/** @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(m_1,...,m_k),lst(x_1,...,x_k))
* nielsen's generalized polylogarithm S(n,p,x)
* 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
- * [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
+ * [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
+ * 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.
- *
+ * 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.
- *
+ *
* - 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 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
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;
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
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();
}
conv *= x2.op(i);
- if ((conv > 1) || ((conv == 1) && (x1.op(0) == 1))) {
- return Li(x1,x2).hold();
+ if (conv >= 1) {
+ return Li(x1, x2).hold();
}
}
}
-static ex Li_eval(const ex& x1, const ex& x2)
+static ex Li_eval(const ex& m_, const ex& x_)
{
- 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();
+ 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;
}
- return Li(x1,x2).evalf();
}
- return Li(x1,x2).hold();
+ 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& x1, const ex& x2, const relational& rel, int order, unsigned options)
+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);
}
}
REGISTER_FUNCTION(Li,
- 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());
+ 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());
//////////////////////////////////////////////////////////////////////
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));
}
}
}
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);
//////////////////////////////////////////////////////////////////////
-static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
+static ex S_evalf(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) && is_a<numeric>(x)) {
+ 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_eval(const ex& x1, const ex& x2, const ex& x3)
+static ex S_eval(const ex& n, const ex& p, const ex& x)
{
- if (x2 == 1) {
- return Li(x1+1,x3);
- }
- 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));
+ 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);
}
}
REGISTER_FUNCTION(S,
- 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());
+ 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());
//////////////////////////////////////////////////////////////////////
if (name == "H") {
lst parameter;
if (is_a<lst>(e.op(0))) {
- parameter = ex_to<lst>(e.op(0));
+ parameter = ex_to<lst>(e.op(0));
} else {
parameter = lst(e.op(0));
}
// returns an expression with zeta functions corresponding to the parameter list for H
-ex convert_H_to_zeta(const lst& l)
+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(l, xtemp).hold())).subs(xtemp == 1);
+ return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
}
// convert signs form Li to H representation
-// not used yet!
-lst convert_parameter_Li_to_H(const lst& l, ex& pf)
+lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
{
lst res;
- lst::const_iterator it = l.begin();
- ex signum = *it;
- pf = *it;
- res.append(*it);
- it++;
- while (it != l.end()) {
- signum = *it * signum;
- res.append(signum);
+ 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;
- it++;
+ res.append((*itm) * signum);
+ itm++;
+ itx++;
}
-
return res;
}
}
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());
+ 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 (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 unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, parameter.nops())
+ / factorial(parameter.nops())).expand());
}
}
}
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());
+ 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 (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());
+ 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 (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());
+ 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());
}
}
}
-static ex H_eval(const ex& x1, const ex& x2)
+static ex H_eval(const ex& m_, const ex& x)
{
- if (x2 == 0) {
- return 0;
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
}
-//TODO
-// 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();
+ 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();
+ }
}
- return H(x1,x2).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& x1, const ex& x2, const relational& rel, int order, unsigned options)
+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;
}
}
REGISTER_FUNCTION(H,
- 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());
+ 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& parameterlst, const ex& arg)
+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>(parameterlst)) {
- return filter2(filter(H(parameterlst, arg).hold())).eval();
+ if (is_a<lst>(m)) {
+ return filter2(filter(H(m, x).hold())).eval();
} else {
- return filter2(filter(H(lst(parameterlst), arg).hold())).eval();
+ return filter2(filter(H(lst(m), x).hold())).eval();
}
}
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;
}
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& x, const print_context& c)
+static void zeta1_print_latex(const ex& m_, const print_context& c)
{
c.s << "\\zeta(";
- if (is_a<lst>(x)) {
- lst arg;
- arg = ex_to<lst>(x);
- lst::const_iterator it = arg.begin();
+ if (is_a<lst>(m_)) {
+ const lst& m = ex_to<lst>(m_);
+ lst::const_iterator it = m.begin();
(*it).print(c);
it++;
- for (; it != arg.end(); it++) {
+ for (; it != m.end(); it++) {
c.s << ",";
(*it).print(c);
}
} else {
- x.print(c);
+ m_.print(c);
}
c.s << ")";
}
-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));
+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));
//////////////////////////////////////////////////////////////////////
// use Hoelder convolution
return numeric(zeta_do_Hoelder_convolution(xi, si));
}
-
+
return zeta(x, s).hold();
}
-static ex zeta2_eval(const ex& x, const ex& s)
+static ex zeta2_eval(const ex& m, const ex& s_)
{
- if (is_exactly_a<lst>(s)) {
- const lst& l = ex_to<lst>(s);
- lst::const_iterator it = l.begin();
- while (it != l.end()) {
- if ((*it).info(info_flags::negative)) {
- return zeta(x, s).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;
}
- it++;
- }
- return zeta(x);
- } else {
- if (s.info(info_flags::positive)) {
- return zeta(x);
+ return zeta(m, s_).hold();
}
+ return zeta(m);
+ } else if (s_.info(info_flags::positive)) {
+ return zeta(m);
}
- return zeta(x, s).hold();
+ return zeta(m, s_).hold();
}
-static ex zeta2_deriv(const ex& x, const ex& s, unsigned deriv_param)
+static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param==0);
- if (is_exactly_a<lst>(x)) {
+ if (is_exactly_a<lst>(m)) {
return _ex0;
} else {
- if ((is_exactly_a<lst>(s) && (s.op(0) > 0)) || (s > 0)) {
- return zeta(_ex1, x);
+ if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
+ return zetaderiv(_ex1, m);
}
return _ex0;
}
}
-static void zeta2_print_latex(const ex& x, const ex& s, const print_context& c)
+static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
{
- lst arg;
- if (is_a<lst>(x)) {
- arg = ex_to<lst>(x);
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
} else {
- arg = lst(x);
+ m = lst(m_);
}
- lst sig;
- if (is_a<lst>(s)) {
- sig = ex_to<lst>(s);
+ lst s;
+ if (is_a<lst>(s_)) {
+ s = ex_to<lst>(s_);
} else {
- sig = lst(s);
+ s = lst(s_);
}
c.s << "\\zeta(";
- lst::const_iterator itarg = arg.begin();
- lst::const_iterator itsig = sig.begin();
- if (*itsig < 0) {
+ lst::const_iterator itm = m.begin();
+ lst::const_iterator its = s.begin();
+ if (*its < 0) {
c.s << "\\overline{";
- (*itarg).print(c);
+ (*itm).print(c);
c.s << "}";
} else {
- (*itarg).print(c);
+ (*itm).print(c);
}
- itsig++;
- itarg++;
- for (; itarg != arg.end(); itarg++, itsig++) {
+ its++;
+ itm++;
+ for (; itm != m.end(); itm++, its++) {
c.s << ",";
- if (*itsig < 0) {
+ if (*its < 0) {
c.s << "\\overline{";
- (*itarg).print(c);
+ (*itm).print(c);
c.s << "}";
} else {
- (*itarg).print(c);
+ (*itm).print(c);
}
}
c.s << ")";
}
-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));
+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