* Implementation of GiNaC's initially known functions. */
/*
- * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2001 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
#include "lst.h"
#include "matrix.h"
#include "mul.h"
-#include "ncmul.h"
-#include "numeric.h"
#include "power.h"
#include "relational.h"
#include "pseries.h"
#include "symbol.h"
#include "utils.h"
-#ifndef NO_NAMESPACE_GINAC
namespace GiNaC {
-#endif // ndef NO_NAMESPACE_GINAC
//////////
// absolute value
//////////
-static ex abs_evalf(const ex & x)
+static ex abs_evalf(const ex & arg)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(abs(x))
-
- return abs(ex_to_numeric(x));
+ BEGIN_TYPECHECK
+ TYPECHECK(arg,numeric)
+ END_TYPECHECK(abs(arg))
+
+ return abs(ex_to<numeric>(arg));
}
-static ex abs_eval(const ex & x)
+static ex abs_eval(const ex & arg)
{
- if (is_ex_exactly_of_type(x, numeric))
- return abs(ex_to_numeric(x));
- else
- return abs(x).hold();
+ if (is_ex_exactly_of_type(arg, numeric))
+ return abs(ex_to<numeric>(arg));
+ else
+ return abs(arg).hold();
}
REGISTER_FUNCTION(abs, eval_func(abs_eval).
evalf_func(abs_evalf));
+
+//////////
+// Complex sign
+//////////
+
+static ex csgn_evalf(const ex & arg)
+{
+ BEGIN_TYPECHECK
+ TYPECHECK(arg,numeric)
+ END_TYPECHECK(csgn(arg))
+
+ return csgn(ex_to<numeric>(arg));
+}
+
+static ex csgn_eval(const ex & arg)
+{
+ if (is_ex_exactly_of_type(arg, numeric))
+ return csgn(ex_to<numeric>(arg));
+
+ else if (is_ex_of_type(arg, mul) &&
+ is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
+ numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
+ if (oc.is_real()) {
+ if (oc > 0)
+ // csgn(42*x) -> csgn(x)
+ return csgn(arg/oc).hold();
+ else
+ // csgn(-42*x) -> -csgn(x)
+ return -csgn(arg/oc).hold();
+ }
+ if (oc.real().is_zero()) {
+ if (oc.imag() > 0)
+ // csgn(42*I*x) -> csgn(I*x)
+ return csgn(I*arg/oc).hold();
+ else
+ // csgn(-42*I*x) -> -csgn(I*x)
+ return -csgn(I*arg/oc).hold();
+ }
+ }
+
+ return csgn(arg).hold();
+}
+
+static ex csgn_series(const ex & arg,
+ const relational & rel,
+ int order,
+ unsigned options)
+{
+ const ex arg_pt = arg.subs(rel);
+ if (arg_pt.info(info_flags::numeric)
+ && ex_to<numeric>(arg_pt).real().is_zero()
+ && !(options & series_options::suppress_branchcut))
+ throw (std::domain_error("csgn_series(): on imaginary axis"));
+
+ epvector seq;
+ seq.push_back(expair(csgn(arg_pt), _ex0()));
+ return pseries(rel,seq);
+}
+
+REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
+ evalf_func(csgn_evalf).
+ series_func(csgn_series));
+
+
+//////////
+// Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
+//////////
+
+static ex eta_evalf(const ex & x, const ex & y)
+{
+ BEGIN_TYPECHECK
+ TYPECHECK(x,numeric)
+ TYPECHECK(y,numeric)
+ END_TYPECHECK(eta(x,y))
+
+ numeric xim = imag(ex_to<numeric>(x));
+ numeric yim = imag(ex_to<numeric>(y));
+ numeric xyim = imag(ex_to<numeric>(x*y));
+ return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
+}
+
+static ex eta_eval(const ex & x, const ex & y)
+{
+ if (is_ex_exactly_of_type(x, numeric) &&
+ is_ex_exactly_of_type(y, numeric)) {
+ // don't call eta_evalf here because it would call Pi.evalf()!
+ numeric xim = imag(ex_to<numeric>(x));
+ numeric yim = imag(ex_to<numeric>(y));
+ numeric xyim = imag(ex_to<numeric>(x*y));
+ return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
+ }
+
+ return eta(x,y).hold();
+}
+
+static ex eta_series(const ex & arg1,
+ const ex & arg2,
+ const relational & rel,
+ int order,
+ unsigned options)
+{
+ const ex arg1_pt = arg1.subs(rel);
+ const ex arg2_pt = arg2.subs(rel);
+ if (ex_to<numeric>(arg1_pt).imag().is_zero() ||
+ ex_to<numeric>(arg2_pt).imag().is_zero() ||
+ ex_to<numeric>(arg1_pt*arg2_pt).imag().is_zero()) {
+ throw (std::domain_error("eta_series(): on discontinuity"));
+ }
+ epvector seq;
+ seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
+ return pseries(rel,seq);
+}
+
+REGISTER_FUNCTION(eta, eval_func(eta_eval).
+ evalf_func(eta_evalf).
+ series_func(eta_series).
+ latex_name("\\eta"));
+
+
//////////
// dilogarithm
//////////
+static ex Li2_evalf(const ex & x)
+{
+ BEGIN_TYPECHECK
+ TYPECHECK(x,numeric)
+ END_TYPECHECK(Li2(x))
+
+ return Li2(ex_to<numeric>(x)); // -> numeric Li2(numeric)
+}
+
static ex Li2_eval(const ex & x)
{
- if (x.is_zero())
- return x;
- if (x.is_equal(_ex1()))
- return power(Pi, _ex2()) / _ex6();
- if (x.is_equal(_ex_1()))
- return -power(Pi, _ex2()) / _ex12();
- return Li2(x).hold();
+ if (x.info(info_flags::numeric)) {
+ // Li2(0) -> 0
+ if (x.is_zero())
+ return _ex0();
+ // Li2(1) -> Pi^2/6
+ if (x.is_equal(_ex1()))
+ return power(Pi,_ex2())/_ex6();
+ // Li2(1/2) -> Pi^2/12 - log(2)^2/2
+ if (x.is_equal(_ex1_2()))
+ return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
+ // Li2(-1) -> -Pi^2/12
+ if (x.is_equal(_ex_1()))
+ return -power(Pi,_ex2())/_ex12();
+ // Li2(I) -> -Pi^2/48+Catalan*I
+ if (x.is_equal(I))
+ return power(Pi,_ex2())/_ex_48() + Catalan*I;
+ // Li2(-I) -> -Pi^2/48-Catalan*I
+ if (x.is_equal(-I))
+ return power(Pi,_ex2())/_ex_48() - Catalan*I;
+ // Li2(float)
+ if (!x.info(info_flags::crational))
+ return Li2_evalf(x);
+ }
+
+ return Li2(x).hold();
}
-REGISTER_FUNCTION(Li2, eval_func(Li2_eval));
+static ex Li2_deriv(const ex & x, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param==0);
+
+ // d/dx Li2(x) -> -log(1-x)/x
+ return -log(1-x)/x;
+}
+
+static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
+{
+ const ex x_pt = x.subs(rel);
+ if (x_pt.info(info_flags::numeric)) {
+ // First special case: x==0 (derivatives have poles)
+ if (x_pt.is_zero()) {
+ // method:
+ // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
+ // simply substitute x==0. The limit, however, exists: it is 1.
+ // We also know all higher derivatives' limits:
+ // (d/dx)^n Li2(x) == n!/n^2.
+ // So the primitive series expansion is
+ // Li2(x==0) == x + x^2/4 + x^3/9 + ...
+ // and so on.
+ // We first construct such a primitive series expansion manually in
+ // a dummy symbol s and then insert the argument's series expansion
+ // for s. Reexpanding the resulting series returns the desired
+ // result.
+ const symbol s;
+ ex ser;
+ // manually construct the primitive expansion
+ for (int i=1; i<order; ++i)
+ ser += pow(s,i) / pow(numeric(i), _num2());
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel, order));
+ // maybe that was terminating, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1()), order));
+ ser += pseries(rel, nseq);
+ // reexpanding it will collapse the series again
+ return ser.series(rel, order);
+ // NB: Of course, this still does not allow us to compute anything
+ // like sin(Li2(x)).series(x==0,2), since then this code here is
+ // not reached and the derivative of sin(Li2(x)) doesn't allow the
+ // substitution x==0. Probably limits *are* needed for the general
+ // cases. In case L'Hospital's rule is implemented for limits and
+ // basic::series() takes care of this, this whole block is probably
+ // obsolete!
+ }
+ // second special case: x==1 (branch point)
+ if (x_pt == _ex1()) {
+ // method:
+ // construct series manually in a dummy symbol s
+ const symbol s;
+ ex ser = zeta(2);
+ // manually construct the primitive expansion
+ for (int i=1; i<order; ++i)
+ ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel, order));
+ // maybe that was terminating, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1()), order));
+ ser += pseries(rel, nseq);
+ // reexpanding it will collapse the series again
+ return ser.series(rel, order);
+ }
+ // third special case: x real, >=1 (branch cut)
+ if (!(options & series_options::suppress_branchcut) &&
+ ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
+ // method:
+ // This is the branch cut: assemble the primitive series manually
+ // and then add the corresponding complex step function.
+ const symbol *s = static_cast<symbol *>(rel.lhs().bp);
+ const ex point = rel.rhs();
+ const symbol foo;
+ epvector seq;
+ // zeroth order term:
+ seq.push_back(expair(Li2(x_pt), _ex0()));
+ // compute the intermediate terms:
+ ex replarg = series(Li2(x), *s==foo, order);
+ for (unsigned i=1; i<replarg.nops()-1; ++i)
+ seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
+ // append an order term:
+ seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
+ return pseries(rel, seq);
+ }
+ }
+ // all other cases should be safe, by now:
+ throw do_taylor(); // caught by function::series()
+}
+
+REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
+ evalf_func(Li2_evalf).
+ derivative_func(Li2_deriv).
+ series_func(Li2_series).
+ latex_name("\\mbox{Li}_2"));
//////////
// trilogarithm
static ex Li3_eval(const ex & x)
{
- if (x.is_zero())
- return x;
- return Li3(x).hold();
+ if (x.is_zero())
+ return x;
+ return Li3(x).hold();
}
-REGISTER_FUNCTION(Li3, eval_func(Li3_eval));
+REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
+ latex_name("\\mbox{Li}_3"));
//////////
// factorial
static ex factorial_evalf(const ex & x)
{
- return factorial(x).hold();
+ return factorial(x).hold();
}
static ex factorial_eval(const ex & x)
{
- if (is_ex_exactly_of_type(x, numeric))
- return factorial(ex_to_numeric(x));
- else
- return factorial(x).hold();
+ if (is_ex_exactly_of_type(x, numeric))
+ return factorial(ex_to<numeric>(x));
+ else
+ return factorial(x).hold();
}
REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
static ex binomial_evalf(const ex & x, const ex & y)
{
- return binomial(x, y).hold();
+ return binomial(x, y).hold();
}
static ex binomial_eval(const ex & x, const ex &y)
{
- if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
- return binomial(ex_to_numeric(x), ex_to_numeric(y));
- else
- return binomial(x, y).hold();
+ if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
+ return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
+ else
+ return binomial(x, y).hold();
}
REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
static ex Order_eval(const ex & x)
{
if (is_ex_exactly_of_type(x, numeric)) {
-
- // O(c)=O(1)
- return Order(_ex1()).hold();
-
+ // O(c) -> O(1) or 0
+ if (!x.is_zero())
+ return Order(_ex1()).hold();
+ else
+ return _ex0();
} else if (is_ex_exactly_of_type(x, mul)) {
-
- mul *m = static_cast<mul *>(x.bp);
- if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric)) {
-
- // O(c*expr)=O(expr)
+ mul *m = static_cast<mul *>(x.bp);
+ // O(c*expr) -> O(expr)
+ if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
return Order(x / m->op(m->nops() - 1)).hold();
- }
}
return Order(x).hold();
}
-static ex Order_series(const ex & x, const symbol & s, const ex & point, int order)
+static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
{
// Just wrap the function into a pseries object
epvector new_seq;
- new_seq.push_back(expair(Order(_ex1()), numeric(min(x.ldegree(s), order))));
- return pseries(s, point, new_seq);
+ GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
+ const symbol *s = static_cast<symbol *>(r.lhs().bp);
+ new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
+ return pseries(r, new_seq);
}
// Differentiation is handled in function::derivative because of its special requirements
REGISTER_FUNCTION(Order, eval_func(Order_eval).
- series_func(Order_series));
+ series_func(Order_series).
+ latex_name("\\mathcal{O}"));
//////////
// Inert partial differentiation operator
static ex Derivative_eval(const ex & f, const ex & l)
{
if (!is_ex_exactly_of_type(f, function)) {
- throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
+ throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
+ }
+ if (!is_ex_exactly_of_type(l, lst)) {
+ throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
}
- if (!is_ex_exactly_of_type(l, lst)) {
- throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
- }
return Derivative(f, l).hold();
}
ex lsolve(const ex &eqns, const ex &symbols)
{
- // solve a system of linear equations
- if (eqns.info(info_flags::relation_equal)) {
- if (!symbols.info(info_flags::symbol)) {
- throw(std::invalid_argument("lsolve: 2nd argument must be a symbol"));
- }
- ex sol=lsolve(lst(eqns),lst(symbols));
-
- GINAC_ASSERT(sol.nops()==1);
- GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
-
- return sol.op(0).op(1); // return rhs of first solution
- }
-
- // syntax checks
- if (!eqns.info(info_flags::list)) {
- throw(std::invalid_argument("lsolve: 1st argument must be a list"));
- }
- for (unsigned i=0; i<eqns.nops(); i++) {
- if (!eqns.op(i).info(info_flags::relation_equal)) {
- throw(std::invalid_argument("lsolve: 1st argument must be a list of equations"));
- }
- }
- if (!symbols.info(info_flags::list)) {
- throw(std::invalid_argument("lsolve: 2nd argument must be a list"));
- }
- for (unsigned i=0; i<symbols.nops(); i++) {
- if (!symbols.op(i).info(info_flags::symbol)) {
- throw(std::invalid_argument("lsolve: 2nd argument must be a list of symbols"));
- }
- }
-
- // build matrix from equation system
- matrix sys(eqns.nops(),symbols.nops());
- matrix rhs(eqns.nops(),1);
- matrix vars(symbols.nops(),1);
-
- for (unsigned r=0; r<eqns.nops(); r++) {
- ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
- ex linpart = eq;
- for (unsigned c=0; c<symbols.nops(); c++) {
- ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
- linpart -= co*symbols.op(c);
- sys.set(r,c,co);
- }
- linpart=linpart.expand();
- rhs.set(r,0,-linpart);
- }
-
- // test if system is linear and fill vars matrix
- for (unsigned i=0; i<symbols.nops(); i++) {
- vars.set(i,0,symbols.op(i));
- if (sys.has(symbols.op(i)))
- throw(std::logic_error("lsolve: system is not linear"));
- if (rhs.has(symbols.op(i)))
- throw(std::logic_error("lsolve: system is not linear"));
- }
-
- //matrix solution=sys.solve(rhs);
- matrix solution;
- try {
- solution = sys.fraction_free_elim(vars,rhs);
- } catch (const runtime_error & e) {
- // probably singular matrix (or other error)
- // return empty solution list
- // cerr << e.what() << endl;
- return lst();
- }
-
- // return a list of equations
- if (solution.cols()!=1) {
- throw(std::runtime_error("lsolve: strange number of columns returned from matrix::solve"));
- }
- if (solution.rows()!=symbols.nops()) {
- cout << "symbols.nops()=" << symbols.nops() << endl;
- cout << "solution.rows()=" << solution.rows() << endl;
- throw(std::runtime_error("lsolve: strange number of rows returned from matrix::solve"));
- }
-
- // return list of the form lst(var1==sol1,var2==sol2,...)
- lst sollist;
- for (unsigned i=0; i<symbols.nops(); i++) {
- sollist.append(symbols.op(i)==solution(i,0));
- }
-
- return sollist;
-}
-
-/** non-commutative power. */
-ex ncpower(const ex &basis, unsigned exponent)
-{
- if (exponent==0) {
- return _ex1();
- }
-
- exvector v;
- v.reserve(exponent);
- for (unsigned i=0; i<exponent; ++i) {
- v.push_back(basis);
- }
-
- return ncmul(v,1);
+ // solve a system of linear equations
+ if (eqns.info(info_flags::relation_equal)) {
+ if (!symbols.info(info_flags::symbol))
+ throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
+ ex sol=lsolve(lst(eqns),lst(symbols));
+
+ GINAC_ASSERT(sol.nops()==1);
+ GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
+
+ return sol.op(0).op(1); // return rhs of first solution
+ }
+
+ // syntax checks
+ if (!eqns.info(info_flags::list)) {
+ throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
+ }
+ for (unsigned i=0; i<eqns.nops(); i++) {
+ if (!eqns.op(i).info(info_flags::relation_equal)) {
+ throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
+ }
+ }
+ if (!symbols.info(info_flags::list)) {
+ throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
+ }
+ for (unsigned i=0; i<symbols.nops(); i++) {
+ if (!symbols.op(i).info(info_flags::symbol)) {
+ throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
+ }
+ }
+
+ // build matrix from equation system
+ matrix sys(eqns.nops(),symbols.nops());
+ matrix rhs(eqns.nops(),1);
+ matrix vars(symbols.nops(),1);
+
+ for (unsigned r=0; r<eqns.nops(); r++) {
+ ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
+ ex linpart = eq;
+ for (unsigned c=0; c<symbols.nops(); c++) {
+ ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
+ linpart -= co*symbols.op(c);
+ sys(r,c) = co;
+ }
+ linpart = linpart.expand();
+ rhs(r,0) = -linpart;
+ }
+
+ // test if system is linear and fill vars matrix
+ for (unsigned i=0; i<symbols.nops(); i++) {
+ vars(i,0) = symbols.op(i);
+ if (sys.has(symbols.op(i)))
+ throw(std::logic_error("lsolve: system is not linear"));
+ if (rhs.has(symbols.op(i)))
+ throw(std::logic_error("lsolve: system is not linear"));
+ }
+
+ matrix solution;
+ try {
+ solution = sys.solve(vars,rhs);
+ } catch (const std::runtime_error & e) {
+ // Probably singular matrix or otherwise overdetermined system:
+ // It is consistent to return an empty list
+ return lst();
+ }
+ GINAC_ASSERT(solution.cols()==1);
+ GINAC_ASSERT(solution.rows()==symbols.nops());
+
+ // return list of equations of the form lst(var1==sol1,var2==sol2,...)
+ lst sollist;
+ for (unsigned i=0; i<symbols.nops(); i++)
+ sollist.append(symbols.op(i)==solution(i,0));
+
+ return sollist;
}
-/** Force inclusion of functions from initcns_gamma and inifcns_zeta
- * for static lib (so ginsh will see them). */
-unsigned force_include_gamma = function_index_Gamma;
+/* Force inclusion of functions from inifcns_gamma and inifcns_zeta
+ * for static lib (so ginsh will see them). */
+unsigned force_include_tgamma = function_index_tgamma;
unsigned force_include_zeta1 = function_index_zeta1;
-#ifndef NO_NAMESPACE_GINAC
} // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff