--- /dev/null
+/** @file inifcns_nstdsums.cpp
+ *
+ * Implementation of some special functions that have a representation as nested sums.
+ * The functions are:
+ * classical polylogarithm Li(n,x)
+ * multiple polylogarithm Li(lst(n_1,...,n_k),lst(x_1,...,x_k)
+ * nielsen's generalized polylogarithm S(n,p,x)
+ * harmonic polylogarithm H(lst(m_1,...,m_k),x)
+ * multiple zeta value mZeta(lst(m_1,...,m_k))
+ *
+ * Some remarks:
+ * - All formulae used can be looked up in the following publication:
+ * Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
+ * This document will be referenced as [Kol] throughout this source code.
+ * - Classical polylogarithms (Li) and nielsen's generalized polylogarithms (S) can be numerically
+ * evaluated in the whole complex plane except for S(n,p,-1) when p is not unit (no formula yet
+ * to tackle these points). And of course, there is still room for speed optimizations ;-).
+ * - The remaining functions can only be numerically evaluated with arguments lying in the unit sphere
+ * at the moment. Sorry. The evaluation especially for mZeta is very slow ... better not use it
+ * right now.
+ * - The functions have no series expansion. To do it, 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.
+ *
+ */
+
+/*
+ * GiNaC Copyright (C) 1999-2003 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
+ * the Free Software Foundation; either version 2 of the License, or
+ * (at your option) any later version.
+ *
+ * This program is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * 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
+ */
+
+#include <stdexcept>
+#include <vector>
+#include <cln/cln.h>
+
+#include "inifcns.h"
+#include "lst.h"
+#include "numeric.h"
+#include "operators.h"
+#include "relational.h"
+
+
+namespace GiNaC {
+
+
+//////////////////////
+// helper functions //
+//////////////////////
+
+
+// helper function for classical polylog Li
+static cln::cl_N Li_series(int n, const cln::cl_N& x, const cln::float_format_t& prec)
+{
+ // Note: argument must be in the unit circle
+ cln::cl_N aug, acc;
+ cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
+ cln::cl_N den = 0;
+ int i = 1;
+ do {
+ num = num * x;
+ cln::cl_R ii = i;
+ den = cln::expt(ii, n);
+ i++;
+ aug = num / den;
+ acc = acc + aug;
+ } while (acc != acc+aug);
+ return acc;
+}
+
+
+// helper function for classical polylog Li
+static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
+{
+ return Li_series(n, x, prec);
+}
+
+
+// helper function for classical polylog Li
+static numeric Li_num(int n, const numeric& x)
+{
+ if (n == 1) {
+ // just a log
+ return -cln::log(1-x.to_cl_N());
+ }
+ if (x.is_zero()) {
+ return 0;
+ }
+ if (x == 1) {
+ // [Kol] (2.22)
+ return cln::zeta(n);
+ }
+ else if (x == -1) {
+ // [Kol] (2.22)
+ return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
+ }
+
+ // what is the desired float format?
+ // first guess: default format
+ cln::float_format_t prec = cln::default_float_format;
+ const cln::cl_N value = x.to_cl_N();
+ // second guess: the argument's format
+ if (!x.real().is_rational())
+ prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
+ else if (!x.imag().is_rational())
+ prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
+
+ // [Kol] (5.15)
+ if (cln::abs(value) > 1) {
+ cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
+ // check if argument is complex. if it is real, the new polylog has to be conjugated.
+ if (cln::zerop(cln::imagpart(value))) {
+ if (n & 1) {
+ result = result + conjugate(Li_projection(n, cln::recip(value), prec));
+ }
+ else {
+ result = result - conjugate(Li_projection(n, cln::recip(value), prec));
+ }
+ }
+ else {
+ if (n & 1) {
+ result = result + Li_projection(n, cln::recip(value), prec);
+ }
+ else {
+ result = result - Li_projection(n, cln::recip(value), prec);
+ }
+ }
+ 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);
+ }
+ result = result - add;
+ return result;
+ }
+ else {
+ return Li_projection(n, value, prec);
+ }
+}
+
+
+// helper function for S(n,p,x)
+static cln::cl_N numeric_nielsen(int n, int step)
+{
+ if (step) {
+ cln::cl_N res;
+ for (int i=1; i<n; i++) {
+ res = res + numeric_nielsen(i, step-1) / cln::cl_I(i);
+ }
+ return res;
+ }
+ else {
+ return 1;
+ }
+}
+
+
+// forward declaration needed by function C below
+static numeric S_num(int n, int p, const numeric& x);
+
+
+// helper function for S(n,p,x)
+// [Kol] (7.2)
+static cln::cl_N C(int n, int p)
+{
+ cln::cl_N result;
+
+ for (int k=0; k<p; k++) {
+ for (int j=0; j<=(n+k-1)/2; j++) {
+ if (k == 0) {
+ if (n & 1) {
+ if (j & 1) {
+ result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
+ }
+ else {
+ result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
+ }
+ }
+ }
+ else {
+ if (k & 1) {
+ if (j & 1) {
+ result = result + cln::factorial(n+k-1)
+ * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ else {
+ result = result - cln::factorial(n+k-1)
+ * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ }
+ else {
+ if (j & 1) {
+ result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ else {
+ result = result + cln::factorial(n+k-1)
+ * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ }
+ }
+ }
+ }
+ int np = n+p;
+ if ((np-1) & 1) {
+ if (((np)/2+n) & 1) {
+ result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
+ }
+ else {
+ result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
+ }
+ }
+
+ return result;
+}
+
+
+// helper function for S(n,p,x)
+// [Kol] remark to (9.1)
+static cln::cl_N a_k(int k)
+{
+ cln::cl_N result;
+
+ if (k == 0) {
+ return 1;
+ }
+
+ result = result;
+ for (int m=2; m<=k; m++) {
+ result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
+ }
+
+ return -result / k;
+}
+
+
+// helper function for S(n,p,x)
+// [Kol] remark to (9.1)
+static cln::cl_N b_k(int k)
+{
+ cln::cl_N result;
+
+ if (k == 0) {
+ return 1;
+ }
+
+ result = result;
+ for (int m=2; m<=k; m++) {
+ result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
+ }
+
+ return result / k;
+}
+
+
+// helper function for S(n,p,x)
+static cln::cl_N S_series(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+{
+ n++;
+ int i = p;
+ p--;
+ cln::cl_N aug, acc;
+ cln::cl_N num = cln::expt(x,p);
+ cln::cl_N converter = cln::complex(cln::cl_float(1, prec), 0);
+ cln::cl_N den = 0;
+ do {
+ num = num * x;
+ den = cln::expt(cln::cl_I(i), n);
+ aug = num / den * numeric_nielsen(i, p);
+ i++;
+ acc = acc + aug;
+ } while (acc != acc+aug);
+
+ return acc;
+}
+
+
+// helper function for S(n,p,x)
+static cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+{
+ // [Kol] (5.3)
+ if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
+
+ cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
+ * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
+
+ for (int s=0; s<n; s++) {
+ cln::cl_N res2;
+ for (int r=0; r<p; r++) {
+ res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
+ * S_series(p-r,n-s,1-x,prec) / cln::factorial(r);
+ }
+ result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
+ }
+
+ return result;
+ }
+
+ return S_series(n, p, x, prec);
+}
+
+
+// helper function for S(n,p,x)
+static numeric S_num(int n, int p, const numeric& x)
+{
+ if (x == 1) {
+ if (n == 1) {
+ // [Kol] (2.22) with (2.21)
+ return cln::zeta(p+1);
+ }
+
+ if (p == 1) {
+ // [Kol] (2.22)
+ return cln::zeta(n+1);
+ }
+
+ // [Kol] (9.1)
+ cln::cl_N result;
+ for (int nu=0; nu<n; nu++) {
+ for (int rho=0; rho<=p; rho++) {
+ result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
+ * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
+ }
+ }
+ result = result * cln::expt(cln::cl_I(-1),n+p-1);
+
+ return result;
+ }
+ else if (x == -1) {
+ // [Kol] (2.22)
+ if (p == 1) {
+ return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
+ }
+ throw std::runtime_error("don't know how to evaluate this function!");
+ }
+
+ // what is the desired float format?
+ // first guess: default format
+ cln::float_format_t prec = cln::default_float_format;
+ const cln::cl_N value = x.to_cl_N();
+ // second guess: the argument's format
+ if (!x.real().is_rational())
+ prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
+ else if (!x.imag().is_rational())
+ prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
+
+
+ // [Kol] (5.3)
+ if (cln::realpart(value) < -0.5) {
+
+ 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);
+
+ for (int s=0; s<n; s++) {
+ cln::cl_N res2;
+ for (int r=0; r<p; r++) {
+ res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
+ * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
+ }
+ result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
+ }
+
+ return result;
+
+ }
+ // [Kol] (5.12)
+ else if (cln::abs(value) > 1) {
+
+ cln::cl_N result;
+
+ for (int s=0; s<p; s++) {
+ for (int r=0; r<=s; r++) {
+ result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
+ / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
+ * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
+ }
+ }
+ result = result * cln::expt(cln::cl_I(-1),n);
+
+ cln::cl_N res2;
+ for (int r=0; r<n; r++) {
+ res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
+ }
+ res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
+
+ result = result + cln::expt(cln::cl_I(-1),p) * res2;
+
+ return result;
+ }
+ else {
+ return S_projection(n, p, value, prec);
+ }
+}
+
+
+// helper function for multiple polylogarithm
+static cln::cl_N numeric_zsum(int n, std::vector<cln::cl_N>& x, std::vector<cln::cl_N>& m)
+{
+ cln::cl_N res;
+ if (x.empty()) {
+ return 1;
+ }
+ for (int i=1; i<n; i++) {
+ std::vector<cln::cl_N>::iterator be;
+ std::vector<cln::cl_N>::iterator en;
+ be = x.begin();
+ be++;
+ en = x.end();
+ std::vector<cln::cl_N> xbuf(be, en);
+ be = m.begin();
+ be++;
+ en = m.end();
+ std::vector<cln::cl_N> mbuf(be, en);
+ res = res + cln::expt(x[0],i) / cln::expt(i,m[0]) * numeric_zsum(i, xbuf, mbuf);
+ }
+ return res;
+}
+
+
+// helper function for harmonic polylogarithm
+static cln::cl_N numeric_harmonic(int n, std::vector<cln::cl_N>& m)
+{
+ cln::cl_N res;
+ if (m.empty()) {
+ return 1;
+ }
+ for (int i=1; i<n; i++) {
+ std::vector<cln::cl_N>::iterator be;
+ std::vector<cln::cl_N>::iterator en;
+ be = m.begin();
+ be++;
+ en = m.end();
+ std::vector<cln::cl_N> mbuf(be, en);
+ res = res + cln::recip(cln::expt(i,m[0])) * numeric_harmonic(i, mbuf);
+ }
+ return res;
+}
+
+
+/////////////////////////////
+// end of helper functions //
+/////////////////////////////
+
+
+// Polylogarithm and multiple polylogarithm
+
+static ex Li_eval(const ex& x1, const ex& x2)
+{
+ if (x2.is_zero()) {
+ return 0;
+ }
+ else {
+ return Li(x1,x2).hold();
+ }
+}
+
+static ex Li_evalf(const ex& x1, const ex& x2)
+{
+ // classical polylogs
+ if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
+ return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
+ }
+ // multiple polylogs
+ else if (is_a<lst>(x1) && is_a<lst>(x2)) {
+ for (int i=0; i<x1.nops(); i++) {
+ if (!is_a<numeric>(x1.op(i)))
+ return Li(x1,x2).hold();
+ if (!is_a<numeric>(x2.op(i)))
+ return Li(x1,x2).hold();
+ if (x2 >= 1)
+ return Li(x1,x2).hold();
+ }
+
+ cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).to_cl_N();
+ cln::cl_N x_1 = ex_to<numeric>(x2.op(x2.nops()-1)).to_cl_N();
+ std::vector<cln::cl_N> x;
+ std::vector<cln::cl_N> m;
+ const int nops = ex_to<numeric>(x1.nops()).to_int();
+ for (int i=nops-2; i>=0; i--) {
+ m.push_back(ex_to<numeric>(x1.op(i)).to_cl_N());
+ x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
+ }
+
+ cln::cl_N res;
+ cln::cl_N resbuf;
+ for (int i=nops; true; i++) {
+ resbuf = res;
+ res = res + cln::expt(x_1,i) / cln::expt(i,m_1) * numeric_zsum(i, x, m);
+ if (cln::zerop(res-resbuf))
+ break;
+ }
+
+ return numeric(res);
+
+ }
+
+ return Li(x1,x2).hold();
+}
+
+REGISTER_FUNCTION(Li, eval_func(Li_eval).evalf_func(Li_evalf).do_not_evalf_params());
+
+
+// Nielsen's generalized polylogarithm
+
+static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
+{
+ if (x2 == 1) {
+ return Li(x1+1,x3);
+ }
+ return S(x1,x2,x3).hold();
+}
+
+static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
+{
+ if (is_a<numeric>(x1) && is_a<numeric>(x2) && is_a<numeric>(x3)) {
+ if ((x3 == -1) && (x2 != 1)) {
+ // no formula to evaluate this ... sorry
+ return S(x1,x2,x3).hold();
+ }
+ return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
+ }
+ return S(x1,x2,x3).hold();
+}
+
+REGISTER_FUNCTION(S, eval_func(S_eval).evalf_func(S_evalf).do_not_evalf_params());
+
+
+// Harmonic polylogarithm
+
+static ex H_eval(const ex& x1, const ex& x2)
+{
+ return H(x1,x2).hold();
+}
+
+static ex H_evalf(const ex& x1, const ex& x2)
+{
+ if (is_a<lst>(x1) && is_a<numeric>(x2)) {
+ for (int i=0; i<x1.nops(); i++) {
+ if (!is_a<numeric>(x1.op(i)))
+ return H(x1,x2).hold();
+ }
+
+ cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).to_cl_N();
+ cln::cl_N x_1 = ex_to<numeric>(x2).to_cl_N();
+ std::vector<cln::cl_N> m;
+ const int nops = ex_to<numeric>(x1.nops()).to_int();
+ for (int i=nops-2; i>=0; i--) {
+ m.push_back(ex_to<numeric>(x1.op(i)).to_cl_N());
+ }
+
+ cln::cl_N res;
+ cln::cl_N resbuf;
+ for (int i=nops; true; i++) {
+ resbuf = res;
+ res = res + cln::expt(x_1,i) / cln::expt(i,m_1) * numeric_harmonic(i, m);
+ if (cln::zerop(res-resbuf))
+ break;
+ }
+
+ return numeric(res);
+
+ }
+
+ return H(x1,x2).hold();
+}
+
+REGISTER_FUNCTION(H, eval_func(H_eval).evalf_func(H_evalf).do_not_evalf_params());
+
+
+// Multiple zeta value
+
+static ex mZeta_eval(const ex& x1)
+{
+ return mZeta(x1).hold();
+}
+
+static ex mZeta_evalf(const ex& x1)
+{
+ if (is_a<lst>(x1)) {
+ for (int i=0; i<x1.nops(); i++) {
+ if (!is_a<numeric>(x1.op(i)))
+ return mZeta(x1).hold();
+ }
+
+ cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).to_cl_N();
+ std::vector<cln::cl_N> m;
+ const int nops = ex_to<numeric>(x1.nops()).to_int();
+ for (int i=nops-2; i>=0; i--) {
+ m.push_back(ex_to<numeric>(x1.op(i)).to_cl_N());
+ }
+
+ cln::float_format_t prec = cln::default_float_format;
+ cln::cl_N res = cln::complex(cln::cl_float(0, prec), 0);
+ cln::cl_N resbuf;
+ for (int i=nops; true; i++) {
+ // to infinity and beyond ... timewise
+ resbuf = res;
+ res = res + cln::recip(cln::expt(i,m_1)) * numeric_harmonic(i, m);
+ if (cln::zerop(res-resbuf))
+ break;
+ }
+
+ return numeric(res);
+
+ }
+
+ return mZeta(x1).hold();
+}
+
+REGISTER_FUNCTION(mZeta, eval_func(mZeta_eval).evalf_func(mZeta_evalf).do_not_evalf_params());
+
+
+} // namespace GiNaC
+
projects / ginac.git / commitdiff