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 H, Li and zeta is defined according to their order in the
22 * nested sums representation.
24 * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
25 * the whole complex plane. Multiple polylogarithms evaluate only if each argument x_i is smaller than
26 * one. The parameters for every function (n, p or n_i) must be positive integers.
28 * - The calculation of classical polylogarithms is speed up by using Bernoulli numbers and
29 * look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
30 * [Cra] and [BBB] for speed up.
32 * - The functions have no series expansion as nested sums. To do it, you have to convert these functions
33 * into the appropriate objects from the nestedsums library, do the expansion and convert the
36 * - Numerical testing of this implementation has been performed by doing a comparison of results
37 * between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
38 * by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
39 * comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
40 * around |x|=1 along with comparisons to corresponding zeta functions.
45 * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
47 * This program is free software; you can redistribute it and/or modify
48 * it under the terms of the GNU General Public License as published by
49 * the Free Software Foundation; either version 2 of the License, or
50 * (at your option) any later version.
52 * This program is distributed in the hope that it will be useful,
53 * but WITHOUT ANY WARRANTY; without even the implied warranty of
54 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
55 * GNU General Public License for more details.
57 * You should have received a copy of the GNU General Public License
58 * along with this program; if not, write to the Free Software
59 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
73 #include "operators.h"
76 #include "relational.h"
90 //////////////////////////////////////////////////////////////////////
92 // Classical polylogarithm Li(n,x)
96 //////////////////////////////////////////////////////////////////////
99 // anonymous namespace for helper functions
103 // lookup table for factors built from Bernoulli numbers
105 std::vector<std::vector<cln::cl_N> > Xn;
109 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
110 // With these numbers the polylogs can be calculated as follows:
111 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
112 // X_0(n) = B_n (Bernoulli numbers)
113 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
114 // The calculation of Xn depends on X0 and X{n-1}.
115 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
116 // This results in a slightly more complicated algorithm for the X_n.
117 // The first index in Xn corresponds to the index of the polylog minus 2.
118 // The second index in Xn corresponds to the index from the actual sum.
121 // rule of thumb. needs to be improved. TODO
122 const int initsize = Digits * 3 / 2;
125 // calculate X_2 and higher (corresponding to Li_4 and higher)
126 std::vector<cln::cl_N> buf(initsize);
127 std::vector<cln::cl_N>::iterator it = buf.begin();
129 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
131 for (int i=2; i<=initsize; i++) {
133 result = 0; // k == 0
135 result = Xn[0][i/2-1]; // k == 0
137 for (int k=1; k<i-1; k++) {
138 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
139 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
142 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
143 result = result + Xn[n-1][i-1] / (i+1); // k == i
150 // special case to handle the X_0 correct
151 std::vector<cln::cl_N> buf(initsize);
152 std::vector<cln::cl_N>::iterator it = buf.begin();
154 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
156 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
158 for (int i=3; i<=initsize; i++) {
160 result = -Xn[0][(i-3)/2]/2;
161 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
164 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
165 for (int k=1; k<i/2; k++) {
166 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
175 std::vector<cln::cl_N> buf(initsize/2);
176 std::vector<cln::cl_N>::iterator it = buf.begin();
177 for (int i=1; i<=initsize/2; i++) {
178 *it = bernoulli(i*2).to_cl_N();
188 // calculates Li(2,x) without Xn
189 cln::cl_N Li2_do_sum(const cln::cl_N& x)
194 cln::cl_I den = 1; // n^2 = 1
199 den = den + i; // n^2 = 4, 9, 16, ...
201 res = res + num / den;
202 } while (res != resbuf);
207 // calculates Li(2,x) with Xn
208 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
210 std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
211 cln::cl_N u = -cln::log(1-x);
212 cln::cl_N factor = u;
213 cln::cl_N res = u - u*u/4;
218 factor = factor * u*u / (2*i * (2*i+1));
219 res = res + (*it) * factor;
220 it++; // should we check it? or rely on initsize? ...
222 } while (res != resbuf);
227 // calculates Li(n,x), n>2 without Xn
228 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
230 cln::cl_N factor = x;
237 res = res + factor / cln::expt(cln::cl_I(i),n);
239 } while (res != resbuf);
244 // calculates Li(n,x), n>2 with Xn
245 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
247 std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
248 cln::cl_N u = -cln::log(1-x);
249 cln::cl_N factor = u;
255 factor = factor * u / i;
256 res = res + (*it) * factor;
257 it++; // should we check it? or rely on initsize? ...
259 } while (res != resbuf);
264 // forward declaration needed by function Li_projection and C below
265 numeric S_num(int n, int p, const numeric& x);
268 // helper function for classical polylog Li
269 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
271 // treat n=2 as special case
273 // check if precalculated X0 exists
278 if (cln::realpart(x) < 0.5) {
279 // choose the faster algorithm
280 // the switching point was empirically determined. the optimal point
281 // depends on hardware, Digits, ... so an approx value is okay.
282 // it solves also the problem with precision due to the u=-log(1-x) transformation
283 if (cln::abs(cln::realpart(x)) < 0.25) {
285 return Li2_do_sum(x);
287 return Li2_do_sum_Xn(x);
290 // choose the faster algorithm
291 if (cln::abs(cln::realpart(x)) > 0.75) {
292 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
294 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
298 // check if precalculated Xn exist
300 for (int i=xnsize; i<n-1; i++) {
305 if (cln::realpart(x) < 0.5) {
306 // choose the faster algorithm
307 // with n>=12 the "normal" summation always wins against the method with Xn
308 if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
309 return Lin_do_sum(n, x);
311 return Lin_do_sum_Xn(n, x);
314 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
315 for (int j=0; j<n-1; j++) {
316 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
317 * cln::expt(cln::log(x), j) / cln::factorial(j);
325 // helper function for classical polylog Li
326 numeric Li_num(int n, const numeric& x)
330 return -cln::log(1-x.to_cl_N());
341 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
344 // what is the desired float format?
345 // first guess: default format
346 cln::float_format_t prec = cln::default_float_format;
347 const cln::cl_N value = x.to_cl_N();
348 // second guess: the argument's format
349 if (!x.real().is_rational())
350 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
351 else if (!x.imag().is_rational())
352 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
355 if (cln::abs(value) > 1) {
356 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
357 // check if argument is complex. if it is real, the new polylog has to be conjugated.
358 if (cln::zerop(cln::imagpart(value))) {
360 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
363 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
368 result = result + Li_projection(n, cln::recip(value), prec);
371 result = result - Li_projection(n, cln::recip(value), prec);
375 for (int j=0; j<n-1; j++) {
376 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
377 * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
379 result = result - add;
383 return Li_projection(n, value, prec);
388 } // end of anonymous namespace
391 //////////////////////////////////////////////////////////////////////
393 // Multiple polylogarithm Li(n,x)
397 //////////////////////////////////////////////////////////////////////
400 // anonymous namespace for helper function
404 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
406 const int j = s.size();
410 for (int i=0; i<s.size(); i++) {
415 for (int i=0; i<x.size(); i++) {
420 std::vector<cln::cl_N> t(j);
421 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
429 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
430 for (int k=j-2; k>=0; k--) {
431 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]);
433 // ... and do it again (to avoid premature drop out due to special arguments)
435 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
436 for (int k=j-2; k>=0; k--) {
437 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]);
439 } while (t[0] != t0buf);
442 cout << "end " << q << " " << t[0] << endl;
448 } // end of anonymous namespace
451 //////////////////////////////////////////////////////////////////////
453 // Classical polylogarithm and multiple polylogarithm Li(n,x)
457 //////////////////////////////////////////////////////////////////////
460 static ex Li_eval(const ex& x1, const ex& x2)
466 if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational)))
467 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
469 for (int i=0; i<x2.nops(); i++) {
470 if (!is_a<numeric>(x2.op(i))) {
471 return Li(x1,x2).hold();
474 return Li(x1,x2).evalf();
476 return Li(x1,x2).hold();
481 static ex Li_evalf(const ex& x1, const ex& x2)
483 // classical polylogs
484 if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
485 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
488 else if (is_a<lst>(x1) && is_a<lst>(x2)) {
490 for (int i=0; i<x1.nops(); i++) {
491 if (!x1.op(i).info(info_flags::posint)) {
492 return Li(x1,x2).hold();
494 if (!is_a<numeric>(x2.op(i))) {
495 return Li(x1,x2).hold();
498 if ((conv > 1) || ((conv == 1) && (x1.op(0) == 1))) {
499 return Li(x1,x2).hold();
504 std::vector<cln::cl_N> x;
505 for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
506 m.push_back(ex_to<numeric>(x1.op(i)).to_int());
507 x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
510 return numeric(multipleLi_do_sum(m, x));
513 return Li(x1,x2).hold();
517 static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
520 seq.push_back(expair(Li(x1,x2), 0));
521 return pseries(rel,seq);
525 static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
527 GINAC_ASSERT(deriv_param < 2);
528 if (deriv_param == 0) {
532 return Li(x1-1, x2) / x2;
539 REGISTER_FUNCTION(Li,
541 evalf_func(Li_evalf).
542 do_not_evalf_params().
543 series_func(Li_series).
544 derivative_func(Li_deriv));
547 //////////////////////////////////////////////////////////////////////
549 // Nielsen's generalized polylogarithm S(n,p,x)
553 //////////////////////////////////////////////////////////////////////
556 // anonymous namespace for helper functions
560 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
562 std::vector<std::vector<cln::cl_N> > Yn;
563 int ynsize = 0; // number of Yn[]
564 int ynlength = 100; // initial length of all Yn[i]
567 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
568 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
569 // representing S_{n,p}(x).
570 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
572 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
573 // representing S_{n,p}(x).
574 // The calculation of Y_n uses the values from Y_{n-1}.
575 void fill_Yn(int n, const cln::float_format_t& prec)
577 const int initsize = ynlength;
578 //const int initsize = initsize_Yn;
579 cln::cl_N one = cln::cl_float(1, prec);
582 std::vector<cln::cl_N> buf(initsize);
583 std::vector<cln::cl_N>::iterator it = buf.begin();
584 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
585 *it = (*itprev) / cln::cl_N(n+1) * one;
588 // sums with an index smaller than the depth are zero and need not to be calculated.
589 // calculation starts with depth, which is n+2)
590 for (int i=n+2; i<=initsize+n; i++) {
591 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
597 std::vector<cln::cl_N> buf(initsize);
598 std::vector<cln::cl_N>::iterator it = buf.begin();
601 for (int i=2; i<=initsize; i++) {
602 *it = *(it-1) + 1 / cln::cl_N(i) * one;
611 // make Yn longer ...
612 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
615 cln::cl_N one = cln::cl_float(1, prec);
617 Yn[0].resize(newsize);
618 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
620 for (int i=ynlength+1; i<=newsize; i++) {
621 *it = *(it-1) + 1 / cln::cl_N(i) * one;
625 for (int n=1; n<ynsize; n++) {
626 Yn[n].resize(newsize);
627 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
628 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
631 for (int i=ynlength+n+1; i<=newsize+n; i++) {
632 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
642 // helper function for S(n,p,x)
644 cln::cl_N C(int n, int p)
648 for (int k=0; k<p; k++) {
649 for (int j=0; j<=(n+k-1)/2; j++) {
653 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
656 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
663 result = result + cln::factorial(n+k-1)
664 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
665 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
668 result = result - cln::factorial(n+k-1)
669 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
670 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
675 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()
676 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
679 result = result + cln::factorial(n+k-1)
680 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
681 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
689 if (((np)/2+n) & 1) {
690 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
693 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
701 // helper function for S(n,p,x)
702 // [Kol] remark to (9.1)
712 for (int m=2; m<=k; m++) {
713 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
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) * b_k(k-m);
739 // helper function for S(n,p,x)
740 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
743 return Li_projection(n+1, x, prec);
746 // check if precalculated values are sufficient
748 for (int i=ynsize; i<p-1; i++) {
753 // should be done otherwise
754 cln::cl_N xf = x * cln::cl_float(1, prec);
758 cln::cl_N factor = cln::expt(xf, p);
762 if (i-p >= ynlength) {
764 make_Yn_longer(ynlength*2, prec);
766 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? ...
767 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
768 factor = factor * xf;
770 } while (res != resbuf);
776 // helper function for S(n,p,x)
777 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
780 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
782 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
783 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
785 for (int s=0; s<n; s++) {
787 for (int r=0; r<p; r++) {
788 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
789 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
791 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
797 return S_do_sum(n, p, x, prec);
801 // helper function for S(n,p,x)
802 numeric S_num(int n, int p, const numeric& x)
806 // [Kol] (2.22) with (2.21)
807 return cln::zeta(p+1);
812 return cln::zeta(n+1);
817 for (int nu=0; nu<n; nu++) {
818 for (int rho=0; rho<=p; rho++) {
819 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
820 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
823 result = result * cln::expt(cln::cl_I(-1),n+p-1);
830 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
832 // throw std::runtime_error("don't know how to evaluate this function!");
835 // what is the desired float format?
836 // first guess: default format
837 cln::float_format_t prec = cln::default_float_format;
838 const cln::cl_N value = x.to_cl_N();
839 // second guess: the argument's format
840 if (!x.real().is_rational())
841 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
842 else if (!x.imag().is_rational())
843 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
847 if (cln::realpart(value) < -0.5) {
849 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
850 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
852 for (int s=0; s<n; s++) {
854 for (int r=0; r<p; r++) {
855 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
856 * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
858 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
865 if (cln::abs(value) > 1) {
869 for (int s=0; s<p; s++) {
870 for (int r=0; r<=s; r++) {
871 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
872 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
873 * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
876 result = result * cln::expt(cln::cl_I(-1),n);
879 for (int r=0; r<n; r++) {
880 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
882 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
884 result = result + cln::expt(cln::cl_I(-1),p) * res2;
889 return S_projection(n, p, value, prec);
894 } // end of anonymous namespace
897 //////////////////////////////////////////////////////////////////////
899 // Nielsen's generalized polylogarithm S(n,p,x)
903 //////////////////////////////////////////////////////////////////////
906 static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
911 if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) &&
912 x1.info(info_flags::posint) && x2.info(info_flags::posint)) {
913 return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
915 return S(x1,x2,x3).hold();
919 static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
921 if (is_a<numeric>(x1) && is_a<numeric>(x2) && is_a<numeric>(x3)) {
922 return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
924 return S(x1,x2,x3).hold();
928 static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options)
931 seq.push_back(expair(S(x1,x2,x3), 0));
932 return pseries(rel,seq);
936 static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param)
938 GINAC_ASSERT(deriv_param < 3);
939 if (deriv_param < 2) {
943 return S(x1-1, x2, x3) / x3;
945 return S(x1, x2-1, x3) / (1-x3);
953 do_not_evalf_params().
954 series_func(S_series).
955 derivative_func(S_deriv));
958 //////////////////////////////////////////////////////////////////////
960 // Harmonic polylogarithm H(m,x) and H(m,s,x)
964 //////////////////////////////////////////////////////////////////////
967 // anonymous namespace for helper functions
971 // forward declaration
972 ex convert_from_RV(const lst& parameterlst, const ex& arg);
975 // multiplies an one-dimensional H with another H
977 ex trafo_H_mult(const ex& h1, const ex& h2)
982 ex h1nops = h1.op(0).nops();
983 ex h2nops = h2.op(0).nops();
985 hshort = h2.op(0).op(0);
986 hlong = ex_to<lst>(h1.op(0));
988 hshort = h1.op(0).op(0);
990 hlong = ex_to<lst>(h2.op(0));
992 hlong = h2.op(0).op(0);
995 for (int i=0; i<=hlong.nops(); i++) {
999 newparameter.append(hlong[j]);
1001 newparameter.append(hshort);
1002 for (; j<hlong.nops(); j++) {
1003 newparameter.append(hlong[j]);
1005 res += H(newparameter, h1.op(1)).hold();
1011 // applies trafo_H_mult recursively on expressions
1012 struct map_trafo_H_mult : public map_function
1014 ex operator()(const ex& e)
1017 return e.map(*this);
1025 for (int pos=0; pos<e.nops(); pos++) {
1026 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1027 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1029 for (ex i=0; i<e.op(pos).op(1); i++) {
1030 Hlst.append(e.op(pos).op(0));
1034 } else if (is_a<function>(e.op(pos))) {
1035 std::string name = ex_to<function>(e.op(pos)).get_name();
1037 if (e.op(pos).op(0).nops() > 1) {
1040 Hlst.append(e.op(pos));
1045 result *= e.op(pos);
1048 if (Hlst.nops() > 0) {
1049 firstH = Hlst[Hlst.nops()-1];
1056 if (Hlst.nops() > 0) {
1057 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1059 for (int i=1; i<Hlst.nops(); i++) {
1060 result *= Hlst.op(i);
1062 result = result.expand();
1063 map_trafo_H_mult recursion;
1064 return recursion(result);
1075 // do integration [ReV] (49)
1076 // put parameter 1 in front of existing parameters
1077 ex trafo_H_prepend_one(const ex& e, const ex& arg)
1081 if (is_a<function>(e)) {
1082 name = ex_to<function>(e).get_name();
1087 for (int i=0; i<e.nops(); i++) {
1088 if (is_a<function>(e.op(i))) {
1089 std::string name = ex_to<function>(e.op(i)).get_name();
1097 lst newparameter = ex_to<lst>(h.op(0));
1098 newparameter.prepend(1);
1099 return e.subs(h == H(newparameter, h.op(1)).hold());
1101 return e * H(lst(1),1-arg).hold();
1106 // do integration [ReV] (55)
1107 // put parameter 0 in front of existing parameters
1108 ex trafo_H_prepend_zero(const ex& e, const ex& arg)
1112 if (is_a<function>(e)) {
1113 name = ex_to<function>(e).get_name();
1118 for (int i=0; i<e.nops(); i++) {
1119 if (is_a<function>(e.op(i))) {
1120 std::string name = ex_to<function>(e.op(i)).get_name();
1128 lst newparameter = ex_to<lst>(h.op(0));
1129 newparameter.prepend(0);
1130 ex addzeta = convert_from_RV(newparameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
1131 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1133 return e * (-H(lst(0),1/arg).hold());
1138 // do x -> 1-x transformation
1139 struct map_trafo_H_1mx : public map_function
1141 ex operator()(const ex& e)
1143 if (is_a<add>(e) || is_a<mul>(e)) {
1144 return e.map(*this);
1147 if (is_a<function>(e)) {
1148 std::string name = ex_to<function>(e).get_name();
1151 lst parameter = ex_to<lst>(e.op(0));
1154 // if all parameters are either zero or one return the transformed function
1155 if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) {
1157 for (int i=parameter.nops(); i>0; i--) {
1158 newparameter.append(0);
1160 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
1161 } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) {
1163 for (int i=parameter.nops(); i>0; i--) {
1164 newparameter.append(1);
1166 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
1169 lst newparameter = parameter;
1170 newparameter.remove_first();
1172 if (parameter.op(0) == 0) {
1175 ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
1176 map_trafo_H_1mx recursion;
1177 ex buffer = recursion(H(newparameter, arg).hold());
1178 if (is_a<add>(buffer)) {
1179 for (int i=0; i<buffer.nops(); i++) {
1180 res -= trafo_H_prepend_one(buffer.op(i), arg);
1183 res -= trafo_H_prepend_one(buffer, arg);
1190 map_trafo_H_1mx recursion;
1191 map_trafo_H_mult unify;
1194 while (parameter.op(firstzero) == 1) {
1197 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1201 newparameter.append(parameter[j+1]);
1203 newparameter.append(1);
1204 for (; j<parameter.nops()-1; j++) {
1205 newparameter.append(parameter[j+1]);
1207 res -= H(newparameter, arg).hold();
1209 return (unify((-H(lst(0), 1-arg).hold() * recursion(H(newparameter, arg).hold())).expand()) +
1210 recursion(res)) / firstzero;
1221 // do x -> 1/x transformation
1222 struct map_trafo_H_1overx : public map_function
1224 ex operator()(const ex& e)
1226 if (is_a<add>(e) || is_a<mul>(e)) {
1227 return e.map(*this);
1230 if (is_a<function>(e)) {
1231 std::string name = ex_to<function>(e).get_name();
1234 lst parameter = ex_to<lst>(e.op(0));
1237 // if all parameters are either zero or one return the transformed function
1238 if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) {
1239 map_trafo_H_mult unify;
1240 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, parameter.nops()) /
1241 factorial(parameter.nops())).expand());
1242 } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) {
1243 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1246 lst newparameter = parameter;
1247 newparameter.remove_first();
1249 if (parameter.op(0) == 0) {
1252 ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
1253 map_trafo_H_1overx recursion;
1254 ex buffer = recursion(H(newparameter, arg).hold());
1255 if (is_a<add>(buffer)) {
1256 for (int i=0; i<buffer.nops(); i++) {
1257 res += trafo_H_prepend_zero(buffer.op(i), arg);
1260 res += trafo_H_prepend_zero(buffer, arg);
1267 map_trafo_H_1overx recursion;
1268 map_trafo_H_mult unify;
1269 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1271 while (parameter.op(firstzero) == 1) {
1274 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1278 newparameter.append(parameter[j+1]);
1280 newparameter.append(1);
1281 for (; j<parameter.nops()-1; j++) {
1282 newparameter.append(parameter[j+1]);
1284 res -= H(newparameter, arg).hold();
1286 res = recursion(res).expand() / firstzero;
1298 // remove trailing zeros from H-parameters
1299 struct map_trafo_H_reduce_trailing_zeros : public map_function
1301 ex operator()(const ex& e)
1303 if (is_a<add>(e) || is_a<mul>(e)) {
1304 return e.map(*this);
1306 if (is_a<function>(e)) {
1307 std::string name = ex_to<function>(e).get_name();
1310 if (is_a<lst>(e.op(0))) {
1311 parameter = ex_to<lst>(e.op(0));
1313 parameter = lst(e.op(0));
1316 if (parameter.op(parameter.nops()-1) == 0) {
1319 if (parameter.nops() == 1) {
1324 lst::const_iterator it = parameter.begin();
1325 while ((it != parameter.end()) && (*it == 0)) {
1328 if (it == parameter.end()) {
1329 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1333 parameter.remove_last();
1334 int lastentry = parameter.nops();
1335 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1340 ex result = log(arg) * H(parameter,arg).hold();
1341 for (ex i=0; i<lastentry; i++) {
1342 if (parameter[i] > 0) {
1344 result -= (parameter[i]-1) * H(parameter, arg).hold();
1348 result -= abs(parameter[i]+1) * H(parameter, arg).hold();
1353 if (lastentry < parameter.nops()) {
1354 result = result / (parameter.nops()-lastentry+1);
1355 return result.map(*this);
1367 // transform H(m,x) with signed m to H(m,s,x)
1368 struct map_trafo_H_convert_signed_m : public map_function
1370 ex operator()(const ex& e)
1372 if (is_a<add>(e) || is_a<mul>(e)) {
1373 return e.map(*this);
1375 if (is_a<function>(e)) {
1376 std::string name = ex_to<function>(e).get_name();
1379 if (is_a<lst>(e.op(0))) {
1380 parameter = ex_to<lst>(e.op(0));
1382 parameter = lst(e.op(0));
1385 bool signedflag = false;
1386 for (int i=0; i<parameter.nops(); i++) {
1387 if (parameter.op(i) < 0) {
1394 for (int i=0; i<parameter.nops(); i++) {
1395 if (parameter.op(i) > 0) {
1399 parameter.let_op(i) = -parameter.op(i);
1402 return H(parameter, signs, arg).hold();
1411 // recursively call convert_from_RV on expression
1412 struct map_trafo_H_convert : public map_function
1414 ex operator()(const ex& e)
1416 if (is_a<add>(e) || is_a<mul>(e) || is_a<power>(e)) {
1417 return e.map(*this);
1419 if (is_a<function>(e)) {
1420 std::string name = ex_to<function>(e).get_name();
1422 lst parameter = ex_to<lst>(e.op(0));
1424 return convert_from_RV(parameter, arg);
1432 // translate notation from nested sums to Remiddi/Vermaseren
1433 lst convert_to_RV(const lst& o)
1436 for (lst::const_iterator it = o.begin(); it != o.end(); it++) {
1438 for (ex i=0; i<(*it)-1; i++) {
1443 for (ex i=0; i<(-*it)-1; i++) {
1453 // translate notation from Remiddi/Vermaseren to nested sums
1454 ex convert_from_RV(const lst& parameterlst, const ex& arg)
1456 lst newparameterlst;
1458 lst::const_iterator it = parameterlst.begin();
1460 while (it != parameterlst.end()) {
1464 newparameterlst.append((*it>0) ? count : -count);
1469 for (int i=1; i<count; i++) {
1470 newparameterlst.append(0);
1473 map_trafo_H_reduce_trailing_zeros filter1;
1474 map_trafo_H_convert_signed_m filter2;
1475 return filter2(filter1(H(newparameterlst, arg).hold()));
1479 // do the actual summation.
1480 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1482 const int j = m.size();
1484 std::vector<cln::cl_N> t(j);
1486 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1487 cln::cl_N factor = cln::expt(x, j) * one;
1493 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1494 for (int k=j-2; k>=1; k--) {
1495 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1497 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1498 factor = factor * x;
1499 } while (t[0] != t0buf);
1505 } // end of anonymous namespace
1508 //////////////////////////////////////////////////////////////////////
1510 // Harmonic polylogarithm H(m,x)
1514 //////////////////////////////////////////////////////////////////////
1517 static ex H2_eval(const ex& x1, const ex& x2)
1525 if (x1.nops() == 1) {
1526 return Li(x1.op(0), x2);
1528 if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) {
1529 return H(x1,x2).evalf();
1531 return H(x1,x2).hold();
1535 static ex H2_evalf(const ex& x1, const ex& x2)
1537 if (is_a<lst>(x1) && is_a<numeric>(x2)) {
1538 for (int i=0; i<x1.nops(); i++) {
1539 if (!x1.op(i).info(info_flags::posint)) {
1540 return H(x1,x2).hold();
1543 if (x1.nops() < 1) {
1546 if (x1.nops() == 1) {
1547 return Li(x1.op(0), x2).evalf();
1549 cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
1551 return zeta(x1).evalf();
1555 if (cln::abs(x) > 1) {
1556 symbol xtemp("xtemp");
1557 map_trafo_H_1overx trafo;
1558 ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
1559 map_trafo_H_convert converter;
1560 res = converter(res);
1561 return res.subs(xtemp==x2).evalf();
1564 // since the x->1-x transformation produces a lot of terms, it is only
1565 // efficient for argument near one.
1566 if (cln::realpart(x) > 0.95) {
1567 symbol xtemp("xtemp");
1568 map_trafo_H_1mx trafo;
1569 ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
1570 map_trafo_H_convert converter;
1571 res = converter(res);
1572 return res.subs(xtemp==x2).evalf();
1575 // no trafo -> do summation
1576 int count = x1.nops();
1577 std::vector<int> r(count);
1578 for (int i=0; i<count; i++) {
1579 r[i] = ex_to<numeric>(x1.op(i)).to_int();
1582 return numeric(H_do_sum(r,x));
1585 return H(x1,x2).hold();
1589 static ex H2_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
1592 seq.push_back(expair(H(x1,x2), 0));
1593 return pseries(rel,seq);
1597 static ex H2_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
1599 GINAC_ASSERT(deriv_param < 2);
1600 if (deriv_param == 0) {
1603 if (is_a<lst>(x1)) {
1604 lst newparameter = ex_to<lst>(x1);
1605 if (x1.op(0) == 1) {
1606 newparameter.remove_first();
1607 return 1/(1-x2) * H(newparameter, x2);
1610 return H(newparameter, x2).hold() / x2;
1616 return H(x1-1, x2).hold() / x2;
1622 unsigned H2_SERIAL::serial =
1623 function::register_new(function_options("H").
1625 evalf_func(H2_evalf).
1626 do_not_evalf_params().
1627 derivative_func(H2_deriv).
1628 latex_name("\\mbox{H}").
1632 //////////////////////////////////////////////////////////////////////
1634 // Harmonic polylogarithm H(m,s,x)
1638 //////////////////////////////////////////////////////////////////////
1641 static ex H3_eval(const ex& x1, const ex& x2, const ex& x3)
1647 return zeta(x1, x2);
1649 if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational))) {
1650 return H(x1, x2, x3).evalf();
1652 return H(x1, x2, x3).hold();
1656 static ex H3_evalf(const ex& x1, const ex& x2, const ex& x3)
1658 if (is_a<lst>(x1) && is_a<numeric>(x3)) {
1659 for (int i=0; i<x1.nops(); i++) {
1660 if (!x1.op(i).info(info_flags::posint)) {
1661 return H(x1, x2, x3).hold();
1664 if (x1.nops() < 1) {
1667 if (x1.nops() == 1) {
1668 return x2.op(0) * Li(x1.op(0), x2.op(0)*x3).evalf();
1670 cln::cl_N x = ex_to<numeric>(x3).to_cl_N();
1672 return zeta(x1, x2).evalf();
1676 if (cln::abs(x) > 1) {
1678 return H(x1, x2, x3).hold();
1679 // symbol xtemp("xtemp");
1680 // lst para = ex_to<lst>(x1);
1681 // for (int i=0; i<para.nops(); i++) {
1682 // para.let_op(i) = para.op(i) * x2.op(i);
1684 // map_trafo_H_1overx trafo;
1685 // ex res = trafo(H(convert_to_RV(para), xtemp));
1686 // map_trafo_H_convert converter;
1687 // res = converter(res);
1688 // return res.subs(xtemp==x3).evalf();
1691 // // since the x->1-x transformation produces a lot of terms, it is only
1692 // // efficient for argument near one.
1693 // if (cln::realpart(x) > 0.95) {
1694 // symbol xtemp("xtemp");
1695 // map_trafo_H_1mx trafo;
1696 // ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
1697 // map_trafo_H_convert converter;
1698 // res = converter(res);
1699 // return res.subs(xtemp==x2).evalf();
1702 // no trafo -> do summation
1703 int count = x1.nops();
1704 std::vector<int> m(count);
1705 std::vector<cln::cl_N> s(count);
1706 cln::cl_N signbuf = 1;
1707 for (int i=0; i<count; i++) {
1708 m[i] = ex_to<numeric>(x1.op(i)).to_int();
1709 signbuf = signbuf * ex_to<numeric>(x2.op(i)).to_cl_N();
1712 s[0] = s[0] * ex_to<numeric>(x3).to_cl_N();
1714 return numeric(signbuf * multipleLi_do_sum(m, s));
1717 return H(x1, x2, x3).hold();
1721 static ex H3_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options)
1724 seq.push_back(expair(H(x1, x2, x3), 0));
1725 return pseries(rel, seq);
1729 static ex H3_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param)
1733 GINAC_ASSERT(deriv_param < 2);
1734 if (deriv_param == 0) {
1737 if (is_a<lst>(x1)) {
1738 lst newparameter = ex_to<lst>(x1);
1739 if (x1.op(0) == 1) {
1740 newparameter.remove_first();
1741 return 1/(1-x2) * H(newparameter, x2);
1744 return H(newparameter, x2).hold() / x2;
1750 return H(x1-1, x2).hold() / x2;
1756 unsigned H3_SERIAL::serial =
1757 function::register_new(function_options("H").
1759 evalf_func(H3_evalf).
1760 do_not_evalf_params().
1761 derivative_func(H3_deriv).
1762 latex_name("\\mbox{H}").
1766 ex convert_H_notation(const ex& parameterlst, const ex& arg)
1768 if (is_a<lst>(parameterlst)) {
1769 for (int i=0; i<parameterlst.nops(); i++) {
1770 if (parameterlst.op(i) == 1) continue;
1771 if (parameterlst.op(i) == 0) continue;
1772 if (parameterlst.op(i) == -1) continue;
1773 throw std::runtime_error("first parameter has to be a list containing only 0, 1 or -1!");
1775 return convert_from_RV(ex_to<lst>(parameterlst), arg).eval();
1777 if (parameterlst == 1) {
1780 if (parameterlst == 0) {
1783 if (parameterlst == -1) {
1786 throw std::runtime_error("first parameter has to be a list containing only 0, 1 or -1!");
1790 //////////////////////////////////////////////////////////////////////
1792 // Multiple zeta values zeta(x) and zeta(x,s)
1796 //////////////////////////////////////////////////////////////////////
1799 // anonymous namespace for helper functions
1803 // parameters and data for [Cra] algorithm
1804 const cln::cl_N lambda = cln::cl_N("319/320");
1807 std::vector<std::vector<cln::cl_N> > f_kj;
1808 std::vector<cln::cl_N> crB;
1809 std::vector<std::vector<cln::cl_N> > crG;
1810 std::vector<cln::cl_N> crX;
1813 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
1815 const int size = a.size();
1816 for (int n=0; n<size; n++) {
1818 for (int m=0; m<=n; m++) {
1819 c[n] = c[n] + a[m]*b[n-m];
1826 void initcX(const std::vector<int>& s)
1828 const int k = s.size();
1834 for (int i=0; i<=L2; i++) {
1835 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
1840 for (int m=0; m<k-1; m++) {
1841 std::vector<cln::cl_N> crGbuf;
1844 for (int i=0; i<=L2; i++) {
1845 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
1847 crG.push_back(crGbuf);
1852 for (int m=0; m<k-1; m++) {
1853 std::vector<cln::cl_N> Xbuf;
1854 for (int i=0; i<=L2; i++) {
1855 Xbuf.push_back(crX[i] * crG[m][i]);
1857 halfcyclic_convolute(Xbuf, crB, crX);
1863 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
1865 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1866 cln::cl_N factor = cln::expt(lambda, Sqk);
1867 cln::cl_N res = factor / Sqk * crX[0] * one;
1872 factor = factor * lambda;
1874 res = res + crX[N] * factor / (N+Sqk);
1875 } while ((res != resbuf) || cln::zerop(crX[N]));
1881 void calc_f(int maxr)
1886 cln::cl_N t0, t1, t2, t3, t4;
1888 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
1889 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1891 t0 = cln::exp(-lambda);
1893 for (k=1; k<=L1; k++) {
1896 for (j=1; j<=maxr; j++) {
1899 for (i=2; i<=j; i++) {
1903 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
1911 cln::cl_N crandall_Z(const std::vector<int>& s)
1913 const int j = s.size();
1922 t0 = t0 + f_kj[q+j-2][s[0]-1];
1923 } while (t0 != t0buf);
1925 return t0 / cln::factorial(s[0]-1);
1928 std::vector<cln::cl_N> t(j);
1935 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
1936 for (int k=j-2; k>=1; k--) {
1937 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
1939 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
1940 } while (t[0] != t0buf);
1942 return t[0] / cln::factorial(s[0]-1);
1947 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
1949 std::vector<int> r = s;
1950 const int j = r.size();
1952 // decide on maximal size of f_kj for crandall_Z
1956 L1 = Digits * 3 + j*2;
1959 // decide on maximal size of crX for crandall_Y
1962 } else if (Digits < 86) {
1964 } else if (Digits < 192) {
1966 } else if (Digits < 394) {
1968 } else if (Digits < 808) {
1978 for (int i=0; i<j; i++) {
1987 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
1989 std::vector<int> rz;
1992 for (int k=r.size()-1; k>0; k--) {
1994 rz.insert(rz.begin(), r.back());
1995 skp1buf = rz.front();
2001 for (int q=0; q<skp1buf; q++) {
2003 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2004 cln::cl_N pp2 = crandall_Z(rz);
2009 res = res - pp1 * pp2 / cln::factorial(q);
2011 res = res + pp1 * pp2 / cln::factorial(q);
2014 rz.front() = skp1buf;
2016 rz.insert(rz.begin(), r.back());
2020 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2026 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2028 const int j = r.size();
2030 // buffer for subsums
2031 std::vector<cln::cl_N> t(j);
2032 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2039 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2040 for (int k=j-2; k>=0; k--) {
2041 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2043 } while (t[0] != t0buf);
2049 // does Hoelder convolution. see [BBB] (7.0)
2050 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2052 // prepare parameters
2053 // holds Li arguments in [BBB] notation
2054 std::vector<int> s = s_;
2055 std::vector<int> m_p = m_;
2056 std::vector<int> m_q;
2057 // holds Li arguments in nested sums notation
2058 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2059 s_p[0] = s_p[0] * cln::cl_N("1/2");
2060 // convert notations
2062 for (int i=0; i<s_.size(); i++) {
2067 s[i] = sig * std::abs(s[i]);
2069 std::vector<cln::cl_N> s_q;
2070 cln::cl_N signum = 1;
2073 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2078 // change parameters
2079 if (s.front() > 0) {
2080 if (m_p.front() == 1) {
2081 m_p.erase(m_p.begin());
2082 s_p.erase(s_p.begin());
2083 if (s_p.size() > 0) {
2084 s_p.front() = s_p.front() * cln::cl_N("1/2");
2090 m_q.insert(m_q.begin(), 1);
2091 if (s_q.size() > 0) {
2092 s_q.front() = s_q.front() * 2;
2094 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2097 if (m_p.front() == 1) {
2098 m_p.erase(m_p.begin());
2099 cln::cl_N spbuf = s_p.front();
2100 s_p.erase(s_p.begin());
2101 if (s_p.size() > 0) {
2102 s_p.front() = s_p.front() * spbuf;
2105 m_q.insert(m_q.begin(), 1);
2106 if (s_q.size() > 0) {
2107 s_q.front() = s_q.front() * 4;
2109 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2113 m_q.insert(m_q.begin(), 1);
2114 if (s_q.size() > 0) {
2115 s_q.front() = s_q.front() * 2;
2117 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2122 if (m_p.size() == 0) break;
2124 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2129 res = res + signum * multipleLi_do_sum(m_q, s_q);
2135 } // end of anonymous namespace
2138 //////////////////////////////////////////////////////////////////////
2140 // Multiple zeta values zeta(x)
2144 //////////////////////////////////////////////////////////////////////
2147 static ex zeta1_evalf(const ex& x)
2149 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2151 // multiple zeta value
2152 const int count = x.nops();
2153 const lst& xlst = ex_to<lst>(x);
2154 std::vector<int> r(count);
2156 // check parameters and convert them
2157 lst::const_iterator it1 = xlst.begin();
2158 std::vector<int>::iterator it2 = r.begin();
2160 if (!(*it1).info(info_flags::posint)) {
2161 return zeta(x).hold();
2163 *it2 = ex_to<numeric>(*it1).to_int();
2166 } while (it2 != r.end());
2168 // check for divergence
2170 return zeta(x).hold();
2173 // decide on summation algorithm
2174 // this is still a bit clumsy
2175 int limit = (Digits>17) ? 10 : 6;
2176 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2177 return numeric(zeta_do_sum_Crandall(r));
2179 return numeric(zeta_do_sum_simple(r));
2183 // single zeta value
2184 if (is_exactly_a<numeric>(x) && (x != 1)) {
2186 return zeta(ex_to<numeric>(x));
2187 } catch (const dunno &e) { }
2190 return zeta(x).hold();
2194 static ex zeta1_eval(const ex& x)
2196 if (is_exactly_a<lst>(x)) {
2197 if (x.nops() == 1) {
2198 return zeta(x.op(0));
2200 return zeta(x).hold();
2203 if (x.info(info_flags::numeric)) {
2204 const numeric& y = ex_to<numeric>(x);
2205 // trap integer arguments:
2206 if (y.is_integer()) {
2210 if (y.is_equal(_num1)) {
2211 return zeta(x).hold();
2213 if (y.info(info_flags::posint)) {
2214 if (y.info(info_flags::odd)) {
2215 return zeta(x).hold();
2217 return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2220 if (y.info(info_flags::odd)) {
2221 return -bernoulli(_num1-y) / (_num1-y);
2228 if (y.info(info_flags::numeric) && !y.info(info_flags::crational))
2229 return zeta1_evalf(x);
2231 return zeta(x).hold();
2235 static ex zeta1_deriv(const ex& x, unsigned deriv_param)
2237 GINAC_ASSERT(deriv_param==0);
2239 if (is_exactly_a<lst>(x)) {
2242 return zeta(_ex1, x);
2247 unsigned zeta1_SERIAL::serial =
2248 function::register_new(function_options("zeta").
2249 eval_func(zeta1_eval).
2250 evalf_func(zeta1_evalf).
2251 do_not_evalf_params().
2252 derivative_func(zeta1_deriv).
2253 latex_name("\\zeta").
2257 //////////////////////////////////////////////////////////////////////
2259 // Alternating Euler sum zeta(x,s)
2263 //////////////////////////////////////////////////////////////////////
2266 static ex zeta2_evalf(const ex& x, const ex& s)
2268 if (is_exactly_a<lst>(x)) {
2270 // alternating Euler sum
2271 const int count = x.nops();
2272 const lst& xlst = ex_to<lst>(x);
2273 const lst& slst = ex_to<lst>(s);
2274 std::vector<int> xi(count);
2275 std::vector<int> si(count);
2277 // check parameters and convert them
2278 lst::const_iterator it_xread = xlst.begin();
2279 lst::const_iterator it_sread = slst.begin();
2280 std::vector<int>::iterator it_xwrite = xi.begin();
2281 std::vector<int>::iterator it_swrite = si.begin();
2283 if (!(*it_xread).info(info_flags::posint)) {
2284 return zeta(x, s).hold();
2286 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2287 if (*it_sread > 0) {
2296 } while (it_xwrite != xi.end());
2298 // check for divergence
2299 if ((xi[0] == 1) && (si[0] == 1)) {
2300 return zeta(x, s).hold();
2303 // use Hoelder convolution
2304 return numeric(zeta_do_Hoelder_convolution(xi, si));
2307 return zeta(x, s).hold();
2311 static ex zeta2_eval(const ex& x, const ex& s)
2313 if (is_exactly_a<lst>(s)) {
2314 const lst& l = ex_to<lst>(s);
2315 lst::const_iterator it = l.begin();
2316 while (it != l.end()) {
2317 if ((*it).info(info_flags::negative)) {
2318 return zeta(x, s).hold();
2324 if (s.info(info_flags::positive)) {
2329 return zeta(x, s).hold();
2333 static ex zeta2_deriv(const ex& x, const ex& s, unsigned deriv_param)
2335 GINAC_ASSERT(deriv_param==0);
2337 if (is_exactly_a<lst>(x)) {
2340 if ((is_exactly_a<lst>(s) && (s.op(0) > 0)) || (s > 0)) {
2341 return zeta(_ex1, x);
2348 unsigned zeta2_SERIAL::serial =
2349 function::register_new(function_options("zeta").
2350 eval_func(zeta2_eval).
2351 evalf_func(zeta2_evalf).
2352 do_not_evalf_params().
2353 derivative_func(zeta2_deriv).
2354 latex_name("\\zeta").
2358 } // namespace GiNaC