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 16737 : sumVga(GEN Vga) { return real_i(vecsum(Vga)); }
27 :
28 : /* ac != 0 */
29 : static double
30 524561 : lemma526_i(double ac, double c, double t, double B)
31 : {
32 524561 : double D = -B/ac; /* sgn(t) = sgn(a) = - sgn(D) */
33 524561 : if (D <= 0)
34 : {
35 458160 : if (D > -100)
36 : {
37 67263 : D = -exp(D) / t;
38 67263 : if (D < - 1/M_E) return 0;
39 66934 : D = dbllambertW_1(D);
40 : }
41 : else
42 : { /* avoid underflow, use asymptotic expansion */
43 390897 : double U = D - log(t);
44 390897 : D = U - log(-U);
45 : }
46 457831 : return pow(maxdd(t, -t * D), c);
47 : }
48 : else
49 : {
50 66401 : if (D < 100)
51 9310 : D = dbllambertW0(-exp(D) / t);
52 : else
53 : { /* avoid overflow, use asymptotic expansion */
54 57091 : double U = D - log(-t);
55 57091 : D = U - log(U);
56 : }
57 66401 : 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 1741229 : dblcoro526(double a, double c, double B)
72 : {
73 1741229 : if (!a) return B <= 0? 0: pow(B/(2*M_PI*c), c);
74 524547 : if (B < 0) B = 1e-9;
75 524547 : 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 8771 : 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 8771 : MOD2(GEN x)
86 8771 : { return typ(x) == t_COMPLEX? mkcomplex(RMOD2(gel(x,1)), gel(x,2)): RMOD2(x); }
87 : static GEN
88 2723 : RgV_MOD2(GEN x)
89 11494 : { 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 2723 : gammapoles(GEN Vga, long *pdV, long bit)
95 : {
96 2723 : long i, m, emax, l = lg(Vga);
97 2723 : GEN P, B = RgV_MOD2(Vga), V = cgetg(l, t_VEC);
98 2723 : P = gen_indexsort(B, (void*)lexcmp, cmp_nodata);
99 7630 : for (i = m = 1; i < l;)
100 : {
101 4907 : GEN u = gel(B, P[i]);
102 : long k;
103 8771 : for(k = i+1; k < l; k++)
104 : {
105 6048 : GEN v = gsub(u, gel(B, P[k]));
106 6048 : if (!gequal0(v) && (!isinexactreal(v) || gexpo(v) > -bit)) break;
107 : }
108 4907 : gel(V, m++) = vecslice(P,i,k-1);
109 4907 : i = k;
110 : }
111 2723 : setlg(V, m); emax = 0;
112 7630 : for (i = 1; i < m; i++)
113 : {
114 4907 : long j, e = 0, li = lg(gel(V,i))-1;
115 4907 : GEN b = gel(B, gel(V,i)[1]);
116 15120 : for (j = 1; j < m; j++)
117 10213 : if (j != i) e -= gexpo(gsub(gel(B, gel(V,j)[1]), b));
118 4907 : emax = maxss(emax, e * li);
119 : }
120 7630 : for (i = 1; i < m; i++) gel(V,i) = vecpermute(Vga, gel(V,i));
121 2723 : *pdV = emax; return V;
122 : }
123 :
124 : static GEN
125 282869 : sercoeff(GEN x, long n, long prec)
126 : {
127 282869 : long N = n - valp(x);
128 282869 : 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 10213 : get_gamma(GEN *pt, GEN LA, GEN m, int round, long precdl, long prec)
134 : {
135 10213 : long i, l = lg(LA);
136 10213 : GEN pr = NULL, t = *pt;
137 26803 : for (i = 1; i < l; i++)
138 : {
139 16590 : GEN u, g, a = gmul2n(gadd(m, gel(LA,i)), -1);
140 16590 : if (round) a = ground(a);
141 16590 : u = deg1pol_shallow(ghalf, a, 0);
142 16590 : g = ggamma(RgX_to_ser(u, precdl), prec);
143 16590 : pr = pr? gmul(pr, g): g;
144 16590 : t = t? gmul(t, u): u;
145 : }
146 10213 : *pt = t; return pr;
147 : }
148 : /* generalized power series expansion of inverse Mellin around x = 0;
149 : * m-th derivative */
150 : static GEN
151 2723 : Kderivsmallinit(GEN ldata, GEN Vga, long m, long bit)
152 : {
153 2723 : const double C2 = MELLININV_CUTOFF;
154 2723 : long prec2, N, j, l, dLA, limn, d = lg(Vga)-1;
155 : GEN piA, LA, L, M, mat;
156 :
157 2723 : LA = gammapoles(Vga, &dLA, bit); N = lg(LA)-1;
158 2723 : prec2 = nbits2prec(dLA + bit * (1 + M_PI*d/C2));
159 : #if BITS_IN_LONG == 32
160 389 : prec2 += prec2 & 1L;
161 : #endif
162 2723 : if (ldata) Vga = ldata_get_gammavec(ldata_newprec(ldata, prec2));
163 2723 : L = cgetg(N+1, t_VECSMALL);
164 2723 : M = cgetg(N+1, t_VEC);
165 2723 : mat = cgetg(N+1, t_VEC);
166 2723 : limn = ceil(2*M_LN2*bit / (d * dbllambertW0(C2/(M_PI*M_E))));
167 2723 : l = limn + 2;
168 7630 : for (j = 1; j <= N; j++)
169 : {
170 4907 : GEN S = gel(LA,j);
171 4907 : GEN C, c, mj, G = NULL, t = NULL, tj = NULL;
172 4907 : long i, k, n, jj, lj = L[j] = lg(S)-1, precdl = lj+3;
173 :
174 4907 : gel(M,j) = mj = gsubsg(2, gel(S, vecindexmin(real_i(S))));
175 15120 : for (jj = 1; jj <= N; jj++)
176 : {
177 : GEN g;
178 10213 : if (jj == j) /* poles come from this class only */
179 4907 : g = get_gamma(&tj, gel(LA,jj), mj, 1, precdl, prec2);
180 : else
181 5306 : g = get_gamma(&t, gel(LA,jj), mj, 0, precdl, prec2);
182 10213 : G = G? gmul(G, g): g;
183 : }
184 4907 : c = cgetg(limn+2,t_COL); gel(c,1) = G;
185 170953 : for (n=1; n <= limn; n++)
186 : {
187 166046 : GEN A = utoineg(2*n), T = RgX_translate(tj, A);
188 : /* T = exact polynomial, may vanish at 0 (=> pole in c[n+1]) */
189 166046 : if (t) T = RgX_mul(T, RgX_translate(t, A)); /* no pole here */
190 166046 : gel(c,n+1) = gdiv(gel(c,n), T);
191 : }
192 4907 : gel(mat, j) = C = cgetg(lj+1, t_COL);
193 13678 : for (k = 1; k <= lj; k++)
194 : {
195 8771 : GEN L = cgetg(l, t_POL);
196 291640 : for (n = 2; n < l; n++) gel(L,n) = sercoeff(gel(c,n), -k, prec2);
197 8771 : L[1] = evalsigne(1)|evalvarn(0); gel(C,k) = L;
198 : }
199 : /* C[k] = \sum_n c_{j,k} t^n =: C_k(t) in Dokchitser's Algo 3.3
200 : * m-th derivative of t^(-M+2) sum_k (-ln t)^k/k! C_k(t^2) */
201 4907 : if (m)
202 : {
203 119 : mj = gsubgs(mj, 2);
204 238 : for (i = 1; i <= m; i++, mj = gaddgs(mj,1))
205 322 : for (k = 1; k <= lj; k++)
206 : {
207 203 : GEN c = gel(C,k), d = RgX_shift_shallow(gmul2n(RgX_deriv(c),1), 1);
208 203 : c = RgX_Rg_mul(c, mj);
209 203 : if (k < lj) c = RgX_add(c, gel(C,k+1));
210 203 : gel(C,k) = RgX_sub(d, c);
211 : }
212 119 : gel(M,j) = gaddgs(mj,2);
213 : }
214 13678 : for (k = 1; k <= lj; k++)
215 : {
216 8771 : GEN c = gel(C,k);
217 8771 : if (k > 2) c = RgX_Rg_div(c, mpfact(k-1));
218 8771 : gel(C,k) = RgX_to_RgC(c, lgpol(c));
219 : }
220 : }
221 : /* Algo 3.3: * \phi^(m)(t) = sum_j t^m_j sum_k (-ln t)^k mat[j,k](t^2) */
222 2723 : piA = gsubsg(m*d, sumVga(Vga));
223 2723 : if (!gequal0(piA)) piA = powPis(gmul2n(piA,-1), prec2);
224 2723 : return mkvec5(L, RgV_neg(M), mat, mkvecsmall(prec2), piA);
225 : }
226 :
227 : /* Evaluate a vector considered as a polynomial using Horner. */
228 : static GEN
229 239830 : evalvec(GEN vec, long N, GEN u)
230 : {
231 239830 : GEN S = gen_0;
232 : long n;
233 239830 : N = minss(N, lg(vec)-1);
234 7095485 : for (n = N; n >= 1; n--) S = gmul(u, gadd(gel(vec,n), S));
235 239847 : return S;
236 : }
237 :
238 : /* gammamellininvinit accessors */
239 : static double
240 4145229 : get_tmax(long bitprec)
241 4145229 : { return (M_LN2 / MELLININV_CUTOFF) * bitprec ; }
242 : static GEN
243 5726 : GMi_get_Vga(GEN K) { return gel(K,2); }
244 : static long
245 9030094 : GMi_get_degree(GEN K) { return lg(gel(K,2))-1; }
246 : static long
247 4478031 : GMi_get_m(GEN K) { return itos( gel(K,3) ); }
248 : static GEN /* [lj,mj,mat,prec2], Kderivsmall only */
249 4616110 : GMi_get_VS(GEN K) { return gel(K,4); }
250 : static GEN /* [Ms,cd,A2], Kderivlarge only */
251 8955333 : GMi_get_VL(GEN K) { return gel(K,5); }
252 : static double
253 4546971 : GMi_get_tmax(GEN K, long bitprec)
254 4546971 : { return (typ(GMi_get_VS(K)) == t_INT)? -1.0 : get_tmax(bitprec); }
255 :
256 : /* Compute m-th derivative of inverse Mellin at x by generalized power series
257 : * around x = 0; x2d = x^(2/d), x is possibly NULL (don't bother about
258 : * complex branches). Assume |x|^(2/d) <= tmax = M_LN2*bitprec/MELLININV_CUTOFF*/
259 : static GEN
260 69068 : Kderivsmall(GEN K, GEN x, GEN x2d, long bitprec)
261 : {
262 69068 : GEN VS = GMi_get_VS(K), L = gel(VS,1), M = gel(VS,2), mat = gel(VS,3);
263 69068 : GEN d2, Lx, x2, S, pi, piA = gel(VS,5);
264 69068 : long prec = gel(VS,4)[1], d = GMi_get_degree(K);
265 69069 : long j, k, limn, N = lg(L)-1;
266 69069 : double xd, Wd, Ed = M_LN2*bitprec / d;
267 :
268 69069 : xd = maxdd(M_PI*dblmodulus(x2d), 1E-13); /* pi |x|^2/d unless x tiny */
269 69069 : if (xd > Ed) pari_err_BUG("Kderivsmall (x2d too large)");
270 : /* Lemma 5.2.6 (2), a = 1 + log(Pi x^(2/d)) = log(e / xd),
271 : * B = log(2)*bitprec / d = Ed */
272 69069 : Wd = dbllambertW0( Ed / (M_E*xd) ); /* solution of w exp(w) = B exp(-a)*/
273 69069 : limn = (long) ceil(2*Ed/Wd);
274 69069 : pi = mppi(prec);
275 69067 : d2 = gdivsg(d,gen_2);
276 69067 : if (x)
277 1122 : x = gmul(gtofp(x,prec), gpow(pi,d2,prec));
278 : else
279 67945 : x = gpow(gmul(gtofp(x2d,prec),pi), d2, prec);
280 : /* at this stage, x has been replaced by pi^(d/2) x */
281 69068 : x2 = gsqr(x);
282 69071 : Lx = gpowers(gneg(glog(x,prec)), vecsmall_max(L));
283 69069 : S = gen_0;
284 191589 : for (j = 1; j <= N; ++j)
285 : {
286 122521 : long lj = L[j];
287 122521 : GEN s = gen_0;
288 362355 : for (k = 1; k <= lj; k++)
289 239832 : s = gadd(s, gmul(gel(Lx,k), evalvec(gmael(mat,j,k), limn, x2)));
290 122523 : S = gadd(S, gmul(gpow(x, gel(M,j), prec), s));
291 : }
292 69068 : if (!gequal0(piA)) S = gmul(S, piA);
293 69069 : return S;
294 : }
295 :
296 : /* In Klarge, we conpute K(t) as (asymptotic) * F(z), where F ~ 1 is given by
297 : * a continued fraction and z = Pi t^(2/d). If we take 2n terms in F (n terms
298 : * in Euler form), F_n(z) - F(z) is experimentally in exp(- C sqrt(n*z))
299 : * where C ~ 8 for d > 2 [HEURISTIC] and C = 4 (theorem) for d = 1 or d = 2
300 : * and vga = [0,1]. For e^(-E) absolute error, we want
301 : * exp(-C sqrt(nz)) < e^-(E+a), where a ~ ln(asymptotic)
302 : * i.e. 2n > (E+a)^2 / t^(2/d) * 2/(C^2 Pi); C^2*Pi/2 ~ 100.5 ~ 101
303 : *
304 : * In fact, this model becomes wrong for z large: we use instead
305 : *
306 : * exp(- sqrt(D * nz/log(z+1))) < e^-(E+a),
307 : * i.e. 2n > (E+a)^2 * log(1 + Pi t^(2/d))/ t^(2/d) * 2/(D Pi); */
308 : static double
309 4491713 : get_D(long d) { return d <= 2 ? 157. : 180.; }
310 : /* if (abs), absolute error rather than relative */
311 : static void
312 4478152 : Kderivlarge_optim(GEN K, long abs, GEN t2d,GEN gcd, long *pbitprec, long *pnlim)
313 : {
314 4478152 : GEN VL = GMi_get_VL(K), A2 = gel(VL,3);
315 4477949 : long bitprec = *pbitprec, d = GMi_get_degree(K);
316 4477990 : const double D = get_D(d), td = dblmodulus(t2d), cd = gtodouble(gcd);
317 4477550 : double a, rtd, E = M_LN2*bitprec;
318 :
319 4477550 : rtd = (typ(t2d) == t_COMPLEX)? gtodouble(gel(t2d,1)): td;
320 : /* A2/2 = A, log(td) = (2/d)*log t */
321 4477550 : a = d*gtodouble(A2)*log2(td)/2 - (M_PI/M_LN2)*d*rtd + log2(cd);/*log2 K(t)~a*/
322 : /* if bitprec <= 0, caller should return K(t) ~ 0 */
323 4477233 : bitprec += 64;
324 4477233 : if (abs)
325 : {
326 4472635 : bitprec += ceil(a);
327 4472635 : if (a <= -65) E = M_LN2*bitprec; /* guarantees E <= initial E */
328 : }
329 4477233 : *pbitprec = bitprec;
330 4477233 : *pnlim = ceil(E*E * log2(1+M_PI*td) / (D*td));
331 4477233 : }
332 :
333 : /* Compute m-th derivative of inverse Mellin at t by continued fraction of
334 : * asymptotic expansion; t2d = t^(2/d). If t is NULL, "lfun" mode: don't
335 : * bother about complex branches + use absolute (rather than relative)
336 : * accuracy */
337 : static GEN
338 4478220 : Kderivlarge(GEN K, GEN t, GEN t2d, long bitprec0)
339 : {
340 : GEN tdA, P, S, pi, z;
341 4478220 : const long d = GMi_get_degree(K);
342 4477975 : GEN M, VL = GMi_get_VL(K), Ms = gel(VL,1), cd = gel(VL,2), A2 = gel(VL,3);
343 4477951 : long status, prec, nlim, m = GMi_get_m(K), bitprec = bitprec0;
344 :
345 4477977 : Kderivlarge_optim(K, !t, t2d, cd, &bitprec, &nlim);
346 4477725 : if (bitprec <= 0) return gen_0;
347 4392591 : prec = nbits2prec(bitprec);
348 4392526 : t2d = gtofp(t2d, prec);
349 4392600 : if (t)
350 4695 : tdA = gpow(t, gdivgu(A2,d), prec);
351 : else
352 4387905 : tdA = gpow(t2d, gdivgu(A2,2), prec);
353 4392551 : tdA = gmul(cd, tdA);
354 :
355 4392304 : pi = mppi(prec);
356 4392357 : z = gmul(pi, t2d);
357 4392947 : P = gmul(tdA, gexp(gmulsg(-d, z), prec));
358 4392886 : if (m) P = gmul(P, gpowgs(mulsr(-2, pi), m));
359 4392886 : M = gel(Ms,1); status = itos(gel(Ms,2));
360 4392777 : if (status == 2)
361 413205 : S = (lg(M) == 2)? gel(M,1) /* constant continued fraction */
362 413207 : : poleval(RgV_to_RgX(M, 0), ginv(z));
363 : else
364 : {
365 3979570 : S = contfraceval_inv(M, z, nlim/2);
366 3979796 : if (DEBUGLEVEL>3)
367 : {
368 0 : GEN S0 = contfraceval_inv(M, z, nlim/2 + 1);
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 3979796 : if (status == 1) S = gmul(S, gsubsg(1, ginv(gmul(z, pi))));
374 : }
375 4393001 : 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 803178 : vp(long p, long c, GEN SMd, GEN sh)
382 : {
383 803178 : GEN s, ve = cgetg(p+2, t_VEC);
384 : long m, j, k;
385 :
386 803178 : gel(ve,1) = gen_1; gel(ve,2) = utoipos(c);
387 2268633 : for (j = 2; j <= p; j++) gel(ve,j+1) = gdivgs(gmulgu(gel(ve,j), c), j);
388 803178 : s = gel(SMd, 1);
389 3071811 : for (m = 1; m <= p; m++)
390 : {
391 2268633 : GEN t, c = gel(SMd, m+1);
392 2268633 : if (gequal0(c)) continue;
393 1225040 : t = gel(ve, m+1);
394 2379835 : for (k = 1; k <= m/2; k++)
395 1154795 : t = gadd(t, gmul(gel(ve, m-2*k+1), RgX_coeff(sh, k)));
396 1225040 : s = gadd(s, gmul(c, t));
397 : }
398 803178 : return s;
399 : }
400 :
401 : static GEN
402 2758 : get_SM(GEN Vga)
403 : {
404 2758 : long k, m, d = lg(Vga)-1;
405 2758 : GEN pol, nS1, SM, C, t = vecsum(Vga);
406 :
407 2758 : pol = roots_to_pol(gmulgs(Vga, -d), 0); /* deg(pol) = d */
408 2758 : SM = cgetg(d+2, t_VEC); gel(SM,1) = gen_1;
409 2758 : if (gequal0(t))
410 : { /* shortcut */
411 4606 : for (m = 1; m <= d; m++) gel(SM,m+1) = gel(pol,d+2-m);
412 1148 : return SM;
413 : }
414 1610 : nS1 = gpowers(gneg(t), d); C = matpascal(d);
415 7007 : for (m = 1; m <= d; m++)
416 : {
417 5397 : pari_sp av = avma;
418 5397 : GEN s = gmul(gel(nS1, m+1), gcoeff(C, d+1, m+1));
419 17920 : for (k = 1; k <= m; k++)
420 : {
421 12523 : GEN e = gmul(gel(nS1, m-k+1), gcoeff(C, d-k+1, m-k+1));
422 12523 : s = gadd(s, gmul(e, RgX_coeff(pol, d-k)));
423 : }
424 5397 : gel(SM, m+1) = gerepileupto(av, s);
425 : }
426 1610 : return SM;
427 : }
428 :
429 : static GEN
430 2758 : get_SMd(GEN Vga)
431 : {
432 2758 : GEN M, SM = get_SM(Vga);
433 2758 : long p, m, d = lg(Vga)-1;
434 :
435 2758 : M = cgetg(d, t_VEC);
436 8855 : for (p = 2; p <= d; p++)
437 : {
438 6097 : GEN a = gen_1, c;
439 6097 : long D = d - p;
440 6097 : gel(M, p-1) = c = cgetg(p+2, t_COL);
441 6097 : gel(c, 1) = gel(SM, p+1);
442 23471 : for (m = 1; m <= p; m++)
443 : {
444 17374 : a = muliu(a, D + m);
445 17374 : gel(c, m+1) = gmul(gel(SM, p-m+1), a);
446 : }
447 : }
448 2758 : 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 13783 : Klargeinit(GEN Vga, long nlimmax, long *status, long prec)
459 : {
460 13783 : long d = lg(Vga) - 1, p, n, cnt;
461 : GEN M, SMd, se, vsinh, vd;
462 :
463 13783 : if (Vgaeasytheta(Vga)) { *status = 2; return mkvec(gen_1); }
464 : /* d >= 2 */
465 2758 : *status = 0; prec += prec >> 1;
466 2758 : SMd = get_SMd(Vga);
467 2758 : se = gsinh(RgX_to_ser(pol_x(0), d+2), 0); setvalp(se,0);
468 2758 : se = gdeflate(se, 0, 2); /* se(x^2) = sinh(x)/x */
469 2758 : vsinh = gpowers(se, d);
470 2758 : vd = gpowers(utoipos(2*d), d);
471 2758 : M = cgetg(nlimmax + d + 1, t_VEC); gel(M,1) = gen_1;
472 367148 : for (n = 2, cnt = 0; n <= nlimmax || cnt; n++)
473 : {
474 367148 : pari_sp av = avma;
475 367148 : long ld = minss(d, n);
476 367148 : GEN s = gen_0;
477 1170326 : for (p = 2; p <= ld; p++)
478 : {
479 803178 : GEN z = vp(p, 2*n-1-p, gel(SMd, p-1), gel(vsinh, d-p+1));
480 803178 : s = gadd(s, gmul(gdiv(z, gel(vd, p+1)), gel(M, n+1-p)));
481 : }
482 367148 : if (prec && !isinexact(s)) s = gtofp(s, prec);
483 367148 : gel(M,n) = s = gerepileupto(av, gdivgs(s, 1-n));
484 367148 : if (gequal0(s))
485 : {
486 48 : cnt++; *status = 1;
487 48 : if (cnt >= d-1) { *status = 2; n -= d-2; break; }
488 : }
489 : else
490 : {
491 367100 : if (n >= nlimmax) { n++; break; }
492 364370 : cnt = 0;
493 : }
494 : }
495 2758 : setlg(M, n); return M;
496 : }
497 :
498 : /* remove trailing zeros from vector. */
499 : static void
500 252 : stripzeros(GEN M)
501 : {
502 : long i;
503 441 : for(i = lg(M)-1; i >= 1; --i)
504 441 : if (!gequal0(gel(M, i))) break;
505 252 : setlg(M, i+1);
506 252 : }
507 :
508 : /* Asymptotic expansion of the m-th derivative of inverse Mellin, to length
509 : * nlimmax. If status = 2, the asymptotic expansion is finite so return only
510 : * the necessary number of terms nlim <= nlimmax + d. */
511 : static GEN
512 13783 : gammamellininvasymp_i(GEN Vga, long nlimmax, long m, long *status, long prec)
513 : {
514 : GEN M, A, Aadd;
515 : long d, i, nlim, n;
516 :
517 13783 : M = Klargeinit(Vga, nlimmax, status, prec);
518 13783 : if (!m) return M;
519 252 : d = lg(Vga)-1;
520 : /* half the exponent of t in asymptotic expansion. */
521 252 : A = gdivgu(gaddsg(1-d, sumVga(Vga)), 2*d);
522 252 : if (*status == 2) M = shallowconcat(M, zerovec(m));
523 252 : nlim = lg(M)-1;
524 252 : Aadd = sstoQ(2-d, 2*d); /* (1/d) - (1/2) */
525 532 : for (i = 1; i <= m; i++, A = gadd(A,Aadd))
526 8022 : for (n = nlim-1; n >= 1; --n)
527 15484 : gel(M, n+1) = gsub(gel(M, n+1),
528 15484 : gmul(gel(M, n), gsub(A, uutoQ(n-1, d))));
529 252 : stripzeros(M); return M;
530 : }
531 : static GEN
532 13797 : get_Vga(GEN x, GEN *ldata)
533 : {
534 13797 : *ldata = lfunmisc_to_ldata_shallow_i(x);
535 13797 : if (*ldata) x = ldata_get_gammavec(*ldata);
536 13797 : return x;
537 : }
538 : GEN
539 28 : gammamellininvasymp(GEN Vga, long nlim, long m)
540 : {
541 28 : pari_sp av = avma;
542 : long status;
543 : GEN ldata;
544 28 : Vga = get_Vga(Vga, &ldata);
545 28 : if (!is_vec_t(typ(Vga)) || lg(Vga) == 1)
546 7 : pari_err_TYPE("gammamellininvasymp",Vga);
547 21 : return gerepilecopy(av, gammamellininvasymp_i(Vga, nlim, m, &status, 0));
548 : }
549 :
550 : /* Does the continued fraction of the asymptotic expansion M at oo of inverse
551 : * Mellin transform attached to Vga have zero Hankel determinants ? */
552 : static long
553 2717 : ishankelspec(GEN Vga, GEN M)
554 : {
555 2717 : long status, i, d = lg(Vga)-1;
556 :
557 2717 : if (d == 5 || d == 7)
558 : { /* known bad cases: a x 5 and a x 7 */
559 127 : GEN v1 = gel(Vga, 1);
560 576 : for (i = 2; i <= d; ++i)
561 464 : if (!gequal(gel(Vga,i), v1)) break;
562 127 : if (i > d) return 1;
563 : }
564 2605 : status = 0;
565 : /* Heuristic: if 6 first terms in contfracinit don't fail, assume it's OK */
566 2605 : pari_CATCH(e_INV) { status = 1; }
567 2605 : pari_TRY { contfracinit(M, minss(lg(M)-2,6)); }
568 2605 : pari_ENDCATCH; return status;
569 : }
570 :
571 : /* Initialize data for computing m-th derivative of inverse Mellin */
572 : GEN
573 13769 : gammamellininvinit(GEN Vga, long m, long bitprec)
574 : {
575 13769 : const double C2 = MELLININV_CUTOFF;
576 13769 : pari_sp ltop = avma;
577 : GEN A2, M, VS, VL, cd, ldata;
578 13769 : long nlimmax, status, d, prec = nbits2prec((4*bitprec)/3);
579 13769 : double E = M_LN2*bitprec, tmax = get_tmax(bitprec); /* = E/C2 */
580 :
581 13769 : Vga = get_Vga(Vga, &ldata); d = lg(Vga)-1;
582 13769 : if (!is_vec_t(typ(Vga)) || !d) pari_err_TYPE("gammamellininvinit",Vga);
583 13762 : nlimmax = ceil(E * log2(1+M_PI*tmax) * C2 / get_D(d));
584 13762 : A2 = gaddsg(m*(2-d) + 1-d, sumVga(Vga));
585 13762 : cd = (d <= 2)? gen_2: gsqrt(gdivgu(int2n(d+1), d), nbits2prec(bitprec));
586 : /* if in Klarge, we have |t| > tmax = E/C2, thus nlim < E*C2/D. */
587 13762 : M = gammamellininvasymp_i(Vga, nlimmax, m, &status, prec);
588 13762 : if (status == 2)
589 : {
590 11039 : tmax = -1.; /* only use Klarge */
591 11039 : VS = gen_0;
592 : }
593 : else
594 : {
595 2723 : VS = Kderivsmallinit(ldata, Vga, m, bitprec);
596 2723 : if (status == 0 && ishankelspec(Vga, M)) status = 1;
597 2723 : if (status == 1)
598 : { /* a Hankel determinant vanishes => contfracinit is undefined.
599 : So compute K(t) / (1 - 1/(pi^2*t)) instead of K(t)*/
600 119 : GEN t = ginv(mppi(prec));
601 : long i;
602 18018 : for (i = 2; i < lg(M); ++i)
603 17899 : gel(M, i) = gadd(gel(M, i), gmul(gel(M, i-1), t));
604 : }
605 : else
606 2604 : M = RgC_gtofp(M, prec); /* convert from rationals to t_REAL: faster */
607 2723 : M = contfracinit(M, lg(M)-2);
608 : }
609 13762 : VL = mkvec3(mkvec2(M, stoi(status)), cd, A2);
610 13762 : return gerepilecopy(ltop, mkvec5(dbltor(tmax), Vga, stoi(m), VS, VL));
611 : }
612 :
613 : /* Compute m-th derivative of inverse Mellin at s2d = s^(d/2) using
614 : * initialization data. Use Taylor expansion at 0 for |s2d| < tmax, and
615 : * asymptotic expansion at oo otherwise. WARNING: assume that accuracy
616 : * has been increased according to tmax by the CALLING program. */
617 : static GEN
618 4547328 : gammamellininvrt_i(GEN K, GEN s, GEN s2d, long bit)
619 : {
620 4547328 : if (dblmodulus(s2d) < GMi_get_tmax(K, bit))
621 69068 : return Kderivsmall(K, s, s2d, bit);
622 : else
623 4478211 : return Kderivlarge(K, s, s2d, bit);
624 : }
625 : GEN
626 4541665 : gammamellininvrt(GEN K, GEN s2d, long bit)
627 4541665 : { return gammamellininvrt_i(K, NULL, s2d, bit); }
628 :
629 : /* Compute inverse Mellin at s. K from gammamellininv OR a Vga, in which
630 : * case the initialization data is computed. */
631 : GEN
632 5817 : gammamellininv(GEN K, GEN s, long m, long bitprec)
633 : {
634 5817 : pari_sp av = avma;
635 : GEN s2d;
636 : long d;
637 :
638 5817 : if (!is_vec_t(typ(K)) || lg(K) != 6 || !is_vec_t(typ(GMi_get_Vga(K))))
639 91 : K = gammamellininvinit(K, m, bitprec);
640 5817 : d = GMi_get_degree(K);
641 5817 : s2d = gpow(s, gdivgu(gen_2, d), nbits2prec(bitprec));
642 5817 : return gerepileupto(av, gammamellininvrt_i(K, s, s2d, bitprec));
643 : }
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