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"
85 //////////////////////////////////////////////////////////////////////
87 // Classical polylogarithm Li(n,x)
91 //////////////////////////////////////////////////////////////////////
94 // anonymous namespace for helper functions
98 // lookup table for factors built from Bernoulli numbers
100 std::vector<std::vector<cln::cl_N> > Xn;
104 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
105 // With these numbers the polylogs can be calculated as follows:
106 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
107 // X_0(n) = B_n (Bernoulli numbers)
108 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
109 // The calculation of Xn depends on X0 and X{n-1}.
110 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
111 // This results in a slightly more complicated algorithm for the X_n.
112 // The first index in Xn corresponds to the index of the polylog minus 2.
113 // The second index in Xn corresponds to the index from the actual sum.
116 // rule of thumb. needs to be improved. TODO
117 const int initsize = Digits * 3 / 2;
120 // calculate X_2 and higher (corresponding to Li_4 and higher)
121 std::vector<cln::cl_N> buf(initsize);
122 std::vector<cln::cl_N>::iterator it = buf.begin();
124 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
126 for (int i=2; i<=initsize; i++) {
128 result = 0; // k == 0
130 result = Xn[0][i/2-1]; // k == 0
132 for (int k=1; k<i-1; k++) {
133 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
134 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
137 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
138 result = result + Xn[n-1][i-1] / (i+1); // k == i
145 // special case to handle the X_0 correct
146 std::vector<cln::cl_N> buf(initsize);
147 std::vector<cln::cl_N>::iterator it = buf.begin();
149 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
151 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
153 for (int i=3; i<=initsize; i++) {
155 result = -Xn[0][(i-3)/2]/2;
156 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
159 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
160 for (int k=1; k<i/2; k++) {
161 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
170 std::vector<cln::cl_N> buf(initsize/2);
171 std::vector<cln::cl_N>::iterator it = buf.begin();
172 for (int i=1; i<=initsize/2; i++) {
173 *it = bernoulli(i*2).to_cl_N();
183 // calculates Li(2,x) without Xn
184 cln::cl_N Li2_do_sum(const cln::cl_N& x)
189 cln::cl_I den = 1; // n^2 = 1
194 den = den + i; // n^2 = 4, 9, 16, ...
196 res = res + num / den;
197 } while (res != resbuf);
202 // calculates Li(2,x) with Xn
203 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
205 std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
206 cln::cl_N u = -cln::log(1-x);
207 cln::cl_N factor = u;
208 cln::cl_N res = u - u*u/4;
213 factor = factor * u*u / (2*i * (2*i+1));
214 res = res + (*it) * factor;
215 it++; // should we check it? or rely on initsize? ...
217 } while (res != resbuf);
222 // calculates Li(n,x), n>2 without Xn
223 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
225 cln::cl_N factor = x;
232 res = res + factor / cln::expt(cln::cl_I(i),n);
234 } while (res != resbuf);
239 // calculates Li(n,x), n>2 with Xn
240 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
242 std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
243 cln::cl_N u = -cln::log(1-x);
244 cln::cl_N factor = u;
250 factor = factor * u / i;
251 res = res + (*it) * factor;
252 it++; // should we check it? or rely on initsize? ...
254 } while (res != resbuf);
259 // forward declaration needed by function Li_projection and C below
260 numeric S_num(int n, int p, const numeric& x);
263 // helper function for classical polylog Li
264 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
266 // treat n=2 as special case
268 // check if precalculated X0 exists
273 if (cln::realpart(x) < 0.5) {
274 // choose the faster algorithm
275 // the switching point was empirically determined. the optimal point
276 // depends on hardware, Digits, ... so an approx value is okay.
277 // it solves also the problem with precision due to the u=-log(1-x) transformation
278 if (cln::abs(cln::realpart(x)) < 0.25) {
280 return Li2_do_sum(x);
282 return Li2_do_sum_Xn(x);
285 // choose the faster algorithm
286 if (cln::abs(cln::realpart(x)) > 0.75) {
287 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
289 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
293 // check if precalculated Xn exist
295 for (int i=xnsize; i<n-1; i++) {
300 if (cln::realpart(x) < 0.5) {
301 // choose the faster algorithm
302 // with n>=12 the "normal" summation always wins against the method with Xn
303 if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
304 return Lin_do_sum(n, x);
306 return Lin_do_sum_Xn(n, x);
309 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
310 for (int j=0; j<n-1; j++) {
311 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
312 * cln::expt(cln::log(x), j) / cln::factorial(j);
320 // helper function for classical polylog Li
321 numeric Li_num(int n, const numeric& x)
325 return -cln::log(1-x.to_cl_N());
336 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
339 // what is the desired float format?
340 // first guess: default format
341 cln::float_format_t prec = cln::default_float_format;
342 const cln::cl_N value = x.to_cl_N();
343 // second guess: the argument's format
344 if (!x.real().is_rational())
345 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
346 else if (!x.imag().is_rational())
347 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
350 if (cln::abs(value) > 1) {
351 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
352 // check if argument is complex. if it is real, the new polylog has to be conjugated.
353 if (cln::zerop(cln::imagpart(value))) {
355 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
358 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
363 result = result + Li_projection(n, cln::recip(value), prec);
366 result = result - Li_projection(n, cln::recip(value), prec);
370 for (int j=0; j<n-1; j++) {
371 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
372 * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
374 result = result - add;
378 return Li_projection(n, value, prec);
383 } // end of anonymous namespace
386 //////////////////////////////////////////////////////////////////////
388 // Multiple polylogarithm Li(n,x)
392 //////////////////////////////////////////////////////////////////////
395 // anonymous namespace for helper function
399 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
401 const int j = s.size();
403 std::vector<cln::cl_N> t(j);
404 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
412 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
413 for (int k=j-2; k>=0; k--) {
414 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]);
416 // ... and do it again (to avoid premature drop out due to special arguments)
418 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
419 for (int k=j-2; k>=0; k--) {
420 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]);
422 } while (t[0] != t0buf);
428 } // end of anonymous namespace
431 //////////////////////////////////////////////////////////////////////
433 // Classical polylogarithm and multiple polylogarithm Li(n,x)
437 //////////////////////////////////////////////////////////////////////
440 static ex Li_eval(const ex& x1, const ex& x2)
446 if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational)))
447 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
449 for (int i=0; i<x2.nops(); i++) {
450 if (!is_a<numeric>(x2.op(i))) {
451 return Li(x1,x2).hold();
454 return Li(x1,x2).evalf();
456 return Li(x1,x2).hold();
461 static ex Li_evalf(const ex& x1, const ex& x2)
463 // classical polylogs
464 if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
465 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
468 else if (is_a<lst>(x1) && is_a<lst>(x2)) {
470 for (int i=0; i<x1.nops(); i++) {
471 if (!x1.op(i).info(info_flags::posint)) {
472 return Li(x1,x2).hold();
474 if (!is_a<numeric>(x2.op(i))) {
475 return Li(x1,x2).hold();
478 if ((conv > 1) || ((conv == 1) && (x1.op(0) == 1))) {
479 return Li(x1,x2).hold();
484 std::vector<cln::cl_N> x;
485 for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
486 m.push_back(ex_to<numeric>(x1.op(i)).to_int());
487 x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
490 return numeric(multipleLi_do_sum(m, x));
493 return Li(x1,x2).hold();
497 static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
500 seq.push_back(expair(Li(x1,x2), 0));
501 return pseries(rel,seq);
505 static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
507 GINAC_ASSERT(deriv_param < 2);
508 if (deriv_param == 0) {
512 return Li(x1-1, x2) / x2;
519 REGISTER_FUNCTION(Li,
521 evalf_func(Li_evalf).
522 do_not_evalf_params().
523 series_func(Li_series).
524 derivative_func(Li_deriv));
527 //////////////////////////////////////////////////////////////////////
529 // Nielsen's generalized polylogarithm S(n,p,x)
533 //////////////////////////////////////////////////////////////////////
536 // anonymous namespace for helper functions
540 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
542 std::vector<std::vector<cln::cl_N> > Yn;
543 int ynsize = 0; // number of Yn[]
544 int ynlength = 100; // initial length of all Yn[i]
547 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
548 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
549 // representing S_{n,p}(x).
550 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
552 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
553 // representing S_{n,p}(x).
554 // The calculation of Y_n uses the values from Y_{n-1}.
555 void fill_Yn(int n, const cln::float_format_t& prec)
557 const int initsize = ynlength;
558 //const int initsize = initsize_Yn;
559 cln::cl_N one = cln::cl_float(1, prec);
562 std::vector<cln::cl_N> buf(initsize);
563 std::vector<cln::cl_N>::iterator it = buf.begin();
564 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
565 *it = (*itprev) / cln::cl_N(n+1) * one;
568 // sums with an index smaller than the depth are zero and need not to be calculated.
569 // calculation starts with depth, which is n+2)
570 for (int i=n+2; i<=initsize+n; i++) {
571 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
577 std::vector<cln::cl_N> buf(initsize);
578 std::vector<cln::cl_N>::iterator it = buf.begin();
581 for (int i=2; i<=initsize; i++) {
582 *it = *(it-1) + 1 / cln::cl_N(i) * one;
591 // make Yn longer ...
592 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
595 cln::cl_N one = cln::cl_float(1, prec);
597 Yn[0].resize(newsize);
598 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
600 for (int i=ynlength+1; i<=newsize; i++) {
601 *it = *(it-1) + 1 / cln::cl_N(i) * one;
605 for (int n=1; n<ynsize; n++) {
606 Yn[n].resize(newsize);
607 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
608 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
611 for (int i=ynlength+n+1; i<=newsize+n; i++) {
612 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
622 // helper function for S(n,p,x)
624 cln::cl_N C(int n, int p)
628 for (int k=0; k<p; k++) {
629 for (int j=0; j<=(n+k-1)/2; j++) {
633 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
636 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
643 result = result + cln::factorial(n+k-1)
644 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
645 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
648 result = result - cln::factorial(n+k-1)
649 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
650 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
655 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()
656 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
659 result = result + cln::factorial(n+k-1)
660 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
661 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
669 if (((np)/2+n) & 1) {
670 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
673 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
681 // helper function for S(n,p,x)
682 // [Kol] remark to (9.1)
692 for (int m=2; m<=k; m++) {
693 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
700 // helper function for S(n,p,x)
701 // [Kol] remark to (9.1)
711 for (int m=2; m<=k; m++) {
712 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
719 // helper function for S(n,p,x)
720 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
723 return Li_projection(n+1, x, prec);
726 // check if precalculated values are sufficient
728 for (int i=ynsize; i<p-1; i++) {
733 // should be done otherwise
734 cln::cl_N xf = x * cln::cl_float(1, prec);
738 cln::cl_N factor = cln::expt(xf, p);
742 if (i-p >= ynlength) {
744 make_Yn_longer(ynlength*2, prec);
746 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? ...
747 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
748 factor = factor * xf;
750 } while (res != resbuf);
756 // helper function for S(n,p,x)
757 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
760 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
762 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
763 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
765 for (int s=0; s<n; s++) {
767 for (int r=0; r<p; r++) {
768 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
769 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
771 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
777 return S_do_sum(n, p, x, prec);
781 // helper function for S(n,p,x)
782 numeric S_num(int n, int p, const numeric& x)
786 // [Kol] (2.22) with (2.21)
787 return cln::zeta(p+1);
792 return cln::zeta(n+1);
797 for (int nu=0; nu<n; nu++) {
798 for (int rho=0; rho<=p; rho++) {
799 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
800 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
803 result = result * cln::expt(cln::cl_I(-1),n+p-1);
810 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
812 // throw std::runtime_error("don't know how to evaluate this function!");
815 // what is the desired float format?
816 // first guess: default format
817 cln::float_format_t prec = cln::default_float_format;
818 const cln::cl_N value = x.to_cl_N();
819 // second guess: the argument's format
820 if (!x.real().is_rational())
821 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
822 else if (!x.imag().is_rational())
823 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
827 if (cln::realpart(value) < -0.5) {
829 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
830 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
832 for (int s=0; s<n; s++) {
834 for (int r=0; r<p; r++) {
835 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
836 * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
838 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
845 if (cln::abs(value) > 1) {
849 for (int s=0; s<p; s++) {
850 for (int r=0; r<=s; r++) {
851 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
852 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
853 * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
856 result = result * cln::expt(cln::cl_I(-1),n);
859 for (int r=0; r<n; r++) {
860 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
862 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
864 result = result + cln::expt(cln::cl_I(-1),p) * res2;
869 return S_projection(n, p, value, prec);
874 } // end of anonymous namespace
877 //////////////////////////////////////////////////////////////////////
879 // Nielsen's generalized polylogarithm S(n,p,x)
883 //////////////////////////////////////////////////////////////////////
886 static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
891 if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) &&
892 x1.info(info_flags::posint) && x2.info(info_flags::posint)) {
893 return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
895 return S(x1,x2,x3).hold();
899 static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
901 if (is_a<numeric>(x1) && is_a<numeric>(x2) && is_a<numeric>(x3)) {
902 return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
904 return S(x1,x2,x3).hold();
908 static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options)
911 seq.push_back(expair(S(x1,x2,x3), 0));
912 return pseries(rel,seq);
916 static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param)
918 GINAC_ASSERT(deriv_param < 3);
919 if (deriv_param < 2) {
923 return S(x1-1, x2, x3) / x3;
925 return S(x1, x2-1, x3) / (1-x3);
933 do_not_evalf_params().
934 series_func(S_series).
935 derivative_func(S_deriv));
938 //////////////////////////////////////////////////////////////////////
940 // Harmonic polylogarithm H(m,x)
944 //////////////////////////////////////////////////////////////////////
947 // anonymous namespace for helper functions
951 // forward declaration
952 ex convert_from_RV(const lst& parameterlst, const ex& arg);
955 // multiplies an one-dimensional H with another H
957 ex trafo_H_mult(const ex& h1, const ex& h2)
962 ex h1nops = h1.op(0).nops();
963 ex h2nops = h2.op(0).nops();
965 hshort = h2.op(0).op(0);
966 hlong = ex_to<lst>(h1.op(0));
968 hshort = h1.op(0).op(0);
970 hlong = ex_to<lst>(h2.op(0));
972 hlong = h2.op(0).op(0);
975 for (int i=0; i<=hlong.nops(); i++) {
979 newparameter.append(hlong[j]);
981 newparameter.append(hshort);
982 for (; j<hlong.nops(); j++) {
983 newparameter.append(hlong[j]);
985 res += H(newparameter, h1.op(1)).hold();
991 // applies trafo_H_mult recursively on expressions
992 struct map_trafo_H_mult : public map_function
994 ex operator()(const ex& e)
1005 for (int pos=0; pos<e.nops(); pos++) {
1006 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1007 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1009 for (ex i=0; i<e.op(pos).op(1); i++) {
1010 Hlst.append(e.op(pos).op(0));
1014 } else if (is_a<function>(e.op(pos))) {
1015 std::string name = ex_to<function>(e.op(pos)).get_name();
1017 if (e.op(pos).op(0).nops() > 1) {
1020 Hlst.append(e.op(pos));
1025 result *= e.op(pos);
1028 if (Hlst.nops() > 0) {
1029 firstH = Hlst[Hlst.nops()-1];
1036 if (Hlst.nops() > 0) {
1037 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1039 for (int i=1; i<Hlst.nops(); i++) {
1040 result *= Hlst.op(i);
1042 result = result.expand();
1043 map_trafo_H_mult recursion;
1044 return recursion(result);
1055 // do integration [ReV] (49)
1056 // put parameter 1 in front of existing parameters
1057 ex trafo_H_prepend_one(const ex& e, const ex& arg)
1061 if (is_a<function>(e)) {
1062 name = ex_to<function>(e).get_name();
1067 for (int i=0; i<e.nops(); i++) {
1068 if (is_a<function>(e.op(i))) {
1069 std::string name = ex_to<function>(e.op(i)).get_name();
1077 lst newparameter = ex_to<lst>(h.op(0));
1078 newparameter.prepend(1);
1079 return e.subs(h == H(newparameter, h.op(1)).hold());
1081 return e * H(lst(1),1-arg).hold();
1086 // do integration [ReV] (55)
1087 // put parameter 0 in front of existing parameters
1088 ex trafo_H_prepend_zero(const ex& e, const ex& arg)
1092 if (is_a<function>(e)) {
1093 name = ex_to<function>(e).get_name();
1098 for (int i=0; i<e.nops(); i++) {
1099 if (is_a<function>(e.op(i))) {
1100 std::string name = ex_to<function>(e.op(i)).get_name();
1108 lst newparameter = ex_to<lst>(h.op(0));
1109 newparameter.prepend(0);
1110 ex addzeta = convert_from_RV(newparameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
1111 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1113 return e * (-H(lst(0),1/arg).hold());
1118 // do x -> 1-x transformation
1119 struct map_trafo_H_1mx : public map_function
1121 ex operator()(const ex& e)
1123 if (is_a<add>(e) || is_a<mul>(e)) {
1124 return e.map(*this);
1127 if (is_a<function>(e)) {
1128 std::string name = ex_to<function>(e).get_name();
1131 lst parameter = ex_to<lst>(e.op(0));
1134 // if all parameters are either zero or one return the transformed function
1135 if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) {
1137 for (int i=parameter.nops(); i>0; i--) {
1138 newparameter.append(0);
1140 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
1141 } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) {
1143 for (int i=parameter.nops(); i>0; i--) {
1144 newparameter.append(1);
1146 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
1149 lst newparameter = parameter;
1150 newparameter.remove_first();
1152 if (parameter.op(0) == 0) {
1155 ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
1156 map_trafo_H_1mx recursion;
1157 ex buffer = recursion(H(newparameter, arg).hold());
1158 if (is_a<add>(buffer)) {
1159 for (int i=0; i<buffer.nops(); i++) {
1160 res -= trafo_H_prepend_one(buffer.op(i), arg);
1163 res -= trafo_H_prepend_one(buffer, arg);
1170 map_trafo_H_1mx recursion;
1171 map_trafo_H_mult unify;
1174 while (parameter.op(firstzero) == 1) {
1177 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1181 newparameter.append(parameter[j+1]);
1183 newparameter.append(1);
1184 for (; j<parameter.nops()-1; j++) {
1185 newparameter.append(parameter[j+1]);
1187 res -= H(newparameter, arg).hold();
1189 return (unify((-H(lst(0), 1-arg).hold() * recursion(H(newparameter, arg).hold())).expand()) +
1190 recursion(res)) / firstzero;
1201 // do x -> 1/x transformation
1202 struct map_trafo_H_1overx : public map_function
1204 ex operator()(const ex& e)
1206 if (is_a<add>(e) || is_a<mul>(e)) {
1207 return e.map(*this);
1210 if (is_a<function>(e)) {
1211 std::string name = ex_to<function>(e).get_name();
1214 lst parameter = ex_to<lst>(e.op(0));
1217 // if all parameters are either zero or one return the transformed function
1218 if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) {
1219 map_trafo_H_mult unify;
1220 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, parameter.nops()) /
1221 factorial(parameter.nops())).expand());
1222 } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) {
1223 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1226 lst newparameter = parameter;
1227 newparameter.remove_first();
1229 if (parameter.op(0) == 0) {
1232 ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
1233 map_trafo_H_1overx recursion;
1234 ex buffer = recursion(H(newparameter, arg).hold());
1235 if (is_a<add>(buffer)) {
1236 for (int i=0; i<buffer.nops(); i++) {
1237 res += trafo_H_prepend_zero(buffer.op(i), arg);
1240 res += trafo_H_prepend_zero(buffer, arg);
1247 map_trafo_H_1overx recursion;
1248 map_trafo_H_mult unify;
1249 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1251 while (parameter.op(firstzero) == 1) {
1254 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1258 newparameter.append(parameter[j+1]);
1260 newparameter.append(1);
1261 for (; j<parameter.nops()-1; j++) {
1262 newparameter.append(parameter[j+1]);
1264 res -= H(newparameter, arg).hold();
1266 res = recursion(res).expand() / firstzero;
1278 // remove trailing zeros from H-parameters
1279 struct map_trafo_H_reduce_trailing_zeros : public map_function
1281 ex operator()(const ex& e)
1283 if (is_a<add>(e) || is_a<mul>(e)) {
1284 return e.map(*this);
1286 if (is_a<function>(e)) {
1287 std::string name = ex_to<function>(e).get_name();
1290 if (is_a<lst>(e.op(0))) {
1291 parameter = ex_to<lst>(e.op(0));
1293 parameter = lst(e.op(0));
1296 if (parameter.op(parameter.nops()-1) == 0) {
1299 if (parameter.nops() == 1) {
1304 lst::const_iterator it = parameter.begin();
1305 while ((it != parameter.end()) && (*it == 0)) {
1308 if (it == parameter.end()) {
1309 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1313 parameter.remove_last();
1314 int lastentry = parameter.nops();
1315 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1320 ex result = log(arg) * H(parameter,arg).hold();
1321 for (ex i=0; i<lastentry; i++) {
1323 result -= (parameter[i]-1) * H(parameter, arg).hold();
1327 if (lastentry < parameter.nops()) {
1328 result = result / (parameter.nops()-lastentry+1);
1329 return result.map(*this);
1341 // recursively call convert_from_RV on expression
1342 struct map_trafo_H_convert : public map_function
1344 ex operator()(const ex& e)
1346 if (is_a<add>(e) || is_a<mul>(e) || is_a<power>(e)) {
1347 return e.map(*this);
1349 if (is_a<function>(e)) {
1350 std::string name = ex_to<function>(e).get_name();
1352 lst parameter = ex_to<lst>(e.op(0));
1354 return convert_from_RV(parameter, arg);
1362 // translate notation from nested sums to Remiddi/Vermaseren
1363 lst convert_to_RV(const lst& o)
1366 for (lst::const_iterator it = o.begin(); it != o.end(); it++) {
1367 for (ex i=0; i<(*it)-1; i++) {
1376 // translate notation from Remiddi/Vermaseren to nested sums
1377 ex convert_from_RV(const lst& parameterlst, const ex& arg)
1379 lst newparameterlst;
1381 lst::const_iterator it = parameterlst.begin();
1383 while (it != parameterlst.end()) {
1387 newparameterlst.append((*it>0) ? count : -count);
1392 for (int i=1; i<count; i++) {
1393 newparameterlst.append(0);
1396 map_trafo_H_reduce_trailing_zeros filter;
1397 return filter(H(newparameterlst, arg).hold());
1401 // do the actual summation.
1402 cln::cl_N H_do_sum(const std::vector<int>& s, const cln::cl_N& x)
1404 const int j = s.size();
1406 std::vector<cln::cl_N> t(j);
1408 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1409 cln::cl_N factor = cln::expt(x, j) * one;
1415 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
1416 for (int k=j-2; k>=1; k--) {
1417 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
1419 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), s[0]);
1420 factor = factor * x;
1421 } while (t[0] != t0buf);
1427 } // end of anonymous namespace
1430 //////////////////////////////////////////////////////////////////////
1432 // Harmonic polylogarithm H(m,x)
1436 //////////////////////////////////////////////////////////////////////
1439 static ex H_eval(const ex& x1, const ex& x2)
1447 if (x1.nops() == 1) {
1448 return Li(x1.op(0), x2);
1450 if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) {
1451 return H(x1,x2).evalf();
1453 return H(x1,x2).hold();
1457 static ex H_evalf(const ex& x1, const ex& x2)
1459 if (is_a<lst>(x1) && is_a<numeric>(x2)) {
1460 for (int i=0; i<x1.nops(); i++) {
1461 if (!x1.op(i).info(info_flags::posint)) {
1462 return H(x1,x2).hold();
1465 if (x1.nops() < 1) {
1468 if (x1.nops() == 1) {
1469 return Li(x1.op(0), x2).evalf();
1471 cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
1473 return zeta(x1).evalf();
1477 if (cln::abs(x) > 1) {
1478 symbol xtemp("xtemp");
1479 map_trafo_H_1overx trafo;
1480 ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
1481 map_trafo_H_convert converter;
1482 res = converter(res);
1483 return res.subs(xtemp==x2).evalf();
1486 // since the x->1-x transformation produces a lot of terms, it is only
1487 // efficient for argument near one.
1488 if (cln::realpart(x) > 0.95) {
1489 symbol xtemp("xtemp");
1490 map_trafo_H_1mx trafo;
1491 ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
1492 map_trafo_H_convert converter;
1493 res = converter(res);
1494 return res.subs(xtemp==x2).evalf();
1497 // no trafo -> do summation
1498 int count = x1.nops();
1499 std::vector<int> r(count);
1500 for (int i=0; i<count; i++) {
1501 r[i] = ex_to<numeric>(x1.op(i)).to_int();
1504 return numeric(H_do_sum(r,x));
1507 return H(x1,x2).hold();
1511 static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
1514 seq.push_back(expair(H(x1,x2), 0));
1515 return pseries(rel,seq);
1519 static ex H_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
1521 GINAC_ASSERT(deriv_param < 2);
1522 if (deriv_param == 0) {
1525 if (is_a<lst>(x1)) {
1526 lst newparameter = ex_to<lst>(x1);
1527 if (x1.op(0) == 1) {
1528 newparameter.remove_first();
1529 return 1/(1-x2) * H(newparameter, x2);
1532 return H(newparameter, x2).hold() / x2;
1538 return H(x1-1, x2).hold() / x2;
1544 REGISTER_FUNCTION(H,
1546 evalf_func(H_evalf).
1547 do_not_evalf_params().
1548 series_func(H_series).
1549 derivative_func(H_deriv));
1552 //////////////////////////////////////////////////////////////////////
1554 // Multiple zeta values zeta(x) and zeta(x,s)
1558 //////////////////////////////////////////////////////////////////////
1561 // anonymous namespace for helper functions
1565 // parameters and data for [Cra] algorithm
1566 const cln::cl_N lambda = cln::cl_N("319/320");
1569 std::vector<std::vector<cln::cl_N> > f_kj;
1570 std::vector<cln::cl_N> crB;
1571 std::vector<std::vector<cln::cl_N> > crG;
1572 std::vector<cln::cl_N> crX;
1575 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
1577 const int size = a.size();
1578 for (int n=0; n<size; n++) {
1580 for (int m=0; m<=n; m++) {
1581 c[n] = c[n] + a[m]*b[n-m];
1588 void initcX(const std::vector<int>& s)
1590 const int k = s.size();
1596 for (int i=0; i<=L2; i++) {
1597 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
1602 for (int m=0; m<k-1; m++) {
1603 std::vector<cln::cl_N> crGbuf;
1606 for (int i=0; i<=L2; i++) {
1607 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
1609 crG.push_back(crGbuf);
1614 for (int m=0; m<k-1; m++) {
1615 std::vector<cln::cl_N> Xbuf;
1616 for (int i=0; i<=L2; i++) {
1617 Xbuf.push_back(crX[i] * crG[m][i]);
1619 halfcyclic_convolute(Xbuf, crB, crX);
1625 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
1627 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1628 cln::cl_N factor = cln::expt(lambda, Sqk);
1629 cln::cl_N res = factor / Sqk * crX[0] * one;
1634 factor = factor * lambda;
1636 res = res + crX[N] * factor / (N+Sqk);
1637 } while ((res != resbuf) || cln::zerop(crX[N]));
1643 void calc_f(int maxr)
1648 cln::cl_N t0, t1, t2, t3, t4;
1650 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
1651 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1653 t0 = cln::exp(-lambda);
1655 for (k=1; k<=L1; k++) {
1658 for (j=1; j<=maxr; j++) {
1661 for (i=2; i<=j; i++) {
1665 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
1673 cln::cl_N crandall_Z(const std::vector<int>& s)
1675 const int j = s.size();
1684 t0 = t0 + f_kj[q+j-2][s[0]-1];
1685 } while (t0 != t0buf);
1687 return t0 / cln::factorial(s[0]-1);
1690 std::vector<cln::cl_N> t(j);
1697 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
1698 for (int k=j-2; k>=1; k--) {
1699 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
1701 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
1702 } while (t[0] != t0buf);
1704 return t[0] / cln::factorial(s[0]-1);
1709 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
1711 std::vector<int> r = s;
1712 const int j = r.size();
1714 // decide on maximal size of f_kj for crandall_Z
1718 L1 = Digits * 3 + j*2;
1721 // decide on maximal size of crX for crandall_Y
1724 } else if (Digits < 86) {
1726 } else if (Digits < 192) {
1728 } else if (Digits < 394) {
1730 } else if (Digits < 808) {
1740 for (int i=0; i<j; i++) {
1749 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
1751 std::vector<int> rz;
1754 for (int k=r.size()-1; k>0; k--) {
1756 rz.insert(rz.begin(), r.back());
1757 skp1buf = rz.front();
1763 for (int q=0; q<skp1buf; q++) {
1765 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
1766 cln::cl_N pp2 = crandall_Z(rz);
1771 res = res - pp1 * pp2 / cln::factorial(q);
1773 res = res + pp1 * pp2 / cln::factorial(q);
1776 rz.front() = skp1buf;
1778 rz.insert(rz.begin(), r.back());
1782 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
1788 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
1790 const int j = r.size();
1792 // buffer for subsums
1793 std::vector<cln::cl_N> t(j);
1794 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1801 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
1802 for (int k=j-2; k>=0; k--) {
1803 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
1805 } while (t[0] != t0buf);
1811 // does Hoelder convolution. see [BBB] (7.0)
1812 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
1814 // prepare parameters
1815 // holds Li arguments in [BBB] notation
1816 std::vector<int> s = s_;
1817 std::vector<int> m_p = m_;
1818 std::vector<int> m_q;
1819 // holds Li arguments in nested sums notation
1820 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
1821 s_p[0] = s_p[0] * cln::cl_N("1/2");
1822 // convert notations
1824 for (int i=0; i<s_.size(); i++) {
1829 s[i] = sig * std::abs(s[i]);
1831 std::vector<cln::cl_N> s_q;
1832 cln::cl_N signum = 1;
1835 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
1840 // change parameters
1841 if (s.front() > 0) {
1842 if (m_p.front() == 1) {
1843 m_p.erase(m_p.begin());
1844 s_p.erase(s_p.begin());
1845 if (s_p.size() > 0) {
1846 s_p.front() = s_p.front() * cln::cl_N("1/2");
1852 m_q.insert(m_q.begin(), 1);
1853 if (s_q.size() > 0) {
1854 s_q.front() = s_q.front() * 2;
1856 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
1859 if (m_p.front() == 1) {
1860 m_p.erase(m_p.begin());
1861 cln::cl_N spbuf = s_p.front();
1862 s_p.erase(s_p.begin());
1863 if (s_p.size() > 0) {
1864 s_p.front() = s_p.front() * spbuf;
1867 m_q.insert(m_q.begin(), 1);
1868 if (s_q.size() > 0) {
1869 s_q.front() = s_q.front() * 4;
1871 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
1875 m_q.insert(m_q.begin(), 1);
1876 if (s_q.size() > 0) {
1877 s_q.front() = s_q.front() * 2;
1879 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
1884 if (m_p.size() == 0) break;
1886 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
1891 res = res + signum * multipleLi_do_sum(m_q, s_q);
1897 } // end of anonymous namespace
1900 //////////////////////////////////////////////////////////////////////
1902 // Multiple zeta values zeta(x)
1906 //////////////////////////////////////////////////////////////////////
1909 static ex zeta1_evalf(const ex& x)
1911 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
1913 // multiple zeta value
1914 const int count = x.nops();
1915 const lst& xlst = ex_to<lst>(x);
1916 std::vector<int> r(count);
1918 // check parameters and convert them
1919 lst::const_iterator it1 = xlst.begin();
1920 std::vector<int>::iterator it2 = r.begin();
1922 if (!(*it1).info(info_flags::posint)) {
1923 return zeta(x).hold();
1925 *it2 = ex_to<numeric>(*it1).to_int();
1928 } while (it2 != r.end());
1930 // check for divergence
1932 return zeta(x).hold();
1935 // decide on summation algorithm
1936 // this is still a bit clumsy
1937 int limit = (Digits>17) ? 10 : 6;
1938 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
1939 return numeric(zeta_do_sum_Crandall(r));
1941 return numeric(zeta_do_sum_simple(r));
1945 // single zeta value
1946 if (is_exactly_a<numeric>(x) && (x != 1)) {
1948 return zeta(ex_to<numeric>(x));
1949 } catch (const dunno &e) { }
1952 return zeta(x).hold();
1956 static ex zeta1_eval(const ex& x)
1958 if (is_exactly_a<lst>(x)) {
1959 if (x.nops() == 1) {
1960 return zeta(x.op(0));
1962 return zeta(x).hold();
1965 if (x.info(info_flags::numeric)) {
1966 const numeric& y = ex_to<numeric>(x);
1967 // trap integer arguments:
1968 if (y.is_integer()) {
1972 if (y.is_equal(_num1)) {
1973 return zeta(x).hold();
1975 if (y.info(info_flags::posint)) {
1976 if (y.info(info_flags::odd)) {
1977 return zeta(x).hold();
1979 return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
1982 if (y.info(info_flags::odd)) {
1983 return -bernoulli(_num1-y) / (_num1-y);
1990 if (y.info(info_flags::numeric) && !y.info(info_flags::crational))
1991 return zeta1_evalf(x);
1993 return zeta(x).hold();
1997 static ex zeta1_deriv(const ex& x, unsigned deriv_param)
1999 GINAC_ASSERT(deriv_param==0);
2001 if (is_exactly_a<lst>(x)) {
2004 return zeta(_ex1, x);
2009 unsigned zeta1_SERIAL::serial =
2010 function::register_new(function_options("zeta").
2011 eval_func(zeta1_eval).
2012 evalf_func(zeta1_evalf).
2013 do_not_evalf_params().
2014 derivative_func(zeta1_deriv).
2015 latex_name("\\zeta").
2019 //////////////////////////////////////////////////////////////////////
2021 // Alternating Euler sum zeta(x,s)
2025 //////////////////////////////////////////////////////////////////////
2028 static ex zeta2_evalf(const ex& x, const ex& s)
2030 if (is_exactly_a<lst>(x)) {
2032 // alternating Euler sum
2033 const int count = x.nops();
2034 const lst& xlst = ex_to<lst>(x);
2035 const lst& slst = ex_to<lst>(s);
2036 std::vector<int> xi(count);
2037 std::vector<int> si(count);
2039 // check parameters and convert them
2040 lst::const_iterator it_xread = xlst.begin();
2041 lst::const_iterator it_sread = slst.begin();
2042 std::vector<int>::iterator it_xwrite = xi.begin();
2043 std::vector<int>::iterator it_swrite = si.begin();
2045 if (!(*it_xread).info(info_flags::posint)) {
2046 return zeta(x, s).hold();
2048 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2049 if (*it_sread > 0) {
2058 } while (it_xwrite != xi.end());
2060 // check for divergence
2061 if ((xi[0] == 1) && (si[0] == 1)) {
2062 return zeta(x, s).hold();
2065 // use Hoelder convolution
2066 return numeric(zeta_do_Hoelder_convolution(xi, si));
2069 return zeta(x, s).hold();
2073 static ex zeta2_eval(const ex& x, const ex& s)
2075 if (is_exactly_a<lst>(s)) {
2076 const lst& l = ex_to<lst>(s);
2077 lst::const_iterator it = l.begin();
2078 while (it != l.end()) {
2079 if ((*it).info(info_flags::negative)) {
2080 return zeta(x, s).hold();
2086 if (s.info(info_flags::positive)) {
2091 return zeta(x, s).hold();
2095 static ex zeta2_deriv(const ex& x, const ex& s, unsigned deriv_param)
2097 GINAC_ASSERT(deriv_param==0);
2099 if (is_exactly_a<lst>(x)) {
2102 if ((is_exactly_a<lst>(s) && (s.op(0) > 0)) || (s > 0)) {
2103 return zeta(_ex1, x);
2110 unsigned zeta2_SERIAL::serial =
2111 function::register_new(function_options("zeta").
2112 eval_func(zeta2_eval).
2113 evalf_func(zeta2_evalf).
2114 do_not_evalf_params().
2115 derivative_func(zeta2_deriv).
2116 latex_name("\\zeta").
2120 } // namespace GiNaC