* The functions are:
* classical polylogarithm Li(n,x)
* multiple polylogarithm Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
- * nielsen's generalized polylogarithm S(n,p,x)
+ * G(lst(a_1,...,a_k),y) or G(lst(a_1,...,a_k),lst(s_1,...,s_k),y)
+ * Nielsen's generalized polylogarithm S(n,p,x)
* harmonic polylogarithm H(m,x) or H(lst(m_1,...,m_k),x)
* multiple zeta value zeta(m) or zeta(lst(m_1,...,m_k))
* alternating Euler sum zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
* [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
* [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
* [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
+ * [VSW] Numerical evaluation of multiple polylogarithms, J.Vollinga, S.Weinzierl, hep-ph/0410259
*
* - The order of parameters and arguments of Li and zeta is defined according to the nested sums
* representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
* 0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
* number --- notation.
*
- * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
- * the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product
- * x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers.
- * If you want to have an alternating Euler sum, you have to give the signs of the parameters as a
- * second argument s to zeta(m,s) containing 1 and -1.
+ * - All functions can be nummerically evaluated with arguments in the whole complex plane. The parameters
+ * for Li, zeta and S must be positive integers. If you want to have an alternating Euler sum, you have
+ * to give the signs of the parameters as a second argument s to zeta(m,s) containing 1 and -1.
*
* - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and
* look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
- * [Cra] and [BBB] for speed up.
+ * [Cra] and [BBB] for speed up. Multiple polylogarithms use Hoelder convolution [BBB].
*
- * - The functions have no series expansion into nested sums. To do this, you have to convert these functions
- * into the appropriate objects from the nestedsums library, do the expansion and convert the
- * result back.
+ * - The functions have no means to do a series expansion into nested sums. To do this, you have to convert
+ * these functions into the appropriate objects from the nestedsums library, do the expansion and convert
+ * the result back.
*
* - Numerical testing of this implementation has been performed by doing a comparison of results
* between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
* by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
* comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
- * around |x|=1 along with comparisons to corresponding zeta functions.
+ * around |x|=1 along with comparisons to corresponding zeta functions. Multiple polylogarithms were
+ * checked against H and zeta and by means of shuffle and quasi-shuffle relations.
*
*/
/*
- * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
+#include <sstream>
#include <stdexcept>
#include <vector>
#include <cln/cln.h>
xnsize++;
}
-
// doubles the number of entries in each Xn[]
void double_Xn()
{
for (int i=1; i<=xninitsizestep/2; ++i) {
Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
}
- if (Xn.size() > 0) {
+ if (Xn.size() > 1) {
int xend = xninitsize + xninitsizestep;
cln::cl_N result;
// X_1
std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
cln::cl_N u = -cln::log(1-x);
cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
- cln::cl_N res = u - u*u/4;
+ cln::cl_N uu = cln::square(u);
+ cln::cl_N res = u - uu/4;
cln::cl_N resbuf;
unsigned i = 1;
do {
resbuf = res;
- factor = factor * u*u / (2*i * (2*i+1));
+ factor = factor * uu / (2*i * (2*i+1));
res = res + (*it) * factor;
i++;
if (++it == xend) {
// helper function for classical polylog Li
-numeric Li_num(int n, const numeric& x)
+numeric Lin_numeric(int n, const numeric& x)
{
if (n == 1) {
// just a log
// [Kol] (2.22)
return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
}
-
+ if (abs(x.real()) < 0.4 && abs(abs(x)-1) < 0.01) {
+ cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
+ cln::cl_N result = -cln::expt(cln::log(x_), n-1) * cln::log(1-x_) / cln::factorial(n-1);
+ for (int j=0; j<n-1; j++) {
+ result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x_).to_cl_N())
+ * cln::expt(cln::log(x_), j) / cln::factorial(j);
+ }
+ return result;
+ }
+
// what is the desired float format?
// first guess: default format
cln::float_format_t prec = cln::default_float_format;
cln::cl_N add;
for (int j=0; j<n-1; j++) {
add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
- * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
+ * Lin_numeric(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
}
result = result - add;
return result;
namespace {
+// performs the actual series summation for multiple polylogarithms
cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
{
+ // ensure all x <> 0.
+ for (std::vector<cln::cl_N>::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits));
+ }
+
const int j = s.size();
+ bool flag_accidental_zero = false;
std::vector<cln::cl_N> t(j);
cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
int q = 0;
do {
t0buf = t[0];
- // do it once ...
- q++;
- t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
- for (int k=j-2; k>=0; k--) {
- 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]);
- }
- // ... and do it again (to avoid premature drop out due to special arguments)
q++;
t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
for (int k=j-2; k>=0; k--) {
+ flag_accidental_zero = cln::zerop(t[k+1]);
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]);
}
- } while (t[0] != t0buf);
+ } while ( (t[0] != t0buf) || flag_accidental_zero );
return t[0];
}
+
+// converts parameter types and calls multipleLi_do_sum (convenience function for G_numeric)
+cln::cl_N mLi_do_summation(const lst& m, const lst& x)
+{
+ std::vector<int> m_int;
+ std::vector<cln::cl_N> x_cln;
+ for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+ m_int.push_back(ex_to<numeric>(*itm).to_int());
+ x_cln.push_back(ex_to<numeric>(*itx).to_cl_N());
+ }
+ return multipleLi_do_sum(m_int, x_cln);
+}
+
+
// forward declaration for Li_eval()
lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
+// holding dummy-symbols for the G/Li transformations
+std::vector<ex> gsyms;
+
+
+// type used by the transformation functions for G
+typedef std::vector<int> Gparameter;
+
+
+// G_eval1-function for G transformations
+ex G_eval1(int a, int scale)
+{
+ if (a != 0) {
+ const ex& scs = gsyms[std::abs(scale)];
+ const ex& as = gsyms[std::abs(a)];
+ if (as != scs) {
+ return -log(1 - scs/as);
+ } else {
+ return -zeta(1);
+ }
+ } else {
+ return log(gsyms[std::abs(scale)]);
+ }
+}
+
+
+// G_eval-function for G transformations
+ex G_eval(const Gparameter& a, int scale)
+{
+ // check for properties of G
+ ex sc = gsyms[std::abs(scale)];
+ lst newa;
+ bool all_zero = true;
+ bool all_ones = true;
+ int count_ones = 0;
+ for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
+ if (*it != 0) {
+ const ex sym = gsyms[std::abs(*it)];
+ newa.append(sym);
+ all_zero = false;
+ if (sym != sc) {
+ all_ones = false;
+ }
+ if (all_ones) {
+ ++count_ones;
+ }
+ } else {
+ all_ones = false;
+ }
+ }
+
+ // care about divergent G: shuffle to separate divergencies that will be canceled
+ // later on in the transformation
+ if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
+ // do shuffle
+ Gparameter short_a;
+ Gparameter::const_iterator it = a.begin();
+ ++it;
+ for (; it != a.end(); ++it) {
+ short_a.push_back(*it);
+ }
+ ex result = G_eval1(a.front(), scale) * G_eval(short_a, scale);
+ it = short_a.begin();
+ for (int i=1; i<count_ones; ++i) {
+ ++it;
+ }
+ for (; it != short_a.end(); ++it) {
+
+ Gparameter newa;
+ Gparameter::const_iterator it2 = short_a.begin();
+ for (--it2; it2 != it;) {
+ ++it2;
+ newa.push_back(*it2);
+ }
+ newa.push_back(a[0]);
+ ++it2;
+ for (; it2 != short_a.end(); ++it2) {
+ newa.push_back(*it2);
+ }
+ result -= G_eval(newa, scale);
+ }
+ return result / count_ones;
+ }
+
+ // G({1,...,1};y) -> G({1};y)^k / k!
+ if (all_ones && a.size() > 1) {
+ return pow(G_eval1(a.front(),scale), count_ones) / factorial(count_ones);
+ }
+
+ // G({0,...,0};y) -> log(y)^k / k!
+ if (all_zero) {
+ return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
+ }
+
+ // no special cases anymore -> convert it into Li
+ lst m;
+ lst x;
+ ex argbuf = gsyms[std::abs(scale)];
+ ex mval = _ex1;
+ for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) {
+ if (*it != 0) {
+ const ex& sym = gsyms[std::abs(*it)];
+ x.append(argbuf / sym);
+ m.append(mval);
+ mval = _ex1;
+ argbuf = sym;
+ } else {
+ ++mval;
+ }
+ }
+ return pow(-1, x.nops()) * Li(m, x);
+}
+
+
+// converts data for G: pending_integrals -> a
+Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
+{
+ GINAC_ASSERT(pending_integrals.size() != 1);
+
+ if (pending_integrals.size() > 0) {
+ // get rid of the first element, which would stand for the new upper limit
+ Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
+ return new_a;
+ } else {
+ // just return empty parameter list
+ Gparameter new_a;
+ return new_a;
+ }
+}
+
+
+// check the parameters a and scale for G and return information about convergence, depth, etc.
+// convergent : true if G(a,scale) is convergent
+// depth : depth of G(a,scale)
+// trailing_zeros : number of trailing zeros of a
+// min_it : iterator of a pointing on the smallest element in a
+Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
+ bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
+{
+ convergent = true;
+ depth = 0;
+ trailing_zeros = 0;
+ min_it = a.end();
+ Gparameter::const_iterator lastnonzero = a.end();
+ for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
+ if (std::abs(*it) > 0) {
+ ++depth;
+ trailing_zeros = 0;
+ lastnonzero = it;
+ if (std::abs(*it) < scale) {
+ convergent = false;
+ if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
+ min_it = it;
+ }
+ }
+ } else {
+ ++trailing_zeros;
+ }
+ }
+ return ++lastnonzero;
+}
+
+
+// add scale to pending_integrals if pending_integrals is empty
+Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
+{
+ GINAC_ASSERT(pending_integrals.size() != 1);
+
+ if (pending_integrals.size() > 0) {
+ return pending_integrals;
+ } else {
+ Gparameter new_pending_integrals;
+ new_pending_integrals.push_back(scale);
+ return new_pending_integrals;
+ }
+}
+
+
+// handles trailing zeroes for an otherwise convergent integral
+ex trailing_zeros_G(const Gparameter& a, int scale)
+{
+ bool convergent;
+ int depth, trailing_zeros;
+ Gparameter::const_iterator last, dummyit;
+ last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
+
+ GINAC_ASSERT(convergent);
+
+ if ((trailing_zeros > 0) && (depth > 0)) {
+ ex result;
+ Gparameter new_a(a.begin(), a.end()-1);
+ result += G_eval1(0, scale) * trailing_zeros_G(new_a, scale);
+ for (Gparameter::const_iterator it = a.begin(); it != last; ++it) {
+ Gparameter new_a(a.begin(), it);
+ new_a.push_back(0);
+ new_a.insert(new_a.end(), it, a.end()-1);
+ result -= trailing_zeros_G(new_a, scale);
+ }
+
+ return result / trailing_zeros;
+ } else {
+ return G_eval(a, scale);
+ }
+}
+
+
+// G transformation [VSW] (57),(58)
+ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale)
+{
+ // pendint = ( y1, b1, ..., br )
+ // a = ( 0, ..., 0, amin )
+ // scale = y2
+ //
+ // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
+ // where sr replaces amin
+
+ GINAC_ASSERT(a.back() != 0);
+ GINAC_ASSERT(a.size() > 0);
+
+ ex result;
+ Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
+ const int psize = pending_integrals.size();
+
+ // length == 1
+ // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
+
+ if (a.size() == 1) {
+
+ // ln(-y2_{-+})
+ result += log(gsyms[ex_to<numeric>(scale).to_int()]);
+ if (a.back() > 0) {
+ new_pending_integrals.push_back(-scale);
+ result += I*Pi;
+ } else {
+ new_pending_integrals.push_back(scale);
+ result -= I*Pi;
+ }
+ if (psize) {
+ result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front());
+ }
+
+ // G(y2_{-+}; sr)
+ result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front());
+
+ // G(0; sr)
+ new_pending_integrals.back() = 0;
+ result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front());
+
+ return result;
+ }
+
+ // length > 1
+ // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+ // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+
+ //term zeta_m
+ result -= zeta(a.size());
+ if (psize) {
+ result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front());
+ }
+
+ // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+ // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
+ Gparameter new_a(a.begin()+1, a.end());
+ new_pending_integrals.push_back(0);
+ result -= depth_one_trafo_G(new_pending_integrals, new_a, scale);
+
+ // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+ // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
+ Gparameter new_pending_integrals_2;
+ new_pending_integrals_2.push_back(scale);
+ new_pending_integrals_2.push_back(0);
+ if (psize) {
+ result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front())
+ * depth_one_trafo_G(new_pending_integrals_2, new_a, scale);
+ } else {
+ result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale);
+ }
+
+ return result;
+}
+
+
+// forward declaration
+ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
+ const Gparameter& pendint, const Gparameter& a_old, int scale);
+
+
+// G transformation [VSW]
+ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale)
+{
+ // main recursion routine
+ //
+ // pendint = ( y1, b1, ..., br )
+ // a = ( a1, ..., amin, ..., aw )
+ // scale = y2
+ //
+ // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
+ // where sr replaces amin
+
+ // find smallest alpha, determine depth and trailing zeros, and check for convergence
+ bool convergent;
+ int depth, trailing_zeros;
+ Gparameter::const_iterator min_it;
+ Gparameter::const_iterator firstzero =
+ check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
+ int min_it_pos = min_it - a.begin();
+
+ // special case: all a's are zero
+ if (depth == 0) {
+ ex result;
+
+ if (a.size() == 0) {
+ result = 1;
+ } else {
+ result = G_eval(a, scale);
+ }
+ if (pendint.size() > 0) {
+ result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front());
+ }
+ return result;
+ }
+
+ // handle trailing zeros
+ if (trailing_zeros > 0) {
+ ex result;
+ Gparameter new_a(a.begin(), a.end()-1);
+ result += G_eval1(0, scale) * G_transform(pendint, new_a, scale);
+ for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) {
+ Gparameter new_a(a.begin(), it);
+ new_a.push_back(0);
+ new_a.insert(new_a.end(), it, a.end()-1);
+ result -= G_transform(pendint, new_a, scale);
+ }
+ return result / trailing_zeros;
+ }
+
+ // convergence case
+ if (convergent) {
+ if (pendint.size() > 0) {
+ return G_eval(convert_pending_integrals_G(pendint), pendint.front()) * G_eval(a, scale);
+ } else {
+ return G_eval(a, scale);
+ }
+ }
+
+ // call basic transformation for depth equal one
+ if (depth == 1) {
+ return depth_one_trafo_G(pendint, a, scale);
+ }
+
+ // do recursion
+ // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
+ // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
+ // + int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) int_0^{sr} ds_{r+1} d/ds_{r+1} G(a1,...,s_{r+1},...,aw,y2)
+
+ // smallest element in last place
+ if (min_it + 1 == a.end()) {
+ do { --min_it; } while (*min_it == 0);
+ Gparameter empty;
+ Gparameter a1(a.begin(),min_it+1);
+ Gparameter a2(min_it+1,a.end());
+
+ ex result = G_transform(pendint,a2,scale)*G_transform(empty,a1,scale);
+
+ result -= shuffle_G(empty,a1,a2,pendint,a,scale);
+ return result;
+ }
+
+ Gparameter empty;
+ Gparameter::iterator changeit;
+
+ // first term G(a_1,..,0,...,a_w;a_0)
+ Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
+ Gparameter new_a = a;
+ new_a[min_it_pos] = 0;
+ ex result = G_transform(empty, new_a, scale);
+ if (pendint.size() > 0) {
+ result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front());
+ }
+
+ // other terms
+ changeit = new_a.begin() + min_it_pos;
+ changeit = new_a.erase(changeit);
+ if (changeit != new_a.begin()) {
+ // smallest in the middle
+ new_pendint.push_back(*changeit);
+ result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
+ * G_transform(empty, new_a, scale);
+ int buffer = *changeit;
+ *changeit = *min_it;
+ result += G_transform(new_pendint, new_a, scale);
+ *changeit = buffer;
+ new_pendint.pop_back();
+ --changeit;
+ new_pendint.push_back(*changeit);
+ result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
+ * G_transform(empty, new_a, scale);
+ *changeit = *min_it;
+ result -= G_transform(new_pendint, new_a, scale);
+ } else {
+ // smallest at the front
+ new_pendint.push_back(scale);
+ result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
+ * G_transform(empty, new_a, scale);
+ new_pendint.back() = *changeit;
+ result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
+ * G_transform(empty, new_a, scale);
+ *changeit = *min_it;
+ result += G_transform(new_pendint, new_a, scale);
+ }
+ return result;
+}
+
+
+// shuffles the two parameter list a1 and a2 and calls G_transform for every term except
+// for the one that is equal to a_old
+ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
+ const Gparameter& pendint, const Gparameter& a_old, int scale)
+{
+ if (a1.size()==0 && a2.size()==0) {
+ // veto the one configuration we don't want
+ if ( a0 == a_old ) return 0;
+
+ return G_transform(pendint,a0,scale);
+ }
+
+ if (a2.size()==0) {
+ Gparameter empty;
+ Gparameter aa0 = a0;
+ aa0.insert(aa0.end(),a1.begin(),a1.end());
+ return shuffle_G(aa0,empty,empty,pendint,a_old,scale);
+ }
+
+ if (a1.size()==0) {
+ Gparameter empty;
+ Gparameter aa0 = a0;
+ aa0.insert(aa0.end(),a2.begin(),a2.end());
+ return shuffle_G(aa0,empty,empty,pendint,a_old,scale);
+ }
+
+ Gparameter a1_removed(a1.begin()+1,a1.end());
+ Gparameter a2_removed(a2.begin()+1,a2.end());
+
+ Gparameter a01 = a0;
+ Gparameter a02 = a0;
+
+ a01.push_back( a1[0] );
+ a02.push_back( a2[0] );
+
+ return shuffle_G(a01,a1_removed,a2,pendint,a_old,scale)
+ + shuffle_G(a02,a1,a2_removed,pendint,a_old,scale);
+}
+
+
+// handles the transformations and the numerical evaluation of G
+// the parameter x, s and y must only contain numerics
+ex G_numeric(const lst& x, const lst& s, const ex& y)
+{
+ // check for convergence and necessary accelerations
+ bool need_trafo = false;
+ bool need_hoelder = false;
+ int depth = 0;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if (!(*it).is_zero()) {
+ ++depth;
+ if (abs(*it) - y < -pow(10,-Digits+2)) {
+ need_trafo = true;
+ break;
+ }
+ if (abs((abs(*it) - y)/y) < 0.01) {
+ need_hoelder = true;
+ }
+ }
+ }
+ if (x.op(x.nops()-1).is_zero()) {
+ need_trafo = true;
+ }
+ if (depth == 1 && !need_trafo) {
+ return -Li(x.nops(), y / x.op(x.nops()-1)).evalf();
+ }
+
+ // convergence transformation
+ if (need_trafo) {
+
+ // sort (|x|<->position) to determine indices
+ std::multimap<ex,int> sortmap;
+ int size = 0;
+ for (int i=0; i<x.nops(); ++i) {
+ if (!x[i].is_zero()) {
+ sortmap.insert(std::pair<ex,int>(abs(x[i]), i));
+ ++size;
+ }
+ }
+ // include upper limit (scale)
+ sortmap.insert(std::pair<ex,int>(abs(y), x.nops()));
+
+ // generate missing dummy-symbols
+ int i = 1;
+ gsyms.clear();
+ gsyms.push_back(symbol("GSYMS_ERROR"));
+ ex lastentry;
+ for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
+ if (it != sortmap.begin()) {
+ if (it->second < x.nops()) {
+ if (x[it->second] == lastentry) {
+ gsyms.push_back(gsyms.back());
+ continue;
+ }
+ } else {
+ if (y == lastentry) {
+ gsyms.push_back(gsyms.back());
+ continue;
+ }
+ }
+ }
+ std::ostringstream os;
+ os << "a" << i;
+ gsyms.push_back(symbol(os.str()));
+ ++i;
+ if (it->second < x.nops()) {
+ lastentry = x[it->second];
+ } else {
+ lastentry = y;
+ }
+ }
+
+ // fill position data according to sorted indices and prepare substitution list
+ Gparameter a(x.nops());
+ lst subslst;
+ int pos = 1;
+ int scale;
+ for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
+ if (it->second < x.nops()) {
+ if (s[it->second] > 0) {
+ a[it->second] = pos;
+ } else {
+ a[it->second] = -pos;
+ }
+ subslst.append(gsyms[pos] == x[it->second]);
+ } else {
+ scale = pos;
+ subslst.append(gsyms[pos] == y);
+ }
+ ++pos;
+ }
+
+ // do transformation
+ Gparameter pendint;
+ ex result = G_transform(pendint, a, scale);
+ // replace dummy symbols with their values
+ result = result.eval().expand();
+ result = result.subs(subslst).evalf();
+
+ return result;
+ }
+
+ // do acceleration transformation (hoelder convolution [BBB])
+ if (need_hoelder) {
+
+ ex result;
+ const int size = x.nops();
+ lst newx;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ newx.append(*it / y);
+ }
+
+ for (int r=0; r<=size; ++r) {
+ ex buffer = pow(-1, r);
+ ex p = 2;
+ bool adjustp;
+ do {
+ adjustp = false;
+ for (lst::const_iterator it = newx.begin(); it != newx.end(); ++it) {
+ if (*it == 1/p) {
+ p += (3-p)/2;
+ adjustp = true;
+ continue;
+ }
+ }
+ } while (adjustp);
+ ex q = p / (p-1);
+ lst qlstx;
+ lst qlsts;
+ for (int j=r; j>=1; --j) {
+ qlstx.append(1-newx.op(j-1));
+ if (newx.op(j-1).info(info_flags::real) && newx.op(j-1) > 1 && newx.op(j-1) <= 2) {
+ qlsts.append( s.op(j-1));
+ } else {
+ qlsts.append( -s.op(j-1));
+ }
+ }
+ if (qlstx.nops() > 0) {
+ buffer *= G_numeric(qlstx, qlsts, 1/q);
+ }
+ lst plstx;
+ lst plsts;
+ for (int j=r+1; j<=size; ++j) {
+ plstx.append(newx.op(j-1));
+ plsts.append(s.op(j-1));
+ }
+ if (plstx.nops() > 0) {
+ buffer *= G_numeric(plstx, plsts, 1/p);
+ }
+ result += buffer;
+ }
+ return result;
+ }
+
+ // do summation
+ lst newx;
+ lst m;
+ int mcount = 1;
+ ex sign = 1;
+ ex factor = y;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if ((*it).is_zero()) {
+ ++mcount;
+ } else {
+ newx.append(factor / (*it));
+ factor = *it;
+ m.append(mcount);
+ mcount = 1;
+ sign = -sign;
+ }
+ }
+
+ return sign * numeric(mLi_do_summation(m, newx));
+}
+
+
+ex mLi_numeric(const lst& m, const lst& x)
+{
+ // let G_numeric do the transformation
+ lst newx;
+ lst s;
+ ex factor = 1;
+ for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+ for (int i = 1; i < *itm; ++i) {
+ newx.append(0);
+ s.append(1);
+ }
+ newx.append(factor / *itx);
+ factor /= *itx;
+ s.append(1);
+ }
+ return pow(-1, m.nops()) * G_numeric(newx, s, _ex1);
+}
+
+
} // end of anonymous namespace
//////////////////////////////////////////////////////////////////////
//
-// Classical polylogarithm and multiple polylogarithm Li(n,x)
+// Generalized multiple polylogarithm G(x, y) and G(x, s, y)
//
// GiNaC function
//
//////////////////////////////////////////////////////////////////////
-static ex Li_evalf(const ex& x1, const ex& x2)
+static ex G2_evalf(const ex& x_, const ex& y)
{
- // classical polylogs
- if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
- return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
+ if (!y.info(info_flags::positive)) {
+ return G(x_, y).hold();
}
- // multiple polylogs
- else if (is_a<lst>(x1) && is_a<lst>(x2)) {
- ex conv = 1;
- for (int i=0; i<x1.nops(); i++) {
- if (!x1.op(i).info(info_flags::posint)) {
- return Li(x1, x2).hold();
- }
- if (!is_a<numeric>(x2.op(i))) {
- return Li(x1, x2).hold();
- }
- conv *= x2.op(i);
- if (abs(conv) >= 1) {
- return Li(x1, x2).hold();
- }
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, y).hold();
+ }
+ lst s;
+ bool all_zero = true;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if (!(*it).info(info_flags::numeric)) {
+ return G(x_, y).hold();
}
-
- std::vector<int> m;
- std::vector<cln::cl_N> x;
- for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
- m.push_back(ex_to<numeric>(x1.op(i)).to_int());
- x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
+ if (*it != _ex0) {
+ all_zero = false;
}
-
- return numeric(multipleLi_do_sum(m, x));
+ s.append(+1);
}
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ return G_numeric(x, s, y);
+}
- return Li(x1,x2).hold();
+
+static ex G2_eval(const ex& x_, const ex& y)
+{
+ //TODO eval to MZV or H or S or Lin
+
+ if (!y.info(info_flags::positive)) {
+ return G(x_, y).hold();
+ }
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, y).hold();
+ }
+ lst s;
+ bool all_zero = true;
+ bool crational = true;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if (!(*it).info(info_flags::numeric)) {
+ return G(x_, y).hold();
+ }
+ if (!(*it).info(info_flags::crational)) {
+ crational = false;
+ }
+ if (*it != _ex0) {
+ all_zero = false;
+ }
+ s.append(+1);
+ }
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ if (!y.info(info_flags::crational)) {
+ crational = false;
+ }
+ if (crational) {
+ return G(x_, y).hold();
+ }
+ return G_numeric(x, s, y);
}
-static ex Li_eval(const ex& m_, const ex& x_)
+unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
+ evalf_func(G2_evalf).
+ eval_func(G2_eval).
+ do_not_evalf_params().
+ overloaded(2));
+//TODO
+// derivative_func(G2_deriv).
+// print_func<print_latex>(G2_print_latex).
+
+
+static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
{
- if (m_.nops() < 2) {
- ex m;
- if (is_a<lst>(m_)) {
- m = m_.op(0);
- } else {
- m = m_;
+ if (!y.info(info_flags::positive)) {
+ return G(x_, s_, y).hold();
+ }
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
+ if (x.nops() != s.nops()) {
+ return G(x_, s_, y).hold();
+ }
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, s_, y).hold();
+ }
+ lst sn;
+ bool all_zero = true;
+ for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
+ if (!(*itx).info(info_flags::numeric)) {
+ return G(x_, y).hold();
}
- ex x;
- if (is_a<lst>(x_)) {
- x = x_.op(0);
+ if (!(*its).info(info_flags::real)) {
+ return G(x_, y).hold();
+ }
+ if (*itx != _ex0) {
+ all_zero = false;
+ }
+ if (*its >= 0) {
+ sn.append(+1);
} else {
- x = x_;
+ sn.append(-1);
}
- if (x == _ex0) {
- return _ex0;
+ }
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ return G_numeric(x, sn, y);
+}
+
+
+static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
+{
+ //TODO eval to MZV or H or S or Lin
+
+ if (!y.info(info_flags::positive)) {
+ return G(x_, s_, y).hold();
+ }
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
+ if (x.nops() != s.nops()) {
+ return G(x_, s_, y).hold();
+ }
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, s_, y).hold();
+ }
+ lst sn;
+ bool all_zero = true;
+ bool crational = true;
+ for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
+ if (!(*itx).info(info_flags::numeric)) {
+ return G(x_, s_, y).hold();
}
- if (x == _ex1) {
- return zeta(m);
+ if (!(*its).info(info_flags::real)) {
+ return G(x_, s_, y).hold();
}
- if (x == _ex_1) {
- return (pow(2,1-m)-1) * zeta(m);
+ if (!(*itx).info(info_flags::crational)) {
+ crational = false;
}
- if (m == _ex1) {
- return -log(1-x);
+ if (*itx != _ex0) {
+ all_zero = false;
}
- if (m.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
- return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
+ if (*its >= 0) {
+ sn.append(+1);
+ } else {
+ sn.append(-1);
}
- } else {
- bool ish = true;
- bool iszeta = true;
- bool iszero = false;
- bool doevalf = false;
- bool doevalfveto = true;
+ }
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ if (!y.info(info_flags::crational)) {
+ crational = false;
+ }
+ if (crational) {
+ return G(x_, s_, y).hold();
+ }
+ return G_numeric(x, sn, y);
+}
+
+
+unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
+ evalf_func(G3_evalf).
+ eval_func(G3_eval).
+ do_not_evalf_params().
+ overloaded(2));
+//TODO
+// derivative_func(G3_deriv).
+// print_func<print_latex>(G3_print_latex).
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Classical polylogarithm and multiple polylogarithm Li(m,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex Li_evalf(const ex& m_, const ex& x_)
+{
+ // classical polylogs
+ if (m_.info(info_flags::posint)) {
+ if (x_.info(info_flags::numeric)) {
+ return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_));
+ } else {
+ // try to numerically evaluate second argument
+ ex x_val = x_.evalf();
+ if (x_val.info(info_flags::numeric)) {
+ return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_val));
+ }
+ }
+ }
+ // multiple polylogs
+ if (is_a<lst>(m_) && is_a<lst>(x_)) {
+
const lst& m = ex_to<lst>(m_);
const lst& x = ex_to<lst>(x_);
- lst::const_iterator itm = m.begin();
- lst::const_iterator itx = x.begin();
- for (; itm != m.end(); itm++, itx++) {
+ if (m.nops() != x.nops()) {
+ return Li(m_,x_).hold();
+ }
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
+ return Li(m_,x_).hold();
+ }
+
+ for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
if (!(*itm).info(info_flags::posint)) {
return Li(m_, x_).hold();
}
- if ((*itx != _ex1) && (*itx != _ex_1)) {
- if (itx != x.begin()) {
- ish = false;
- }
- iszeta = false;
+ if (!(*itx).info(info_flags::numeric)) {
+ return Li(m_, x_).hold();
}
if (*itx == _ex0) {
- iszero = true;
+ return _ex0;
}
- if (!(*itx).info(info_flags::numeric)) {
- doevalfveto = false;
+ }
+
+ return mLi_numeric(m, x);
+ }
+
+ return Li(m_,x_).hold();
+}
+
+
+static ex Li_eval(const ex& m_, const ex& x_)
+{
+ if (is_a<lst>(m_)) {
+ if (is_a<lst>(x_)) {
+ // multiple polylogs
+ const lst& m = ex_to<lst>(m_);
+ const lst& x = ex_to<lst>(x_);
+ if (m.nops() != x.nops()) {
+ return Li(m_,x_).hold();
}
- if (!(*itx).info(info_flags::crational)) {
- doevalf = true;
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ bool is_H = true;
+ bool is_zeta = true;
+ bool do_evalf = true;
+ bool crational = true;
+ for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+ if (!(*itm).info(info_flags::posint)) {
+ return Li(m_,x_).hold();
+ }
+ if ((*itx != _ex1) && (*itx != _ex_1)) {
+ if (itx != x.begin()) {
+ is_H = false;
+ }
+ is_zeta = false;
+ }
+ if (*itx == _ex0) {
+ return _ex0;
+ }
+ if (!(*itx).info(info_flags::numeric)) {
+ do_evalf = false;
+ }
+ if (!(*itx).info(info_flags::crational)) {
+ crational = false;
+ }
+ }
+ if (is_zeta) {
+ return zeta(m_,x_);
+ }
+ if (is_H) {
+ ex prefactor;
+ lst newm = convert_parameter_Li_to_H(m, x, prefactor);
+ return prefactor * H(newm, x[0]);
+ }
+ if (do_evalf && !crational) {
+ return mLi_numeric(m,x);
}
}
- if (iszeta) {
- return zeta(m_, x_);
- }
- if (iszero) {
- return _ex0;
- }
- if (ish) {
- ex pf;
- lst newm = convert_parameter_Li_to_H(m, x, pf);
- return pf * H(newm, x[0]);
+ return Li(m_, x_).hold();
+ } else if (is_a<lst>(x_)) {
+ return Li(m_, x_).hold();
+ }
+
+ // classical polylogs
+ if (x_ == _ex0) {
+ return _ex0;
+ }
+ if (x_ == _ex1) {
+ return zeta(m_);
+ }
+ if (x_ == _ex_1) {
+ return (pow(2,1-m_)-1) * zeta(m_);
+ }
+ if (m_ == _ex1) {
+ return -log(1-x_);
+ }
+ if (m_ == _ex2) {
+ if (x_.is_equal(I)) {
+ return power(Pi,_ex2)/_ex_48 + Catalan*I;
}
- if (doevalfveto && doevalf) {
- return Li(m_, x_).evalf();
+ if (x_.is_equal(-I)) {
+ return power(Pi,_ex2)/_ex_48 - Catalan*I;
}
}
+ if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
+ return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_));
+ }
+
return Li(m_, x_).hold();
}
static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
{
- epvector seq;
- seq.push_back(expair(Li(m, x), 0));
- return pseries(rel, seq);
+ if (is_a<lst>(m) || is_a<lst>(x)) {
+ // multiple polylog
+ epvector seq;
+ seq.push_back(expair(Li(m, x), 0));
+ return pseries(rel, seq);
+ }
+
+ // classical polylog
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
+ // First special case: x==0 (derivatives have poles)
+ if (x_pt.is_zero()) {
+ const symbol s;
+ ex ser;
+ // manually construct the primitive expansion
+ for (int i=1; i<order; ++i)
+ ser += pow(s,i) / pow(numeric(i), m);
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+ // maybe that was terminating, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1), order));
+ ser += pseries(rel, nseq);
+ // reexpanding it will collapse the series again
+ return ser.series(rel, order);
+ }
+ // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
+ throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
+ }
+ // all other cases should be safe, by now:
+ throw do_taylor(); // caught by function::series()
}
static ex S_evalf(const ex& n, const ex& p, const ex& x)
{
- if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a<numeric>(x)) {
- return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
+ if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
+ if (is_a<numeric>(x)) {
+ return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
+ } else {
+ ex x_val = x.evalf();
+ if (is_a<numeric>(x_val)) {
+ return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x_val));
+ }
+ }
}
return S(n, p, x).hold();
}
static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
{
- epvector seq;
- seq.push_back(expair(S(n, p, x), 0));
- return pseries(rel, seq);
+ if (p == _ex1) {
+ return Li(n+1, x).series(rel, order, options);
+ }
+
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
+ // First special case: x==0 (derivatives have poles)
+ if (x_pt.is_zero()) {
+ const symbol s;
+ ex ser;
+ // manually construct the primitive expansion
+ // subsum = Euler-Zagier-Sum is needed
+ // dirty hack (slow ...) calculation of subsum:
+ std::vector<ex> presubsum, subsum;
+ subsum.push_back(0);
+ for (int i=1; i<order-1; ++i) {
+ subsum.push_back(subsum[i-1] + numeric(1, i));
+ }
+ for (int depth=2; depth<p; ++depth) {
+ presubsum = subsum;
+ for (int i=1; i<order-1; ++i) {
+ subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
+ }
+ }
+
+ for (int i=1; i<order; ++i) {
+ ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
+ }
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+ // maybe that was terminating, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1), order));
+ ser += pseries(rel, nseq);
+ // reexpanding it will collapse the series again
+ return ser.series(rel, order);
+ }
+ // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
+ throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
+ }
+ // all other cases should be safe, by now:
+ throw do_taylor(); // caught by function::series()
}
lst res;
lst::const_iterator itm = m.begin();
lst::const_iterator itx = ++x.begin();
- ex signum = _ex1;
+ int signum = 1;
pf = _ex1;
res.append(*itm);
itm++;
while (itx != x.end()) {
- signum *= *itx;
+ signum *= (*itx > 0) ? 1 : -1;
pf *= signum;
res.append((*itm) * signum);
itm++;
}
+// do integration [ReV] (49)
+// put parameter 1 in front of existing parameters
+ex trafo_H_prepend_one(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (int i=0; i<e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(1);
+ return e.subs(h == H(newparameter, h.op(1)).hold());
+ } else {
+ return e * H(lst(1),1-arg).hold();
+ }
+}
+
+
// do integration [ReV] (55)
// put parameter -1 in front of existing parameters
ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
}
+// do x -> 1-x transformation
+struct map_trafo_H_1mx : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+
+ lst parameter = ex_to<lst>(e.op(0));
+ ex arg = e.op(1);
+
+ // special cases if all parameters are either 0, 1 or -1
+ bool allthesame = true;
+ if (parameter.op(0) == 0) {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 0) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ lst newparameter;
+ for (int i=parameter.nops(); i>0; i--) {
+ newparameter.append(0);
+ }
+ return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
+ }
+ } else if (parameter.op(0) == -1) {
+ throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
+ } else {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ lst newparameter;
+ for (int i=parameter.nops(); i>0; i--) {
+ newparameter.append(1);
+ }
+ return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
+ }
+ }
+
+ lst newparameter = parameter;
+ newparameter.remove_first();
+
+ if (parameter.op(0) == 0) {
+
+ // leading zero
+ ex res = convert_H_to_zeta(parameter);
+ //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
+ map_trafo_H_1mx recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (int i=0; i<buffer.nops(); i++) {
+ res -= trafo_H_prepend_one(buffer.op(i), arg);
+ }
+ } else {
+ res -= trafo_H_prepend_one(buffer, arg);
+ }
+ return res;
+
+ } else {
+
+ // leading one
+ map_trafo_H_1mx recursion;
+ map_trafo_H_mult unify;
+ ex res;
+ int firstzero = 0;
+ while (parameter.op(firstzero) == 1) {
+ firstzero++;
+ }
+ for (int i=firstzero-1; i<parameter.nops()-1; i++) {
+ lst newparameter;
+ int j=0;
+ for (; j<=i; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ newparameter.append(1);
+ for (; j<parameter.nops()-1; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ res -= H(newparameter, arg).hold();
+ }
+ return (unify((-H(lst(0), 1-arg).hold() * recursion(H(newparameter, arg).hold())).expand()) +
+ recursion(res)) / firstzero;
+
+ }
+
+ }
+ }
+ return e;
+ }
+};
+
+
// do x -> 1/x transformation
struct map_trafo_H_1overx : public map_function
{
static ex H_evalf(const ex& x1, const ex& x2)
{
- if (is_a<lst>(x1) && is_a<numeric>(x2)) {
+ if (is_a<lst>(x1)) {
+
+ cln::cl_N x;
+ if (is_a<numeric>(x2)) {
+ x = ex_to<numeric>(x2).to_cl_N();
+ } else {
+ ex x2_val = x2.evalf();
+ if (is_a<numeric>(x2_val)) {
+ x = ex_to<numeric>(x2_val).to_cl_N();
+ }
+ }
+
for (int i=0; i<x1.nops(); i++) {
if (!x1.op(i).info(info_flags::integer)) {
- return H(x1,x2).hold();
+ return H(x1, x2).hold();
}
}
if (x1.nops() < 1) {
- return H(x1,x2).hold();
+ return H(x1, x2).hold();
}
- cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
-
const lst& morg = ex_to<lst>(x1);
// remove trailing zeros ...
if (*(--morg.end()) == 0) {
return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
}
// ... and expand parameter notation
+ bool has_minus_one = false;
lst m;
for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
if (*it > 1) {
m.append(0);
}
m.append(1);
- } else if (*it < -1) {
+ } else if (*it <= -1) {
for (ex count=*it+1; count < 0; count++) {
m.append(0);
}
m.append(-1);
+ has_minus_one = true;
} else {
m.append(*it);
}
}
- // since the transformations produce a lot of terms, they are only efficient for
- // argument near one.
- // no transformation needed -> do summation
+ // do summation
if (cln::abs(x) < 0.95) {
lst m_lst;
lst s_lst;
}
}
+ symbol xtemp("xtemp");
ex res = 1;
// ensure that the realpart of the argument is positive
}
}
- // choose transformations
- symbol xtemp("xtemp");
- if (cln::abs(x-1) < 1.4142) {
- // x -> (1-x)/(1+x)
- map_trafo_H_1mxt1px trafo;
- res *= trafo(H(m, xtemp));
- } else {
- // x -> 1/x
+ // x -> 1/x
+ if (cln::abs(x) >= 2.0) {
map_trafo_H_1overx trafo;
res *= trafo(H(m, xtemp));
if (cln::imagpart(x) <= 0) {
} else {
res = res.subs(H_polesign == I*Pi);
}
+ return res.subs(xtemp == numeric(x)).evalf();
+ }
+
+ // check transformations for 0.95 <= |x| < 2.0
+
+ // |(1-x)/(1+x)| < 0.9 -> circular area with center=9,53+0i and radius=9.47
+ if (cln::abs(x-9.53) <= 9.47) {
+ // x -> (1-x)/(1+x)
+ map_trafo_H_1mxt1px trafo;
+ res *= trafo(H(m, xtemp));
+ } else {
+ // x -> 1-x
+ if (has_minus_one) {
+ map_trafo_H_convert_to_Li filter;
+ return filter(H(m, numeric(x)).hold()).evalf();
+ }
+ map_trafo_H_1mx trafo;
+ res *= trafo(H(m, xtemp));
}
-
- // simplify result
-// TODO
-// map_trafo_H_convert converter;
-// res = converter(res).expand();
-// lst ll;
-// res.find(H(wild(1),wild(2)), ll);
-// res.find(zeta(wild(1)), ll);
-// res.find(zeta(wild(1),wild(2)), ll);
-// res = res.collect(ll);
return res.subs(xtemp == numeric(x)).evalf();
}
if (y.is_zero()) {
return _ex_1_2;
}
- if (y.is_equal(_num1)) {
+ if (y.is_equal(*_num1_p)) {
return zeta(m).hold();
}
if (y.info(info_flags::posint)) {
if (y.info(info_flags::odd)) {
return zeta(m).hold();
} else {
- return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
+ return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
}
} else {
if (y.info(info_flags::odd)) {
- return -bernoulli(_num1-y) / (_num1-y);
+ return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
} else {
return _ex0;
}
git://www.ginac.de / ginac.git / blobdiff