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 nummerically 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) 1999-2011 Johannes Gutenberg University Mainz, Germany
52 * This program is free software; you can redistribute it and/or modify
53 * it under the terms of the GNU General Public License as published by
54 * the Free Software Foundation; either version 2 of the License, or
55 * (at your option) any later version.
57 * This program is distributed in the hope that it will be useful,
58 * but WITHOUT ANY WARRANTY; without even the implied warranty of
59 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
60 * GNU General Public License for more details.
62 * You should have received a copy of the GNU General Public License
63 * along with this program; if not, write to the Free Software
64 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
74 #include "operators.h"
77 #include "relational.h"
90 //////////////////////////////////////////////////////////////////////
92 // Classical polylogarithm Li(n,x)
96 //////////////////////////////////////////////////////////////////////
99 // anonymous namespace for helper functions
103 // lookup table for factors built from Bernoulli numbers
105 std::vector<std::vector<cln::cl_N> > Xn;
106 // initial size of Xn that should suffice for 32bit machines (must be even)
107 const int xninitsizestep = 26;
108 int xninitsize = xninitsizestep;
112 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
113 // With these numbers the polylogs can be calculated as follows:
114 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
115 // X_0(n) = B_n (Bernoulli numbers)
116 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
117 // The calculation of Xn depends on X0 and X{n-1}.
118 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
119 // This results in a slightly more complicated algorithm for the X_n.
120 // The first index in Xn corresponds to the index of the polylog minus 2.
121 // The second index in Xn corresponds to the index from the actual sum.
125 // calculate X_2 and higher (corresponding to Li_4 and higher)
126 std::vector<cln::cl_N> buf(xninitsize);
127 std::vector<cln::cl_N>::iterator it = buf.begin();
129 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
131 for (int i=2; i<=xninitsize; i++) {
133 result = 0; // k == 0
135 result = Xn[0][i/2-1]; // k == 0
137 for (int k=1; k<i-1; k++) {
138 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
139 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
142 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
143 result = result + Xn[n-1][i-1] / (i+1); // k == i
150 // special case to handle the X_0 correct
151 std::vector<cln::cl_N> buf(xninitsize);
152 std::vector<cln::cl_N>::iterator it = buf.begin();
154 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
156 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
158 for (int i=3; i<=xninitsize; i++) {
160 result = -Xn[0][(i-3)/2]/2;
161 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
164 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
165 for (int k=1; k<i/2; k++) {
166 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
175 std::vector<cln::cl_N> buf(xninitsize/2);
176 std::vector<cln::cl_N>::iterator it = buf.begin();
177 for (int i=1; i<=xninitsize/2; i++) {
178 *it = bernoulli(i*2).to_cl_N();
187 // doubles the number of entries in each Xn[]
190 const int pos0 = xninitsize / 2;
192 for (int i=1; i<=xninitsizestep/2; ++i) {
193 Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
196 int xend = xninitsize + xninitsizestep;
199 for (int i=xninitsize+1; i<=xend; ++i) {
201 result = -Xn[0][(i-3)/2]/2;
202 Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result);
204 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
205 for (int k=1; k<i/2; k++) {
206 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
208 Xn[1].push_back(result);
212 for (size_t n=2; n<Xn.size(); ++n) {
213 for (int i=xninitsize+1; i<=xend; ++i) {
215 result = 0; // k == 0
217 result = Xn[0][i/2-1]; // k == 0
219 for (int k=1; k<i-1; ++k) {
220 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
221 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
224 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
225 result = result + Xn[n-1][i-1] / (i+1); // k == i
226 Xn[n].push_back(result);
230 xninitsize += xninitsizestep;
234 // calculates Li(2,x) without Xn
235 cln::cl_N Li2_do_sum(const cln::cl_N& x)
239 cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
240 cln::cl_I den = 1; // n^2 = 1
245 den = den + i; // n^2 = 4, 9, 16, ...
247 res = res + num / den;
248 } while (res != resbuf);
253 // calculates Li(2,x) with Xn
254 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
256 std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
257 std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
258 cln::cl_N u = -cln::log(1-x);
259 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
260 cln::cl_N uu = cln::square(u);
261 cln::cl_N res = u - uu/4;
266 factor = factor * uu / (2*i * (2*i+1));
267 res = res + (*it) * factor;
271 it = Xn[0].begin() + (i-1);
274 } while (res != resbuf);
279 // calculates Li(n,x), n>2 without Xn
280 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
282 cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
289 res = res + factor / cln::expt(cln::cl_I(i),n);
291 } while (res != resbuf);
296 // calculates Li(n,x), n>2 with Xn
297 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
299 std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
300 std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
301 cln::cl_N u = -cln::log(1-x);
302 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
308 factor = factor * u / i;
309 res = res + (*it) * factor;
313 it = Xn[n-2].begin() + (i-2);
314 xend = Xn[n-2].end();
316 } while (res != resbuf);
321 // forward declaration needed by function Li_projection and C below
322 const cln::cl_N S_num(int n, int p, const cln::cl_N& x);
325 // helper function for classical polylog Li
326 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
328 // treat n=2 as special case
330 // check if precalculated X0 exists
335 if (cln::realpart(x) < 0.5) {
336 // choose the faster algorithm
337 // the switching point was empirically determined. the optimal point
338 // depends on hardware, Digits, ... so an approx value is okay.
339 // it solves also the problem with precision due to the u=-log(1-x) transformation
340 if (cln::abs(cln::realpart(x)) < 0.25) {
342 return Li2_do_sum(x);
344 return Li2_do_sum_Xn(x);
347 // choose the faster algorithm
348 if (cln::abs(cln::realpart(x)) > 0.75) {
352 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
355 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
359 // check if precalculated Xn exist
361 for (int i=xnsize; i<n-1; i++) {
366 if (cln::realpart(x) < 0.5) {
367 // choose the faster algorithm
368 // with n>=12 the "normal" summation always wins against the method with Xn
369 if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
370 return Lin_do_sum(n, x);
372 return Lin_do_sum_Xn(n, x);
375 cln::cl_N result = 0;
376 if ( x != 1 ) result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
377 for (int j=0; j<n-1; j++) {
378 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
379 * cln::expt(cln::log(x), j) / cln::factorial(j);
386 // helper function for classical polylog Li
387 const cln::cl_N Lin_numeric(const int n, const cln::cl_N& x)
391 return -cln::log(1-x);
402 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
404 if (cln::abs(realpart(x)) < 0.4 && cln::abs(cln::abs(x)-1) < 0.01) {
405 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
406 for (int j=0; j<n-1; j++) {
407 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
408 * cln::expt(cln::log(x), j) / cln::factorial(j);
413 // what is the desired float format?
414 // first guess: default format
415 cln::float_format_t prec = cln::default_float_format;
416 const cln::cl_N value = x;
417 // second guess: the argument's format
418 if (!instanceof(realpart(x), cln::cl_RA_ring))
419 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
420 else if (!instanceof(imagpart(x), cln::cl_RA_ring))
421 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
424 if (cln::abs(value) > 1) {
425 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
426 // check if argument is complex. if it is real, the new polylog has to be conjugated.
427 if (cln::zerop(cln::imagpart(value))) {
429 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
432 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
437 result = result + Li_projection(n, cln::recip(value), prec);
440 result = result - Li_projection(n, cln::recip(value), prec);
444 for (int j=0; j<n-1; j++) {
445 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
446 * Lin_numeric(n-j,1) * cln::expt(cln::log(-value),j) / cln::factorial(j);
448 result = result - add;
452 return Li_projection(n, value, prec);
457 } // end of anonymous namespace
460 //////////////////////////////////////////////////////////////////////
462 // Multiple polylogarithm Li(n,x)
466 //////////////////////////////////////////////////////////////////////
469 // anonymous namespace for helper function
473 // performs the actual series summation for multiple polylogarithms
474 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
476 // ensure all x <> 0.
477 for (std::vector<cln::cl_N>::const_iterator it = x.begin(); it != x.end(); ++it) {
478 if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits));
481 const int j = s.size();
482 bool flag_accidental_zero = false;
484 std::vector<cln::cl_N> t(j);
485 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
492 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
493 for (int k=j-2; k>=0; k--) {
494 t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
497 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
498 for (int k=j-2; k>=0; k--) {
499 flag_accidental_zero = cln::zerop(t[k+1]);
500 t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
502 } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero );
508 // forward declaration for Li_eval()
509 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
512 // type used by the transformation functions for G
513 typedef std::vector<int> Gparameter;
516 // G_eval1-function for G transformations
517 ex G_eval1(int a, int scale, const exvector& gsyms)
520 const ex& scs = gsyms[std::abs(scale)];
521 const ex& as = gsyms[std::abs(a)];
523 return -log(1 - scs/as);
528 return log(gsyms[std::abs(scale)]);
533 // G_eval-function for G transformations
534 ex G_eval(const Gparameter& a, int scale, const exvector& gsyms)
536 // check for properties of G
537 ex sc = gsyms[std::abs(scale)];
539 bool all_zero = true;
540 bool all_ones = true;
542 for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
544 const ex sym = gsyms[std::abs(*it)];
558 // care about divergent G: shuffle to separate divergencies that will be canceled
559 // later on in the transformation
560 if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
563 Gparameter::const_iterator it = a.begin();
565 for (; it != a.end(); ++it) {
566 short_a.push_back(*it);
568 ex result = G_eval1(a.front(), scale, gsyms) * G_eval(short_a, scale, gsyms);
569 it = short_a.begin();
570 for (int i=1; i<count_ones; ++i) {
573 for (; it != short_a.end(); ++it) {
576 Gparameter::const_iterator it2 = short_a.begin();
577 for (; it2 != it; ++it2) {
578 newa.push_back(*it2);
581 newa.push_back(a[0]);
584 for (; it2 != short_a.end(); ++it2) {
585 newa.push_back(*it2);
587 result -= G_eval(newa, scale, gsyms);
589 return result / count_ones;
592 // G({1,...,1};y) -> G({1};y)^k / k!
593 if (all_ones && a.size() > 1) {
594 return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones);
597 // G({0,...,0};y) -> log(y)^k / k!
599 return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
602 // no special cases anymore -> convert it into Li
605 ex argbuf = gsyms[std::abs(scale)];
607 for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) {
609 const ex& sym = gsyms[std::abs(*it)];
610 x.append(argbuf / sym);
618 return pow(-1, x.nops()) * Li(m, x);
622 // converts data for G: pending_integrals -> a
623 Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
625 GINAC_ASSERT(pending_integrals.size() != 1);
627 if (pending_integrals.size() > 0) {
628 // get rid of the first element, which would stand for the new upper limit
629 Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
632 // just return empty parameter list
639 // check the parameters a and scale for G and return information about convergence, depth, etc.
640 // convergent : true if G(a,scale) is convergent
641 // depth : depth of G(a,scale)
642 // trailing_zeros : number of trailing zeros of a
643 // min_it : iterator of a pointing on the smallest element in a
644 Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
645 bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
651 Gparameter::const_iterator lastnonzero = a.end();
652 for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
653 if (std::abs(*it) > 0) {
657 if (std::abs(*it) < scale) {
659 if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
667 if (lastnonzero == a.end())
669 return ++lastnonzero;
673 // add scale to pending_integrals if pending_integrals is empty
674 Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
676 GINAC_ASSERT(pending_integrals.size() != 1);
678 if (pending_integrals.size() > 0) {
679 return pending_integrals;
681 Gparameter new_pending_integrals;
682 new_pending_integrals.push_back(scale);
683 return new_pending_integrals;
688 // handles trailing zeroes for an otherwise convergent integral
689 ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms)
692 int depth, trailing_zeros;
693 Gparameter::const_iterator last, dummyit;
694 last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
696 GINAC_ASSERT(convergent);
698 if ((trailing_zeros > 0) && (depth > 0)) {
700 Gparameter new_a(a.begin(), a.end()-1);
701 result += G_eval1(0, scale, gsyms) * trailing_zeros_G(new_a, scale, gsyms);
702 for (Gparameter::const_iterator it = a.begin(); it != last; ++it) {
703 Gparameter new_a(a.begin(), it);
705 new_a.insert(new_a.end(), it, a.end()-1);
706 result -= trailing_zeros_G(new_a, scale, gsyms);
709 return result / trailing_zeros;
711 return G_eval(a, scale, gsyms);
716 // G transformation [VSW] (57),(58)
717 ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale, const exvector& gsyms)
719 // pendint = ( y1, b1, ..., br )
720 // a = ( 0, ..., 0, amin )
723 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
724 // where sr replaces amin
726 GINAC_ASSERT(a.back() != 0);
727 GINAC_ASSERT(a.size() > 0);
730 Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
731 const int psize = pending_integrals.size();
734 // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
739 result += log(gsyms[ex_to<numeric>(scale).to_int()]);
741 new_pending_integrals.push_back(-scale);
744 new_pending_integrals.push_back(scale);
748 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
749 pending_integrals.front(),
754 result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
755 new_pending_integrals.front(),
759 new_pending_integrals.back() = 0;
760 result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
761 new_pending_integrals.front(),
768 // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
769 // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
772 result -= zeta(a.size());
774 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
775 pending_integrals.front(),
779 // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
780 // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
781 Gparameter new_a(a.begin()+1, a.end());
782 new_pending_integrals.push_back(0);
783 result -= depth_one_trafo_G(new_pending_integrals, new_a, scale, gsyms);
785 // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
786 // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
787 Gparameter new_pending_integrals_2;
788 new_pending_integrals_2.push_back(scale);
789 new_pending_integrals_2.push_back(0);
791 result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
792 pending_integrals.front(),
794 * depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
796 result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
803 // forward declaration
804 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
805 const Gparameter& pendint, const Gparameter& a_old, int scale,
806 const exvector& gsyms);
809 // G transformation [VSW]
810 ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale,
811 const exvector& gsyms)
813 // main recursion routine
815 // pendint = ( y1, b1, ..., br )
816 // a = ( a1, ..., amin, ..., aw )
819 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
820 // where sr replaces amin
822 // find smallest alpha, determine depth and trailing zeros, and check for convergence
824 int depth, trailing_zeros;
825 Gparameter::const_iterator min_it;
826 Gparameter::const_iterator firstzero =
827 check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
828 int min_it_pos = min_it - a.begin();
830 // special case: all a's are zero
837 result = G_eval(a, scale, gsyms);
839 if (pendint.size() > 0) {
840 result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
847 // handle trailing zeros
848 if (trailing_zeros > 0) {
850 Gparameter new_a(a.begin(), a.end()-1);
851 result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms);
852 for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) {
853 Gparameter new_a(a.begin(), it);
855 new_a.insert(new_a.end(), it, a.end()-1);
856 result -= G_transform(pendint, new_a, scale, gsyms);
858 return result / trailing_zeros;
863 if (pendint.size() > 0) {
864 return G_eval(convert_pending_integrals_G(pendint),
865 pendint.front(), gsyms)*
866 G_eval(a, scale, gsyms);
868 return G_eval(a, scale, gsyms);
872 // call basic transformation for depth equal one
874 return depth_one_trafo_G(pendint, a, scale, gsyms);
878 // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
879 // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
880 // + 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)
882 // smallest element in last place
883 if (min_it + 1 == a.end()) {
884 do { --min_it; } while (*min_it == 0);
886 Gparameter a1(a.begin(),min_it+1);
887 Gparameter a2(min_it+1,a.end());
889 ex result = G_transform(pendint, a2, scale, gsyms)*
890 G_transform(empty, a1, scale, gsyms);
892 result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms);
897 Gparameter::iterator changeit;
899 // first term G(a_1,..,0,...,a_w;a_0)
900 Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
901 Gparameter new_a = a;
902 new_a[min_it_pos] = 0;
903 ex result = G_transform(empty, new_a, scale, gsyms);
904 if (pendint.size() > 0) {
905 result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
906 pendint.front(), gsyms);
910 changeit = new_a.begin() + min_it_pos;
911 changeit = new_a.erase(changeit);
912 if (changeit != new_a.begin()) {
913 // smallest in the middle
914 new_pendint.push_back(*changeit);
915 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
916 new_pendint.front(), gsyms)*
917 G_transform(empty, new_a, scale, gsyms);
918 int buffer = *changeit;
920 result += G_transform(new_pendint, new_a, scale, gsyms);
922 new_pendint.pop_back();
924 new_pendint.push_back(*changeit);
925 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
926 new_pendint.front(), gsyms)*
927 G_transform(empty, new_a, scale, gsyms);
929 result -= G_transform(new_pendint, new_a, scale, gsyms);
931 // smallest at the front
932 new_pendint.push_back(scale);
933 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
934 new_pendint.front(), gsyms)*
935 G_transform(empty, new_a, scale, gsyms);
936 new_pendint.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);
941 result += G_transform(new_pendint, new_a, scale, gsyms);
947 // shuffles the two parameter list a1 and a2 and calls G_transform for every term except
948 // for the one that is equal to a_old
949 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
950 const Gparameter& pendint, const Gparameter& a_old, int scale,
951 const exvector& gsyms)
953 if (a1.size()==0 && a2.size()==0) {
954 // veto the one configuration we don't want
955 if ( a0 == a_old ) return 0;
957 return G_transform(pendint, a0, scale, gsyms);
963 aa0.insert(aa0.end(),a1.begin(),a1.end());
964 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms);
970 aa0.insert(aa0.end(),a2.begin(),a2.end());
971 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms);
974 Gparameter a1_removed(a1.begin()+1,a1.end());
975 Gparameter a2_removed(a2.begin()+1,a2.end());
980 a01.push_back( a1[0] );
981 a02.push_back( a2[0] );
983 return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms)
984 + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms);
987 // handles the transformations and the numerical evaluation of G
988 // the parameter x, s and y must only contain numerics
990 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
993 // do acceleration transformation (hoelder convolution [BBB])
994 // the parameter x, s and y must only contain numerics
996 G_do_hoelder(std::vector<cln::cl_N> x, /* yes, it's passed by value */
997 const std::vector<int>& s, const cln::cl_N& y)
1000 const std::size_t size = x.size();
1001 for (std::size_t i = 0; i < size; ++i)
1004 for (std::size_t r = 0; r <= size; ++r) {
1005 cln::cl_N buffer(1 & r ? -1 : 1);
1010 for (std::size_t i = 0; i < size; ++i) {
1011 if (x[i] == cln::cl_RA(1)/p) {
1012 p = p/2 + cln::cl_RA(3)/2;
1018 cln::cl_RA q = p/(p-1);
1019 std::vector<cln::cl_N> qlstx;
1020 std::vector<int> qlsts;
1021 for (std::size_t j = r; j >= 1; --j) {
1022 qlstx.push_back(cln::cl_N(1) - x[j-1]);
1023 if (instanceof(x[j-1], cln::cl_R_ring) && realpart(x[j-1]) > 1) {
1026 qlsts.push_back(-s[j-1]);
1029 if (qlstx.size() > 0) {
1030 buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
1032 std::vector<cln::cl_N> plstx;
1033 std::vector<int> plsts;
1034 for (std::size_t j = r+1; j <= size; ++j) {
1035 plstx.push_back(x[j-1]);
1036 plsts.push_back(s[j-1]);
1038 if (plstx.size() > 0) {
1039 buffer = buffer*G_numeric(plstx, plsts, 1/p);
1041 result = result + buffer;
1046 // convergence transformation, used for numerical evaluation of G function.
1047 // the parameter x, s and y must only contain numerics
1049 G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1052 // sort (|x|<->position) to determine indices
1053 typedef std::multimap<cln::cl_R, std::size_t> sortmap_t;
1055 std::size_t size = 0;
1056 for (std::size_t i = 0; i < x.size(); ++i) {
1058 sortmap.insert(std::make_pair(abs(x[i]), i));
1062 // include upper limit (scale)
1063 sortmap.insert(std::make_pair(abs(y), x.size()));
1065 // generate missing dummy-symbols
1067 // holding dummy-symbols for the G/Li transformations
1069 gsyms.push_back(symbol("GSYMS_ERROR"));
1070 cln::cl_N lastentry(0);
1071 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1072 if (it != sortmap.begin()) {
1073 if (it->second < x.size()) {
1074 if (x[it->second] == lastentry) {
1075 gsyms.push_back(gsyms.back());
1079 if (y == lastentry) {
1080 gsyms.push_back(gsyms.back());
1085 std::ostringstream os;
1087 gsyms.push_back(symbol(os.str()));
1089 if (it->second < x.size()) {
1090 lastentry = x[it->second];
1096 // fill position data according to sorted indices and prepare substitution list
1097 Gparameter a(x.size());
1099 std::size_t pos = 1;
1101 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1102 if (it->second < x.size()) {
1103 if (s[it->second] > 0) {
1104 a[it->second] = pos;
1106 a[it->second] = -int(pos);
1108 subslst[gsyms[pos]] = numeric(x[it->second]);
1111 subslst[gsyms[pos]] = numeric(y);
1116 // do transformation
1118 ex result = G_transform(pendint, a, scale, gsyms);
1119 // replace dummy symbols with their values
1120 result = result.eval().expand();
1121 result = result.subs(subslst).evalf();
1122 if (!is_a<numeric>(result))
1123 throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
1125 cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
1129 // handles the transformations and the numerical evaluation of G
1130 // the parameter x, s and y must only contain numerics
1132 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1135 // check for convergence and necessary accelerations
1136 bool need_trafo = false;
1137 bool need_hoelder = false;
1138 bool have_trailing_zero = false;
1139 std::size_t depth = 0;
1140 for (std::size_t i = 0; i < x.size(); ++i) {
1143 const cln::cl_N x_y = abs(x[i]) - y;
1144 if (instanceof(x_y, cln::cl_R_ring) &&
1145 realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2)))
1148 if (abs(abs(x[i]/y) - 1) < 0.01)
1149 need_hoelder = true;
1152 have_trailing_zero = zerop(x.back());
1153 if (have_trailing_zero) {
1156 need_hoelder = false;
1160 if (depth == 1 && x.size() == 2 && !need_trafo)
1161 return - Li_projection(2, y/x[1], cln::float_format(Digits));
1163 // do acceleration transformation (hoelder convolution [BBB])
1165 return G_do_hoelder(x, s, y);
1167 // convergence transformation
1169 return G_do_trafo(x, s, y);
1172 std::vector<cln::cl_N> newx;
1173 newx.reserve(x.size());
1175 m.reserve(x.size());
1178 cln::cl_N factor = y;
1179 for (std::size_t i = 0; i < x.size(); ++i) {
1183 newx.push_back(factor/x[i]);
1185 m.push_back(mcount);
1191 return sign*multipleLi_do_sum(m, newx);
1195 ex mLi_numeric(const lst& m, const lst& x)
1197 // let G_numeric do the transformation
1198 std::vector<cln::cl_N> newx;
1199 newx.reserve(x.nops());
1201 s.reserve(x.nops());
1202 cln::cl_N factor(1);
1203 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1204 for (int i = 1; i < *itm; ++i) {
1205 newx.push_back(cln::cl_N(0));
1208 const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
1210 newx.push_back(factor);
1211 if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) {
1218 return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
1222 } // end of anonymous namespace
1225 //////////////////////////////////////////////////////////////////////
1227 // Generalized multiple polylogarithm G(x, y) and G(x, s, y)
1231 //////////////////////////////////////////////////////////////////////
1234 static ex G2_evalf(const ex& x_, const ex& y)
1236 if (!y.info(info_flags::positive)) {
1237 return G(x_, y).hold();
1239 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1240 if (x.nops() == 0) {
1244 return G(x_, y).hold();
1247 s.reserve(x.nops());
1248 bool all_zero = true;
1249 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1250 if (!(*it).info(info_flags::numeric)) {
1251 return G(x_, y).hold();
1256 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1264 return pow(log(y), x.nops()) / factorial(x.nops());
1266 std::vector<cln::cl_N> xv;
1267 xv.reserve(x.nops());
1268 for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1269 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1270 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1271 return numeric(result);
1275 static ex G2_eval(const ex& x_, const ex& y)
1277 //TODO eval to MZV or H or S or Lin
1279 if (!y.info(info_flags::positive)) {
1280 return G(x_, y).hold();
1282 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1283 if (x.nops() == 0) {
1287 return G(x_, y).hold();
1290 s.reserve(x.nops());
1291 bool all_zero = true;
1292 bool crational = true;
1293 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1294 if (!(*it).info(info_flags::numeric)) {
1295 return G(x_, y).hold();
1297 if (!(*it).info(info_flags::crational)) {
1303 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1311 return pow(log(y), x.nops()) / factorial(x.nops());
1313 if (!y.info(info_flags::crational)) {
1317 return G(x_, y).hold();
1319 std::vector<cln::cl_N> xv;
1320 xv.reserve(x.nops());
1321 for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1322 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1323 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1324 return numeric(result);
1328 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1329 evalf_func(G2_evalf).
1331 do_not_evalf_params().
1334 // derivative_func(G2_deriv).
1335 // print_func<print_latex>(G2_print_latex).
1338 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1340 if (!y.info(info_flags::positive)) {
1341 return G(x_, s_, y).hold();
1343 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1344 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1345 if (x.nops() != s.nops()) {
1346 return G(x_, s_, y).hold();
1348 if (x.nops() == 0) {
1352 return G(x_, s_, y).hold();
1354 std::vector<int> sn;
1355 sn.reserve(s.nops());
1356 bool all_zero = true;
1357 for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1358 if (!(*itx).info(info_flags::numeric)) {
1359 return G(x_, y).hold();
1361 if (!(*its).info(info_flags::real)) {
1362 return G(x_, y).hold();
1367 if ( ex_to<numeric>(*itx).is_real() ) {
1368 if ( ex_to<numeric>(*itx).is_positive() ) {
1380 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1389 return pow(log(y), x.nops()) / factorial(x.nops());
1391 std::vector<cln::cl_N> xn;
1392 xn.reserve(x.nops());
1393 for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1394 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1395 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1396 return numeric(result);
1400 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1402 //TODO eval to MZV or H or S or Lin
1404 if (!y.info(info_flags::positive)) {
1405 return G(x_, s_, y).hold();
1407 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1408 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1409 if (x.nops() != s.nops()) {
1410 return G(x_, s_, y).hold();
1412 if (x.nops() == 0) {
1416 return G(x_, s_, y).hold();
1418 std::vector<int> sn;
1419 sn.reserve(s.nops());
1420 bool all_zero = true;
1421 bool crational = true;
1422 for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1423 if (!(*itx).info(info_flags::numeric)) {
1424 return G(x_, s_, y).hold();
1426 if (!(*its).info(info_flags::real)) {
1427 return G(x_, s_, y).hold();
1429 if (!(*itx).info(info_flags::crational)) {
1435 if ( ex_to<numeric>(*itx).is_real() ) {
1436 if ( ex_to<numeric>(*itx).is_positive() ) {
1448 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1457 return pow(log(y), x.nops()) / factorial(x.nops());
1459 if (!y.info(info_flags::crational)) {
1463 return G(x_, s_, y).hold();
1465 std::vector<cln::cl_N> xn;
1466 xn.reserve(x.nops());
1467 for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1468 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1469 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1470 return numeric(result);
1474 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1475 evalf_func(G3_evalf).
1477 do_not_evalf_params().
1480 // derivative_func(G3_deriv).
1481 // print_func<print_latex>(G3_print_latex).
1484 //////////////////////////////////////////////////////////////////////
1486 // Classical polylogarithm and multiple polylogarithm Li(m,x)
1490 //////////////////////////////////////////////////////////////////////
1493 static ex Li_evalf(const ex& m_, const ex& x_)
1495 // classical polylogs
1496 if (m_.info(info_flags::posint)) {
1497 if (x_.info(info_flags::numeric)) {
1498 int m__ = ex_to<numeric>(m_).to_int();
1499 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1500 const cln::cl_N result = Lin_numeric(m__, x__);
1501 return numeric(result);
1503 // try to numerically evaluate second argument
1504 ex x_val = x_.evalf();
1505 if (x_val.info(info_flags::numeric)) {
1506 int m__ = ex_to<numeric>(m_).to_int();
1507 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1508 const cln::cl_N result = Lin_numeric(m__, x__);
1509 return numeric(result);
1513 // multiple polylogs
1514 if (is_a<lst>(m_) && is_a<lst>(x_)) {
1516 const lst& m = ex_to<lst>(m_);
1517 const lst& x = ex_to<lst>(x_);
1518 if (m.nops() != x.nops()) {
1519 return Li(m_,x_).hold();
1521 if (x.nops() == 0) {
1524 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1525 return Li(m_,x_).hold();
1528 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1529 if (!(*itm).info(info_flags::posint)) {
1530 return Li(m_, x_).hold();
1532 if (!(*itx).info(info_flags::numeric)) {
1533 return Li(m_, x_).hold();
1540 return mLi_numeric(m, x);
1543 return Li(m_,x_).hold();
1547 static ex Li_eval(const ex& m_, const ex& x_)
1549 if (is_a<lst>(m_)) {
1550 if (is_a<lst>(x_)) {
1551 // multiple polylogs
1552 const lst& m = ex_to<lst>(m_);
1553 const lst& x = ex_to<lst>(x_);
1554 if (m.nops() != x.nops()) {
1555 return Li(m_,x_).hold();
1557 if (x.nops() == 0) {
1561 bool is_zeta = true;
1562 bool do_evalf = true;
1563 bool crational = true;
1564 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1565 if (!(*itm).info(info_flags::posint)) {
1566 return Li(m_,x_).hold();
1568 if ((*itx != _ex1) && (*itx != _ex_1)) {
1569 if (itx != x.begin()) {
1577 if (!(*itx).info(info_flags::numeric)) {
1580 if (!(*itx).info(info_flags::crational)) {
1586 for (lst::const_iterator itx = x.begin(); itx != x.end(); ++itx) {
1587 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
1588 // XXX: 1 + 0.0*I is considered equal to 1. However
1589 // the former is a not automatically converted
1590 // to a real number. Do the conversion explicitly
1591 // to avoid the "numeric::operator>(): complex inequality"
1592 // exception (and similar problems).
1593 newx.append(*itx != _ex_1 ? _ex1 : _ex_1);
1595 return zeta(m_, newx);
1599 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1600 return prefactor * H(newm, x[0]);
1602 if (do_evalf && !crational) {
1603 return mLi_numeric(m,x);
1606 return Li(m_, x_).hold();
1607 } else if (is_a<lst>(x_)) {
1608 return Li(m_, x_).hold();
1611 // classical polylogs
1619 return (pow(2,1-m_)-1) * zeta(m_);
1625 if (x_.is_equal(I)) {
1626 return power(Pi,_ex2)/_ex_48 + Catalan*I;
1628 if (x_.is_equal(-I)) {
1629 return power(Pi,_ex2)/_ex_48 - Catalan*I;
1632 if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1633 int m__ = ex_to<numeric>(m_).to_int();
1634 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1635 const cln::cl_N result = Lin_numeric(m__, x__);
1636 return numeric(result);
1639 return Li(m_, x_).hold();
1643 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1645 if (is_a<lst>(m) || is_a<lst>(x)) {
1648 seq.push_back(expair(Li(m, x), 0));
1649 return pseries(rel, seq);
1652 // classical polylog
1653 const ex x_pt = x.subs(rel, subs_options::no_pattern);
1654 if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1655 // First special case: x==0 (derivatives have poles)
1656 if (x_pt.is_zero()) {
1659 // manually construct the primitive expansion
1660 for (int i=1; i<order; ++i)
1661 ser += pow(s,i) / pow(numeric(i), m);
1662 // substitute the argument's series expansion
1663 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1664 // maybe that was terminating, so add a proper order term
1666 nseq.push_back(expair(Order(_ex1), order));
1667 ser += pseries(rel, nseq);
1668 // reexpanding it will collapse the series again
1669 return ser.series(rel, order);
1671 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1672 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1674 // all other cases should be safe, by now:
1675 throw do_taylor(); // caught by function::series()
1679 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1681 GINAC_ASSERT(deriv_param < 2);
1682 if (deriv_param == 0) {
1685 if (m_.nops() > 1) {
1686 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1689 if (is_a<lst>(m_)) {
1695 if (is_a<lst>(x_)) {
1701 return Li(m-1, x) / x;
1708 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1711 if (is_a<lst>(m_)) {
1717 if (is_a<lst>(x_)) {
1722 c.s << "\\mathrm{Li}_{";
1723 lst::const_iterator itm = m.begin();
1726 for (; itm != m.end(); itm++) {
1731 lst::const_iterator itx = x.begin();
1734 for (; itx != x.end(); itx++) {
1742 REGISTER_FUNCTION(Li,
1743 evalf_func(Li_evalf).
1745 series_func(Li_series).
1746 derivative_func(Li_deriv).
1747 print_func<print_latex>(Li_print_latex).
1748 do_not_evalf_params());
1751 //////////////////////////////////////////////////////////////////////
1753 // Nielsen's generalized polylogarithm S(n,p,x)
1757 //////////////////////////////////////////////////////////////////////
1760 // anonymous namespace for helper functions
1764 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1766 std::vector<std::vector<cln::cl_N> > Yn;
1767 int ynsize = 0; // number of Yn[]
1768 int ynlength = 100; // initial length of all Yn[i]
1771 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1772 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1773 // representing S_{n,p}(x).
1774 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1775 // equivalent Z-sum.
1776 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1777 // representing S_{n,p}(x).
1778 // The calculation of Y_n uses the values from Y_{n-1}.
1779 void fill_Yn(int n, const cln::float_format_t& prec)
1781 const int initsize = ynlength;
1782 //const int initsize = initsize_Yn;
1783 cln::cl_N one = cln::cl_float(1, prec);
1786 std::vector<cln::cl_N> buf(initsize);
1787 std::vector<cln::cl_N>::iterator it = buf.begin();
1788 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1789 *it = (*itprev) / cln::cl_N(n+1) * one;
1792 // sums with an index smaller than the depth are zero and need not to be calculated.
1793 // calculation starts with depth, which is n+2)
1794 for (int i=n+2; i<=initsize+n; i++) {
1795 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1801 std::vector<cln::cl_N> buf(initsize);
1802 std::vector<cln::cl_N>::iterator it = buf.begin();
1805 for (int i=2; i<=initsize; i++) {
1806 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1815 // make Yn longer ...
1816 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1819 cln::cl_N one = cln::cl_float(1, prec);
1821 Yn[0].resize(newsize);
1822 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
1824 for (int i=ynlength+1; i<=newsize; i++) {
1825 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1829 for (int n=1; n<ynsize; n++) {
1830 Yn[n].resize(newsize);
1831 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
1832 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1835 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1836 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1846 // helper function for S(n,p,x)
1848 cln::cl_N C(int n, int p)
1852 for (int k=0; k<p; k++) {
1853 for (int j=0; j<=(n+k-1)/2; j++) {
1857 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1860 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1867 result = result + cln::factorial(n+k-1)
1868 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1869 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1872 result = result - cln::factorial(n+k-1)
1873 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1874 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1879 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1880 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1883 result = result + cln::factorial(n+k-1)
1884 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1885 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1893 if (((np)/2+n) & 1) {
1894 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1897 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1905 // helper function for S(n,p,x)
1906 // [Kol] remark to (9.1)
1907 cln::cl_N a_k(int k)
1916 for (int m=2; m<=k; m++) {
1917 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1924 // helper function for S(n,p,x)
1925 // [Kol] remark to (9.1)
1926 cln::cl_N b_k(int k)
1935 for (int m=2; m<=k; m++) {
1936 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1943 // helper function for S(n,p,x)
1944 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1946 static cln::float_format_t oldprec = cln::default_float_format;
1949 return Li_projection(n+1, x, prec);
1952 // precision has changed, we need to clear lookup table Yn
1953 if ( oldprec != prec ) {
1960 // check if precalculated values are sufficient
1962 for (int i=ynsize; i<p-1; i++) {
1967 // should be done otherwise
1968 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1969 cln::cl_N xf = x * one;
1970 //cln::cl_N xf = x * cln::cl_float(1, prec);
1974 cln::cl_N factor = cln::expt(xf, p);
1978 if (i-p >= ynlength) {
1980 make_Yn_longer(ynlength*2, prec);
1982 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? ...
1983 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1984 factor = factor * xf;
1986 } while (res != resbuf);
1992 // helper function for S(n,p,x)
1993 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1996 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
1998 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
1999 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
2001 for (int s=0; s<n; s++) {
2003 for (int r=0; r<p; r++) {
2004 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
2005 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
2007 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2013 return S_do_sum(n, p, x, prec);
2017 // helper function for S(n,p,x)
2018 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
2022 // [Kol] (2.22) with (2.21)
2023 return cln::zeta(p+1);
2028 return cln::zeta(n+1);
2033 for (int nu=0; nu<n; nu++) {
2034 for (int rho=0; rho<=p; rho++) {
2035 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2036 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2039 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2046 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2048 // throw std::runtime_error("don't know how to evaluate this function!");
2051 // what is the desired float format?
2052 // first guess: default format
2053 cln::float_format_t prec = cln::default_float_format;
2054 const cln::cl_N value = x;
2055 // second guess: the argument's format
2056 if (!instanceof(realpart(value), cln::cl_RA_ring))
2057 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2058 else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2059 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2062 // 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
2063 // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
2064 if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
2066 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2067 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2069 for (int s=0; s<n; s++) {
2071 for (int r=0; r<p; r++) {
2072 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2073 * S_num(p-r,n-s,1-value) / cln::factorial(r);
2075 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2082 if (cln::abs(value) > 1) {
2086 for (int s=0; s<p; s++) {
2087 for (int r=0; r<=s; r++) {
2088 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2089 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2090 * S_num(n+s-r,p-s,cln::recip(value));
2093 result = result * cln::expt(cln::cl_I(-1),n);
2096 for (int r=0; r<n; r++) {
2097 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2099 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2101 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2106 if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
2109 for (int s=0; s<p-1; s++)
2112 ex res = H(m,numeric(value)).evalf();
2113 return ex_to<numeric>(res).to_cl_N();
2116 return S_projection(n, p, value, prec);
2121 } // end of anonymous namespace
2124 //////////////////////////////////////////////////////////////////////
2126 // Nielsen's generalized polylogarithm S(n,p,x)
2130 //////////////////////////////////////////////////////////////////////
2133 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2135 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2136 const int n_ = ex_to<numeric>(n).to_int();
2137 const int p_ = ex_to<numeric>(p).to_int();
2138 if (is_a<numeric>(x)) {
2139 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2140 const cln::cl_N result = S_num(n_, p_, x_);
2141 return numeric(result);
2143 ex x_val = x.evalf();
2144 if (is_a<numeric>(x_val)) {
2145 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2146 const cln::cl_N result = S_num(n_, p_, x_val_);
2147 return numeric(result);
2151 return S(n, p, x).hold();
2155 static ex S_eval(const ex& n, const ex& p, const ex& x)
2157 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2163 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2171 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2172 int n_ = ex_to<numeric>(n).to_int();
2173 int p_ = ex_to<numeric>(p).to_int();
2174 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2175 const cln::cl_N result = S_num(n_, p_, x_);
2176 return numeric(result);
2181 return pow(-log(1-x), p) / factorial(p);
2183 return S(n, p, x).hold();
2187 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2190 return Li(n+1, x).series(rel, order, options);
2193 const ex x_pt = x.subs(rel, subs_options::no_pattern);
2194 if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2195 // First special case: x==0 (derivatives have poles)
2196 if (x_pt.is_zero()) {
2199 // manually construct the primitive expansion
2200 // subsum = Euler-Zagier-Sum is needed
2201 // dirty hack (slow ...) calculation of subsum:
2202 std::vector<ex> presubsum, subsum;
2203 subsum.push_back(0);
2204 for (int i=1; i<order-1; ++i) {
2205 subsum.push_back(subsum[i-1] + numeric(1, i));
2207 for (int depth=2; depth<p; ++depth) {
2209 for (int i=1; i<order-1; ++i) {
2210 subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2214 for (int i=1; i<order; ++i) {
2215 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2217 // substitute the argument's series expansion
2218 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2219 // maybe that was terminating, so add a proper order term
2221 nseq.push_back(expair(Order(_ex1), order));
2222 ser += pseries(rel, nseq);
2223 // reexpanding it will collapse the series again
2224 return ser.series(rel, order);
2226 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2227 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2229 // all other cases should be safe, by now:
2230 throw do_taylor(); // caught by function::series()
2234 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2236 GINAC_ASSERT(deriv_param < 3);
2237 if (deriv_param < 2) {
2241 return S(n-1, p, x) / x;
2243 return S(n, p-1, x) / (1-x);
2248 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2250 c.s << "\\mathrm{S}_{";
2260 REGISTER_FUNCTION(S,
2261 evalf_func(S_evalf).
2263 series_func(S_series).
2264 derivative_func(S_deriv).
2265 print_func<print_latex>(S_print_latex).
2266 do_not_evalf_params());
2269 //////////////////////////////////////////////////////////////////////
2271 // Harmonic polylogarithm H(m,x)
2275 //////////////////////////////////////////////////////////////////////
2278 // anonymous namespace for helper functions
2282 // regulates the pole (used by 1/x-transformation)
2283 symbol H_polesign("IMSIGN");
2286 // convert parameters from H to Li representation
2287 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2288 // returns true if some parameters are negative
2289 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2291 // expand parameter list
2293 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
2295 for (ex count=*it-1; count > 0; count--) {
2299 } else if (*it < -1) {
2300 for (ex count=*it+1; count < 0; count++) {
2311 bool has_negative_parameters = false;
2313 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
2319 m.append((*it+acc-1) * signum);
2321 m.append((*it-acc+1) * signum);
2327 has_negative_parameters = true;
2330 if (has_negative_parameters) {
2331 for (std::size_t i=0; i<m.nops(); i++) {
2333 m.let_op(i) = -m.op(i);
2341 return has_negative_parameters;
2345 // recursivly transforms H to corresponding multiple polylogarithms
2346 struct map_trafo_H_convert_to_Li : public map_function
2348 ex operator()(const ex& e)
2350 if (is_a<add>(e) || is_a<mul>(e)) {
2351 return e.map(*this);
2353 if (is_a<function>(e)) {
2354 std::string name = ex_to<function>(e).get_name();
2357 if (is_a<lst>(e.op(0))) {
2358 parameter = ex_to<lst>(e.op(0));
2360 parameter = lst(e.op(0));
2367 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2368 s.let_op(0) = s.op(0) * arg;
2369 return pf * Li(m, s).hold();
2371 for (std::size_t i=0; i<m.nops(); i++) {
2374 s.let_op(0) = s.op(0) * arg;
2375 return Li(m, s).hold();
2384 // recursivly transforms H to corresponding zetas
2385 struct map_trafo_H_convert_to_zeta : public map_function
2387 ex operator()(const ex& e)
2389 if (is_a<add>(e) || is_a<mul>(e)) {
2390 return e.map(*this);
2392 if (is_a<function>(e)) {
2393 std::string name = ex_to<function>(e).get_name();
2396 if (is_a<lst>(e.op(0))) {
2397 parameter = ex_to<lst>(e.op(0));
2399 parameter = lst(e.op(0));
2405 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2406 return pf * zeta(m, s);
2417 // remove trailing zeros from H-parameters
2418 struct map_trafo_H_reduce_trailing_zeros : public map_function
2420 ex operator()(const ex& e)
2422 if (is_a<add>(e) || is_a<mul>(e)) {
2423 return e.map(*this);
2425 if (is_a<function>(e)) {
2426 std::string name = ex_to<function>(e).get_name();
2429 if (is_a<lst>(e.op(0))) {
2430 parameter = ex_to<lst>(e.op(0));
2432 parameter = lst(e.op(0));
2435 if (parameter.op(parameter.nops()-1) == 0) {
2438 if (parameter.nops() == 1) {
2443 lst::const_iterator it = parameter.begin();
2444 while ((it != parameter.end()) && (*it == 0)) {
2447 if (it == parameter.end()) {
2448 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2452 parameter.remove_last();
2453 std::size_t lastentry = parameter.nops();
2454 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2459 ex result = log(arg) * H(parameter,arg).hold();
2461 for (ex i=0; i<lastentry; i++) {
2462 if (parameter[i] > 0) {
2464 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2467 } else if (parameter[i] < 0) {
2469 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2477 if (lastentry < parameter.nops()) {
2478 result = result / (parameter.nops()-lastentry+1);
2479 return result.map(*this);
2491 // returns an expression with zeta functions corresponding to the parameter list for H
2492 ex convert_H_to_zeta(const lst& m)
2494 symbol xtemp("xtemp");
2495 map_trafo_H_reduce_trailing_zeros filter;
2496 map_trafo_H_convert_to_zeta filter2;
2497 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2501 // convert signs form Li to H representation
2502 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2505 lst::const_iterator itm = m.begin();
2506 lst::const_iterator itx = ++x.begin();
2511 while (itx != x.end()) {
2512 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
2513 // XXX: 1 + 0.0*I is considered equal to 1. However the former
2514 // is not automatically converted to a real number.
2515 // Do the conversion explicitly to avoid the
2516 // "numeric::operator>(): complex inequality" exception.
2517 signum *= (*itx != _ex_1) ? 1 : -1;
2519 res.append((*itm) * signum);
2527 // multiplies an one-dimensional H with another H
2529 ex trafo_H_mult(const ex& h1, const ex& h2)
2534 ex h1nops = h1.op(0).nops();
2535 ex h2nops = h2.op(0).nops();
2537 hshort = h2.op(0).op(0);
2538 hlong = ex_to<lst>(h1.op(0));
2540 hshort = h1.op(0).op(0);
2542 hlong = ex_to<lst>(h2.op(0));
2544 hlong = h2.op(0).op(0);
2547 for (std::size_t i=0; i<=hlong.nops(); i++) {
2551 newparameter.append(hlong[j]);
2553 newparameter.append(hshort);
2554 for (; j<hlong.nops(); j++) {
2555 newparameter.append(hlong[j]);
2557 res += H(newparameter, h1.op(1)).hold();
2563 // applies trafo_H_mult recursively on expressions
2564 struct map_trafo_H_mult : public map_function
2566 ex operator()(const ex& e)
2569 return e.map(*this);
2577 for (std::size_t pos=0; pos<e.nops(); pos++) {
2578 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2579 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2581 for (ex i=0; i<e.op(pos).op(1); i++) {
2582 Hlst.append(e.op(pos).op(0));
2586 } else if (is_a<function>(e.op(pos))) {
2587 std::string name = ex_to<function>(e.op(pos)).get_name();
2589 if (e.op(pos).op(0).nops() > 1) {
2592 Hlst.append(e.op(pos));
2597 result *= e.op(pos);
2600 if (Hlst.nops() > 0) {
2601 firstH = Hlst[Hlst.nops()-1];
2608 if (Hlst.nops() > 0) {
2609 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2611 for (std::size_t i=1; i<Hlst.nops(); i++) {
2612 result *= Hlst.op(i);
2614 result = result.expand();
2615 map_trafo_H_mult recursion;
2616 return recursion(result);
2627 // do integration [ReV] (55)
2628 // put parameter 0 in front of existing parameters
2629 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2633 if (is_a<function>(e)) {
2634 name = ex_to<function>(e).get_name();
2639 for (std::size_t i=0; i<e.nops(); i++) {
2640 if (is_a<function>(e.op(i))) {
2641 std::string name = ex_to<function>(e.op(i)).get_name();
2649 lst newparameter = ex_to<lst>(h.op(0));
2650 newparameter.prepend(0);
2651 ex addzeta = convert_H_to_zeta(newparameter);
2652 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2654 return e * (-H(lst(ex(0)),1/arg).hold());
2659 // do integration [ReV] (49)
2660 // put parameter 1 in front of existing parameters
2661 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2665 if (is_a<function>(e)) {
2666 name = ex_to<function>(e).get_name();
2671 for (std::size_t i=0; i<e.nops(); i++) {
2672 if (is_a<function>(e.op(i))) {
2673 std::string name = ex_to<function>(e.op(i)).get_name();
2681 lst newparameter = ex_to<lst>(h.op(0));
2682 newparameter.prepend(1);
2683 return e.subs(h == H(newparameter, h.op(1)).hold());
2685 return e * H(lst(ex(1)),1-arg).hold();
2690 // do integration [ReV] (55)
2691 // put parameter -1 in front of existing parameters
2692 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2696 if (is_a<function>(e)) {
2697 name = ex_to<function>(e).get_name();
2702 for (std::size_t i=0; i<e.nops(); i++) {
2703 if (is_a<function>(e.op(i))) {
2704 std::string name = ex_to<function>(e.op(i)).get_name();
2712 lst newparameter = ex_to<lst>(h.op(0));
2713 newparameter.prepend(-1);
2714 ex addzeta = convert_H_to_zeta(newparameter);
2715 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2717 ex addzeta = convert_H_to_zeta(lst(ex(-1)));
2718 return (e * (addzeta - H(lst(ex(-1)),1/arg).hold())).expand();
2723 // do integration [ReV] (55)
2724 // put parameter -1 in front of existing parameters
2725 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2729 if (is_a<function>(e)) {
2730 name = ex_to<function>(e).get_name();
2735 for (std::size_t i = 0; i < e.nops(); i++) {
2736 if (is_a<function>(e.op(i))) {
2737 std::string name = ex_to<function>(e.op(i)).get_name();
2745 lst newparameter = ex_to<lst>(h.op(0));
2746 newparameter.prepend(-1);
2747 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2749 return (e * H(lst(ex(-1)),(1-arg)/(1+arg)).hold()).expand();
2754 // do integration [ReV] (55)
2755 // put parameter 1 in front of existing parameters
2756 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2760 if (is_a<function>(e)) {
2761 name = ex_to<function>(e).get_name();
2766 for (std::size_t i = 0; i < e.nops(); i++) {
2767 if (is_a<function>(e.op(i))) {
2768 std::string name = ex_to<function>(e.op(i)).get_name();
2776 lst newparameter = ex_to<lst>(h.op(0));
2777 newparameter.prepend(1);
2778 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2780 return (e * H(lst(ex(1)),(1-arg)/(1+arg)).hold()).expand();
2785 // do x -> 1-x transformation
2786 struct map_trafo_H_1mx : public map_function
2788 ex operator()(const ex& e)
2790 if (is_a<add>(e) || is_a<mul>(e)) {
2791 return e.map(*this);
2794 if (is_a<function>(e)) {
2795 std::string name = ex_to<function>(e).get_name();
2798 lst parameter = ex_to<lst>(e.op(0));
2801 // special cases if all parameters are either 0, 1 or -1
2802 bool allthesame = true;
2803 if (parameter.op(0) == 0) {
2804 for (std::size_t i = 1; i < parameter.nops(); i++) {
2805 if (parameter.op(i) != 0) {
2812 for (int i=parameter.nops(); i>0; i--) {
2813 newparameter.append(1);
2815 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2817 } else if (parameter.op(0) == -1) {
2818 throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2820 for (std::size_t i = 1; i < parameter.nops(); i++) {
2821 if (parameter.op(i) != 1) {
2828 for (int i=parameter.nops(); i>0; i--) {
2829 newparameter.append(0);
2831 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2835 lst newparameter = parameter;
2836 newparameter.remove_first();
2838 if (parameter.op(0) == 0) {
2841 ex res = convert_H_to_zeta(parameter);
2842 //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2843 map_trafo_H_1mx recursion;
2844 ex buffer = recursion(H(newparameter, arg).hold());
2845 if (is_a<add>(buffer)) {
2846 for (std::size_t i = 0; i < buffer.nops(); i++) {
2847 res -= trafo_H_prepend_one(buffer.op(i), arg);
2850 res -= trafo_H_prepend_one(buffer, arg);
2857 map_trafo_H_1mx recursion;
2858 map_trafo_H_mult unify;
2859 ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2860 std::size_t firstzero = 0;
2861 while (parameter.op(firstzero) == 1) {
2864 for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2868 newparameter.append(parameter[j+1]);
2870 newparameter.append(1);
2871 for (; j<parameter.nops()-1; j++) {
2872 newparameter.append(parameter[j+1]);
2874 res -= H(newparameter, arg).hold();
2876 res = recursion(res).expand() / firstzero;
2886 // do x -> 1/x transformation
2887 struct map_trafo_H_1overx : public map_function
2889 ex operator()(const ex& e)
2891 if (is_a<add>(e) || is_a<mul>(e)) {
2892 return e.map(*this);
2895 if (is_a<function>(e)) {
2896 std::string name = ex_to<function>(e).get_name();
2899 lst parameter = ex_to<lst>(e.op(0));
2902 // special cases if all parameters are either 0, 1 or -1
2903 bool allthesame = true;
2904 if (parameter.op(0) == 0) {
2905 for (std::size_t i = 1; i < parameter.nops(); i++) {
2906 if (parameter.op(i) != 0) {
2912 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2914 } else if (parameter.op(0) == -1) {
2915 for (std::size_t i = 1; i < parameter.nops(); i++) {
2916 if (parameter.op(i) != -1) {
2922 map_trafo_H_mult unify;
2923 return unify((pow(H(lst(ex(-1)),1/arg).hold() - H(lst(ex(0)),1/arg).hold(), parameter.nops())
2924 / factorial(parameter.nops())).expand());
2927 for (std::size_t i = 1; i < parameter.nops(); i++) {
2928 if (parameter.op(i) != 1) {
2934 map_trafo_H_mult unify;
2935 return unify((pow(H(lst(ex(1)),1/arg).hold() + H(lst(ex(0)),1/arg).hold() + H_polesign, parameter.nops())
2936 / factorial(parameter.nops())).expand());
2940 lst newparameter = parameter;
2941 newparameter.remove_first();
2943 if (parameter.op(0) == 0) {
2946 ex res = convert_H_to_zeta(parameter);
2947 map_trafo_H_1overx recursion;
2948 ex buffer = recursion(H(newparameter, arg).hold());
2949 if (is_a<add>(buffer)) {
2950 for (std::size_t i = 0; i < buffer.nops(); i++) {
2951 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2954 res += trafo_H_1tx_prepend_zero(buffer, arg);
2958 } else if (parameter.op(0) == -1) {
2960 // leading negative one
2961 ex res = convert_H_to_zeta(parameter);
2962 map_trafo_H_1overx recursion;
2963 ex buffer = recursion(H(newparameter, arg).hold());
2964 if (is_a<add>(buffer)) {
2965 for (std::size_t i = 0; i < buffer.nops(); i++) {
2966 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2969 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2976 map_trafo_H_1overx recursion;
2977 map_trafo_H_mult unify;
2978 ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2979 std::size_t firstzero = 0;
2980 while (parameter.op(firstzero) == 1) {
2983 for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
2987 newparameter.append(parameter[j+1]);
2989 newparameter.append(1);
2990 for (; j<parameter.nops()-1; j++) {
2991 newparameter.append(parameter[j+1]);
2993 res -= H(newparameter, arg).hold();
2995 res = recursion(res).expand() / firstzero;
3007 // do x -> (1-x)/(1+x) transformation
3008 struct map_trafo_H_1mxt1px : public map_function
3010 ex operator()(const ex& e)
3012 if (is_a<add>(e) || is_a<mul>(e)) {
3013 return e.map(*this);
3016 if (is_a<function>(e)) {
3017 std::string name = ex_to<function>(e).get_name();
3020 lst parameter = ex_to<lst>(e.op(0));
3023 // special cases if all parameters are either 0, 1 or -1
3024 bool allthesame = true;
3025 if (parameter.op(0) == 0) {
3026 for (std::size_t i = 1; i < parameter.nops(); i++) {
3027 if (parameter.op(i) != 0) {
3033 map_trafo_H_mult unify;
3034 return unify((pow(-H(lst(ex(1)),(1-arg)/(1+arg)).hold() - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3035 / factorial(parameter.nops())).expand());
3037 } else if (parameter.op(0) == -1) {
3038 for (std::size_t i = 1; i < parameter.nops(); i++) {
3039 if (parameter.op(i) != -1) {
3045 map_trafo_H_mult unify;
3046 return unify((pow(log(2) - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3047 / factorial(parameter.nops())).expand());
3050 for (std::size_t i = 1; i < parameter.nops(); i++) {
3051 if (parameter.op(i) != 1) {
3057 map_trafo_H_mult unify;
3058 return unify((pow(-log(2) - H(lst(ex(0)),(1-arg)/(1+arg)).hold() + H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3059 / factorial(parameter.nops())).expand());
3063 lst newparameter = parameter;
3064 newparameter.remove_first();
3066 if (parameter.op(0) == 0) {
3069 ex res = convert_H_to_zeta(parameter);
3070 map_trafo_H_1mxt1px recursion;
3071 ex buffer = recursion(H(newparameter, arg).hold());
3072 if (is_a<add>(buffer)) {
3073 for (std::size_t i = 0; i < buffer.nops(); i++) {
3074 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3077 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3081 } else if (parameter.op(0) == -1) {
3083 // leading negative one
3084 ex res = convert_H_to_zeta(parameter);
3085 map_trafo_H_1mxt1px recursion;
3086 ex buffer = recursion(H(newparameter, arg).hold());
3087 if (is_a<add>(buffer)) {
3088 for (std::size_t i = 0; i < buffer.nops(); i++) {
3089 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3092 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3099 map_trafo_H_1mxt1px recursion;
3100 map_trafo_H_mult unify;
3101 ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
3102 std::size_t firstzero = 0;
3103 while (parameter.op(firstzero) == 1) {
3106 for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3110 newparameter.append(parameter[j+1]);
3112 newparameter.append(1);
3113 for (; j<parameter.nops()-1; j++) {
3114 newparameter.append(parameter[j+1]);
3116 res -= H(newparameter, arg).hold();
3118 res = recursion(res).expand() / firstzero;
3130 // do the actual summation.
3131 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3133 const int j = m.size();
3135 std::vector<cln::cl_N> t(j);
3137 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3138 cln::cl_N factor = cln::expt(x, j) * one;
3144 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3145 for (int k=j-2; k>=1; k--) {
3146 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3148 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3149 factor = factor * x;
3150 } while (t[0] != t0buf);
3156 } // end of anonymous namespace
3159 //////////////////////////////////////////////////////////////////////
3161 // Harmonic polylogarithm H(m,x)
3165 //////////////////////////////////////////////////////////////////////
3168 static ex H_evalf(const ex& x1, const ex& x2)
3170 if (is_a<lst>(x1)) {
3173 if (is_a<numeric>(x2)) {
3174 x = ex_to<numeric>(x2).to_cl_N();
3176 ex x2_val = x2.evalf();
3177 if (is_a<numeric>(x2_val)) {
3178 x = ex_to<numeric>(x2_val).to_cl_N();
3182 for (std::size_t i = 0; i < x1.nops(); i++) {
3183 if (!x1.op(i).info(info_flags::integer)) {
3184 return H(x1, x2).hold();
3187 if (x1.nops() < 1) {
3188 return H(x1, x2).hold();
3191 const lst& morg = ex_to<lst>(x1);
3192 // remove trailing zeros ...
3193 if (*(--morg.end()) == 0) {
3194 symbol xtemp("xtemp");
3195 map_trafo_H_reduce_trailing_zeros filter;
3196 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3198 // ... and expand parameter notation
3199 bool has_minus_one = false;
3201 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
3203 for (ex count=*it-1; count > 0; count--) {
3207 } else if (*it <= -1) {
3208 for (ex count=*it+1; count < 0; count++) {
3212 has_minus_one = true;
3219 if (cln::abs(x) < 0.95) {
3223 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3224 // negative parameters -> s_lst is filled
3225 std::vector<int> m_int;
3226 std::vector<cln::cl_N> x_cln;
3227 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
3228 it_int != m_lst.end(); it_int++, it_cln++) {
3229 m_int.push_back(ex_to<numeric>(*it_int).to_int());
3230 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3232 x_cln.front() = x_cln.front() * x;
3233 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3235 // only positive parameters
3237 if (m_lst.nops() == 1) {
3238 return Li(m_lst.op(0), x2).evalf();
3240 std::vector<int> m_int;
3241 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
3242 m_int.push_back(ex_to<numeric>(*it).to_int());
3244 return numeric(H_do_sum(m_int, x));
3248 symbol xtemp("xtemp");
3251 // ensure that the realpart of the argument is positive
3252 if (cln::realpart(x) < 0) {
3254 for (std::size_t i = 0; i < m.nops(); i++) {
3256 m.let_op(i) = -m.op(i);
3263 if (cln::abs(x) >= 2.0) {
3264 map_trafo_H_1overx trafo;
3265 res *= trafo(H(m, xtemp).hold());
3266 if (cln::imagpart(x) <= 0) {
3267 res = res.subs(H_polesign == -I*Pi);
3269 res = res.subs(H_polesign == I*Pi);
3271 return res.subs(xtemp == numeric(x)).evalf();
3274 // check transformations for 0.95 <= |x| < 2.0
3276 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3277 if (cln::abs(x-9.53) <= 9.47) {
3279 map_trafo_H_1mxt1px trafo;
3280 res *= trafo(H(m, xtemp).hold());
3283 if (has_minus_one) {
3284 map_trafo_H_convert_to_Li filter;
3285 return filter(H(m, numeric(x)).hold()).evalf();
3287 map_trafo_H_1mx trafo;
3288 res *= trafo(H(m, xtemp).hold());
3291 return res.subs(xtemp == numeric(x)).evalf();
3294 return H(x1,x2).hold();
3298 static ex H_eval(const ex& m_, const ex& x)
3301 if (is_a<lst>(m_)) {
3306 if (m.nops() == 0) {
3314 if (*m.begin() > _ex1) {
3320 } else if (*m.begin() < _ex_1) {
3326 } else if (*m.begin() == _ex0) {
3333 for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
3334 if ((*it).info(info_flags::integer)) {
3345 } else if (*it < _ex_1) {
3365 } else if (step == 1) {
3377 // if some m_i is not an integer
3378 return H(m_, x).hold();
3381 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3382 return convert_H_to_zeta(m);
3388 return H(m_, x).hold();
3390 return pow(log(x), m.nops()) / factorial(m.nops());
3393 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3395 } else if ((step == 1) && (pos1 == _ex0)){
3400 return pow(-1, p) * S(n, p, -x);
3406 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3407 return H(m_, x).evalf();
3409 return H(m_, x).hold();
3413 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3416 seq.push_back(expair(H(m, x), 0));
3417 return pseries(rel, seq);
3421 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3423 GINAC_ASSERT(deriv_param < 2);
3424 if (deriv_param == 0) {
3428 if (is_a<lst>(m_)) {
3444 return 1/(1-x) * H(m, x);
3445 } else if (mb == _ex_1) {
3446 return 1/(1+x) * H(m, x);
3453 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3456 if (is_a<lst>(m_)) {
3461 c.s << "\\mathrm{H}_{";
3462 lst::const_iterator itm = m.begin();
3465 for (; itm != m.end(); itm++) {
3475 REGISTER_FUNCTION(H,
3476 evalf_func(H_evalf).
3478 series_func(H_series).
3479 derivative_func(H_deriv).
3480 print_func<print_latex>(H_print_latex).
3481 do_not_evalf_params());
3484 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3485 ex convert_H_to_Li(const ex& m, const ex& x)
3487 map_trafo_H_reduce_trailing_zeros filter;
3488 map_trafo_H_convert_to_Li filter2;
3490 return filter2(filter(H(m, x).hold()));
3492 return filter2(filter(H(lst(m), x).hold()));
3497 //////////////////////////////////////////////////////////////////////
3499 // Multiple zeta values zeta(x) and zeta(x,s)
3503 //////////////////////////////////////////////////////////////////////
3506 // anonymous namespace for helper functions
3510 // parameters and data for [Cra] algorithm
3511 const cln::cl_N lambda = cln::cl_N("319/320");
3513 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3515 const int size = a.size();
3516 for (int n=0; n<size; n++) {
3518 for (int m=0; m<=n; m++) {
3519 c[n] = c[n] + a[m]*b[n-m];
3526 static void initcX(std::vector<cln::cl_N>& crX,
3527 const std::vector<int>& s,
3530 std::vector<cln::cl_N> crB(L2 + 1);
3531 for (int i=0; i<=L2; i++)
3532 crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i);
3536 std::vector<std::vector<cln::cl_N> > crG(s.size() - 1, std::vector<cln::cl_N>(L2 + 1));
3537 for (int m=0; m < (int)s.size() - 1; m++) {
3540 for (int i = 0; i <= L2; i++)
3541 crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2);
3546 for (std::size_t m = 0; m < s.size() - 1; m++) {
3547 std::vector<cln::cl_N> Xbuf(L2 + 1);
3548 for (int i = 0; i <= L2; i++)
3549 Xbuf[i] = crX[i] * crG[m][i];
3551 halfcyclic_convolute(Xbuf, crB, crX);
3557 static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk,
3558 const std::vector<cln::cl_N>& crX)
3560 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3561 cln::cl_N factor = cln::expt(lambda, Sqk);
3562 cln::cl_N res = factor / Sqk * crX[0] * one;
3567 factor = factor * lambda;
3569 res = res + crX[N] * factor / (N+Sqk);
3570 } while ((res != resbuf) || cln::zerop(crX[N]));
3576 static void calc_f(std::vector<std::vector<cln::cl_N> >& f_kj,
3577 const int maxr, const int L1)
3579 cln::cl_N t0, t1, t2, t3, t4;
3581 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
3582 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3584 t0 = cln::exp(-lambda);
3586 for (k=1; k<=L1; k++) {
3589 for (j=1; j<=maxr; j++) {
3592 for (i=2; i<=j; i++) {
3596 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3604 static cln::cl_N crandall_Z(const std::vector<int>& s,
3605 const std::vector<std::vector<cln::cl_N> >& f_kj)
3607 const int j = s.size();
3616 t0 = t0 + f_kj[q+j-2][s[0]-1];
3617 } while (t0 != t0buf);
3619 return t0 / cln::factorial(s[0]-1);
3622 std::vector<cln::cl_N> t(j);
3629 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3630 for (int k=j-2; k>=1; k--) {
3631 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3633 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3634 } while (t[0] != t0buf);
3636 return t[0] / cln::factorial(s[0]-1);
3641 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3643 std::vector<int> r = s;
3644 const int j = r.size();
3648 // decide on maximal size of f_kj for crandall_Z
3652 L1 = Digits * 3 + j*2;
3656 // decide on maximal size of crX for crandall_Y
3659 } else if (Digits < 86) {
3661 } else if (Digits < 192) {
3663 } else if (Digits < 394) {
3665 } else if (Digits < 808) {
3675 for (int i=0; i<j; i++) {
3682 std::vector<std::vector<cln::cl_N> > f_kj(L1);
3683 calc_f(f_kj, maxr, L1);
3685 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3687 std::vector<int> rz;
3690 for (int k=r.size()-1; k>0; k--) {
3692 rz.insert(rz.begin(), r.back());
3693 skp1buf = rz.front();
3697 std::vector<cln::cl_N> crX;
3700 for (int q=0; q<skp1buf; q++) {
3702 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k, crX);
3703 cln::cl_N pp2 = crandall_Z(rz, f_kj);
3708 res = res - pp1 * pp2 / cln::factorial(q);
3710 res = res + pp1 * pp2 / cln::factorial(q);
3713 rz.front() = skp1buf;
3715 rz.insert(rz.begin(), r.back());
3717 std::vector<cln::cl_N> crX;
3718 initcX(crX, rz, L2);
3720 res = (res + crandall_Y_loop(S-j, crX)) / r0factorial
3721 + crandall_Z(rz, f_kj);
3727 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3729 const int j = r.size();
3731 // buffer for subsums
3732 std::vector<cln::cl_N> t(j);
3733 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3740 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3741 for (int k=j-2; k>=0; k--) {
3742 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3744 } while (t[0] != t0buf);
3750 // does Hoelder convolution. see [BBB] (7.0)
3751 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3753 // prepare parameters
3754 // holds Li arguments in [BBB] notation
3755 std::vector<int> s = s_;
3756 std::vector<int> m_p = m_;
3757 std::vector<int> m_q;
3758 // holds Li arguments in nested sums notation
3759 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3760 s_p[0] = s_p[0] * cln::cl_N("1/2");
3761 // convert notations
3763 for (std::size_t i = 0; i < s_.size(); i++) {
3768 s[i] = sig * std::abs(s[i]);
3770 std::vector<cln::cl_N> s_q;
3771 cln::cl_N signum = 1;
3774 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3779 // change parameters
3780 if (s.front() > 0) {
3781 if (m_p.front() == 1) {
3782 m_p.erase(m_p.begin());
3783 s_p.erase(s_p.begin());
3784 if (s_p.size() > 0) {
3785 s_p.front() = s_p.front() * cln::cl_N("1/2");
3791 m_q.insert(m_q.begin(), 1);
3792 if (s_q.size() > 0) {
3793 s_q.front() = s_q.front() * 2;
3795 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3798 if (m_p.front() == 1) {
3799 m_p.erase(m_p.begin());
3800 cln::cl_N spbuf = s_p.front();
3801 s_p.erase(s_p.begin());
3802 if (s_p.size() > 0) {
3803 s_p.front() = s_p.front() * spbuf;
3806 m_q.insert(m_q.begin(), 1);
3807 if (s_q.size() > 0) {
3808 s_q.front() = s_q.front() * 4;
3810 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3814 m_q.insert(m_q.begin(), 1);
3815 if (s_q.size() > 0) {
3816 s_q.front() = s_q.front() * 2;
3818 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3823 if (m_p.size() == 0) break;
3825 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3830 res = res + signum * multipleLi_do_sum(m_q, s_q);
3836 } // end of anonymous namespace
3839 //////////////////////////////////////////////////////////////////////
3841 // Multiple zeta values zeta(x)
3845 //////////////////////////////////////////////////////////////////////
3848 static ex zeta1_evalf(const ex& x)
3850 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3852 // multiple zeta value
3853 const int count = x.nops();
3854 const lst& xlst = ex_to<lst>(x);
3855 std::vector<int> r(count);
3857 // check parameters and convert them
3858 lst::const_iterator it1 = xlst.begin();
3859 std::vector<int>::iterator it2 = r.begin();
3861 if (!(*it1).info(info_flags::posint)) {
3862 return zeta(x).hold();
3864 *it2 = ex_to<numeric>(*it1).to_int();
3867 } while (it2 != r.end());
3869 // check for divergence
3871 return zeta(x).hold();
3874 // decide on summation algorithm
3875 // this is still a bit clumsy
3876 int limit = (Digits>17) ? 10 : 6;
3877 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3878 return numeric(zeta_do_sum_Crandall(r));
3880 return numeric(zeta_do_sum_simple(r));
3884 // single zeta value
3885 if (is_exactly_a<numeric>(x) && (x != 1)) {
3887 return zeta(ex_to<numeric>(x));
3888 } catch (const dunno &e) { }
3891 return zeta(x).hold();
3895 static ex zeta1_eval(const ex& m)
3897 if (is_exactly_a<lst>(m)) {
3898 if (m.nops() == 1) {
3899 return zeta(m.op(0));
3901 return zeta(m).hold();
3904 if (m.info(info_flags::numeric)) {
3905 const numeric& y = ex_to<numeric>(m);
3906 // trap integer arguments:
3907 if (y.is_integer()) {
3911 if (y.is_equal(*_num1_p)) {
3912 return zeta(m).hold();
3914 if (y.info(info_flags::posint)) {
3915 if (y.info(info_flags::odd)) {
3916 return zeta(m).hold();
3918 return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3921 if (y.info(info_flags::odd)) {
3922 return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3929 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3930 return zeta1_evalf(m);
3933 return zeta(m).hold();
3937 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3939 GINAC_ASSERT(deriv_param==0);
3941 if (is_exactly_a<lst>(m)) {
3944 return zetaderiv(_ex1, m);
3949 static void zeta1_print_latex(const ex& m_, const print_context& c)
3952 if (is_a<lst>(m_)) {
3953 const lst& m = ex_to<lst>(m_);
3954 lst::const_iterator it = m.begin();
3957 for (; it != m.end(); it++) {
3968 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
3969 evalf_func(zeta1_evalf).
3970 eval_func(zeta1_eval).
3971 derivative_func(zeta1_deriv).
3972 print_func<print_latex>(zeta1_print_latex).
3973 do_not_evalf_params().
3977 //////////////////////////////////////////////////////////////////////
3979 // Alternating Euler sum zeta(x,s)
3983 //////////////////////////////////////////////////////////////////////
3986 static ex zeta2_evalf(const ex& x, const ex& s)
3988 if (is_exactly_a<lst>(x)) {
3990 // alternating Euler sum
3991 const int count = x.nops();
3992 const lst& xlst = ex_to<lst>(x);
3993 const lst& slst = ex_to<lst>(s);
3994 std::vector<int> xi(count);
3995 std::vector<int> si(count);
3997 // check parameters and convert them
3998 lst::const_iterator it_xread = xlst.begin();
3999 lst::const_iterator it_sread = slst.begin();
4000 std::vector<int>::iterator it_xwrite = xi.begin();
4001 std::vector<int>::iterator it_swrite = si.begin();
4003 if (!(*it_xread).info(info_flags::posint)) {
4004 return zeta(x, s).hold();
4006 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
4007 if (*it_sread > 0) {
4016 } while (it_xwrite != xi.end());
4018 // check for divergence
4019 if ((xi[0] == 1) && (si[0] == 1)) {
4020 return zeta(x, s).hold();
4023 // use Hoelder convolution
4024 return numeric(zeta_do_Hoelder_convolution(xi, si));
4027 return zeta(x, s).hold();
4031 static ex zeta2_eval(const ex& m, const ex& s_)
4033 if (is_exactly_a<lst>(s_)) {
4034 const lst& s = ex_to<lst>(s_);
4035 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
4036 if ((*it).info(info_flags::positive)) {
4039 return zeta(m, s_).hold();
4042 } else if (s_.info(info_flags::positive)) {
4046 return zeta(m, s_).hold();
4050 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
4052 GINAC_ASSERT(deriv_param==0);
4054 if (is_exactly_a<lst>(m)) {
4057 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
4058 return zetaderiv(_ex1, m);
4065 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
4068 if (is_a<lst>(m_)) {
4074 if (is_a<lst>(s_)) {
4080 lst::const_iterator itm = m.begin();
4081 lst::const_iterator its = s.begin();
4083 c.s << "\\overline{";
4091 for (; itm != m.end(); itm++, its++) {
4094 c.s << "\\overline{";
4105 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
4106 evalf_func(zeta2_evalf).
4107 eval_func(zeta2_eval).
4108 derivative_func(zeta2_deriv).
4109 print_func<print_latex>(zeta2_print_latex).
4110 do_not_evalf_params().
4114 } // namespace GiNaC