 : return gc_GEN(ltop, mkvec5(dbltor(tmax), Vga, stoi(m), VS, VL));
641 : }
642 :
643 : /* Compute m-th derivative of inverse Mellin at s2d = s^(d/2) using
644 : * initialization data. Use Taylor expansion at 0 for |s2d| < tmax, and
645 : * asymptotic expansion at oo otherwise. WARNING: assume that accuracy
646 : * has been increased according to tmax by the CALLING program. */
647 : static GEN
648 7053539 : gammamellininvrt_i(GEN K, GEN s, GEN s2d, long bit)
649 : {
650 7053539 : if (dblmodulus(s2d) < GMi_get_tmax(K, bit))
651 89335 : return Kderivsmall(K, s, s2d, bit);
652 : else
653 6964204 : return Kderivlarge(K, s, s2d, bit);
654 : }
655 : GEN
656 7047715 : gammamellininvrt(GEN K, GEN s2d, long bit)
657 7047715 : { return gammamellininvrt_i(K, NULL, s2d, bit); }
658 :
659 : /* Compute inverse Mellin at s. K from gammamellininv OR a Vga, in which
660 : * case the initialization data is computed. */
661 : GEN
662 5824 : gammamellininv(GEN K, GEN s, long m, long bitprec)
663 : {
664 5824 : pari_sp av = avma;
665 : GEN s2d;
666 : long d;
667 :
668 5824 : if (!is_vec_t(typ(K)) || lg(K) != 6 || !is_vec_t(typ(GMi_get_Vga(K))))
669 98 : K = gammamellininvinit(K, m, bitprec);
670 5824 : d = GMi_get_degree(K);
671 5824 : s2d = gpow(s, gdivgu(gen_2, d), nbits2prec(bitprec));
672 5824 : return gc_upto(av, gammamellininvrt_i(K, s, s2d, bitprec));
673 : }
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