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) 1999-2018 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(x) < 0.25) {
341 return Li2_do_sum(x);
343 // Li2_do_sum practically doesn't converge near x == ±I
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(x) < 0.3) || (n >= 12)) {
370 return Lin_do_sum(n, x);
372 // Li2_do_sum practically doesn't converge near x == ±I
373 return Lin_do_sum_Xn(n, x);
376 cln::cl_N result = 0;
377 if ( x != 1 ) result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
378 for (int j=0; j<n-1; j++) {
379 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
380 * cln::expt(cln::log(x), j) / cln::factorial(j);
387 // helper function for classical polylog Li
388 const cln::cl_N Lin_numeric(const int n, const cln::cl_N& x)
392 return -cln::log(1-x);
403 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
405 if (cln::abs(realpart(x)) < 0.4 && cln::abs(cln::abs(x)-1) < 0.01) {
406 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
407 for (int j=0; j<n-1; j++) {
408 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
409 * cln::expt(cln::log(x), j) / cln::factorial(j);
414 // what is the desired float format?
415 // first guess: default format
416 cln::float_format_t prec = cln::default_float_format;
417 const cln::cl_N value = x;
418 // second guess: the argument's format
419 if (!instanceof(realpart(x), cln::cl_RA_ring))
420 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
421 else if (!instanceof(imagpart(x), cln::cl_RA_ring))
422 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
425 if (cln::abs(value) > 1) {
426 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
427 // check if argument is complex. if it is real, the new polylog has to be conjugated.
428 if (cln::zerop(cln::imagpart(value))) {
430 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
433 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
438 result = result + Li_projection(n, cln::recip(value), prec);
441 result = result - Li_projection(n, cln::recip(value), prec);
445 for (int j=0; j<n-1; j++) {
446 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
447 * Lin_numeric(n-j,1) * cln::expt(cln::log(-value),j) / cln::factorial(j);
449 result = result - add;
453 return Li_projection(n, value, prec);
458 } // end of anonymous namespace
461 //////////////////////////////////////////////////////////////////////
463 // Multiple polylogarithm Li(n,x)
467 //////////////////////////////////////////////////////////////////////
470 // anonymous namespace for helper function
474 // performs the actual series summation for multiple polylogarithms
475 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
477 // ensure all x <> 0.
478 for (const auto & it : x) {
479 if (it == 0) return cln::cl_float(0, cln::float_format(Digits));
482 const int j = s.size();
483 bool flag_accidental_zero = false;
485 std::vector<cln::cl_N> t(j);
486 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
493 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
494 for (int k=j-2; k>=0; k--) {
495 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]);
498 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
499 for (int k=j-2; k>=0; k--) {
500 flag_accidental_zero = cln::zerop(t[k+1]);
501 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]);
503 } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero );
509 // forward declaration for Li_eval()
510 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
513 // type used by the transformation functions for G
514 typedef std::vector<int> Gparameter;
517 // G_eval1-function for G transformations
518 ex G_eval1(int a, int scale, const exvector& gsyms)
521 const ex& scs = gsyms[std::abs(scale)];
522 const ex& as = gsyms[std::abs(a)];
524 return -log(1 - scs/as);
529 return log(gsyms[std::abs(scale)]);
534 // G_eval-function for G transformations
535 ex G_eval(const Gparameter& a, int scale, const exvector& gsyms)
537 // check for properties of G
538 ex sc = gsyms[std::abs(scale)];
540 bool all_zero = true;
541 bool all_ones = true;
543 for (const auto & it : a) {
545 const ex sym = gsyms[std::abs(it)];
559 // care about divergent G: shuffle to separate divergencies that will be canceled
560 // later on in the transformation
561 if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
563 Gparameter short_a(a.begin()+1, a.end());
564 ex result = G_eval1(a.front(), scale, gsyms) * G_eval(short_a, scale, gsyms);
566 auto it = short_a.begin();
567 advance(it, count_ones-1);
568 for (; it != short_a.end(); ++it) {
570 Gparameter newa(short_a.begin(), it);
572 newa.push_back(a[0]);
573 newa.insert(newa.end(), it+1, short_a.end());
574 result -= G_eval(newa, scale, gsyms);
576 return result / count_ones;
579 // G({1,...,1};y) -> G({1};y)^k / k!
580 if (all_ones && a.size() > 1) {
581 return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones);
584 // G({0,...,0};y) -> log(y)^k / k!
586 return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
589 // no special cases anymore -> convert it into Li
592 ex argbuf = gsyms[std::abs(scale)];
594 for (const auto & it : a) {
596 const ex& sym = gsyms[std::abs(it)];
597 x.append(argbuf / sym);
605 return pow(-1, x.nops()) * Li(m, x);
609 // converts data for G: pending_integrals -> a
610 Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
612 GINAC_ASSERT(pending_integrals.size() != 1);
614 if (pending_integrals.size() > 0) {
615 // get rid of the first element, which would stand for the new upper limit
616 Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
619 // just return empty parameter list
626 // check the parameters a and scale for G and return information about convergence, depth, etc.
627 // convergent : true if G(a,scale) is convergent
628 // depth : depth of G(a,scale)
629 // trailing_zeros : number of trailing zeros of a
630 // min_it : iterator of a pointing on the smallest element in a
631 Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
632 bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
638 auto lastnonzero = a.end();
639 for (auto it = a.begin(); it != a.end(); ++it) {
640 if (std::abs(*it) > 0) {
644 if (std::abs(*it) < scale) {
646 if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
654 if (lastnonzero == a.end())
656 return ++lastnonzero;
660 // add scale to pending_integrals if pending_integrals is empty
661 Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
663 GINAC_ASSERT(pending_integrals.size() != 1);
665 if (pending_integrals.size() > 0) {
666 return pending_integrals;
668 Gparameter new_pending_integrals;
669 new_pending_integrals.push_back(scale);
670 return new_pending_integrals;
675 // handles trailing zeroes for an otherwise convergent integral
676 ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms)
679 int depth, trailing_zeros;
680 Gparameter::const_iterator last, dummyit;
681 last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
683 GINAC_ASSERT(convergent);
685 if ((trailing_zeros > 0) && (depth > 0)) {
687 Gparameter new_a(a.begin(), a.end()-1);
688 result += G_eval1(0, scale, gsyms) * trailing_zeros_G(new_a, scale, gsyms);
689 for (auto it = a.begin(); it != last; ++it) {
690 Gparameter new_a(a.begin(), it);
692 new_a.insert(new_a.end(), it, a.end()-1);
693 result -= trailing_zeros_G(new_a, scale, gsyms);
696 return result / trailing_zeros;
698 return G_eval(a, scale, gsyms);
703 // G transformation [VSW] (57),(58)
704 ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale, const exvector& gsyms)
706 // pendint = ( y1, b1, ..., br )
707 // a = ( 0, ..., 0, amin )
710 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
711 // where sr replaces amin
713 GINAC_ASSERT(a.back() != 0);
714 GINAC_ASSERT(a.size() > 0);
717 Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
718 const int psize = pending_integrals.size();
721 // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
726 result += log(gsyms[ex_to<numeric>(scale).to_int()]);
728 new_pending_integrals.push_back(-scale);
731 new_pending_integrals.push_back(scale);
735 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
736 pending_integrals.front(),
741 result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
742 new_pending_integrals.front(),
746 new_pending_integrals.back() = 0;
747 result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
748 new_pending_integrals.front(),
755 // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
756 // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
759 result -= zeta(a.size());
761 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
762 pending_integrals.front(),
766 // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
767 // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
768 Gparameter new_a(a.begin()+1, a.end());
769 new_pending_integrals.push_back(0);
770 result -= depth_one_trafo_G(new_pending_integrals, new_a, scale, gsyms);
772 // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
773 // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
774 Gparameter new_pending_integrals_2;
775 new_pending_integrals_2.push_back(scale);
776 new_pending_integrals_2.push_back(0);
778 result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
779 pending_integrals.front(),
781 * depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
783 result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
790 // forward declaration
791 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
792 const Gparameter& pendint, const Gparameter& a_old, int scale,
793 const exvector& gsyms, bool flag_trailing_zeros_only);
796 // G transformation [VSW]
797 ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale,
798 const exvector& gsyms, bool flag_trailing_zeros_only)
800 // main recursion routine
802 // pendint = ( y1, b1, ..., br )
803 // a = ( a1, ..., amin, ..., aw )
806 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
807 // where sr replaces amin
809 // find smallest alpha, determine depth and trailing zeros, and check for convergence
811 int depth, trailing_zeros;
812 Gparameter::const_iterator min_it;
813 auto firstzero = check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
814 int min_it_pos = distance(a.begin(), min_it);
816 // special case: all a's are zero
823 result = G_eval(a, scale, gsyms);
825 if (pendint.size() > 0) {
826 result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
833 // handle trailing zeros
834 if (trailing_zeros > 0) {
836 Gparameter new_a(a.begin(), a.end()-1);
837 result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
838 for (auto it = a.begin(); it != firstzero; ++it) {
839 Gparameter new_a(a.begin(), it);
841 new_a.insert(new_a.end(), it, a.end()-1);
842 result -= G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
844 return result / trailing_zeros;
847 // convergence case or flag_trailing_zeros_only
848 if (convergent || flag_trailing_zeros_only) {
849 if (pendint.size() > 0) {
850 return G_eval(convert_pending_integrals_G(pendint),
851 pendint.front(), gsyms) *
852 G_eval(a, scale, gsyms);
854 return G_eval(a, scale, gsyms);
858 // call basic transformation for depth equal one
860 return depth_one_trafo_G(pendint, a, scale, gsyms);
864 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
865 // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
866 // + 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)
868 // smallest element in last place
869 if (min_it + 1 == a.end()) {
870 do { --min_it; } while (*min_it == 0);
872 Gparameter a1(a.begin(),min_it+1);
873 Gparameter a2(min_it+1,a.end());
875 ex result = G_transform(pendint, a2, scale, gsyms, flag_trailing_zeros_only)*
876 G_transform(empty, a1, scale, gsyms, flag_trailing_zeros_only);
878 result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms, flag_trailing_zeros_only);
883 Gparameter::iterator changeit;
885 // first term G(a_1,..,0,...,a_w;a_0)
886 Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
887 Gparameter new_a = a;
888 new_a[min_it_pos] = 0;
889 ex result = G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
890 if (pendint.size() > 0) {
891 result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
892 pendint.front(), gsyms);
896 changeit = new_a.begin() + min_it_pos;
897 changeit = new_a.erase(changeit);
898 if (changeit != new_a.begin()) {
899 // smallest in the middle
900 new_pendint.push_back(*changeit);
901 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
902 new_pendint.front(), gsyms)*
903 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
904 int buffer = *changeit;
906 result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
908 new_pendint.pop_back();
910 new_pendint.push_back(*changeit);
911 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
912 new_pendint.front(), gsyms)*
913 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
915 result -= G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
917 // smallest at the front
918 new_pendint.push_back(scale);
919 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
920 new_pendint.front(), gsyms)*
921 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
922 new_pendint.back() = *changeit;
923 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
924 new_pendint.front(), gsyms)*
925 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
927 result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
933 // shuffles the two parameter list a1 and a2 and calls G_transform for every term except
934 // for the one that is equal to a_old
935 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
936 const Gparameter& pendint, const Gparameter& a_old, int scale,
937 const exvector& gsyms, bool flag_trailing_zeros_only)
939 if (a1.size()==0 && a2.size()==0) {
940 // veto the one configuration we don't want
941 if ( a0 == a_old ) return 0;
943 return G_transform(pendint, a0, scale, gsyms, flag_trailing_zeros_only);
949 aa0.insert(aa0.end(),a1.begin(),a1.end());
950 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
956 aa0.insert(aa0.end(),a2.begin(),a2.end());
957 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
960 Gparameter a1_removed(a1.begin()+1,a1.end());
961 Gparameter a2_removed(a2.begin()+1,a2.end());
966 a01.push_back( a1[0] );
967 a02.push_back( a2[0] );
969 return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms, flag_trailing_zeros_only)
970 + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
973 // handles the transformations and the numerical evaluation of G
974 // the parameter x, s and y must only contain numerics
976 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
979 // do acceleration transformation (hoelder convolution [BBB])
980 // the parameter x, s and y must only contain numerics
982 G_do_hoelder(std::vector<cln::cl_N> x, /* yes, it's passed by value */
983 const std::vector<int>& s, const cln::cl_N& y)
986 const std::size_t size = x.size();
987 for (std::size_t i = 0; i < size; ++i)
990 for (std::size_t r = 0; r <= size; ++r) {
991 cln::cl_N buffer(1 & r ? -1 : 1);
996 for (std::size_t i = 0; i < size; ++i) {
997 if (x[i] == cln::cl_RA(1)/p) {
998 p = p/2 + cln::cl_RA(3)/2;
1004 cln::cl_RA q = p/(p-1);
1005 std::vector<cln::cl_N> qlstx;
1006 std::vector<int> qlsts;
1007 for (std::size_t j = r; j >= 1; --j) {
1008 qlstx.push_back(cln::cl_N(1) - x[j-1]);
1009 if (imagpart(x[j-1])==0 && realpart(x[j-1]) >= 1) {
1012 qlsts.push_back(-s[j-1]);
1015 if (qlstx.size() > 0) {
1016 buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
1018 std::vector<cln::cl_N> plstx;
1019 std::vector<int> plsts;
1020 for (std::size_t j = r+1; j <= size; ++j) {
1021 plstx.push_back(x[j-1]);
1022 plsts.push_back(s[j-1]);
1024 if (plstx.size() > 0) {
1025 buffer = buffer*G_numeric(plstx, plsts, 1/p);
1027 result = result + buffer;
1032 class less_object_for_cl_N
1035 bool operator() (const cln::cl_N & a, const cln::cl_N & b) const
1038 if (abs(a) != abs(b))
1039 return (abs(a) < abs(b)) ? true : false;
1042 if (phase(a) != phase(b))
1043 return (phase(a) < phase(b)) ? true : false;
1045 // equal, therefore "less" is not true
1051 // convergence transformation, used for numerical evaluation of G function.
1052 // the parameter x, s and y must only contain numerics
1054 G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1055 const cln::cl_N& y, bool flag_trailing_zeros_only)
1057 // sort (|x|<->position) to determine indices
1058 typedef std::multimap<cln::cl_N, std::size_t, less_object_for_cl_N> sortmap_t;
1060 std::size_t size = 0;
1061 for (std::size_t i = 0; i < x.size(); ++i) {
1063 sortmap.insert(std::make_pair(x[i], i));
1067 // include upper limit (scale)
1068 sortmap.insert(std::make_pair(y, x.size()));
1070 // generate missing dummy-symbols
1072 // holding dummy-symbols for the G/Li transformations
1074 gsyms.push_back(symbol("GSYMS_ERROR"));
1075 cln::cl_N lastentry(0);
1076 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1077 if (it != sortmap.begin()) {
1078 if (it->second < x.size()) {
1079 if (x[it->second] == lastentry) {
1080 gsyms.push_back(gsyms.back());
1084 if (y == lastentry) {
1085 gsyms.push_back(gsyms.back());
1090 std::ostringstream os;
1092 gsyms.push_back(symbol(os.str()));
1094 if (it->second < x.size()) {
1095 lastentry = x[it->second];
1101 // fill position data according to sorted indices and prepare substitution list
1102 Gparameter a(x.size());
1104 std::size_t pos = 1;
1106 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1107 if (it->second < x.size()) {
1108 if (s[it->second] > 0) {
1109 a[it->second] = pos;
1111 a[it->second] = -int(pos);
1113 subslst[gsyms[pos]] = numeric(x[it->second]);
1116 subslst[gsyms[pos]] = numeric(y);
1121 // do transformation
1123 ex result = G_transform(pendint, a, scale, gsyms, flag_trailing_zeros_only);
1124 // replace dummy symbols with their values
1125 result = result.expand();
1126 result = result.subs(subslst).evalf();
1127 if (!is_a<numeric>(result))
1128 throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
1130 cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
1134 // handles the transformations and the numerical evaluation of G
1135 // the parameter x, s and y must only contain numerics
1137 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1140 // check for convergence and necessary accelerations
1141 bool need_trafo = false;
1142 bool need_hoelder = false;
1143 bool have_trailing_zero = false;
1144 std::size_t depth = 0;
1145 for (auto & xi : x) {
1148 const cln::cl_N x_y = abs(xi) - y;
1149 if (instanceof(x_y, cln::cl_R_ring) &&
1150 realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2)))
1153 if (abs(abs(xi/y) - 1) < 0.01)
1154 need_hoelder = true;
1157 if (zerop(x.back())) {
1158 have_trailing_zero = true;
1162 if (depth == 1 && x.size() == 2 && !need_trafo)
1163 return - Li_projection(2, y/x[1], cln::float_format(Digits));
1165 // do acceleration transformation (hoelder convolution [BBB])
1166 if (need_hoelder && !have_trailing_zero)
1167 return G_do_hoelder(x, s, y);
1169 // convergence transformation
1171 return G_do_trafo(x, s, y, have_trailing_zero);
1174 std::vector<cln::cl_N> newx;
1175 newx.reserve(x.size());
1177 m.reserve(x.size());
1180 cln::cl_N factor = y;
1181 for (auto & xi : x) {
1185 newx.push_back(factor/xi);
1187 m.push_back(mcount);
1193 return sign*multipleLi_do_sum(m, newx);
1197 ex mLi_numeric(const lst& m, const lst& x)
1199 // let G_numeric do the transformation
1200 std::vector<cln::cl_N> newx;
1201 newx.reserve(x.nops());
1203 s.reserve(x.nops());
1204 cln::cl_N factor(1);
1205 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1206 for (int i = 1; i < *itm; ++i) {
1207 newx.push_back(cln::cl_N(0));
1210 const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
1212 newx.push_back(factor);
1213 if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) {
1220 return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
1224 } // end of anonymous namespace
1227 //////////////////////////////////////////////////////////////////////
1229 // Generalized multiple polylogarithm G(x, y) and G(x, s, y)
1233 //////////////////////////////////////////////////////////////////////
1236 static ex G2_evalf(const ex& x_, const ex& y)
1238 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1239 return G(x_, y).hold();
1241 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1242 if (x.nops() == 0) {
1246 return G(x_, y).hold();
1249 s.reserve(x.nops());
1250 bool all_zero = true;
1251 for (const auto & it : x) {
1252 if (!it.info(info_flags::numeric)) {
1253 return G(x_, y).hold();
1258 if ( !ex_to<numeric>(it).is_real() && ex_to<numeric>(it).imag() < 0 ) {
1266 return pow(log(y), x.nops()) / factorial(x.nops());
1268 std::vector<cln::cl_N> xv;
1269 xv.reserve(x.nops());
1270 for (const auto & it : x)
1271 xv.push_back(ex_to<numeric>(it).to_cl_N());
1272 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1273 return numeric(result);
1277 static ex G2_eval(const ex& x_, const ex& y)
1279 //TODO eval to MZV or H or S or Lin
1281 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1282 return G(x_, y).hold();
1284 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1285 if (x.nops() == 0) {
1289 return G(x_, y).hold();
1292 s.reserve(x.nops());
1293 bool all_zero = true;
1294 bool crational = true;
1295 for (const auto & it : x) {
1296 if (!it.info(info_flags::numeric)) {
1297 return G(x_, y).hold();
1299 if (!it.info(info_flags::crational)) {
1305 if ( !ex_to<numeric>(it).is_real() && ex_to<numeric>(it).imag() < 0 ) {
1313 return pow(log(y), x.nops()) / factorial(x.nops());
1315 if (!y.info(info_flags::crational)) {
1319 return G(x_, y).hold();
1321 std::vector<cln::cl_N> xv;
1322 xv.reserve(x.nops());
1323 for (const auto & it : x)
1324 xv.push_back(ex_to<numeric>(it).to_cl_N());
1325 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1326 return numeric(result);
1330 // option do_not_evalf_params() removed.
1331 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1332 evalf_func(G2_evalf).
1336 // derivative_func(G2_deriv).
1337 // print_func<print_latex>(G2_print_latex).
1340 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1342 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1343 return G(x_, s_, y).hold();
1345 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1346 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst{s_};
1347 if (x.nops() != s.nops()) {
1348 return G(x_, s_, y).hold();
1350 if (x.nops() == 0) {
1354 return G(x_, s_, y).hold();
1356 std::vector<int> sn;
1357 sn.reserve(s.nops());
1358 bool all_zero = true;
1359 for (auto itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1360 if (!(*itx).info(info_flags::numeric)) {
1361 return G(x_, y).hold();
1363 if (!(*its).info(info_flags::real)) {
1364 return G(x_, y).hold();
1369 if ( ex_to<numeric>(*itx).is_real() ) {
1370 if ( ex_to<numeric>(*itx).is_positive() ) {
1382 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1391 return pow(log(y), x.nops()) / factorial(x.nops());
1393 std::vector<cln::cl_N> xn;
1394 xn.reserve(x.nops());
1395 for (const auto & it : x)
1396 xn.push_back(ex_to<numeric>(it).to_cl_N());
1397 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1398 return numeric(result);
1402 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1404 //TODO eval to MZV or H or S or Lin
1406 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1407 return G(x_, s_, y).hold();
1409 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1410 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst{s_};
1411 if (x.nops() != s.nops()) {
1412 return G(x_, s_, y).hold();
1414 if (x.nops() == 0) {
1418 return G(x_, s_, y).hold();
1420 std::vector<int> sn;
1421 sn.reserve(s.nops());
1422 bool all_zero = true;
1423 bool crational = true;
1424 for (auto itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1425 if (!(*itx).info(info_flags::numeric)) {
1426 return G(x_, s_, y).hold();
1428 if (!(*its).info(info_flags::real)) {
1429 return G(x_, s_, y).hold();
1431 if (!(*itx).info(info_flags::crational)) {
1437 if ( ex_to<numeric>(*itx).is_real() ) {
1438 if ( ex_to<numeric>(*itx).is_positive() ) {
1450 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1459 return pow(log(y), x.nops()) / factorial(x.nops());
1461 if (!y.info(info_flags::crational)) {
1465 return G(x_, s_, y).hold();
1467 std::vector<cln::cl_N> xn;
1468 xn.reserve(x.nops());
1469 for (const auto & it : x)
1470 xn.push_back(ex_to<numeric>(it).to_cl_N());
1471 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1472 return numeric(result);
1476 // option do_not_evalf_params() removed.
1477 // This is safe: in the code above it only matters if s_ > 0 or s_ < 0,
1478 // s_ is allowed to be of floating type.
1479 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1480 evalf_func(G3_evalf).
1484 // derivative_func(G3_deriv).
1485 // print_func<print_latex>(G3_print_latex).
1488 //////////////////////////////////////////////////////////////////////
1490 // Classical polylogarithm and multiple polylogarithm Li(m,x)
1494 //////////////////////////////////////////////////////////////////////
1497 static ex Li_evalf(const ex& m_, const ex& x_)
1499 // classical polylogs
1500 if (m_.info(info_flags::posint)) {
1501 if (x_.info(info_flags::numeric)) {
1502 int m__ = ex_to<numeric>(m_).to_int();
1503 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1504 const cln::cl_N result = Lin_numeric(m__, x__);
1505 return numeric(result);
1507 // try to numerically evaluate second argument
1508 ex x_val = x_.evalf();
1509 if (x_val.info(info_flags::numeric)) {
1510 int m__ = ex_to<numeric>(m_).to_int();
1511 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1512 const cln::cl_N result = Lin_numeric(m__, x__);
1513 return numeric(result);
1517 // multiple polylogs
1518 if (is_a<lst>(m_) && is_a<lst>(x_)) {
1520 const lst& m = ex_to<lst>(m_);
1521 const lst& x = ex_to<lst>(x_);
1522 if (m.nops() != x.nops()) {
1523 return Li(m_,x_).hold();
1525 if (x.nops() == 0) {
1528 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1529 return Li(m_,x_).hold();
1532 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1533 if (!(*itm).info(info_flags::posint)) {
1534 return Li(m_, x_).hold();
1536 if (!(*itx).info(info_flags::numeric)) {
1537 return Li(m_, x_).hold();
1544 return mLi_numeric(m, x);
1547 return Li(m_,x_).hold();
1551 static ex Li_eval(const ex& m_, const ex& x_)
1553 if (is_a<lst>(m_)) {
1554 if (is_a<lst>(x_)) {
1555 // multiple polylogs
1556 const lst& m = ex_to<lst>(m_);
1557 const lst& x = ex_to<lst>(x_);
1558 if (m.nops() != x.nops()) {
1559 return Li(m_,x_).hold();
1561 if (x.nops() == 0) {
1565 bool is_zeta = true;
1566 bool do_evalf = true;
1567 bool crational = true;
1568 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1569 if (!(*itm).info(info_flags::posint)) {
1570 return Li(m_,x_).hold();
1572 if ((*itx != _ex1) && (*itx != _ex_1)) {
1573 if (itx != x.begin()) {
1581 if (!(*itx).info(info_flags::numeric)) {
1584 if (!(*itx).info(info_flags::crational)) {
1590 for (const auto & itx : x) {
1591 GINAC_ASSERT((itx == _ex1) || (itx == _ex_1));
1592 // XXX: 1 + 0.0*I is considered equal to 1. However
1593 // the former is a not automatically converted
1594 // to a real number. Do the conversion explicitly
1595 // to avoid the "numeric::operator>(): complex inequality"
1596 // exception (and similar problems).
1597 newx.append(itx != _ex_1 ? _ex1 : _ex_1);
1599 return zeta(m_, newx);
1603 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1604 return prefactor * H(newm, x[0]);
1606 if (do_evalf && !crational) {
1607 return mLi_numeric(m,x);
1610 return Li(m_, x_).hold();
1611 } else if (is_a<lst>(x_)) {
1612 return Li(m_, x_).hold();
1615 // classical polylogs
1623 return (pow(2,1-m_)-1) * zeta(m_);
1629 if (x_.is_equal(I)) {
1630 return power(Pi,_ex2)/_ex_48 + Catalan*I;
1632 if (x_.is_equal(-I)) {
1633 return power(Pi,_ex2)/_ex_48 - Catalan*I;
1636 if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1637 int m__ = ex_to<numeric>(m_).to_int();
1638 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1639 const cln::cl_N result = Lin_numeric(m__, x__);
1640 return numeric(result);
1643 return Li(m_, x_).hold();
1647 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1649 if (is_a<lst>(m) || is_a<lst>(x)) {
1651 epvector seq { expair(Li(m, x), 0) };
1652 return pseries(rel, std::move(seq));
1655 // classical polylog
1656 const ex x_pt = x.subs(rel, subs_options::no_pattern);
1657 if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1658 // First special case: x==0 (derivatives have poles)
1659 if (x_pt.is_zero()) {
1662 // manually construct the primitive expansion
1663 for (int i=1; i<order; ++i)
1664 ser += pow(s,i) / pow(numeric(i), m);
1665 // substitute the argument's series expansion
1666 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1667 // maybe that was terminating, so add a proper order term
1668 epvector nseq { expair(Order(_ex1), order) };
1669 ser += pseries(rel, std::move(nseq));
1670 // reexpanding it will collapse the series again
1671 return ser.series(rel, order);
1673 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1674 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1676 // all other cases should be safe, by now:
1677 throw do_taylor(); // caught by function::series()
1681 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1683 GINAC_ASSERT(deriv_param < 2);
1684 if (deriv_param == 0) {
1687 if (m_.nops() > 1) {
1688 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1691 if (is_a<lst>(m_)) {
1697 if (is_a<lst>(x_)) {
1703 return Li(m-1, x) / x;
1710 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1713 if (is_a<lst>(m_)) {
1719 if (is_a<lst>(x_)) {
1724 c.s << "\\mathrm{Li}_{";
1725 auto itm = m.begin();
1728 for (; itm != m.end(); itm++) {
1733 auto itx = x.begin();
1736 for (; itx != x.end(); itx++) {
1744 REGISTER_FUNCTION(Li,
1745 evalf_func(Li_evalf).
1747 series_func(Li_series).
1748 derivative_func(Li_deriv).
1749 print_func<print_latex>(Li_print_latex).
1750 do_not_evalf_params());
1753 //////////////////////////////////////////////////////////////////////
1755 // Nielsen's generalized polylogarithm S(n,p,x)
1759 //////////////////////////////////////////////////////////////////////
1762 // anonymous namespace for helper functions
1766 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1768 std::vector<std::vector<cln::cl_N>> Yn;
1769 int ynsize = 0; // number of Yn[]
1770 int ynlength = 100; // initial length of all Yn[i]
1773 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1774 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1775 // representing S_{n,p}(x).
1776 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1777 // equivalent Z-sum.
1778 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1779 // representing S_{n,p}(x).
1780 // The calculation of Y_n uses the values from Y_{n-1}.
1781 void fill_Yn(int n, const cln::float_format_t& prec)
1783 const int initsize = ynlength;
1784 //const int initsize = initsize_Yn;
1785 cln::cl_N one = cln::cl_float(1, prec);
1788 std::vector<cln::cl_N> buf(initsize);
1789 auto it = buf.begin();
1790 auto itprev = Yn[n-1].begin();
1791 *it = (*itprev) / cln::cl_N(n+1) * one;
1794 // sums with an index smaller than the depth are zero and need not to be calculated.
1795 // calculation starts with depth, which is n+2)
1796 for (int i=n+2; i<=initsize+n; i++) {
1797 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1803 std::vector<cln::cl_N> buf(initsize);
1804 auto it = buf.begin();
1807 for (int i=2; i<=initsize; i++) {
1808 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1817 // make Yn longer ...
1818 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1821 cln::cl_N one = cln::cl_float(1, prec);
1823 Yn[0].resize(newsize);
1824 auto it = Yn[0].begin();
1826 for (int i=ynlength+1; i<=newsize; i++) {
1827 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1831 for (int n=1; n<ynsize; n++) {
1832 Yn[n].resize(newsize);
1833 auto it = Yn[n].begin();
1834 auto itprev = Yn[n-1].begin();
1837 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1838 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1848 // helper function for S(n,p,x)
1850 cln::cl_N C(int n, int p)
1854 for (int k=0; k<p; k++) {
1855 for (int j=0; j<=(n+k-1)/2; j++) {
1859 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1862 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1869 result = result + cln::factorial(n+k-1)
1870 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1871 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1874 result = result - cln::factorial(n+k-1)
1875 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1876 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1881 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1882 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1885 result = result + cln::factorial(n+k-1)
1886 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1887 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1895 if (((np)/2+n) & 1) {
1896 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1899 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1907 // helper function for S(n,p,x)
1908 // [Kol] remark to (9.1)
1909 cln::cl_N a_k(int k)
1918 for (int m=2; m<=k; m++) {
1919 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1926 // helper function for S(n,p,x)
1927 // [Kol] remark to (9.1)
1928 cln::cl_N b_k(int k)
1937 for (int m=2; m<=k; m++) {
1938 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1945 // helper function for S(n,p,x)
1946 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1948 static cln::float_format_t oldprec = cln::default_float_format;
1951 return Li_projection(n+1, x, prec);
1954 // precision has changed, we need to clear lookup table Yn
1955 if ( oldprec != prec ) {
1962 // check if precalculated values are sufficient
1964 for (int i=ynsize; i<p-1; i++) {
1969 // should be done otherwise
1970 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1971 cln::cl_N xf = x * one;
1972 //cln::cl_N xf = x * cln::cl_float(1, prec);
1976 cln::cl_N factor = cln::expt(xf, p);
1980 if (i-p >= ynlength) {
1982 make_Yn_longer(ynlength*2, prec);
1984 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? ...
1985 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1986 factor = factor * xf;
1988 } while (res != resbuf);
1994 // helper function for S(n,p,x)
1995 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1998 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
2000 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
2001 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
2003 for (int s=0; s<n; s++) {
2005 for (int r=0; r<p; r++) {
2006 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
2007 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
2009 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2015 return S_do_sum(n, p, x, prec);
2019 // helper function for S(n,p,x)
2020 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
2024 // [Kol] (2.22) with (2.21)
2025 return cln::zeta(p+1);
2030 return cln::zeta(n+1);
2035 for (int nu=0; nu<n; nu++) {
2036 for (int rho=0; rho<=p; rho++) {
2037 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2038 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2041 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2048 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2050 // throw std::runtime_error("don't know how to evaluate this function!");
2053 // what is the desired float format?
2054 // first guess: default format
2055 cln::float_format_t prec = cln::default_float_format;
2056 const cln::cl_N value = x;
2057 // second guess: the argument's format
2058 if (!instanceof(realpart(value), cln::cl_RA_ring))
2059 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2060 else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2061 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2064 // 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
2065 // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
2066 if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
2068 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2069 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2071 for (int s=0; s<n; s++) {
2073 for (int r=0; r<p; r++) {
2074 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2075 * S_num(p-r,n-s,1-value) / cln::factorial(r);
2077 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2084 if (cln::abs(value) > 1) {
2088 for (int s=0; s<p; s++) {
2089 for (int r=0; r<=s; r++) {
2090 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2091 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2092 * S_num(n+s-r,p-s,cln::recip(value));
2095 result = result * cln::expt(cln::cl_I(-1),n);
2098 for (int r=0; r<n; r++) {
2099 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2101 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2103 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2108 if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
2111 for (int s=0; s<p-1; s++)
2114 ex res = H(m,numeric(value)).evalf();
2115 return ex_to<numeric>(res).to_cl_N();
2118 return S_projection(n, p, value, prec);
2123 } // end of anonymous namespace
2126 //////////////////////////////////////////////////////////////////////
2128 // Nielsen's generalized polylogarithm S(n,p,x)
2132 //////////////////////////////////////////////////////////////////////
2135 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2137 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2138 const int n_ = ex_to<numeric>(n).to_int();
2139 const int p_ = ex_to<numeric>(p).to_int();
2140 if (is_a<numeric>(x)) {
2141 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2142 const cln::cl_N result = S_num(n_, p_, x_);
2143 return numeric(result);
2145 ex x_val = x.evalf();
2146 if (is_a<numeric>(x_val)) {
2147 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2148 const cln::cl_N result = S_num(n_, p_, x_val_);
2149 return numeric(result);
2153 return S(n, p, x).hold();
2157 static ex S_eval(const ex& n, const ex& p, const ex& x)
2159 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2165 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2173 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2174 int n_ = ex_to<numeric>(n).to_int();
2175 int p_ = ex_to<numeric>(p).to_int();
2176 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2177 const cln::cl_N result = S_num(n_, p_, x_);
2178 return numeric(result);
2183 return pow(-log(1-x), p) / factorial(p);
2185 return S(n, p, x).hold();
2189 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2192 return Li(n+1, x).series(rel, order, options);
2195 const ex x_pt = x.subs(rel, subs_options::no_pattern);
2196 if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2197 // First special case: x==0 (derivatives have poles)
2198 if (x_pt.is_zero()) {
2201 // manually construct the primitive expansion
2202 // subsum = Euler-Zagier-Sum is needed
2203 // dirty hack (slow ...) calculation of subsum:
2204 std::vector<ex> presubsum, subsum;
2205 subsum.push_back(0);
2206 for (int i=1; i<order-1; ++i) {
2207 subsum.push_back(subsum[i-1] + numeric(1, i));
2209 for (int depth=2; depth<p; ++depth) {
2211 for (int i=1; i<order-1; ++i) {
2212 subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2216 for (int i=1; i<order; ++i) {
2217 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2219 // substitute the argument's series expansion
2220 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2221 // maybe that was terminating, so add a proper order term
2222 epvector nseq { expair(Order(_ex1), order) };
2223 ser += pseries(rel, std::move(nseq));
2224 // reexpanding it will collapse the series again
2225 return ser.series(rel, order);
2227 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2228 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2230 // all other cases should be safe, by now:
2231 throw do_taylor(); // caught by function::series()
2235 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2237 GINAC_ASSERT(deriv_param < 3);
2238 if (deriv_param < 2) {
2242 return S(n-1, p, x) / x;
2244 return S(n, p-1, x) / (1-x);
2249 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2251 c.s << "\\mathrm{S}_{";
2261 REGISTER_FUNCTION(S,
2262 evalf_func(S_evalf).
2264 series_func(S_series).
2265 derivative_func(S_deriv).
2266 print_func<print_latex>(S_print_latex).
2267 do_not_evalf_params());
2270 //////////////////////////////////////////////////////////////////////
2272 // Harmonic polylogarithm H(m,x)
2276 //////////////////////////////////////////////////////////////////////
2279 // anonymous namespace for helper functions
2283 // regulates the pole (used by 1/x-transformation)
2284 symbol H_polesign("IMSIGN");
2287 // convert parameters from H to Li representation
2288 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2289 // returns true if some parameters are negative
2290 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2292 // expand parameter list
2294 for (const auto & it : l) {
2296 for (ex count=it-1; count > 0; count--) {
2300 } else if (it < -1) {
2301 for (ex count=it+1; count < 0; count++) {
2312 bool has_negative_parameters = false;
2314 for (const auto & it : mexp) {
2320 m.append((it+acc-1) * signum);
2322 m.append((it-acc+1) * signum);
2328 has_negative_parameters = true;
2331 if (has_negative_parameters) {
2332 for (std::size_t i=0; i<m.nops(); i++) {
2334 m.let_op(i) = -m.op(i);
2342 return has_negative_parameters;
2346 // recursivly transforms H to corresponding multiple polylogarithms
2347 struct map_trafo_H_convert_to_Li : public map_function
2349 ex operator()(const ex& e) override
2351 if (is_a<add>(e) || is_a<mul>(e)) {
2352 return e.map(*this);
2354 if (is_a<function>(e)) {
2355 std::string name = ex_to<function>(e).get_name();
2358 if (is_a<lst>(e.op(0))) {
2359 parameter = ex_to<lst>(e.op(0));
2361 parameter = lst{e.op(0)};
2368 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2369 s.let_op(0) = s.op(0) * arg;
2370 return pf * Li(m, s).hold();
2372 for (std::size_t i=0; i<m.nops(); i++) {
2375 s.let_op(0) = s.op(0) * arg;
2376 return Li(m, s).hold();
2385 // recursivly transforms H to corresponding zetas
2386 struct map_trafo_H_convert_to_zeta : public map_function
2388 ex operator()(const ex& e) override
2390 if (is_a<add>(e) || is_a<mul>(e)) {
2391 return e.map(*this);
2393 if (is_a<function>(e)) {
2394 std::string name = ex_to<function>(e).get_name();
2397 if (is_a<lst>(e.op(0))) {
2398 parameter = ex_to<lst>(e.op(0));
2400 parameter = lst{e.op(0)};
2406 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2407 return pf * zeta(m, s);
2418 // remove trailing zeros from H-parameters
2419 struct map_trafo_H_reduce_trailing_zeros : public map_function
2421 ex operator()(const ex& e) override
2423 if (is_a<add>(e) || is_a<mul>(e)) {
2424 return e.map(*this);
2426 if (is_a<function>(e)) {
2427 std::string name = ex_to<function>(e).get_name();
2430 if (is_a<lst>(e.op(0))) {
2431 parameter = ex_to<lst>(e.op(0));
2433 parameter = lst{e.op(0)};
2436 if (parameter.op(parameter.nops()-1) == 0) {
2439 if (parameter.nops() == 1) {
2444 auto it = parameter.begin();
2445 while ((it != parameter.end()) && (*it == 0)) {
2448 if (it == parameter.end()) {
2449 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2453 parameter.remove_last();
2454 std::size_t lastentry = parameter.nops();
2455 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2460 ex result = log(arg) * H(parameter,arg).hold();
2462 for (ex i=0; i<lastentry; i++) {
2463 if (parameter[i] > 0) {
2465 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2468 } else if (parameter[i] < 0) {
2470 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2478 if (lastentry < parameter.nops()) {
2479 result = result / (parameter.nops()-lastentry+1);
2480 return result.map(*this);
2492 // returns an expression with zeta functions corresponding to the parameter list for H
2493 ex convert_H_to_zeta(const lst& m)
2495 symbol xtemp("xtemp");
2496 map_trafo_H_reduce_trailing_zeros filter;
2497 map_trafo_H_convert_to_zeta filter2;
2498 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2502 // convert signs form Li to H representation
2503 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2506 auto itm = m.begin();
2507 auto itx = ++x.begin();
2512 while (itx != x.end()) {
2513 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
2514 // XXX: 1 + 0.0*I is considered equal to 1. However the former
2515 // is not automatically converted to a real number.
2516 // Do the conversion explicitly to avoid the
2517 // "numeric::operator>(): complex inequality" exception.
2518 signum *= (*itx != _ex_1) ? 1 : -1;
2520 res.append((*itm) * signum);
2528 // multiplies an one-dimensional H with another H
2530 ex trafo_H_mult(const ex& h1, const ex& h2)
2535 ex h1nops = h1.op(0).nops();
2536 ex h2nops = h2.op(0).nops();
2538 hshort = h2.op(0).op(0);
2539 hlong = ex_to<lst>(h1.op(0));
2541 hshort = h1.op(0).op(0);
2543 hlong = ex_to<lst>(h2.op(0));
2545 hlong = lst{h2.op(0).op(0)};
2548 for (std::size_t i=0; i<=hlong.nops(); i++) {
2552 newparameter.append(hlong[j]);
2554 newparameter.append(hshort);
2555 for (; j<hlong.nops(); j++) {
2556 newparameter.append(hlong[j]);
2558 res += H(newparameter, h1.op(1)).hold();
2564 // applies trafo_H_mult recursively on expressions
2565 struct map_trafo_H_mult : public map_function
2567 ex operator()(const ex& e) override
2570 return e.map(*this);
2578 for (std::size_t pos=0; pos<e.nops(); pos++) {
2579 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2580 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2582 for (ex i=0; i<e.op(pos).op(1); i++) {
2583 Hlst.append(e.op(pos).op(0));
2587 } else if (is_a<function>(e.op(pos))) {
2588 std::string name = ex_to<function>(e.op(pos)).get_name();
2590 if (e.op(pos).op(0).nops() > 1) {
2593 Hlst.append(e.op(pos));
2598 result *= e.op(pos);
2601 if (Hlst.nops() > 0) {
2602 firstH = Hlst[Hlst.nops()-1];
2609 if (Hlst.nops() > 0) {
2610 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2612 for (std::size_t i=1; i<Hlst.nops(); i++) {
2613 result *= Hlst.op(i);
2615 result = result.expand();
2616 map_trafo_H_mult recursion;
2617 return recursion(result);
2628 // do integration [ReV] (55)
2629 // put parameter 0 in front of existing parameters
2630 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2634 if (is_a<function>(e)) {
2635 name = ex_to<function>(e).get_name();
2640 for (std::size_t i=0; i<e.nops(); i++) {
2641 if (is_a<function>(e.op(i))) {
2642 std::string name = ex_to<function>(e.op(i)).get_name();
2650 lst newparameter = ex_to<lst>(h.op(0));
2651 newparameter.prepend(0);
2652 ex addzeta = convert_H_to_zeta(newparameter);
2653 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2655 return e * (-H(lst{ex(0)},1/arg).hold());
2660 // do integration [ReV] (49)
2661 // put parameter 1 in front of existing parameters
2662 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2666 if (is_a<function>(e)) {
2667 name = ex_to<function>(e).get_name();
2672 for (std::size_t i=0; i<e.nops(); i++) {
2673 if (is_a<function>(e.op(i))) {
2674 std::string name = ex_to<function>(e.op(i)).get_name();
2682 lst newparameter = ex_to<lst>(h.op(0));
2683 newparameter.prepend(1);
2684 return e.subs(h == H(newparameter, h.op(1)).hold());
2686 return e * H(lst{ex(1)},1-arg).hold();
2691 // do integration [ReV] (55)
2692 // put parameter -1 in front of existing parameters
2693 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2697 if (is_a<function>(e)) {
2698 name = ex_to<function>(e).get_name();
2703 for (std::size_t i=0; i<e.nops(); i++) {
2704 if (is_a<function>(e.op(i))) {
2705 std::string name = ex_to<function>(e.op(i)).get_name();
2713 lst newparameter = ex_to<lst>(h.op(0));
2714 newparameter.prepend(-1);
2715 ex addzeta = convert_H_to_zeta(newparameter);
2716 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2718 ex addzeta = convert_H_to_zeta(lst{ex(-1)});
2719 return (e * (addzeta - H(lst{ex(-1)},1/arg).hold())).expand();
2724 // do integration [ReV] (55)
2725 // put parameter -1 in front of existing parameters
2726 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2730 if (is_a<function>(e)) {
2731 name = ex_to<function>(e).get_name();
2736 for (std::size_t i = 0; i < e.nops(); i++) {
2737 if (is_a<function>(e.op(i))) {
2738 std::string name = ex_to<function>(e.op(i)).get_name();
2746 lst newparameter = ex_to<lst>(h.op(0));
2747 newparameter.prepend(-1);
2748 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2750 return (e * H(lst{ex(-1)},(1-arg)/(1+arg)).hold()).expand();
2755 // do integration [ReV] (55)
2756 // put parameter 1 in front of existing parameters
2757 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2761 if (is_a<function>(e)) {
2762 name = ex_to<function>(e).get_name();
2767 for (std::size_t i = 0; i < e.nops(); i++) {
2768 if (is_a<function>(e.op(i))) {
2769 std::string name = ex_to<function>(e.op(i)).get_name();
2777 lst newparameter = ex_to<lst>(h.op(0));
2778 newparameter.prepend(1);
2779 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2781 return (e * H(lst{ex(1)},(1-arg)/(1+arg)).hold()).expand();
2786 // do x -> 1-x transformation
2787 struct map_trafo_H_1mx : public map_function
2789 ex operator()(const ex& e) override
2791 if (is_a<add>(e) || is_a<mul>(e)) {
2792 return e.map(*this);
2795 if (is_a<function>(e)) {
2796 std::string name = ex_to<function>(e).get_name();
2799 lst parameter = ex_to<lst>(e.op(0));
2802 // special cases if all parameters are either 0, 1 or -1
2803 bool allthesame = true;
2804 if (parameter.op(0) == 0) {
2805 for (std::size_t i = 1; i < parameter.nops(); i++) {
2806 if (parameter.op(i) != 0) {
2813 for (int i=parameter.nops(); i>0; i--) {
2814 newparameter.append(1);
2816 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2818 } else if (parameter.op(0) == -1) {
2819 throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2821 for (std::size_t i = 1; i < parameter.nops(); i++) {
2822 if (parameter.op(i) != 1) {
2829 for (int i=parameter.nops(); i>0; i--) {
2830 newparameter.append(0);
2832 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2836 lst newparameter = parameter;
2837 newparameter.remove_first();
2839 if (parameter.op(0) == 0) {
2842 ex res = convert_H_to_zeta(parameter);
2843 //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2844 map_trafo_H_1mx recursion;
2845 ex buffer = recursion(H(newparameter, arg).hold());
2846 if (is_a<add>(buffer)) {
2847 for (std::size_t i = 0; i < buffer.nops(); i++) {
2848 res -= trafo_H_prepend_one(buffer.op(i), arg);
2851 res -= trafo_H_prepend_one(buffer, arg);
2858 map_trafo_H_1mx recursion;
2859 map_trafo_H_mult unify;
2860 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
2861 std::size_t firstzero = 0;
2862 while (parameter.op(firstzero) == 1) {
2865 for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2869 newparameter.append(parameter[j+1]);
2871 newparameter.append(1);
2872 for (; j<parameter.nops()-1; j++) {
2873 newparameter.append(parameter[j+1]);
2875 res -= H(newparameter, arg).hold();
2877 res = recursion(res).expand() / firstzero;
2887 // do x -> 1/x transformation
2888 struct map_trafo_H_1overx : public map_function
2890 ex operator()(const ex& e) override
2892 if (is_a<add>(e) || is_a<mul>(e)) {
2893 return e.map(*this);
2896 if (is_a<function>(e)) {
2897 std::string name = ex_to<function>(e).get_name();
2900 lst parameter = ex_to<lst>(e.op(0));
2903 // special cases if all parameters are either 0, 1 or -1
2904 bool allthesame = true;
2905 if (parameter.op(0) == 0) {
2906 for (std::size_t i = 1; i < parameter.nops(); i++) {
2907 if (parameter.op(i) != 0) {
2913 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2915 } else if (parameter.op(0) == -1) {
2916 for (std::size_t i = 1; i < parameter.nops(); i++) {
2917 if (parameter.op(i) != -1) {
2923 map_trafo_H_mult unify;
2924 return unify((pow(H(lst{ex(-1)},1/arg).hold() - H(lst{ex(0)},1/arg).hold(), parameter.nops())
2925 / factorial(parameter.nops())).expand());
2928 for (std::size_t i = 1; i < parameter.nops(); i++) {
2929 if (parameter.op(i) != 1) {
2935 map_trafo_H_mult unify;
2936 return unify((pow(H(lst{ex(1)},1/arg).hold() + H(lst{ex(0)},1/arg).hold() + H_polesign, parameter.nops())
2937 / factorial(parameter.nops())).expand());
2941 lst newparameter = parameter;
2942 newparameter.remove_first();
2944 if (parameter.op(0) == 0) {
2947 ex res = convert_H_to_zeta(parameter);
2948 map_trafo_H_1overx recursion;
2949 ex buffer = recursion(H(newparameter, arg).hold());
2950 if (is_a<add>(buffer)) {
2951 for (std::size_t i = 0; i < buffer.nops(); i++) {
2952 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2955 res += trafo_H_1tx_prepend_zero(buffer, arg);
2959 } else if (parameter.op(0) == -1) {
2961 // leading negative one
2962 ex res = convert_H_to_zeta(parameter);
2963 map_trafo_H_1overx recursion;
2964 ex buffer = recursion(H(newparameter, arg).hold());
2965 if (is_a<add>(buffer)) {
2966 for (std::size_t i = 0; i < buffer.nops(); i++) {
2967 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2970 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2977 map_trafo_H_1overx recursion;
2978 map_trafo_H_mult unify;
2979 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
2980 std::size_t firstzero = 0;
2981 while (parameter.op(firstzero) == 1) {
2984 for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
2988 newparameter.append(parameter[j+1]);
2990 newparameter.append(1);
2991 for (; j<parameter.nops()-1; j++) {
2992 newparameter.append(parameter[j+1]);
2994 res -= H(newparameter, arg).hold();
2996 res = recursion(res).expand() / firstzero;
3008 // do x -> (1-x)/(1+x) transformation
3009 struct map_trafo_H_1mxt1px : public map_function
3011 ex operator()(const ex& e) override
3013 if (is_a<add>(e) || is_a<mul>(e)) {
3014 return e.map(*this);
3017 if (is_a<function>(e)) {
3018 std::string name = ex_to<function>(e).get_name();
3021 lst parameter = ex_to<lst>(e.op(0));
3024 // special cases if all parameters are either 0, 1 or -1
3025 bool allthesame = true;
3026 if (parameter.op(0) == 0) {
3027 for (std::size_t i = 1; i < parameter.nops(); i++) {
3028 if (parameter.op(i) != 0) {
3034 map_trafo_H_mult unify;
3035 return unify((pow(-H(lst{ex(1)},(1-arg)/(1+arg)).hold() - H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops())
3036 / factorial(parameter.nops())).expand());
3038 } else if (parameter.op(0) == -1) {
3039 for (std::size_t i = 1; i < parameter.nops(); i++) {
3040 if (parameter.op(i) != -1) {
3046 map_trafo_H_mult unify;
3047 return unify((pow(log(2) - H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops())
3048 / factorial(parameter.nops())).expand());
3051 for (std::size_t i = 1; i < parameter.nops(); i++) {
3052 if (parameter.op(i) != 1) {
3058 map_trafo_H_mult unify;
3059 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())
3060 / factorial(parameter.nops())).expand());
3064 lst newparameter = parameter;
3065 newparameter.remove_first();
3067 if (parameter.op(0) == 0) {
3070 ex res = convert_H_to_zeta(parameter);
3071 map_trafo_H_1mxt1px recursion;
3072 ex buffer = recursion(H(newparameter, arg).hold());
3073 if (is_a<add>(buffer)) {
3074 for (std::size_t i = 0; i < buffer.nops(); i++) {
3075 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3078 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3082 } else if (parameter.op(0) == -1) {
3084 // leading negative one
3085 ex res = convert_H_to_zeta(parameter);
3086 map_trafo_H_1mxt1px recursion;
3087 ex buffer = recursion(H(newparameter, arg).hold());
3088 if (is_a<add>(buffer)) {
3089 for (std::size_t i = 0; i < buffer.nops(); i++) {
3090 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3093 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3100 map_trafo_H_1mxt1px recursion;
3101 map_trafo_H_mult unify;
3102 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
3103 std::size_t firstzero = 0;
3104 while (parameter.op(firstzero) == 1) {
3107 for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3111 newparameter.append(parameter[j+1]);
3113 newparameter.append(1);
3114 for (; j<parameter.nops()-1; j++) {
3115 newparameter.append(parameter[j+1]);
3117 res -= H(newparameter, arg).hold();
3119 res = recursion(res).expand() / firstzero;
3131 // do the actual summation.
3132 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3134 const int j = m.size();
3136 std::vector<cln::cl_N> t(j);
3138 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3139 cln::cl_N factor = cln::expt(x, j) * one;
3145 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3146 for (int k=j-2; k>=1; k--) {
3147 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3149 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3150 factor = factor * x;
3151 } while (t[0] != t0buf);
3157 } // end of anonymous namespace
3160 //////////////////////////////////////////////////////////////////////
3162 // Harmonic polylogarithm H(m,x)
3166 //////////////////////////////////////////////////////////////////////
3169 static ex H_evalf(const ex& x1, const ex& x2)
3171 if (is_a<lst>(x1)) {
3174 if (is_a<numeric>(x2)) {
3175 x = ex_to<numeric>(x2).to_cl_N();
3177 ex x2_val = x2.evalf();
3178 if (is_a<numeric>(x2_val)) {
3179 x = ex_to<numeric>(x2_val).to_cl_N();
3183 for (std::size_t i = 0; i < x1.nops(); i++) {
3184 if (!x1.op(i).info(info_flags::integer)) {
3185 return H(x1, x2).hold();
3188 if (x1.nops() < 1) {
3189 return H(x1, x2).hold();
3192 const lst& morg = ex_to<lst>(x1);
3193 // remove trailing zeros ...
3194 if (*(--morg.end()) == 0) {
3195 symbol xtemp("xtemp");
3196 map_trafo_H_reduce_trailing_zeros filter;
3197 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3199 // ... and expand parameter notation
3200 bool has_minus_one = false;
3202 for (const auto & it : morg) {
3204 for (ex count=it-1; count > 0; count--) {
3208 } else if (it <= -1) {
3209 for (ex count=it+1; count < 0; count++) {
3213 has_minus_one = true;
3220 if (cln::abs(x) < 0.95) {
3224 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3225 // negative parameters -> s_lst is filled
3226 std::vector<int> m_int;
3227 std::vector<cln::cl_N> x_cln;
3228 for (auto it_int = m_lst.begin(), it_cln = s_lst.begin();
3229 it_int != m_lst.end(); it_int++, it_cln++) {
3230 m_int.push_back(ex_to<numeric>(*it_int).to_int());
3231 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3233 x_cln.front() = x_cln.front() * x;
3234 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3236 // only positive parameters
3238 if (m_lst.nops() == 1) {
3239 return Li(m_lst.op(0), x2).evalf();
3241 std::vector<int> m_int;
3242 for (const auto & it : m_lst) {
3243 m_int.push_back(ex_to<numeric>(it).to_int());
3245 return numeric(H_do_sum(m_int, x));
3249 symbol xtemp("xtemp");
3252 // ensure that the realpart of the argument is positive
3253 if (cln::realpart(x) < 0) {
3255 for (std::size_t i = 0; i < m.nops(); i++) {
3257 m.let_op(i) = -m.op(i);
3264 if (cln::abs(x) >= 2.0) {
3265 map_trafo_H_1overx trafo;
3266 res *= trafo(H(m, xtemp).hold());
3267 if (cln::imagpart(x) <= 0) {
3268 res = res.subs(H_polesign == -I*Pi);
3270 res = res.subs(H_polesign == I*Pi);
3272 return res.subs(xtemp == numeric(x)).evalf();
3275 // check transformations for 0.95 <= |x| < 2.0
3277 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3278 if (cln::abs(x-9.53) <= 9.47) {
3280 map_trafo_H_1mxt1px trafo;
3281 res *= trafo(H(m, xtemp).hold());
3284 if (has_minus_one) {
3285 map_trafo_H_convert_to_Li filter;
3286 return filter(H(m, numeric(x)).hold()).evalf();
3288 map_trafo_H_1mx trafo;
3289 res *= trafo(H(m, xtemp).hold());
3292 return res.subs(xtemp == numeric(x)).evalf();
3295 return H(x1,x2).hold();
3299 static ex H_eval(const ex& m_, const ex& x)
3302 if (is_a<lst>(m_)) {
3307 if (m.nops() == 0) {
3315 if (*m.begin() > _ex1) {
3321 } else if (*m.begin() < _ex_1) {
3327 } else if (*m.begin() == _ex0) {
3334 for (auto it = ++m.begin(); it != m.end(); it++) {
3335 if (it->info(info_flags::integer)) {
3346 } else if (*it < _ex_1) {
3366 } else if (step == 1) {
3378 // if some m_i is not an integer
3379 return H(m_, x).hold();
3382 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3383 return convert_H_to_zeta(m);
3389 return H(m_, x).hold();
3391 return pow(log(x), m.nops()) / factorial(m.nops());
3394 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3396 } else if ((step == 1) && (pos1 == _ex0)){
3401 return pow(-1, p) * S(n, p, -x);
3407 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3408 return H(m_, x).evalf();
3410 return H(m_, x).hold();
3414 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3416 epvector seq { expair(H(m, x), 0) };
3417 return pseries(rel, std::move(seq));
3421 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3423 GINAC_ASSERT(deriv_param < 2);
3424 if (deriv_param == 0) {
3428 if (is_a<lst>(m_)) {
3444 return 1/(1-x) * H(m, x);
3445 } else if (mb == _ex_1) {
3446 return 1/(1+x) * H(m, x);
3453 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3456 if (is_a<lst>(m_)) {
3461 c.s << "\\mathrm{H}_{";
3462 auto itm = m.begin();
3465 for (; itm != m.end(); itm++) {
3475 REGISTER_FUNCTION(H,
3476 evalf_func(H_evalf).
3478 series_func(H_series).
3479 derivative_func(H_deriv).
3480 print_func<print_latex>(H_print_latex).
3481 do_not_evalf_params());
3484 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3485 ex convert_H_to_Li(const ex& m, const ex& x)
3487 map_trafo_H_reduce_trailing_zeros filter;
3488 map_trafo_H_convert_to_Li filter2;
3490 return filter2(filter(H(m, x).hold()));
3492 return filter2(filter(H(lst{m}, x).hold()));
3497 //////////////////////////////////////////////////////////////////////
3499 // Multiple zeta values zeta(x) and zeta(x,s)
3503 //////////////////////////////////////////////////////////////////////
3506 // anonymous namespace for helper functions
3510 // parameters and data for [Cra] algorithm
3511 const cln::cl_N lambda = cln::cl_N("319/320");
3513 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3515 const int size = a.size();
3516 for (int n=0; n<size; n++) {
3518 for (int m=0; m<=n; m++) {
3519 c[n] = c[n] + a[m]*b[n-m];
3526 static void initcX(std::vector<cln::cl_N>& crX,
3527 const std::vector<int>& s,
3530 std::vector<cln::cl_N> crB(L2 + 1);
3531 for (int i=0; i<=L2; i++)
3532 crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i);
3536 std::vector<std::vector<cln::cl_N>> crG(s.size() - 1, std::vector<cln::cl_N>(L2 + 1));
3537 for (int m=0; m < (int)s.size() - 1; m++) {
3540 for (int i = 0; i <= L2; i++)
3541 crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2);
3546 for (std::size_t m = 0; m < s.size() - 1; m++) {
3547 std::vector<cln::cl_N> Xbuf(L2 + 1);
3548 for (int i = 0; i <= L2; i++)
3549 Xbuf[i] = crX[i] * crG[m][i];
3551 halfcyclic_convolute(Xbuf, crB, crX);
3557 static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk,
3558 const std::vector<cln::cl_N>& crX)
3560 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3561 cln::cl_N factor = cln::expt(lambda, Sqk);
3562 cln::cl_N res = factor / Sqk * crX[0] * one;
3567 factor = factor * lambda;
3569 res = res + crX[N] * factor / (N+Sqk);
3570 } while ((res != resbuf) || cln::zerop(crX[N]));
3576 static void calc_f(std::vector<std::vector<cln::cl_N>>& f_kj,
3577 const int maxr, const int L1)
3579 cln::cl_N t0, t1, t2, t3, t4;
3581 auto it = f_kj.begin();
3582 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3584 t0 = cln::exp(-lambda);
3586 for (k=1; k<=L1; k++) {
3589 for (j=1; j<=maxr; j++) {
3592 for (i=2; i<=j; i++) {
3596 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3604 static cln::cl_N crandall_Z(const std::vector<int>& s,
3605 const std::vector<std::vector<cln::cl_N>>& f_kj)
3607 const int j = s.size();
3616 t0 = t0 + f_kj[q+j-2][s[0]-1];
3617 } while (t0 != t0buf);
3619 return t0 / cln::factorial(s[0]-1);
3622 std::vector<cln::cl_N> t(j);
3629 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3630 for (int k=j-2; k>=1; k--) {
3631 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3633 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3634 } while (t[0] != t0buf);
3636 return t[0] / cln::factorial(s[0]-1);
3641 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3643 std::vector<int> r = s;
3644 const int j = r.size();
3648 // decide on maximal size of f_kj for crandall_Z
3652 L1 = Digits * 3 + j*2;
3656 // decide on maximal size of crX for crandall_Y
3659 } else if (Digits < 86) {
3661 } else if (Digits < 192) {
3663 } else if (Digits < 394) {
3665 } else if (Digits < 808) {
3675 for (int i=0; i<j; i++) {
3682 std::vector<std::vector<cln::cl_N>> f_kj(L1);
3683 calc_f(f_kj, maxr, L1);
3685 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3687 std::vector<int> rz;
3690 for (int k=r.size()-1; k>0; k--) {
3692 rz.insert(rz.begin(), r.back());
3693 skp1buf = rz.front();
3697 std::vector<cln::cl_N> crX;
3700 for (int q=0; q<skp1buf; q++) {
3702 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k, crX);
3703 cln::cl_N pp2 = crandall_Z(rz, f_kj);
3708 res = res - pp1 * pp2 / cln::factorial(q);
3710 res = res + pp1 * pp2 / cln::factorial(q);
3713 rz.front() = skp1buf;
3715 rz.insert(rz.begin(), r.back());
3717 std::vector<cln::cl_N> crX;
3718 initcX(crX, rz, L2);
3720 res = (res + crandall_Y_loop(S-j, crX)) / r0factorial
3721 + crandall_Z(rz, f_kj);
3727 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3729 const int j = r.size();
3731 // buffer for subsums
3732 std::vector<cln::cl_N> t(j);
3733 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3740 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3741 for (int k=j-2; k>=0; k--) {
3742 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3744 } while (t[0] != t0buf);
3750 // does Hoelder convolution. see [BBB] (7.0)
3751 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3753 // prepare parameters
3754 // holds Li arguments in [BBB] notation
3755 std::vector<int> s = s_;
3756 std::vector<int> m_p = m_;
3757 std::vector<int> m_q;
3758 // holds Li arguments in nested sums notation
3759 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3760 s_p[0] = s_p[0] * cln::cl_N("1/2");
3761 // convert notations
3763 for (std::size_t i = 0; i < s_.size(); i++) {
3768 s[i] = sig * std::abs(s[i]);
3770 std::vector<cln::cl_N> s_q;
3771 cln::cl_N signum = 1;
3774 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3779 // change parameters
3780 if (s.front() > 0) {
3781 if (m_p.front() == 1) {
3782 m_p.erase(m_p.begin());
3783 s_p.erase(s_p.begin());
3784 if (s_p.size() > 0) {
3785 s_p.front() = s_p.front() * cln::cl_N("1/2");
3791 m_q.insert(m_q.begin(), 1);
3792 if (s_q.size() > 0) {
3793 s_q.front() = s_q.front() * 2;
3795 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3798 if (m_p.front() == 1) {
3799 m_p.erase(m_p.begin());
3800 cln::cl_N spbuf = s_p.front();
3801 s_p.erase(s_p.begin());
3802 if (s_p.size() > 0) {
3803 s_p.front() = s_p.front() * spbuf;
3806 m_q.insert(m_q.begin(), 1);
3807 if (s_q.size() > 0) {
3808 s_q.front() = s_q.front() * 4;
3810 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3814 m_q.insert(m_q.begin(), 1);
3815 if (s_q.size() > 0) {
3816 s_q.front() = s_q.front() * 2;
3818 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3823 if (m_p.size() == 0) break;
3825 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3830 res = res + signum * multipleLi_do_sum(m_q, s_q);
3836 } // end of anonymous namespace
3839 //////////////////////////////////////////////////////////////////////
3841 // Multiple zeta values zeta(x)
3845 //////////////////////////////////////////////////////////////////////
3848 static ex zeta1_evalf(const ex& x)
3850 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3852 // multiple zeta value
3853 const int count = x.nops();
3854 const lst& xlst = ex_to<lst>(x);
3855 std::vector<int> r(count);
3857 // check parameters and convert them
3858 auto it1 = xlst.begin();
3859 auto it2 = r.begin();
3861 if (!(*it1).info(info_flags::posint)) {
3862 return zeta(x).hold();
3864 *it2 = ex_to<numeric>(*it1).to_int();
3867 } while (it2 != r.end());
3869 // check for divergence
3871 return zeta(x).hold();
3874 // decide on summation algorithm
3875 // this is still a bit clumsy
3876 int limit = (Digits>17) ? 10 : 6;
3877 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3878 return numeric(zeta_do_sum_Crandall(r));
3880 return numeric(zeta_do_sum_simple(r));
3884 // single zeta value
3885 if (is_exactly_a<numeric>(x) && (x != 1)) {
3887 return zeta(ex_to<numeric>(x));
3888 } catch (const dunno &e) { }
3891 return zeta(x).hold();
3895 static ex zeta1_eval(const ex& m)
3897 if (is_exactly_a<lst>(m)) {
3898 if (m.nops() == 1) {
3899 return zeta(m.op(0));
3901 return zeta(m).hold();
3904 if (m.info(info_flags::numeric)) {
3905 const numeric& y = ex_to<numeric>(m);
3906 // trap integer arguments:
3907 if (y.is_integer()) {
3911 if (y.is_equal(*_num1_p)) {
3912 return zeta(m).hold();
3914 if (y.info(info_flags::posint)) {
3915 if (y.info(info_flags::odd)) {
3916 return zeta(m).hold();
3918 return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3921 if (y.info(info_flags::odd)) {
3922 return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3929 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3930 return zeta1_evalf(m);
3933 return zeta(m).hold();
3937 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3939 GINAC_ASSERT(deriv_param==0);
3941 if (is_exactly_a<lst>(m)) {
3944 return zetaderiv(_ex1, m);
3949 static void zeta1_print_latex(const ex& m_, const print_context& c)
3952 if (is_a<lst>(m_)) {
3953 const lst& m = ex_to<lst>(m_);
3954 auto it = m.begin();
3957 for (; it != m.end(); it++) {
3968 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
3969 evalf_func(zeta1_evalf).
3970 eval_func(zeta1_eval).
3971 derivative_func(zeta1_deriv).
3972 print_func<print_latex>(zeta1_print_latex).
3973 do_not_evalf_params().
3977 //////////////////////////////////////////////////////////////////////
3979 // Alternating Euler sum zeta(x,s)
3983 //////////////////////////////////////////////////////////////////////
3986 static ex zeta2_evalf(const ex& x, const ex& s)
3988 if (is_exactly_a<lst>(x)) {
3990 // alternating Euler sum
3991 const int count = x.nops();
3992 const lst& xlst = ex_to<lst>(x);
3993 const lst& slst = ex_to<lst>(s);
3994 std::vector<int> xi(count);
3995 std::vector<int> si(count);
3997 // check parameters and convert them
3998 auto it_xread = xlst.begin();
3999 auto it_sread = slst.begin();
4000 auto it_xwrite = xi.begin();
4001 auto it_swrite = si.begin();
4003 if (!(*it_xread).info(info_flags::posint)) {
4004 return zeta(x, s).hold();
4006 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
4007 if (*it_sread > 0) {
4016 } while (it_xwrite != xi.end());
4018 // check for divergence
4019 if ((xi[0] == 1) && (si[0] == 1)) {
4020 return zeta(x, s).hold();
4023 // use Hoelder convolution
4024 return numeric(zeta_do_Hoelder_convolution(xi, si));
4027 return zeta(x, s).hold();
4031 static ex zeta2_eval(const ex& m, const ex& s_)
4033 if (is_exactly_a<lst>(s_)) {
4034 const lst& s = ex_to<lst>(s_);
4035 for (const auto & it : s) {
4036 if (it.info(info_flags::positive)) {
4039 return zeta(m, s_).hold();
4042 } else if (s_.info(info_flags::positive)) {
4046 return zeta(m, s_).hold();
4050 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
4052 GINAC_ASSERT(deriv_param==0);
4054 if (is_exactly_a<lst>(m)) {
4057 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
4058 return zetaderiv(_ex1, m);
4065 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
4068 if (is_a<lst>(m_)) {
4074 if (is_a<lst>(s_)) {
4080 auto itm = m.begin();
4081 auto its = s.begin();
4083 c.s << "\\overline{";
4091 for (; itm != m.end(); itm++, its++) {
4094 c.s << "\\overline{";
4105 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
4106 evalf_func(zeta2_evalf).
4107 eval_func(zeta2_eval).
4108 derivative_func(zeta2_deriv).
4109 print_func<print_latex>(zeta2_print_latex).
4110 do_not_evalf_params().
4114 } // namespace GiNaC