X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_nstdsums.cpp;h=1fd80aefe2e453b269fcf34e403d9149669e2014;hp=6217d8b5b2901ace0eb1cd547781cdaf03d0f465;hb=db52ae4c832e9a9981ecf78ecc3c9e59f461d8e4;hpb=33265c53dabbd91612a6d5e0f9af30801073ca74 diff --git a/ginac/inifcns_nstdsums.cpp b/ginac/inifcns_nstdsums.cpp index 6217d8b5..1fd80aef 100644 --- a/ginac/inifcns_nstdsums.cpp +++ b/ginac/inifcns_nstdsums.cpp @@ -5,7 +5,8 @@ * 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)) @@ -17,36 +18,36 @@ * [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 speed up by using Bernoulli numbers and + * - 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) 1999-2003 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 @@ -60,9 +61,10 @@ * * 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 #include #include #include @@ -102,6 +104,9 @@ namespace { // lookup table for factors built from Bernoulli numbers // see fill_Xn() std::vector > Xn; +// initial size of Xn that should suffice for 32bit machines (must be even) +const int xninitsizestep = 26; +int xninitsize = xninitsizestep; int xnsize = 0; @@ -117,17 +122,14 @@ int xnsize = 0; // The second index in Xn corresponds to the index from the actual sum. void fill_Xn(int n) { - // rule of thumb. needs to be improved. TODO - const int initsize = Digits * 3 / 2; - if (n>1) { // calculate X_2 and higher (corresponding to Li_4 and higher) - std::vector buf(initsize); + std::vector buf(xninitsize); std::vector::iterator it = buf.begin(); cln::cl_N result; *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1 it++; - for (int i=2; i<=initsize; i++) { + for (int i=2; i<=xninitsize; i++) { if (i&1) { result = 0; // k == 0 } else { @@ -147,14 +149,14 @@ void fill_Xn(int n) Xn.push_back(buf); } else if (n==1) { // special case to handle the X_0 correct - std::vector buf(initsize); + std::vector buf(xninitsize); std::vector::iterator it = buf.begin(); cln::cl_N result; *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1 it++; *it = cln::cl_I(17)/cln::cl_I(36); // i == 2 it++; - for (int i=3; i<=initsize; i++) { + for (int i=3; i<=xninitsize; i++) { if (i & 1) { result = -Xn[0][(i-3)/2]/2; *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result; @@ -171,9 +173,9 @@ void fill_Xn(int n) Xn.push_back(buf); } else { // calculate X_0 - std::vector buf(initsize/2); + std::vector buf(xninitsize/2); std::vector::iterator it = buf.begin(); - for (int i=1; i<=initsize/2; i++) { + for (int i=1; i<=xninitsize/2; i++) { *it = bernoulli(i*2).to_cl_N(); it++; } @@ -183,13 +185,59 @@ void fill_Xn(int n) xnsize++; } +// doubles the number of entries in each Xn[] +void double_Xn() +{ + const int pos0 = xninitsize / 2; + // X_0 + for (int i=1; i<=xninitsizestep/2; ++i) { + Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N()); + } + if (Xn.size() > 1) { + int xend = xninitsize + xninitsizestep; + cln::cl_N result; + // X_1 + for (int i=xninitsize+1; i<=xend; ++i) { + if (i & 1) { + result = -Xn[0][(i-3)/2]/2; + Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result); + } else { + result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1); + for (int k=1; k 1)) ) { + result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1); + } + } + result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1 + result = result + Xn[n-1][i-1] / (i+1); // k == i + Xn[n].push_back(result); + } + } + } + xninitsize += xninitsizestep; +} + // calculates Li(2,x) without Xn cln::cl_N Li2_do_sum(const cln::cl_N& x) { cln::cl_N res = x; cln::cl_N resbuf; - cln::cl_N num = x; + cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits)); cln::cl_I den = 1; // n^2 = 1 unsigned i = 3; do { @@ -207,17 +255,23 @@ cln::cl_N Li2_do_sum(const cln::cl_N& x) cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x) { std::vector::const_iterator it = Xn[0].begin(); + std::vector::const_iterator xend = Xn[0].end(); cln::cl_N u = -cln::log(1-x); - cln::cl_N factor = u; - cln::cl_N res = u - u*u/4; + cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits)); + 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; - it++; // should we check it? or rely on initsize? ... i++; + if (++it == xend) { + double_Xn(); + it = Xn[0].begin() + (i-1); + xend = Xn[0].end(); + } } while (res != resbuf); return res; } @@ -226,7 +280,7 @@ cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x) // calculates Li(n,x), n>2 without Xn cln::cl_N Lin_do_sum(int n, const cln::cl_N& x) { - cln::cl_N factor = x; + cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits)); cln::cl_N res = x; cln::cl_N resbuf; int i=2; @@ -244,8 +298,9 @@ cln::cl_N Lin_do_sum(int n, const cln::cl_N& x) cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x) { std::vector::const_iterator it = Xn[n-2].begin(); + std::vector::const_iterator xend = Xn[n-2].end(); cln::cl_N u = -cln::log(1-x); - cln::cl_N factor = u; + cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits)); cln::cl_N res = u; cln::cl_N resbuf; unsigned i=2; @@ -253,8 +308,12 @@ cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x) resbuf = res; factor = factor * u / i; res = res + (*it) * factor; - it++; // should we check it? or rely on initsize? ... i++; + if (++it == xend) { + double_Xn(); + it = Xn[n-2].begin() + (i-2); + xend = Xn[n-2].end(); + } } while (res != resbuf); return res; } @@ -322,7 +381,7 @@ cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& pr // 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 @@ -339,7 +398,16 @@ numeric Li_num(int n, const numeric& x) // [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(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& s, const std::vector& x) +// performs the actual series summation for multiple polylogarithms +cln::cl_N multipleLi_do_sum(const std::vector& s, const std::vector& x) +{ + const int j = s.size(); + + std::vector t(j); + cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); + + cln::cl_N t0buf; + 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--) { + 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); + + 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 m_int; + std::vector x_cln; + for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { + m_int.push_back(ex_to(*itm).to_int()); + x_cln.push_back(ex_to(*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 gsyms; + + +// type used by the transformation functions for G +typedef std::vector 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 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(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 sortmap; + int size = 0; + for (int i=0; i(abs(x[i]), i)); + ++size; + } + } + // include upper limit (scale) + sortmap.insert(std::pair(abs(y), x.nops())); + + // generate missing dummy-symbols + int i = 1; + gsyms.clear(); + gsyms.push_back(symbol("GSYMS_ERROR")); + ex lastentry; + for (std::multimap::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::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 + + +////////////////////////////////////////////////////////////////////// +// +// Generalized multiple polylogarithm G(x, y) and G(x, s, y) +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// + + +static ex G2_evalf(const ex& x_, const ex& y) +{ + if (!y.info(info_flags::positive)) { + return G(x_, y).hold(); + } + lst x = is_a(x_) ? ex_to(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(); + } + if (*it != _ex0) { + all_zero = false; + } + s.append(+1); + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + return G_numeric(x, s, y); +} + + +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(x_) ? ex_to(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); +} + + +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(G2_print_latex). + + +static ex G3_evalf(const ex& x_, const ex& s_, const ex& y) +{ + if (!y.info(info_flags::positive)) { + return G(x_, s_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst(x_); + lst s = is_a(s_) ? ex_to(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(); + } + if (!(*its).info(info_flags::real)) { + return G(x_, y).hold(); + } + if (*itx != _ex0) { + all_zero = false; + } + if (*its >= 0) { + sn.append(+1); + } else { + sn.append(-1); + } + } + 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) { - const int j = s.size(); - - std::vector t(j); - cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); + //TODO eval to MZV or H or S or Lin - cln::cl_N t0buf; - 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]); + if (!y.info(info_flags::positive)) { + return G(x_, s_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst(x_); + lst s = is_a(s_) ? ex_to(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(); } - // ... 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--) { - 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]); + if (!(*its).info(info_flags::real)) { + return G(x_, s_, y).hold(); } - } while (t[0] != t0buf); - - return t[0]; + if (!(*itx).info(info_flags::crational)) { + crational = false; + } + if (*itx != _ex0) { + all_zero = false; + } + if (*its >= 0) { + sn.append(+1); + } else { + sn.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_, s_, y).hold(); + } + return G_numeric(x, sn, y); } -// forward declaration for Li_eval() -lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf); - -} // end of anonymous namespace +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(G3_print_latex). ////////////////////////////////////////////////////////////////////// // -// Classical polylogarithm and multiple polylogarithm Li(n,x) +// Classical polylogarithm and multiple polylogarithm Li(m,x) // // GiNaC function // ////////////////////////////////////////////////////////////////////// -static ex Li_evalf(const ex& x1, const ex& x2) +static ex Li_evalf(const ex& m_, const ex& x_) { // classical polylogs - if (is_a(x1) && is_a(x2)) { - return Li_num(ex_to(x1).to_int(), ex_to(x2)); + if (m_.info(info_flags::posint)) { + if (x_.info(info_flags::numeric)) { + return Lin_numeric(ex_to(m_).to_int(), ex_to(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(m_).to_int(), ex_to(x_val)); + } + } } // multiple polylogs - else if (is_a(x1) && is_a(x2)) { - ex conv = 1; - for (int i=0; i(m_) && is_a(x_)) { + + const lst& m = ex_to(m_); + const lst& x = ex_to(x_); + 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 (!is_a(x2.op(i))) { - return Li(x1, x2).hold(); + if (!(*itx).info(info_flags::numeric)) { + return Li(m_, x_).hold(); } - conv *= x2.op(i); - if (conv >= 1) { - return Li(x1, x2).hold(); + if (*itx == _ex0) { + return _ex0; } } - std::vector m; - std::vector x; - for (int i=0; i(x1.nops()).to_int(); i++) { - m.push_back(ex_to(x1.op(i)).to_int()); - x.push_back(ex_to(x2.op(i)).to_cl_N()); - } - - return numeric(multipleLi_do_sum(m, x)); + return mLi_numeric(m, x); } - return Li(x1,x2).hold(); + return Li(m_,x_).hold(); } static ex Li_eval(const ex& m_, const ex& x_) { - if (m_.nops() < 2) { - ex m; - if (is_a(m_)) { - m = m_.op(0); - } else { - m = m_; - } - ex x; + if (is_a(m_)) { if (is_a(x_)) { - x = x_.op(0); - } else { - x = x_; - } - 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.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) { - return Li_num(ex_to(m).to_int(), ex_to(x)); - } - } else { - bool ish = true; - bool iszeta = true; - bool iszero = false; - bool doevalf = false; - bool doevalfveto = true; - const lst& m = ex_to(m_); - const lst& x = ex_to(x_); - lst::const_iterator itm = m.begin(); - lst::const_iterator itx = x.begin(); - for (; itm != m.end(); itm++, itx++) { - if (!(*itm).info(info_flags::posint)) { - return Li(m_, x_).hold(); + // multiple polylogs + const lst& m = ex_to(m_); + const lst& x = ex_to(x_); + if (m.nops() != x.nops()) { + return Li(m_,x_).hold(); } - if ((*itx != _ex1) && (*itx != _ex_1)) { - if (itx != x.begin()) { - ish = false; + 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; } - iszeta = false; } - if (*itx == _ex0) { - iszero = true; + if (is_zeta) { + return zeta(m_,x_); } - if (!(*itx).info(info_flags::numeric)) { - doevalfveto = false; + if (is_H) { + ex prefactor; + lst newm = convert_parameter_Li_to_H(m, x, prefactor); + return prefactor * H(newm, x[0]); } - if (!(*itx).info(info_flags::crational)) { - doevalf = true; + 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(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(m_).to_int(), ex_to(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(m) || is_a(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=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() } @@ -846,7 +1817,9 @@ cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& } // should be done otherwise - cln::cl_N xf = x * cln::cl_float(1, prec); + cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); + cln::cl_N xf = x * one; + //cln::cl_N xf = x * cln::cl_float(1, prec); cln::cl_N res; cln::cl_N resbuf; @@ -937,9 +1910,8 @@ numeric S_num(int n, int p, const numeric& x) else if (!x.imag().is_rational()) prec = cln::float_format(cln::the(cln::imagpart(value))); - // [Kol] (5.3) - if (cln::realpart(value) < -0.5) { + if ((cln::realpart(value) < -0.5) || (n == 0)) { cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n) * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p); @@ -1000,8 +1972,15 @@ numeric S_num(int n, int p, const numeric& x) 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(x)) { - return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(x)); + if (n.info(info_flags::posint) && p.info(info_flags::posint)) { + if (is_a(x)) { + return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(x)); + } else { + ex x_val = x.evalf(); + if (is_a(x_val)) { + return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(x_val)); + } + } } return S(n, p, x).hold(); } @@ -1027,15 +2006,58 @@ static ex S_eval(const ex& n, const ex& p, const ex& x) return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(x)); } } + if (n.is_zero()) { + // [Kol] (5.3) + return pow(-log(1-x), p) / factorial(p); + } 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 presubsum, subsum; + subsum.push_back(0); + for (int i=1; i=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() } @@ -1086,6 +2108,10 @@ REGISTER_FUNCTION(S, // anonymous namespace for helper functions namespace { + +// regulates the pole (used by 1/x-transformation) +symbol H_polesign("IMSIGN"); + // convert parameters from H to Li representation // parameters are expected to be in expanded form, i.e. only 0, 1 and -1 @@ -1141,10 +2167,6 @@ bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf) } } } - for (; acc > 1; acc--) { - throw std::runtime_error("ERROR!"); - m.append(0); - } return has_negative_parameters; } @@ -1312,12 +2334,12 @@ lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf) 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++; @@ -1459,6 +2481,37 @@ ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg) } +// 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(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (int i=0; i(e.op(i))) { + std::string name = ex_to(e.op(i)).get_name(); + if (name == "H") { + h = e.op(i); + } + } + } + } + if (h != 0) { + lst newparameter = ex_to(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) @@ -1554,6 +2607,109 @@ ex trafo_H_1mxt1px_prepend_one(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(e) || is_a(e)) { + return e.map(*this); + } + + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + + lst parameter = ex_to(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; i0; 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; i0; 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(buffer)) { + for (int i=0; i 1/x transformation struct map_trafo_H_1overx : public map_function { @@ -1603,7 +2759,7 @@ struct map_trafo_H_1overx : public map_function } if (allthesame) { map_trafo_H_mult unify; - return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, parameter.nops()) + return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops()) / factorial(parameter.nops())).expand()); } } @@ -1838,18 +2994,27 @@ cln::cl_N H_do_sum(const std::vector& m, const cln::cl_N& x) static ex H_evalf(const ex& x1, const ex& x2) { - if (is_a(x1) && is_a(x2)) { + if (is_a(x1)) { + + cln::cl_N x; + if (is_a(x2)) { + x = ex_to(x2).to_cl_N(); + } else { + ex x2_val = x2.evalf(); + if (is_a(x2_val)) { + x = ex_to(x2_val).to_cl_N(); + } + } + for (int i=0; i(x2).to_cl_N(); - const lst& morg = ex_to(x1); // remove trailing zeros ... if (*(--morg.end()) == 0) { @@ -1858,6 +3023,7 @@ static ex H_evalf(const ex& x1, const ex& x2) 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) { @@ -1865,19 +3031,18 @@ static ex H_evalf(const ex& x1, const ex& x2) 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; @@ -1907,6 +3072,7 @@ static ex H_evalf(const ex& x1, const ex& x2) } } + symbol xtemp("xtemp"); ex res = 1; // ensure that the realpart of the argument is positive @@ -1920,28 +3086,35 @@ static ex H_evalf(const ex& x1, const ex& x2) } } - // choose transformations - symbol xtemp("xtemp"); - if (cln::abs(x-1) < 1.4142) { + // x -> 1/x + if (cln::abs(x) >= 2.0) { + map_trafo_H_1overx trafo; + res *= trafo(H(m, xtemp)); + if (cln::imagpart(x) <= 0) { + res = res.subs(H_polesign == -I*Pi); + } 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 - map_trafo_H_1overx trafo; + // 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(); } @@ -2141,9 +3314,9 @@ ex convert_H_to_Li(const ex& m, const ex& x) map_trafo_H_reduce_trailing_zeros filter; map_trafo_H_convert_to_Li filter2; if (is_a(m)) { - return filter2(filter(H(m, x).hold())).eval(); + return filter2(filter(H(m, x).hold())); } else { - return filter2(filter(H(lst(m), x).hold())).eval(); + return filter2(filter(H(lst(m), x).hold())); } } @@ -2568,18 +3741,18 @@ static ex zeta1_eval(const ex& m) 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; } @@ -2625,7 +3798,7 @@ static void zeta1_print_latex(const ex& m_, const print_context& c) } -unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta"). +unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1). evalf_func(zeta1_evalf). eval_func(zeta1_eval). derivative_func(zeta1_deriv). @@ -2762,7 +3935,7 @@ static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c } -unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta"). +unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2). evalf_func(zeta2_evalf). eval_func(zeta2_eval). derivative_func(zeta2_deriv).