/** @file inifcns_nstdsums.cpp
*
* Implementation of some special functions that have a representation as nested sums.
- *
- * The functions are:
+ *
+ * The functions are:
* classical polylogarithm Li(n,x)
- * multiple polylogarithm Li(lst(n_1,...,n_k),lst(x_1,...,x_k))
+ * multiple polylogarithm Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
* nielsen's generalized polylogarithm S(n,p,x)
- * harmonic polylogarithm H(n,x) or H(lst(n_1,...,n_k),x)
- * multiple zeta value zeta(n) or zeta(lst(n_1,...,n_k))
+ * 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))
*
* Some remarks:
- *
+ *
* - All formulae used can be looked up in the following publications:
* [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
- * [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
+ * [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
+ *
+ * - 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.
*
- * - The order of parameters and arguments of H, Li and zeta is defined according to their order in the
- * nested sums representation.
- *
* - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
- * the whole complex plane. Multiple polylogarithms evaluate only if each argument x_i is smaller than
- * one. The parameters for every function (n, p or n_i) must be positive integers.
- *
+ * 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.
+ *
* - The calculation of classical polylogarithms is speed up by using Bernoulli numbers and
- * look-up tables. S uses look-up tables as well. The zeta function applies the algorithm in
- * [Cra] for speed up.
- *
- * - The functions have no series expansion as nested sums. To do it, you have to convert these functions
+ * look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
+ * [Cra] and [BBB] for speed up.
+ *
+ * - 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.
- *
+ * 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
//////////////////////////////////////////////////////////////////////
//
-// Classical polylogarithm Li
+// Classical polylogarithm Li(n,x)
//
// helper functions
//
{
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 {
{
std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
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 - u*u/4;
cln::cl_N resbuf;
unsigned i = 1;
// 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;
{
std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
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;
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);
+ * cln::expt(cln::log(x), j) / cln::factorial(j);
}
return result;
}
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);
+ * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
}
result = result - add;
return result;
//////////////////////////////////////////////////////////////////////
//
-// Multiple polylogarithm Li
+// Multiple polylogarithm Li(n,x)
//
// helper function
//
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];
}
+// forward declaration for Li_eval()
+lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
+
} // end of anonymous namespace
//////////////////////////////////////////////////////////////////////
//
-// Classical polylogarithm and multiple polylogarithm Li
+// Classical polylogarithm and multiple polylogarithm Li(n,x)
//
// GiNaC function
//
//////////////////////////////////////////////////////////////////////
-static ex Li_eval(const ex& x1, const ex& x2)
-{
- if (x2.is_zero()) {
- return _ex0;
- }
- else {
- if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational)))
- return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
- if (is_a<lst>(x2)) {
- for (int i=0; i<x2.nops(); i++) {
- if (!is_a<numeric>(x2.op(i))) {
- return Li(x1,x2).hold();
- }
- }
- return Li(x1,x2).evalf();
- }
- return Li(x1,x2).hold();
- }
-}
-
-
static ex Li_evalf(const ex& x1, const ex& x2)
{
// classical polylogs
}
// 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();
+ return Li(x1, x2).hold();
}
if (!is_a<numeric>(x2.op(i))) {
- return Li(x1,x2).hold();
+ return Li(x1, x2).hold();
}
- if (x2.op(i) >= 1) {
- return Li(x1,x2).hold();
+ conv *= x2.op(i);
+ if (abs(conv) >= 1) {
+ return Li(x1, x2).hold();
}
}
}
-static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
+static ex Li_eval(const ex& m_, const ex& x_)
+{
+ if (m_.nops() < 2) {
+ ex m;
+ if (is_a<lst>(m_)) {
+ m = m_.op(0);
+ } else {
+ m = m_;
+ }
+ ex x;
+ if (is_a<lst>(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<numeric>(m).to_int(), ex_to<numeric>(x));
+ }
+ } else {
+ bool ish = true;
+ bool iszeta = true;
+ bool iszero = false;
+ bool doevalf = false;
+ bool doevalfveto = true;
+ 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 (!(*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 == _ex0) {
+ iszero = true;
+ }
+ if (!(*itx).info(info_flags::numeric)) {
+ doevalfveto = false;
+ }
+ if (!(*itx).info(info_flags::crational)) {
+ doevalf = true;
+ }
+ }
+ 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]);
+ }
+ if (doevalfveto && doevalf) {
+ return Li(m_, x_).evalf();
+ }
+ }
+ 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(x1,x2), 0));
- return pseries(rel,seq);
+ seq.push_back(expair(Li(m, x), 0));
+ return pseries(rel, seq);
}
-static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
+static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param < 2);
if (deriv_param == 0) {
return _ex0;
}
- if (x1 > 0) {
- return Li(x1-1, x2) / x2;
+ if (m_.nops() > 1) {
+ throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
+ }
+ ex m;
+ if (is_a<lst>(m_)) {
+ m = m_.op(0);
+ } else {
+ m = m_;
+ }
+ ex x;
+ if (is_a<lst>(x_)) {
+ x = x_.op(0);
+ } else {
+ x = x_;
+ }
+ if (m > 0) {
+ return Li(m-1, x) / x;
} else {
- return 1/(1-x2);
+ return 1/(1-x);
}
}
+static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
+{
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ lst x;
+ if (is_a<lst>(x_)) {
+ x = ex_to<lst>(x_);
+ } else {
+ x = lst(x_);
+ }
+ c.s << "\\mbox{Li}_{";
+ lst::const_iterator itm = m.begin();
+ (*itm).print(c);
+ itm++;
+ for (; itm != m.end(); itm++) {
+ c.s << ",";
+ (*itm).print(c);
+ }
+ c.s << "}(";
+ lst::const_iterator itx = x.begin();
+ (*itx).print(c);
+ itx++;
+ for (; itx != x.end(); itx++) {
+ c.s << ",";
+ (*itx).print(c);
+ }
+ c.s << ")";
+}
+
+
REGISTER_FUNCTION(Li,
- eval_func(Li_eval).
- evalf_func(Li_evalf).
- do_not_evalf_params().
- series_func(Li_series).
- derivative_func(Li_deriv));
+ evalf_func(Li_evalf).
+ eval_func(Li_eval).
+ series_func(Li_series).
+ derivative_func(Li_deriv).
+ print_func<print_latex>(Li_print_latex).
+ do_not_evalf_params());
//////////////////////////////////////////////////////////////////////
//
-// Nielsen's generalized polylogarithm S
+// Nielsen's generalized polylogarithm S(n,p,x)
//
// helper functions
//
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));
+ * 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));
+ * 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));
+ / (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));
+ * 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));
}
}
}
}
// 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;
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);
+ * 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_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
+ * S_do_sum(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);
}
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);
+ * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
}
}
result = result * cln::expt(cln::cl_I(-1),n+p-1);
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);
+ * 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);
+ * 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);
}
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();
+ / 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);
//////////////////////////////////////////////////////////////////////
//
-// Nielsen's generalized polylogarithm S
+// Nielsen's generalized polylogarithm S(n,p,x)
//
// GiNaC function
//
//////////////////////////////////////////////////////////////////////
-static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
+static ex S_evalf(const ex& n, const ex& p, const ex& x)
{
- if (x2 == 1) {
- return Li(x1+1,x3);
+ 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 (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) &&
- x1.info(info_flags::posint) && x2.info(info_flags::posint)) {
- 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();
+ return S(n, p, x).hold();
}
-static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
+static ex S_eval(const ex& n, const ex& p, const ex& x)
{
- if (is_a<numeric>(x1) && is_a<numeric>(x2) && is_a<numeric>(x3)) {
- return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
+ if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
+ if (x == 0) {
+ return _ex0;
+ }
+ if (x == 1) {
+ lst m(n+1);
+ for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
+ m.append(1);
+ }
+ return zeta(m);
+ }
+ if (p == 1) {
+ return Li(n+1, x);
+ }
+ if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
+ return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
+ }
}
- return S(x1,x2,x3).hold();
+ return S(n, p, x).hold();
}
-static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options)
+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(x1,x2,x3), 0));
- return pseries(rel,seq);
+ seq.push_back(expair(S(n, p, x), 0));
+ return pseries(rel, seq);
}
-static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param)
+static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param < 3);
if (deriv_param < 2) {
return _ex0;
}
- if (x1 > 0) {
- return S(x1-1, x2, x3) / x3;
+ if (n > 0) {
+ return S(n-1, p, x) / x;
} else {
- return S(x1, x2-1, x3) / (1-x3);
+ return S(n, p-1, x) / (1-x);
}
}
+static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
+{
+ c.s << "\\mbox{S}_{";
+ n.print(c);
+ c.s << ",";
+ p.print(c);
+ c.s << "}(";
+ x.print(c);
+ c.s << ")";
+}
+
+
REGISTER_FUNCTION(S,
- eval_func(S_eval).
- evalf_func(S_evalf).
- do_not_evalf_params().
- series_func(S_series).
- derivative_func(S_deriv));
+ evalf_func(S_evalf).
+ eval_func(S_eval).
+ series_func(S_series).
+ derivative_func(S_deriv).
+ print_func<print_latex>(S_print_latex).
+ do_not_evalf_params());
//////////////////////////////////////////////////////////////////////
//
-// Harmonic polylogarithm H
+// Harmonic polylogarithm H(m,x)
//
-// helper function
+// helper functions
//
//////////////////////////////////////////////////////////////////////
namespace {
-// forward declaration
-ex convert_from_RV(const lst& parameterlst, const ex& arg);
+// convert parameters from H to Li representation
+// parameters are expected to be in expanded form, i.e. only 0, 1 and -1
+// returns true if some parameters are negative
+bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
+{
+ // expand parameter list
+ lst mexp;
+ for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
+ if (*it > 1) {
+ for (ex count=*it-1; count > 0; count--) {
+ mexp.append(0);
+ }
+ mexp.append(1);
+ } else if (*it < -1) {
+ for (ex count=*it+1; count < 0; count++) {
+ mexp.append(0);
+ }
+ mexp.append(-1);
+ } else {
+ mexp.append(*it);
+ }
+ }
+
+ ex signum = 1;
+ pf = 1;
+ bool has_negative_parameters = false;
+ ex acc = 1;
+ for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
+ if (*it == 0) {
+ acc++;
+ continue;
+ }
+ if (*it > 0) {
+ m.append((*it+acc-1) * signum);
+ } else {
+ m.append((*it-acc+1) * signum);
+ }
+ acc = 1;
+ signum = *it;
+ pf *= *it;
+ if (pf < 0) {
+ has_negative_parameters = true;
+ }
+ }
+ if (has_negative_parameters) {
+ for (int i=0; i<m.nops(); i++) {
+ if (m.op(i) < 0) {
+ m.let_op(i) = -m.op(i);
+ s.append(-1);
+ } else {
+ s.append(1);
+ }
+ }
+ }
+
+ return has_negative_parameters;
+}
+
+
+// recursivly transforms H to corresponding multiple polylogarithms
+struct map_trafo_H_convert_to_Li : 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;
+ if (is_a<lst>(e.op(0))) {
+ parameter = ex_to<lst>(e.op(0));
+ } else {
+ parameter = lst(e.op(0));
+ }
+ ex arg = e.op(1);
+
+ lst m;
+ lst s;
+ ex pf;
+ if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
+ s.let_op(0) = s.op(0) * arg;
+ return pf * Li(m, s).hold();
+ } else {
+ for (int i=0; i<m.nops(); i++) {
+ s.append(1);
+ }
+ s.let_op(0) = s.op(0) * arg;
+ return Li(m, s).hold();
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
+// recursivly transforms H to corresponding zetas
+struct map_trafo_H_convert_to_zeta : 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;
+ if (is_a<lst>(e.op(0))) {
+ parameter = ex_to<lst>(e.op(0));
+ } else {
+ parameter = lst(e.op(0));
+ }
+
+ lst m;
+ lst s;
+ ex pf;
+ if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
+ return pf * zeta(m, s);
+ } else {
+ return zeta(m);
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
+// remove trailing zeros from H-parameters
+struct map_trafo_H_reduce_trailing_zeros : 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;
+ if (is_a<lst>(e.op(0))) {
+ parameter = ex_to<lst>(e.op(0));
+ } else {
+ parameter = lst(e.op(0));
+ }
+ ex arg = e.op(1);
+ if (parameter.op(parameter.nops()-1) == 0) {
+
+ //
+ if (parameter.nops() == 1) {
+ return log(arg);
+ }
+
+ //
+ lst::const_iterator it = parameter.begin();
+ while ((it != parameter.end()) && (*it == 0)) {
+ it++;
+ }
+ if (it == parameter.end()) {
+ return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
+ }
+
+ //
+ parameter.remove_last();
+ int lastentry = parameter.nops();
+ while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
+ lastentry--;
+ }
+
+ //
+ ex result = log(arg) * H(parameter,arg).hold();
+ ex acc = 0;
+ for (ex i=0; i<lastentry; i++) {
+ if (parameter[i] > 0) {
+ parameter[i]++;
+ result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
+ parameter[i]--;
+ acc = 0;
+ } else if (parameter[i] < 0) {
+ parameter[i]--;
+ result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
+ parameter[i]++;
+ acc = 0;
+ } else {
+ acc++;
+ }
+ }
+
+ if (lastentry < parameter.nops()) {
+ result = result / (parameter.nops()-lastentry+1);
+ return result.map(*this);
+ } else {
+ return result;
+ }
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
+// returns an expression with zeta functions corresponding to the parameter list for H
+ex convert_H_to_zeta(const lst& m)
+{
+ symbol xtemp("xtemp");
+ map_trafo_H_reduce_trailing_zeros filter;
+ map_trafo_H_convert_to_zeta filter2;
+ return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
+}
+
+
+// convert signs form Li to H representation
+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;
+ pf = _ex1;
+ res.append(*itm);
+ itm++;
+ while (itx != x.end()) {
+ signum *= *itx;
+ pf *= signum;
+ res.append((*itm) * signum);
+ itm++;
+ itx++;
+ }
+ return res;
+}
// multiplies an one-dimensional H with another H
};
-// do integration [ReV] (49)
-// put parameter 1 in front of existing parameters
-ex trafo_H_prepend_one(const ex& e, const ex& arg)
+// do integration [ReV] (55)
+// put parameter 0 in front of existing parameters
+ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
{
ex h;
std::string name;
}
if (h != 0) {
lst newparameter = ex_to<lst>(h.op(0));
- newparameter.prepend(1);
- return e.subs(h == H(newparameter, h.op(1)).hold());
+ newparameter.prepend(0);
+ ex addzeta = convert_H_to_zeta(newparameter);
+ return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
} else {
- return e * H(lst(1),1-arg).hold();
+ return e * (-H(lst(0),1/arg).hold());
}
}
// do integration [ReV] (55)
-// put parameter 0 in front of existing parameters
-ex trafo_H_prepend_zero(const ex& e, const ex& arg)
+// put parameter -1 in front of existing parameters
+ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
{
ex h;
std::string name;
}
if (h != 0) {
lst newparameter = ex_to<lst>(h.op(0));
- newparameter.prepend(0);
- ex addzeta = convert_from_RV(newparameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
+ newparameter.prepend(-1);
+ ex addzeta = convert_H_to_zeta(newparameter);
return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
} else {
- return e * (-H(lst(0),1/arg).hold());
+ ex addzeta = convert_H_to_zeta(lst(-1));
+ return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
}
}
-// do x -> 1-x transformation
-struct map_trafo_H_1mx : public map_function
+// do integration [ReV] (55)
+// put parameter -1 in front of existing parameters
+ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
{
- ex operator()(const ex& e)
- {
- if (is_a<add>(e) || is_a<mul>(e)) {
- return e.map(*this);
+ 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 (is_a<function>(e)) {
- std::string name = ex_to<function>(e).get_name();
- if (name == "H") {
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(-1);
+ return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
+ } else {
+ return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
+ }
+}
+
+
+// do integration [ReV] (55)
+// put parameter 1 in front of existing parameters
+ex trafo_H_1mxt1px_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()).expand();
+ } else {
+ return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
+ }
+}
+
+
+// do x -> 1/x transformation
+struct map_trafo_H_1overx : 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);
- // if all parameters are either zero or one return the transformed function
- if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) {
- lst newparameter;
- for (int i=parameter.nops(); i>0; i--) {
- newparameter.append(0);
+ // 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;
+ }
}
- return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
- } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) {
- lst newparameter;
- for (int i=parameter.nops(); i>0; i--) {
- newparameter.append(1);
+ if (allthesame) {
+ return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
+ }
+ } else if (parameter.op(0) == -1) {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != -1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 1) {
+ allthesame = false;
+ break;
+ }
+ }
+ 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())
+ / factorial(parameter.nops())).expand());
}
- 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_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
- map_trafo_H_1mx recursion;
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1overx recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (int i=0; i<buffer.nops(); i++) {
+ res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
+ }
+ } else {
+ res += trafo_H_1tx_prepend_zero(buffer, arg);
+ }
+ return res;
+
+ } else if (parameter.op(0) == -1) {
+
+ // leading negative one
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1overx 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);
+ res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
}
} else {
- res -= trafo_H_prepend_one(buffer, arg);
+ res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
}
return res;
} else {
// leading one
- map_trafo_H_1mx recursion;
+ map_trafo_H_1overx recursion;
map_trafo_H_mult unify;
- ex res;
+ ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
int firstzero = 0;
while (parameter.op(firstzero) == 1) {
firstzero++;
}
res -= H(newparameter, arg).hold();
}
- return (unify((-H(lst(0), 1-arg).hold() * recursion(H(newparameter, arg).hold())).expand()) +
- recursion(res)) / firstzero;
+ res = recursion(res).expand() / firstzero;
+ return unify(res);
}
};
-// do x -> 1/x transformation
-struct map_trafo_H_1overx : public map_function
+// do x -> (1-x)/(1+x) transformation
+struct map_trafo_H_1mxt1px : public map_function
{
ex operator()(const ex& e)
{
lst parameter = ex_to<lst>(e.op(0));
ex arg = e.op(1);
- // if all parameters are either zero or one return the transformed function
- if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) {
- map_trafo_H_mult unify;
- return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, parameter.nops()) /
- factorial(parameter.nops())).expand());
- } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) {
- return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
+ // 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) {
+ map_trafo_H_mult unify;
+ return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else if (parameter.op(0) == -1) {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != -1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else {
+ for (int i=1; i<parameter.nops(); i++) {
+ if (parameter.op(i) != 1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
}
lst newparameter = parameter;
newparameter.remove_first();
if (parameter.op(0) == 0) {
-
+
// leading zero
- ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
- map_trafo_H_1overx recursion;
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1mxt1px 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_zero(buffer.op(i), arg);
+ res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
}
} else {
- res += trafo_H_prepend_zero(buffer, arg);
+ res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
+ }
+ return res;
+
+ } else if (parameter.op(0) == -1) {
+
+ // leading negative one
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1mxt1px recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (int i=0; i<buffer.nops(); i++) {
+ res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
+ }
+ } else {
+ res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
}
return res;
} else {
// leading one
- map_trafo_H_1overx recursion;
+ map_trafo_H_1mxt1px recursion;
map_trafo_H_mult unify;
ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
int firstzero = 0;
};
-// remove trailing zeros from H-parameters
-struct map_trafo_H_reduce_trailing_zeros : 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;
- if (is_a<lst>(e.op(0))) {
- parameter = ex_to<lst>(e.op(0));
- } else {
- parameter = lst(e.op(0));
- }
- ex arg = e.op(1);
- if (parameter.op(parameter.nops()-1) == 0) {
-
- //
- if (parameter.nops() == 1) {
- return log(arg);
- }
-
- //
- lst::const_iterator it = parameter.begin();
- while ((it != parameter.end()) && (*it == 0)) {
- it++;
- }
- if (it == parameter.end()) {
- return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
- }
-
- //
- parameter.remove_last();
- int lastentry = parameter.nops();
- while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
- lastentry--;
- }
-
- //
- ex result = log(arg) * H(parameter,arg).hold();
- for (ex i=0; i<lastentry; i++) {
- parameter[i]++;
- result -= (parameter[i]-1) * H(parameter, arg).hold();
- parameter[i]--;
- }
-
- if (lastentry < parameter.nops()) {
- result = result / (parameter.nops()-lastentry+1);
- return result.map(*this);
- } else {
- return result;
- }
- }
- }
- }
- return e;
- }
-};
-
-
-// recursively call convert_from_RV on expression
-struct map_trafo_H_convert : public map_function
-{
- ex operator()(const ex& e)
- {
- if (is_a<add>(e) || is_a<mul>(e) || is_a<power>(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);
- return convert_from_RV(parameter, arg);
- }
- }
- return e;
- }
-};
-
-
-// translate notation from nested sums to Remiddi/Vermaseren
-lst convert_to_RV(const lst& o)
-{
- lst res;
- for (lst::const_iterator it = o.begin(); it != o.end(); it++) {
- for (ex i=0; i<(*it)-1; i++) {
- res.append(0);
- }
- res.append(1);
- }
- return res;
-}
-
-
-// translate notation from Remiddi/Vermaseren to nested sums
-ex convert_from_RV(const lst& parameterlst, const ex& arg)
-{
- lst newparameterlst;
-
- lst::const_iterator it = parameterlst.begin();
- int count = 1;
- while (it != parameterlst.end()) {
- if (*it == 0) {
- count++;
- } else {
- newparameterlst.append((*it>0) ? count : -count);
- count = 1;
- }
- it++;
- }
- for (int i=1; i<count; i++) {
- newparameterlst.append(0);
- }
-
- map_trafo_H_reduce_trailing_zeros filter;
- return filter(H(newparameterlst, arg).hold());
-}
-
-
// do the actual summation.
-cln::cl_N H_do_sum(const std::vector<int>& s, const cln::cl_N& x)
+cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
{
- const int j = s.size();
+ const int j = m.size();
std::vector<cln::cl_N> t(j);
do {
t0buf = t[0];
q++;
- t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
+ t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
for (int k=j-2; k>=1; k--) {
- t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
+ t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
}
- t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), s[0]);
+ t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
factor = factor * x;
} while (t[0] != t0buf);
-
+
return t[0];
}
//////////////////////////////////////////////////////////////////////
//
-// Harmonic polylogarithm H
+// Harmonic polylogarithm H(m,x)
//
// GiNaC function
//
//////////////////////////////////////////////////////////////////////
-static ex H_eval(const ex& x1, const ex& x2)
-{
- if (x2 == 0) {
- return 0;
- }
- if (x2 == 1) {
- return zeta(x1);
- }
- if (x1.nops() == 1) {
- return Li(x1.op(0), x2);
- }
- if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) {
- return H(x1,x2).evalf();
- }
- 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 (!x1.op(i).info(info_flags::posint)) {
+ if (!x1.op(i).info(info_flags::integer)) {
return H(x1,x2).hold();
}
}
if (x1.nops() < 1) {
- return _ex1;
- }
- if (x1.nops() == 1) {
- return Li(x1.op(0), x2).evalf();
+ return H(x1,x2).hold();
}
+
cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
- if (x == 1) {
- return zeta(x1).evalf();
+
+ const lst& morg = ex_to<lst>(x1);
+ // remove trailing zeros ...
+ if (*(--morg.end()) == 0) {
+ symbol xtemp("xtemp");
+ map_trafo_H_reduce_trailing_zeros filter;
+ return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
+ }
+ // ... and expand parameter notation
+ lst m;
+ for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
+ if (*it > 1) {
+ for (ex count=*it-1; count > 0; count--) {
+ m.append(0);
+ }
+ m.append(1);
+ } else if (*it < -1) {
+ for (ex count=*it+1; count < 0; count++) {
+ m.append(0);
+ }
+ m.append(-1);
+ } else {
+ m.append(*it);
+ }
}
- // choose trafo
- if (cln::abs(x) > 1) {
- symbol xtemp("xtemp");
- map_trafo_H_1overx trafo;
- ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
- map_trafo_H_convert converter;
- res = converter(res);
- return res.subs(xtemp==x2).evalf();
+ // since the transformations produce a lot of terms, they are only efficient for
+ // argument near one.
+ // no transformation needed -> do summation
+ if (cln::abs(x) < 0.95) {
+ lst m_lst;
+ lst s_lst;
+ ex pf;
+ if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
+ // negative parameters -> s_lst is filled
+ std::vector<int> m_int;
+ std::vector<cln::cl_N> x_cln;
+ for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
+ it_int != m_lst.end(); it_int++, it_cln++) {
+ m_int.push_back(ex_to<numeric>(*it_int).to_int());
+ x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
+ }
+ x_cln.front() = x_cln.front() * x;
+ return pf * numeric(multipleLi_do_sum(m_int, x_cln));
+ } else {
+ // only positive parameters
+ //TODO
+ if (m_lst.nops() == 1) {
+ return Li(m_lst.op(0), x2).evalf();
+ }
+ std::vector<int> m_int;
+ for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
+ m_int.push_back(ex_to<numeric>(*it).to_int());
+ }
+ return numeric(H_do_sum(m_int, x));
+ }
}
- // since the x->1-x transformation produces a lot of terms, it is only
- // efficient for argument near one.
- if (cln::realpart(x) > 0.95) {
- symbol xtemp("xtemp");
- map_trafo_H_1mx trafo;
- ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
- map_trafo_H_convert converter;
- res = converter(res);
- return res.subs(xtemp==x2).evalf();
+ ex res = 1;
+
+ // ensure that the realpart of the argument is positive
+ if (cln::realpart(x) < 0) {
+ x = -x;
+ for (int i=0; i<m.nops(); i++) {
+ if (m.op(i) != 0) {
+ m.let_op(i) = -m.op(i);
+ res *= -1;
+ }
+ }
}
- // no trafo -> do summation
- int count = x1.nops();
- std::vector<int> r(count);
- for (int i=0; i<count; i++) {
- r[i] = ex_to<numeric>(x1.op(i)).to_int();
+ // 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
+ map_trafo_H_1overx trafo;
+ res *= trafo(H(m, xtemp));
}
- return numeric(H_do_sum(r,x));
+ // 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();
}
return H(x1,x2).hold();
}
-static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
+static ex H_eval(const ex& m_, const ex& x)
+{
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ if (m.nops() == 0) {
+ return _ex1;
+ }
+ ex pos1;
+ ex pos2;
+ ex n;
+ ex p;
+ int step = 0;
+ if (*m.begin() > _ex1) {
+ step++;
+ pos1 = _ex0;
+ pos2 = _ex1;
+ n = *m.begin()-1;
+ p = _ex1;
+ } else if (*m.begin() < _ex_1) {
+ step++;
+ pos1 = _ex0;
+ pos2 = _ex_1;
+ n = -*m.begin()-1;
+ p = _ex1;
+ } else if (*m.begin() == _ex0) {
+ pos1 = _ex0;
+ n = _ex1;
+ } else {
+ pos1 = *m.begin();
+ p = _ex1;
+ }
+ for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
+ if ((*it).info(info_flags::integer)) {
+ if (step == 0) {
+ if (*it > _ex1) {
+ if (pos1 == _ex0) {
+ step = 1;
+ pos2 = _ex1;
+ n += *it-1;
+ p = _ex1;
+ } else {
+ step = 2;
+ }
+ } else if (*it < _ex_1) {
+ if (pos1 == _ex0) {
+ step = 1;
+ pos2 = _ex_1;
+ n += -*it-1;
+ p = _ex1;
+ } else {
+ step = 2;
+ }
+ } else {
+ if (*it != pos1) {
+ step = 1;
+ pos2 = *it;
+ }
+ if (*it == _ex0) {
+ n++;
+ } else {
+ p++;
+ }
+ }
+ } else if (step == 1) {
+ if (*it != pos2) {
+ step = 2;
+ } else {
+ if (*it == _ex0) {
+ n++;
+ } else {
+ p++;
+ }
+ }
+ }
+ } else {
+ // if some m_i is not an integer
+ return H(m_, x).hold();
+ }
+ }
+ if ((x == _ex1) && (*(--m.end()) != _ex0)) {
+ return convert_H_to_zeta(m);
+ }
+ if (step == 0) {
+ if (pos1 == _ex0) {
+ // all zero
+ if (x == _ex0) {
+ return H(m_, x).hold();
+ }
+ return pow(log(x), m.nops()) / factorial(m.nops());
+ } else {
+ // all (minus) one
+ return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
+ }
+ } else if ((step == 1) && (pos1 == _ex0)){
+ // convertible to S
+ if (pos2 == _ex1) {
+ return S(n, p, x);
+ } else {
+ return pow(-1, p) * S(n, p, -x);
+ }
+ }
+ if (x == _ex0) {
+ return _ex0;
+ }
+ if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
+ return H(m_, x).evalf();
+ }
+ return H(m_, x).hold();
+}
+
+
+static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
{
epvector seq;
- seq.push_back(expair(H(x1,x2), 0));
- return pseries(rel,seq);
+ seq.push_back(expair(H(m, x), 0));
+ return pseries(rel, seq);
}
-static ex H_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
+static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param < 2);
if (deriv_param == 0) {
return _ex0;
}
- if (is_a<lst>(x1)) {
- lst newparameter = ex_to<lst>(x1);
- if (x1.op(0) == 1) {
- newparameter.remove_first();
- return 1/(1-x2) * H(newparameter, x2);
- } else {
- newparameter[0]--;
- return H(newparameter, x2).hold() / x2;
- }
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
} else {
- if (x1 == 1) {
- return 1/(1-x2);
- } else {
- return H(x1-1, x2).hold() / x2;
- }
+ m = lst(m_);
+ }
+ ex mb = *m.begin();
+ if (mb > _ex1) {
+ m[0]--;
+ return H(m, x) / x;
+ }
+ if (mb < _ex_1) {
+ m[0]++;
+ return H(m, x) / x;
+ }
+ m.remove_first();
+ if (mb == _ex1) {
+ return 1/(1-x) * H(m, x);
+ } else if (mb == _ex_1) {
+ return 1/(1+x) * H(m, x);
+ } else {
+ return H(m, x) / x;
}
}
+static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
+{
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ c.s << "\\mbox{H}_{";
+ lst::const_iterator itm = m.begin();
+ (*itm).print(c);
+ itm++;
+ for (; itm != m.end(); itm++) {
+ c.s << ",";
+ (*itm).print(c);
+ }
+ c.s << "}(";
+ x.print(c);
+ c.s << ")";
+}
+
+
REGISTER_FUNCTION(H,
- eval_func(H_eval).
- evalf_func(H_evalf).
- do_not_evalf_params().
- series_func(H_series).
- derivative_func(H_deriv));
+ evalf_func(H_evalf).
+ eval_func(H_eval).
+ series_func(H_series).
+ derivative_func(H_deriv).
+ print_func<print_latex>(H_print_latex).
+ do_not_evalf_params());
+
+
+// takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
+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<lst>(m)) {
+ return filter2(filter(H(m, x).hold()));
+ } else {
+ return filter2(filter(H(lst(m), x).hold()));
+ }
+}
//////////////////////////////////////////////////////////////////////
//
-// Multiple zeta values zeta
+// Multiple zeta values zeta(x) and zeta(x,s)
//
// helper functions
//
}
+// does Hoelder convolution. see [BBB] (7.0)
+cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
+{
+ // prepare parameters
+ // holds Li arguments in [BBB] notation
+ std::vector<int> s = s_;
+ std::vector<int> m_p = m_;
+ std::vector<int> m_q;
+ // holds Li arguments in nested sums notation
+ std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
+ s_p[0] = s_p[0] * cln::cl_N("1/2");
+ // convert notations
+ int sig = 1;
+ for (int i=0; i<s_.size(); i++) {
+ if (s_[i] < 0) {
+ sig = -sig;
+ s_p[i] = -s_p[i];
+ }
+ s[i] = sig * std::abs(s[i]);
+ }
+ std::vector<cln::cl_N> s_q;
+ cln::cl_N signum = 1;
+
+ // first term
+ cln::cl_N res = multipleLi_do_sum(m_p, s_p);
+
+ // middle terms
+ do {
+
+ // change parameters
+ if (s.front() > 0) {
+ if (m_p.front() == 1) {
+ m_p.erase(m_p.begin());
+ s_p.erase(s_p.begin());
+ if (s_p.size() > 0) {
+ s_p.front() = s_p.front() * cln::cl_N("1/2");
+ }
+ s.erase(s.begin());
+ m_q.front()++;
+ } else {
+ m_p.front()--;
+ m_q.insert(m_q.begin(), 1);
+ if (s_q.size() > 0) {
+ s_q.front() = s_q.front() * 2;
+ }
+ s_q.insert(s_q.begin(), cln::cl_N("1/2"));
+ }
+ } else {
+ if (m_p.front() == 1) {
+ m_p.erase(m_p.begin());
+ cln::cl_N spbuf = s_p.front();
+ s_p.erase(s_p.begin());
+ if (s_p.size() > 0) {
+ s_p.front() = s_p.front() * spbuf;
+ }
+ s.erase(s.begin());
+ m_q.insert(m_q.begin(), 1);
+ if (s_q.size() > 0) {
+ s_q.front() = s_q.front() * 4;
+ }
+ s_q.insert(s_q.begin(), cln::cl_N("1/4"));
+ signum = -signum;
+ } else {
+ m_p.front()--;
+ m_q.insert(m_q.begin(), 1);
+ if (s_q.size() > 0) {
+ s_q.front() = s_q.front() * 2;
+ }
+ s_q.insert(s_q.begin(), cln::cl_N("1/2"));
+ }
+ }
+
+ // exiting the loop
+ if (m_p.size() == 0) break;
+
+ res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
+
+ } while (true);
+
+ // last term
+ res = res + signum * multipleLi_do_sum(m_q, s_q);
+
+ return res;
+}
+
+
} // end of anonymous namespace
//////////////////////////////////////////////////////////////////////
//
-// Multiple zeta values zeta
+// Multiple zeta values zeta(x)
//
// GiNaC function
//
return numeric(zeta_do_sum_simple(r));
}
}
-
+
// single zeta value
if (is_exactly_a<numeric>(x) && (x != 1)) {
try {
}
-static ex zeta1_eval(const ex& x)
+static ex zeta1_eval(const ex& m)
{
- if (is_exactly_a<lst>(x)) {
- if (x.nops() == 1) {
- return zeta(x.op(0));
+ if (is_exactly_a<lst>(m)) {
+ if (m.nops() == 1) {
+ return zeta(m.op(0));
}
- return zeta(x).hold();
+ return zeta(m).hold();
}
- if (x.info(info_flags::numeric)) {
- const numeric& y = ex_to<numeric>(x);
+ if (m.info(info_flags::numeric)) {
+ const numeric& y = ex_to<numeric>(m);
// trap integer arguments:
if (y.is_integer()) {
if (y.is_zero()) {
return _ex_1_2;
}
if (y.is_equal(_num1)) {
- return zeta(x).hold();
+ return zeta(m).hold();
}
if (y.info(info_flags::posint)) {
if (y.info(info_flags::odd)) {
- return zeta(x).hold();
+ return zeta(m).hold();
} else {
return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
}
}
}
// zeta(float)
- if (y.info(info_flags::numeric) && !y.info(info_flags::crational))
- return zeta1_evalf(x);
+ if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
+ return zeta1_evalf(m);
+ }
}
- return zeta(x).hold();
+ return zeta(m).hold();
}
-static ex zeta1_deriv(const ex& x, unsigned deriv_param)
+static ex zeta1_deriv(const ex& m, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param==0);
- if (is_exactly_a<lst>(x)) {
+ if (is_exactly_a<lst>(m)) {
return _ex0;
} else {
- return zeta(_ex1, x);
+ return zetaderiv(_ex1, m);
+ }
+}
+
+
+static void zeta1_print_latex(const ex& m_, const print_context& c)
+{
+ c.s << "\\zeta(";
+ if (is_a<lst>(m_)) {
+ const lst& m = ex_to<lst>(m_);
+ lst::const_iterator it = m.begin();
+ (*it).print(c);
+ it++;
+ for (; it != m.end(); it++) {
+ c.s << ",";
+ (*it).print(c);
+ }
+ } else {
+ m_.print(c);
}
+ c.s << ")";
}
-unsigned zeta1_SERIAL::serial =
- function::register_new(function_options("zeta").
- eval_func(zeta1_eval).
- evalf_func(zeta1_evalf).
- do_not_evalf_params().
- derivative_func(zeta1_deriv).
- latex_name("\\zeta").
- overloaded(2));
+unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta").
+ evalf_func(zeta1_evalf).
+ eval_func(zeta1_eval).
+ derivative_func(zeta1_deriv).
+ print_func<print_latex>(zeta1_print_latex).
+ do_not_evalf_params().
+ overloaded(2));
//////////////////////////////////////////////////////////////////////
//
-// Multiple zeta values mZeta
-//
-// The use of mZeta is deprecated! This function will be removed
-// from GiNaC source soon. Use zeta instead!!
+// Alternating Euler sum zeta(x,s)
//
// GiNaC function
//
//////////////////////////////////////////////////////////////////////
-static ex mZeta_eval(const ex& x1)
+static ex zeta2_evalf(const ex& x, const ex& s)
{
- return mZeta(x1).hold();
+ if (is_exactly_a<lst>(x)) {
+
+ // alternating Euler sum
+ const int count = x.nops();
+ const lst& xlst = ex_to<lst>(x);
+ const lst& slst = ex_to<lst>(s);
+ std::vector<int> xi(count);
+ std::vector<int> si(count);
+
+ // check parameters and convert them
+ lst::const_iterator it_xread = xlst.begin();
+ lst::const_iterator it_sread = slst.begin();
+ std::vector<int>::iterator it_xwrite = xi.begin();
+ std::vector<int>::iterator it_swrite = si.begin();
+ do {
+ if (!(*it_xread).info(info_flags::posint)) {
+ return zeta(x, s).hold();
+ }
+ *it_xwrite = ex_to<numeric>(*it_xread).to_int();
+ if (*it_sread > 0) {
+ *it_swrite = 1;
+ } else {
+ *it_swrite = -1;
+ }
+ it_xread++;
+ it_sread++;
+ it_xwrite++;
+ it_swrite++;
+ } while (it_xwrite != xi.end());
+
+ // check for divergence
+ if ((xi[0] == 1) && (si[0] == 1)) {
+ return zeta(x, s).hold();
+ }
+
+ // use Hoelder convolution
+ return numeric(zeta_do_Hoelder_convolution(xi, si));
+ }
+
+ return zeta(x, s).hold();
}
-static ex mZeta_evalf(const ex& x1)
+static ex zeta2_eval(const ex& m, const ex& s_)
{
- if (is_a<lst>(x1)) {
- for (int i=0; i<x1.nops(); i++) {
- if (!x1.op(i).info(info_flags::posint))
- return mZeta(x1).hold();
+ if (is_exactly_a<lst>(s_)) {
+ const lst& s = ex_to<lst>(s_);
+ for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
+ if ((*it).info(info_flags::positive)) {
+ continue;
+ }
+ return zeta(m, s_).hold();
}
+ return zeta(m);
+ } else if (s_.info(info_flags::positive)) {
+ return zeta(m);
+ }
- const int j = x1.nops();
+ return zeta(m, s_).hold();
+}
- std::vector<int> r(j);
- for (int i=0; i<j; i++) {
- r[j-1-i] = ex_to<numeric>(x1.op(i)).to_int();
- }
- // check for divergence
- if (r[0] == 1) {
- return mZeta(x1).hold();
- }
+static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param==0);
- // if only one argument, use cln::zeta
- if (j == 1) {
- return numeric(cln::zeta(r[0]));
- }
-
- // decide on summation algorithm
- // this is still a bit clumsy
- int limit = (Digits>17) ? 10 : 6;
- if (r[0]<limit || (j>3 && r[1]<limit/2)) {
- return numeric(zeta_do_sum_Crandall(r));
- } else {
- return numeric(zeta_do_sum_simple(r));
+ if (is_exactly_a<lst>(m)) {
+ return _ex0;
+ } else {
+ if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
+ return zetaderiv(_ex1, m);
}
- } else if (x1.info(info_flags::posint) && (x1 != 1)) {
- return numeric(cln::zeta(ex_to<numeric>(x1).to_int()));
+ return _ex0;
}
-
- return mZeta(x1).hold();
}
-static ex mZeta_deriv(const ex& x, unsigned deriv_param)
+static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
{
- return 0;
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ lst s;
+ if (is_a<lst>(s_)) {
+ s = ex_to<lst>(s_);
+ } else {
+ s = lst(s_);
+ }
+ c.s << "\\zeta(";
+ lst::const_iterator itm = m.begin();
+ lst::const_iterator its = s.begin();
+ if (*its < 0) {
+ c.s << "\\overline{";
+ (*itm).print(c);
+ c.s << "}";
+ } else {
+ (*itm).print(c);
+ }
+ its++;
+ itm++;
+ for (; itm != m.end(); itm++, its++) {
+ c.s << ",";
+ if (*its < 0) {
+ c.s << "\\overline{";
+ (*itm).print(c);
+ c.s << "}";
+ } else {
+ (*itm).print(c);
+ }
+ }
+ c.s << ")";
}
-REGISTER_FUNCTION(mZeta,
- eval_func(mZeta_eval).
- evalf_func(mZeta_evalf).
- do_not_evalf_params().
- derivative_func(mZeta_deriv));
+unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta").
+ evalf_func(zeta2_evalf).
+ eval_func(zeta2_eval).
+ derivative_func(zeta2_deriv).
+ print_func<print_latex>(zeta2_print_latex).
+ do_not_evalf_params().
+ overloaded(2));
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff