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 speed 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) 1999-2003 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;
108 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
109 // With these numbers the polylogs can be calculated as follows:
110 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
111 // X_0(n) = B_n (Bernoulli numbers)
112 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
113 // The calculation of Xn depends on X0 and X{n-1}.
114 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
115 // This results in a slightly more complicated algorithm for the X_n.
116 // The first index in Xn corresponds to the index of the polylog minus 2.
117 // The second index in Xn corresponds to the index from the actual sum.
120 // rule of thumb. needs to be improved. TODO
121 const int initsize = Digits * 3 / 2;
124 // calculate X_2 and higher (corresponding to Li_4 and higher)
125 std::vector<cln::cl_N> buf(initsize);
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<=initsize; 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(initsize);
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<=initsize; 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(initsize/2);
175 std::vector<cln::cl_N>::iterator it = buf.begin();
176 for (int i=1; i<=initsize/2; i++) {
177 *it = bernoulli(i*2).to_cl_N();
187 // calculates Li(2,x) without Xn
188 cln::cl_N Li2_do_sum(const cln::cl_N& x)
193 cln::cl_I den = 1; // n^2 = 1
198 den = den + i; // n^2 = 4, 9, 16, ...
200 res = res + num / den;
201 } while (res != resbuf);
206 // calculates Li(2,x) with Xn
207 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
209 std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
210 cln::cl_N u = -cln::log(1-x);
211 cln::cl_N factor = u;
212 cln::cl_N res = u - u*u/4;
217 factor = factor * u*u / (2*i * (2*i+1));
218 res = res + (*it) * factor;
219 it++; // should we check it? or rely on initsize? ...
221 } while (res != resbuf);
226 // calculates Li(n,x), n>2 without Xn
227 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
229 cln::cl_N factor = x;
236 res = res + factor / cln::expt(cln::cl_I(i),n);
238 } while (res != resbuf);
243 // calculates Li(n,x), n>2 with Xn
244 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
246 std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
247 cln::cl_N u = -cln::log(1-x);
248 cln::cl_N factor = u;
254 factor = factor * u / i;
255 res = res + (*it) * factor;
256 it++; // should we check it? or rely on initsize? ...
258 } while (res != resbuf);
263 // forward declaration needed by function Li_projection and C below
264 numeric S_num(int n, int p, const numeric& x);
267 // helper function for classical polylog Li
268 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
270 // treat n=2 as special case
272 // check if precalculated X0 exists
277 if (cln::realpart(x) < 0.5) {
278 // choose the faster algorithm
279 // the switching point was empirically determined. the optimal point
280 // depends on hardware, Digits, ... so an approx value is okay.
281 // it solves also the problem with precision due to the u=-log(1-x) transformation
282 if (cln::abs(cln::realpart(x)) < 0.25) {
284 return Li2_do_sum(x);
286 return Li2_do_sum_Xn(x);
289 // choose the faster algorithm
290 if (cln::abs(cln::realpart(x)) > 0.75) {
291 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
293 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
297 // check if precalculated Xn exist
299 for (int i=xnsize; i<n-1; i++) {
304 if (cln::realpart(x) < 0.5) {
305 // choose the faster algorithm
306 // with n>=12 the "normal" summation always wins against the method with Xn
307 if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
308 return Lin_do_sum(n, x);
310 return Lin_do_sum_Xn(n, x);
313 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
314 for (int j=0; j<n-1; j++) {
315 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
316 * cln::expt(cln::log(x), j) / cln::factorial(j);
324 // helper function for classical polylog Li
325 numeric Li_num(int n, const numeric& x)
329 return -cln::log(1-x.to_cl_N());
340 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
343 // what is the desired float format?
344 // first guess: default format
345 cln::float_format_t prec = cln::default_float_format;
346 const cln::cl_N value = x.to_cl_N();
347 // second guess: the argument's format
348 if (!x.real().is_rational())
349 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
350 else if (!x.imag().is_rational())
351 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
354 if (cln::abs(value) > 1) {
355 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
356 // check if argument is complex. if it is real, the new polylog has to be conjugated.
357 if (cln::zerop(cln::imagpart(value))) {
359 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
362 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
367 result = result + Li_projection(n, cln::recip(value), prec);
370 result = result - Li_projection(n, cln::recip(value), prec);
374 for (int j=0; j<n-1; j++) {
375 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
376 * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
378 result = result - add;
382 return Li_projection(n, value, prec);
387 } // end of anonymous namespace
390 //////////////////////////////////////////////////////////////////////
392 // Multiple polylogarithm Li(n,x)
396 //////////////////////////////////////////////////////////////////////
399 // anonymous namespace for helper function
403 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
405 const int j = s.size();
407 std::vector<cln::cl_N> t(j);
408 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
416 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
417 for (int k=j-2; k>=0; k--) {
418 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]);
420 // ... and do it again (to avoid premature drop out due to special arguments)
422 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
423 for (int k=j-2; k>=0; k--) {
424 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]);
426 } while (t[0] != t0buf);
432 } // end of anonymous namespace
435 //////////////////////////////////////////////////////////////////////
437 // Classical polylogarithm and multiple polylogarithm Li(n,x)
441 //////////////////////////////////////////////////////////////////////
444 static ex Li_evalf(const ex& x1, const ex& x2)
446 // classical polylogs
447 if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
448 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
451 else if (is_a<lst>(x1) && is_a<lst>(x2)) {
453 for (int i=0; i<x1.nops(); i++) {
454 if (!x1.op(i).info(info_flags::posint)) {
455 return Li(x1,x2).hold();
457 if (!is_a<numeric>(x2.op(i))) {
458 return Li(x1,x2).hold();
461 if ((conv > 1) || ((conv == 1) && (x1.op(0) == 1))) {
462 return Li(x1,x2).hold();
467 std::vector<cln::cl_N> x;
468 for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
469 m.push_back(ex_to<numeric>(x1.op(i)).to_int());
470 x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
473 return numeric(multipleLi_do_sum(m, x));
476 return Li(x1,x2).hold();
480 static ex Li_eval(const ex& x1, const ex& x2)
486 if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational)))
487 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
489 for (int i=0; i<x2.nops(); i++) {
490 if (!is_a<numeric>(x2.op(i))) {
491 return Li(x1,x2).hold();
494 return Li(x1,x2).evalf();
496 return Li(x1,x2).hold();
501 static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
504 seq.push_back(expair(Li(x1,x2), 0));
505 return pseries(rel,seq);
509 static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
511 GINAC_ASSERT(deriv_param < 2);
512 if (deriv_param == 0) {
516 return Li(x1-1, x2) / x2;
523 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
537 c.s << "\\mbox{Li}_{";
538 lst::const_iterator itm = m.begin();
541 for (; itm != m.end(); itm++) {
546 lst::const_iterator itx = x.begin();
549 for (; itx != x.end(); itx++) {
557 REGISTER_FUNCTION(Li,
558 evalf_func(Li_evalf).
560 series_func(Li_series).
561 derivative_func(Li_deriv).
562 print_func<print_latex>(Li_print_latex).
563 do_not_evalf_params());
566 //////////////////////////////////////////////////////////////////////
568 // Nielsen's generalized polylogarithm S(n,p,x)
572 //////////////////////////////////////////////////////////////////////
575 // anonymous namespace for helper functions
579 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
581 std::vector<std::vector<cln::cl_N> > Yn;
582 int ynsize = 0; // number of Yn[]
583 int ynlength = 100; // initial length of all Yn[i]
586 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
587 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
588 // representing S_{n,p}(x).
589 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
591 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
592 // representing S_{n,p}(x).
593 // The calculation of Y_n uses the values from Y_{n-1}.
594 void fill_Yn(int n, const cln::float_format_t& prec)
596 const int initsize = ynlength;
597 //const int initsize = initsize_Yn;
598 cln::cl_N one = cln::cl_float(1, prec);
601 std::vector<cln::cl_N> buf(initsize);
602 std::vector<cln::cl_N>::iterator it = buf.begin();
603 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
604 *it = (*itprev) / cln::cl_N(n+1) * one;
607 // sums with an index smaller than the depth are zero and need not to be calculated.
608 // calculation starts with depth, which is n+2)
609 for (int i=n+2; i<=initsize+n; i++) {
610 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
616 std::vector<cln::cl_N> buf(initsize);
617 std::vector<cln::cl_N>::iterator it = buf.begin();
620 for (int i=2; i<=initsize; i++) {
621 *it = *(it-1) + 1 / cln::cl_N(i) * one;
630 // make Yn longer ...
631 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
634 cln::cl_N one = cln::cl_float(1, prec);
636 Yn[0].resize(newsize);
637 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
639 for (int i=ynlength+1; i<=newsize; i++) {
640 *it = *(it-1) + 1 / cln::cl_N(i) * one;
644 for (int n=1; n<ynsize; n++) {
645 Yn[n].resize(newsize);
646 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
647 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
650 for (int i=ynlength+n+1; i<=newsize+n; i++) {
651 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
661 // helper function for S(n,p,x)
663 cln::cl_N C(int n, int p)
667 for (int k=0; k<p; k++) {
668 for (int j=0; j<=(n+k-1)/2; j++) {
672 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
675 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
682 result = result + cln::factorial(n+k-1)
683 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
684 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
687 result = result - cln::factorial(n+k-1)
688 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
689 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
694 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()
695 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
698 result = result + cln::factorial(n+k-1)
699 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
700 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
708 if (((np)/2+n) & 1) {
709 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
712 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
720 // helper function for S(n,p,x)
721 // [Kol] remark to (9.1)
731 for (int m=2; m<=k; m++) {
732 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
739 // helper function for S(n,p,x)
740 // [Kol] remark to (9.1)
750 for (int m=2; m<=k; m++) {
751 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
758 // helper function for S(n,p,x)
759 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
762 return Li_projection(n+1, x, prec);
765 // check if precalculated values are sufficient
767 for (int i=ynsize; i<p-1; i++) {
772 // should be done otherwise
773 cln::cl_N xf = x * cln::cl_float(1, prec);
777 cln::cl_N factor = cln::expt(xf, p);
781 if (i-p >= ynlength) {
783 make_Yn_longer(ynlength*2, prec);
785 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? ...
786 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
787 factor = factor * xf;
789 } while (res != resbuf);
795 // helper function for S(n,p,x)
796 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
799 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
801 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
802 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
804 for (int s=0; s<n; s++) {
806 for (int r=0; r<p; r++) {
807 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
808 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
810 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
816 return S_do_sum(n, p, x, prec);
820 // helper function for S(n,p,x)
821 numeric S_num(int n, int p, const numeric& x)
825 // [Kol] (2.22) with (2.21)
826 return cln::zeta(p+1);
831 return cln::zeta(n+1);
836 for (int nu=0; nu<n; nu++) {
837 for (int rho=0; rho<=p; rho++) {
838 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
839 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
842 result = result * cln::expt(cln::cl_I(-1),n+p-1);
849 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
851 // throw std::runtime_error("don't know how to evaluate this function!");
854 // what is the desired float format?
855 // first guess: default format
856 cln::float_format_t prec = cln::default_float_format;
857 const cln::cl_N value = x.to_cl_N();
858 // second guess: the argument's format
859 if (!x.real().is_rational())
860 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
861 else if (!x.imag().is_rational())
862 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
866 if (cln::realpart(value) < -0.5) {
868 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
869 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
871 for (int s=0; s<n; s++) {
873 for (int r=0; r<p; r++) {
874 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
875 * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
877 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
884 if (cln::abs(value) > 1) {
888 for (int s=0; s<p; s++) {
889 for (int r=0; r<=s; r++) {
890 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
891 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
892 * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
895 result = result * cln::expt(cln::cl_I(-1),n);
898 for (int r=0; r<n; r++) {
899 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
901 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
903 result = result + cln::expt(cln::cl_I(-1),p) * res2;
908 return S_projection(n, p, value, prec);
913 } // end of anonymous namespace
916 //////////////////////////////////////////////////////////////////////
918 // Nielsen's generalized polylogarithm S(n,p,x)
922 //////////////////////////////////////////////////////////////////////
925 static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
927 if (is_a<numeric>(x1) && is_a<numeric>(x2) && is_a<numeric>(x3)) {
928 return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
930 return S(x1,x2,x3).hold();
934 static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
939 if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) &&
940 x1.info(info_flags::posint) && x2.info(info_flags::posint)) {
941 return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
943 return S(x1,x2,x3).hold();
947 static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options)
950 seq.push_back(expair(S(x1,x2,x3), 0));
951 return pseries(rel,seq);
955 static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param)
957 GINAC_ASSERT(deriv_param < 3);
958 if (deriv_param < 2) {
962 return S(x1-1, x2, x3) / x3;
964 return S(x1, x2-1, x3) / (1-x3);
969 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
971 c.s << "\\mbox{S}_{";
984 series_func(S_series).
985 derivative_func(S_deriv).
986 print_func<print_latex>(S_print_latex).
987 do_not_evalf_params());
990 //////////////////////////////////////////////////////////////////////
992 // Harmonic polylogarithm H(m,x)
996 //////////////////////////////////////////////////////////////////////
999 // anonymous namespace for helper functions
1003 // convert parameters from H to Li representation
1004 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
1005 // returns true if some parameters are negative
1006 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
1008 // expand parameter list
1010 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
1012 for (ex count=*it-1; count > 0; count--) {
1016 } else if (*it < -1) {
1017 for (ex count=*it+1; count < 0; count++) {
1028 bool has_negative_parameters = false;
1030 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
1036 m.append((*it+acc-1) * signum);
1038 m.append((*it-acc+1) * signum);
1044 has_negative_parameters = true;
1047 if (has_negative_parameters) {
1048 for (int i=0; i<m.nops(); i++) {
1050 m.let_op(i) = -m.op(i);
1057 for (; acc > 1; acc--) {
1058 throw std::runtime_error("ERROR!");
1062 return has_negative_parameters;
1066 // recursivly transforms H to corresponding multiple polylogarithms
1067 struct map_trafo_H_convert_to_Li : public map_function
1069 ex operator()(const ex& e)
1071 if (is_a<add>(e) || is_a<mul>(e)) {
1072 return e.map(*this);
1074 if (is_a<function>(e)) {
1075 std::string name = ex_to<function>(e).get_name();
1078 if (is_a<lst>(e.op(0))) {
1079 parameter = ex_to<lst>(e.op(0));
1081 parameter = lst(e.op(0));
1088 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1089 s.let_op(0) = s.op(0) * arg;
1090 return pf * Li(m, s).hold();
1092 for (int i=0; i<m.nops(); i++) {
1095 s.let_op(0) = s.op(0) * arg;
1096 return Li(m, s).hold();
1105 // recursivly transforms H to corresponding zetas
1106 struct map_trafo_H_convert_to_zeta : public map_function
1108 ex operator()(const ex& e)
1110 if (is_a<add>(e) || is_a<mul>(e)) {
1111 return e.map(*this);
1113 if (is_a<function>(e)) {
1114 std::string name = ex_to<function>(e).get_name();
1117 if (is_a<lst>(e.op(0))) {
1118 parameter = ex_to<lst>(e.op(0));
1120 parameter = lst(e.op(0));
1126 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1127 return pf * zeta(m, s);
1138 // remove trailing zeros from H-parameters
1139 struct map_trafo_H_reduce_trailing_zeros : public map_function
1141 ex operator()(const ex& e)
1143 if (is_a<add>(e) || is_a<mul>(e)) {
1144 return e.map(*this);
1146 if (is_a<function>(e)) {
1147 std::string name = ex_to<function>(e).get_name();
1150 if (is_a<lst>(e.op(0))) {
1151 parameter = ex_to<lst>(e.op(0));
1153 parameter = lst(e.op(0));
1156 if (parameter.op(parameter.nops()-1) == 0) {
1159 if (parameter.nops() == 1) {
1164 lst::const_iterator it = parameter.begin();
1165 while ((it != parameter.end()) && (*it == 0)) {
1168 if (it == parameter.end()) {
1169 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1173 parameter.remove_last();
1174 int lastentry = parameter.nops();
1175 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1180 ex result = log(arg) * H(parameter,arg).hold();
1182 for (ex i=0; i<lastentry; i++) {
1183 if (parameter[i] > 0) {
1185 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
1188 } else if (parameter[i] < 0) {
1190 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
1198 if (lastentry < parameter.nops()) {
1199 result = result / (parameter.nops()-lastentry+1);
1200 return result.map(*this);
1212 // returns an expression with zeta functions corresponding to the parameter list for H
1213 ex convert_H_to_zeta(const lst& l)
1215 symbol xtemp("xtemp");
1216 map_trafo_H_reduce_trailing_zeros filter;
1217 map_trafo_H_convert_to_zeta filter2;
1218 return filter2(filter(H(l, xtemp).hold())).subs(xtemp == 1);
1222 // convert signs form Li to H representation
1224 lst convert_parameter_Li_to_H(const lst& l, ex& pf)
1227 lst::const_iterator it = l.begin();
1232 while (it != l.end()) {
1233 signum = *it * signum;
1243 // multiplies an one-dimensional H with another H
1245 ex trafo_H_mult(const ex& h1, const ex& h2)
1250 ex h1nops = h1.op(0).nops();
1251 ex h2nops = h2.op(0).nops();
1253 hshort = h2.op(0).op(0);
1254 hlong = ex_to<lst>(h1.op(0));
1256 hshort = h1.op(0).op(0);
1258 hlong = ex_to<lst>(h2.op(0));
1260 hlong = h2.op(0).op(0);
1263 for (int i=0; i<=hlong.nops(); i++) {
1267 newparameter.append(hlong[j]);
1269 newparameter.append(hshort);
1270 for (; j<hlong.nops(); j++) {
1271 newparameter.append(hlong[j]);
1273 res += H(newparameter, h1.op(1)).hold();
1279 // applies trafo_H_mult recursively on expressions
1280 struct map_trafo_H_mult : public map_function
1282 ex operator()(const ex& e)
1285 return e.map(*this);
1293 for (int pos=0; pos<e.nops(); pos++) {
1294 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1295 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1297 for (ex i=0; i<e.op(pos).op(1); i++) {
1298 Hlst.append(e.op(pos).op(0));
1302 } else if (is_a<function>(e.op(pos))) {
1303 std::string name = ex_to<function>(e.op(pos)).get_name();
1305 if (e.op(pos).op(0).nops() > 1) {
1308 Hlst.append(e.op(pos));
1313 result *= e.op(pos);
1316 if (Hlst.nops() > 0) {
1317 firstH = Hlst[Hlst.nops()-1];
1324 if (Hlst.nops() > 0) {
1325 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1327 for (int i=1; i<Hlst.nops(); i++) {
1328 result *= Hlst.op(i);
1330 result = result.expand();
1331 map_trafo_H_mult recursion;
1332 return recursion(result);
1343 // do integration [ReV] (55)
1344 // put parameter 0 in front of existing parameters
1345 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
1349 if (is_a<function>(e)) {
1350 name = ex_to<function>(e).get_name();
1355 for (int i=0; i<e.nops(); i++) {
1356 if (is_a<function>(e.op(i))) {
1357 std::string name = ex_to<function>(e.op(i)).get_name();
1365 lst newparameter = ex_to<lst>(h.op(0));
1366 newparameter.prepend(0);
1367 ex addzeta = convert_H_to_zeta(newparameter);
1368 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1370 return e * (-H(lst(0),1/arg).hold());
1375 // do integration [ReV] (55)
1376 // put parameter -1 in front of existing parameters
1377 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
1381 if (is_a<function>(e)) {
1382 name = ex_to<function>(e).get_name();
1387 for (int i=0; i<e.nops(); i++) {
1388 if (is_a<function>(e.op(i))) {
1389 std::string name = ex_to<function>(e.op(i)).get_name();
1397 lst newparameter = ex_to<lst>(h.op(0));
1398 newparameter.prepend(-1);
1399 ex addzeta = convert_H_to_zeta(newparameter);
1400 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1402 ex addzeta = convert_H_to_zeta(lst(-1));
1403 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
1408 // do integration [ReV] (55)
1409 // put parameter -1 in front of existing parameters
1410 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
1414 if (is_a<function>(e)) {
1415 name = ex_to<function>(e).get_name();
1420 for (int i=0; i<e.nops(); i++) {
1421 if (is_a<function>(e.op(i))) {
1422 std::string name = ex_to<function>(e.op(i)).get_name();
1430 lst newparameter = ex_to<lst>(h.op(0));
1431 newparameter.prepend(-1);
1432 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1434 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
1439 // do integration [ReV] (55)
1440 // put parameter 1 in front of existing parameters
1441 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
1445 if (is_a<function>(e)) {
1446 name = ex_to<function>(e).get_name();
1451 for (int i=0; i<e.nops(); i++) {
1452 if (is_a<function>(e.op(i))) {
1453 std::string name = ex_to<function>(e.op(i)).get_name();
1461 lst newparameter = ex_to<lst>(h.op(0));
1462 newparameter.prepend(1);
1463 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1465 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
1470 // do x -> 1/x transformation
1471 struct map_trafo_H_1overx : public map_function
1473 ex operator()(const ex& e)
1475 if (is_a<add>(e) || is_a<mul>(e)) {
1476 return e.map(*this);
1479 if (is_a<function>(e)) {
1480 std::string name = ex_to<function>(e).get_name();
1483 lst parameter = ex_to<lst>(e.op(0));
1486 // special cases if all parameters are either 0, 1 or -1
1487 bool allthesame = true;
1488 if (parameter.op(0) == 0) {
1489 for (int i=1; i<parameter.nops(); i++) {
1490 if (parameter.op(i) != 0) {
1496 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1498 } else if (parameter.op(0) == -1) {
1499 for (int i=1; i<parameter.nops(); i++) {
1500 if (parameter.op(i) != -1) {
1506 map_trafo_H_mult unify;
1507 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops()) /
1508 factorial(parameter.nops())).expand());
1511 for (int i=1; i<parameter.nops(); i++) {
1512 if (parameter.op(i) != 1) {
1518 map_trafo_H_mult unify;
1519 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, parameter.nops()) /
1520 factorial(parameter.nops())).expand());
1524 lst newparameter = parameter;
1525 newparameter.remove_first();
1527 if (parameter.op(0) == 0) {
1530 ex res = convert_H_to_zeta(parameter);
1531 map_trafo_H_1overx recursion;
1532 ex buffer = recursion(H(newparameter, arg).hold());
1533 if (is_a<add>(buffer)) {
1534 for (int i=0; i<buffer.nops(); i++) {
1535 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
1538 res += trafo_H_1tx_prepend_zero(buffer, arg);
1542 } else if (parameter.op(0) == -1) {
1544 // leading negative one
1545 ex res = convert_H_to_zeta(parameter);
1546 map_trafo_H_1overx recursion;
1547 ex buffer = recursion(H(newparameter, arg).hold());
1548 if (is_a<add>(buffer)) {
1549 for (int i=0; i<buffer.nops(); i++) {
1550 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
1553 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
1560 map_trafo_H_1overx recursion;
1561 map_trafo_H_mult unify;
1562 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1564 while (parameter.op(firstzero) == 1) {
1567 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1571 newparameter.append(parameter[j+1]);
1573 newparameter.append(1);
1574 for (; j<parameter.nops()-1; j++) {
1575 newparameter.append(parameter[j+1]);
1577 res -= H(newparameter, arg).hold();
1579 res = recursion(res).expand() / firstzero;
1591 // do x -> (1-x)/(1+x) transformation
1592 struct map_trafo_H_1mxt1px : public map_function
1594 ex operator()(const ex& e)
1596 if (is_a<add>(e) || is_a<mul>(e)) {
1597 return e.map(*this);
1600 if (is_a<function>(e)) {
1601 std::string name = ex_to<function>(e).get_name();
1604 lst parameter = ex_to<lst>(e.op(0));
1607 // special cases if all parameters are either 0, 1 or -1
1608 bool allthesame = true;
1609 if (parameter.op(0) == 0) {
1610 for (int i=1; i<parameter.nops(); i++) {
1611 if (parameter.op(i) != 0) {
1617 map_trafo_H_mult unify;
1618 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops()) /
1619 factorial(parameter.nops())).expand());
1621 } else if (parameter.op(0) == -1) {
1622 for (int i=1; i<parameter.nops(); i++) {
1623 if (parameter.op(i) != -1) {
1629 map_trafo_H_mult unify;
1630 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops()) /
1631 factorial(parameter.nops())).expand());
1634 for (int i=1; i<parameter.nops(); i++) {
1635 if (parameter.op(i) != 1) {
1641 map_trafo_H_mult unify;
1642 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops()) /
1643 factorial(parameter.nops())).expand());
1647 lst newparameter = parameter;
1648 newparameter.remove_first();
1650 if (parameter.op(0) == 0) {
1653 ex res = convert_H_to_zeta(parameter);
1654 map_trafo_H_1mxt1px recursion;
1655 ex buffer = recursion(H(newparameter, arg).hold());
1656 if (is_a<add>(buffer)) {
1657 for (int i=0; i<buffer.nops(); i++) {
1658 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1661 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1665 } else if (parameter.op(0) == -1) {
1667 // leading negative one
1668 ex res = convert_H_to_zeta(parameter);
1669 map_trafo_H_1mxt1px recursion;
1670 ex buffer = recursion(H(newparameter, arg).hold());
1671 if (is_a<add>(buffer)) {
1672 for (int i=0; i<buffer.nops(); i++) {
1673 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1676 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1683 map_trafo_H_1mxt1px recursion;
1684 map_trafo_H_mult unify;
1685 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1687 while (parameter.op(firstzero) == 1) {
1690 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1694 newparameter.append(parameter[j+1]);
1696 newparameter.append(1);
1697 for (; j<parameter.nops()-1; j++) {
1698 newparameter.append(parameter[j+1]);
1700 res -= H(newparameter, arg).hold();
1702 res = recursion(res).expand() / firstzero;
1714 // do the actual summation.
1715 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1717 const int j = m.size();
1719 std::vector<cln::cl_N> t(j);
1721 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1722 cln::cl_N factor = cln::expt(x, j) * one;
1728 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1729 for (int k=j-2; k>=1; k--) {
1730 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1732 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1733 factor = factor * x;
1734 } while (t[0] != t0buf);
1740 } // end of anonymous namespace
1743 //////////////////////////////////////////////////////////////////////
1745 // Harmonic polylogarithm H(m,x)
1749 //////////////////////////////////////////////////////////////////////
1752 static ex H_evalf(const ex& x1, const ex& x2)
1754 if (is_a<lst>(x1) && is_a<numeric>(x2)) {
1755 for (int i=0; i<x1.nops(); i++) {
1756 if (!x1.op(i).info(info_flags::integer)) {
1757 return H(x1,x2).hold();
1760 if (x1.nops() < 1) {
1761 return H(x1,x2).hold();
1764 cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
1766 const lst& morg = ex_to<lst>(x1);
1767 // remove trailing zeros ...
1768 if (*(--morg.end()) == 0) {
1769 symbol xtemp("xtemp");
1770 map_trafo_H_reduce_trailing_zeros filter;
1771 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
1773 // ... and expand parameter notation
1775 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
1777 for (ex count=*it-1; count > 0; count--) {
1781 } else if (*it < -1) {
1782 for (ex count=*it+1; count < 0; count++) {
1791 // since the transformations produce a lot of terms, they are only efficient for
1792 // argument near one.
1793 // no transformation needed -> do summation
1794 if (cln::abs(x) < 0.95) {
1798 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
1799 // negative parameters -> s_lst is filled
1800 std::vector<int> m_int;
1801 std::vector<cln::cl_N> x_cln;
1802 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
1803 it_int != m_lst.end(); it_int++, it_cln++) {
1804 m_int.push_back(ex_to<numeric>(*it_int).to_int());
1805 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
1807 x_cln.front() = x_cln.front() * x;
1808 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
1810 // only positive parameters
1812 if (m_lst.nops() == 1) {
1813 return Li(m_lst.op(0), x2).evalf();
1815 std::vector<int> m_int;
1816 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
1817 m_int.push_back(ex_to<numeric>(*it).to_int());
1819 return numeric(H_do_sum(m_int, x));
1825 // ensure that the realpart of the argument is positive
1826 if (cln::realpart(x) < 0) {
1828 for (int i=0; i<m.nops(); i++) {
1830 m.let_op(i) = -m.op(i);
1836 // choose transformations
1837 symbol xtemp("xtemp");
1838 if (cln::abs(x-1) < 1.4142) {
1840 map_trafo_H_1mxt1px trafo;
1841 res *= trafo(H(m, xtemp));
1844 map_trafo_H_1overx trafo;
1845 res *= trafo(H(m, xtemp));
1850 // map_trafo_H_convert converter;
1851 // res = converter(res).expand();
1853 // res.find(H(wild(1),wild(2)), ll);
1854 // res.find(zeta(wild(1)), ll);
1855 // res.find(zeta(wild(1),wild(2)), ll);
1856 // res = res.collect(ll);
1858 return res.subs(xtemp == numeric(x)).evalf();
1861 return H(x1,x2).hold();
1865 static ex H_eval(const ex& x1, const ex& x2)
1874 // if (x1.nops() == 1) {
1875 // return Li(x1.op(0), x2);
1877 if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) {
1878 return H(x1,x2).evalf();
1880 return H(x1,x2).hold();
1884 static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
1887 seq.push_back(expair(H(x1,x2), 0));
1888 return pseries(rel,seq);
1892 static ex H_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
1894 GINAC_ASSERT(deriv_param < 2);
1895 if (deriv_param == 0) {
1898 if (is_a<lst>(x1)) {
1899 lst newparameter = ex_to<lst>(x1);
1900 if (x1.op(0) == 1) {
1901 newparameter.remove_first();
1902 return 1/(1-x2) * H(newparameter, x2);
1905 return H(newparameter, x2).hold() / x2;
1911 return H(x1-1, x2).hold() / x2;
1917 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
1920 if (is_a<lst>(m_)) {
1925 c.s << "\\mbox{H}_{";
1926 lst::const_iterator itm = m.begin();
1929 for (; itm != m.end(); itm++) {
1939 REGISTER_FUNCTION(H,
1940 evalf_func(H_evalf).
1942 series_func(H_series).
1943 derivative_func(H_deriv).
1944 print_func<print_latex>(H_print_latex).
1945 do_not_evalf_params());
1948 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
1949 ex convert_H_to_Li(const ex& parameterlst, const ex& arg)
1951 map_trafo_H_reduce_trailing_zeros filter;
1952 map_trafo_H_convert_to_Li filter2;
1953 if (is_a<lst>(parameterlst)) {
1954 return filter2(filter(H(parameterlst, arg).hold())).eval();
1956 return filter2(filter(H(lst(parameterlst), arg).hold())).eval();
1961 //////////////////////////////////////////////////////////////////////
1963 // Multiple zeta values zeta(x) and zeta(x,s)
1967 //////////////////////////////////////////////////////////////////////
1970 // anonymous namespace for helper functions
1974 // parameters and data for [Cra] algorithm
1975 const cln::cl_N lambda = cln::cl_N("319/320");
1978 std::vector<std::vector<cln::cl_N> > f_kj;
1979 std::vector<cln::cl_N> crB;
1980 std::vector<std::vector<cln::cl_N> > crG;
1981 std::vector<cln::cl_N> crX;
1984 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
1986 const int size = a.size();
1987 for (int n=0; n<size; n++) {
1989 for (int m=0; m<=n; m++) {
1990 c[n] = c[n] + a[m]*b[n-m];
1997 void initcX(const std::vector<int>& s)
1999 const int k = s.size();
2005 for (int i=0; i<=L2; i++) {
2006 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
2011 for (int m=0; m<k-1; m++) {
2012 std::vector<cln::cl_N> crGbuf;
2015 for (int i=0; i<=L2; i++) {
2016 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
2018 crG.push_back(crGbuf);
2023 for (int m=0; m<k-1; m++) {
2024 std::vector<cln::cl_N> Xbuf;
2025 for (int i=0; i<=L2; i++) {
2026 Xbuf.push_back(crX[i] * crG[m][i]);
2028 halfcyclic_convolute(Xbuf, crB, crX);
2034 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
2036 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2037 cln::cl_N factor = cln::expt(lambda, Sqk);
2038 cln::cl_N res = factor / Sqk * crX[0] * one;
2043 factor = factor * lambda;
2045 res = res + crX[N] * factor / (N+Sqk);
2046 } while ((res != resbuf) || cln::zerop(crX[N]));
2052 void calc_f(int maxr)
2057 cln::cl_N t0, t1, t2, t3, t4;
2059 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
2060 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2062 t0 = cln::exp(-lambda);
2064 for (k=1; k<=L1; k++) {
2067 for (j=1; j<=maxr; j++) {
2070 for (i=2; i<=j; i++) {
2074 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
2082 cln::cl_N crandall_Z(const std::vector<int>& s)
2084 const int j = s.size();
2093 t0 = t0 + f_kj[q+j-2][s[0]-1];
2094 } while (t0 != t0buf);
2096 return t0 / cln::factorial(s[0]-1);
2099 std::vector<cln::cl_N> t(j);
2106 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
2107 for (int k=j-2; k>=1; k--) {
2108 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
2110 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
2111 } while (t[0] != t0buf);
2113 return t[0] / cln::factorial(s[0]-1);
2118 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
2120 std::vector<int> r = s;
2121 const int j = r.size();
2123 // decide on maximal size of f_kj for crandall_Z
2127 L1 = Digits * 3 + j*2;
2130 // decide on maximal size of crX for crandall_Y
2133 } else if (Digits < 86) {
2135 } else if (Digits < 192) {
2137 } else if (Digits < 394) {
2139 } else if (Digits < 808) {
2149 for (int i=0; i<j; i++) {
2158 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
2160 std::vector<int> rz;
2163 for (int k=r.size()-1; k>0; k--) {
2165 rz.insert(rz.begin(), r.back());
2166 skp1buf = rz.front();
2172 for (int q=0; q<skp1buf; q++) {
2174 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2175 cln::cl_N pp2 = crandall_Z(rz);
2180 res = res - pp1 * pp2 / cln::factorial(q);
2182 res = res + pp1 * pp2 / cln::factorial(q);
2185 rz.front() = skp1buf;
2187 rz.insert(rz.begin(), r.back());
2191 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2197 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2199 const int j = r.size();
2201 // buffer for subsums
2202 std::vector<cln::cl_N> t(j);
2203 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2210 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2211 for (int k=j-2; k>=0; k--) {
2212 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2214 } while (t[0] != t0buf);
2220 // does Hoelder convolution. see [BBB] (7.0)
2221 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2223 // prepare parameters
2224 // holds Li arguments in [BBB] notation
2225 std::vector<int> s = s_;
2226 std::vector<int> m_p = m_;
2227 std::vector<int> m_q;
2228 // holds Li arguments in nested sums notation
2229 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2230 s_p[0] = s_p[0] * cln::cl_N("1/2");
2231 // convert notations
2233 for (int i=0; i<s_.size(); i++) {
2238 s[i] = sig * std::abs(s[i]);
2240 std::vector<cln::cl_N> s_q;
2241 cln::cl_N signum = 1;
2244 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2249 // change parameters
2250 if (s.front() > 0) {
2251 if (m_p.front() == 1) {
2252 m_p.erase(m_p.begin());
2253 s_p.erase(s_p.begin());
2254 if (s_p.size() > 0) {
2255 s_p.front() = s_p.front() * cln::cl_N("1/2");
2261 m_q.insert(m_q.begin(), 1);
2262 if (s_q.size() > 0) {
2263 s_q.front() = s_q.front() * 2;
2265 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2268 if (m_p.front() == 1) {
2269 m_p.erase(m_p.begin());
2270 cln::cl_N spbuf = s_p.front();
2271 s_p.erase(s_p.begin());
2272 if (s_p.size() > 0) {
2273 s_p.front() = s_p.front() * spbuf;
2276 m_q.insert(m_q.begin(), 1);
2277 if (s_q.size() > 0) {
2278 s_q.front() = s_q.front() * 4;
2280 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2284 m_q.insert(m_q.begin(), 1);
2285 if (s_q.size() > 0) {
2286 s_q.front() = s_q.front() * 2;
2288 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2293 if (m_p.size() == 0) break;
2295 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2300 res = res + signum * multipleLi_do_sum(m_q, s_q);
2306 } // end of anonymous namespace
2309 //////////////////////////////////////////////////////////////////////
2311 // Multiple zeta values zeta(x)
2315 //////////////////////////////////////////////////////////////////////
2318 static ex zeta1_evalf(const ex& x)
2320 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2322 // multiple zeta value
2323 const int count = x.nops();
2324 const lst& xlst = ex_to<lst>(x);
2325 std::vector<int> r(count);
2327 // check parameters and convert them
2328 lst::const_iterator it1 = xlst.begin();
2329 std::vector<int>::iterator it2 = r.begin();
2331 if (!(*it1).info(info_flags::posint)) {
2332 return zeta(x).hold();
2334 *it2 = ex_to<numeric>(*it1).to_int();
2337 } while (it2 != r.end());
2339 // check for divergence
2341 return zeta(x).hold();
2344 // decide on summation algorithm
2345 // this is still a bit clumsy
2346 int limit = (Digits>17) ? 10 : 6;
2347 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2348 return numeric(zeta_do_sum_Crandall(r));
2350 return numeric(zeta_do_sum_simple(r));
2354 // single zeta value
2355 if (is_exactly_a<numeric>(x) && (x != 1)) {
2357 return zeta(ex_to<numeric>(x));
2358 } catch (const dunno &e) { }
2361 return zeta(x).hold();
2365 static ex zeta1_eval(const ex& x)
2367 if (is_exactly_a<lst>(x)) {
2368 if (x.nops() == 1) {
2369 return zeta(x.op(0));
2371 return zeta(x).hold();
2374 if (x.info(info_flags::numeric)) {
2375 const numeric& y = ex_to<numeric>(x);
2376 // trap integer arguments:
2377 if (y.is_integer()) {
2381 if (y.is_equal(_num1)) {
2382 return zeta(x).hold();
2384 if (y.info(info_flags::posint)) {
2385 if (y.info(info_flags::odd)) {
2386 return zeta(x).hold();
2388 return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2391 if (y.info(info_flags::odd)) {
2392 return -bernoulli(_num1-y) / (_num1-y);
2399 if (y.info(info_flags::numeric) && !y.info(info_flags::crational))
2400 return zeta1_evalf(x);
2402 return zeta(x).hold();
2406 static ex zeta1_deriv(const ex& x, unsigned deriv_param)
2408 GINAC_ASSERT(deriv_param==0);
2410 if (is_exactly_a<lst>(x)) {
2413 return zeta(_ex1, x);
2418 static void zeta1_print_latex(const ex& x, const print_context& c)
2423 arg = ex_to<lst>(x);
2424 lst::const_iterator it = arg.begin();
2427 for (; it != arg.end(); it++) {
2438 unsigned zeta1_SERIAL::serial =
2439 function::register_new(function_options("zeta").
2440 evalf_func(zeta1_evalf).
2441 eval_func(zeta1_eval).
2442 derivative_func(zeta1_deriv).
2443 print_func<print_latex>(zeta1_print_latex).
2444 do_not_evalf_params().
2448 //////////////////////////////////////////////////////////////////////
2450 // Alternating Euler sum zeta(x,s)
2454 //////////////////////////////////////////////////////////////////////
2457 static ex zeta2_evalf(const ex& x, const ex& s)
2459 if (is_exactly_a<lst>(x)) {
2461 // alternating Euler sum
2462 const int count = x.nops();
2463 const lst& xlst = ex_to<lst>(x);
2464 const lst& slst = ex_to<lst>(s);
2465 std::vector<int> xi(count);
2466 std::vector<int> si(count);
2468 // check parameters and convert them
2469 lst::const_iterator it_xread = xlst.begin();
2470 lst::const_iterator it_sread = slst.begin();
2471 std::vector<int>::iterator it_xwrite = xi.begin();
2472 std::vector<int>::iterator it_swrite = si.begin();
2474 if (!(*it_xread).info(info_flags::posint)) {
2475 return zeta(x, s).hold();
2477 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2478 if (*it_sread > 0) {
2487 } while (it_xwrite != xi.end());
2489 // check for divergence
2490 if ((xi[0] == 1) && (si[0] == 1)) {
2491 return zeta(x, s).hold();
2494 // use Hoelder convolution
2495 return numeric(zeta_do_Hoelder_convolution(xi, si));
2498 return zeta(x, s).hold();
2502 static ex zeta2_eval(const ex& x, const ex& s)
2504 if (is_exactly_a<lst>(s)) {
2505 const lst& l = ex_to<lst>(s);
2506 lst::const_iterator it = l.begin();
2507 while (it != l.end()) {
2508 if ((*it).info(info_flags::negative)) {
2509 return zeta(x, s).hold();
2515 if (s.info(info_flags::positive)) {
2520 return zeta(x, s).hold();
2524 static ex zeta2_deriv(const ex& x, const ex& s, unsigned deriv_param)
2526 GINAC_ASSERT(deriv_param==0);
2528 if (is_exactly_a<lst>(x)) {
2531 if ((is_exactly_a<lst>(s) && (s.op(0) > 0)) || (s > 0)) {
2532 return zeta(_ex1, x);
2539 static void zeta2_print_latex(const ex& x, const ex& s, const print_context& c)
2543 arg = ex_to<lst>(x);
2549 sig = ex_to<lst>(s);
2554 lst::const_iterator itarg = arg.begin();
2555 lst::const_iterator itsig = sig.begin();
2557 c.s << "\\overline{";
2565 for (; itarg != arg.end(); itarg++, itsig++) {
2568 c.s << "\\overline{";
2579 unsigned zeta2_SERIAL::serial =
2580 function::register_new(function_options("zeta").
2581 evalf_func(zeta2_evalf).
2582 eval_func(zeta2_eval).
2583 derivative_func(zeta2_deriv).
2584 print_func<print_latex>(zeta2_print_latex).
2585 do_not_evalf_params().
2589 } // namespace GiNaC