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 * nielsen's generalized polylogarithm S(n,p,x)
9 * harmonic polylogarithm H(m,x) or H(lst(m_1,...,m_k),x)
10 * multiple zeta value zeta(m) or zeta(lst(m_1,...,m_k))
11 * alternating Euler sum zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
15 * - All formulae used can be looked up in the following publications:
16 * [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
17 * [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
18 * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
19 * [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 * - The order of parameters and arguments of Li and zeta is defined according to the nested sums
22 * representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
23 * 0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
24 * number --- notation.
26 * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
27 * the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product
28 * x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers.
29 * If you want to have an alternating Euler sum, you have to give the signs of the parameters as a
30 * 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.
36 * - The functions have no series expansion into nested sums. To do this, you have to convert these functions
37 * into the appropriate objects from the nestedsums library, do the expansion and convert the
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.
49 * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
51 * This program is free software; you can redistribute it and/or modify
52 * it under the terms of the GNU General Public License as published by
53 * the Free Software Foundation; either version 2 of the License, or
54 * (at your option) any later version.
56 * This program is distributed in the hope that it will be useful,
57 * but WITHOUT ANY WARRANTY; without even the implied warranty of
58 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
59 * GNU General Public License for more details.
61 * You should have received a copy of the GNU General Public License
62 * along with this program; if not, write to the Free Software
63 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
77 #include "operators.h"
80 #include "relational.h"
89 //////////////////////////////////////////////////////////////////////
91 // Classical polylogarithm Li(n,x)
95 //////////////////////////////////////////////////////////////////////
98 // anonymous namespace for helper functions
102 // lookup table for factors built from Bernoulli numbers
104 std::vector<std::vector<cln::cl_N> > Xn;
105 // initial size of Xn that should suffice for 32bit machines (must be even)
106 const int xninitsizestep = 26;
107 int xninitsize = xninitsizestep;
111 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
112 // With these numbers the polylogs can be calculated as follows:
113 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
114 // X_0(n) = B_n (Bernoulli numbers)
115 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
116 // The calculation of Xn depends on X0 and X{n-1}.
117 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
118 // This results in a slightly more complicated algorithm for the X_n.
119 // The first index in Xn corresponds to the index of the polylog minus 2.
120 // The second index in Xn corresponds to the index from the actual sum.
124 // calculate X_2 and higher (corresponding to Li_4 and higher)
125 std::vector<cln::cl_N> buf(xninitsize);
126 std::vector<cln::cl_N>::iterator it = buf.begin();
128 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
130 for (int i=2; i<=xninitsize; i++) {
132 result = 0; // k == 0
134 result = Xn[0][i/2-1]; // k == 0
136 for (int k=1; k<i-1; k++) {
137 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
138 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
141 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
142 result = result + Xn[n-1][i-1] / (i+1); // k == i
149 // special case to handle the X_0 correct
150 std::vector<cln::cl_N> buf(xninitsize);
151 std::vector<cln::cl_N>::iterator it = buf.begin();
153 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
155 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
157 for (int i=3; i<=xninitsize; i++) {
159 result = -Xn[0][(i-3)/2]/2;
160 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
163 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
164 for (int k=1; k<i/2; k++) {
165 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
174 std::vector<cln::cl_N> buf(xninitsize/2);
175 std::vector<cln::cl_N>::iterator it = buf.begin();
176 for (int i=1; i<=xninitsize/2; i++) {
177 *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 (int 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 numeric S_num(int n, int p, const numeric& 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) {
349 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
351 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
355 // check if precalculated Xn exist
357 for (int i=xnsize; i<n-1; i++) {
362 if (cln::realpart(x) < 0.5) {
363 // choose the faster algorithm
364 // with n>=12 the "normal" summation always wins against the method with Xn
365 if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
366 return Lin_do_sum(n, x);
368 return Lin_do_sum_Xn(n, x);
371 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
372 for (int j=0; j<n-1; j++) {
373 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
374 * cln::expt(cln::log(x), j) / cln::factorial(j);
382 // helper function for classical polylog Li
383 numeric Li_num(int n, const numeric& x)
387 return -cln::log(1-x.to_cl_N());
398 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
401 // what is the desired float format?
402 // first guess: default format
403 cln::float_format_t prec = cln::default_float_format;
404 const cln::cl_N value = x.to_cl_N();
405 // second guess: the argument's format
406 if (!x.real().is_rational())
407 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
408 else if (!x.imag().is_rational())
409 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
412 if (cln::abs(value) > 1) {
413 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
414 // check if argument is complex. if it is real, the new polylog has to be conjugated.
415 if (cln::zerop(cln::imagpart(value))) {
417 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
420 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
425 result = result + Li_projection(n, cln::recip(value), prec);
428 result = result - Li_projection(n, cln::recip(value), prec);
432 for (int j=0; j<n-1; j++) {
433 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
434 * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
436 result = result - add;
440 return Li_projection(n, value, prec);
445 } // end of anonymous namespace
448 //////////////////////////////////////////////////////////////////////
450 // Multiple polylogarithm Li(n,x)
454 //////////////////////////////////////////////////////////////////////
457 // anonymous namespace for helper function
461 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
463 const int j = s.size();
465 std::vector<cln::cl_N> t(j);
466 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
474 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
475 for (int k=j-2; k>=0; k--) {
476 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]);
478 // ... and do it again (to avoid premature drop out due to special arguments)
480 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
481 for (int k=j-2; k>=0; k--) {
482 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]);
484 } while (t[0] != t0buf);
489 // forward declaration for Li_eval()
490 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
493 } // end of anonymous namespace
496 //////////////////////////////////////////////////////////////////////
498 // Classical polylogarithm and multiple polylogarithm Li(n,x)
502 //////////////////////////////////////////////////////////////////////
505 static ex Li_evalf(const ex& x1, const ex& x2)
507 // classical polylogs
508 if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
509 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
511 if (is_a<numeric>(x1) && !is_a<lst>(x2)) {
512 // try to numerically evaluate second argument
513 ex x2_val = x2.evalf();
514 if (is_a<numeric>(x2_val)) {
515 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2_val));
517 return Li(x1, x2).hold();
521 else if (is_a<lst>(x1) && is_a<lst>(x2)) {
523 for (int i=0; i<x1.nops(); i++) {
524 if (!x1.op(i).info(info_flags::posint)) {
525 return Li(x1, x2).hold();
527 if (!is_a<numeric>(x2.op(i))) {
528 return Li(x1, x2).hold();
531 if (abs(conv) >= 1) {
532 return Li(x1, x2).hold();
537 std::vector<cln::cl_N> x;
538 for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
539 m.push_back(ex_to<numeric>(x1.op(i)).to_int());
540 x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
543 return numeric(multipleLi_do_sum(m, x));
546 return Li(x1,x2).hold();
550 static ex Li_eval(const ex& m_, const ex& x_)
572 return (pow(2,1-m)-1) * zeta(m);
577 if (m.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
578 return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
584 bool doevalf = false;
585 bool doevalfveto = true;
586 const lst& m = ex_to<lst>(m_);
587 const lst& x = ex_to<lst>(x_);
588 lst::const_iterator itm = m.begin();
589 lst::const_iterator itx = x.begin();
590 for (; itm != m.end(); itm++, itx++) {
591 if (!(*itm).info(info_flags::posint)) {
592 return Li(m_, x_).hold();
594 if ((*itx != _ex1) && (*itx != _ex_1)) {
595 if (itx != x.begin()) {
603 if (!(*itx).info(info_flags::numeric)) {
606 if (!(*itx).info(info_flags::crational)) {
618 lst newm = convert_parameter_Li_to_H(m, x, pf);
619 return pf * H(newm, x[0]);
621 if (doevalfveto && doevalf) {
622 return Li(m_, x_).evalf();
625 return Li(m_, x_).hold();
629 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
632 seq.push_back(expair(Li(m, x), 0));
633 return pseries(rel, seq);
637 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
639 GINAC_ASSERT(deriv_param < 2);
640 if (deriv_param == 0) {
644 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
659 return Li(m-1, x) / x;
666 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
680 c.s << "\\mbox{Li}_{";
681 lst::const_iterator itm = m.begin();
684 for (; itm != m.end(); itm++) {
689 lst::const_iterator itx = x.begin();
692 for (; itx != x.end(); itx++) {
700 REGISTER_FUNCTION(Li,
701 evalf_func(Li_evalf).
703 series_func(Li_series).
704 derivative_func(Li_deriv).
705 print_func<print_latex>(Li_print_latex).
706 do_not_evalf_params());
709 //////////////////////////////////////////////////////////////////////
711 // Nielsen's generalized polylogarithm S(n,p,x)
715 //////////////////////////////////////////////////////////////////////
718 // anonymous namespace for helper functions
722 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
724 std::vector<std::vector<cln::cl_N> > Yn;
725 int ynsize = 0; // number of Yn[]
726 int ynlength = 100; // initial length of all Yn[i]
729 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
730 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
731 // representing S_{n,p}(x).
732 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
734 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
735 // representing S_{n,p}(x).
736 // The calculation of Y_n uses the values from Y_{n-1}.
737 void fill_Yn(int n, const cln::float_format_t& prec)
739 const int initsize = ynlength;
740 //const int initsize = initsize_Yn;
741 cln::cl_N one = cln::cl_float(1, prec);
744 std::vector<cln::cl_N> buf(initsize);
745 std::vector<cln::cl_N>::iterator it = buf.begin();
746 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
747 *it = (*itprev) / cln::cl_N(n+1) * one;
750 // sums with an index smaller than the depth are zero and need not to be calculated.
751 // calculation starts with depth, which is n+2)
752 for (int i=n+2; i<=initsize+n; i++) {
753 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
759 std::vector<cln::cl_N> buf(initsize);
760 std::vector<cln::cl_N>::iterator it = buf.begin();
763 for (int i=2; i<=initsize; i++) {
764 *it = *(it-1) + 1 / cln::cl_N(i) * one;
773 // make Yn longer ...
774 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
777 cln::cl_N one = cln::cl_float(1, prec);
779 Yn[0].resize(newsize);
780 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
782 for (int i=ynlength+1; i<=newsize; i++) {
783 *it = *(it-1) + 1 / cln::cl_N(i) * one;
787 for (int n=1; n<ynsize; n++) {
788 Yn[n].resize(newsize);
789 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
790 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
793 for (int i=ynlength+n+1; i<=newsize+n; i++) {
794 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
804 // helper function for S(n,p,x)
806 cln::cl_N C(int n, int p)
810 for (int k=0; k<p; k++) {
811 for (int j=0; j<=(n+k-1)/2; j++) {
815 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
818 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
825 result = result + cln::factorial(n+k-1)
826 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
827 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
830 result = result - cln::factorial(n+k-1)
831 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
832 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
837 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
838 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
841 result = result + cln::factorial(n+k-1)
842 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
843 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
851 if (((np)/2+n) & 1) {
852 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
855 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
863 // helper function for S(n,p,x)
864 // [Kol] remark to (9.1)
874 for (int m=2; m<=k; m++) {
875 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
882 // helper function for S(n,p,x)
883 // [Kol] remark to (9.1)
893 for (int m=2; m<=k; m++) {
894 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
901 // helper function for S(n,p,x)
902 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
905 return Li_projection(n+1, x, prec);
908 // check if precalculated values are sufficient
910 for (int i=ynsize; i<p-1; i++) {
915 // should be done otherwise
916 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
917 cln::cl_N xf = x * one;
918 //cln::cl_N xf = x * cln::cl_float(1, prec);
922 cln::cl_N factor = cln::expt(xf, p);
926 if (i-p >= ynlength) {
928 make_Yn_longer(ynlength*2, prec);
930 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? ...
931 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
932 factor = factor * xf;
934 } while (res != resbuf);
940 // helper function for S(n,p,x)
941 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
944 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
946 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
947 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
949 for (int s=0; s<n; s++) {
951 for (int r=0; r<p; r++) {
952 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
953 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
955 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
961 return S_do_sum(n, p, x, prec);
965 // helper function for S(n,p,x)
966 numeric S_num(int n, int p, const numeric& x)
970 // [Kol] (2.22) with (2.21)
971 return cln::zeta(p+1);
976 return cln::zeta(n+1);
981 for (int nu=0; nu<n; nu++) {
982 for (int rho=0; rho<=p; rho++) {
983 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
984 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
987 result = result * cln::expt(cln::cl_I(-1),n+p-1);
994 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
996 // throw std::runtime_error("don't know how to evaluate this function!");
999 // what is the desired float format?
1000 // first guess: default format
1001 cln::float_format_t prec = cln::default_float_format;
1002 const cln::cl_N value = x.to_cl_N();
1003 // second guess: the argument's format
1004 if (!x.real().is_rational())
1005 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1006 else if (!x.imag().is_rational())
1007 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1010 if ((cln::realpart(value) < -0.5) || (n == 0)) {
1012 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
1013 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
1015 for (int s=0; s<n; s++) {
1017 for (int r=0; r<p; r++) {
1018 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
1019 * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
1021 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
1028 if (cln::abs(value) > 1) {
1032 for (int s=0; s<p; s++) {
1033 for (int r=0; r<=s; r++) {
1034 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
1035 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
1036 * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
1039 result = result * cln::expt(cln::cl_I(-1),n);
1042 for (int r=0; r<n; r++) {
1043 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
1045 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
1047 result = result + cln::expt(cln::cl_I(-1),p) * res2;
1052 return S_projection(n, p, value, prec);
1057 } // end of anonymous namespace
1060 //////////////////////////////////////////////////////////////////////
1062 // Nielsen's generalized polylogarithm S(n,p,x)
1066 //////////////////////////////////////////////////////////////////////
1069 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1071 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1072 if (is_a<numeric>(x)) {
1073 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1075 ex x_val = x.evalf();
1076 if (is_a<numeric>(x_val)) {
1077 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x_val));
1081 return S(n, p, x).hold();
1085 static ex S_eval(const ex& n, const ex& p, const ex& x)
1087 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1093 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
1101 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
1102 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1107 return pow(-log(1-x), p) / factorial(p);
1109 return S(n, p, x).hold();
1113 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
1116 seq.push_back(expair(S(n, p, x), 0));
1117 return pseries(rel, seq);
1121 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
1123 GINAC_ASSERT(deriv_param < 3);
1124 if (deriv_param < 2) {
1128 return S(n-1, p, x) / x;
1130 return S(n, p-1, x) / (1-x);
1135 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
1137 c.s << "\\mbox{S}_{";
1147 REGISTER_FUNCTION(S,
1148 evalf_func(S_evalf).
1150 series_func(S_series).
1151 derivative_func(S_deriv).
1152 print_func<print_latex>(S_print_latex).
1153 do_not_evalf_params());
1156 //////////////////////////////////////////////////////////////////////
1158 // Harmonic polylogarithm H(m,x)
1162 //////////////////////////////////////////////////////////////////////
1165 // anonymous namespace for helper functions
1169 // regulates the pole (used by 1/x-transformation)
1170 symbol H_polesign("IMSIGN");
1173 // convert parameters from H to Li representation
1174 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
1175 // returns true if some parameters are negative
1176 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
1178 // expand parameter list
1180 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
1182 for (ex count=*it-1; count > 0; count--) {
1186 } else if (*it < -1) {
1187 for (ex count=*it+1; count < 0; count++) {
1198 bool has_negative_parameters = false;
1200 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
1206 m.append((*it+acc-1) * signum);
1208 m.append((*it-acc+1) * signum);
1214 has_negative_parameters = true;
1217 if (has_negative_parameters) {
1218 for (int i=0; i<m.nops(); i++) {
1220 m.let_op(i) = -m.op(i);
1228 return has_negative_parameters;
1232 // recursivly transforms H to corresponding multiple polylogarithms
1233 struct map_trafo_H_convert_to_Li : public map_function
1235 ex operator()(const ex& e)
1237 if (is_a<add>(e) || is_a<mul>(e)) {
1238 return e.map(*this);
1240 if (is_a<function>(e)) {
1241 std::string name = ex_to<function>(e).get_name();
1244 if (is_a<lst>(e.op(0))) {
1245 parameter = ex_to<lst>(e.op(0));
1247 parameter = lst(e.op(0));
1254 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1255 s.let_op(0) = s.op(0) * arg;
1256 return pf * Li(m, s).hold();
1258 for (int i=0; i<m.nops(); i++) {
1261 s.let_op(0) = s.op(0) * arg;
1262 return Li(m, s).hold();
1271 // recursivly transforms H to corresponding zetas
1272 struct map_trafo_H_convert_to_zeta : public map_function
1274 ex operator()(const ex& e)
1276 if (is_a<add>(e) || is_a<mul>(e)) {
1277 return e.map(*this);
1279 if (is_a<function>(e)) {
1280 std::string name = ex_to<function>(e).get_name();
1283 if (is_a<lst>(e.op(0))) {
1284 parameter = ex_to<lst>(e.op(0));
1286 parameter = lst(e.op(0));
1292 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1293 return pf * zeta(m, s);
1304 // remove trailing zeros from H-parameters
1305 struct map_trafo_H_reduce_trailing_zeros : public map_function
1307 ex operator()(const ex& e)
1309 if (is_a<add>(e) || is_a<mul>(e)) {
1310 return e.map(*this);
1312 if (is_a<function>(e)) {
1313 std::string name = ex_to<function>(e).get_name();
1316 if (is_a<lst>(e.op(0))) {
1317 parameter = ex_to<lst>(e.op(0));
1319 parameter = lst(e.op(0));
1322 if (parameter.op(parameter.nops()-1) == 0) {
1325 if (parameter.nops() == 1) {
1330 lst::const_iterator it = parameter.begin();
1331 while ((it != parameter.end()) && (*it == 0)) {
1334 if (it == parameter.end()) {
1335 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1339 parameter.remove_last();
1340 int lastentry = parameter.nops();
1341 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1346 ex result = log(arg) * H(parameter,arg).hold();
1348 for (ex i=0; i<lastentry; i++) {
1349 if (parameter[i] > 0) {
1351 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
1354 } else if (parameter[i] < 0) {
1356 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
1364 if (lastentry < parameter.nops()) {
1365 result = result / (parameter.nops()-lastentry+1);
1366 return result.map(*this);
1378 // returns an expression with zeta functions corresponding to the parameter list for H
1379 ex convert_H_to_zeta(const lst& m)
1381 symbol xtemp("xtemp");
1382 map_trafo_H_reduce_trailing_zeros filter;
1383 map_trafo_H_convert_to_zeta filter2;
1384 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
1388 // convert signs form Li to H representation
1389 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
1392 lst::const_iterator itm = m.begin();
1393 lst::const_iterator itx = ++x.begin();
1398 while (itx != x.end()) {
1401 res.append((*itm) * signum);
1409 // multiplies an one-dimensional H with another H
1411 ex trafo_H_mult(const ex& h1, const ex& h2)
1416 ex h1nops = h1.op(0).nops();
1417 ex h2nops = h2.op(0).nops();
1419 hshort = h2.op(0).op(0);
1420 hlong = ex_to<lst>(h1.op(0));
1422 hshort = h1.op(0).op(0);
1424 hlong = ex_to<lst>(h2.op(0));
1426 hlong = h2.op(0).op(0);
1429 for (int i=0; i<=hlong.nops(); i++) {
1433 newparameter.append(hlong[j]);
1435 newparameter.append(hshort);
1436 for (; j<hlong.nops(); j++) {
1437 newparameter.append(hlong[j]);
1439 res += H(newparameter, h1.op(1)).hold();
1445 // applies trafo_H_mult recursively on expressions
1446 struct map_trafo_H_mult : public map_function
1448 ex operator()(const ex& e)
1451 return e.map(*this);
1459 for (int pos=0; pos<e.nops(); pos++) {
1460 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1461 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1463 for (ex i=0; i<e.op(pos).op(1); i++) {
1464 Hlst.append(e.op(pos).op(0));
1468 } else if (is_a<function>(e.op(pos))) {
1469 std::string name = ex_to<function>(e.op(pos)).get_name();
1471 if (e.op(pos).op(0).nops() > 1) {
1474 Hlst.append(e.op(pos));
1479 result *= e.op(pos);
1482 if (Hlst.nops() > 0) {
1483 firstH = Hlst[Hlst.nops()-1];
1490 if (Hlst.nops() > 0) {
1491 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1493 for (int i=1; i<Hlst.nops(); i++) {
1494 result *= Hlst.op(i);
1496 result = result.expand();
1497 map_trafo_H_mult recursion;
1498 return recursion(result);
1509 // do integration [ReV] (55)
1510 // put parameter 0 in front of existing parameters
1511 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
1515 if (is_a<function>(e)) {
1516 name = ex_to<function>(e).get_name();
1521 for (int i=0; i<e.nops(); i++) {
1522 if (is_a<function>(e.op(i))) {
1523 std::string name = ex_to<function>(e.op(i)).get_name();
1531 lst newparameter = ex_to<lst>(h.op(0));
1532 newparameter.prepend(0);
1533 ex addzeta = convert_H_to_zeta(newparameter);
1534 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1536 return e * (-H(lst(0),1/arg).hold());
1541 // do integration [ReV] (55)
1542 // put parameter -1 in front of existing parameters
1543 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
1547 if (is_a<function>(e)) {
1548 name = ex_to<function>(e).get_name();
1553 for (int i=0; i<e.nops(); i++) {
1554 if (is_a<function>(e.op(i))) {
1555 std::string name = ex_to<function>(e.op(i)).get_name();
1563 lst newparameter = ex_to<lst>(h.op(0));
1564 newparameter.prepend(-1);
1565 ex addzeta = convert_H_to_zeta(newparameter);
1566 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1568 ex addzeta = convert_H_to_zeta(lst(-1));
1569 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
1574 // do integration [ReV] (55)
1575 // put parameter -1 in front of existing parameters
1576 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
1580 if (is_a<function>(e)) {
1581 name = ex_to<function>(e).get_name();
1586 for (int i=0; i<e.nops(); i++) {
1587 if (is_a<function>(e.op(i))) {
1588 std::string name = ex_to<function>(e.op(i)).get_name();
1596 lst newparameter = ex_to<lst>(h.op(0));
1597 newparameter.prepend(-1);
1598 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1600 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
1605 // do integration [ReV] (55)
1606 // put parameter 1 in front of existing parameters
1607 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
1611 if (is_a<function>(e)) {
1612 name = ex_to<function>(e).get_name();
1617 for (int i=0; i<e.nops(); i++) {
1618 if (is_a<function>(e.op(i))) {
1619 std::string name = ex_to<function>(e.op(i)).get_name();
1627 lst newparameter = ex_to<lst>(h.op(0));
1628 newparameter.prepend(1);
1629 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1631 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
1636 // do x -> 1/x transformation
1637 struct map_trafo_H_1overx : public map_function
1639 ex operator()(const ex& e)
1641 if (is_a<add>(e) || is_a<mul>(e)) {
1642 return e.map(*this);
1645 if (is_a<function>(e)) {
1646 std::string name = ex_to<function>(e).get_name();
1649 lst parameter = ex_to<lst>(e.op(0));
1652 // special cases if all parameters are either 0, 1 or -1
1653 bool allthesame = true;
1654 if (parameter.op(0) == 0) {
1655 for (int i=1; i<parameter.nops(); i++) {
1656 if (parameter.op(i) != 0) {
1662 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1664 } else if (parameter.op(0) == -1) {
1665 for (int i=1; i<parameter.nops(); i++) {
1666 if (parameter.op(i) != -1) {
1672 map_trafo_H_mult unify;
1673 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
1674 / factorial(parameter.nops())).expand());
1677 for (int i=1; i<parameter.nops(); i++) {
1678 if (parameter.op(i) != 1) {
1684 map_trafo_H_mult unify;
1685 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
1686 / factorial(parameter.nops())).expand());
1690 lst newparameter = parameter;
1691 newparameter.remove_first();
1693 if (parameter.op(0) == 0) {
1696 ex res = convert_H_to_zeta(parameter);
1697 map_trafo_H_1overx recursion;
1698 ex buffer = recursion(H(newparameter, arg).hold());
1699 if (is_a<add>(buffer)) {
1700 for (int i=0; i<buffer.nops(); i++) {
1701 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
1704 res += trafo_H_1tx_prepend_zero(buffer, arg);
1708 } else if (parameter.op(0) == -1) {
1710 // leading negative one
1711 ex res = convert_H_to_zeta(parameter);
1712 map_trafo_H_1overx recursion;
1713 ex buffer = recursion(H(newparameter, arg).hold());
1714 if (is_a<add>(buffer)) {
1715 for (int i=0; i<buffer.nops(); i++) {
1716 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
1719 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
1726 map_trafo_H_1overx recursion;
1727 map_trafo_H_mult unify;
1728 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1730 while (parameter.op(firstzero) == 1) {
1733 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1737 newparameter.append(parameter[j+1]);
1739 newparameter.append(1);
1740 for (; j<parameter.nops()-1; j++) {
1741 newparameter.append(parameter[j+1]);
1743 res -= H(newparameter, arg).hold();
1745 res = recursion(res).expand() / firstzero;
1757 // do x -> (1-x)/(1+x) transformation
1758 struct map_trafo_H_1mxt1px : public map_function
1760 ex operator()(const ex& e)
1762 if (is_a<add>(e) || is_a<mul>(e)) {
1763 return e.map(*this);
1766 if (is_a<function>(e)) {
1767 std::string name = ex_to<function>(e).get_name();
1770 lst parameter = ex_to<lst>(e.op(0));
1773 // special cases if all parameters are either 0, 1 or -1
1774 bool allthesame = true;
1775 if (parameter.op(0) == 0) {
1776 for (int i=1; i<parameter.nops(); i++) {
1777 if (parameter.op(i) != 0) {
1783 map_trafo_H_mult unify;
1784 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1785 / factorial(parameter.nops())).expand());
1787 } else if (parameter.op(0) == -1) {
1788 for (int i=1; i<parameter.nops(); i++) {
1789 if (parameter.op(i) != -1) {
1795 map_trafo_H_mult unify;
1796 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1797 / factorial(parameter.nops())).expand());
1800 for (int i=1; i<parameter.nops(); i++) {
1801 if (parameter.op(i) != 1) {
1807 map_trafo_H_mult unify;
1808 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1809 / factorial(parameter.nops())).expand());
1813 lst newparameter = parameter;
1814 newparameter.remove_first();
1816 if (parameter.op(0) == 0) {
1819 ex res = convert_H_to_zeta(parameter);
1820 map_trafo_H_1mxt1px recursion;
1821 ex buffer = recursion(H(newparameter, arg).hold());
1822 if (is_a<add>(buffer)) {
1823 for (int i=0; i<buffer.nops(); i++) {
1824 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1827 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1831 } else if (parameter.op(0) == -1) {
1833 // leading negative one
1834 ex res = convert_H_to_zeta(parameter);
1835 map_trafo_H_1mxt1px recursion;
1836 ex buffer = recursion(H(newparameter, arg).hold());
1837 if (is_a<add>(buffer)) {
1838 for (int i=0; i<buffer.nops(); i++) {
1839 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1842 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1849 map_trafo_H_1mxt1px recursion;
1850 map_trafo_H_mult unify;
1851 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1853 while (parameter.op(firstzero) == 1) {
1856 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1860 newparameter.append(parameter[j+1]);
1862 newparameter.append(1);
1863 for (; j<parameter.nops()-1; j++) {
1864 newparameter.append(parameter[j+1]);
1866 res -= H(newparameter, arg).hold();
1868 res = recursion(res).expand() / firstzero;
1880 // do the actual summation.
1881 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1883 const int j = m.size();
1885 std::vector<cln::cl_N> t(j);
1887 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1888 cln::cl_N factor = cln::expt(x, j) * one;
1894 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1895 for (int k=j-2; k>=1; k--) {
1896 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1898 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1899 factor = factor * x;
1900 } while (t[0] != t0buf);
1906 } // end of anonymous namespace
1909 //////////////////////////////////////////////////////////////////////
1911 // Harmonic polylogarithm H(m,x)
1915 //////////////////////////////////////////////////////////////////////
1918 static ex H_evalf(const ex& x1, const ex& x2)
1920 if (is_a<lst>(x1)) {
1923 if (is_a<numeric>(x2)) {
1924 x = ex_to<numeric>(x2).to_cl_N();
1926 ex x2_val = x2.evalf();
1927 if (is_a<numeric>(x2_val)) {
1928 x = ex_to<numeric>(x2_val).to_cl_N();
1932 for (int i=0; i<x1.nops(); i++) {
1933 if (!x1.op(i).info(info_flags::integer)) {
1934 return H(x1, x2).hold();
1937 if (x1.nops() < 1) {
1938 return H(x1, x2).hold();
1941 const lst& morg = ex_to<lst>(x1);
1942 // remove trailing zeros ...
1943 if (*(--morg.end()) == 0) {
1944 symbol xtemp("xtemp");
1945 map_trafo_H_reduce_trailing_zeros filter;
1946 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
1948 // ... and expand parameter notation
1950 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
1952 for (ex count=*it-1; count > 0; count--) {
1956 } else if (*it < -1) {
1957 for (ex count=*it+1; count < 0; count++) {
1966 // since the transformations produce a lot of terms, they are only efficient for
1967 // argument near one.
1968 // no transformation needed -> do summation
1969 if (cln::abs(x) < 0.95) {
1973 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
1974 // negative parameters -> s_lst is filled
1975 std::vector<int> m_int;
1976 std::vector<cln::cl_N> x_cln;
1977 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
1978 it_int != m_lst.end(); it_int++, it_cln++) {
1979 m_int.push_back(ex_to<numeric>(*it_int).to_int());
1980 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
1982 x_cln.front() = x_cln.front() * x;
1983 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
1985 // only positive parameters
1987 if (m_lst.nops() == 1) {
1988 return Li(m_lst.op(0), x2).evalf();
1990 std::vector<int> m_int;
1991 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
1992 m_int.push_back(ex_to<numeric>(*it).to_int());
1994 return numeric(H_do_sum(m_int, x));
2000 // ensure that the realpart of the argument is positive
2001 if (cln::realpart(x) < 0) {
2003 for (int i=0; i<m.nops(); i++) {
2005 m.let_op(i) = -m.op(i);
2011 // choose transformations
2012 symbol xtemp("xtemp");
2013 if (cln::abs(x-1) < 1.4142) {
2015 map_trafo_H_1mxt1px trafo;
2016 res *= trafo(H(m, xtemp));
2019 map_trafo_H_1overx trafo;
2020 res *= trafo(H(m, xtemp));
2021 if (cln::imagpart(x) <= 0) {
2022 res = res.subs(H_polesign == -I*Pi);
2024 res = res.subs(H_polesign == I*Pi);
2030 // map_trafo_H_convert converter;
2031 // res = converter(res).expand();
2033 // res.find(H(wild(1),wild(2)), ll);
2034 // res.find(zeta(wild(1)), ll);
2035 // res.find(zeta(wild(1),wild(2)), ll);
2036 // res = res.collect(ll);
2038 return res.subs(xtemp == numeric(x)).evalf();
2041 return H(x1,x2).hold();
2045 static ex H_eval(const ex& m_, const ex& x)
2048 if (is_a<lst>(m_)) {
2053 if (m.nops() == 0) {
2061 if (*m.begin() > _ex1) {
2067 } else if (*m.begin() < _ex_1) {
2073 } else if (*m.begin() == _ex0) {
2080 for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
2081 if ((*it).info(info_flags::integer)) {
2092 } else if (*it < _ex_1) {
2112 } else if (step == 1) {
2124 // if some m_i is not an integer
2125 return H(m_, x).hold();
2128 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
2129 return convert_H_to_zeta(m);
2135 return H(m_, x).hold();
2137 return pow(log(x), m.nops()) / factorial(m.nops());
2140 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
2142 } else if ((step == 1) && (pos1 == _ex0)){
2147 return pow(-1, p) * S(n, p, -x);
2153 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2154 return H(m_, x).evalf();
2156 return H(m_, x).hold();
2160 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
2163 seq.push_back(expair(H(m, x), 0));
2164 return pseries(rel, seq);
2168 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
2170 GINAC_ASSERT(deriv_param < 2);
2171 if (deriv_param == 0) {
2175 if (is_a<lst>(m_)) {
2191 return 1/(1-x) * H(m, x);
2192 } else if (mb == _ex_1) {
2193 return 1/(1+x) * H(m, x);
2200 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
2203 if (is_a<lst>(m_)) {
2208 c.s << "\\mbox{H}_{";
2209 lst::const_iterator itm = m.begin();
2212 for (; itm != m.end(); itm++) {
2222 REGISTER_FUNCTION(H,
2223 evalf_func(H_evalf).
2225 series_func(H_series).
2226 derivative_func(H_deriv).
2227 print_func<print_latex>(H_print_latex).
2228 do_not_evalf_params());
2231 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
2232 ex convert_H_to_Li(const ex& m, const ex& x)
2234 map_trafo_H_reduce_trailing_zeros filter;
2235 map_trafo_H_convert_to_Li filter2;
2237 return filter2(filter(H(m, x).hold()));
2239 return filter2(filter(H(lst(m), x).hold()));
2244 //////////////////////////////////////////////////////////////////////
2246 // Multiple zeta values zeta(x) and zeta(x,s)
2250 //////////////////////////////////////////////////////////////////////
2253 // anonymous namespace for helper functions
2257 // parameters and data for [Cra] algorithm
2258 const cln::cl_N lambda = cln::cl_N("319/320");
2261 std::vector<std::vector<cln::cl_N> > f_kj;
2262 std::vector<cln::cl_N> crB;
2263 std::vector<std::vector<cln::cl_N> > crG;
2264 std::vector<cln::cl_N> crX;
2267 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
2269 const int size = a.size();
2270 for (int n=0; n<size; n++) {
2272 for (int m=0; m<=n; m++) {
2273 c[n] = c[n] + a[m]*b[n-m];
2280 void initcX(const std::vector<int>& s)
2282 const int k = s.size();
2288 for (int i=0; i<=L2; i++) {
2289 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
2294 for (int m=0; m<k-1; m++) {
2295 std::vector<cln::cl_N> crGbuf;
2298 for (int i=0; i<=L2; i++) {
2299 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
2301 crG.push_back(crGbuf);
2306 for (int m=0; m<k-1; m++) {
2307 std::vector<cln::cl_N> Xbuf;
2308 for (int i=0; i<=L2; i++) {
2309 Xbuf.push_back(crX[i] * crG[m][i]);
2311 halfcyclic_convolute(Xbuf, crB, crX);
2317 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
2319 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2320 cln::cl_N factor = cln::expt(lambda, Sqk);
2321 cln::cl_N res = factor / Sqk * crX[0] * one;
2326 factor = factor * lambda;
2328 res = res + crX[N] * factor / (N+Sqk);
2329 } while ((res != resbuf) || cln::zerop(crX[N]));
2335 void calc_f(int maxr)
2340 cln::cl_N t0, t1, t2, t3, t4;
2342 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
2343 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2345 t0 = cln::exp(-lambda);
2347 for (k=1; k<=L1; k++) {
2350 for (j=1; j<=maxr; j++) {
2353 for (i=2; i<=j; i++) {
2357 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
2365 cln::cl_N crandall_Z(const std::vector<int>& s)
2367 const int j = s.size();
2376 t0 = t0 + f_kj[q+j-2][s[0]-1];
2377 } while (t0 != t0buf);
2379 return t0 / cln::factorial(s[0]-1);
2382 std::vector<cln::cl_N> t(j);
2389 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
2390 for (int k=j-2; k>=1; k--) {
2391 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
2393 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
2394 } while (t[0] != t0buf);
2396 return t[0] / cln::factorial(s[0]-1);
2401 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
2403 std::vector<int> r = s;
2404 const int j = r.size();
2406 // decide on maximal size of f_kj for crandall_Z
2410 L1 = Digits * 3 + j*2;
2413 // decide on maximal size of crX for crandall_Y
2416 } else if (Digits < 86) {
2418 } else if (Digits < 192) {
2420 } else if (Digits < 394) {
2422 } else if (Digits < 808) {
2432 for (int i=0; i<j; i++) {
2441 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
2443 std::vector<int> rz;
2446 for (int k=r.size()-1; k>0; k--) {
2448 rz.insert(rz.begin(), r.back());
2449 skp1buf = rz.front();
2455 for (int q=0; q<skp1buf; q++) {
2457 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2458 cln::cl_N pp2 = crandall_Z(rz);
2463 res = res - pp1 * pp2 / cln::factorial(q);
2465 res = res + pp1 * pp2 / cln::factorial(q);
2468 rz.front() = skp1buf;
2470 rz.insert(rz.begin(), r.back());
2474 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2480 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2482 const int j = r.size();
2484 // buffer for subsums
2485 std::vector<cln::cl_N> t(j);
2486 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2493 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2494 for (int k=j-2; k>=0; k--) {
2495 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2497 } while (t[0] != t0buf);
2503 // does Hoelder convolution. see [BBB] (7.0)
2504 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2506 // prepare parameters
2507 // holds Li arguments in [BBB] notation
2508 std::vector<int> s = s_;
2509 std::vector<int> m_p = m_;
2510 std::vector<int> m_q;
2511 // holds Li arguments in nested sums notation
2512 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2513 s_p[0] = s_p[0] * cln::cl_N("1/2");
2514 // convert notations
2516 for (int i=0; i<s_.size(); i++) {
2521 s[i] = sig * std::abs(s[i]);
2523 std::vector<cln::cl_N> s_q;
2524 cln::cl_N signum = 1;
2527 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2532 // change parameters
2533 if (s.front() > 0) {
2534 if (m_p.front() == 1) {
2535 m_p.erase(m_p.begin());
2536 s_p.erase(s_p.begin());
2537 if (s_p.size() > 0) {
2538 s_p.front() = s_p.front() * cln::cl_N("1/2");
2544 m_q.insert(m_q.begin(), 1);
2545 if (s_q.size() > 0) {
2546 s_q.front() = s_q.front() * 2;
2548 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2551 if (m_p.front() == 1) {
2552 m_p.erase(m_p.begin());
2553 cln::cl_N spbuf = s_p.front();
2554 s_p.erase(s_p.begin());
2555 if (s_p.size() > 0) {
2556 s_p.front() = s_p.front() * spbuf;
2559 m_q.insert(m_q.begin(), 1);
2560 if (s_q.size() > 0) {
2561 s_q.front() = s_q.front() * 4;
2563 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2567 m_q.insert(m_q.begin(), 1);
2568 if (s_q.size() > 0) {
2569 s_q.front() = s_q.front() * 2;
2571 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2576 if (m_p.size() == 0) break;
2578 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2583 res = res + signum * multipleLi_do_sum(m_q, s_q);
2589 } // end of anonymous namespace
2592 //////////////////////////////////////////////////////////////////////
2594 // Multiple zeta values zeta(x)
2598 //////////////////////////////////////////////////////////////////////
2601 static ex zeta1_evalf(const ex& x)
2603 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2605 // multiple zeta value
2606 const int count = x.nops();
2607 const lst& xlst = ex_to<lst>(x);
2608 std::vector<int> r(count);
2610 // check parameters and convert them
2611 lst::const_iterator it1 = xlst.begin();
2612 std::vector<int>::iterator it2 = r.begin();
2614 if (!(*it1).info(info_flags::posint)) {
2615 return zeta(x).hold();
2617 *it2 = ex_to<numeric>(*it1).to_int();
2620 } while (it2 != r.end());
2622 // check for divergence
2624 return zeta(x).hold();
2627 // decide on summation algorithm
2628 // this is still a bit clumsy
2629 int limit = (Digits>17) ? 10 : 6;
2630 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2631 return numeric(zeta_do_sum_Crandall(r));
2633 return numeric(zeta_do_sum_simple(r));
2637 // single zeta value
2638 if (is_exactly_a<numeric>(x) && (x != 1)) {
2640 return zeta(ex_to<numeric>(x));
2641 } catch (const dunno &e) { }
2644 return zeta(x).hold();
2648 static ex zeta1_eval(const ex& m)
2650 if (is_exactly_a<lst>(m)) {
2651 if (m.nops() == 1) {
2652 return zeta(m.op(0));
2654 return zeta(m).hold();
2657 if (m.info(info_flags::numeric)) {
2658 const numeric& y = ex_to<numeric>(m);
2659 // trap integer arguments:
2660 if (y.is_integer()) {
2664 if (y.is_equal(_num1)) {
2665 return zeta(m).hold();
2667 if (y.info(info_flags::posint)) {
2668 if (y.info(info_flags::odd)) {
2669 return zeta(m).hold();
2671 return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2674 if (y.info(info_flags::odd)) {
2675 return -bernoulli(_num1-y) / (_num1-y);
2682 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
2683 return zeta1_evalf(m);
2686 return zeta(m).hold();
2690 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
2692 GINAC_ASSERT(deriv_param==0);
2694 if (is_exactly_a<lst>(m)) {
2697 return zetaderiv(_ex1, m);
2702 static void zeta1_print_latex(const ex& m_, const print_context& c)
2705 if (is_a<lst>(m_)) {
2706 const lst& m = ex_to<lst>(m_);
2707 lst::const_iterator it = m.begin();
2710 for (; it != m.end(); it++) {
2721 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
2722 evalf_func(zeta1_evalf).
2723 eval_func(zeta1_eval).
2724 derivative_func(zeta1_deriv).
2725 print_func<print_latex>(zeta1_print_latex).
2726 do_not_evalf_params().
2730 //////////////////////////////////////////////////////////////////////
2732 // Alternating Euler sum zeta(x,s)
2736 //////////////////////////////////////////////////////////////////////
2739 static ex zeta2_evalf(const ex& x, const ex& s)
2741 if (is_exactly_a<lst>(x)) {
2743 // alternating Euler sum
2744 const int count = x.nops();
2745 const lst& xlst = ex_to<lst>(x);
2746 const lst& slst = ex_to<lst>(s);
2747 std::vector<int> xi(count);
2748 std::vector<int> si(count);
2750 // check parameters and convert them
2751 lst::const_iterator it_xread = xlst.begin();
2752 lst::const_iterator it_sread = slst.begin();
2753 std::vector<int>::iterator it_xwrite = xi.begin();
2754 std::vector<int>::iterator it_swrite = si.begin();
2756 if (!(*it_xread).info(info_flags::posint)) {
2757 return zeta(x, s).hold();
2759 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2760 if (*it_sread > 0) {
2769 } while (it_xwrite != xi.end());
2771 // check for divergence
2772 if ((xi[0] == 1) && (si[0] == 1)) {
2773 return zeta(x, s).hold();
2776 // use Hoelder convolution
2777 return numeric(zeta_do_Hoelder_convolution(xi, si));
2780 return zeta(x, s).hold();
2784 static ex zeta2_eval(const ex& m, const ex& s_)
2786 if (is_exactly_a<lst>(s_)) {
2787 const lst& s = ex_to<lst>(s_);
2788 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
2789 if ((*it).info(info_flags::positive)) {
2792 return zeta(m, s_).hold();
2795 } else if (s_.info(info_flags::positive)) {
2799 return zeta(m, s_).hold();
2803 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
2805 GINAC_ASSERT(deriv_param==0);
2807 if (is_exactly_a<lst>(m)) {
2810 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
2811 return zetaderiv(_ex1, m);
2818 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
2821 if (is_a<lst>(m_)) {
2827 if (is_a<lst>(s_)) {
2833 lst::const_iterator itm = m.begin();
2834 lst::const_iterator its = s.begin();
2836 c.s << "\\overline{";
2844 for (; itm != m.end(); itm++, its++) {
2847 c.s << "\\overline{";
2858 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
2859 evalf_func(zeta2_evalf).
2860 eval_func(zeta2_eval).
2861 derivative_func(zeta2_deriv).
2862 print_func<print_latex>(zeta2_print_latex).
2863 do_not_evalf_params().
2867 } // namespace GiNaC