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) 1999-2020 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 for (std::size_t r = 0; r <= size; ++r) {
1017 cln::cl_N buffer(1 & r ? -1 : 1);
1022 for (std::size_t i = 0; i < size; ++i) {
1023 if (x[i] == cln::cl_RA(1)/p) {
1024 p = p/2 + cln::cl_RA(3)/2;
1030 cln::cl_RA q = p/(p-1);
1031 std::vector<cln::cl_N> qlstx;
1032 std::vector<int> qlsts;
1033 for (std::size_t j = r; j >= 1; --j) {
1034 qlstx.push_back(cln::cl_N(1) - x[j-1]);
1035 if (imagpart(x[j-1])==0 && realpart(x[j-1]) >= 1) {
1038 qlsts.push_back(-s[j-1]);
1041 if (qlstx.size() > 0) {
1042 buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
1044 std::vector<cln::cl_N> plstx;
1045 std::vector<int> plsts;
1046 for (std::size_t j = r+1; j <= size; ++j) {
1047 plstx.push_back(x[j-1]);
1048 plsts.push_back(s[j-1]);
1050 if (plstx.size() > 0) {
1051 buffer = buffer*G_numeric(plstx, plsts, 1/p);
1053 result = result + buffer;
1058 class less_object_for_cl_N
1061 bool operator() (const cln::cl_N & a, const cln::cl_N & b) const
1064 if (abs(a) != abs(b))
1065 return (abs(a) < abs(b)) ? true : false;
1068 if (phase(a) != phase(b))
1069 return (phase(a) < phase(b)) ? true : false;
1071 // equal, therefore "less" is not true
1077 // convergence transformation, used for numerical evaluation of G function.
1078 // the parameter x, s and y must only contain numerics
1080 G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1081 const cln::cl_N& y, bool flag_trailing_zeros_only)
1083 // sort (|x|<->position) to determine indices
1084 typedef std::multimap<cln::cl_N, std::size_t, less_object_for_cl_N> sortmap_t;
1086 std::size_t size = 0;
1087 for (std::size_t i = 0; i < x.size(); ++i) {
1089 sortmap.insert(std::make_pair(x[i], i));
1093 // include upper limit (scale)
1094 sortmap.insert(std::make_pair(y, x.size()));
1096 // generate missing dummy-symbols
1098 // holding dummy-symbols for the G/Li transformations
1100 gsyms.push_back(symbol("GSYMS_ERROR"));
1101 cln::cl_N lastentry(0);
1102 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1103 if (it != sortmap.begin()) {
1104 if (it->second < x.size()) {
1105 if (x[it->second] == lastentry) {
1106 gsyms.push_back(gsyms.back());
1110 if (y == lastentry) {
1111 gsyms.push_back(gsyms.back());
1116 std::ostringstream os;
1118 gsyms.push_back(symbol(os.str()));
1120 if (it->second < x.size()) {
1121 lastentry = x[it->second];
1127 // fill position data according to sorted indices and prepare substitution list
1128 Gparameter a(x.size());
1130 std::size_t pos = 1;
1132 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1133 if (it->second < x.size()) {
1134 if (s[it->second] > 0) {
1135 a[it->second] = pos;
1137 a[it->second] = -int(pos);
1139 subslst[gsyms[pos]] = numeric(x[it->second]);
1142 subslst[gsyms[pos]] = numeric(y);
1147 // do transformation
1149 ex result = G_transform(pendint, a, scale, gsyms, flag_trailing_zeros_only);
1150 // replace dummy symbols with their values
1151 result = result.expand();
1152 result = result.subs(subslst).evalf();
1153 if (!is_a<numeric>(result))
1154 throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
1156 cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
1160 // handles the transformations and the numerical evaluation of G
1161 // the parameter x, s and y must only contain numerics
1163 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1166 // check for convergence and necessary accelerations
1167 bool need_trafo = false;
1168 bool need_hoelder = false;
1169 bool have_trailing_zero = false;
1170 std::size_t depth = 0;
1171 for (auto & xi : x) {
1174 const cln::cl_N x_y = abs(xi) - y;
1175 if (instanceof(x_y, cln::cl_R_ring) &&
1176 realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2)))
1179 if (abs(abs(xi/y) - 1) < 0.01)
1180 need_hoelder = true;
1183 if (zerop(x.back())) {
1184 have_trailing_zero = true;
1188 if (depth == 1 && x.size() == 2 && !need_trafo)
1189 return - Li_projection(2, y/x[1], cln::float_format(Digits));
1191 // do acceleration transformation (hoelder convolution [BBB])
1192 if (need_hoelder && !have_trailing_zero)
1193 return G_do_hoelder(x, s, y);
1195 // convergence transformation
1197 return G_do_trafo(x, s, y, have_trailing_zero);
1200 std::vector<cln::cl_N> newx;
1201 newx.reserve(x.size());
1203 m.reserve(x.size());
1206 cln::cl_N factor = y;
1207 for (auto & xi : x) {
1211 newx.push_back(factor/xi);
1213 m.push_back(mcount);
1219 return sign*multipleLi_do_sum(m, newx);
1223 ex mLi_numeric(const lst& m, const lst& x)
1225 // let G_numeric do the transformation
1226 std::vector<cln::cl_N> newx;
1227 newx.reserve(x.nops());
1229 s.reserve(x.nops());
1230 cln::cl_N factor(1);
1231 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1232 for (int i = 1; i < *itm; ++i) {
1233 newx.push_back(cln::cl_N(0));
1236 const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
1238 newx.push_back(factor);
1239 if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) {
1246 return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
1250 } // end of anonymous namespace
1253 //////////////////////////////////////////////////////////////////////
1255 // Generalized multiple polylogarithm G(x, y) and G(x, s, y)
1259 //////////////////////////////////////////////////////////////////////
1262 static ex G2_evalf(const ex& x_, const ex& y)
1264 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1265 return G(x_, y).hold();
1267 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1268 if (x.nops() == 0) {
1272 return G(x_, y).hold();
1275 s.reserve(x.nops());
1276 bool all_zero = true;
1277 for (const auto & it : x) {
1278 if (!it.info(info_flags::numeric)) {
1279 return G(x_, y).hold();
1284 if ( !ex_to<numeric>(it).is_real() && ex_to<numeric>(it).imag() < 0 ) {
1292 return pow(log(y), x.nops()) / factorial(x.nops());
1294 std::vector<cln::cl_N> xv;
1295 xv.reserve(x.nops());
1296 for (const auto & it : x)
1297 xv.push_back(ex_to<numeric>(it).to_cl_N());
1298 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1299 return numeric(result);
1303 static ex G2_eval(const ex& x_, const ex& y)
1305 //TODO eval to MZV or H or S or Lin
1307 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1308 return G(x_, y).hold();
1310 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1311 if (x.nops() == 0) {
1315 return G(x_, y).hold();
1318 s.reserve(x.nops());
1319 bool all_zero = true;
1320 bool crational = true;
1321 for (const auto & it : x) {
1322 if (!it.info(info_flags::numeric)) {
1323 return G(x_, y).hold();
1325 if (!it.info(info_flags::crational)) {
1331 if ( !ex_to<numeric>(it).is_real() && ex_to<numeric>(it).imag() < 0 ) {
1339 return pow(log(y), x.nops()) / factorial(x.nops());
1341 if (!y.info(info_flags::crational)) {
1345 return G(x_, y).hold();
1347 std::vector<cln::cl_N> xv;
1348 xv.reserve(x.nops());
1349 for (const auto & it : x)
1350 xv.push_back(ex_to<numeric>(it).to_cl_N());
1351 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1352 return numeric(result);
1356 // option do_not_evalf_params() removed.
1357 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1358 evalf_func(G2_evalf).
1362 // derivative_func(G2_deriv).
1363 // print_func<print_latex>(G2_print_latex).
1366 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1368 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1369 return G(x_, s_, y).hold();
1371 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1372 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst{s_};
1373 if (x.nops() != s.nops()) {
1374 return G(x_, s_, y).hold();
1376 if (x.nops() == 0) {
1380 return G(x_, s_, y).hold();
1382 std::vector<int> sn;
1383 sn.reserve(s.nops());
1384 bool all_zero = true;
1385 for (auto itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1386 if (!(*itx).info(info_flags::numeric)) {
1387 return G(x_, y).hold();
1389 if (!(*its).info(info_flags::real)) {
1390 return G(x_, y).hold();
1395 if ( ex_to<numeric>(*itx).is_real() ) {
1396 if ( ex_to<numeric>(*itx).is_positive() ) {
1408 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1417 return pow(log(y), x.nops()) / factorial(x.nops());
1419 std::vector<cln::cl_N> xn;
1420 xn.reserve(x.nops());
1421 for (const auto & it : x)
1422 xn.push_back(ex_to<numeric>(it).to_cl_N());
1423 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1424 return numeric(result);
1428 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1430 //TODO eval to MZV or H or S or Lin
1432 if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1433 return G(x_, s_, y).hold();
1435 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst{x_};
1436 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst{s_};
1437 if (x.nops() != s.nops()) {
1438 return G(x_, s_, y).hold();
1440 if (x.nops() == 0) {
1444 return G(x_, s_, y).hold();
1446 std::vector<int> sn;
1447 sn.reserve(s.nops());
1448 bool all_zero = true;
1449 bool crational = true;
1450 for (auto itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1451 if (!(*itx).info(info_flags::numeric)) {
1452 return G(x_, s_, y).hold();
1454 if (!(*its).info(info_flags::real)) {
1455 return G(x_, s_, y).hold();
1457 if (!(*itx).info(info_flags::crational)) {
1463 if ( ex_to<numeric>(*itx).is_real() ) {
1464 if ( ex_to<numeric>(*itx).is_positive() ) {
1476 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1485 return pow(log(y), x.nops()) / factorial(x.nops());
1487 if (!y.info(info_flags::crational)) {
1491 return G(x_, s_, y).hold();
1493 std::vector<cln::cl_N> xn;
1494 xn.reserve(x.nops());
1495 for (const auto & it : x)
1496 xn.push_back(ex_to<numeric>(it).to_cl_N());
1497 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1498 return numeric(result);
1502 // option do_not_evalf_params() removed.
1503 // This is safe: in the code above it only matters if s_ > 0 or s_ < 0,
1504 // s_ is allowed to be of floating type.
1505 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1506 evalf_func(G3_evalf).
1510 // derivative_func(G3_deriv).
1511 // print_func<print_latex>(G3_print_latex).
1514 //////////////////////////////////////////////////////////////////////
1516 // Classical polylogarithm and multiple polylogarithm Li(m,x)
1520 //////////////////////////////////////////////////////////////////////
1523 static ex Li_evalf(const ex& m_, const ex& x_)
1525 // classical polylogs
1526 if (m_.info(info_flags::posint)) {
1527 if (x_.info(info_flags::numeric)) {
1528 int m__ = ex_to<numeric>(m_).to_int();
1529 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1530 const cln::cl_N result = Lin_numeric(m__, x__);
1531 return numeric(result);
1533 // try to numerically evaluate second argument
1534 ex x_val = x_.evalf();
1535 if (x_val.info(info_flags::numeric)) {
1536 int m__ = ex_to<numeric>(m_).to_int();
1537 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1538 const cln::cl_N result = Lin_numeric(m__, x__);
1539 return numeric(result);
1543 // multiple polylogs
1544 if (is_a<lst>(m_) && is_a<lst>(x_)) {
1546 const lst& m = ex_to<lst>(m_);
1547 const lst& x = ex_to<lst>(x_);
1548 if (m.nops() != x.nops()) {
1549 return Li(m_,x_).hold();
1551 if (x.nops() == 0) {
1554 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1555 return Li(m_,x_).hold();
1558 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1559 if (!(*itm).info(info_flags::posint)) {
1560 return Li(m_, x_).hold();
1562 if (!(*itx).info(info_flags::numeric)) {
1563 return Li(m_, x_).hold();
1570 return mLi_numeric(m, x);
1573 return Li(m_,x_).hold();
1577 static ex Li_eval(const ex& m_, const ex& x_)
1579 if (is_a<lst>(m_)) {
1580 if (is_a<lst>(x_)) {
1581 // multiple polylogs
1582 const lst& m = ex_to<lst>(m_);
1583 const lst& x = ex_to<lst>(x_);
1584 if (m.nops() != x.nops()) {
1585 return Li(m_,x_).hold();
1587 if (x.nops() == 0) {
1591 bool is_zeta = true;
1592 bool do_evalf = true;
1593 bool crational = true;
1594 for (auto itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1595 if (!(*itm).info(info_flags::posint)) {
1596 return Li(m_,x_).hold();
1598 if ((*itx != _ex1) && (*itx != _ex_1)) {
1599 if (itx != x.begin()) {
1607 if (!(*itx).info(info_flags::numeric)) {
1610 if (!(*itx).info(info_flags::crational)) {
1616 for (const auto & itx : x) {
1617 GINAC_ASSERT((itx == _ex1) || (itx == _ex_1));
1618 // XXX: 1 + 0.0*I is considered equal to 1. However
1619 // the former is a not automatically converted
1620 // to a real number. Do the conversion explicitly
1621 // to avoid the "numeric::operator>(): complex inequality"
1622 // exception (and similar problems).
1623 newx.append(itx != _ex_1 ? _ex1 : _ex_1);
1625 return zeta(m_, newx);
1629 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1630 return prefactor * H(newm, x[0]);
1632 if (do_evalf && !crational) {
1633 return mLi_numeric(m,x);
1636 return Li(m_, x_).hold();
1637 } else if (is_a<lst>(x_)) {
1638 return Li(m_, x_).hold();
1641 // classical polylogs
1649 return (pow(2,1-m_)-1) * zeta(m_);
1655 if (x_.is_equal(I)) {
1656 return power(Pi,_ex2)/_ex_48 + Catalan*I;
1658 if (x_.is_equal(-I)) {
1659 return power(Pi,_ex2)/_ex_48 - Catalan*I;
1662 if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1663 int m__ = ex_to<numeric>(m_).to_int();
1664 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1665 const cln::cl_N result = Lin_numeric(m__, x__);
1666 return numeric(result);
1669 return Li(m_, x_).hold();
1673 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1675 if (is_a<lst>(m) || is_a<lst>(x)) {
1677 epvector seq { expair(Li(m, x), 0) };
1678 return pseries(rel, std::move(seq));
1681 // classical polylog
1682 const ex x_pt = x.subs(rel, subs_options::no_pattern);
1683 if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1684 // First special case: x==0 (derivatives have poles)
1685 if (x_pt.is_zero()) {
1688 // manually construct the primitive expansion
1689 for (int i=1; i<order; ++i)
1690 ser += pow(s,i) / pow(numeric(i), m);
1691 // substitute the argument's series expansion
1692 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1693 // maybe that was terminating, so add a proper order term
1694 epvector nseq { expair(Order(_ex1), order) };
1695 ser += pseries(rel, std::move(nseq));
1696 // reexpanding it will collapse the series again
1697 return ser.series(rel, order);
1699 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1700 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1702 // all other cases should be safe, by now:
1703 throw do_taylor(); // caught by function::series()
1707 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1709 GINAC_ASSERT(deriv_param < 2);
1710 if (deriv_param == 0) {
1713 if (m_.nops() > 1) {
1714 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1717 if (is_a<lst>(m_)) {
1723 if (is_a<lst>(x_)) {
1729 return Li(m-1, x) / x;
1736 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1739 if (is_a<lst>(m_)) {
1745 if (is_a<lst>(x_)) {
1750 c.s << "\\mathrm{Li}_{";
1751 auto itm = m.begin();
1754 for (; itm != m.end(); itm++) {
1759 auto itx = x.begin();
1762 for (; itx != x.end(); itx++) {
1770 REGISTER_FUNCTION(Li,
1771 evalf_func(Li_evalf).
1773 series_func(Li_series).
1774 derivative_func(Li_deriv).
1775 print_func<print_latex>(Li_print_latex).
1776 do_not_evalf_params());
1779 //////////////////////////////////////////////////////////////////////
1781 // Nielsen's generalized polylogarithm S(n,p,x)
1785 //////////////////////////////////////////////////////////////////////
1788 // anonymous namespace for helper functions
1792 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1794 std::vector<std::vector<cln::cl_N>> Yn;
1795 int ynsize = 0; // number of Yn[]
1796 int ynlength = 100; // initial length of all Yn[i]
1799 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1800 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1801 // representing S_{n,p}(x).
1802 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1803 // equivalent Z-sum.
1804 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1805 // representing S_{n,p}(x).
1806 // The calculation of Y_n uses the values from Y_{n-1}.
1807 void fill_Yn(int n, const cln::float_format_t& prec)
1809 const int initsize = ynlength;
1810 //const int initsize = initsize_Yn;
1811 cln::cl_N one = cln::cl_float(1, prec);
1814 std::vector<cln::cl_N> buf(initsize);
1815 auto it = buf.begin();
1816 auto itprev = Yn[n-1].begin();
1817 *it = (*itprev) / cln::cl_N(n+1) * one;
1820 // sums with an index smaller than the depth are zero and need not to be calculated.
1821 // calculation starts with depth, which is n+2)
1822 for (int i=n+2; i<=initsize+n; i++) {
1823 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1829 std::vector<cln::cl_N> buf(initsize);
1830 auto it = buf.begin();
1833 for (int i=2; i<=initsize; i++) {
1834 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1843 // make Yn longer ...
1844 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1847 cln::cl_N one = cln::cl_float(1, prec);
1849 Yn[0].resize(newsize);
1850 auto it = Yn[0].begin();
1852 for (int i=ynlength+1; i<=newsize; i++) {
1853 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1857 for (int n=1; n<ynsize; n++) {
1858 Yn[n].resize(newsize);
1859 auto it = Yn[n].begin();
1860 auto itprev = Yn[n-1].begin();
1863 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1864 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1874 // helper function for S(n,p,x)
1876 cln::cl_N C(int n, int p)
1880 for (int k=0; k<p; k++) {
1881 for (int j=0; j<=(n+k-1)/2; j++) {
1885 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1888 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1895 result = result + cln::factorial(n+k-1)
1896 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1897 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1900 result = result - cln::factorial(n+k-1)
1901 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1902 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1907 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1908 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1911 result = result + cln::factorial(n+k-1)
1912 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1913 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1921 if (((np)/2+n) & 1) {
1922 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1925 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1933 // helper function for S(n,p,x)
1934 // [Kol] remark to (9.1)
1935 cln::cl_N a_k(int k)
1944 for (int m=2; m<=k; m++) {
1945 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1952 // helper function for S(n,p,x)
1953 // [Kol] remark to (9.1)
1954 cln::cl_N b_k(int k)
1963 for (int m=2; m<=k; m++) {
1964 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1971 // helper function for S(n,p,x)
1972 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1974 static cln::float_format_t oldprec = cln::default_float_format;
1977 return Li_projection(n+1, x, prec);
1980 // precision has changed, we need to clear lookup table Yn
1981 if ( oldprec != prec ) {
1988 // check if precalculated values are sufficient
1990 for (int i=ynsize; i<p-1; i++) {
1995 // should be done otherwise
1996 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1997 cln::cl_N xf = x * one;
1998 //cln::cl_N xf = x * cln::cl_float(1, prec);
2002 cln::cl_N factor = cln::expt(xf, p);
2006 if (i-p >= ynlength) {
2008 make_Yn_longer(ynlength*2, prec);
2010 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? ...
2011 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
2012 factor = factor * xf;
2014 } while (res != resbuf);
2020 // helper function for S(n,p,x)
2021 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
2024 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
2026 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
2027 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
2029 for (int s=0; s<n; s++) {
2031 for (int r=0; r<p; r++) {
2032 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
2033 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
2035 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2041 return S_do_sum(n, p, x, prec);
2045 // helper function for S(n,p,x)
2046 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
2050 // [Kol] (2.22) with (2.21)
2051 return cln::zeta(p+1);
2056 return cln::zeta(n+1);
2061 for (int nu=0; nu<n; nu++) {
2062 for (int rho=0; rho<=p; rho++) {
2063 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2064 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2067 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2074 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2076 // throw std::runtime_error("don't know how to evaluate this function!");
2079 // what is the desired float format?
2080 // first guess: default format
2081 cln::float_format_t prec = cln::default_float_format;
2082 const cln::cl_N value = x;
2083 // second guess: the argument's format
2084 if (!instanceof(realpart(value), cln::cl_RA_ring))
2085 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2086 else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2087 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2090 // 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
2091 // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
2092 if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
2094 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2095 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2097 for (int s=0; s<n; s++) {
2099 for (int r=0; r<p; r++) {
2100 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2101 * S_num(p-r,n-s,1-value) / cln::factorial(r);
2103 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2110 if (cln::abs(value) > 1) {
2114 for (int s=0; s<p; s++) {
2115 for (int r=0; r<=s; r++) {
2116 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2117 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2118 * S_num(n+s-r,p-s,cln::recip(value));
2121 result = result * cln::expt(cln::cl_I(-1),n);
2124 for (int r=0; r<n; r++) {
2125 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2127 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2129 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2134 if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
2137 for (int s=0; s<p-1; s++)
2140 ex res = H(m,numeric(value)).evalf();
2141 return ex_to<numeric>(res).to_cl_N();
2144 return S_projection(n, p, value, prec);
2149 } // end of anonymous namespace
2152 //////////////////////////////////////////////////////////////////////
2154 // Nielsen's generalized polylogarithm S(n,p,x)
2158 //////////////////////////////////////////////////////////////////////
2161 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2163 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2164 const int n_ = ex_to<numeric>(n).to_int();
2165 const int p_ = ex_to<numeric>(p).to_int();
2166 if (is_a<numeric>(x)) {
2167 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2168 const cln::cl_N result = S_num(n_, p_, x_);
2169 return numeric(result);
2171 ex x_val = x.evalf();
2172 if (is_a<numeric>(x_val)) {
2173 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2174 const cln::cl_N result = S_num(n_, p_, x_val_);
2175 return numeric(result);
2179 return S(n, p, x).hold();
2183 static ex S_eval(const ex& n, const ex& p, const ex& x)
2185 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2191 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2199 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2200 int n_ = ex_to<numeric>(n).to_int();
2201 int p_ = ex_to<numeric>(p).to_int();
2202 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2203 const cln::cl_N result = S_num(n_, p_, x_);
2204 return numeric(result);
2209 return pow(-log(1-x), p) / factorial(p);
2211 return S(n, p, x).hold();
2215 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2218 return Li(n+1, x).series(rel, order, options);
2221 const ex x_pt = x.subs(rel, subs_options::no_pattern);
2222 if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2223 // First special case: x==0 (derivatives have poles)
2224 if (x_pt.is_zero()) {
2227 // manually construct the primitive expansion
2228 // subsum = Euler-Zagier-Sum is needed
2229 // dirty hack (slow ...) calculation of subsum:
2230 std::vector<ex> presubsum, subsum;
2231 subsum.push_back(0);
2232 for (int i=1; i<order-1; ++i) {
2233 subsum.push_back(subsum[i-1] + numeric(1, i));
2235 for (int depth=2; depth<p; ++depth) {
2237 for (int i=1; i<order-1; ++i) {
2238 subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2242 for (int i=1; i<order; ++i) {
2243 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2245 // substitute the argument's series expansion
2246 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2247 // maybe that was terminating, so add a proper order term
2248 epvector nseq { expair(Order(_ex1), order) };
2249 ser += pseries(rel, std::move(nseq));
2250 // reexpanding it will collapse the series again
2251 return ser.series(rel, order);
2253 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2254 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2256 // all other cases should be safe, by now:
2257 throw do_taylor(); // caught by function::series()
2261 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2263 GINAC_ASSERT(deriv_param < 3);
2264 if (deriv_param < 2) {
2268 return S(n-1, p, x) / x;
2270 return S(n, p-1, x) / (1-x);
2275 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2277 c.s << "\\mathrm{S}_{";
2287 REGISTER_FUNCTION(S,
2288 evalf_func(S_evalf).
2290 series_func(S_series).
2291 derivative_func(S_deriv).
2292 print_func<print_latex>(S_print_latex).
2293 do_not_evalf_params());
2296 //////////////////////////////////////////////////////////////////////
2298 // Harmonic polylogarithm H(m,x)
2302 //////////////////////////////////////////////////////////////////////
2305 // anonymous namespace for helper functions
2309 // regulates the pole (used by 1/x-transformation)
2310 symbol H_polesign("IMSIGN");
2313 // convert parameters from H to Li representation
2314 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2315 // returns true if some parameters are negative
2316 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2318 // expand parameter list
2320 for (const auto & it : l) {
2322 for (ex count=it-1; count > 0; count--) {
2326 } else if (it < -1) {
2327 for (ex count=it+1; count < 0; count++) {
2338 bool has_negative_parameters = false;
2340 for (const auto & it : mexp) {
2346 m.append((it+acc-1) * signum);
2348 m.append((it-acc+1) * signum);
2354 has_negative_parameters = true;
2357 if (has_negative_parameters) {
2358 for (std::size_t i=0; i<m.nops(); i++) {
2360 m.let_op(i) = -m.op(i);
2368 return has_negative_parameters;
2372 // recursivly transforms H to corresponding multiple polylogarithms
2373 struct map_trafo_H_convert_to_Li : public map_function
2375 ex operator()(const ex& e) override
2377 if (is_a<add>(e) || is_a<mul>(e)) {
2378 return e.map(*this);
2380 if (is_a<function>(e)) {
2381 std::string name = ex_to<function>(e).get_name();
2384 if (is_a<lst>(e.op(0))) {
2385 parameter = ex_to<lst>(e.op(0));
2387 parameter = lst{e.op(0)};
2394 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2395 s.let_op(0) = s.op(0) * arg;
2396 return pf * Li(m, s).hold();
2398 for (std::size_t i=0; i<m.nops(); i++) {
2401 s.let_op(0) = s.op(0) * arg;
2402 return Li(m, s).hold();
2411 // recursivly transforms H to corresponding zetas
2412 struct map_trafo_H_convert_to_zeta : public map_function
2414 ex operator()(const ex& e) override
2416 if (is_a<add>(e) || is_a<mul>(e)) {
2417 return e.map(*this);
2419 if (is_a<function>(e)) {
2420 std::string name = ex_to<function>(e).get_name();
2423 if (is_a<lst>(e.op(0))) {
2424 parameter = ex_to<lst>(e.op(0));
2426 parameter = lst{e.op(0)};
2432 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2433 return pf * zeta(m, s);
2444 // remove trailing zeros from H-parameters
2445 struct map_trafo_H_reduce_trailing_zeros : public map_function
2447 ex operator()(const ex& e) override
2449 if (is_a<add>(e) || is_a<mul>(e)) {
2450 return e.map(*this);
2452 if (is_a<function>(e)) {
2453 std::string name = ex_to<function>(e).get_name();
2456 if (is_a<lst>(e.op(0))) {
2457 parameter = ex_to<lst>(e.op(0));
2459 parameter = lst{e.op(0)};
2462 if (parameter.op(parameter.nops()-1) == 0) {
2465 if (parameter.nops() == 1) {
2470 auto it = parameter.begin();
2471 while ((it != parameter.end()) && (*it == 0)) {
2474 if (it == parameter.end()) {
2475 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2479 parameter.remove_last();
2480 std::size_t lastentry = parameter.nops();
2481 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2486 ex result = log(arg) * H(parameter,arg).hold();
2488 for (ex i=0; i<lastentry; i++) {
2489 if (parameter[i] > 0) {
2491 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2494 } else if (parameter[i] < 0) {
2496 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2504 if (lastentry < parameter.nops()) {
2505 result = result / (parameter.nops()-lastentry+1);
2506 return result.map(*this);
2518 // returns an expression with zeta functions corresponding to the parameter list for H
2519 ex convert_H_to_zeta(const lst& m)
2521 symbol xtemp("xtemp");
2522 map_trafo_H_reduce_trailing_zeros filter;
2523 map_trafo_H_convert_to_zeta filter2;
2524 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2528 // convert signs form Li to H representation
2529 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2532 auto itm = m.begin();
2533 auto itx = ++x.begin();
2538 while (itx != x.end()) {
2539 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
2540 // XXX: 1 + 0.0*I is considered equal to 1. However the former
2541 // is not automatically converted to a real number.
2542 // Do the conversion explicitly to avoid the
2543 // "numeric::operator>(): complex inequality" exception.
2544 signum *= (*itx != _ex_1) ? 1 : -1;
2546 res.append((*itm) * signum);
2554 // multiplies an one-dimensional H with another H
2556 ex trafo_H_mult(const ex& h1, const ex& h2)
2561 ex h1nops = h1.op(0).nops();
2562 ex h2nops = h2.op(0).nops();
2564 hshort = h2.op(0).op(0);
2565 hlong = ex_to<lst>(h1.op(0));
2567 hshort = h1.op(0).op(0);
2569 hlong = ex_to<lst>(h2.op(0));
2571 hlong = lst{h2.op(0).op(0)};
2574 for (std::size_t i=0; i<=hlong.nops(); i++) {
2578 newparameter.append(hlong[j]);
2580 newparameter.append(hshort);
2581 for (; j<hlong.nops(); j++) {
2582 newparameter.append(hlong[j]);
2584 res += H(newparameter, h1.op(1)).hold();
2590 // applies trafo_H_mult recursively on expressions
2591 struct map_trafo_H_mult : public map_function
2593 ex operator()(const ex& e) override
2596 return e.map(*this);
2604 for (std::size_t pos=0; pos<e.nops(); pos++) {
2605 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2606 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2608 for (ex i=0; i<e.op(pos).op(1); i++) {
2609 Hlst.append(e.op(pos).op(0));
2613 } else if (is_a<function>(e.op(pos))) {
2614 std::string name = ex_to<function>(e.op(pos)).get_name();
2616 if (e.op(pos).op(0).nops() > 1) {
2619 Hlst.append(e.op(pos));
2624 result *= e.op(pos);
2627 if (Hlst.nops() > 0) {
2628 firstH = Hlst[Hlst.nops()-1];
2635 if (Hlst.nops() > 0) {
2636 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2638 for (std::size_t i=1; i<Hlst.nops(); i++) {
2639 result *= Hlst.op(i);
2641 result = result.expand();
2642 map_trafo_H_mult recursion;
2643 return recursion(result);
2654 // do integration [ReV] (55)
2655 // put parameter 0 in front of existing parameters
2656 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2660 if (is_a<function>(e)) {
2661 name = ex_to<function>(e).get_name();
2666 for (std::size_t i=0; i<e.nops(); i++) {
2667 if (is_a<function>(e.op(i))) {
2668 std::string name = ex_to<function>(e.op(i)).get_name();
2676 lst newparameter = ex_to<lst>(h.op(0));
2677 newparameter.prepend(0);
2678 ex addzeta = convert_H_to_zeta(newparameter);
2679 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2681 return e * (-H(lst{ex(0)},1/arg).hold());
2686 // do integration [ReV] (49)
2687 // put parameter 1 in front of existing parameters
2688 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2692 if (is_a<function>(e)) {
2693 name = ex_to<function>(e).get_name();
2698 for (std::size_t i=0; i<e.nops(); i++) {
2699 if (is_a<function>(e.op(i))) {
2700 std::string name = ex_to<function>(e.op(i)).get_name();
2708 lst newparameter = ex_to<lst>(h.op(0));
2709 newparameter.prepend(1);
2710 return e.subs(h == H(newparameter, h.op(1)).hold());
2712 return e * H(lst{ex(1)},1-arg).hold();
2717 // do integration [ReV] (55)
2718 // put parameter -1 in front of existing parameters
2719 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2723 if (is_a<function>(e)) {
2724 name = ex_to<function>(e).get_name();
2729 for (std::size_t i=0; i<e.nops(); i++) {
2730 if (is_a<function>(e.op(i))) {
2731 std::string name = ex_to<function>(e.op(i)).get_name();
2739 lst newparameter = ex_to<lst>(h.op(0));
2740 newparameter.prepend(-1);
2741 ex addzeta = convert_H_to_zeta(newparameter);
2742 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2744 ex addzeta = convert_H_to_zeta(lst{ex(-1)});
2745 return (e * (addzeta - H(lst{ex(-1)},1/arg).hold())).expand();
2750 // do integration [ReV] (55)
2751 // put parameter -1 in front of existing parameters
2752 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2756 if (is_a<function>(e)) {
2757 name = ex_to<function>(e).get_name();
2762 for (std::size_t i = 0; i < e.nops(); i++) {
2763 if (is_a<function>(e.op(i))) {
2764 std::string name = ex_to<function>(e.op(i)).get_name();
2772 lst newparameter = ex_to<lst>(h.op(0));
2773 newparameter.prepend(-1);
2774 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2776 return (e * H(lst{ex(-1)},(1-arg)/(1+arg)).hold()).expand();
2781 // do integration [ReV] (55)
2782 // put parameter 1 in front of existing parameters
2783 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2787 if (is_a<function>(e)) {
2788 name = ex_to<function>(e).get_name();
2793 for (std::size_t i = 0; i < e.nops(); i++) {
2794 if (is_a<function>(e.op(i))) {
2795 std::string name = ex_to<function>(e.op(i)).get_name();
2803 lst newparameter = ex_to<lst>(h.op(0));
2804 newparameter.prepend(1);
2805 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2807 return (e * H(lst{ex(1)},(1-arg)/(1+arg)).hold()).expand();
2812 // do x -> 1-x transformation
2813 struct map_trafo_H_1mx : public map_function
2815 ex operator()(const ex& e) override
2817 if (is_a<add>(e) || is_a<mul>(e)) {
2818 return e.map(*this);
2821 if (is_a<function>(e)) {
2822 std::string name = ex_to<function>(e).get_name();
2825 lst parameter = ex_to<lst>(e.op(0));
2828 // special cases if all parameters are either 0, 1 or -1
2829 bool allthesame = true;
2830 if (parameter.op(0) == 0) {
2831 for (std::size_t i = 1; i < parameter.nops(); i++) {
2832 if (parameter.op(i) != 0) {
2839 for (int i=parameter.nops(); i>0; i--) {
2840 newparameter.append(1);
2842 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2844 } else if (parameter.op(0) == -1) {
2845 throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2847 for (std::size_t i = 1; i < parameter.nops(); i++) {
2848 if (parameter.op(i) != 1) {
2855 for (int i=parameter.nops(); i>0; i--) {
2856 newparameter.append(0);
2858 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2862 lst newparameter = parameter;
2863 newparameter.remove_first();
2865 if (parameter.op(0) == 0) {
2868 ex res = convert_H_to_zeta(parameter);
2869 //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2870 map_trafo_H_1mx recursion;
2871 ex buffer = recursion(H(newparameter, arg).hold());
2872 if (is_a<add>(buffer)) {
2873 for (std::size_t i = 0; i < buffer.nops(); i++) {
2874 res -= trafo_H_prepend_one(buffer.op(i), arg);
2877 res -= trafo_H_prepend_one(buffer, arg);
2884 map_trafo_H_1mx recursion;
2885 map_trafo_H_mult unify;
2886 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
2887 std::size_t firstzero = 0;
2888 while (parameter.op(firstzero) == 1) {
2891 for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2895 newparameter.append(parameter[j+1]);
2897 newparameter.append(1);
2898 for (; j<parameter.nops()-1; j++) {
2899 newparameter.append(parameter[j+1]);
2901 res -= H(newparameter, arg).hold();
2903 res = recursion(res).expand() / firstzero;
2913 // do x -> 1/x transformation
2914 struct map_trafo_H_1overx : public map_function
2916 ex operator()(const ex& e) override
2918 if (is_a<add>(e) || is_a<mul>(e)) {
2919 return e.map(*this);
2922 if (is_a<function>(e)) {
2923 std::string name = ex_to<function>(e).get_name();
2926 lst parameter = ex_to<lst>(e.op(0));
2929 // special cases if all parameters are either 0, 1 or -1
2930 bool allthesame = true;
2931 if (parameter.op(0) == 0) {
2932 for (std::size_t i = 1; i < parameter.nops(); i++) {
2933 if (parameter.op(i) != 0) {
2939 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2941 } else if (parameter.op(0) == -1) {
2942 for (std::size_t i = 1; i < parameter.nops(); i++) {
2943 if (parameter.op(i) != -1) {
2949 map_trafo_H_mult unify;
2950 return unify((pow(H(lst{ex(-1)},1/arg).hold() - H(lst{ex(0)},1/arg).hold(), parameter.nops())
2951 / factorial(parameter.nops())).expand());
2954 for (std::size_t i = 1; i < parameter.nops(); i++) {
2955 if (parameter.op(i) != 1) {
2961 map_trafo_H_mult unify;
2962 return unify((pow(H(lst{ex(1)},1/arg).hold() + H(lst{ex(0)},1/arg).hold() + H_polesign, parameter.nops())
2963 / factorial(parameter.nops())).expand());
2967 lst newparameter = parameter;
2968 newparameter.remove_first();
2970 if (parameter.op(0) == 0) {
2973 ex res = convert_H_to_zeta(parameter);
2974 map_trafo_H_1overx recursion;
2975 ex buffer = recursion(H(newparameter, arg).hold());
2976 if (is_a<add>(buffer)) {
2977 for (std::size_t i = 0; i < buffer.nops(); i++) {
2978 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2981 res += trafo_H_1tx_prepend_zero(buffer, arg);
2985 } else if (parameter.op(0) == -1) {
2987 // leading negative one
2988 ex res = convert_H_to_zeta(parameter);
2989 map_trafo_H_1overx recursion;
2990 ex buffer = recursion(H(newparameter, arg).hold());
2991 if (is_a<add>(buffer)) {
2992 for (std::size_t i = 0; i < buffer.nops(); i++) {
2993 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2996 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
3003 map_trafo_H_1overx recursion;
3004 map_trafo_H_mult unify;
3005 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
3006 std::size_t firstzero = 0;
3007 while (parameter.op(firstzero) == 1) {
3010 for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
3014 newparameter.append(parameter[j+1]);
3016 newparameter.append(1);
3017 for (; j<parameter.nops()-1; j++) {
3018 newparameter.append(parameter[j+1]);
3020 res -= H(newparameter, arg).hold();
3022 res = recursion(res).expand() / firstzero;
3034 // do x -> (1-x)/(1+x) transformation
3035 struct map_trafo_H_1mxt1px : public map_function
3037 ex operator()(const ex& e) override
3039 if (is_a<add>(e) || is_a<mul>(e)) {
3040 return e.map(*this);
3043 if (is_a<function>(e)) {
3044 std::string name = ex_to<function>(e).get_name();
3047 lst parameter = ex_to<lst>(e.op(0));
3050 // special cases if all parameters are either 0, 1 or -1
3051 bool allthesame = true;
3052 if (parameter.op(0) == 0) {
3053 for (std::size_t i = 1; i < parameter.nops(); i++) {
3054 if (parameter.op(i) != 0) {
3060 map_trafo_H_mult unify;
3061 return unify((pow(-H(lst{ex(1)},(1-arg)/(1+arg)).hold() - H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops())
3062 / factorial(parameter.nops())).expand());
3064 } else if (parameter.op(0) == -1) {
3065 for (std::size_t i = 1; i < parameter.nops(); i++) {
3066 if (parameter.op(i) != -1) {
3072 map_trafo_H_mult unify;
3073 return unify((pow(log(2) - H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops())
3074 / factorial(parameter.nops())).expand());
3077 for (std::size_t i = 1; i < parameter.nops(); i++) {
3078 if (parameter.op(i) != 1) {
3084 map_trafo_H_mult unify;
3085 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())
3086 / factorial(parameter.nops())).expand());
3090 lst newparameter = parameter;
3091 newparameter.remove_first();
3093 if (parameter.op(0) == 0) {
3096 ex res = convert_H_to_zeta(parameter);
3097 map_trafo_H_1mxt1px recursion;
3098 ex buffer = recursion(H(newparameter, arg).hold());
3099 if (is_a<add>(buffer)) {
3100 for (std::size_t i = 0; i < buffer.nops(); i++) {
3101 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3104 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3108 } else if (parameter.op(0) == -1) {
3110 // leading negative one
3111 ex res = convert_H_to_zeta(parameter);
3112 map_trafo_H_1mxt1px recursion;
3113 ex buffer = recursion(H(newparameter, arg).hold());
3114 if (is_a<add>(buffer)) {
3115 for (std::size_t i = 0; i < buffer.nops(); i++) {
3116 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3119 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3126 map_trafo_H_1mxt1px recursion;
3127 map_trafo_H_mult unify;
3128 ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold();
3129 std::size_t firstzero = 0;
3130 while (parameter.op(firstzero) == 1) {
3133 for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3137 newparameter.append(parameter[j+1]);
3139 newparameter.append(1);
3140 for (; j<parameter.nops()-1; j++) {
3141 newparameter.append(parameter[j+1]);
3143 res -= H(newparameter, arg).hold();
3145 res = recursion(res).expand() / firstzero;
3157 // do the actual summation.
3158 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3160 const int j = m.size();
3162 std::vector<cln::cl_N> t(j);
3164 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3165 cln::cl_N factor = cln::expt(x, j) * one;
3171 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3172 for (int k=j-2; k>=1; k--) {
3173 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3175 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3176 factor = factor * x;
3177 } while (t[0] != t0buf);
3183 } // end of anonymous namespace
3186 //////////////////////////////////////////////////////////////////////
3188 // Harmonic polylogarithm H(m,x)
3192 //////////////////////////////////////////////////////////////////////
3195 static ex H_evalf(const ex& x1, const ex& x2)
3197 if (is_a<lst>(x1)) {
3200 if (is_a<numeric>(x2)) {
3201 x = ex_to<numeric>(x2).to_cl_N();
3203 ex x2_val = x2.evalf();
3204 if (is_a<numeric>(x2_val)) {
3205 x = ex_to<numeric>(x2_val).to_cl_N();
3209 for (std::size_t i = 0; i < x1.nops(); i++) {
3210 if (!x1.op(i).info(info_flags::integer)) {
3211 return H(x1, x2).hold();
3214 if (x1.nops() < 1) {
3215 return H(x1, x2).hold();
3218 const lst& morg = ex_to<lst>(x1);
3219 // remove trailing zeros ...
3220 if (*(--morg.end()) == 0) {
3221 symbol xtemp("xtemp");
3222 map_trafo_H_reduce_trailing_zeros filter;
3223 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3225 // ... and expand parameter notation
3227 for (const auto & it : morg) {
3229 for (ex count=it-1; count > 0; count--) {
3233 } else if (it <= -1) {
3234 for (ex count=it+1; count < 0; count++) {
3244 if (cln::abs(x) < 0.95) {
3248 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3249 // negative parameters -> s_lst is filled
3250 std::vector<int> m_int;
3251 std::vector<cln::cl_N> x_cln;
3252 for (auto it_int = m_lst.begin(), it_cln = s_lst.begin();
3253 it_int != m_lst.end(); it_int++, it_cln++) {
3254 m_int.push_back(ex_to<numeric>(*it_int).to_int());
3255 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3257 x_cln.front() = x_cln.front() * x;
3258 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3260 // only positive parameters
3262 if (m_lst.nops() == 1) {
3263 return Li(m_lst.op(0), x2).evalf();
3265 std::vector<int> m_int;
3266 for (const auto & it : m_lst) {
3267 m_int.push_back(ex_to<numeric>(it).to_int());
3269 return numeric(H_do_sum(m_int, x));
3273 symbol xtemp("xtemp");
3276 // ensure that the realpart of the argument is positive
3277 if (cln::realpart(x) < 0) {
3279 for (std::size_t i = 0; i < m.nops(); i++) {
3281 m.let_op(i) = -m.op(i);
3288 if (cln::abs(x) >= 2.0) {
3289 map_trafo_H_1overx trafo;
3290 res *= trafo(H(m, xtemp).hold());
3291 if (cln::imagpart(x) <= 0) {
3292 res = res.subs(H_polesign == -I*Pi);
3294 res = res.subs(H_polesign == I*Pi);
3296 return res.subs(xtemp == numeric(x)).evalf();
3299 // check for letters (-1)
3300 bool has_minus_one = false;
3301 for (const auto & it : m) {
3303 has_minus_one = true;
3306 // check transformations for 0.95 <= |x| < 2.0
3308 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3309 if (cln::abs(x-9.53) <= 9.47) {
3311 map_trafo_H_1mxt1px trafo;
3312 res *= trafo(H(m, xtemp).hold());
3315 if (has_minus_one) {
3316 map_trafo_H_convert_to_Li filter;
3317 return filter(H(m, numeric(x)).hold()).evalf();
3319 map_trafo_H_1mx trafo;
3320 res *= trafo(H(m, xtemp).hold());
3323 return res.subs(xtemp == numeric(x)).evalf();
3326 return H(x1,x2).hold();
3330 static ex H_eval(const ex& m_, const ex& x)
3333 if (is_a<lst>(m_)) {
3338 if (m.nops() == 0) {
3346 if (*m.begin() > _ex1) {
3352 } else if (*m.begin() < _ex_1) {
3358 } else if (*m.begin() == _ex0) {
3365 for (auto it = ++m.begin(); it != m.end(); it++) {
3366 if (it->info(info_flags::integer)) {
3377 } else if (*it < _ex_1) {
3397 } else if (step == 1) {
3409 // if some m_i is not an integer
3410 return H(m_, x).hold();
3413 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3414 return convert_H_to_zeta(m);
3420 return H(m_, x).hold();
3422 return pow(log(x), m.nops()) / factorial(m.nops());
3425 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3427 } else if ((step == 1) && (pos1 == _ex0)){
3432 return pow(-1, p) * S(n, p, -x);
3438 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3439 return H(m_, x).evalf();
3441 return H(m_, x).hold();
3445 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3447 epvector seq { expair(H(m, x), 0) };
3448 return pseries(rel, std::move(seq));
3452 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3454 GINAC_ASSERT(deriv_param < 2);
3455 if (deriv_param == 0) {
3459 if (is_a<lst>(m_)) {
3475 return 1/(1-x) * H(m, x);
3476 } else if (mb == _ex_1) {
3477 return 1/(1+x) * H(m, x);
3484 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3487 if (is_a<lst>(m_)) {
3492 c.s << "\\mathrm{H}_{";
3493 auto itm = m.begin();
3496 for (; itm != m.end(); itm++) {
3506 REGISTER_FUNCTION(H,
3507 evalf_func(H_evalf).
3509 series_func(H_series).
3510 derivative_func(H_deriv).
3511 print_func<print_latex>(H_print_latex).
3512 do_not_evalf_params());
3515 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3516 ex convert_H_to_Li(const ex& m, const ex& x)
3518 map_trafo_H_reduce_trailing_zeros filter;
3519 map_trafo_H_convert_to_Li filter2;
3521 return filter2(filter(H(m, x).hold()));
3523 return filter2(filter(H(lst{m}, x).hold()));
3528 //////////////////////////////////////////////////////////////////////
3530 // Multiple zeta values zeta(x) and zeta(x,s)
3534 //////////////////////////////////////////////////////////////////////
3537 // anonymous namespace for helper functions
3541 // parameters and data for [Cra] algorithm
3542 const cln::cl_N lambda = cln::cl_N("319/320");
3544 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3546 const int size = a.size();
3547 for (int n=0; n<size; n++) {
3549 for (int m=0; m<=n; m++) {
3550 c[n] = c[n] + a[m]*b[n-m];
3557 static void initcX(std::vector<cln::cl_N>& crX,
3558 const std::vector<int>& s,
3561 std::vector<cln::cl_N> crB(L2 + 1);
3562 for (int i=0; i<=L2; i++)
3563 crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i);
3567 std::vector<std::vector<cln::cl_N>> crG(s.size() - 1, std::vector<cln::cl_N>(L2 + 1));
3568 for (int m=0; m < (int)s.size() - 1; m++) {
3571 for (int i = 0; i <= L2; i++)
3572 crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2);
3577 for (std::size_t m = 0; m < s.size() - 1; m++) {
3578 std::vector<cln::cl_N> Xbuf(L2 + 1);
3579 for (int i = 0; i <= L2; i++)
3580 Xbuf[i] = crX[i] * crG[m][i];
3582 halfcyclic_convolute(Xbuf, crB, crX);
3588 static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk,
3589 const std::vector<cln::cl_N>& crX)
3591 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3592 cln::cl_N factor = cln::expt(lambda, Sqk);
3593 cln::cl_N res = factor / Sqk * crX[0] * one;
3598 factor = factor * lambda;
3600 res = res + crX[N] * factor / (N+Sqk);
3601 } while (((res != resbuf) || cln::zerop(crX[N])) && (N+1 < crX.size()));
3607 static void calc_f(std::vector<std::vector<cln::cl_N>>& f_kj,
3608 const int maxr, const int L1)
3610 cln::cl_N t0, t1, t2, t3, t4;
3612 auto it = f_kj.begin();
3613 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3615 t0 = cln::exp(-lambda);
3617 for (k=1; k<=L1; k++) {
3620 for (j=1; j<=maxr; j++) {
3623 for (i=2; i<=j; i++) {
3627 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3635 static cln::cl_N crandall_Z(const std::vector<int>& s,
3636 const std::vector<std::vector<cln::cl_N>>& f_kj)
3638 const int j = s.size();
3647 t0 = t0 + f_kj[q+j-2][s[0]-1];
3648 } while ((t0 != t0buf) && (q+j-1 < f_kj.size()));
3650 return t0 / cln::factorial(s[0]-1);
3653 std::vector<cln::cl_N> t(j);
3660 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3661 for (int k=j-2; k>=1; k--) {
3662 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3664 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3665 } while ((t[0] != t0buf) && (q+j-1 < f_kj.size()));
3667 return t[0] / cln::factorial(s[0]-1);
3672 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3674 std::vector<int> r = s;
3675 const int j = r.size();
3679 // decide on maximal size of f_kj for crandall_Z
3683 L1 = Digits * 3 + j*2;
3687 // decide on maximal size of crX for crandall_Y
3690 } else if (Digits < 86) {
3692 } else if (Digits < 192) {
3694 } else if (Digits < 394) {
3696 } else if (Digits < 808) {
3698 } else if (Digits < 1636) {
3701 // [Cra] section 6, log10(lambda/2/Pi) approx -0.79 for lambda=319/320, add some extra digits
3702 L2 = std::pow(2, ceil( std::log2((long(Digits))/0.79 + 40 )) ) - 1;
3709 for (int i=0; i<j; i++) {
3716 std::vector<std::vector<cln::cl_N>> f_kj(L1);
3717 calc_f(f_kj, maxr, L1);
3719 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3721 std::vector<int> rz;
3724 for (int k=r.size()-1; k>0; k--) {
3726 rz.insert(rz.begin(), r.back());
3727 skp1buf = rz.front();
3731 std::vector<cln::cl_N> crX;
3734 for (int q=0; q<skp1buf; q++) {
3736 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k, crX);
3737 cln::cl_N pp2 = crandall_Z(rz, f_kj);
3742 res = res - pp1 * pp2 / cln::factorial(q);
3744 res = res + pp1 * pp2 / cln::factorial(q);
3747 rz.front() = skp1buf;
3749 rz.insert(rz.begin(), r.back());
3751 std::vector<cln::cl_N> crX;
3752 initcX(crX, rz, L2);
3754 res = (res + crandall_Y_loop(S-j, crX)) / r0factorial
3755 + crandall_Z(rz, f_kj);
3761 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3763 const int j = r.size();
3765 // buffer for subsums
3766 std::vector<cln::cl_N> t(j);
3767 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3774 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3775 for (int k=j-2; k>=0; k--) {
3776 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3778 } while (t[0] != t0buf);
3784 // does Hoelder convolution. see [BBB] (7.0)
3785 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3787 // prepare parameters
3788 // holds Li arguments in [BBB] notation
3789 std::vector<int> s = s_;
3790 std::vector<int> m_p = m_;
3791 std::vector<int> m_q;
3792 // holds Li arguments in nested sums notation
3793 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3794 s_p[0] = s_p[0] * cln::cl_N("1/2");
3795 // convert notations
3797 for (std::size_t i = 0; i < s_.size(); i++) {
3802 s[i] = sig * std::abs(s[i]);
3804 std::vector<cln::cl_N> s_q;
3805 cln::cl_N signum = 1;
3808 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3813 // change parameters
3814 if (s.front() > 0) {
3815 if (m_p.front() == 1) {
3816 m_p.erase(m_p.begin());
3817 s_p.erase(s_p.begin());
3818 if (s_p.size() > 0) {
3819 s_p.front() = s_p.front() * cln::cl_N("1/2");
3825 m_q.insert(m_q.begin(), 1);
3826 if (s_q.size() > 0) {
3827 s_q.front() = s_q.front() * 2;
3829 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3832 if (m_p.front() == 1) {
3833 m_p.erase(m_p.begin());
3834 cln::cl_N spbuf = s_p.front();
3835 s_p.erase(s_p.begin());
3836 if (s_p.size() > 0) {
3837 s_p.front() = s_p.front() * spbuf;
3840 m_q.insert(m_q.begin(), 1);
3841 if (s_q.size() > 0) {
3842 s_q.front() = s_q.front() * 4;
3844 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3848 m_q.insert(m_q.begin(), 1);
3849 if (s_q.size() > 0) {
3850 s_q.front() = s_q.front() * 2;
3852 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3857 if (m_p.size() == 0) break;
3859 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3864 res = res + signum * multipleLi_do_sum(m_q, s_q);
3870 } // end of anonymous namespace
3873 //////////////////////////////////////////////////////////////////////
3875 // Multiple zeta values zeta(x)
3879 //////////////////////////////////////////////////////////////////////
3882 static ex zeta1_evalf(const ex& x)
3884 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3886 // multiple zeta value
3887 const int count = x.nops();
3888 const lst& xlst = ex_to<lst>(x);
3889 std::vector<int> r(count);
3890 std::vector<int> si(count);
3892 // check parameters and convert them
3893 auto it1 = xlst.begin();
3894 auto it2 = r.begin();
3895 auto it_swrite = si.begin();
3897 if (!(*it1).info(info_flags::posint)) {
3898 return zeta(x).hold();
3900 *it2 = ex_to<numeric>(*it1).to_int();
3905 } while (it2 != r.end());
3907 // check for divergence
3909 return zeta(x).hold();
3912 // use Hoelder convolution if Digits is large
3914 return numeric(zeta_do_Hoelder_convolution(r, si));
3916 // decide on summation algorithm
3917 // this is still a bit clumsy
3918 int limit = (Digits>17) ? 10 : 6;
3919 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3920 return numeric(zeta_do_sum_Crandall(r));
3922 return numeric(zeta_do_sum_simple(r));
3926 // single zeta value
3927 if (is_exactly_a<numeric>(x) && (x != 1)) {
3929 return zeta(ex_to<numeric>(x));
3930 } catch (const dunno &e) { }
3933 return zeta(x).hold();
3937 static ex zeta1_eval(const ex& m)
3939 if (is_exactly_a<lst>(m)) {
3940 if (m.nops() == 1) {
3941 return zeta(m.op(0));
3943 return zeta(m).hold();
3946 if (m.info(info_flags::numeric)) {
3947 const numeric& y = ex_to<numeric>(m);
3948 // trap integer arguments:
3949 if (y.is_integer()) {
3953 if (y.is_equal(*_num1_p)) {
3954 return zeta(m).hold();
3956 if (y.info(info_flags::posint)) {
3957 if (y.info(info_flags::odd)) {
3958 return zeta(m).hold();
3960 return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3963 if (y.info(info_flags::odd)) {
3964 return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3971 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3972 return zeta1_evalf(m);
3975 return zeta(m).hold();
3979 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3981 GINAC_ASSERT(deriv_param==0);
3983 if (is_exactly_a<lst>(m)) {
3986 return zetaderiv(_ex1, m);
3991 static void zeta1_print_latex(const ex& m_, const print_context& c)
3994 if (is_a<lst>(m_)) {
3995 const lst& m = ex_to<lst>(m_);
3996 auto it = m.begin();
3999 for (; it != m.end(); it++) {
4010 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
4011 evalf_func(zeta1_evalf).
4012 eval_func(zeta1_eval).
4013 derivative_func(zeta1_deriv).
4014 print_func<print_latex>(zeta1_print_latex).
4015 do_not_evalf_params().
4019 //////////////////////////////////////////////////////////////////////
4021 // Alternating Euler sum zeta(x,s)
4025 //////////////////////////////////////////////////////////////////////
4028 static ex zeta2_evalf(const ex& x, const ex& s)
4030 if (is_exactly_a<lst>(x)) {
4032 // alternating Euler sum
4033 const int count = x.nops();
4034 const lst& xlst = ex_to<lst>(x);
4035 const lst& slst = ex_to<lst>(s);
4036 std::vector<int> xi(count);
4037 std::vector<int> si(count);
4039 // check parameters and convert them
4040 auto it_xread = xlst.begin();
4041 auto it_sread = slst.begin();
4042 auto it_xwrite = xi.begin();
4043 auto it_swrite = si.begin();
4045 if (!(*it_xread).info(info_flags::posint)) {
4046 return zeta(x, s).hold();
4048 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
4049 if (*it_sread > 0) {
4058 } while (it_xwrite != xi.end());
4060 // check for divergence
4061 if ((xi[0] == 1) && (si[0] == 1)) {
4062 return zeta(x, s).hold();
4065 // use Hoelder convolution
4066 return numeric(zeta_do_Hoelder_convolution(xi, si));
4069 // x and s are not lists: convert to lists
4070 return zeta(lst{x}, lst{s}).evalf();
4074 static ex zeta2_eval(const ex& m, const ex& s_)
4076 if (is_exactly_a<lst>(s_)) {
4077 const lst& s = ex_to<lst>(s_);
4078 for (const auto & it : s) {
4079 if (it.info(info_flags::positive)) {
4082 return zeta(m, s_).hold();
4085 } else if (s_.info(info_flags::positive)) {
4089 return zeta(m, s_).hold();
4093 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
4095 GINAC_ASSERT(deriv_param==0);
4097 if (is_exactly_a<lst>(m)) {
4100 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
4101 return zetaderiv(_ex1, m);
4108 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
4111 if (is_a<lst>(m_)) {
4117 if (is_a<lst>(s_)) {
4123 auto itm = m.begin();
4124 auto its = s.begin();
4126 c.s << "\\overline{";
4134 for (; itm != m.end(); itm++, its++) {
4137 c.s << "\\overline{";
4148 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
4149 evalf_func(zeta2_evalf).
4150 eval_func(zeta2_eval).
4151 derivative_func(zeta2_deriv).
4152 print_func<print_latex>(zeta2_print_latex).
4153 do_not_evalf_params().
4157 } // namespace GiNaC