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) 1999-2022 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"
91 //////////////////////////////////////////////////////////////////////
93 // Classical polylogarithm Li(n,x)
97 //////////////////////////////////////////////////////////////////////
100 // anonymous namespace for helper functions
104 // lookup table for factors built from Bernoulli numbers
106 std::vector<std::vector<cln::cl_N>> Xn;
107 // initial size of Xn that should suffice for 32bit machines (must be even)
108 const int xninitsizestep = 26;
109 int xninitsize = xninitsizestep;
113 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
114 // With these numbers the polylogs can be calculated as follows:
115 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
116 // X_0(n) = B_n (Bernoulli numbers)
117 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
118 // The calculation of Xn depends on X0 and X{n-1}.
119 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
120 // This results in a slightly more complicated algorithm for the X_n.
121 // The first index in Xn corresponds to the index of the polylog minus 2.
122 // The second index in Xn corresponds to the index from the actual sum.
126 // calculate X_2 and higher (corresponding to Li_4 and higher)
127 std::vector<cln::cl_N> buf(xninitsize);
128 auto it = buf.begin();
130 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
132 for (int i=2; i<=xninitsize; i++) {
134 result = 0; // k == 0
136 result = Xn[0][i/2-1]; // k == 0
138 for (int k=1; k<i-1; k++) {
139 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
140 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
143 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
144 result = result + Xn[n-1][i-1] / (i+1); // k == i
151 // special case to handle the X_0 correct
152 std::vector<cln::cl_N> buf(xninitsize);
153 auto it = buf.begin();
155 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
157 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
159 for (int i=3; i<=xninitsize; i++) {
161 result = -Xn[0][(i-3)/2]/2;
162 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
165 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
166 for (int k=1; k<i/2; k++) {
167 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
176 std::vector<cln::cl_N> buf(xninitsize/2);
177 auto it = buf.begin();
178 for (int i=1; i<=xninitsize/2; i++) {
179 *it = bernoulli(i*2).to_cl_N();
188 // doubles the number of entries in each Xn[]
191 const int pos0 = xninitsize / 2;
193 for (int i=1; i<=xninitsizestep/2; ++i) {
194 Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
197 int xend = xninitsize + xninitsizestep;
200 for (int i=xninitsize+1; i<=xend; ++i) {
202 result = -Xn[0][(i-3)/2]/2;
203 Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result);
205 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
206 for (int k=1; k<i/2; k++) {
207 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
209 Xn[1].push_back(result);
213 for (size_t n=2; n<Xn.size(); ++n) {
214 for (int i=xninitsize+1; i<=xend; ++i) {
216 result = 0; // k == 0
218 result = Xn[0][i/2-1]; // k == 0
220 for (int k=1; k<i-1; ++k) {
221 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
222 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
225 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
226 result = result + Xn[n-1][i-1] / (i+1); // k == i
227 Xn[n].push_back(result);
231 xninitsize += xninitsizestep;
235 // calculates Li(2,x) without Xn
236 cln::cl_N Li2_do_sum(const cln::cl_N& x)
240 cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
241 cln::cl_I den = 1; // n^2 = 1
246 den = den + i; // n^2 = 4, 9, 16, ...
248 res = res + num / den;
249 } while (res != resbuf);
254 // calculates Li(2,x) with Xn
255 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
257 std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
258 std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
259 cln::cl_N u = -cln::log(1-x);
260 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
261 cln::cl_N uu = cln::square(u);
262 cln::cl_N res = u - uu/4;
267 factor = factor * uu / (2*i * (2*i+1));
268 res = res + (*it) * factor;
272 it = Xn[0].begin() + (i-1);
275 } while (res != resbuf);
280 // calculates Li(n,x), n>2 without Xn
281 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
283 cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
290 res = res + factor / cln::expt(cln::cl_I(i),n);
292 } while (res != resbuf);
297 // calculates Li(n,x), n>2 with Xn
298 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
300 std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
301 std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
302 cln::cl_N u = -cln::log(1-x);
303 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
309 factor = factor * u / i;
310 res = res + (*it) * factor;
314 it = Xn[n-2].begin() + (i-2);
315 xend = Xn[n-2].end();
317 } while (res != resbuf);
322 // forward declaration needed by function Li_projection and C below
323 const cln::cl_N S_num(int n, int p, const cln::cl_N& x);
326 // helper function for classical polylog Li
327 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
329 // treat n=2 as special case
331 // check if precalculated X0 exists
336 if (cln::realpart(x) < 0.5) {
337 // choose the faster algorithm
338 // the switching point was empirically determined. the optimal point
339 // depends on hardware, Digits, ... so an approx value is okay.
340 // it solves also the problem with precision due to the u=-log(1-x) transformation
341 if (cln::abs(x) < 0.25) {
342 return Li2_do_sum(x);
344 // Li2_do_sum practically doesn't converge near x == ±I
345 return Li2_do_sum_Xn(x);
348 // choose the faster algorithm
349 if (cln::abs(cln::realpart(x)) > 0.75) {
353 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
356 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
360 // check if precalculated Xn exist
362 for (int i=xnsize; i<n-1; i++) {
367 if (cln::realpart(x) < 0.5) {
368 // choose the faster algorithm
369 // with n>=12 the "normal" summation always wins against the method with Xn
370 if ((cln::abs(x) < 0.3) || (n >= 12)) {
371 return Lin_do_sum(n, x);
373 // Li2_do_sum practically doesn't converge near x == ±I
374 return Lin_do_sum_Xn(n, x);
377 cln::cl_N result = 0;
378 if ( x != 1 ) result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
379 for (int j=0; j<n-1; j++) {
380 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
381 * cln::expt(cln::log(x), j) / cln::factorial(j);
388 // helper function for classical polylog Li
389 const cln::cl_N Lin_numeric(const int n, const cln::cl_N& x)
393 return -cln::log(1-x);
404 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
406 if (cln::abs(realpart(x)) < 0.4 && cln::abs(cln::abs(x)-1) < 0.01) {
407 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
408 for (int j=0; j<n-1; j++) {
409 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
410 * cln::expt(cln::log(x), j) / cln::factorial(j);
415 // what is the desired float format?
416 // first guess: default format
417 cln::float_format_t prec = cln::default_float_format;
418 const cln::cl_N value = x;
419 // second guess: the argument's format
420 if (!instanceof(realpart(x), cln::cl_RA_ring))
421 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
422 else if (!instanceof(imagpart(x), cln::cl_RA_ring))
423 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
426 if (cln::abs(value) > 1) {
427 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
428 // check if argument is complex. if it is real, the new polylog has to be conjugated.
429 if (cln::zerop(cln::imagpart(value))) {
431 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
434 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
439 result = result + Li_projection(n, cln::recip(value), prec);
442 result = result - Li_projection(n, cln::recip(value), prec);
446 for (int j=0; j<n-1; j++) {
447 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
448 * Lin_numeric(n-j,1) * cln::expt(cln::log(-value),j) / cln::factorial(j);
450 result = result - add;
454 return Li_projection(n, value, prec);
459 } // end of anonymous namespace
462 //////////////////////////////////////////////////////////////////////
464 // Multiple polylogarithm Li(n,x)
468 //////////////////////////////////////////////////////////////////////
471 // anonymous namespace for helper function
475 // performs the actual series summation for multiple polylogarithms
476 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
478 // ensure all x <> 0.
479 for (const auto & it : x) {
480 if (it == 0) return cln::cl_float(0, cln::float_format(Digits));
483 const int j = s.size();
484 bool flag_accidental_zero = false;
486 std::vector<cln::cl_N> t(j);
487 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
494 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
495 for (int k=j-2; k>=0; k--) {
496 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]);
499 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
500 for (int k=j-2; k>=0; k--) {
501 flag_accidental_zero = cln::zerop(t[k+1]);
502 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]);
504 } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero );
510 // forward declaration for Li_eval()
511 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
514 // type used by the transformation functions for G
515 typedef std::vector<int> Gparameter;
518 // G_eval1-function for G transformations
519 ex G_eval1(int a, int scale, const exvector& gsyms)
522 const ex& scs = gsyms[std::abs(scale)];
523 const ex& as = gsyms[std::abs(a)];
525 return -log(1 - scs/as);
530 return log(gsyms[std::abs(scale)]);
535 // G_eval-function for G transformations
536 ex G_eval(const Gparameter& a, int scale, const exvector& gsyms)
538 // check for properties of G
539 ex sc = gsyms[std::abs(scale)];
541 bool all_zero = true;
542 bool all_ones = true;
544 for (const auto & it : a) {
546 const ex sym = gsyms[std::abs(it)];
560 // care about divergent G: shuffle to separate divergencies that will be canceled
561 // later on in the transformation
562 if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
564 Gparameter short_a(a.begin()+1, a.end());
565 ex result = G_eval1(a.front(), scale, gsyms) * G_eval(short_a, scale, gsyms);
567 auto it = short_a.begin();
568 advance(it, count_ones-1);
569 for (; it != short_a.end(); ++it) {
571 Gparameter newa(short_a.begin(), it);
573 newa.push_back(a[0]);
574 newa.insert(newa.end(), it+1, short_a.end());
575 result -= G_eval(newa, scale, gsyms);
577 return result / count_ones;
580 // G({1,...,1};y) -> G({1};y)^k / k!
581 if (all_ones && a.size() > 1) {
582 return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones);
585 // G({0,...,0};y) -> log(y)^k / k!
587 return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
590 // no special cases anymore -> convert it into Li
593 ex argbuf = gsyms[std::abs(scale)];
595 for (const auto & it : a) {
597 const ex& sym = gsyms[std::abs(it)];
598 x.append(argbuf / sym);
606 return pow(-1, x.nops()) * Li(m, x);
609 // convert back to standard G-function, keep information on small imaginary parts
610 ex G_eval_to_G(const Gparameter& a, int scale, const exvector& gsyms)
614 for (const auto & it : a) {
616 z.append(gsyms[std::abs(it)]);
627 return G(z,s,gsyms[std::abs(scale)]);
631 // converts data for G: pending_integrals -> a
632 Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
634 GINAC_ASSERT(pending_integrals.size() != 1);
636 if (pending_integrals.size() > 0) {
637 // get rid of the first element, which would stand for the new upper limit
638 Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
641 // just return empty parameter list
648 // check the parameters a and scale for G and return information about convergence, depth, etc.
649 // convergent : true if G(a,scale) is convergent
650 // depth : depth of G(a,scale)
651 // trailing_zeros : number of trailing zeros of a
652 // min_it : iterator of a pointing on the smallest element in a
653 Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
654 bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
660 auto lastnonzero = a.end();
661 for (auto it = a.begin(); it != a.end(); ++it) {
662 if (std::abs(*it) > 0) {
666 if (std::abs(*it) < scale) {
668 if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
676 if (lastnonzero == a.end())
678 return ++lastnonzero;
682 // add scale to pending_integrals if pending_integrals is empty
683 Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
685 GINAC_ASSERT(pending_integrals.size() != 1);
687 if (pending_integrals.size() > 0) {
688 return pending_integrals;
690 Gparameter new_pending_integrals;
691 new_pending_integrals.push_back(scale);
692 return new_pending_integrals;
697 // handles trailing zeroes for an otherwise convergent integral
698 ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms)
701 int depth, trailing_zeros;
702 Gparameter::const_iterator last, dummyit;
703 last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
705 GINAC_ASSERT(convergent);
707 if ((trailing_zeros > 0) && (depth > 0)) {
709 Gparameter new_a(a.begin(), a.end()-1);
710 result += G_eval1(0, scale, gsyms) * trailing_zeros_G(new_a, scale, gsyms);
711 for (auto it = a.begin(); it != last; ++it) {
712 Gparameter new_a(a.begin(), it);
714 new_a.insert(new_a.end(), it, a.end()-1);
715 result -= trailing_zeros_G(new_a, scale, gsyms);
718 return result / trailing_zeros;
720 return G_eval(a, scale, gsyms);
725 // G transformation [VSW] (57),(58)
726 ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale, const exvector& gsyms)
728 // pendint = ( y1, b1, ..., br )
729 // a = ( 0, ..., 0, amin )
732 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
733 // where sr replaces amin
735 GINAC_ASSERT(a.back() != 0);
736 GINAC_ASSERT(a.size() > 0);
739 Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
740 const int psize = pending_integrals.size();
743 // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
748 result += log(gsyms[ex_to<numeric>(scale).to_int()]);
750 new_pending_integrals.push_back(-scale);
753 new_pending_integrals.push_back(scale);
757 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
758 pending_integrals.front(),
763 result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
764 new_pending_integrals.front(),
768 new_pending_integrals.back() = 0;
769 result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
770 new_pending_integrals.front(),
777 // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
778 // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
781 result -= zeta(a.size());
783 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
784 pending_integrals.front(),
788 // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
789 // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
790 Gparameter new_a(a.begin()+1, a.end());
791 new_pending_integrals.push_back(0);
792 result -= depth_one_trafo_G(new_pending_integrals, new_a, scale, gsyms);
794 // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
795 // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
796 Gparameter new_pending_integrals_2;
797 new_pending_integrals_2.push_back(scale);
798 new_pending_integrals_2.push_back(0);
800 result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
801 pending_integrals.front(),
803 * depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
805 result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
812 // forward declaration
813 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
814 const Gparameter& pendint, const Gparameter& a_old, int scale,
815 const exvector& gsyms, bool flag_trailing_zeros_only);
818 // G transformation [VSW]
819 ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale,
820 const exvector& gsyms, bool flag_trailing_zeros_only)
822 // main recursion routine
824 // pendint = ( y1, b1, ..., br )
825 // a = ( a1, ..., amin, ..., aw )
828 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
829 // where sr replaces amin
831 // find smallest alpha, determine depth and trailing zeros, and check for convergence
833 int depth, trailing_zeros;
834 Gparameter::const_iterator min_it;
835 auto firstzero = check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
836 int min_it_pos = distance(a.begin(), min_it);
838 // special case: all a's are zero
845 result = G_eval(a, scale, gsyms);
847 if (pendint.size() > 0) {
848 result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
855 // handle trailing zeros
856 if (trailing_zeros > 0) {
858 Gparameter new_a(a.begin(), a.end()-1);
859 result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
860 for (auto it = a.begin(); it != firstzero; ++it) {
861 Gparameter new_a(a.begin(), it);
863 new_a.insert(new_a.end(), it, a.end()-1);
864 result -= G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
866 return result / trailing_zeros;
869 // flag_trailing_zeros_only: in this case we don't have pending integrals
870 if (flag_trailing_zeros_only)
871 return G_eval_to_G(a, scale, gsyms);
875 if (pendint.size() > 0) {
876 return G_eval(convert_pending_integrals_G(pendint),
877 pendint.front(), gsyms) *
878 G_eval(a, scale, gsyms);
880 return G_eval(a, scale, gsyms);
884 // call basic transformation for depth equal one
886 return depth_one_trafo_G(pendint, a, scale, gsyms);
890 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
891 // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
892 // + 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)
894 // smallest element in last place
895 if (min_it + 1 == a.end()) {
896 do { --min_it; } while (*min_it == 0);
898 Gparameter a1(a.begin(),min_it+1);
899 Gparameter a2(min_it+1,a.end());
901 ex result = G_transform(pendint, a2, scale, gsyms, flag_trailing_zeros_only)*
902 G_transform(empty, a1, scale, gsyms, flag_trailing_zeros_only);
904 result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms, flag_trailing_zeros_only);
909 Gparameter::iterator changeit;
911 // first term G(a_1,..,0,...,a_w;a_0)
912 Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
913 Gparameter new_a = a;
914 new_a[min_it_pos] = 0;
915 ex result = G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
916 if (pendint.size() > 0) {
917 result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
918 pendint.front(), gsyms);
922 changeit = new_a.begin() + min_it_pos;
923 changeit = new_a.erase(changeit);
924 if (changeit != new_a.begin()) {
925 // smallest in the middle
926 new_pendint.push_back(*changeit);
927 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
928 new_pendint.front(), gsyms)*
929 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
930 int buffer = *changeit;
932 result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
934 new_pendint.pop_back();
936 new_pendint.push_back(*changeit);
937 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
938 new_pendint.front(), gsyms)*
939 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
941 result -= G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
943 // smallest at the front
944 new_pendint.push_back(scale);
945 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
946 new_pendint.front(), gsyms)*
947 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
948 new_pendint.back() = *changeit;
949 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
950 new_pendint.front(), gsyms)*
951 G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
953 result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
959 // shuffles the two parameter list a1 and a2 and calls G_transform for every term except
960 // for the one that is equal to a_old
961 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
962 const Gparameter& pendint, const Gparameter& a_old, int scale,
963 const exvector& gsyms, bool flag_trailing_zeros_only)
965 if (a1.size()==0 && a2.size()==0) {
966 // veto the one configuration we don't want
967 if ( a0 == a_old ) return 0;
969 return G_transform(pendint, a0, scale, gsyms, flag_trailing_zeros_only);
975 aa0.insert(aa0.end(),a1.begin(),a1.end());
976 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
982 aa0.insert(aa0.end(),a2.begin(),a2.end());
983 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
986 Gparameter a1_removed(a1.begin()+1,a1.end());
987 Gparameter a2_removed(a2.begin()+1,a2.end());
992 a01.push_back( a1[0] );
993 a02.push_back( a2[0] );
995 return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms, flag_trailing_zeros_only)
996 + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
999 // handles the transformations and the numerical evaluation of G
1000 // the parameter x, s and y must only contain numerics
1002 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1003 const cln::cl_N& y);
1005 // do acceleration transformation (hoelder convolution [BBB])
1006 // the parameter x, s and y must only contain numerics
1008 G_do_hoelder(std::vector<cln::cl_N> x, /* yes, it's passed by value */
1009 const std::vector<int>& s, const cln::cl_N& y)
1012 const std::size_t size = x.size();
1013 for (std::size_t i = 0; i < size; ++i)
1016 // 24.03.2021: this block can be outside the loop over r
1021 for (std::size_t i = 0; i < size; ++i) {
1022 // 24.03.2021: replaced (x[i] == cln::cl_RA(1)/p) by (cln::zerop(x[i] - cln::cl_RA(1)/p)
1023 // in the case where we compare a float with a rational, CLN behaves differently in the two versions
1024 if (cln::zerop(x[i] - cln::cl_RA(1)/p) ) {
1025 p = p/2 + cln::cl_RA(3)/2;
1031 cln::cl_RA q = p/(p-1);
1033 for (std::size_t r = 0; r <= size; ++r) {
1034 cln::cl_N buffer(1 & r ? -1 : 1);
1035 std::vector<cln::cl_N> qlstx;
1036 std::vector<int> qlsts;
1037 for (std::size_t j = r; j >= 1; --j) {
1038 qlstx.push_back(cln::cl_N(1) - x[j-1]);
1039 if (imagpart(x[j-1])==0 && realpart(x[j-1]) >= 1) {
1042 qlsts.push_back(-s[j-1]);
1045 if (qlstx.size() > 0) {
1046 buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
1048 std::vector<cln::cl_N> plstx;
1049 std::vector<int> plsts;
1050 for (std::size_t j = r+1; j <= size; ++j) {
1051 plstx.push_back(x[j-1]);
1052 plsts.push_back(s[j-1]);
1054 if (plstx.size() > 0) {
1055 buffer = buffer*G_numeric(plstx, plsts, 1/p);
1057 result = result + buffer;
1062 class less_object_for_cl_N
1065 bool operator() (const cln::cl_N & a, const cln::cl_N & b) const
1068 if (abs(a) != abs(b))
1069 return (abs(a) < abs(b)) ? true : false;
1072 if (phase(a) != phase(b))
1073 return (phase(a) < phase(b)) ? true : false;
1075 // equal, therefore "less" is not true
1081 // convergence transformation, used for numerical evaluation of G function.
1082 // the parameter x, s and y must only contain numerics
1084 G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1085 const cln::cl_N& y, bool flag_trailing_zeros_only)
1087 // sort (|x|<->position) to determine indices
1088 typedef std::multimap<cln::cl_N, std::size_t, less_object_for_cl_N> sortmap_t;
1090 std::size_t size = 0;
1091 for (std::size_t i = 0; i < x.size(); ++i) {
1093 sortmap.insert(std::make_pair(x[i], i));
1097 // include upper limit (scale)
1098 sortmap.insert(std::make_pair(y, x.size()));
1100 // generate missing dummy-symbols
1102 // holding dummy-symbols for the G/Li transformations
1104 gsyms.push_back(symbol("GSYMS_ERROR"));
1105 cln::cl_N lastentry(0);
1106 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1107 if (it != sortmap.begin()) {
1108 if (it->second < x.size()) {
1109 if (x[it->second] == lastentry) {
1110 gsyms.push_back(gsyms.back());
1114 if (y == lastentry) {
1115 gsyms.push_back(gsyms.back());
1120 std::ostringstream os;
1122 gsyms.push_back(symbol(os.str()));
1124 if (it->second < x.size()) {
1125 lastentry = x[it->second];
1131 // fill position data according to sorted indices and prepare substitution list
1132 Gparameter a(x.size());
1134 std::size_t pos = 1;
1136 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1137 if (it->second < x.size()) {
1138 if (s[it->second] > 0) {
1139 a[it->second] = pos;
1141 a[it->second] = -int(pos);
1143 subslst[gsyms[pos]] = numeric(x[it->second]);
1146 subslst[gsyms[pos]] = numeric(y);
1151 // do transformation
1153 ex result = G_transform(pendint, a, scale, gsyms, flag_trailing_zeros_only);
1154 // replace dummy symbols with their values
1155 result = result.expand();
1156 result = result.subs(subslst).evalf();
1157 if (!is_a<numeric>(result))
1158 throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
1160 cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
1164 // handles the transformations and the numerical evaluation of G
1165 // the parameter x, s and y must only contain numerics
1167 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1170 // check for convergence and necessary accelerations
1171 bool need_trafo = false;
1172 bool need_hoelder = false;
1173 bool have_trailing_zero = false;
1174 std::size_t depth = 0;
1175 for (auto & xi : x) {
1178 const cln::cl_N x_y = abs(xi) - y;
1179 if (instanceof(x_y, cln::cl_R_ring) &&
1180 realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2)))
1183 if (abs(abs(xi/y) - 1) < 0.01)
1184 need_hoelder = true;
1187 if (zerop(x.back())) {
1188 have_trailing_zero = true;
1192 if (depth == 1 && x.size() == 2 && !need_trafo)
1193 return - Li_projection(2, y/x[1], cln::float_format(Digits));
1195 // do acceleration transformation (hoelder convolution [BBB])
1196 if (need_hoelder && !have_trailing_zero)
1197 return G_do_hoelder(x, s, y);
1199 // convergence transformation
1201 return G_do_trafo(x, s, y, have_trailing_zero);
1204 std::vector<cln::cl_N> newx;
1205 newx.reserve(x.size());
1207 m.reserve(x.size());
1210 cln::cl_N factor = y;
1211 for (auto & xi : x) {
1215 newx.push_back(factor/xi);
1217 m.push_back(mcount);
1223 return sign*multipleLi_do_sum(m, newx);
1227 ex mLi_numeric(const lst& m, const lst& x)
1229 // let G_numeric do the transformation
1230 std::vector<cln::cl_N> newx;
1231 newx.reserve(x.nops());
1233 s.reserve(x.nops());
1234 cln::cl_N factor(1);
1235 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1236 for (int i = 1; i < *itm; ++i) {
1237 newx.push_back(cln::cl_N(0));
1240 const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
1242 newx.push_back(factor);
1243 if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) {
1250 return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
1254 } // end of anonymous namespace
1257 //////////////////////////////////////////////////////////////////////
1259 // Generalized multiple polylogarithm G(x, y) and G(x, s, y)
1263 //////////////////////////////////////////////////////////////////////
1266 static ex G2_evalf(const ex& x_, const ex& y)
1268 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1269 return G(x_, y).hold();
1271 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1272 if (x.nops() == 0) {
1276 return G(x_, y).hold();
1279 s.reserve(x.nops());
1280 bool all_zero = true;
1281 for (const auto & it : x) {
1282 if (!it.info(info_flags::numeric)) {
1283 return G(x_, y).hold();
1288 if ( !ex_to<numeric>(it).is_real() && ex_to<numeric>(it).imag() < 0 ) {
1296 return pow(log(y), x.nops()) / factorial(x.nops());
1298 std::vector<cln::cl_N> xv;
1299 xv.reserve(x.nops());
1300 for (const auto & it : x)
1301 xv.push_back(ex_to<numeric>(it).to_cl_N());
1302 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1303 return numeric(result);
1307 static ex G2_eval(const ex& x_, const ex& y)
1309 //TODO eval to MZV or H or S or Lin
1311 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1312 return G(x_, y).hold();
1314 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1315 if (x.nops() == 0) {
1319 return G(x_, y).hold();
1322 s.reserve(x.nops());
1323 bool all_zero = true;
1324 bool crational = true;
1325 for (const auto & it : x) {
1326 if (!it.info(info_flags::numeric)) {
1327 return G(x_, y).hold();
1329 if (!it.info(info_flags::crational)) {
1335 if ( !ex_to<numeric>(it).is_real() && ex_to<numeric>(it).imag() < 0 ) {
1343 return pow(log(y), x.nops()) / factorial(x.nops());
1345 if (!y.info(info_flags::crational)) {
1349 return G(x_, y).hold();
1351 std::vector<cln::cl_N> xv;
1352 xv.reserve(x.nops());
1353 for (const auto & it : x)
1354 xv.push_back(ex_to<numeric>(it).to_cl_N());
1355 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1356 return numeric(result);
1360 // option do_not_evalf_params() removed.
1361 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1362 evalf_func(G2_evalf).
1366 // derivative_func(G2_deriv).
1367 // print_func<print_latex>(G2_print_latex).
1370 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1372 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1373 return G(x_, s_, y).hold();
1375 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1376 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst{s_};
1377 if (x.nops() != s.nops()) {
1378 return G(x_, s_, y).hold();
1380 if (x.nops() == 0) {
1384 return G(x_, s_, y).hold();
1386 std::vector<int> sn;
1387 sn.reserve(s.nops());
1388 bool all_zero = true;
1389 for (auto itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1390 if (!(*itx).info(info_flags::numeric)) {
1391 return G(x_, y).hold();
1393 if (!(*its).info(info_flags::real)) {
1394 return G(x_, y).hold();
1399 if ( ex_to<numeric>(*itx).is_real() ) {
1400 if ( ex_to<numeric>(*itx).is_positive() ) {
1412 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1421 return pow(log(y), x.nops()) / factorial(x.nops());
1423 std::vector<cln::cl_N> xn;
1424 xn.reserve(x.nops());
1425 for (const auto & it : x)
1426 xn.push_back(ex_to<numeric>(it).to_cl_N());
1427 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1428 return numeric(result);
1432 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1434 //TODO eval to MZV or H or S or Lin
1436 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1437 return G(x_, s_, y).hold();
1439 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1440 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst{s_};
1441 if (x.nops() != s.nops()) {
1442 return G(x_, s_, y).hold();
1444 if (x.nops() == 0) {
1448 return G(x_, s_, y).hold();
1450 std::vector<int> sn;
1451 sn.reserve(s.nops());
1452 bool all_zero = true;
1453 bool crational = true;
1454 for (auto itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1455 if (!(*itx).info(info_flags::numeric)) {
1456 return G(x_, s_, y).hold();
1458 if (!(*its).info(info_flags::real)) {
1459 return G(x_, s_, y).hold();
1461 if (!(*itx).info(info_flags::crational)) {
1467 if ( ex_to<numeric>(*itx).is_real() ) {
1468 if ( ex_to<numeric>(*itx).is_positive() ) {
1480 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1489 return pow(log(y), x.nops()) / factorial(x.nops());
1491 if (!y.info(info_flags::crational)) {
1495 return G(x_, s_, y).hold();
1497 std::vector<cln::cl_N> xn;
1498 xn.reserve(x.nops());
1499 for (const auto & it : x)
1500 xn.push_back(ex_to<numeric>(it).to_cl_N());
1501 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1502 return numeric(result);
1506 // option do_not_evalf_params() removed.
1507 // This is safe: in the code above it only matters if s_ > 0 or s_ < 0,
1508 // s_ is allowed to be of floating type.
1509 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1510 evalf_func(G3_evalf).
1514 // derivative_func(G3_deriv).
1515 // print_func<print_latex>(G3_print_latex).
1518 //////////////////////////////////////////////////////////////////////
1520 // Classical polylogarithm and multiple polylogarithm Li(m,x)
1524 //////////////////////////////////////////////////////////////////////
1527 static ex Li_evalf(const ex& m_, const ex& x_)
1529 // classical polylogs
1530 if (m_.info(info_flags::posint)) {
1531 if (x_.info(info_flags::numeric)) {
1532 int m__ = ex_to<numeric>(m_).to_int();
1533 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1534 const cln::cl_N result = Lin_numeric(m__, x__);
1535 return numeric(result);
1537 // try to numerically evaluate second argument
1538 ex x_val = x_.evalf();
1539 if (x_val.info(info_flags::numeric)) {
1540 int m__ = ex_to<numeric>(m_).to_int();
1541 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1542 const cln::cl_N result = Lin_numeric(m__, x__);
1543 return numeric(result);
1547 // multiple polylogs
1548 if (is_a<lst>(m_) && is_a<lst>(x_)) {
1550 const lst& m = ex_to<lst>(m_);
1551 const lst& x = ex_to<lst>(x_);
1552 if (m.nops() != x.nops()) {
1553 return Li(m_,x_).hold();
1555 if (x.nops() == 0) {
1558 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1559 return Li(m_,x_).hold();
1562 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1563 if (!(*itm).info(info_flags::posint)) {
1564 return Li(m_, x_).hold();
1566 if (!(*itx).info(info_flags::numeric)) {
1567 return Li(m_, x_).hold();
1574 return mLi_numeric(m, x);
1577 return Li(m_,x_).hold();
1581 static ex Li_eval(const ex& m_, const ex& x_)
1583 if (is_a<lst>(m_)) {
1584 if (is_a<lst>(x_)) {
1585 // multiple polylogs
1586 const lst& m = ex_to<lst>(m_);
1587 const lst& x = ex_to<lst>(x_);
1588 if (m.nops() != x.nops()) {
1589 return Li(m_,x_).hold();
1591 if (x.nops() == 0) {
1595 bool is_zeta = true;
1596 bool do_evalf = true;
1597 bool crational = true;
1598 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1599 if (!(*itm).info(info_flags::posint)) {
1600 return Li(m_,x_).hold();
1602 if ((*itx != _ex1) && (*itx != _ex_1)) {
1603 if (itx != x.begin()) {
1611 if (!(*itx).info(info_flags::numeric)) {
1614 if (!(*itx).info(info_flags::crational)) {
1620 for (const auto & itx : x) {
1621 GINAC_ASSERT((itx == _ex1) || (itx == _ex_1));
1622 // XXX: 1 + 0.0*I is considered equal to 1. However
1623 // the former is a not automatically converted
1624 // to a real number. Do the conversion explicitly
1625 // to avoid the "numeric::operator>(): complex inequality"
1626 // exception (and similar problems).
1627 newx.append(itx != _ex_1 ? _ex1 : _ex_1);
1629 return zeta(m_, newx);
1633 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1634 return prefactor * H(newm, x[0]);
1636 if (do_evalf && !crational) {
1637 return mLi_numeric(m,x);
1640 return Li(m_, x_).hold();
1641 } else if (is_a<lst>(x_)) {
1642 return Li(m_, x_).hold();
1645 // classical polylogs
1653 return (pow(2,1-m_)-1) * zeta(m_);
1659 if (x_.is_equal(I)) {
1660 return power(Pi,_ex2)/_ex_48 + Catalan*I;
1662 if (x_.is_equal(-I)) {
1663 return power(Pi,_ex2)/_ex_48 - Catalan*I;
1666 if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1667 int m__ = ex_to<numeric>(m_).to_int();
1668 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1669 const cln::cl_N result = Lin_numeric(m__, x__);
1670 return numeric(result);
1673 return Li(m_, x_).hold();
1677 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1679 if (is_a<lst>(m) || is_a<lst>(x)) {
1681 epvector seq { expair(Li(m, x), 0) };
1682 return pseries(rel, std::move(seq));
1685 // classical polylog
1686 const ex x_pt = x.subs(rel, subs_options::no_pattern);
1687 if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1688 // First special case: x==0 (derivatives have poles)
1689 if (x_pt.is_zero()) {
1692 // manually construct the primitive expansion
1693 for (int i=1; i<order; ++i)
1694 ser += pow(s,i) / pow(numeric(i), m);
1695 // substitute the argument's series expansion
1696 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1697 // maybe that was terminating, so add a proper order term
1698 epvector nseq { expair(Order(_ex1), order) };
1699 ser += pseries(rel, std::move(nseq));
1700 // reexpanding it will collapse the series again
1701 return ser.series(rel, order);
1703 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1704 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1706 // all other cases should be safe, by now:
1707 throw do_taylor(); // caught by function::series()
1711 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1713 GINAC_ASSERT(deriv_param < 2);
1714 if (deriv_param == 0) {
1717 if (m_.nops() > 1) {
1718 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1721 if (is_a<lst>(m_)) {
1727 if (is_a<lst>(x_)) {
1733 return Li(m-1, x) / x;
1740 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1743 if (is_a<lst>(m_)) {
1749 if (is_a<lst>(x_)) {
1754 c.s << "\\mathrm{Li}_{";
1755 auto itm = m.begin();
1758 for (; itm != m.end(); itm++) {
1763 auto itx = x.begin();
1766 for (; itx != x.end(); itx++) {
1774 REGISTER_FUNCTION(Li,
1775 evalf_func(Li_evalf).
1777 series_func(Li_series).
1778 derivative_func(Li_deriv).
1779 print_func<print_latex>(Li_print_latex).
1780 do_not_evalf_params());
1783 //////////////////////////////////////////////////////////////////////
1785 // Nielsen's generalized polylogarithm S(n,p,x)
1789 //////////////////////////////////////////////////////////////////////
1792 // anonymous namespace for helper functions
1796 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1798 std::vector<std::vector<cln::cl_N>> Yn;
1799 int ynsize = 0; // number of Yn[]
1800 int ynlength = 100; // initial length of all Yn[i]
1803 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1804 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1805 // representing S_{n,p}(x).
1806 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1807 // equivalent Z-sum.
1808 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1809 // representing S_{n,p}(x).
1810 // The calculation of Y_n uses the values from Y_{n-1}.
1811 void fill_Yn(int n, const cln::float_format_t& prec)
1813 const int initsize = ynlength;
1814 //const int initsize = initsize_Yn;
1815 cln::cl_N one = cln::cl_float(1, prec);
1818 std::vector<cln::cl_N> buf(initsize);
1819 auto it = buf.begin();
1820 auto itprev = Yn[n-1].begin();
1821 *it = (*itprev) / cln::cl_N(n+1) * one;
1824 // sums with an index smaller than the depth are zero and need not to be calculated.
1825 // calculation starts with depth, which is n+2)
1826 for (int i=n+2; i<=initsize+n; i++) {
1827 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1833 std::vector<cln::cl_N> buf(initsize);
1834 auto it = buf.begin();
1837 for (int i=2; i<=initsize; i++) {
1838 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1847 // make Yn longer ...
1848 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1851 cln::cl_N one = cln::cl_float(1, prec);
1853 Yn[0].resize(newsize);
1854 auto it = Yn[0].begin();
1856 for (int i=ynlength+1; i<=newsize; i++) {
1857 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1861 for (int n=1; n<ynsize; n++) {
1862 Yn[n].resize(newsize);
1863 auto it = Yn[n].begin();
1864 auto itprev = Yn[n-1].begin();
1867 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1868 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1878 // helper function for S(n,p,x)
1880 cln::cl_N C(int n, int p)
1884 for (int k=0; k<p; k++) {
1885 for (int j=0; j<=(n+k-1)/2; j++) {
1889 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1892 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1899 result = result + cln::factorial(n+k-1)
1900 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1901 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1904 result = result - cln::factorial(n+k-1)
1905 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1906 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1911 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1912 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1915 result = result + cln::factorial(n+k-1)
1916 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1917 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1925 if (((np)/2+n) & 1) {
1926 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1929 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1937 // helper function for S(n,p,x)
1938 // [Kol] remark to (9.1)
1939 cln::cl_N a_k(int k)
1948 for (int m=2; m<=k; m++) {
1949 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1956 // helper function for S(n,p,x)
1957 // [Kol] remark to (9.1)
1958 cln::cl_N b_k(int k)
1967 for (int m=2; m<=k; m++) {
1968 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1975 // helper function for S(n,p,x)
1976 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1978 static cln::float_format_t oldprec = cln::default_float_format;
1981 return Li_projection(n+1, x, prec);
1984 // precision has changed, we need to clear lookup table Yn
1985 if ( oldprec != prec ) {
1992 // check if precalculated values are sufficient
1994 for (int i=ynsize; i<p-1; i++) {
1999 // should be done otherwise
2000 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2001 cln::cl_N xf = x * one;
2002 //cln::cl_N xf = x * cln::cl_float(1, prec);
2006 cln::cl_N factor = cln::expt(xf, p);
2010 if (i-p >= ynlength) {
2012 make_Yn_longer(ynlength*2, prec);
2014 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? ...
2015 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
2016 factor = factor * xf;
2018 } while (res != resbuf);
2024 // helper function for S(n,p,x)
2025 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
2028 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
2030 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
2031 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
2033 for (int s=0; s<n; s++) {
2035 for (int r=0; r<p; r++) {
2036 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
2037 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
2039 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2045 return S_do_sum(n, p, x, prec);
2049 // helper function for S(n,p,x)
2050 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
2054 // [Kol] (2.22) with (2.21)
2055 return cln::zeta(p+1);
2060 return cln::zeta(n+1);
2065 for (int nu=0; nu<n; nu++) {
2066 for (int rho=0; rho<=p; rho++) {
2067 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2068 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2071 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2078 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2080 // throw std::runtime_error("don't know how to evaluate this function!");
2083 // what is the desired float format?
2084 // first guess: default format
2085 cln::float_format_t prec = cln::default_float_format;
2086 const cln::cl_N value = x;
2087 // second guess: the argument's format
2088 if (!instanceof(realpart(value), cln::cl_RA_ring))
2089 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2090 else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2091 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2094 // 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
2095 // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
2096 if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
2098 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2099 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2101 for (int s=0; s<n; s++) {
2103 for (int r=0; r<p; r++) {
2104 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2105 * S_num(p-r,n-s,1-value) / cln::factorial(r);
2107 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2114 if (cln::abs(value) > 1) {
2118 for (int s=0; s<p; s++) {
2119 for (int r=0; r<=s; r++) {
2120 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2121 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2122 * S_num(n+s-r,p-s,cln::recip(value));
2125 result = result * cln::expt(cln::cl_I(-1),n);
2128 for (int r=0; r<n; r++) {
2129 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2131 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2133 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2138 if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
2141 for (int s=0; s<p-1; s++)
2144 ex res = H(m,numeric(value)).evalf();
2145 return ex_to<numeric>(res).to_cl_N();
2148 return S_projection(n, p, value, prec);
2153 } // end of anonymous namespace
2156 //////////////////////////////////////////////////////////////////////
2158 // Nielsen's generalized polylogarithm S(n,p,x)
2162 //////////////////////////////////////////////////////////////////////
2165 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2167 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2168 const int n_ = ex_to<numeric>(n).to_int();
2169 const int p_ = ex_to<numeric>(p).to_int();
2170 if (is_a<numeric>(x)) {
2171 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2172 const cln::cl_N result = S_num(n_, p_, x_);
2173 return numeric(result);
2175 ex x_val = x.evalf();
2176 if (is_a<numeric>(x_val)) {
2177 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2178 const cln::cl_N result = S_num(n_, p_, x_val_);
2179 return numeric(result);
2183 return S(n, p, x).hold();
2187 static ex S_eval(const ex& n, const ex& p, const ex& x)
2189 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2195 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2203 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2204 int n_ = ex_to<numeric>(n).to_int();
2205 int p_ = ex_to<numeric>(p).to_int();
2206 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2207 const cln::cl_N result = S_num(n_, p_, x_);
2208 return numeric(result);
2213 return pow(-log(1-x), p) / factorial(p);
2215 return S(n, p, x).hold();
2219 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2222 return Li(n+1, x).series(rel, order, options);
2225 const ex x_pt = x.subs(rel, subs_options::no_pattern);
2226 if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2227 // First special case: x==0 (derivatives have poles)
2228 if (x_pt.is_zero()) {
2231 // manually construct the primitive expansion
2232 // subsum = Euler-Zagier-Sum is needed
2233 // dirty hack (slow ...) calculation of subsum:
2234 std::vector<ex> presubsum, subsum;
2235 subsum.push_back(0);
2236 for (int i=1; i<order-1; ++i) {
2237 subsum.push_back(subsum[i-1] + numeric(1, i));
2239 for (int depth=2; depth<p; ++depth) {
2241 for (int i=1; i<order-1; ++i) {
2242 subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2246 for (int i=1; i<order; ++i) {
2247 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2249 // substitute the argument's series expansion
2250 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2251 // maybe that was terminating, so add a proper order term
2252 epvector nseq { expair(Order(_ex1), order) };
2253 ser += pseries(rel, std::move(nseq));
2254 // reexpanding it will collapse the series again
2255 return ser.series(rel, order);
2257 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2258 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2260 // all other cases should be safe, by now:
2261 throw do_taylor(); // caught by function::series()
2265 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2267 GINAC_ASSERT(deriv_param < 3);
2268 if (deriv_param < 2) {
2272 return S(n-1, p, x) / x;
2274 return S(n, p-1, x) / (1-x);
2279 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2281 c.s << "\\mathrm{S}_{";
2291 REGISTER_FUNCTION(S,
2292 evalf_func(S_evalf).
2294 series_func(S_series).
2295 derivative_func(S_deriv).
2296 print_func<print_latex>(S_print_latex).
2297 do_not_evalf_params());
2300 //////////////////////////////////////////////////////////////////////
2302 // Harmonic polylogarithm H(m,x)
2306 //////////////////////////////////////////////////////////////////////
2309 // anonymous namespace for helper functions
2313 // regulates the pole (used by 1/x-transformation)
2314 symbol H_polesign("IMSIGN");
2317 // convert parameters from H to Li representation
2318 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2319 // returns true if some parameters are negative
2320 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2322 // expand parameter list
2324 for (const auto & it : l) {
2326 for (ex count=it-1; count > 0; count--) {
2330 } else if (it < -1) {
2331 for (ex count=it+1; count < 0; count++) {
2342 bool has_negative_parameters = false;
2344 for (const auto & it : mexp) {
2350 m.append((it+acc-1) * signum);
2352 m.append((it-acc+1) * signum);
2358 has_negative_parameters = true;
2361 if (has_negative_parameters) {
2362 for (std::size_t i=0; i<m.nops(); i++) {
2364 m.let_op(i) = -m.op(i);
2372 return has_negative_parameters;
2376 // recursivly transforms H to corresponding multiple polylogarithms
2377 struct map_trafo_H_convert_to_Li : public map_function
2379 ex operator()(const ex& e) override
2381 if (is_a<add>(e) || is_a<mul>(e)) {
2382 return e.map(*this);
2384 if (is_a<function>(e)) {
2385 std::string name = ex_to<function>(e).get_name();
2388 if (is_a<lst>(e.op(0))) {
2389 parameter = ex_to<lst>(e.op(0));
2391 parameter = lst{e.op(0)};
2398 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2399 s.let_op(0) = s.op(0) * arg;
2400 return pf * Li(m, s).hold();
2402 for (std::size_t i=0; i<m.nops(); i++) {
2405 s.let_op(0) = s.op(0) * arg;
2406 return Li(m, s).hold();
2415 // recursivly transforms H to corresponding zetas
2416 struct map_trafo_H_convert_to_zeta : public map_function
2418 ex operator()(const ex& e) override
2420 if (is_a<add>(e) || is_a<mul>(e)) {
2421 return e.map(*this);
2423 if (is_a<function>(e)) {
2424 std::string name = ex_to<function>(e).get_name();
2427 if (is_a<lst>(e.op(0))) {
2428 parameter = ex_to<lst>(e.op(0));
2430 parameter = lst{e.op(0)};
2436 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2437 return pf * zeta(m, s);
2448 // remove trailing zeros from H-parameters
2449 struct map_trafo_H_reduce_trailing_zeros : public map_function
2451 ex operator()(const ex& e) override
2453 if (is_a<add>(e) || is_a<mul>(e)) {
2454 return e.map(*this);
2456 if (is_a<function>(e)) {
2457 std::string name = ex_to<function>(e).get_name();
2460 if (is_a<lst>(e.op(0))) {
2461 parameter = ex_to<lst>(e.op(0));
2463 parameter = lst{e.op(0)};
2466 if (parameter.op(parameter.nops()-1) == 0) {
2469 if (parameter.nops() == 1) {
2474 auto it = parameter.begin();
2475 while ((it != parameter.end()) && (*it == 0)) {
2478 if (it == parameter.end()) {
2479 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2483 parameter.remove_last();
2484 std::size_t lastentry = parameter.nops();
2485 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2490 ex result = log(arg) * H(parameter,arg).hold();
2492 for (ex i=0; i<lastentry; i++) {
2493 if (parameter[i] > 0) {
2495 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2498 } else if (parameter[i] < 0) {
2500 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2508 if (lastentry < parameter.nops()) {
2509 result = result / (parameter.nops()-lastentry+1);
2510 return result.map(*this);
2522 // returns an expression with zeta functions corresponding to the parameter list for H
2523 ex convert_H_to_zeta(const lst& m)
2525 symbol xtemp("xtemp");
2526 map_trafo_H_reduce_trailing_zeros filter;
2527 map_trafo_H_convert_to_zeta filter2;
2528 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2532 // convert signs form Li to H representation
2533 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2536 auto itm = m.begin();
2537 auto itx = ++x.begin();
2542 while (itx != x.end()) {
2543 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
2544 // XXX: 1 + 0.0*I is considered equal to 1. However the former
2545 // is not automatically converted to a real number.
2546 // Do the conversion explicitly to avoid the
2547 // "numeric::operator>(): complex inequality" exception.
2548 signum *= (*itx != _ex_1) ? 1 : -1;
2550 res.append((*itm) * signum);
2558 // multiplies an one-dimensional H with another H
2560 ex trafo_H_mult(const ex& h1, const ex& h2)
2565 ex h1nops = h1.op(0).nops();
2566 ex h2nops = h2.op(0).nops();
2568 hshort = h2.op(0).op(0);
2569 hlong = ex_to<lst>(h1.op(0));
2571 hshort = h1.op(0).op(0);
2573 hlong = ex_to<lst>(h2.op(0));
2575 hlong = lst{h2.op(0).op(0)};
2578 for (std::size_t i=0; i<=hlong.nops(); i++) {
2582 newparameter.append(hlong[j]);
2584 newparameter.append(hshort);
2585 for (; j<hlong.nops(); j++) {
2586 newparameter.append(hlong[j]);
2588 res += H(newparameter, h1.op(1)).hold();
2594 // applies trafo_H_mult recursively on expressions
2595 struct map_trafo_H_mult : public map_function
2597 ex operator()(const ex& e) override
2600 return e.map(*this);
2608 for (std::size_t pos=0; pos<e.nops(); pos++) {
2609 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2610 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2612 for (ex i=0; i<e.op(pos).op(1); i++) {
2613 Hlst.append(e.op(pos).op(0));
2617 } else if (is_a<function>(e.op(pos))) {
2618 std::string name = ex_to<function>(e.op(pos)).get_name();
2620 if (e.op(pos).op(0).nops() > 1) {
2623 Hlst.append(e.op(pos));
2628 result *= e.op(pos);
2631 if (Hlst.nops() > 0) {
2632 firstH = Hlst[Hlst.nops()-1];
2639 if (Hlst.nops() > 0) {
2640 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2642 for (std::size_t i=1; i<Hlst.nops(); i++) {
2643 result *= Hlst.op(i);
2645 result = result.expand();
2646 map_trafo_H_mult recursion;
2647 return recursion(result);
2658 // do integration [ReV] (55)
2659 // put parameter 0 in front of existing parameters
2660 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2664 if (is_a<function>(e)) {
2665 name = ex_to<function>(e).get_name();
2670 for (std::size_t i=0; i<e.nops(); i++) {
2671 if (is_a<function>(e.op(i))) {
2672 std::string name = ex_to<function>(e.op(i)).get_name();
2680 lst newparameter = ex_to<lst>(h.op(0));
2681 newparameter.prepend(0);
2682 ex addzeta = convert_H_to_zeta(newparameter);
2683 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2685 return e * (-H(lst{ex(0)},1/arg).hold());
2690 // do integration [ReV] (49)
2691 // put parameter 1 in front of existing parameters
2692 ex trafo_H_prepend_one(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 return e.subs(h == H(newparameter, h.op(1)).hold());
2716 return e * H(lst{ex(1)},1-arg).hold();
2721 // do integration [ReV] (55)
2722 // put parameter -1 in front of existing parameters
2723 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2727 if (is_a<function>(e)) {
2728 name = ex_to<function>(e).get_name();
2733 for (std::size_t i=0; i<e.nops(); i++) {
2734 if (is_a<function>(e.op(i))) {
2735 std::string name = ex_to<function>(e.op(i)).get_name();
2743 lst newparameter = ex_to<lst>(h.op(0));
2744 newparameter.prepend(-1);
2745 ex addzeta = convert_H_to_zeta(newparameter);
2746 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2748 ex addzeta = convert_H_to_zeta(lst{ex(-1)});
2749 return (e * (addzeta - H(lst{ex(-1)},1/arg).hold())).expand();
2754 // do integration [ReV] (55)
2755 // put parameter -1 in front of existing parameters
2756 ex trafo_H_1mxt1px_prepend_minusone(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 integration [ReV] (55)
2786 // put parameter 1 in front of existing parameters
2787 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2791 if (is_a<function>(e)) {
2792 name = ex_to<function>(e).get_name();
2797 for (std::size_t i = 0; i < e.nops(); i++) {
2798 if (is_a<function>(e.op(i))) {
2799 std::string name = ex_to<function>(e.op(i)).get_name();
2807 lst newparameter = ex_to<lst>(h.op(0));
2808 newparameter.prepend(1);
2809 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2811 return (e * H(lst{ex(1)},(1-arg)/(1+arg)).hold()).expand();
2816 // do x -> 1-x transformation
2817 struct map_trafo_H_1mx : public map_function
2819 ex operator()(const ex& e) override
2821 if (is_a<add>(e) || is_a<mul>(e)) {
2822 return e.map(*this);
2825 if (is_a<function>(e)) {
2826 std::string name = ex_to<function>(e).get_name();
2829 lst parameter = ex_to<lst>(e.op(0));
2832 // special cases if all parameters are either 0, 1 or -1
2833 bool allthesame = true;
2834 if (parameter.op(0) == 0) {
2835 for (std::size_t i = 1; i < parameter.nops(); i++) {
2836 if (parameter.op(i) != 0) {
2843 for (int i=parameter.nops(); i>0; i--) {
2844 newparameter.append(1);
2846 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2848 } else if (parameter.op(0) == -1) {
2849 throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2851 for (std::size_t i = 1; i < parameter.nops(); i++) {
2852 if (parameter.op(i) != 1) {
2859 for (int i=parameter.nops(); i>0; i--) {
2860 newparameter.append(0);
2862 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2866 lst newparameter = parameter;
2867 newparameter.remove_first();
2869 if (parameter.op(0) == 0) {
2872 ex res = convert_H_to_zeta(parameter);
2873 //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2874 map_trafo_H_1mx recursion;
2875 ex buffer = recursion(H(newparameter, arg).hold());
2876 if (is_a<add>(buffer)) {
2877 for (std::size_t i = 0; i < buffer.nops(); i++) {
2878 res -= trafo_H_prepend_one(buffer.op(i), arg);
2881 res -= trafo_H_prepend_one(buffer, arg);
2888 map_trafo_H_1mx recursion;
2889 map_trafo_H_mult unify;
2890 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
2891 std::size_t firstzero = 0;
2892 while (parameter.op(firstzero) == 1) {
2895 for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2899 newparameter.append(parameter[j+1]);
2901 newparameter.append(1);
2902 for (; j<parameter.nops()-1; j++) {
2903 newparameter.append(parameter[j+1]);
2905 res -= H(newparameter, arg).hold();
2907 res = recursion(res).expand() / firstzero;
2917 // do x -> 1/x transformation
2918 struct map_trafo_H_1overx : public map_function
2920 ex operator()(const ex& e) override
2922 if (is_a<add>(e) || is_a<mul>(e)) {
2923 return e.map(*this);
2926 if (is_a<function>(e)) {
2927 std::string name = ex_to<function>(e).get_name();
2930 lst parameter = ex_to<lst>(e.op(0));
2933 // special cases if all parameters are either 0, 1 or -1
2934 bool allthesame = true;
2935 if (parameter.op(0) == 0) {
2936 for (std::size_t i = 1; i < parameter.nops(); i++) {
2937 if (parameter.op(i) != 0) {
2943 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2945 } else if (parameter.op(0) == -1) {
2946 for (std::size_t i = 1; i < parameter.nops(); i++) {
2947 if (parameter.op(i) != -1) {
2953 map_trafo_H_mult unify;
2954 return unify((pow(H(lst{ex(-1)},1/arg).hold() - H(lst{ex(0)},1/arg).hold(), parameter.nops())
2955 / factorial(parameter.nops())).expand());
2958 for (std::size_t i = 1; i < parameter.nops(); i++) {
2959 if (parameter.op(i) != 1) {
2965 map_trafo_H_mult unify;
2966 return unify((pow(H(lst{ex(1)},1/arg).hold() + H(lst{ex(0)},1/arg).hold() + H_polesign, parameter.nops())
2967 / factorial(parameter.nops())).expand());
2971 lst newparameter = parameter;
2972 newparameter.remove_first();
2974 if (parameter.op(0) == 0) {
2977 ex res = convert_H_to_zeta(parameter);
2978 map_trafo_H_1overx recursion;
2979 ex buffer = recursion(H(newparameter, arg).hold());
2980 if (is_a<add>(buffer)) {
2981 for (std::size_t i = 0; i < buffer.nops(); i++) {
2982 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2985 res += trafo_H_1tx_prepend_zero(buffer, arg);
2989 } else if (parameter.op(0) == -1) {
2991 // leading negative one
2992 ex res = convert_H_to_zeta(parameter);
2993 map_trafo_H_1overx recursion;
2994 ex buffer = recursion(H(newparameter, arg).hold());
2995 if (is_a<add>(buffer)) {
2996 for (std::size_t i = 0; i < buffer.nops(); i++) {
2997 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
3000 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
3007 map_trafo_H_1overx recursion;
3008 map_trafo_H_mult unify;
3009 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
3010 std::size_t firstzero = 0;
3011 while (parameter.op(firstzero) == 1) {
3014 for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
3018 newparameter.append(parameter[j+1]);
3020 newparameter.append(1);
3021 for (; j<parameter.nops()-1; j++) {
3022 newparameter.append(parameter[j+1]);
3024 res -= H(newparameter, arg).hold();
3026 res = recursion(res).expand() / firstzero;
3038 // do x -> (1-x)/(1+x) transformation
3039 struct map_trafo_H_1mxt1px : public map_function
3041 ex operator()(const ex& e) override
3043 if (is_a<add>(e) || is_a<mul>(e)) {
3044 return e.map(*this);
3047 if (is_a<function>(e)) {
3048 std::string name = ex_to<function>(e).get_name();
3051 lst parameter = ex_to<lst>(e.op(0));
3054 // special cases if all parameters are either 0, 1 or -1
3055 bool allthesame = true;
3056 if (parameter.op(0) == 0) {
3057 for (std::size_t i = 1; i < parameter.nops(); i++) {
3058 if (parameter.op(i) != 0) {
3064 map_trafo_H_mult unify;
3065 return unify((pow(-H(lst{ex(1)},(1-arg)/(1+arg)).hold() - H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops())
3066 / factorial(parameter.nops())).expand());
3068 } else if (parameter.op(0) == -1) {
3069 for (std::size_t i = 1; i < parameter.nops(); i++) {
3070 if (parameter.op(i) != -1) {
3076 map_trafo_H_mult unify;
3077 return unify((pow(log(2) - H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops())
3078 / factorial(parameter.nops())).expand());
3081 for (std::size_t i = 1; i < parameter.nops(); i++) {
3082 if (parameter.op(i) != 1) {
3088 map_trafo_H_mult unify;
3089 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())
3090 / factorial(parameter.nops())).expand());
3094 lst newparameter = parameter;
3095 newparameter.remove_first();
3097 if (parameter.op(0) == 0) {
3100 ex res = convert_H_to_zeta(parameter);
3101 map_trafo_H_1mxt1px recursion;
3102 ex buffer = recursion(H(newparameter, arg).hold());
3103 if (is_a<add>(buffer)) {
3104 for (std::size_t i = 0; i < buffer.nops(); i++) {
3105 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3108 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3112 } else if (parameter.op(0) == -1) {
3114 // leading negative one
3115 ex res = convert_H_to_zeta(parameter);
3116 map_trafo_H_1mxt1px recursion;
3117 ex buffer = recursion(H(newparameter, arg).hold());
3118 if (is_a<add>(buffer)) {
3119 for (std::size_t i = 0; i < buffer.nops(); i++) {
3120 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3123 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3130 map_trafo_H_1mxt1px recursion;
3131 map_trafo_H_mult unify;
3132 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
3133 std::size_t firstzero = 0;
3134 while (parameter.op(firstzero) == 1) {
3137 for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3141 newparameter.append(parameter[j+1]);
3143 newparameter.append(1);
3144 for (; j<parameter.nops()-1; j++) {
3145 newparameter.append(parameter[j+1]);
3147 res -= H(newparameter, arg).hold();
3149 res = recursion(res).expand() / firstzero;
3161 // do the actual summation.
3162 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3164 const int j = m.size();
3166 std::vector<cln::cl_N> t(j);
3168 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3169 cln::cl_N factor = cln::expt(x, j) * one;
3175 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3176 for (int k=j-2; k>=1; k--) {
3177 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3179 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3180 factor = factor * x;
3181 } while (t[0] != t0buf);
3187 } // end of anonymous namespace
3190 //////////////////////////////////////////////////////////////////////
3192 // Harmonic polylogarithm H(m,x)
3196 //////////////////////////////////////////////////////////////////////
3199 static ex H_evalf(const ex& x1, const ex& x2)
3201 if (is_a<lst>(x1)) {
3204 if (is_a<numeric>(x2)) {
3205 x = ex_to<numeric>(x2).to_cl_N();
3207 ex x2_val = x2.evalf();
3208 if (is_a<numeric>(x2_val)) {
3209 x = ex_to<numeric>(x2_val).to_cl_N();
3213 for (std::size_t i = 0; i < x1.nops(); i++) {
3214 if (!x1.op(i).info(info_flags::integer)) {
3215 return H(x1, x2).hold();
3218 if (x1.nops() < 1) {
3219 return H(x1, x2).hold();
3222 const lst& morg = ex_to<lst>(x1);
3223 // remove trailing zeros ...
3224 if (*(--morg.end()) == 0) {
3225 symbol xtemp("xtemp");
3226 map_trafo_H_reduce_trailing_zeros filter;
3227 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3229 // ... and expand parameter notation
3231 for (const auto & it : morg) {
3233 for (ex count=it-1; count > 0; count--) {
3237 } else if (it <= -1) {
3238 for (ex count=it+1; count < 0; count++) {
3248 if (cln::abs(x) < 0.95) {
3252 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3253 // negative parameters -> s_lst is filled
3254 std::vector<int> m_int;
3255 std::vector<cln::cl_N> x_cln;
3256 for (auto it_int = m_lst.begin(), it_cln = s_lst.begin();
3257 it_int != m_lst.end(); it_int++, it_cln++) {
3258 m_int.push_back(ex_to<numeric>(*it_int).to_int());
3259 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3261 x_cln.front() = x_cln.front() * x;
3262 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3264 // only positive parameters
3266 if (m_lst.nops() == 1) {
3267 return Li(m_lst.op(0), x2).evalf();
3269 std::vector<int> m_int;
3270 for (const auto & it : m_lst) {
3271 m_int.push_back(ex_to<numeric>(it).to_int());
3273 return numeric(H_do_sum(m_int, x));
3277 symbol xtemp("xtemp");
3280 // ensure that the realpart of the argument is positive
3281 if (cln::realpart(x) < 0) {
3283 for (std::size_t i = 0; i < m.nops(); i++) {
3285 m.let_op(i) = -m.op(i);
3292 if (cln::abs(x) >= 2.0) {
3293 map_trafo_H_1overx trafo;
3294 res *= trafo(H(m, xtemp).hold());
3295 if (cln::imagpart(x) <= 0) {
3296 res = res.subs(H_polesign == -I*Pi);
3298 res = res.subs(H_polesign == I*Pi);
3300 return res.subs(xtemp == numeric(x)).evalf();
3303 // check for letters (-1)
3304 bool has_minus_one = false;
3305 for (const auto & it : m) {
3307 has_minus_one = true;
3310 // check transformations for 0.95 <= |x| < 2.0
3312 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3313 if (cln::abs(x-9.53) <= 9.47) {
3315 map_trafo_H_1mxt1px trafo;
3316 res *= trafo(H(m, xtemp).hold());
3319 if (has_minus_one) {
3320 map_trafo_H_convert_to_Li filter;
3321 // 09.06.2021: bug fix: don't forget a possible minus sign from the case realpart(x) < 0
3322 res *= filter(H(m, numeric(x)).hold()).evalf();
3325 map_trafo_H_1mx trafo;
3326 res *= trafo(H(m, xtemp).hold());
3329 return res.subs(xtemp == numeric(x)).evalf();
3332 return H(x1,x2).hold();
3336 static ex H_eval(const ex& m_, const ex& x)
3339 if (is_a<lst>(m_)) {
3344 if (m.nops() == 0) {
3352 if (*m.begin() > _ex1) {
3358 } else if (*m.begin() < _ex_1) {
3364 } else if (*m.begin() == _ex0) {
3371 for (auto it = ++m.begin(); it != m.end(); it++) {
3372 if (it->info(info_flags::integer)) {
3383 } else if (*it < _ex_1) {
3403 } else if (step == 1) {
3415 // if some m_i is not an integer
3416 return H(m_, x).hold();
3419 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3420 return convert_H_to_zeta(m);
3426 return H(m_, x).hold();
3428 return pow(log(x), m.nops()) / factorial(m.nops());
3431 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3433 } else if ((step == 1) && (pos1 == _ex0)){
3438 return pow(-1, p) * S(n, p, -x);
3444 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3445 return H(m_, x).evalf();
3447 return H(m_, x).hold();
3451 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3453 epvector seq { expair(H(m, x), 0) };
3454 return pseries(rel, std::move(seq));
3458 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3460 GINAC_ASSERT(deriv_param < 2);
3461 if (deriv_param == 0) {
3465 if (is_a<lst>(m_)) {
3481 return 1/(1-x) * H(m, x);
3482 } else if (mb == _ex_1) {
3483 return 1/(1+x) * H(m, x);
3490 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3493 if (is_a<lst>(m_)) {
3498 c.s << "\\mathrm{H}_{";
3499 auto itm = m.begin();
3502 for (; itm != m.end(); itm++) {
3512 REGISTER_FUNCTION(H,
3513 evalf_func(H_evalf).
3515 series_func(H_series).
3516 derivative_func(H_deriv).
3517 print_func<print_latex>(H_print_latex).
3518 do_not_evalf_params());
3521 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3522 ex convert_H_to_Li(const ex& m, const ex& x)
3524 map_trafo_H_reduce_trailing_zeros filter;
3525 map_trafo_H_convert_to_Li filter2;
3527 return filter2(filter(H(m, x).hold()));
3529 return filter2(filter(H(lst{m}, x).hold()));
3534 //////////////////////////////////////////////////////////////////////
3536 // Multiple zeta values zeta(x) and zeta(x,s)
3540 //////////////////////////////////////////////////////////////////////
3543 // anonymous namespace for helper functions
3547 // parameters and data for [Cra] algorithm
3548 const cln::cl_N lambda = cln::cl_N("319/320");
3550 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3552 const int size = a.size();
3553 for (int n=0; n<size; n++) {
3555 for (int m=0; m<=n; m++) {
3556 c[n] = c[n] + a[m]*b[n-m];
3563 static void initcX(std::vector<cln::cl_N>& crX,
3564 const std::vector<int>& s,
3567 std::vector<cln::cl_N> crB(L2 + 1);
3568 for (int i=0; i<=L2; i++)
3569 crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i);
3573 std::vector<std::vector<cln::cl_N>> crG(s.size() - 1, std::vector<cln::cl_N>(L2 + 1));
3574 for (int m=0; m < (int)s.size() - 1; m++) {
3577 for (int i = 0; i <= L2; i++)
3578 crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2);
3583 for (std::size_t m = 0; m < s.size() - 1; m++) {
3584 std::vector<cln::cl_N> Xbuf(L2 + 1);
3585 for (int i = 0; i <= L2; i++)
3586 Xbuf[i] = crX[i] * crG[m][i];
3588 halfcyclic_convolute(Xbuf, crB, crX);
3594 static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk,
3595 const std::vector<cln::cl_N>& crX)
3597 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3598 cln::cl_N factor = cln::expt(lambda, Sqk);
3599 cln::cl_N res = factor / Sqk * crX[0] * one;
3604 factor = factor * lambda;
3606 res = res + crX[N] * factor / (N+Sqk);
3607 } while (((res != resbuf) || cln::zerop(crX[N])) && (N+1 < crX.size()));
3613 static void calc_f(std::vector<std::vector<cln::cl_N>>& f_kj,
3614 const int maxr, const int L1)
3616 cln::cl_N t0, t1, t2, t3, t4;
3618 auto it = f_kj.begin();
3619 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3621 t0 = cln::exp(-lambda);
3623 for (k=1; k<=L1; k++) {
3626 for (j=1; j<=maxr; j++) {
3629 for (i=2; i<=j; i++) {
3633 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3641 static cln::cl_N crandall_Z(const std::vector<int>& s,
3642 const std::vector<std::vector<cln::cl_N>>& f_kj)
3644 const int j = s.size();
3653 t0 = t0 + f_kj[q+j-2][s[0]-1];
3654 } while ((t0 != t0buf) && (q+j-1 < f_kj.size()));
3656 return t0 / cln::factorial(s[0]-1);
3659 std::vector<cln::cl_N> t(j);
3666 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3667 for (int k=j-2; k>=1; k--) {
3668 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3670 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3671 } while ((t[0] != t0buf) && (q+j-1 < f_kj.size()));
3673 return t[0] / cln::factorial(s[0]-1);
3678 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3680 std::vector<int> r = s;
3681 const int j = r.size();
3685 // decide on maximal size of f_kj for crandall_Z
3689 L1 = Digits * 3 + j*2;
3693 // decide on maximal size of crX for crandall_Y
3696 } else if (Digits < 86) {
3698 } else if (Digits < 192) {
3700 } else if (Digits < 394) {
3702 } else if (Digits < 808) {
3704 } else if (Digits < 1636) {
3707 // [Cra] section 6, log10(lambda/2/Pi) approx -0.79 for lambda=319/320, add some extra digits
3708 L2 = std::pow(2, ceil( std::log2((long(Digits))/0.79 + 40 )) ) - 1;
3715 for (int i=0; i<j; i++) {
3722 std::vector<std::vector<cln::cl_N>> f_kj(L1);
3723 calc_f(f_kj, maxr, L1);
3725 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3727 std::vector<int> rz;
3730 for (int k=r.size()-1; k>0; k--) {
3732 rz.insert(rz.begin(), r.back());
3733 skp1buf = rz.front();
3737 std::vector<cln::cl_N> crX;
3740 for (int q=0; q<skp1buf; q++) {
3742 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k, crX);
3743 cln::cl_N pp2 = crandall_Z(rz, f_kj);
3748 res = res - pp1 * pp2 / cln::factorial(q);
3750 res = res + pp1 * pp2 / cln::factorial(q);
3753 rz.front() = skp1buf;
3755 rz.insert(rz.begin(), r.back());
3757 std::vector<cln::cl_N> crX;
3758 initcX(crX, rz, L2);
3760 res = (res + crandall_Y_loop(S-j, crX)) / r0factorial
3761 + crandall_Z(rz, f_kj);
3767 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3769 const int j = r.size();
3771 // buffer for subsums
3772 std::vector<cln::cl_N> t(j);
3773 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3780 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3781 for (int k=j-2; k>=0; k--) {
3782 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3784 } while (t[0] != t0buf);
3790 // does Hoelder convolution. see [BBB] (7.0)
3791 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3793 // prepare parameters
3794 // holds Li arguments in [BBB] notation
3795 std::vector<int> s = s_;
3796 std::vector<int> m_p = m_;
3797 std::vector<int> m_q;
3798 // holds Li arguments in nested sums notation
3799 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3800 s_p[0] = s_p[0] * cln::cl_N("1/2");
3801 // convert notations
3803 for (std::size_t i = 0; i < s_.size(); i++) {
3808 s[i] = sig * std::abs(s[i]);
3810 std::vector<cln::cl_N> s_q;
3811 cln::cl_N signum = 1;
3814 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3819 // change parameters
3820 if (s.front() > 0) {
3821 if (m_p.front() == 1) {
3822 m_p.erase(m_p.begin());
3823 s_p.erase(s_p.begin());
3824 if (s_p.size() > 0) {
3825 s_p.front() = s_p.front() * cln::cl_N("1/2");
3831 m_q.insert(m_q.begin(), 1);
3832 if (s_q.size() > 0) {
3833 s_q.front() = s_q.front() * 2;
3835 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3838 if (m_p.front() == 1) {
3839 m_p.erase(m_p.begin());
3840 cln::cl_N spbuf = s_p.front();
3841 s_p.erase(s_p.begin());
3842 if (s_p.size() > 0) {
3843 s_p.front() = s_p.front() * spbuf;
3846 m_q.insert(m_q.begin(), 1);
3847 if (s_q.size() > 0) {
3848 s_q.front() = s_q.front() * 4;
3850 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3854 m_q.insert(m_q.begin(), 1);
3855 if (s_q.size() > 0) {
3856 s_q.front() = s_q.front() * 2;
3858 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3863 if (m_p.size() == 0) break;
3865 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3870 res = res + signum * multipleLi_do_sum(m_q, s_q);
3876 } // end of anonymous namespace
3879 //////////////////////////////////////////////////////////////////////
3881 // Multiple zeta values zeta(x)
3885 //////////////////////////////////////////////////////////////////////
3888 static ex zeta1_evalf(const ex& x)
3890 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3892 // multiple zeta value
3893 const int count = x.nops();
3894 const lst& xlst = ex_to<lst>(x);
3895 std::vector<int> r(count);
3896 std::vector<int> si(count);
3898 // check parameters and convert them
3899 auto it1 = xlst.begin();
3900 auto it2 = r.begin();
3901 auto it_swrite = si.begin();
3903 if (!(*it1).info(info_flags::posint)) {
3904 return zeta(x).hold();
3906 *it2 = ex_to<numeric>(*it1).to_int();
3911 } while (it2 != r.end());
3913 // check for divergence
3915 return zeta(x).hold();
3918 // use Hoelder convolution if Digits is large
3920 return numeric(zeta_do_Hoelder_convolution(r, si));
3922 // decide on summation algorithm
3923 // this is still a bit clumsy
3924 int limit = (Digits>17) ? 10 : 6;
3925 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3926 return numeric(zeta_do_sum_Crandall(r));
3928 return numeric(zeta_do_sum_simple(r));
3932 // single zeta value
3933 if (is_exactly_a<numeric>(x) && (x != 1)) {
3935 return zeta(ex_to<numeric>(x));
3936 } catch (const dunno &e) { }
3939 return zeta(x).hold();
3943 static ex zeta1_eval(const ex& m)
3945 if (is_exactly_a<lst>(m)) {
3946 if (m.nops() == 1) {
3947 return zeta(m.op(0));
3949 return zeta(m).hold();
3952 if (m.info(info_flags::numeric)) {
3953 const numeric& y = ex_to<numeric>(m);
3954 // trap integer arguments:
3955 if (y.is_integer()) {
3959 if (y.is_equal(*_num1_p)) {
3960 return zeta(m).hold();
3962 if (y.info(info_flags::posint)) {
3963 if (y.info(info_flags::odd)) {
3964 return zeta(m).hold();
3966 return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3969 if (y.info(info_flags::odd)) {
3970 return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3977 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3978 return zeta1_evalf(m);
3981 return zeta(m).hold();
3985 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3987 GINAC_ASSERT(deriv_param==0);
3989 if (is_exactly_a<lst>(m)) {
3992 return zetaderiv(_ex1, m);
3997 static void zeta1_print_latex(const ex& m_, const print_context& c)
4000 if (is_a<lst>(m_)) {
4001 const lst& m = ex_to<lst>(m_);
4002 auto it = m.begin();
4005 for (; it != m.end(); it++) {
4016 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
4017 evalf_func(zeta1_evalf).
4018 eval_func(zeta1_eval).
4019 derivative_func(zeta1_deriv).
4020 print_func<print_latex>(zeta1_print_latex).
4021 do_not_evalf_params().
4025 //////////////////////////////////////////////////////////////////////
4027 // Alternating Euler sum zeta(x,s)
4031 //////////////////////////////////////////////////////////////////////
4034 static ex zeta2_evalf(const ex& x, const ex& s)
4036 if (is_exactly_a<lst>(x)) {
4038 // alternating Euler sum
4039 const int count = x.nops();
4040 const lst& xlst = ex_to<lst>(x);
4041 const lst& slst = ex_to<lst>(s);
4042 std::vector<int> xi(count);
4043 std::vector<int> si(count);
4045 // check parameters and convert them
4046 auto it_xread = xlst.begin();
4047 auto it_sread = slst.begin();
4048 auto it_xwrite = xi.begin();
4049 auto it_swrite = si.begin();
4051 if (!(*it_xread).info(info_flags::posint)) {
4052 return zeta(x, s).hold();
4054 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
4055 if (*it_sread > 0) {
4064 } while (it_xwrite != xi.end());
4066 // check for divergence
4067 if ((xi[0] == 1) && (si[0] == 1)) {
4068 return zeta(x, s).hold();
4071 // use Hoelder convolution
4072 return numeric(zeta_do_Hoelder_convolution(xi, si));
4075 // x and s are not lists: convert to lists
4076 return zeta(lst{x}, lst{s}).evalf();
4080 static ex zeta2_eval(const ex& m, const ex& s_)
4082 if (is_exactly_a<lst>(s_)) {
4083 const lst& s = ex_to<lst>(s_);
4084 for (const auto & it : s) {
4085 if (it.info(info_flags::positive)) {
4088 return zeta(m, s_).hold();
4091 } else if (s_.info(info_flags::positive)) {
4095 return zeta(m, s_).hold();
4099 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
4101 GINAC_ASSERT(deriv_param==0);
4103 if (is_exactly_a<lst>(m)) {
4106 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
4107 return zetaderiv(_ex1, m);
4114 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
4117 if (is_a<lst>(m_)) {
4123 if (is_a<lst>(s_)) {
4129 auto itm = m.begin();
4130 auto its = s.begin();
4132 c.s << "\\overline{";
4140 for (; itm != m.end(); itm++, its++) {
4143 c.s << "\\overline{";
4154 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
4155 evalf_func(zeta2_evalf).
4156 eval_func(zeta2_eval).
4157 derivative_func(zeta2_deriv).
4158 print_func<print_latex>(zeta2_print_latex).
4159 do_not_evalf_params().
4163 } // namespace GiNaC