1 /** @file inifcns_nstdsums.cpp
3 * Implementation of some special functions that have a representation as nested sums.
6 * classical polylogarithm Li(n,x)
7 * multiple polylogarithm Li(lst{m_1,...,m_k},lst{x_1,...,x_k})
8 * G(lst{a_1,...,a_k},y) or G(lst{a_1,...,a_k},lst{s_1,...,s_k},y)
9 * Nielsen's generalized polylogarithm S(n,p,x)
10 * harmonic polylogarithm H(m,x) or H(lst{m_1,...,m_k},x)
11 * multiple zeta value zeta(m) or zeta(lst{m_1,...,m_k})
12 * alternating Euler sum zeta(m,s) or zeta(lst{m_1,...,m_k},lst{s_1,...,s_k})
16 * - All formulae used can be looked up in the following publications:
17 * [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
18 * [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
19 * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
20 * [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
21 * [VSW] Numerical evaluation of multiple polylogarithms, J.Vollinga, S.Weinzierl, hep-ph/0410259
23 * - The order of parameters and arguments of Li and zeta is defined according to the nested sums
24 * representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
25 * 0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
26 * number --- notation.
28 * - All functions can be numerically evaluated with arguments in the whole complex plane. The parameters
29 * for Li, zeta and S must be positive integers. If you want to have an alternating Euler sum, you have
30 * to give the signs of the parameters as a second argument s to zeta(m,s) containing 1 and -1.
32 * - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and
33 * look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
34 * [Cra] and [BBB] for speed up. Multiple polylogarithms use Hoelder convolution [BBB].
36 * - The functions have no means to do a series expansion into nested sums. To do this, you have to convert
37 * these functions into the appropriate objects from the nestedsums library, do the expansion and convert
40 * - Numerical testing of this implementation has been performed by doing a comparison of results
41 * between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
42 * by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
43 * comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
44 * around |x|=1 along with comparisons to corresponding zeta functions. Multiple polylogarithms were
45 * checked against H and zeta and by means of shuffle and quasi-shuffle relations.
50 * GiNaC Copyright (C) 1999-2016 Johannes Gutenberg University Mainz, Germany
52 * This program is free software; you can redistribute it and/or modify
53 * it under the terms of the GNU General Public License as published by
54 * the Free Software Foundation; either version 2 of the License, or
55 * (at your option) any later version.
57 * This program is distributed in the hope that it will be useful,
58 * but WITHOUT ANY WARRANTY; without even the implied warranty of
59 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
60 * GNU General Public License for more details.
62 * You should have received a copy of the GNU General Public License
63 * along with this program; if not, write to the Free Software
64 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
74 #include "operators.h"
77 #include "relational.h"
90 //////////////////////////////////////////////////////////////////////
92 // Classical polylogarithm Li(n,x)
96 //////////////////////////////////////////////////////////////////////
99 // anonymous namespace for helper functions
103 // lookup table for factors built from Bernoulli numbers
105 std::vector<std::vector<cln::cl_N>> Xn;
106 // initial size of Xn that should suffice for 32bit machines (must be even)
107 const int xninitsizestep = 26;
108 int xninitsize = xninitsizestep;
112 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
113 // With these numbers the polylogs can be calculated as follows:
114 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
115 // X_0(n) = B_n (Bernoulli numbers)
116 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
117 // The calculation of Xn depends on X0 and X{n-1}.
118 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
119 // This results in a slightly more complicated algorithm for the X_n.
120 // The first index in Xn corresponds to the index of the polylog minus 2.
121 // The second index in Xn corresponds to the index from the actual sum.
125 // calculate X_2 and higher (corresponding to Li_4 and higher)
126 std::vector<cln::cl_N> buf(xninitsize);
127 auto it = buf.begin();
129 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
131 for (int i=2; i<=xninitsize; i++) {
133 result = 0; // k == 0
135 result = Xn[0][i/2-1]; // k == 0
137 for (int k=1; k<i-1; k++) {
138 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
139 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
142 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
143 result = result + Xn[n-1][i-1] / (i+1); // k == i
150 // special case to handle the X_0 correct
151 std::vector<cln::cl_N> buf(xninitsize);
152 auto it = buf.begin();
154 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
156 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
158 for (int i=3; i<=xninitsize; i++) {
160 result = -Xn[0][(i-3)/2]/2;
161 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
164 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
165 for (int k=1; k<i/2; k++) {
166 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
175 std::vector<cln::cl_N> buf(xninitsize/2);
176 auto it = buf.begin();
177 for (int i=1; i<=xninitsize/2; i++) {
178 *it = bernoulli(i*2).to_cl_N();
187 // doubles the number of entries in each Xn[]
190 const int pos0 = xninitsize / 2;
192 for (int i=1; i<=xninitsizestep/2; ++i) {
193 Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
196 int xend = xninitsize + xninitsizestep;
199 for (int i=xninitsize+1; i<=xend; ++i) {
201 result = -Xn[0][(i-3)/2]/2;
202 Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result);
204 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
205 for (int k=1; k<i/2; k++) {
206 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
208 Xn[1].push_back(result);
212 for (size_t n=2; n<Xn.size(); ++n) {
213 for (int i=xninitsize+1; i<=xend; ++i) {
215 result = 0; // k == 0
217 result = Xn[0][i/2-1]; // k == 0
219 for (int k=1; k<i-1; ++k) {
220 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
221 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
224 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
225 result = result + Xn[n-1][i-1] / (i+1); // k == i
226 Xn[n].push_back(result);
230 xninitsize += xninitsizestep;
234 // calculates Li(2,x) without Xn
235 cln::cl_N Li2_do_sum(const cln::cl_N& x)
239 cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
240 cln::cl_I den = 1; // n^2 = 1
245 den = den + i; // n^2 = 4, 9, 16, ...
247 res = res + num / den;
248 } while (res != resbuf);
253 // calculates Li(2,x) with Xn
254 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
256 std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
257 std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
258 cln::cl_N u = -cln::log(1-x);
259 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
260 cln::cl_N uu = cln::square(u);
261 cln::cl_N res = u - uu/4;
266 factor = factor * uu / (2*i * (2*i+1));
267 res = res + (*it) * factor;
271 it = Xn[0].begin() + (i-1);
274 } while (res != resbuf);
279 // calculates Li(n,x), n>2 without Xn
280 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
282 cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
289 res = res + factor / cln::expt(cln::cl_I(i),n);
291 } while (res != resbuf);
296 // calculates Li(n,x), n>2 with Xn
297 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
299 std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
300 std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
301 cln::cl_N u = -cln::log(1-x);
302 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
308 factor = factor * u / i;
309 res = res + (*it) * factor;
313 it = Xn[n-2].begin() + (i-2);
314 xend = Xn[n-2].end();
316 } while (res != resbuf);
321 // forward declaration needed by function Li_projection and C below
322 const cln::cl_N S_num(int n, int p, const cln::cl_N& x);
325 // helper function for classical polylog Li
326 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
328 // treat n=2 as special case
330 // check if precalculated X0 exists
335 if (cln::realpart(x) < 0.5) {
336 // choose the faster algorithm
337 // the switching point was empirically determined. the optimal point
338 // depends on hardware, Digits, ... so an approx value is okay.
339 // it solves also the problem with precision due to the u=-log(1-x) transformation
340 if (cln::abs(cln::realpart(x)) < 0.25) {
342 return Li2_do_sum(x);
344 return Li2_do_sum_Xn(x);
347 // choose the faster algorithm
348 if (cln::abs(cln::realpart(x)) > 0.75) {
352 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
355 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
359 // check if precalculated Xn exist
361 for (int i=xnsize; i<n-1; i++) {
366 if (cln::realpart(x) < 0.5) {
367 // choose the faster algorithm
368 // with n>=12 the "normal" summation always wins against the method with Xn
369 if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
370 return Lin_do_sum(n, x);
372 return Lin_do_sum_Xn(n, x);
375 cln::cl_N result = 0;
376 if ( x != 1 ) result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
377 for (int j=0; j<n-1; j++) {
378 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
379 * cln::expt(cln::log(x), j) / cln::factorial(j);
386 // helper function for classical polylog Li
387 const cln::cl_N Lin_numeric(const int n, const cln::cl_N& x)
391 return -cln::log(1-x);
402 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
404 if (cln::abs(realpart(x)) < 0.4 && cln::abs(cln::abs(x)-1) < 0.01) {
405 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
406 for (int j=0; j<n-1; j++) {
407 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
408 * cln::expt(cln::log(x), j) / cln::factorial(j);
413 // what is the desired float format?
414 // first guess: default format
415 cln::float_format_t prec = cln::default_float_format;
416 const cln::cl_N value = x;
417 // second guess: the argument's format
418 if (!instanceof(realpart(x), cln::cl_RA_ring))
419 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
420 else if (!instanceof(imagpart(x), cln::cl_RA_ring))
421 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
424 if (cln::abs(value) > 1) {
425 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
426 // check if argument is complex. if it is real, the new polylog has to be conjugated.
427 if (cln::zerop(cln::imagpart(value))) {
429 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
432 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
437 result = result + Li_projection(n, cln::recip(value), prec);
440 result = result - Li_projection(n, cln::recip(value), prec);
444 for (int j=0; j<n-1; j++) {
445 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
446 * Lin_numeric(n-j,1) * cln::expt(cln::log(-value),j) / cln::factorial(j);
448 result = result - add;
452 return Li_projection(n, value, prec);
457 } // end of anonymous namespace
460 //////////////////////////////////////////////////////////////////////
462 // Multiple polylogarithm Li(n,x)
466 //////////////////////////////////////////////////////////////////////
469 // anonymous namespace for helper function
473 // performs the actual series summation for multiple polylogarithms
474 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
476 // ensure all x <> 0.
477 for (const auto & it : x) {
478 if (it == 0) return cln::cl_float(0, cln::float_format(Digits));
481 const int j = s.size();
482 bool flag_accidental_zero = false;
484 std::vector<cln::cl_N> t(j);
485 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
492 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
493 for (int k=j-2; k>=0; k--) {
494 t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
497 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
498 for (int k=j-2; k>=0; k--) {
499 flag_accidental_zero = cln::zerop(t[k+1]);
500 t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
502 } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero );
508 // forward declaration for Li_eval()
509 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
512 // type used by the transformation functions for G
513 typedef std::vector<int> Gparameter;
516 // G_eval1-function for G transformations
517 ex G_eval1(int a, int scale, const exvector& gsyms)
520 const ex& scs = gsyms[std::abs(scale)];
521 const ex& as = gsyms[std::abs(a)];
523 return -log(1 - scs/as);
528 return log(gsyms[std::abs(scale)]);
533 // G_eval-function for G transformations
534 ex G_eval(const Gparameter& a, int scale, const exvector& gsyms)
536 // check for properties of G
537 ex sc = gsyms[std::abs(scale)];
539 bool all_zero = true;
540 bool all_ones = true;
542 for (const auto & it : a) {
544 const ex sym = gsyms[std::abs(it)];
558 // care about divergent G: shuffle to separate divergencies that will be canceled
559 // later on in the transformation
560 if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
562 Gparameter short_a(a.begin()+1, a.end());
563 ex result = G_eval1(a.front(), scale, gsyms) * G_eval(short_a, scale, gsyms);
565 auto it = short_a.begin();
566 advance(it, count_ones-1);
567 for (; it != short_a.end(); ++it) {
569 Gparameter newa(short_a.begin(), it);
571 newa.push_back(a[0]);
572 newa.insert(newa.end(), it+1, short_a.end());
573 result -= G_eval(newa, scale, gsyms);
575 return result / count_ones;
578 // G({1,...,1};y) -> G({1};y)^k / k!
579 if (all_ones && a.size() > 1) {
580 return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones);
583 // G({0,...,0};y) -> log(y)^k / k!
585 return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
588 // no special cases anymore -> convert it into Li
591 ex argbuf = gsyms[std::abs(scale)];
593 for (const auto & it : a) {
595 const ex& sym = gsyms[std::abs(it)];
596 x.append(argbuf / sym);
604 return pow(-1, x.nops()) * Li(m, x);
608 // converts data for G: pending_integrals -> a
609 Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
611 GINAC_ASSERT(pending_integrals.size() != 1);
613 if (pending_integrals.size() > 0) {
614 // get rid of the first element, which would stand for the new upper limit
615 Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
618 // just return empty parameter list
625 // check the parameters a and scale for G and return information about convergence, depth, etc.
626 // convergent : true if G(a,scale) is convergent
627 // depth : depth of G(a,scale)
628 // trailing_zeros : number of trailing zeros of a
629 // min_it : iterator of a pointing on the smallest element in a
630 Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
631 bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
637 auto lastnonzero = a.end();
638 for (auto it = a.begin(); it != a.end(); ++it) {
639 if (std::abs(*it) > 0) {
643 if (std::abs(*it) < scale) {
645 if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
653 if (lastnonzero == a.end())
655 return ++lastnonzero;
659 // add scale to pending_integrals if pending_integrals is empty
660 Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
662 GINAC_ASSERT(pending_integrals.size() != 1);
664 if (pending_integrals.size() > 0) {
665 return pending_integrals;
667 Gparameter new_pending_integrals;
668 new_pending_integrals.push_back(scale);
669 return new_pending_integrals;
674 // handles trailing zeroes for an otherwise convergent integral
675 ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms)
678 int depth, trailing_zeros;
679 Gparameter::const_iterator last, dummyit;
680 last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
682 GINAC_ASSERT(convergent);
684 if ((trailing_zeros > 0) && (depth > 0)) {
686 Gparameter new_a(a.begin(), a.end()-1);
687 result += G_eval1(0, scale, gsyms) * trailing_zeros_G(new_a, scale, gsyms);
688 for (auto it = a.begin(); it != last; ++it) {
689 Gparameter new_a(a.begin(), it);
691 new_a.insert(new_a.end(), it, a.end()-1);
692 result -= trailing_zeros_G(new_a, scale, gsyms);
695 return result / trailing_zeros;
697 return G_eval(a, scale, gsyms);
702 // G transformation [VSW] (57),(58)
703 ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale, const exvector& gsyms)
705 // pendint = ( y1, b1, ..., br )
706 // a = ( 0, ..., 0, amin )
709 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
710 // where sr replaces amin
712 GINAC_ASSERT(a.back() != 0);
713 GINAC_ASSERT(a.size() > 0);
716 Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
717 const int psize = pending_integrals.size();
720 // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
725 result += log(gsyms[ex_to<numeric>(scale).to_int()]);
727 new_pending_integrals.push_back(-scale);
730 new_pending_integrals.push_back(scale);
734 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
735 pending_integrals.front(),
740 result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
741 new_pending_integrals.front(),
745 new_pending_integrals.back() = 0;
746 result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
747 new_pending_integrals.front(),
754 // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
755 // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
758 result -= zeta(a.size());
760 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
761 pending_integrals.front(),
765 // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
766 // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
767 Gparameter new_a(a.begin()+1, a.end());
768 new_pending_integrals.push_back(0);
769 result -= depth_one_trafo_G(new_pending_integrals, new_a, scale, gsyms);
771 // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
772 // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
773 Gparameter new_pending_integrals_2;
774 new_pending_integrals_2.push_back(scale);
775 new_pending_integrals_2.push_back(0);
777 result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
778 pending_integrals.front(),
780 * depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
782 result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
789 // forward declaration
790 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
791 const Gparameter& pendint, const Gparameter& a_old, int scale,
792 const exvector& gsyms, bool flag_trailing_zeros_only);
795 // G transformation [VSW]
796 ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale,
797 const exvector& gsyms, bool flag_trailing_zeros_only)
799 // main recursion routine
801 // pendint = ( y1, b1, ..., br )
802 // a = ( a1, ..., amin, ..., aw )
805 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
806 // where sr replaces amin
808 // find smallest alpha, determine depth and trailing zeros, and check for convergence
810 int depth, trailing_zeros;
811 Gparameter::const_iterator min_it;
812 auto firstzero = check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
813 int min_it_pos = distance(a.begin(), min_it);
815 // special case: all a's are zero
822 result = G_eval(a, scale, gsyms);
824 if (pendint.size() > 0) {
825 result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
832 // handle trailing zeros
833 if (trailing_zeros > 0) {
835 Gparameter new_a(a.begin(), a.end()-1);
836 result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
837 for (auto it = a.begin(); it != firstzero; ++it) {
838 Gparameter new_a(a.begin(), it);
840 new_a.insert(new_a.end(), it, a.end()-1);
841 result -= G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
843 return result / trailing_zeros;
846 // convergence case or flag_trailing_zeros_only
847 if (convergent || flag_trailing_zeros_only) {
848 if (pendint.size() > 0) {
849 return G_eval(convert_pending_integrals_G(pendint),
850 pendint.front(), gsyms) *
851 G_eval(a, scale, gsyms);
853 return G_eval(a, scale, gsyms);
857 // call basic transformation for depth equal one
859 return depth_one_trafo_G(pendint, a, scale, gsyms);
863 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
864 // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
865 // + int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) int_0^{sr} ds_{r+1} d/ds_{r+1} G(a1,...,s_{r+1},...,aw,y2)
867 // smallest element in last place
868 if (min_it + 1 == a.end()) {
869 do { --min_it; } while (*min_it == 0);
871 Gparameter a1(a.begin(),min_it+1);
872 Gparameter a2(min_it+1,a.end());
874 ex result = G_transform(pendint, a2, scale, gsyms, flag_trailing_zeros_only)*
875 G_transform(empty, a1, scale, gsyms, flag_trailing_zeros_only);
877 result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms, flag_trailing_zeros_only);
882 Gparameter::iterator changeit;
884 // first term G(a_1,..,0,...,a_w;a_0)
885 Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
886 Gparameter new_a = a;
887 new_a[min_it_pos] = 0;
888 ex result = G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
889 if (pendint.size() > 0) {
890 result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
891 pendint.front(), gsyms);
895 changeit = new_a.begin() + min_it_pos;
896 changeit = new_a.erase(changeit);
897 if (changeit != new_a.begin()) {
898 // smallest in the middle
899 new_pendint.push_back(*changeit);
900 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
901 new_pendint.front(), gsyms)*
902 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
903 int buffer = *changeit;
905 result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
907 new_pendint.pop_back();
909 new_pendint.push_back(*changeit);
910 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
911 new_pendint.front(), gsyms)*
912 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
914 result -= G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
916 // smallest at the front
917 new_pendint.push_back(scale);
918 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
919 new_pendint.front(), gsyms)*
920 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
921 new_pendint.back() = *changeit;
922 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
923 new_pendint.front(), gsyms)*
924 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
926 result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
932 // shuffles the two parameter list a1 and a2 and calls G_transform for every term except
933 // for the one that is equal to a_old
934 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
935 const Gparameter& pendint, const Gparameter& a_old, int scale,
936 const exvector& gsyms, bool flag_trailing_zeros_only)
938 if (a1.size()==0 && a2.size()==0) {
939 // veto the one configuration we don't want
940 if ( a0 == a_old ) return 0;
942 return G_transform(pendint, a0, scale, gsyms, flag_trailing_zeros_only);
948 aa0.insert(aa0.end(),a1.begin(),a1.end());
949 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
955 aa0.insert(aa0.end(),a2.begin(),a2.end());
956 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
959 Gparameter a1_removed(a1.begin()+1,a1.end());
960 Gparameter a2_removed(a2.begin()+1,a2.end());
965 a01.push_back( a1[0] );
966 a02.push_back( a2[0] );
968 return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms, flag_trailing_zeros_only)
969 + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
972 // handles the transformations and the numerical evaluation of G
973 // the parameter x, s and y must only contain numerics
975 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
978 // do acceleration transformation (hoelder convolution [BBB])
979 // the parameter x, s and y must only contain numerics
981 G_do_hoelder(std::vector<cln::cl_N> x, /* yes, it's passed by value */
982 const std::vector<int>& s, const cln::cl_N& y)
985 const std::size_t size = x.size();
986 for (std::size_t i = 0; i < size; ++i)
989 for (std::size_t r = 0; r <= size; ++r) {
990 cln::cl_N buffer(1 & r ? -1 : 1);
995 for (std::size_t i = 0; i < size; ++i) {
996 if (x[i] == cln::cl_RA(1)/p) {
997 p = p/2 + cln::cl_RA(3)/2;
1003 cln::cl_RA q = p/(p-1);
1004 std::vector<cln::cl_N> qlstx;
1005 std::vector<int> qlsts;
1006 for (std::size_t j = r; j >= 1; --j) {
1007 qlstx.push_back(cln::cl_N(1) - x[j-1]);
1008 if (instanceof(x[j-1], cln::cl_R_ring) && realpart(x[j-1]) > 1) {
1011 qlsts.push_back(-s[j-1]);
1014 if (qlstx.size() > 0) {
1015 buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
1017 std::vector<cln::cl_N> plstx;
1018 std::vector<int> plsts;
1019 for (std::size_t j = r+1; j <= size; ++j) {
1020 plstx.push_back(x[j-1]);
1021 plsts.push_back(s[j-1]);
1023 if (plstx.size() > 0) {
1024 buffer = buffer*G_numeric(plstx, plsts, 1/p);
1026 result = result + buffer;
1031 class less_object_for_cl_N
1034 bool operator() (const cln::cl_N & a, const cln::cl_N & b) const
1037 if (abs(a) != abs(b))
1038 return (abs(a) < abs(b)) ? true : false;
1041 if (phase(a) != phase(b))
1042 return (phase(a) < phase(b)) ? true : false;
1044 // equal, therefore "less" is not true
1050 // convergence transformation, used for numerical evaluation of G function.
1051 // the parameter x, s and y must only contain numerics
1053 G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1054 const cln::cl_N& y, bool flag_trailing_zeros_only)
1056 // sort (|x|<->position) to determine indices
1057 typedef std::multimap<cln::cl_N, std::size_t, less_object_for_cl_N> sortmap_t;
1059 std::size_t size = 0;
1060 for (std::size_t i = 0; i < x.size(); ++i) {
1062 sortmap.insert(std::make_pair(x[i], i));
1066 // include upper limit (scale)
1067 sortmap.insert(std::make_pair(y, x.size()));
1069 // generate missing dummy-symbols
1071 // holding dummy-symbols for the G/Li transformations
1073 gsyms.push_back(symbol("GSYMS_ERROR"));
1074 cln::cl_N lastentry(0);
1075 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1076 if (it != sortmap.begin()) {
1077 if (it->second < x.size()) {
1078 if (x[it->second] == lastentry) {
1079 gsyms.push_back(gsyms.back());
1083 if (y == lastentry) {
1084 gsyms.push_back(gsyms.back());
1089 std::ostringstream os;
1091 gsyms.push_back(symbol(os.str()));
1093 if (it->second < x.size()) {
1094 lastentry = x[it->second];
1100 // fill position data according to sorted indices and prepare substitution list
1101 Gparameter a(x.size());
1103 std::size_t pos = 1;
1105 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1106 if (it->second < x.size()) {
1107 if (s[it->second] > 0) {
1108 a[it->second] = pos;
1110 a[it->second] = -int(pos);
1112 subslst[gsyms[pos]] = numeric(x[it->second]);
1115 subslst[gsyms[pos]] = numeric(y);
1120 // do transformation
1122 ex result = G_transform(pendint, a, scale, gsyms, flag_trailing_zeros_only);
1123 // replace dummy symbols with their values
1124 result = result.expand();
1125 result = result.subs(subslst).evalf();
1126 if (!is_a<numeric>(result))
1127 throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
1129 cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
1133 // handles the transformations and the numerical evaluation of G
1134 // the parameter x, s and y must only contain numerics
1136 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1139 // check for convergence and necessary accelerations
1140 bool need_trafo = false;
1141 bool need_hoelder = false;
1142 bool have_trailing_zero = false;
1143 std::size_t depth = 0;
1144 for (auto & xi : x) {
1147 const cln::cl_N x_y = abs(xi) - y;
1148 if (instanceof(x_y, cln::cl_R_ring) &&
1149 realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2)))
1152 if (abs(abs(xi/y) - 1) < 0.01)
1153 need_hoelder = true;
1156 if (zerop(x.back())) {
1157 have_trailing_zero = true;
1161 if (depth == 1 && x.size() == 2 && !need_trafo)
1162 return - Li_projection(2, y/x[1], cln::float_format(Digits));
1164 // do acceleration transformation (hoelder convolution [BBB])
1165 if (need_hoelder && !have_trailing_zero)
1166 return G_do_hoelder(x, s, y);
1168 // convergence transformation
1170 return G_do_trafo(x, s, y, have_trailing_zero);
1173 std::vector<cln::cl_N> newx;
1174 newx.reserve(x.size());
1176 m.reserve(x.size());
1179 cln::cl_N factor = y;
1180 for (auto & xi : x) {
1184 newx.push_back(factor/xi);
1186 m.push_back(mcount);
1192 return sign*multipleLi_do_sum(m, newx);
1196 ex mLi_numeric(const lst& m, const lst& x)
1198 // let G_numeric do the transformation
1199 std::vector<cln::cl_N> newx;
1200 newx.reserve(x.nops());
1202 s.reserve(x.nops());
1203 cln::cl_N factor(1);
1204 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1205 for (int i = 1; i < *itm; ++i) {
1206 newx.push_back(cln::cl_N(0));
1209 const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
1211 newx.push_back(factor);
1212 if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) {
1219 return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
1223 } // end of anonymous namespace
1226 //////////////////////////////////////////////////////////////////////
1228 // Generalized multiple polylogarithm G(x, y) and G(x, s, y)
1232 //////////////////////////////////////////////////////////////////////
1235 static ex G2_evalf(const ex& x_, const ex& y)
1237 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1238 return G(x_, y).hold();
1240 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1241 if (x.nops() == 0) {
1245 return G(x_, y).hold();
1248 s.reserve(x.nops());
1249 bool all_zero = true;
1250 for (const auto & it : x) {
1251 if (!it.info(info_flags::numeric)) {
1252 return G(x_, y).hold();
1257 if ( !ex_to<numeric>(it).is_real() && ex_to<numeric>(it).imag() < 0 ) {
1265 return pow(log(y), x.nops()) / factorial(x.nops());
1267 std::vector<cln::cl_N> xv;
1268 xv.reserve(x.nops());
1269 for (const auto & it : x)
1270 xv.push_back(ex_to<numeric>(it).to_cl_N());
1271 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1272 return numeric(result);
1276 static ex G2_eval(const ex& x_, const ex& y)
1278 //TODO eval to MZV or H or S or Lin
1280 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1281 return G(x_, y).hold();
1283 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1284 if (x.nops() == 0) {
1288 return G(x_, y).hold();
1291 s.reserve(x.nops());
1292 bool all_zero = true;
1293 bool crational = true;
1294 for (const auto & it : x) {
1295 if (!it.info(info_flags::numeric)) {
1296 return G(x_, y).hold();
1298 if (!it.info(info_flags::crational)) {
1304 if ( !ex_to<numeric>(it).is_real() && ex_to<numeric>(it).imag() < 0 ) {
1312 return pow(log(y), x.nops()) / factorial(x.nops());
1314 if (!y.info(info_flags::crational)) {
1318 return G(x_, y).hold();
1320 std::vector<cln::cl_N> xv;
1321 xv.reserve(x.nops());
1322 for (const auto & it : x)
1323 xv.push_back(ex_to<numeric>(it).to_cl_N());
1324 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1325 return numeric(result);
1329 // option do_not_evalf_params() removed.
1330 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1331 evalf_func(G2_evalf).
1335 // derivative_func(G2_deriv).
1336 // print_func<print_latex>(G2_print_latex).
1339 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1341 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1342 return G(x_, s_, y).hold();
1344 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1345 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst{s_};
1346 if (x.nops() != s.nops()) {
1347 return G(x_, s_, y).hold();
1349 if (x.nops() == 0) {
1353 return G(x_, s_, y).hold();
1355 std::vector<int> sn;
1356 sn.reserve(s.nops());
1357 bool all_zero = true;
1358 for (auto itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1359 if (!(*itx).info(info_flags::numeric)) {
1360 return G(x_, y).hold();
1362 if (!(*its).info(info_flags::real)) {
1363 return G(x_, y).hold();
1368 if ( ex_to<numeric>(*itx).is_real() ) {
1369 if ( ex_to<numeric>(*itx).is_positive() ) {
1381 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1390 return pow(log(y), x.nops()) / factorial(x.nops());
1392 std::vector<cln::cl_N> xn;
1393 xn.reserve(x.nops());
1394 for (const auto & it : x)
1395 xn.push_back(ex_to<numeric>(it).to_cl_N());
1396 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1397 return numeric(result);
1401 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1403 //TODO eval to MZV or H or S or Lin
1405 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1406 return G(x_, s_, y).hold();
1408 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1409 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst{s_};
1410 if (x.nops() != s.nops()) {
1411 return G(x_, s_, y).hold();
1413 if (x.nops() == 0) {
1417 return G(x_, s_, y).hold();
1419 std::vector<int> sn;
1420 sn.reserve(s.nops());
1421 bool all_zero = true;
1422 bool crational = true;
1423 for (auto itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1424 if (!(*itx).info(info_flags::numeric)) {
1425 return G(x_, s_, y).hold();
1427 if (!(*its).info(info_flags::real)) {
1428 return G(x_, s_, y).hold();
1430 if (!(*itx).info(info_flags::crational)) {
1436 if ( ex_to<numeric>(*itx).is_real() ) {
1437 if ( ex_to<numeric>(*itx).is_positive() ) {
1449 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1458 return pow(log(y), x.nops()) / factorial(x.nops());
1460 if (!y.info(info_flags::crational)) {
1464 return G(x_, s_, y).hold();
1466 std::vector<cln::cl_N> xn;
1467 xn.reserve(x.nops());
1468 for (const auto & it : x)
1469 xn.push_back(ex_to<numeric>(it).to_cl_N());
1470 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1471 return numeric(result);
1475 // option do_not_evalf_params() removed.
1476 // This is safe: in the code above it only matters if s_ > 0 or s_ < 0,
1477 // s_ is allowed to be of floating type.
1478 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1479 evalf_func(G3_evalf).
1483 // derivative_func(G3_deriv).
1484 // print_func<print_latex>(G3_print_latex).
1487 //////////////////////////////////////////////////////////////////////
1489 // Classical polylogarithm and multiple polylogarithm Li(m,x)
1493 //////////////////////////////////////////////////////////////////////
1496 static ex Li_evalf(const ex& m_, const ex& x_)
1498 // classical polylogs
1499 if (m_.info(info_flags::posint)) {
1500 if (x_.info(info_flags::numeric)) {
1501 int m__ = ex_to<numeric>(m_).to_int();
1502 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1503 const cln::cl_N result = Lin_numeric(m__, x__);
1504 return numeric(result);
1506 // try to numerically evaluate second argument
1507 ex x_val = x_.evalf();
1508 if (x_val.info(info_flags::numeric)) {
1509 int m__ = ex_to<numeric>(m_).to_int();
1510 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1511 const cln::cl_N result = Lin_numeric(m__, x__);
1512 return numeric(result);
1516 // multiple polylogs
1517 if (is_a<lst>(m_) && is_a<lst>(x_)) {
1519 const lst& m = ex_to<lst>(m_);
1520 const lst& x = ex_to<lst>(x_);
1521 if (m.nops() != x.nops()) {
1522 return Li(m_,x_).hold();
1524 if (x.nops() == 0) {
1527 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1528 return Li(m_,x_).hold();
1531 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1532 if (!(*itm).info(info_flags::posint)) {
1533 return Li(m_, x_).hold();
1535 if (!(*itx).info(info_flags::numeric)) {
1536 return Li(m_, x_).hold();
1543 return mLi_numeric(m, x);
1546 return Li(m_,x_).hold();
1550 static ex Li_eval(const ex& m_, const ex& x_)
1552 if (is_a<lst>(m_)) {
1553 if (is_a<lst>(x_)) {
1554 // multiple polylogs
1555 const lst& m = ex_to<lst>(m_);
1556 const lst& x = ex_to<lst>(x_);
1557 if (m.nops() != x.nops()) {
1558 return Li(m_,x_).hold();
1560 if (x.nops() == 0) {
1564 bool is_zeta = true;
1565 bool do_evalf = true;
1566 bool crational = true;
1567 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1568 if (!(*itm).info(info_flags::posint)) {
1569 return Li(m_,x_).hold();
1571 if ((*itx != _ex1) && (*itx != _ex_1)) {
1572 if (itx != x.begin()) {
1580 if (!(*itx).info(info_flags::numeric)) {
1583 if (!(*itx).info(info_flags::crational)) {
1589 for (const auto & itx : x) {
1590 GINAC_ASSERT((itx == _ex1) || (itx == _ex_1));
1591 // XXX: 1 + 0.0*I is considered equal to 1. However
1592 // the former is a not automatically converted
1593 // to a real number. Do the conversion explicitly
1594 // to avoid the "numeric::operator>(): complex inequality"
1595 // exception (and similar problems).
1596 newx.append(itx != _ex_1 ? _ex1 : _ex_1);
1598 return zeta(m_, newx);
1602 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1603 return prefactor * H(newm, x[0]);
1605 if (do_evalf && !crational) {
1606 return mLi_numeric(m,x);
1609 return Li(m_, x_).hold();
1610 } else if (is_a<lst>(x_)) {
1611 return Li(m_, x_).hold();
1614 // classical polylogs
1622 return (pow(2,1-m_)-1) * zeta(m_);
1628 if (x_.is_equal(I)) {
1629 return power(Pi,_ex2)/_ex_48 + Catalan*I;
1631 if (x_.is_equal(-I)) {
1632 return power(Pi,_ex2)/_ex_48 - Catalan*I;
1635 if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1636 int m__ = ex_to<numeric>(m_).to_int();
1637 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1638 const cln::cl_N result = Lin_numeric(m__, x__);
1639 return numeric(result);
1642 return Li(m_, x_).hold();
1646 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1648 if (is_a<lst>(m) || is_a<lst>(x)) {
1650 epvector seq { expair(Li(m, x), 0) };
1651 return pseries(rel, std::move(seq));
1654 // classical polylog
1655 const ex x_pt = x.subs(rel, subs_options::no_pattern);
1656 if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1657 // First special case: x==0 (derivatives have poles)
1658 if (x_pt.is_zero()) {
1661 // manually construct the primitive expansion
1662 for (int i=1; i<order; ++i)
1663 ser += pow(s,i) / pow(numeric(i), m);
1664 // substitute the argument's series expansion
1665 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1666 // maybe that was terminating, so add a proper order term
1667 epvector nseq { expair(Order(_ex1), order) };
1668 ser += pseries(rel, std::move(nseq));
1669 // reexpanding it will collapse the series again
1670 return ser.series(rel, order);
1672 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1673 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1675 // all other cases should be safe, by now:
1676 throw do_taylor(); // caught by function::series()
1680 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1682 GINAC_ASSERT(deriv_param < 2);
1683 if (deriv_param == 0) {
1686 if (m_.nops() > 1) {
1687 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1690 if (is_a<lst>(m_)) {
1696 if (is_a<lst>(x_)) {
1702 return Li(m-1, x) / x;
1709 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1712 if (is_a<lst>(m_)) {
1718 if (is_a<lst>(x_)) {
1723 c.s << "\\mathrm{Li}_{";
1724 auto itm = m.begin();
1727 for (; itm != m.end(); itm++) {
1732 auto itx = x.begin();
1735 for (; itx != x.end(); itx++) {
1743 REGISTER_FUNCTION(Li,
1744 evalf_func(Li_evalf).
1746 series_func(Li_series).
1747 derivative_func(Li_deriv).
1748 print_func<print_latex>(Li_print_latex).
1749 do_not_evalf_params());
1752 //////////////////////////////////////////////////////////////////////
1754 // Nielsen's generalized polylogarithm S(n,p,x)
1758 //////////////////////////////////////////////////////////////////////
1761 // anonymous namespace for helper functions
1765 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1767 std::vector<std::vector<cln::cl_N>> Yn;
1768 int ynsize = 0; // number of Yn[]
1769 int ynlength = 100; // initial length of all Yn[i]
1772 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1773 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1774 // representing S_{n,p}(x).
1775 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1776 // equivalent Z-sum.
1777 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1778 // representing S_{n,p}(x).
1779 // The calculation of Y_n uses the values from Y_{n-1}.
1780 void fill_Yn(int n, const cln::float_format_t& prec)
1782 const int initsize = ynlength;
1783 //const int initsize = initsize_Yn;
1784 cln::cl_N one = cln::cl_float(1, prec);
1787 std::vector<cln::cl_N> buf(initsize);
1788 auto it = buf.begin();
1789 auto itprev = Yn[n-1].begin();
1790 *it = (*itprev) / cln::cl_N(n+1) * one;
1793 // sums with an index smaller than the depth are zero and need not to be calculated.
1794 // calculation starts with depth, which is n+2)
1795 for (int i=n+2; i<=initsize+n; i++) {
1796 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1802 std::vector<cln::cl_N> buf(initsize);
1803 auto it = buf.begin();
1806 for (int i=2; i<=initsize; i++) {
1807 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1816 // make Yn longer ...
1817 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1820 cln::cl_N one = cln::cl_float(1, prec);
1822 Yn[0].resize(newsize);
1823 auto it = Yn[0].begin();
1825 for (int i=ynlength+1; i<=newsize; i++) {
1826 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1830 for (int n=1; n<ynsize; n++) {
1831 Yn[n].resize(newsize);
1832 auto it = Yn[n].begin();
1833 auto itprev = Yn[n-1].begin();
1836 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1837 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1847 // helper function for S(n,p,x)
1849 cln::cl_N C(int n, int p)
1853 for (int k=0; k<p; k++) {
1854 for (int j=0; j<=(n+k-1)/2; j++) {
1858 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1861 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1868 result = result + cln::factorial(n+k-1)
1869 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1870 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1873 result = result - cln::factorial(n+k-1)
1874 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1875 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1880 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1881 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1884 result = result + cln::factorial(n+k-1)
1885 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1886 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1894 if (((np)/2+n) & 1) {
1895 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1898 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1906 // helper function for S(n,p,x)
1907 // [Kol] remark to (9.1)
1908 cln::cl_N a_k(int k)
1917 for (int m=2; m<=k; m++) {
1918 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1925 // helper function for S(n,p,x)
1926 // [Kol] remark to (9.1)
1927 cln::cl_N b_k(int k)
1936 for (int m=2; m<=k; m++) {
1937 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1944 // helper function for S(n,p,x)
1945 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1947 static cln::float_format_t oldprec = cln::default_float_format;
1950 return Li_projection(n+1, x, prec);
1953 // precision has changed, we need to clear lookup table Yn
1954 if ( oldprec != prec ) {
1961 // check if precalculated values are sufficient
1963 for (int i=ynsize; i<p-1; i++) {
1968 // should be done otherwise
1969 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1970 cln::cl_N xf = x * one;
1971 //cln::cl_N xf = x * cln::cl_float(1, prec);
1975 cln::cl_N factor = cln::expt(xf, p);
1979 if (i-p >= ynlength) {
1981 make_Yn_longer(ynlength*2, prec);
1983 res = res + factor / cln::expt(cln::cl_I(i),n+1) * Yn[p-2][i-p]; // should we check it? or rely on magic number? ...
1984 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1985 factor = factor * xf;
1987 } while (res != resbuf);
1993 // helper function for S(n,p,x)
1994 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1997 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
1999 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
2000 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
2002 for (int s=0; s<n; s++) {
2004 for (int r=0; r<p; r++) {
2005 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
2006 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
2008 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2014 return S_do_sum(n, p, x, prec);
2018 // helper function for S(n,p,x)
2019 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
2023 // [Kol] (2.22) with (2.21)
2024 return cln::zeta(p+1);
2029 return cln::zeta(n+1);
2034 for (int nu=0; nu<n; nu++) {
2035 for (int rho=0; rho<=p; rho++) {
2036 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2037 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2040 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2047 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2049 // throw std::runtime_error("don't know how to evaluate this function!");
2052 // what is the desired float format?
2053 // first guess: default format
2054 cln::float_format_t prec = cln::default_float_format;
2055 const cln::cl_N value = x;
2056 // second guess: the argument's format
2057 if (!instanceof(realpart(value), cln::cl_RA_ring))
2058 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2059 else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2060 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2063 // the condition abs(1-value)>1 avoids an infinite recursion in the region abs(value)<=1 && abs(value)>0.95 && abs(1-value)<=1 && abs(1-value)>0.95
2064 // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
2065 if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
2067 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2068 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2070 for (int s=0; s<n; s++) {
2072 for (int r=0; r<p; r++) {
2073 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2074 * S_num(p-r,n-s,1-value) / cln::factorial(r);
2076 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2083 if (cln::abs(value) > 1) {
2087 for (int s=0; s<p; s++) {
2088 for (int r=0; r<=s; r++) {
2089 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2090 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2091 * S_num(n+s-r,p-s,cln::recip(value));
2094 result = result * cln::expt(cln::cl_I(-1),n);
2097 for (int r=0; r<n; r++) {
2098 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2100 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2102 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2107 if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
2110 for (int s=0; s<p-1; s++)
2113 ex res = H(m,numeric(value)).evalf();
2114 return ex_to<numeric>(res).to_cl_N();
2117 return S_projection(n, p, value, prec);
2122 } // end of anonymous namespace
2125 //////////////////////////////////////////////////////////////////////
2127 // Nielsen's generalized polylogarithm S(n,p,x)
2131 //////////////////////////////////////////////////////////////////////
2134 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2136 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2137 const int n_ = ex_to<numeric>(n).to_int();
2138 const int p_ = ex_to<numeric>(p).to_int();
2139 if (is_a<numeric>(x)) {
2140 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2141 const cln::cl_N result = S_num(n_, p_, x_);
2142 return numeric(result);
2144 ex x_val = x.evalf();
2145 if (is_a<numeric>(x_val)) {
2146 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2147 const cln::cl_N result = S_num(n_, p_, x_val_);
2148 return numeric(result);
2152 return S(n, p, x).hold();
2156 static ex S_eval(const ex& n, const ex& p, const ex& x)
2158 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2164 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2172 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2173 int n_ = ex_to<numeric>(n).to_int();
2174 int p_ = ex_to<numeric>(p).to_int();
2175 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2176 const cln::cl_N result = S_num(n_, p_, x_);
2177 return numeric(result);
2182 return pow(-log(1-x), p) / factorial(p);
2184 return S(n, p, x).hold();
2188 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2191 return Li(n+1, x).series(rel, order, options);
2194 const ex x_pt = x.subs(rel, subs_options::no_pattern);
2195 if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2196 // First special case: x==0 (derivatives have poles)
2197 if (x_pt.is_zero()) {
2200 // manually construct the primitive expansion
2201 // subsum = Euler-Zagier-Sum is needed
2202 // dirty hack (slow ...) calculation of subsum:
2203 std::vector<ex> presubsum, subsum;
2204 subsum.push_back(0);
2205 for (int i=1; i<order-1; ++i) {
2206 subsum.push_back(subsum[i-1] + numeric(1, i));
2208 for (int depth=2; depth<p; ++depth) {
2210 for (int i=1; i<order-1; ++i) {
2211 subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2215 for (int i=1; i<order; ++i) {
2216 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2218 // substitute the argument's series expansion
2219 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2220 // maybe that was terminating, so add a proper order term
2221 epvector nseq { expair(Order(_ex1), order) };
2222 ser += pseries(rel, std::move(nseq));
2223 // reexpanding it will collapse the series again
2224 return ser.series(rel, order);
2226 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2227 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2229 // all other cases should be safe, by now:
2230 throw do_taylor(); // caught by function::series()
2234 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2236 GINAC_ASSERT(deriv_param < 3);
2237 if (deriv_param < 2) {
2241 return S(n-1, p, x) / x;
2243 return S(n, p-1, x) / (1-x);
2248 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2250 c.s << "\\mathrm{S}_{";
2260 REGISTER_FUNCTION(S,
2261 evalf_func(S_evalf).
2263 series_func(S_series).
2264 derivative_func(S_deriv).
2265 print_func<print_latex>(S_print_latex).
2266 do_not_evalf_params());
2269 //////////////////////////////////////////////////////////////////////
2271 // Harmonic polylogarithm H(m,x)
2275 //////////////////////////////////////////////////////////////////////
2278 // anonymous namespace for helper functions
2282 // regulates the pole (used by 1/x-transformation)
2283 symbol H_polesign("IMSIGN");
2286 // convert parameters from H to Li representation
2287 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2288 // returns true if some parameters are negative
2289 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2291 // expand parameter list
2293 for (const auto & it : l) {
2295 for (ex count=it-1; count > 0; count--) {
2299 } else if (it < -1) {
2300 for (ex count=it+1; count < 0; count++) {
2311 bool has_negative_parameters = false;
2313 for (const auto & it : mexp) {
2319 m.append((it+acc-1) * signum);
2321 m.append((it-acc+1) * signum);
2327 has_negative_parameters = true;
2330 if (has_negative_parameters) {
2331 for (std::size_t i=0; i<m.nops(); i++) {
2333 m.let_op(i) = -m.op(i);
2341 return has_negative_parameters;
2345 // recursivly transforms H to corresponding multiple polylogarithms
2346 struct map_trafo_H_convert_to_Li : public map_function
2348 ex operator()(const ex& e) override
2350 if (is_a<add>(e) || is_a<mul>(e)) {
2351 return e.map(*this);
2353 if (is_a<function>(e)) {
2354 std::string name = ex_to<function>(e).get_name();
2357 if (is_a<lst>(e.op(0))) {
2358 parameter = ex_to<lst>(e.op(0));
2360 parameter = lst{e.op(0)};
2367 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2368 s.let_op(0) = s.op(0) * arg;
2369 return pf * Li(m, s).hold();
2371 for (std::size_t i=0; i<m.nops(); i++) {
2374 s.let_op(0) = s.op(0) * arg;
2375 return Li(m, s).hold();
2384 // recursivly transforms H to corresponding zetas
2385 struct map_trafo_H_convert_to_zeta : public map_function
2387 ex operator()(const ex& e) override
2389 if (is_a<add>(e) || is_a<mul>(e)) {
2390 return e.map(*this);
2392 if (is_a<function>(e)) {
2393 std::string name = ex_to<function>(e).get_name();
2396 if (is_a<lst>(e.op(0))) {
2397 parameter = ex_to<lst>(e.op(0));
2399 parameter = lst{e.op(0)};
2405 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2406 return pf * zeta(m, s);
2417 // remove trailing zeros from H-parameters
2418 struct map_trafo_H_reduce_trailing_zeros : public map_function
2420 ex operator()(const ex& e) override
2422 if (is_a<add>(e) || is_a<mul>(e)) {
2423 return e.map(*this);
2425 if (is_a<function>(e)) {
2426 std::string name = ex_to<function>(e).get_name();
2429 if (is_a<lst>(e.op(0))) {
2430 parameter = ex_to<lst>(e.op(0));
2432 parameter = lst{e.op(0)};
2435 if (parameter.op(parameter.nops()-1) == 0) {
2438 if (parameter.nops() == 1) {
2443 auto it = parameter.begin();
2444 while ((it != parameter.end()) && (*it == 0)) {
2447 if (it == parameter.end()) {
2448 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2452 parameter.remove_last();
2453 std::size_t lastentry = parameter.nops();
2454 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2459 ex result = log(arg) * H(parameter,arg).hold();
2461 for (ex i=0; i<lastentry; i++) {
2462 if (parameter[i] > 0) {
2464 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2467 } else if (parameter[i] < 0) {
2469 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2477 if (lastentry < parameter.nops()) {
2478 result = result / (parameter.nops()-lastentry+1);
2479 return result.map(*this);
2491 // returns an expression with zeta functions corresponding to the parameter list for H
2492 ex convert_H_to_zeta(const lst& m)
2494 symbol xtemp("xtemp");
2495 map_trafo_H_reduce_trailing_zeros filter;
2496 map_trafo_H_convert_to_zeta filter2;
2497 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2501 // convert signs form Li to H representation
2502 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2505 auto itm = m.begin();
2506 auto itx = ++x.begin();
2511 while (itx != x.end()) {
2512 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
2513 // XXX: 1 + 0.0*I is considered equal to 1. However the former
2514 // is not automatically converted to a real number.
2515 // Do the conversion explicitly to avoid the
2516 // "numeric::operator>(): complex inequality" exception.
2517 signum *= (*itx != _ex_1) ? 1 : -1;
2519 res.append((*itm) * signum);
2527 // multiplies an one-dimensional H with another H
2529 ex trafo_H_mult(const ex& h1, const ex& h2)
2534 ex h1nops = h1.op(0).nops();
2535 ex h2nops = h2.op(0).nops();
2537 hshort = h2.op(0).op(0);
2538 hlong = ex_to<lst>(h1.op(0));
2540 hshort = h1.op(0).op(0);
2542 hlong = ex_to<lst>(h2.op(0));
2544 hlong = lst{h2.op(0).op(0)};
2547 for (std::size_t i=0; i<=hlong.nops(); i++) {
2551 newparameter.append(hlong[j]);
2553 newparameter.append(hshort);
2554 for (; j<hlong.nops(); j++) {
2555 newparameter.append(hlong[j]);
2557 res += H(newparameter, h1.op(1)).hold();
2563 // applies trafo_H_mult recursively on expressions
2564 struct map_trafo_H_mult : public map_function
2566 ex operator()(const ex& e) override
2569 return e.map(*this);
2577 for (std::size_t pos=0; pos<e.nops(); pos++) {
2578 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2579 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2581 for (ex i=0; i<e.op(pos).op(1); i++) {
2582 Hlst.append(e.op(pos).op(0));
2586 } else if (is_a<function>(e.op(pos))) {
2587 std::string name = ex_to<function>(e.op(pos)).get_name();
2589 if (e.op(pos).op(0).nops() > 1) {
2592 Hlst.append(e.op(pos));
2597 result *= e.op(pos);
2600 if (Hlst.nops() > 0) {
2601 firstH = Hlst[Hlst.nops()-1];
2608 if (Hlst.nops() > 0) {
2609 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2611 for (std::size_t i=1; i<Hlst.nops(); i++) {
2612 result *= Hlst.op(i);
2614 result = result.expand();
2615 map_trafo_H_mult recursion;
2616 return recursion(result);
2627 // do integration [ReV] (55)
2628 // put parameter 0 in front of existing parameters
2629 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2633 if (is_a<function>(e)) {
2634 name = ex_to<function>(e).get_name();
2639 for (std::size_t i=0; i<e.nops(); i++) {
2640 if (is_a<function>(e.op(i))) {
2641 std::string name = ex_to<function>(e.op(i)).get_name();
2649 lst newparameter = ex_to<lst>(h.op(0));
2650 newparameter.prepend(0);
2651 ex addzeta = convert_H_to_zeta(newparameter);
2652 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2654 return e * (-H(lst{ex(0)},1/arg).hold());
2659 // do integration [ReV] (49)
2660 // put parameter 1 in front of existing parameters
2661 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2665 if (is_a<function>(e)) {
2666 name = ex_to<function>(e).get_name();
2671 for (std::size_t i=0; i<e.nops(); i++) {
2672 if (is_a<function>(e.op(i))) {
2673 std::string name = ex_to<function>(e.op(i)).get_name();
2681 lst newparameter = ex_to<lst>(h.op(0));
2682 newparameter.prepend(1);
2683 return e.subs(h == H(newparameter, h.op(1)).hold());
2685 return e * H(lst{ex(1)},1-arg).hold();
2690 // do integration [ReV] (55)
2691 // put parameter -1 in front of existing parameters
2692 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2696 if (is_a<function>(e)) {
2697 name = ex_to<function>(e).get_name();
2702 for (std::size_t i=0; i<e.nops(); i++) {
2703 if (is_a<function>(e.op(i))) {
2704 std::string name = ex_to<function>(e.op(i)).get_name();
2712 lst newparameter = ex_to<lst>(h.op(0));
2713 newparameter.prepend(-1);
2714 ex addzeta = convert_H_to_zeta(newparameter);
2715 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2717 ex addzeta = convert_H_to_zeta(lst{ex(-1)});
2718 return (e * (addzeta - H(lst{ex(-1)},1/arg).hold())).expand();
2723 // do integration [ReV] (55)
2724 // put parameter -1 in front of existing parameters
2725 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2729 if (is_a<function>(e)) {
2730 name = ex_to<function>(e).get_name();
2735 for (std::size_t i = 0; i < e.nops(); i++) {
2736 if (is_a<function>(e.op(i))) {
2737 std::string name = ex_to<function>(e.op(i)).get_name();
2745 lst newparameter = ex_to<lst>(h.op(0));
2746 newparameter.prepend(-1);
2747 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2749 return (e * H(lst{ex(-1)},(1-arg)/(1+arg)).hold()).expand();
2754 // do integration [ReV] (55)
2755 // put parameter 1 in front of existing parameters
2756 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2760 if (is_a<function>(e)) {
2761 name = ex_to<function>(e).get_name();
2766 for (std::size_t i = 0; i < e.nops(); i++) {
2767 if (is_a<function>(e.op(i))) {
2768 std::string name = ex_to<function>(e.op(i)).get_name();
2776 lst newparameter = ex_to<lst>(h.op(0));
2777 newparameter.prepend(1);
2778 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2780 return (e * H(lst{ex(1)},(1-arg)/(1+arg)).hold()).expand();
2785 // do x -> 1-x transformation
2786 struct map_trafo_H_1mx : public map_function
2788 ex operator()(const ex& e) override
2790 if (is_a<add>(e) || is_a<mul>(e)) {
2791 return e.map(*this);
2794 if (is_a<function>(e)) {
2795 std::string name = ex_to<function>(e).get_name();
2798 lst parameter = ex_to<lst>(e.op(0));
2801 // special cases if all parameters are either 0, 1 or -1
2802 bool allthesame = true;
2803 if (parameter.op(0) == 0) {
2804 for (std::size_t i = 1; i < parameter.nops(); i++) {
2805 if (parameter.op(i) != 0) {
2812 for (int i=parameter.nops(); i>0; i--) {
2813 newparameter.append(1);
2815 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2817 } else if (parameter.op(0) == -1) {
2818 throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2820 for (std::size_t i = 1; i < parameter.nops(); i++) {
2821 if (parameter.op(i) != 1) {
2828 for (int i=parameter.nops(); i>0; i--) {
2829 newparameter.append(0);
2831 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2835 lst newparameter = parameter;
2836 newparameter.remove_first();
2838 if (parameter.op(0) == 0) {
2841 ex res = convert_H_to_zeta(parameter);
2842 //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2843 map_trafo_H_1mx recursion;
2844 ex buffer = recursion(H(newparameter, arg).hold());
2845 if (is_a<add>(buffer)) {
2846 for (std::size_t i = 0; i < buffer.nops(); i++) {
2847 res -= trafo_H_prepend_one(buffer.op(i), arg);
2850 res -= trafo_H_prepend_one(buffer, arg);
2857 map_trafo_H_1mx recursion;
2858 map_trafo_H_mult unify;
2859 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
2860 std::size_t firstzero = 0;
2861 while (parameter.op(firstzero) == 1) {
2864 for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2868 newparameter.append(parameter[j+1]);
2870 newparameter.append(1);
2871 for (; j<parameter.nops()-1; j++) {
2872 newparameter.append(parameter[j+1]);
2874 res -= H(newparameter, arg).hold();
2876 res = recursion(res).expand() / firstzero;
2886 // do x -> 1/x transformation
2887 struct map_trafo_H_1overx : public map_function
2889 ex operator()(const ex& e) override
2891 if (is_a<add>(e) || is_a<mul>(e)) {
2892 return e.map(*this);
2895 if (is_a<function>(e)) {
2896 std::string name = ex_to<function>(e).get_name();
2899 lst parameter = ex_to<lst>(e.op(0));
2902 // special cases if all parameters are either 0, 1 or -1
2903 bool allthesame = true;
2904 if (parameter.op(0) == 0) {
2905 for (std::size_t i = 1; i < parameter.nops(); i++) {
2906 if (parameter.op(i) != 0) {
2912 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2914 } else if (parameter.op(0) == -1) {
2915 for (std::size_t i = 1; i < parameter.nops(); i++) {
2916 if (parameter.op(i) != -1) {
2922 map_trafo_H_mult unify;
2923 return unify((pow(H(lst{ex(-1)},1/arg).hold() - H(lst{ex(0)},1/arg).hold(), parameter.nops())
2924 / factorial(parameter.nops())).expand());
2927 for (std::size_t i = 1; i < parameter.nops(); i++) {
2928 if (parameter.op(i) != 1) {
2934 map_trafo_H_mult unify;
2935 return unify((pow(H(lst{ex(1)},1/arg).hold() + H(lst{ex(0)},1/arg).hold() + H_polesign, parameter.nops())
2936 / factorial(parameter.nops())).expand());
2940 lst newparameter = parameter;
2941 newparameter.remove_first();
2943 if (parameter.op(0) == 0) {
2946 ex res = convert_H_to_zeta(parameter);
2947 map_trafo_H_1overx recursion;
2948 ex buffer = recursion(H(newparameter, arg).hold());
2949 if (is_a<add>(buffer)) {
2950 for (std::size_t i = 0; i < buffer.nops(); i++) {
2951 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2954 res += trafo_H_1tx_prepend_zero(buffer, arg);
2958 } else if (parameter.op(0) == -1) {
2960 // leading negative one
2961 ex res = convert_H_to_zeta(parameter);
2962 map_trafo_H_1overx recursion;
2963 ex buffer = recursion(H(newparameter, arg).hold());
2964 if (is_a<add>(buffer)) {
2965 for (std::size_t i = 0; i < buffer.nops(); i++) {
2966 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2969 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2976 map_trafo_H_1overx recursion;
2977 map_trafo_H_mult unify;
2978 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
2979 std::size_t firstzero = 0;
2980 while (parameter.op(firstzero) == 1) {
2983 for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
2987 newparameter.append(parameter[j+1]);
2989 newparameter.append(1);
2990 for (; j<parameter.nops()-1; j++) {
2991 newparameter.append(parameter[j+1]);
2993 res -= H(newparameter, arg).hold();
2995 res = recursion(res).expand() / firstzero;
3007 // do x -> (1-x)/(1+x) transformation
3008 struct map_trafo_H_1mxt1px : public map_function
3010 ex operator()(const ex& e) override
3012 if (is_a<add>(e) || is_a<mul>(e)) {
3013 return e.map(*this);
3016 if (is_a<function>(e)) {
3017 std::string name = ex_to<function>(e).get_name();
3020 lst parameter = ex_to<lst>(e.op(0));
3023 // special cases if all parameters are either 0, 1 or -1
3024 bool allthesame = true;
3025 if (parameter.op(0) == 0) {
3026 for (std::size_t i = 1; i < parameter.nops(); i++) {
3027 if (parameter.op(i) != 0) {
3033 map_trafo_H_mult unify;
3034 return unify((pow(-H(lst{ex(1)},(1-arg)/(1+arg)).hold() - H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops())
3035 / factorial(parameter.nops())).expand());
3037 } else if (parameter.op(0) == -1) {
3038 for (std::size_t i = 1; i < parameter.nops(); i++) {
3039 if (parameter.op(i) != -1) {
3045 map_trafo_H_mult unify;
3046 return unify((pow(log(2) - H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops())
3047 / factorial(parameter.nops())).expand());
3050 for (std::size_t i = 1; i < parameter.nops(); i++) {
3051 if (parameter.op(i) != 1) {
3057 map_trafo_H_mult unify;
3058 return unify((pow(-log(2) - H(lst{ex(0)},(1-arg)/(1+arg)).hold() + H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops())
3059 / factorial(parameter.nops())).expand());
3063 lst newparameter = parameter;
3064 newparameter.remove_first();
3066 if (parameter.op(0) == 0) {
3069 ex res = convert_H_to_zeta(parameter);
3070 map_trafo_H_1mxt1px recursion;
3071 ex buffer = recursion(H(newparameter, arg).hold());
3072 if (is_a<add>(buffer)) {
3073 for (std::size_t i = 0; i < buffer.nops(); i++) {
3074 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3077 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3081 } else if (parameter.op(0) == -1) {
3083 // leading negative one
3084 ex res = convert_H_to_zeta(parameter);
3085 map_trafo_H_1mxt1px recursion;
3086 ex buffer = recursion(H(newparameter, arg).hold());
3087 if (is_a<add>(buffer)) {
3088 for (std::size_t i = 0; i < buffer.nops(); i++) {
3089 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3092 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3099 map_trafo_H_1mxt1px recursion;
3100 map_trafo_H_mult unify;
3101 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
3102 std::size_t firstzero = 0;
3103 while (parameter.op(firstzero) == 1) {
3106 for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3110 newparameter.append(parameter[j+1]);
3112 newparameter.append(1);
3113 for (; j<parameter.nops()-1; j++) {
3114 newparameter.append(parameter[j+1]);
3116 res -= H(newparameter, arg).hold();
3118 res = recursion(res).expand() / firstzero;
3130 // do the actual summation.
3131 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3133 const int j = m.size();
3135 std::vector<cln::cl_N> t(j);
3137 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3138 cln::cl_N factor = cln::expt(x, j) * one;
3144 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3145 for (int k=j-2; k>=1; k--) {
3146 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3148 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3149 factor = factor * x;
3150 } while (t[0] != t0buf);
3156 } // end of anonymous namespace
3159 //////////////////////////////////////////////////////////////////////
3161 // Harmonic polylogarithm H(m,x)
3165 //////////////////////////////////////////////////////////////////////
3168 static ex H_evalf(const ex& x1, const ex& x2)
3170 if (is_a<lst>(x1)) {
3173 if (is_a<numeric>(x2)) {
3174 x = ex_to<numeric>(x2).to_cl_N();
3176 ex x2_val = x2.evalf();
3177 if (is_a<numeric>(x2_val)) {
3178 x = ex_to<numeric>(x2_val).to_cl_N();
3182 for (std::size_t i = 0; i < x1.nops(); i++) {
3183 if (!x1.op(i).info(info_flags::integer)) {
3184 return H(x1, x2).hold();
3187 if (x1.nops() < 1) {
3188 return H(x1, x2).hold();
3191 const lst& morg = ex_to<lst>(x1);
3192 // remove trailing zeros ...
3193 if (*(--morg.end()) == 0) {
3194 symbol xtemp("xtemp");
3195 map_trafo_H_reduce_trailing_zeros filter;
3196 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3198 // ... and expand parameter notation
3199 bool has_minus_one = false;
3201 for (const auto & it : morg) {
3203 for (ex count=it-1; count > 0; count--) {
3207 } else if (it <= -1) {
3208 for (ex count=it+1; count < 0; count++) {
3212 has_minus_one = true;
3219 if (cln::abs(x) < 0.95) {
3223 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3224 // negative parameters -> s_lst is filled
3225 std::vector<int> m_int;
3226 std::vector<cln::cl_N> x_cln;
3227 for (auto it_int = m_lst.begin(), it_cln = s_lst.begin();
3228 it_int != m_lst.end(); it_int++, it_cln++) {
3229 m_int.push_back(ex_to<numeric>(*it_int).to_int());
3230 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3232 x_cln.front() = x_cln.front() * x;
3233 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3235 // only positive parameters
3237 if (m_lst.nops() == 1) {
3238 return Li(m_lst.op(0), x2).evalf();
3240 std::vector<int> m_int;
3241 for (const auto & it : m_lst) {
3242 m_int.push_back(ex_to<numeric>(it).to_int());
3244 return numeric(H_do_sum(m_int, x));
3248 symbol xtemp("xtemp");
3251 // ensure that the realpart of the argument is positive
3252 if (cln::realpart(x) < 0) {
3254 for (std::size_t i = 0; i < m.nops(); i++) {
3256 m.let_op(i) = -m.op(i);
3263 if (cln::abs(x) >= 2.0) {
3264 map_trafo_H_1overx trafo;
3265 res *= trafo(H(m, xtemp).hold());
3266 if (cln::imagpart(x) <= 0) {
3267 res = res.subs(H_polesign == -I*Pi);
3269 res = res.subs(H_polesign == I*Pi);
3271 return res.subs(xtemp == numeric(x)).evalf();
3274 // check transformations for 0.95 <= |x| < 2.0
3276 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3277 if (cln::abs(x-9.53) <= 9.47) {
3279 map_trafo_H_1mxt1px trafo;
3280 res *= trafo(H(m, xtemp).hold());
3283 if (has_minus_one) {
3284 map_trafo_H_convert_to_Li filter;
3285 return filter(H(m, numeric(x)).hold()).evalf();
3287 map_trafo_H_1mx trafo;
3288 res *= trafo(H(m, xtemp).hold());
3291 return res.subs(xtemp == numeric(x)).evalf();
3294 return H(x1,x2).hold();
3298 static ex H_eval(const ex& m_, const ex& x)
3301 if (is_a<lst>(m_)) {
3306 if (m.nops() == 0) {
3314 if (*m.begin() > _ex1) {
3320 } else if (*m.begin() < _ex_1) {
3326 } else if (*m.begin() == _ex0) {
3333 for (auto it = ++m.begin(); it != m.end(); it++) {
3334 if (it->info(info_flags::integer)) {
3345 } else if (*it < _ex_1) {
3365 } else if (step == 1) {
3377 // if some m_i is not an integer
3378 return H(m_, x).hold();
3381 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3382 return convert_H_to_zeta(m);
3388 return H(m_, x).hold();
3390 return pow(log(x), m.nops()) / factorial(m.nops());
3393 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3395 } else if ((step == 1) && (pos1 == _ex0)){
3400 return pow(-1, p) * S(n, p, -x);
3406 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3407 return H(m_, x).evalf();
3409 return H(m_, x).hold();
3413 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3415 epvector seq { expair(H(m, x), 0) };
3416 return pseries(rel, std::move(seq));
3420 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3422 GINAC_ASSERT(deriv_param < 2);
3423 if (deriv_param == 0) {
3427 if (is_a<lst>(m_)) {
3443 return 1/(1-x) * H(m, x);
3444 } else if (mb == _ex_1) {
3445 return 1/(1+x) * H(m, x);
3452 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3455 if (is_a<lst>(m_)) {
3460 c.s << "\\mathrm{H}_{";
3461 auto itm = m.begin();
3464 for (; itm != m.end(); itm++) {
3474 REGISTER_FUNCTION(H,
3475 evalf_func(H_evalf).
3477 series_func(H_series).
3478 derivative_func(H_deriv).
3479 print_func<print_latex>(H_print_latex).
3480 do_not_evalf_params());
3483 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3484 ex convert_H_to_Li(const ex& m, const ex& x)
3486 map_trafo_H_reduce_trailing_zeros filter;
3487 map_trafo_H_convert_to_Li filter2;
3489 return filter2(filter(H(m, x).hold()));
3491 return filter2(filter(H(lst{m}, x).hold()));
3496 //////////////////////////////////////////////////////////////////////
3498 // Multiple zeta values zeta(x) and zeta(x,s)
3502 //////////////////////////////////////////////////////////////////////
3505 // anonymous namespace for helper functions
3509 // parameters and data for [Cra] algorithm
3510 const cln::cl_N lambda = cln::cl_N("319/320");
3512 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3514 const int size = a.size();
3515 for (int n=0; n<size; n++) {
3517 for (int m=0; m<=n; m++) {
3518 c[n] = c[n] + a[m]*b[n-m];
3525 static void initcX(std::vector<cln::cl_N>& crX,
3526 const std::vector<int>& s,
3529 std::vector<cln::cl_N> crB(L2 + 1);
3530 for (int i=0; i<=L2; i++)
3531 crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i);
3535 std::vector<std::vector<cln::cl_N>> crG(s.size() - 1, std::vector<cln::cl_N>(L2 + 1));
3536 for (int m=0; m < (int)s.size() - 1; m++) {
3539 for (int i = 0; i <= L2; i++)
3540 crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2);
3545 for (std::size_t m = 0; m < s.size() - 1; m++) {
3546 std::vector<cln::cl_N> Xbuf(L2 + 1);
3547 for (int i = 0; i <= L2; i++)
3548 Xbuf[i] = crX[i] * crG[m][i];
3550 halfcyclic_convolute(Xbuf, crB, crX);
3556 static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk,
3557 const std::vector<cln::cl_N>& crX)
3559 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3560 cln::cl_N factor = cln::expt(lambda, Sqk);
3561 cln::cl_N res = factor / Sqk * crX[0] * one;
3566 factor = factor * lambda;
3568 res = res + crX[N] * factor / (N+Sqk);
3569 } while ((res != resbuf) || cln::zerop(crX[N]));
3575 static void calc_f(std::vector<std::vector<cln::cl_N>>& f_kj,
3576 const int maxr, const int L1)
3578 cln::cl_N t0, t1, t2, t3, t4;
3580 auto it = f_kj.begin();
3581 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3583 t0 = cln::exp(-lambda);
3585 for (k=1; k<=L1; k++) {
3588 for (j=1; j<=maxr; j++) {
3591 for (i=2; i<=j; i++) {
3595 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3603 static cln::cl_N crandall_Z(const std::vector<int>& s,
3604 const std::vector<std::vector<cln::cl_N>>& f_kj)
3606 const int j = s.size();
3615 t0 = t0 + f_kj[q+j-2][s[0]-1];
3616 } while (t0 != t0buf);
3618 return t0 / cln::factorial(s[0]-1);
3621 std::vector<cln::cl_N> t(j);
3628 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3629 for (int k=j-2; k>=1; k--) {
3630 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3632 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3633 } while (t[0] != t0buf);
3635 return t[0] / cln::factorial(s[0]-1);
3640 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3642 std::vector<int> r = s;
3643 const int j = r.size();
3647 // decide on maximal size of f_kj for crandall_Z
3651 L1 = Digits * 3 + j*2;
3655 // decide on maximal size of crX for crandall_Y
3658 } else if (Digits < 86) {
3660 } else if (Digits < 192) {
3662 } else if (Digits < 394) {
3664 } else if (Digits < 808) {
3674 for (int i=0; i<j; i++) {
3681 std::vector<std::vector<cln::cl_N>> f_kj(L1);
3682 calc_f(f_kj, maxr, L1);
3684 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3686 std::vector<int> rz;
3689 for (int k=r.size()-1; k>0; k--) {
3691 rz.insert(rz.begin(), r.back());
3692 skp1buf = rz.front();
3696 std::vector<cln::cl_N> crX;
3699 for (int q=0; q<skp1buf; q++) {
3701 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k, crX);
3702 cln::cl_N pp2 = crandall_Z(rz, f_kj);
3707 res = res - pp1 * pp2 / cln::factorial(q);
3709 res = res + pp1 * pp2 / cln::factorial(q);
3712 rz.front() = skp1buf;
3714 rz.insert(rz.begin(), r.back());
3716 std::vector<cln::cl_N> crX;
3717 initcX(crX, rz, L2);
3719 res = (res + crandall_Y_loop(S-j, crX)) / r0factorial
3720 + crandall_Z(rz, f_kj);
3726 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3728 const int j = r.size();
3730 // buffer for subsums
3731 std::vector<cln::cl_N> t(j);
3732 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3739 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3740 for (int k=j-2; k>=0; k--) {
3741 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3743 } while (t[0] != t0buf);
3749 // does Hoelder convolution. see [BBB] (7.0)
3750 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3752 // prepare parameters
3753 // holds Li arguments in [BBB] notation
3754 std::vector<int> s = s_;
3755 std::vector<int> m_p = m_;
3756 std::vector<int> m_q;
3757 // holds Li arguments in nested sums notation
3758 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3759 s_p[0] = s_p[0] * cln::cl_N("1/2");
3760 // convert notations
3762 for (std::size_t i = 0; i < s_.size(); i++) {
3767 s[i] = sig * std::abs(s[i]);
3769 std::vector<cln::cl_N> s_q;
3770 cln::cl_N signum = 1;
3773 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3778 // change parameters
3779 if (s.front() > 0) {
3780 if (m_p.front() == 1) {
3781 m_p.erase(m_p.begin());
3782 s_p.erase(s_p.begin());
3783 if (s_p.size() > 0) {
3784 s_p.front() = s_p.front() * cln::cl_N("1/2");
3790 m_q.insert(m_q.begin(), 1);
3791 if (s_q.size() > 0) {
3792 s_q.front() = s_q.front() * 2;
3794 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3797 if (m_p.front() == 1) {
3798 m_p.erase(m_p.begin());
3799 cln::cl_N spbuf = s_p.front();
3800 s_p.erase(s_p.begin());
3801 if (s_p.size() > 0) {
3802 s_p.front() = s_p.front() * spbuf;
3805 m_q.insert(m_q.begin(), 1);
3806 if (s_q.size() > 0) {
3807 s_q.front() = s_q.front() * 4;
3809 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3813 m_q.insert(m_q.begin(), 1);
3814 if (s_q.size() > 0) {
3815 s_q.front() = s_q.front() * 2;
3817 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3822 if (m_p.size() == 0) break;
3824 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3829 res = res + signum * multipleLi_do_sum(m_q, s_q);
3835 } // end of anonymous namespace
3838 //////////////////////////////////////////////////////////////////////
3840 // Multiple zeta values zeta(x)
3844 //////////////////////////////////////////////////////////////////////
3847 static ex zeta1_evalf(const ex& x)
3849 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3851 // multiple zeta value
3852 const int count = x.nops();
3853 const lst& xlst = ex_to<lst>(x);
3854 std::vector<int> r(count);
3856 // check parameters and convert them
3857 auto it1 = xlst.begin();
3858 auto it2 = r.begin();
3860 if (!(*it1).info(info_flags::posint)) {
3861 return zeta(x).hold();
3863 *it2 = ex_to<numeric>(*it1).to_int();
3866 } while (it2 != r.end());
3868 // check for divergence
3870 return zeta(x).hold();
3873 // decide on summation algorithm
3874 // this is still a bit clumsy
3875 int limit = (Digits>17) ? 10 : 6;
3876 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3877 return numeric(zeta_do_sum_Crandall(r));
3879 return numeric(zeta_do_sum_simple(r));
3883 // single zeta value
3884 if (is_exactly_a<numeric>(x) && (x != 1)) {
3886 return zeta(ex_to<numeric>(x));
3887 } catch (const dunno &e) { }
3890 return zeta(x).hold();
3894 static ex zeta1_eval(const ex& m)
3896 if (is_exactly_a<lst>(m)) {
3897 if (m.nops() == 1) {
3898 return zeta(m.op(0));
3900 return zeta(m).hold();
3903 if (m.info(info_flags::numeric)) {
3904 const numeric& y = ex_to<numeric>(m);
3905 // trap integer arguments:
3906 if (y.is_integer()) {
3910 if (y.is_equal(*_num1_p)) {
3911 return zeta(m).hold();
3913 if (y.info(info_flags::posint)) {
3914 if (y.info(info_flags::odd)) {
3915 return zeta(m).hold();
3917 return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3920 if (y.info(info_flags::odd)) {
3921 return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3928 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3929 return zeta1_evalf(m);
3932 return zeta(m).hold();
3936 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3938 GINAC_ASSERT(deriv_param==0);
3940 if (is_exactly_a<lst>(m)) {
3943 return zetaderiv(_ex1, m);
3948 static void zeta1_print_latex(const ex& m_, const print_context& c)
3951 if (is_a<lst>(m_)) {
3952 const lst& m = ex_to<lst>(m_);
3953 auto it = m.begin();
3956 for (; it != m.end(); it++) {
3967 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
3968 evalf_func(zeta1_evalf).
3969 eval_func(zeta1_eval).
3970 derivative_func(zeta1_deriv).
3971 print_func<print_latex>(zeta1_print_latex).
3972 do_not_evalf_params().
3976 //////////////////////////////////////////////////////////////////////
3978 // Alternating Euler sum zeta(x,s)
3982 //////////////////////////////////////////////////////////////////////
3985 static ex zeta2_evalf(const ex& x, const ex& s)
3987 if (is_exactly_a<lst>(x)) {
3989 // alternating Euler sum
3990 const int count = x.nops();
3991 const lst& xlst = ex_to<lst>(x);
3992 const lst& slst = ex_to<lst>(s);
3993 std::vector<int> xi(count);
3994 std::vector<int> si(count);
3996 // check parameters and convert them
3997 auto it_xread = xlst.begin();
3998 auto it_sread = slst.begin();
3999 auto it_xwrite = xi.begin();
4000 auto it_swrite = si.begin();
4002 if (!(*it_xread).info(info_flags::posint)) {
4003 return zeta(x, s).hold();
4005 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
4006 if (*it_sread > 0) {
4015 } while (it_xwrite != xi.end());
4017 // check for divergence
4018 if ((xi[0] == 1) && (si[0] == 1)) {
4019 return zeta(x, s).hold();
4022 // use Hoelder convolution
4023 return numeric(zeta_do_Hoelder_convolution(xi, si));
4026 return zeta(x, s).hold();
4030 static ex zeta2_eval(const ex& m, const ex& s_)
4032 if (is_exactly_a<lst>(s_)) {
4033 const lst& s = ex_to<lst>(s_);
4034 for (const auto & it : s) {
4035 if (it.info(info_flags::positive)) {
4038 return zeta(m, s_).hold();
4041 } else if (s_.info(info_flags::positive)) {
4045 return zeta(m, s_).hold();
4049 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
4051 GINAC_ASSERT(deriv_param==0);
4053 if (is_exactly_a<lst>(m)) {
4056 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
4057 return zetaderiv(_ex1, m);
4064 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
4067 if (is_a<lst>(m_)) {
4073 if (is_a<lst>(s_)) {
4079 auto itm = m.begin();
4080 auto its = s.begin();
4082 c.s << "\\overline{";
4090 for (; itm != m.end(); itm++, its++) {
4093 c.s << "\\overline{";
4104 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
4105 evalf_func(zeta2_evalf).
4106 eval_func(zeta2_eval).
4107 derivative_func(zeta2_deriv).
4108 print_func<print_latex>(zeta2_print_latex).
4109 do_not_evalf_params().
4113 } // namespace GiNaC