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) &&
1024 realpart(x[j-1]) > 1 && realpart(x[j-1]) <= 2) {
1025 qlsts.push_back(s[j-1]);
1027 qlsts.push_back(-s[j-1]);
1030 if (qlstx.size() > 0) {
1031 buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
1033 std::vector<cln::cl_N> plstx;
1034 std::vector<int> plsts;
1035 for (std::size_t j = r+1; j <= size; ++j) {
1036 plstx.push_back(x[j-1]);
1037 plsts.push_back(s[j-1]);
1039 if (plstx.size() > 0) {
1040 buffer = buffer*G_numeric(plstx, plsts, 1/p);
1042 result = result + buffer;
1047 // convergence transformation, used for numerical evaluation of G function.
1048 // the parameter x, s and y must only contain numerics
1050 G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1053 // sort (|x|<->position) to determine indices
1054 typedef std::multimap<cln::cl_R, std::size_t> sortmap_t;
1056 std::size_t size = 0;
1057 for (std::size_t i = 0; i < x.size(); ++i) {
1059 sortmap.insert(std::make_pair(abs(x[i]), i));
1063 // include upper limit (scale)
1064 sortmap.insert(std::make_pair(abs(y), x.size()));
1066 // generate missing dummy-symbols
1068 // holding dummy-symbols for the G/Li transformations
1070 gsyms.push_back(symbol("GSYMS_ERROR"));
1071 cln::cl_N lastentry(0);
1072 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1073 if (it != sortmap.begin()) {
1074 if (it->second < x.size()) {
1075 if (x[it->second] == lastentry) {
1076 gsyms.push_back(gsyms.back());
1080 if (y == lastentry) {
1081 gsyms.push_back(gsyms.back());
1086 std::ostringstream os;
1088 gsyms.push_back(symbol(os.str()));
1090 if (it->second < x.size()) {
1091 lastentry = x[it->second];
1097 // fill position data according to sorted indices and prepare substitution list
1098 Gparameter a(x.size());
1100 std::size_t pos = 1;
1102 for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1103 if (it->second < x.size()) {
1104 if (s[it->second] > 0) {
1105 a[it->second] = pos;
1107 a[it->second] = -int(pos);
1109 subslst[gsyms[pos]] = numeric(x[it->second]);
1112 subslst[gsyms[pos]] = numeric(y);
1117 // do transformation
1119 ex result = G_transform(pendint, a, scale, gsyms);
1120 // replace dummy symbols with their values
1121 result = result.eval().expand();
1122 result = result.subs(subslst).evalf();
1123 if (!is_a<numeric>(result))
1124 throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
1126 cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
1130 // handles the transformations and the numerical evaluation of G
1131 // the parameter x, s and y must only contain numerics
1133 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1136 // check for convergence and necessary accelerations
1137 bool need_trafo = false;
1138 bool need_hoelder = 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 if (zerop(x[x.size() - 1]))
1155 if (depth == 1 && x.size() == 2 && !need_trafo)
1156 return - Li_projection(2, y/x[1], cln::float_format(Digits));
1158 // do acceleration transformation (hoelder convolution [BBB])
1160 return G_do_hoelder(x, s, y);
1162 // convergence transformation
1164 return G_do_trafo(x, s, y);
1167 std::vector<cln::cl_N> newx;
1168 newx.reserve(x.size());
1170 m.reserve(x.size());
1173 cln::cl_N factor = y;
1174 for (std::size_t i = 0; i < x.size(); ++i) {
1178 newx.push_back(factor/x[i]);
1180 m.push_back(mcount);
1186 return sign*multipleLi_do_sum(m, newx);
1190 ex mLi_numeric(const lst& m, const lst& x)
1192 // let G_numeric do the transformation
1193 std::vector<cln::cl_N> newx;
1194 newx.reserve(x.nops());
1196 s.reserve(x.nops());
1197 cln::cl_N factor(1);
1198 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1199 for (int i = 1; i < *itm; ++i) {
1200 newx.push_back(cln::cl_N(0));
1203 const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
1205 newx.push_back(factor);
1206 if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) {
1213 return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
1217 } // end of anonymous namespace
1220 //////////////////////////////////////////////////////////////////////
1222 // Generalized multiple polylogarithm G(x, y) and G(x, s, y)
1226 //////////////////////////////////////////////////////////////////////
1229 static ex G2_evalf(const ex& x_, const ex& y)
1231 if (!y.info(info_flags::positive)) {
1232 return G(x_, y).hold();
1234 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1235 if (x.nops() == 0) {
1239 return G(x_, y).hold();
1242 s.reserve(x.nops());
1243 bool all_zero = true;
1244 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1245 if (!(*it).info(info_flags::numeric)) {
1246 return G(x_, y).hold();
1251 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1259 return pow(log(y), x.nops()) / factorial(x.nops());
1261 std::vector<cln::cl_N> xv;
1262 xv.reserve(x.nops());
1263 for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1264 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1265 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1266 return numeric(result);
1270 static ex G2_eval(const ex& x_, const ex& y)
1272 //TODO eval to MZV or H or S or Lin
1274 if (!y.info(info_flags::positive)) {
1275 return G(x_, y).hold();
1277 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1278 if (x.nops() == 0) {
1282 return G(x_, y).hold();
1285 s.reserve(x.nops());
1286 bool all_zero = true;
1287 bool crational = true;
1288 for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1289 if (!(*it).info(info_flags::numeric)) {
1290 return G(x_, y).hold();
1292 if (!(*it).info(info_flags::crational)) {
1298 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1306 return pow(log(y), x.nops()) / factorial(x.nops());
1308 if (!y.info(info_flags::crational)) {
1312 return G(x_, y).hold();
1314 std::vector<cln::cl_N> xv;
1315 xv.reserve(x.nops());
1316 for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1317 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1318 cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1319 return numeric(result);
1323 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1324 evalf_func(G2_evalf).
1326 do_not_evalf_params().
1329 // derivative_func(G2_deriv).
1330 // print_func<print_latex>(G2_print_latex).
1333 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1335 if (!y.info(info_flags::positive)) {
1336 return G(x_, s_, y).hold();
1338 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1339 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1340 if (x.nops() != s.nops()) {
1341 return G(x_, s_, y).hold();
1343 if (x.nops() == 0) {
1347 return G(x_, s_, y).hold();
1349 std::vector<int> sn;
1350 sn.reserve(s.nops());
1351 bool all_zero = true;
1352 for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1353 if (!(*itx).info(info_flags::numeric)) {
1354 return G(x_, y).hold();
1356 if (!(*its).info(info_flags::real)) {
1357 return G(x_, y).hold();
1362 if ( ex_to<numeric>(*itx).is_real() ) {
1363 if ( ex_to<numeric>(*itx).is_positive() ) {
1375 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1384 return pow(log(y), x.nops()) / factorial(x.nops());
1386 std::vector<cln::cl_N> xn;
1387 xn.reserve(x.nops());
1388 for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1389 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1390 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1391 return numeric(result);
1395 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1397 //TODO eval to MZV or H or S or Lin
1399 if (!y.info(info_flags::positive)) {
1400 return G(x_, s_, y).hold();
1402 lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1403 lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1404 if (x.nops() != s.nops()) {
1405 return G(x_, s_, y).hold();
1407 if (x.nops() == 0) {
1411 return G(x_, s_, y).hold();
1413 std::vector<int> sn;
1414 sn.reserve(s.nops());
1415 bool all_zero = true;
1416 bool crational = true;
1417 for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1418 if (!(*itx).info(info_flags::numeric)) {
1419 return G(x_, s_, y).hold();
1421 if (!(*its).info(info_flags::real)) {
1422 return G(x_, s_, y).hold();
1424 if (!(*itx).info(info_flags::crational)) {
1430 if ( ex_to<numeric>(*itx).is_real() ) {
1431 if ( ex_to<numeric>(*itx).is_positive() ) {
1443 if ( ex_to<numeric>(*itx).imag() > 0 ) {
1452 return pow(log(y), x.nops()) / factorial(x.nops());
1454 if (!y.info(info_flags::crational)) {
1458 return G(x_, s_, y).hold();
1460 std::vector<cln::cl_N> xn;
1461 xn.reserve(x.nops());
1462 for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1463 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1464 cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1465 return numeric(result);
1469 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1470 evalf_func(G3_evalf).
1472 do_not_evalf_params().
1475 // derivative_func(G3_deriv).
1476 // print_func<print_latex>(G3_print_latex).
1479 //////////////////////////////////////////////////////////////////////
1481 // Classical polylogarithm and multiple polylogarithm Li(m,x)
1485 //////////////////////////////////////////////////////////////////////
1488 static ex Li_evalf(const ex& m_, const ex& x_)
1490 // classical polylogs
1491 if (m_.info(info_flags::posint)) {
1492 if (x_.info(info_flags::numeric)) {
1493 int m__ = ex_to<numeric>(m_).to_int();
1494 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1495 const cln::cl_N result = Lin_numeric(m__, x__);
1496 return numeric(result);
1498 // try to numerically evaluate second argument
1499 ex x_val = x_.evalf();
1500 if (x_val.info(info_flags::numeric)) {
1501 int m__ = ex_to<numeric>(m_).to_int();
1502 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1503 const cln::cl_N result = Lin_numeric(m__, x__);
1504 return numeric(result);
1508 // multiple polylogs
1509 if (is_a<lst>(m_) && is_a<lst>(x_)) {
1511 const lst& m = ex_to<lst>(m_);
1512 const lst& x = ex_to<lst>(x_);
1513 if (m.nops() != x.nops()) {
1514 return Li(m_,x_).hold();
1516 if (x.nops() == 0) {
1519 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1520 return Li(m_,x_).hold();
1523 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1524 if (!(*itm).info(info_flags::posint)) {
1525 return Li(m_, x_).hold();
1527 if (!(*itx).info(info_flags::numeric)) {
1528 return Li(m_, x_).hold();
1535 return mLi_numeric(m, x);
1538 return Li(m_,x_).hold();
1542 static ex Li_eval(const ex& m_, const ex& x_)
1544 if (is_a<lst>(m_)) {
1545 if (is_a<lst>(x_)) {
1546 // multiple polylogs
1547 const lst& m = ex_to<lst>(m_);
1548 const lst& x = ex_to<lst>(x_);
1549 if (m.nops() != x.nops()) {
1550 return Li(m_,x_).hold();
1552 if (x.nops() == 0) {
1556 bool is_zeta = true;
1557 bool do_evalf = true;
1558 bool crational = true;
1559 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1560 if (!(*itm).info(info_flags::posint)) {
1561 return Li(m_,x_).hold();
1563 if ((*itx != _ex1) && (*itx != _ex_1)) {
1564 if (itx != x.begin()) {
1572 if (!(*itx).info(info_flags::numeric)) {
1575 if (!(*itx).info(info_flags::crational)) {
1581 for (lst::const_iterator itx = x.begin(); itx != x.end(); ++itx) {
1582 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
1583 // XXX: 1 + 0.0*I is considered equal to 1. However
1584 // the former is a not automatically converted
1585 // to a real number. Do the conversion explicitly
1586 // to avoid the "numeric::operator>(): complex inequality"
1587 // exception (and similar problems).
1588 newx.append(*itx != _ex_1 ? _ex1 : _ex_1);
1590 return zeta(m_, newx);
1594 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1595 return prefactor * H(newm, x[0]);
1597 if (do_evalf && !crational) {
1598 return mLi_numeric(m,x);
1601 return Li(m_, x_).hold();
1602 } else if (is_a<lst>(x_)) {
1603 return Li(m_, x_).hold();
1606 // classical polylogs
1614 return (pow(2,1-m_)-1) * zeta(m_);
1620 if (x_.is_equal(I)) {
1621 return power(Pi,_ex2)/_ex_48 + Catalan*I;
1623 if (x_.is_equal(-I)) {
1624 return power(Pi,_ex2)/_ex_48 - Catalan*I;
1627 if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1628 int m__ = ex_to<numeric>(m_).to_int();
1629 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1630 const cln::cl_N result = Lin_numeric(m__, x__);
1631 return numeric(result);
1634 return Li(m_, x_).hold();
1638 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1640 if (is_a<lst>(m) || is_a<lst>(x)) {
1643 seq.push_back(expair(Li(m, x), 0));
1644 return pseries(rel, seq);
1647 // classical polylog
1648 const ex x_pt = x.subs(rel, subs_options::no_pattern);
1649 if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1650 // First special case: x==0 (derivatives have poles)
1651 if (x_pt.is_zero()) {
1654 // manually construct the primitive expansion
1655 for (int i=1; i<order; ++i)
1656 ser += pow(s,i) / pow(numeric(i), m);
1657 // substitute the argument's series expansion
1658 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1659 // maybe that was terminating, so add a proper order term
1661 nseq.push_back(expair(Order(_ex1), order));
1662 ser += pseries(rel, nseq);
1663 // reexpanding it will collapse the series again
1664 return ser.series(rel, order);
1666 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1667 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1669 // all other cases should be safe, by now:
1670 throw do_taylor(); // caught by function::series()
1674 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1676 GINAC_ASSERT(deriv_param < 2);
1677 if (deriv_param == 0) {
1680 if (m_.nops() > 1) {
1681 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1684 if (is_a<lst>(m_)) {
1690 if (is_a<lst>(x_)) {
1696 return Li(m-1, x) / x;
1703 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1706 if (is_a<lst>(m_)) {
1712 if (is_a<lst>(x_)) {
1717 c.s << "\\mathrm{Li}_{";
1718 lst::const_iterator itm = m.begin();
1721 for (; itm != m.end(); itm++) {
1726 lst::const_iterator itx = x.begin();
1729 for (; itx != x.end(); itx++) {
1737 REGISTER_FUNCTION(Li,
1738 evalf_func(Li_evalf).
1740 series_func(Li_series).
1741 derivative_func(Li_deriv).
1742 print_func<print_latex>(Li_print_latex).
1743 do_not_evalf_params());
1746 //////////////////////////////////////////////////////////////////////
1748 // Nielsen's generalized polylogarithm S(n,p,x)
1752 //////////////////////////////////////////////////////////////////////
1755 // anonymous namespace for helper functions
1759 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1761 std::vector<std::vector<cln::cl_N> > Yn;
1762 int ynsize = 0; // number of Yn[]
1763 int ynlength = 100; // initial length of all Yn[i]
1766 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1767 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1768 // representing S_{n,p}(x).
1769 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1770 // equivalent Z-sum.
1771 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1772 // representing S_{n,p}(x).
1773 // The calculation of Y_n uses the values from Y_{n-1}.
1774 void fill_Yn(int n, const cln::float_format_t& prec)
1776 const int initsize = ynlength;
1777 //const int initsize = initsize_Yn;
1778 cln::cl_N one = cln::cl_float(1, prec);
1781 std::vector<cln::cl_N> buf(initsize);
1782 std::vector<cln::cl_N>::iterator it = buf.begin();
1783 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1784 *it = (*itprev) / cln::cl_N(n+1) * one;
1787 // sums with an index smaller than the depth are zero and need not to be calculated.
1788 // calculation starts with depth, which is n+2)
1789 for (int i=n+2; i<=initsize+n; i++) {
1790 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1796 std::vector<cln::cl_N> buf(initsize);
1797 std::vector<cln::cl_N>::iterator it = buf.begin();
1800 for (int i=2; i<=initsize; i++) {
1801 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1810 // make Yn longer ...
1811 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1814 cln::cl_N one = cln::cl_float(1, prec);
1816 Yn[0].resize(newsize);
1817 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
1819 for (int i=ynlength+1; i<=newsize; i++) {
1820 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1824 for (int n=1; n<ynsize; n++) {
1825 Yn[n].resize(newsize);
1826 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
1827 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1830 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1831 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1841 // helper function for S(n,p,x)
1843 cln::cl_N C(int n, int p)
1847 for (int k=0; k<p; k++) {
1848 for (int j=0; j<=(n+k-1)/2; j++) {
1852 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1855 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1862 result = result + cln::factorial(n+k-1)
1863 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1864 / (cln::factorial(k) * cln::factorial(n-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));
1874 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1875 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1878 result = result + cln::factorial(n+k-1)
1879 * 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));
1888 if (((np)/2+n) & 1) {
1889 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1892 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1900 // helper function for S(n,p,x)
1901 // [Kol] remark to (9.1)
1902 cln::cl_N a_k(int k)
1911 for (int m=2; m<=k; m++) {
1912 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1919 // helper function for S(n,p,x)
1920 // [Kol] remark to (9.1)
1921 cln::cl_N b_k(int k)
1930 for (int m=2; m<=k; m++) {
1931 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1938 // helper function for S(n,p,x)
1939 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1941 static cln::float_format_t oldprec = cln::default_float_format;
1944 return Li_projection(n+1, x, prec);
1947 // precision has changed, we need to clear lookup table Yn
1948 if ( oldprec != prec ) {
1955 // check if precalculated values are sufficient
1957 for (int i=ynsize; i<p-1; i++) {
1962 // should be done otherwise
1963 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1964 cln::cl_N xf = x * one;
1965 //cln::cl_N xf = x * cln::cl_float(1, prec);
1969 cln::cl_N factor = cln::expt(xf, p);
1973 if (i-p >= ynlength) {
1975 make_Yn_longer(ynlength*2, prec);
1977 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? ...
1978 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1979 factor = factor * xf;
1981 } while (res != resbuf);
1987 // helper function for S(n,p,x)
1988 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1991 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
1993 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
1994 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
1996 for (int s=0; s<n; s++) {
1998 for (int r=0; r<p; r++) {
1999 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
2000 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
2002 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2008 return S_do_sum(n, p, x, prec);
2012 // helper function for S(n,p,x)
2013 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
2017 // [Kol] (2.22) with (2.21)
2018 return cln::zeta(p+1);
2023 return cln::zeta(n+1);
2028 for (int nu=0; nu<n; nu++) {
2029 for (int rho=0; rho<=p; rho++) {
2030 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2031 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2034 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2041 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2043 // throw std::runtime_error("don't know how to evaluate this function!");
2046 // what is the desired float format?
2047 // first guess: default format
2048 cln::float_format_t prec = cln::default_float_format;
2049 const cln::cl_N value = x;
2050 // second guess: the argument's format
2051 if (!instanceof(realpart(value), cln::cl_RA_ring))
2052 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2053 else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2054 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2057 // 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
2058 // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
2059 if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
2061 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2062 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2064 for (int s=0; s<n; s++) {
2066 for (int r=0; r<p; r++) {
2067 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2068 * S_num(p-r,n-s,1-value) / cln::factorial(r);
2070 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2077 if (cln::abs(value) > 1) {
2081 for (int s=0; s<p; s++) {
2082 for (int r=0; r<=s; r++) {
2083 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2084 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2085 * S_num(n+s-r,p-s,cln::recip(value));
2088 result = result * cln::expt(cln::cl_I(-1),n);
2091 for (int r=0; r<n; r++) {
2092 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2094 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2096 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2101 if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
2104 for (int s=0; s<p-1; s++)
2107 ex res = H(m,numeric(value)).evalf();
2108 return ex_to<numeric>(res).to_cl_N();
2111 return S_projection(n, p, value, prec);
2116 } // end of anonymous namespace
2119 //////////////////////////////////////////////////////////////////////
2121 // Nielsen's generalized polylogarithm S(n,p,x)
2125 //////////////////////////////////////////////////////////////////////
2128 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2130 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2131 const int n_ = ex_to<numeric>(n).to_int();
2132 const int p_ = ex_to<numeric>(p).to_int();
2133 if (is_a<numeric>(x)) {
2134 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2135 const cln::cl_N result = S_num(n_, p_, x_);
2136 return numeric(result);
2138 ex x_val = x.evalf();
2139 if (is_a<numeric>(x_val)) {
2140 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2141 const cln::cl_N result = S_num(n_, p_, x_val_);
2142 return numeric(result);
2146 return S(n, p, x).hold();
2150 static ex S_eval(const ex& n, const ex& p, const ex& x)
2152 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2158 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2166 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2167 int n_ = ex_to<numeric>(n).to_int();
2168 int p_ = ex_to<numeric>(p).to_int();
2169 const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2170 const cln::cl_N result = S_num(n_, p_, x_);
2171 return numeric(result);
2176 return pow(-log(1-x), p) / factorial(p);
2178 return S(n, p, x).hold();
2182 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2185 return Li(n+1, x).series(rel, order, options);
2188 const ex x_pt = x.subs(rel, subs_options::no_pattern);
2189 if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2190 // First special case: x==0 (derivatives have poles)
2191 if (x_pt.is_zero()) {
2194 // manually construct the primitive expansion
2195 // subsum = Euler-Zagier-Sum is needed
2196 // dirty hack (slow ...) calculation of subsum:
2197 std::vector<ex> presubsum, subsum;
2198 subsum.push_back(0);
2199 for (int i=1; i<order-1; ++i) {
2200 subsum.push_back(subsum[i-1] + numeric(1, i));
2202 for (int depth=2; depth<p; ++depth) {
2204 for (int i=1; i<order-1; ++i) {
2205 subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2209 for (int i=1; i<order; ++i) {
2210 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2212 // substitute the argument's series expansion
2213 ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2214 // maybe that was terminating, so add a proper order term
2216 nseq.push_back(expair(Order(_ex1), order));
2217 ser += pseries(rel, nseq);
2218 // reexpanding it will collapse the series again
2219 return ser.series(rel, order);
2221 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2222 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2224 // all other cases should be safe, by now:
2225 throw do_taylor(); // caught by function::series()
2229 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2231 GINAC_ASSERT(deriv_param < 3);
2232 if (deriv_param < 2) {
2236 return S(n-1, p, x) / x;
2238 return S(n, p-1, x) / (1-x);
2243 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2245 c.s << "\\mathrm{S}_{";
2255 REGISTER_FUNCTION(S,
2256 evalf_func(S_evalf).
2258 series_func(S_series).
2259 derivative_func(S_deriv).
2260 print_func<print_latex>(S_print_latex).
2261 do_not_evalf_params());
2264 //////////////////////////////////////////////////////////////////////
2266 // Harmonic polylogarithm H(m,x)
2270 //////////////////////////////////////////////////////////////////////
2273 // anonymous namespace for helper functions
2277 // regulates the pole (used by 1/x-transformation)
2278 symbol H_polesign("IMSIGN");
2281 // convert parameters from H to Li representation
2282 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2283 // returns true if some parameters are negative
2284 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2286 // expand parameter list
2288 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
2290 for (ex count=*it-1; count > 0; count--) {
2294 } else if (*it < -1) {
2295 for (ex count=*it+1; count < 0; count++) {
2306 bool has_negative_parameters = false;
2308 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
2314 m.append((*it+acc-1) * signum);
2316 m.append((*it-acc+1) * signum);
2322 has_negative_parameters = true;
2325 if (has_negative_parameters) {
2326 for (std::size_t i=0; i<m.nops(); i++) {
2328 m.let_op(i) = -m.op(i);
2336 return has_negative_parameters;
2340 // recursivly transforms H to corresponding multiple polylogarithms
2341 struct map_trafo_H_convert_to_Li : public map_function
2343 ex operator()(const ex& e)
2345 if (is_a<add>(e) || is_a<mul>(e)) {
2346 return e.map(*this);
2348 if (is_a<function>(e)) {
2349 std::string name = ex_to<function>(e).get_name();
2352 if (is_a<lst>(e.op(0))) {
2353 parameter = ex_to<lst>(e.op(0));
2355 parameter = lst(e.op(0));
2362 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2363 s.let_op(0) = s.op(0) * arg;
2364 return pf * Li(m, s).hold();
2366 for (std::size_t i=0; i<m.nops(); i++) {
2369 s.let_op(0) = s.op(0) * arg;
2370 return Li(m, s).hold();
2379 // recursivly transforms H to corresponding zetas
2380 struct map_trafo_H_convert_to_zeta : public map_function
2382 ex operator()(const ex& e)
2384 if (is_a<add>(e) || is_a<mul>(e)) {
2385 return e.map(*this);
2387 if (is_a<function>(e)) {
2388 std::string name = ex_to<function>(e).get_name();
2391 if (is_a<lst>(e.op(0))) {
2392 parameter = ex_to<lst>(e.op(0));
2394 parameter = lst(e.op(0));
2400 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2401 return pf * zeta(m, s);
2412 // remove trailing zeros from H-parameters
2413 struct map_trafo_H_reduce_trailing_zeros : public map_function
2415 ex operator()(const ex& e)
2417 if (is_a<add>(e) || is_a<mul>(e)) {
2418 return e.map(*this);
2420 if (is_a<function>(e)) {
2421 std::string name = ex_to<function>(e).get_name();
2424 if (is_a<lst>(e.op(0))) {
2425 parameter = ex_to<lst>(e.op(0));
2427 parameter = lst(e.op(0));
2430 if (parameter.op(parameter.nops()-1) == 0) {
2433 if (parameter.nops() == 1) {
2438 lst::const_iterator it = parameter.begin();
2439 while ((it != parameter.end()) && (*it == 0)) {
2442 if (it == parameter.end()) {
2443 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2447 parameter.remove_last();
2448 std::size_t lastentry = parameter.nops();
2449 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2454 ex result = log(arg) * H(parameter,arg).hold();
2456 for (ex i=0; i<lastentry; i++) {
2457 if (parameter[i] > 0) {
2459 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2462 } else if (parameter[i] < 0) {
2464 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2472 if (lastentry < parameter.nops()) {
2473 result = result / (parameter.nops()-lastentry+1);
2474 return result.map(*this);
2486 // returns an expression with zeta functions corresponding to the parameter list for H
2487 ex convert_H_to_zeta(const lst& m)
2489 symbol xtemp("xtemp");
2490 map_trafo_H_reduce_trailing_zeros filter;
2491 map_trafo_H_convert_to_zeta filter2;
2492 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2496 // convert signs form Li to H representation
2497 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2500 lst::const_iterator itm = m.begin();
2501 lst::const_iterator itx = ++x.begin();
2506 while (itx != x.end()) {
2507 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
2508 // XXX: 1 + 0.0*I is considered equal to 1. However the former
2509 // is not automatically converted to a real number.
2510 // Do the conversion explicitly to avoid the
2511 // "numeric::operator>(): complex inequality" exception.
2512 signum *= (*itx != _ex_1) ? 1 : -1;
2514 res.append((*itm) * signum);
2522 // multiplies an one-dimensional H with another H
2524 ex trafo_H_mult(const ex& h1, const ex& h2)
2529 ex h1nops = h1.op(0).nops();
2530 ex h2nops = h2.op(0).nops();
2532 hshort = h2.op(0).op(0);
2533 hlong = ex_to<lst>(h1.op(0));
2535 hshort = h1.op(0).op(0);
2537 hlong = ex_to<lst>(h2.op(0));
2539 hlong = h2.op(0).op(0);
2542 for (std::size_t i=0; i<=hlong.nops(); i++) {
2546 newparameter.append(hlong[j]);
2548 newparameter.append(hshort);
2549 for (; j<hlong.nops(); j++) {
2550 newparameter.append(hlong[j]);
2552 res += H(newparameter, h1.op(1)).hold();
2558 // applies trafo_H_mult recursively on expressions
2559 struct map_trafo_H_mult : public map_function
2561 ex operator()(const ex& e)
2564 return e.map(*this);
2572 for (std::size_t pos=0; pos<e.nops(); pos++) {
2573 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2574 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2576 for (ex i=0; i<e.op(pos).op(1); i++) {
2577 Hlst.append(e.op(pos).op(0));
2581 } else if (is_a<function>(e.op(pos))) {
2582 std::string name = ex_to<function>(e.op(pos)).get_name();
2584 if (e.op(pos).op(0).nops() > 1) {
2587 Hlst.append(e.op(pos));
2592 result *= e.op(pos);
2595 if (Hlst.nops() > 0) {
2596 firstH = Hlst[Hlst.nops()-1];
2603 if (Hlst.nops() > 0) {
2604 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2606 for (std::size_t i=1; i<Hlst.nops(); i++) {
2607 result *= Hlst.op(i);
2609 result = result.expand();
2610 map_trafo_H_mult recursion;
2611 return recursion(result);
2622 // do integration [ReV] (55)
2623 // put parameter 0 in front of existing parameters
2624 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2628 if (is_a<function>(e)) {
2629 name = ex_to<function>(e).get_name();
2634 for (std::size_t i=0; i<e.nops(); i++) {
2635 if (is_a<function>(e.op(i))) {
2636 std::string name = ex_to<function>(e.op(i)).get_name();
2644 lst newparameter = ex_to<lst>(h.op(0));
2645 newparameter.prepend(0);
2646 ex addzeta = convert_H_to_zeta(newparameter);
2647 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2649 return e * (-H(lst(ex(0)),1/arg).hold());
2654 // do integration [ReV] (49)
2655 // put parameter 1 in front of existing parameters
2656 ex trafo_H_prepend_one(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(1);
2678 return e.subs(h == H(newparameter, h.op(1)).hold());
2680 return e * H(lst(ex(1)),1-arg).hold();
2685 // do integration [ReV] (55)
2686 // put parameter -1 in front of existing parameters
2687 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2691 if (is_a<function>(e)) {
2692 name = ex_to<function>(e).get_name();
2697 for (std::size_t i=0; i<e.nops(); i++) {
2698 if (is_a<function>(e.op(i))) {
2699 std::string name = ex_to<function>(e.op(i)).get_name();
2707 lst newparameter = ex_to<lst>(h.op(0));
2708 newparameter.prepend(-1);
2709 ex addzeta = convert_H_to_zeta(newparameter);
2710 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2712 ex addzeta = convert_H_to_zeta(lst(ex(-1)));
2713 return (e * (addzeta - H(lst(ex(-1)),1/arg).hold())).expand();
2718 // do integration [ReV] (55)
2719 // put parameter -1 in front of existing parameters
2720 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2724 if (is_a<function>(e)) {
2725 name = ex_to<function>(e).get_name();
2730 for (std::size_t i = 0; i < e.nops(); i++) {
2731 if (is_a<function>(e.op(i))) {
2732 std::string name = ex_to<function>(e.op(i)).get_name();
2740 lst newparameter = ex_to<lst>(h.op(0));
2741 newparameter.prepend(-1);
2742 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2744 return (e * H(lst(ex(-1)),(1-arg)/(1+arg)).hold()).expand();
2749 // do integration [ReV] (55)
2750 // put parameter 1 in front of existing parameters
2751 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2755 if (is_a<function>(e)) {
2756 name = ex_to<function>(e).get_name();
2761 for (std::size_t i = 0; i < e.nops(); i++) {
2762 if (is_a<function>(e.op(i))) {
2763 std::string name = ex_to<function>(e.op(i)).get_name();
2771 lst newparameter = ex_to<lst>(h.op(0));
2772 newparameter.prepend(1);
2773 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2775 return (e * H(lst(ex(1)),(1-arg)/(1+arg)).hold()).expand();
2780 // do x -> 1-x transformation
2781 struct map_trafo_H_1mx : public map_function
2783 ex operator()(const ex& e)
2785 if (is_a<add>(e) || is_a<mul>(e)) {
2786 return e.map(*this);
2789 if (is_a<function>(e)) {
2790 std::string name = ex_to<function>(e).get_name();
2793 lst parameter = ex_to<lst>(e.op(0));
2796 // special cases if all parameters are either 0, 1 or -1
2797 bool allthesame = true;
2798 if (parameter.op(0) == 0) {
2799 for (std::size_t i = 1; i < parameter.nops(); i++) {
2800 if (parameter.op(i) != 0) {
2807 for (int i=parameter.nops(); i>0; i--) {
2808 newparameter.append(1);
2810 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2812 } else if (parameter.op(0) == -1) {
2813 throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2815 for (std::size_t i = 1; i < parameter.nops(); i++) {
2816 if (parameter.op(i) != 1) {
2823 for (int i=parameter.nops(); i>0; i--) {
2824 newparameter.append(0);
2826 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2830 lst newparameter = parameter;
2831 newparameter.remove_first();
2833 if (parameter.op(0) == 0) {
2836 ex res = convert_H_to_zeta(parameter);
2837 //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2838 map_trafo_H_1mx recursion;
2839 ex buffer = recursion(H(newparameter, arg).hold());
2840 if (is_a<add>(buffer)) {
2841 for (std::size_t i = 0; i < buffer.nops(); i++) {
2842 res -= trafo_H_prepend_one(buffer.op(i), arg);
2845 res -= trafo_H_prepend_one(buffer, arg);
2852 map_trafo_H_1mx recursion;
2853 map_trafo_H_mult unify;
2854 ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2855 std::size_t firstzero = 0;
2856 while (parameter.op(firstzero) == 1) {
2859 for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2863 newparameter.append(parameter[j+1]);
2865 newparameter.append(1);
2866 for (; j<parameter.nops()-1; j++) {
2867 newparameter.append(parameter[j+1]);
2869 res -= H(newparameter, arg).hold();
2871 res = recursion(res).expand() / firstzero;
2881 // do x -> 1/x transformation
2882 struct map_trafo_H_1overx : public map_function
2884 ex operator()(const ex& e)
2886 if (is_a<add>(e) || is_a<mul>(e)) {
2887 return e.map(*this);
2890 if (is_a<function>(e)) {
2891 std::string name = ex_to<function>(e).get_name();
2894 lst parameter = ex_to<lst>(e.op(0));
2897 // special cases if all parameters are either 0, 1 or -1
2898 bool allthesame = true;
2899 if (parameter.op(0) == 0) {
2900 for (std::size_t i = 1; i < parameter.nops(); i++) {
2901 if (parameter.op(i) != 0) {
2907 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2909 } else if (parameter.op(0) == -1) {
2910 for (std::size_t i = 1; i < parameter.nops(); i++) {
2911 if (parameter.op(i) != -1) {
2917 map_trafo_H_mult unify;
2918 return unify((pow(H(lst(ex(-1)),1/arg).hold() - H(lst(ex(0)),1/arg).hold(), parameter.nops())
2919 / factorial(parameter.nops())).expand());
2922 for (std::size_t i = 1; i < parameter.nops(); i++) {
2923 if (parameter.op(i) != 1) {
2929 map_trafo_H_mult unify;
2930 return unify((pow(H(lst(ex(1)),1/arg).hold() + H(lst(ex(0)),1/arg).hold() + H_polesign, parameter.nops())
2931 / factorial(parameter.nops())).expand());
2935 lst newparameter = parameter;
2936 newparameter.remove_first();
2938 if (parameter.op(0) == 0) {
2941 ex res = convert_H_to_zeta(parameter);
2942 map_trafo_H_1overx recursion;
2943 ex buffer = recursion(H(newparameter, arg).hold());
2944 if (is_a<add>(buffer)) {
2945 for (std::size_t i = 0; i < buffer.nops(); i++) {
2946 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2949 res += trafo_H_1tx_prepend_zero(buffer, arg);
2953 } else if (parameter.op(0) == -1) {
2955 // leading negative one
2956 ex res = convert_H_to_zeta(parameter);
2957 map_trafo_H_1overx recursion;
2958 ex buffer = recursion(H(newparameter, arg).hold());
2959 if (is_a<add>(buffer)) {
2960 for (std::size_t i = 0; i < buffer.nops(); i++) {
2961 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2964 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2971 map_trafo_H_1overx recursion;
2972 map_trafo_H_mult unify;
2973 ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2974 std::size_t firstzero = 0;
2975 while (parameter.op(firstzero) == 1) {
2978 for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
2982 newparameter.append(parameter[j+1]);
2984 newparameter.append(1);
2985 for (; j<parameter.nops()-1; j++) {
2986 newparameter.append(parameter[j+1]);
2988 res -= H(newparameter, arg).hold();
2990 res = recursion(res).expand() / firstzero;
3002 // do x -> (1-x)/(1+x) transformation
3003 struct map_trafo_H_1mxt1px : public map_function
3005 ex operator()(const ex& e)
3007 if (is_a<add>(e) || is_a<mul>(e)) {
3008 return e.map(*this);
3011 if (is_a<function>(e)) {
3012 std::string name = ex_to<function>(e).get_name();
3015 lst parameter = ex_to<lst>(e.op(0));
3018 // special cases if all parameters are either 0, 1 or -1
3019 bool allthesame = true;
3020 if (parameter.op(0) == 0) {
3021 for (std::size_t i = 1; i < parameter.nops(); i++) {
3022 if (parameter.op(i) != 0) {
3028 map_trafo_H_mult unify;
3029 return unify((pow(-H(lst(ex(1)),(1-arg)/(1+arg)).hold() - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3030 / factorial(parameter.nops())).expand());
3032 } else if (parameter.op(0) == -1) {
3033 for (std::size_t i = 1; i < parameter.nops(); i++) {
3034 if (parameter.op(i) != -1) {
3040 map_trafo_H_mult unify;
3041 return unify((pow(log(2) - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3042 / factorial(parameter.nops())).expand());
3045 for (std::size_t i = 1; i < parameter.nops(); i++) {
3046 if (parameter.op(i) != 1) {
3052 map_trafo_H_mult unify;
3053 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())
3054 / factorial(parameter.nops())).expand());
3058 lst newparameter = parameter;
3059 newparameter.remove_first();
3061 if (parameter.op(0) == 0) {
3064 ex res = convert_H_to_zeta(parameter);
3065 map_trafo_H_1mxt1px recursion;
3066 ex buffer = recursion(H(newparameter, arg).hold());
3067 if (is_a<add>(buffer)) {
3068 for (std::size_t i = 0; i < buffer.nops(); i++) {
3069 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3072 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3076 } else if (parameter.op(0) == -1) {
3078 // leading negative one
3079 ex res = convert_H_to_zeta(parameter);
3080 map_trafo_H_1mxt1px recursion;
3081 ex buffer = recursion(H(newparameter, arg).hold());
3082 if (is_a<add>(buffer)) {
3083 for (std::size_t i = 0; i < buffer.nops(); i++) {
3084 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3087 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3094 map_trafo_H_1mxt1px recursion;
3095 map_trafo_H_mult unify;
3096 ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
3097 std::size_t firstzero = 0;
3098 while (parameter.op(firstzero) == 1) {
3101 for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3105 newparameter.append(parameter[j+1]);
3107 newparameter.append(1);
3108 for (; j<parameter.nops()-1; j++) {
3109 newparameter.append(parameter[j+1]);
3111 res -= H(newparameter, arg).hold();
3113 res = recursion(res).expand() / firstzero;
3125 // do the actual summation.
3126 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3128 const int j = m.size();
3130 std::vector<cln::cl_N> t(j);
3132 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3133 cln::cl_N factor = cln::expt(x, j) * one;
3139 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3140 for (int k=j-2; k>=1; k--) {
3141 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3143 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3144 factor = factor * x;
3145 } while (t[0] != t0buf);
3151 } // end of anonymous namespace
3154 //////////////////////////////////////////////////////////////////////
3156 // Harmonic polylogarithm H(m,x)
3160 //////////////////////////////////////////////////////////////////////
3163 static ex H_evalf(const ex& x1, const ex& x2)
3165 if (is_a<lst>(x1)) {
3168 if (is_a<numeric>(x2)) {
3169 x = ex_to<numeric>(x2).to_cl_N();
3171 ex x2_val = x2.evalf();
3172 if (is_a<numeric>(x2_val)) {
3173 x = ex_to<numeric>(x2_val).to_cl_N();
3177 for (std::size_t i = 0; i < x1.nops(); i++) {
3178 if (!x1.op(i).info(info_flags::integer)) {
3179 return H(x1, x2).hold();
3182 if (x1.nops() < 1) {
3183 return H(x1, x2).hold();
3186 const lst& morg = ex_to<lst>(x1);
3187 // remove trailing zeros ...
3188 if (*(--morg.end()) == 0) {
3189 symbol xtemp("xtemp");
3190 map_trafo_H_reduce_trailing_zeros filter;
3191 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3193 // ... and expand parameter notation
3194 bool has_minus_one = false;
3196 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
3198 for (ex count=*it-1; count > 0; count--) {
3202 } else if (*it <= -1) {
3203 for (ex count=*it+1; count < 0; count++) {
3207 has_minus_one = true;
3214 if (cln::abs(x) < 0.95) {
3218 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3219 // negative parameters -> s_lst is filled
3220 std::vector<int> m_int;
3221 std::vector<cln::cl_N> x_cln;
3222 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
3223 it_int != m_lst.end(); it_int++, it_cln++) {
3224 m_int.push_back(ex_to<numeric>(*it_int).to_int());
3225 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3227 x_cln.front() = x_cln.front() * x;
3228 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3230 // only positive parameters
3232 if (m_lst.nops() == 1) {
3233 return Li(m_lst.op(0), x2).evalf();
3235 std::vector<int> m_int;
3236 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
3237 m_int.push_back(ex_to<numeric>(*it).to_int());
3239 return numeric(H_do_sum(m_int, x));
3243 symbol xtemp("xtemp");
3246 // ensure that the realpart of the argument is positive
3247 if (cln::realpart(x) < 0) {
3249 for (std::size_t i = 0; i < m.nops(); i++) {
3251 m.let_op(i) = -m.op(i);
3258 if (cln::abs(x) >= 2.0) {
3259 map_trafo_H_1overx trafo;
3260 res *= trafo(H(m, xtemp).hold());
3261 if (cln::imagpart(x) <= 0) {
3262 res = res.subs(H_polesign == -I*Pi);
3264 res = res.subs(H_polesign == I*Pi);
3266 return res.subs(xtemp == numeric(x)).evalf();
3269 // check transformations for 0.95 <= |x| < 2.0
3271 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3272 if (cln::abs(x-9.53) <= 9.47) {
3274 map_trafo_H_1mxt1px trafo;
3275 res *= trafo(H(m, xtemp).hold());
3278 if (has_minus_one) {
3279 map_trafo_H_convert_to_Li filter;
3280 return filter(H(m, numeric(x)).hold()).evalf();
3282 map_trafo_H_1mx trafo;
3283 res *= trafo(H(m, xtemp).hold());
3286 return res.subs(xtemp == numeric(x)).evalf();
3289 return H(x1,x2).hold();
3293 static ex H_eval(const ex& m_, const ex& x)
3296 if (is_a<lst>(m_)) {
3301 if (m.nops() == 0) {
3309 if (*m.begin() > _ex1) {
3315 } else if (*m.begin() < _ex_1) {
3321 } else if (*m.begin() == _ex0) {
3328 for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
3329 if ((*it).info(info_flags::integer)) {
3340 } else if (*it < _ex_1) {
3360 } else if (step == 1) {
3372 // if some m_i is not an integer
3373 return H(m_, x).hold();
3376 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3377 return convert_H_to_zeta(m);
3383 return H(m_, x).hold();
3385 return pow(log(x), m.nops()) / factorial(m.nops());
3388 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3390 } else if ((step == 1) && (pos1 == _ex0)){
3395 return pow(-1, p) * S(n, p, -x);
3401 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3402 return H(m_, x).evalf();
3404 return H(m_, x).hold();
3408 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3411 seq.push_back(expair(H(m, x), 0));
3412 return pseries(rel, seq);
3416 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3418 GINAC_ASSERT(deriv_param < 2);
3419 if (deriv_param == 0) {
3423 if (is_a<lst>(m_)) {
3439 return 1/(1-x) * H(m, x);
3440 } else if (mb == _ex_1) {
3441 return 1/(1+x) * H(m, x);
3448 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3451 if (is_a<lst>(m_)) {
3456 c.s << "\\mathrm{H}_{";
3457 lst::const_iterator itm = m.begin();
3460 for (; itm != m.end(); itm++) {
3470 REGISTER_FUNCTION(H,
3471 evalf_func(H_evalf).
3473 series_func(H_series).
3474 derivative_func(H_deriv).
3475 print_func<print_latex>(H_print_latex).
3476 do_not_evalf_params());
3479 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3480 ex convert_H_to_Li(const ex& m, const ex& x)
3482 map_trafo_H_reduce_trailing_zeros filter;
3483 map_trafo_H_convert_to_Li filter2;
3485 return filter2(filter(H(m, x).hold()));
3487 return filter2(filter(H(lst(m), x).hold()));
3492 //////////////////////////////////////////////////////////////////////
3494 // Multiple zeta values zeta(x) and zeta(x,s)
3498 //////////////////////////////////////////////////////////////////////
3501 // anonymous namespace for helper functions
3505 // parameters and data for [Cra] algorithm
3506 const cln::cl_N lambda = cln::cl_N("319/320");
3508 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3510 const int size = a.size();
3511 for (int n=0; n<size; n++) {
3513 for (int m=0; m<=n; m++) {
3514 c[n] = c[n] + a[m]*b[n-m];
3521 static void initcX(std::vector<cln::cl_N>& crX,
3522 const std::vector<int>& s,
3525 std::vector<cln::cl_N> crB(L2 + 1);
3526 for (int i=0; i<=L2; i++)
3527 crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i);
3531 std::vector<std::vector<cln::cl_N> > crG(s.size() - 1, std::vector<cln::cl_N>(L2 + 1));
3532 for (int m=0; m < (int)s.size() - 1; m++) {
3535 for (int i = 0; i <= L2; i++)
3536 crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2);
3541 for (std::size_t m = 0; m < s.size() - 1; m++) {
3542 std::vector<cln::cl_N> Xbuf(L2 + 1);
3543 for (int i = 0; i <= L2; i++)
3544 Xbuf[i] = crX[i] * crG[m][i];
3546 halfcyclic_convolute(Xbuf, crB, crX);
3552 static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk,
3553 const std::vector<cln::cl_N>& crX)
3555 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3556 cln::cl_N factor = cln::expt(lambda, Sqk);
3557 cln::cl_N res = factor / Sqk * crX[0] * one;
3562 factor = factor * lambda;
3564 res = res + crX[N] * factor / (N+Sqk);
3565 } while ((res != resbuf) || cln::zerop(crX[N]));
3571 static void calc_f(std::vector<std::vector<cln::cl_N> >& f_kj,
3572 const int maxr, const int L1)
3574 cln::cl_N t0, t1, t2, t3, t4;
3576 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
3577 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3579 t0 = cln::exp(-lambda);
3581 for (k=1; k<=L1; k++) {
3584 for (j=1; j<=maxr; j++) {
3587 for (i=2; i<=j; i++) {
3591 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3599 static cln::cl_N crandall_Z(const std::vector<int>& s,
3600 const std::vector<std::vector<cln::cl_N> >& f_kj)
3602 const int j = s.size();
3611 t0 = t0 + f_kj[q+j-2][s[0]-1];
3612 } while (t0 != t0buf);
3614 return t0 / cln::factorial(s[0]-1);
3617 std::vector<cln::cl_N> t(j);
3624 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3625 for (int k=j-2; k>=1; k--) {
3626 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3628 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3629 } while (t[0] != t0buf);
3631 return t[0] / cln::factorial(s[0]-1);
3636 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3638 std::vector<int> r = s;
3639 const int j = r.size();
3643 // decide on maximal size of f_kj for crandall_Z
3647 L1 = Digits * 3 + j*2;
3651 // decide on maximal size of crX for crandall_Y
3654 } else if (Digits < 86) {
3656 } else if (Digits < 192) {
3658 } else if (Digits < 394) {
3660 } else if (Digits < 808) {
3670 for (int i=0; i<j; i++) {
3677 std::vector<std::vector<cln::cl_N> > f_kj(L1);
3678 calc_f(f_kj, maxr, L1);
3680 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3682 std::vector<int> rz;
3685 for (int k=r.size()-1; k>0; k--) {
3687 rz.insert(rz.begin(), r.back());
3688 skp1buf = rz.front();
3692 std::vector<cln::cl_N> crX;
3695 for (int q=0; q<skp1buf; q++) {
3697 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k, crX);
3698 cln::cl_N pp2 = crandall_Z(rz, f_kj);
3703 res = res - pp1 * pp2 / cln::factorial(q);
3705 res = res + pp1 * pp2 / cln::factorial(q);
3708 rz.front() = skp1buf;
3710 rz.insert(rz.begin(), r.back());
3712 std::vector<cln::cl_N> crX;
3713 initcX(crX, rz, L2);
3715 res = (res + crandall_Y_loop(S-j, crX)) / r0factorial
3716 + crandall_Z(rz, f_kj);
3722 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3724 const int j = r.size();
3726 // buffer for subsums
3727 std::vector<cln::cl_N> t(j);
3728 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3735 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3736 for (int k=j-2; k>=0; k--) {
3737 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3739 } while (t[0] != t0buf);
3745 // does Hoelder convolution. see [BBB] (7.0)
3746 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3748 // prepare parameters
3749 // holds Li arguments in [BBB] notation
3750 std::vector<int> s = s_;
3751 std::vector<int> m_p = m_;
3752 std::vector<int> m_q;
3753 // holds Li arguments in nested sums notation
3754 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3755 s_p[0] = s_p[0] * cln::cl_N("1/2");
3756 // convert notations
3758 for (std::size_t i = 0; i < s_.size(); i++) {
3763 s[i] = sig * std::abs(s[i]);
3765 std::vector<cln::cl_N> s_q;
3766 cln::cl_N signum = 1;
3769 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3774 // change parameters
3775 if (s.front() > 0) {
3776 if (m_p.front() == 1) {
3777 m_p.erase(m_p.begin());
3778 s_p.erase(s_p.begin());
3779 if (s_p.size() > 0) {
3780 s_p.front() = s_p.front() * cln::cl_N("1/2");
3786 m_q.insert(m_q.begin(), 1);
3787 if (s_q.size() > 0) {
3788 s_q.front() = s_q.front() * 2;
3790 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3793 if (m_p.front() == 1) {
3794 m_p.erase(m_p.begin());
3795 cln::cl_N spbuf = s_p.front();
3796 s_p.erase(s_p.begin());
3797 if (s_p.size() > 0) {
3798 s_p.front() = s_p.front() * spbuf;
3801 m_q.insert(m_q.begin(), 1);
3802 if (s_q.size() > 0) {
3803 s_q.front() = s_q.front() * 4;
3805 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3809 m_q.insert(m_q.begin(), 1);
3810 if (s_q.size() > 0) {
3811 s_q.front() = s_q.front() * 2;
3813 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3818 if (m_p.size() == 0) break;
3820 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3825 res = res + signum * multipleLi_do_sum(m_q, s_q);
3831 } // end of anonymous namespace
3834 //////////////////////////////////////////////////////////////////////
3836 // Multiple zeta values zeta(x)
3840 //////////////////////////////////////////////////////////////////////
3843 static ex zeta1_evalf(const ex& x)
3845 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3847 // multiple zeta value
3848 const int count = x.nops();
3849 const lst& xlst = ex_to<lst>(x);
3850 std::vector<int> r(count);
3852 // check parameters and convert them
3853 lst::const_iterator it1 = xlst.begin();
3854 std::vector<int>::iterator it2 = r.begin();
3856 if (!(*it1).info(info_flags::posint)) {
3857 return zeta(x).hold();
3859 *it2 = ex_to<numeric>(*it1).to_int();
3862 } while (it2 != r.end());
3864 // check for divergence
3866 return zeta(x).hold();
3869 // decide on summation algorithm
3870 // this is still a bit clumsy
3871 int limit = (Digits>17) ? 10 : 6;
3872 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3873 return numeric(zeta_do_sum_Crandall(r));
3875 return numeric(zeta_do_sum_simple(r));
3879 // single zeta value
3880 if (is_exactly_a<numeric>(x) && (x != 1)) {
3882 return zeta(ex_to<numeric>(x));
3883 } catch (const dunno &e) { }
3886 return zeta(x).hold();
3890 static ex zeta1_eval(const ex& m)
3892 if (is_exactly_a<lst>(m)) {
3893 if (m.nops() == 1) {
3894 return zeta(m.op(0));
3896 return zeta(m).hold();
3899 if (m.info(info_flags::numeric)) {
3900 const numeric& y = ex_to<numeric>(m);
3901 // trap integer arguments:
3902 if (y.is_integer()) {
3906 if (y.is_equal(*_num1_p)) {
3907 return zeta(m).hold();
3909 if (y.info(info_flags::posint)) {
3910 if (y.info(info_flags::odd)) {
3911 return zeta(m).hold();
3913 return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3916 if (y.info(info_flags::odd)) {
3917 return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3924 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3925 return zeta1_evalf(m);
3928 return zeta(m).hold();
3932 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3934 GINAC_ASSERT(deriv_param==0);
3936 if (is_exactly_a<lst>(m)) {
3939 return zetaderiv(_ex1, m);
3944 static void zeta1_print_latex(const ex& m_, const print_context& c)
3947 if (is_a<lst>(m_)) {
3948 const lst& m = ex_to<lst>(m_);
3949 lst::const_iterator it = m.begin();
3952 for (; it != m.end(); it++) {
3963 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
3964 evalf_func(zeta1_evalf).
3965 eval_func(zeta1_eval).
3966 derivative_func(zeta1_deriv).
3967 print_func<print_latex>(zeta1_print_latex).
3968 do_not_evalf_params().
3972 //////////////////////////////////////////////////////////////////////
3974 // Alternating Euler sum zeta(x,s)
3978 //////////////////////////////////////////////////////////////////////
3981 static ex zeta2_evalf(const ex& x, const ex& s)
3983 if (is_exactly_a<lst>(x)) {
3985 // alternating Euler sum
3986 const int count = x.nops();
3987 const lst& xlst = ex_to<lst>(x);
3988 const lst& slst = ex_to<lst>(s);
3989 std::vector<int> xi(count);
3990 std::vector<int> si(count);
3992 // check parameters and convert them
3993 lst::const_iterator it_xread = xlst.begin();
3994 lst::const_iterator it_sread = slst.begin();
3995 std::vector<int>::iterator it_xwrite = xi.begin();
3996 std::vector<int>::iterator it_swrite = si.begin();
3998 if (!(*it_xread).info(info_flags::posint)) {
3999 return zeta(x, s).hold();
4001 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
4002 if (*it_sread > 0) {
4011 } while (it_xwrite != xi.end());
4013 // check for divergence
4014 if ((xi[0] == 1) && (si[0] == 1)) {
4015 return zeta(x, s).hold();
4018 // use Hoelder convolution
4019 return numeric(zeta_do_Hoelder_convolution(xi, si));
4022 return zeta(x, s).hold();
4026 static ex zeta2_eval(const ex& m, const ex& s_)
4028 if (is_exactly_a<lst>(s_)) {
4029 const lst& s = ex_to<lst>(s_);
4030 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
4031 if ((*it).info(info_flags::positive)) {
4034 return zeta(m, s_).hold();
4037 } else if (s_.info(info_flags::positive)) {
4041 return zeta(m, s_).hold();
4045 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
4047 GINAC_ASSERT(deriv_param==0);
4049 if (is_exactly_a<lst>(m)) {
4052 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
4053 return zetaderiv(_ex1, m);
4060 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
4063 if (is_a<lst>(m_)) {
4069 if (is_a<lst>(s_)) {
4075 lst::const_iterator itm = m.begin();
4076 lst::const_iterator its = s.begin();
4078 c.s << "\\overline{";
4086 for (; itm != m.end(); itm++, its++) {
4089 c.s << "\\overline{";
4100 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
4101 evalf_func(zeta2_evalf).
4102 eval_func(zeta2_eval).
4103 derivative_func(zeta2_deriv).
4104 print_func<print_latex>(zeta2_print_latex).
4105 do_not_evalf_params().
4109 } // namespace GiNaC