return abs(x).hold();
}
-REGISTER_FUNCTION(abs, evalf_func(abs_eval).
+REGISTER_FUNCTION(abs, eval_func(abs_eval).
evalf_func(abs_evalf));
+
+//////////
+// Complex sign
+//////////
+
+static ex csgn_evalf(const ex & x)
+{
+ BEGIN_TYPECHECK
+ TYPECHECK(x,numeric)
+ END_TYPECHECK(csgn(x))
+
+ return csgn(ex_to_numeric(x));
+}
+
+static ex csgn_eval(const ex & x)
+{
+ if (is_ex_exactly_of_type(x, numeric))
+ return csgn(ex_to_numeric(x));
+
+ else if (is_ex_exactly_of_type(x, mul)) {
+ numeric oc = ex_to_numeric(x.op(x.nops()-1));
+ if (oc.is_real()) {
+ if (oc > 0)
+ // csgn(42*x) -> csgn(x)
+ return csgn(x/oc).hold();
+ else
+ // csgn(-42*x) -> -csgn(x)
+ return -csgn(x/oc).hold();
+ }
+ if (oc.real().is_zero()) {
+ if (oc.imag() > 0)
+ // csgn(42*I*x) -> csgn(I*x)
+ return csgn(I*x/oc).hold();
+ else
+ // csgn(-42*I*x) -> -csgn(I*x)
+ return -csgn(I*x/oc).hold();
+ }
+ }
+
+ return csgn(x).hold();
+}
+
+static ex csgn_series(const ex & x, const relational & rel, int order)
+{
+ const ex x_pt = x.subs(rel);
+ if (x_pt.info(info_flags::numeric)) {
+ if (ex_to_numeric(x_pt).real().is_zero())
+ throw (std::domain_error("csgn_series(): on imaginary axis"));
+ epvector seq;
+ seq.push_back(expair(csgn(x_pt), _ex0()));
+ return pseries(rel,seq);
+ }
+ epvector seq;
+ seq.push_back(expair(csgn(x_pt), _ex0()));
+ return pseries(rel,seq);
+}
+
+REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
+ evalf_func(csgn_evalf).
+ series_func(csgn_series));
+
//////////
// dilogarithm
//////////
static ex Li2_eval(const ex & x)
{
+ // Li2(0) -> 0
if (x.is_zero())
return x;
+ // Li2(1) -> Pi^2/6
if (x.is_equal(_ex1()))
- return power(Pi, _ex2()) / _ex6();
+ 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();
+ 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;
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)
+{
+ const ex x_pt = x.subs(rel);
+ if (!x_pt.is_zero() && !x_pt.is_equal(_ex1()))
+ throw do_taylor(); // caught by function::series()
+ // First 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;
+ // construct manually the primitive expansion
+ for (int i=1; i<order; ++i)
+ ser += pow(s,i)/pow(numeric(i),numeric(2));
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel,order));
+ // maybe that was terminanting, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1()), numeric(order)));
+ ser += pseries(rel, nseq);
+ // reexpand will collapse the series again
+ ser = ser.series(rel,order);
+ return ser;
+ // NOTE: 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.
+ }
+ // second problematic case: x real, >=1 (branch cut)
+ return pseries();
+ // TODO: Li2_series should do something around branch point?
+ // Careful: may involve logs!
+}
+
+REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
+ derivative_func(Li2_deriv).
+ series_func(Li2_series));
//////////
// trilogarithm
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)
{
// 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(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));
+//////////
+// 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"));
+ }
+ if (!is_ex_exactly_of_type(l, lst)) {
+ throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
+ }
+ return Derivative(f, l).hold();
+}
+
+REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
+
//////////
// Solve linear system
//////////
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;
+ 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);
+ ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
linpart -= co*symbols.op(c);
sys.set(r,c,co);
}
// 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))) {
+ if (sys.has(symbols.op(i)))
throw(std::logic_error("lsolve: system is not linear"));
- }
- if (rhs.has(symbols.op(i))) {
+ 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);
+ solution = sys.fraction_free_elim(vars,rhs);
} catch (const runtime_error & e) {
// probably singular matrix (or other error)
// return empty solution list
/** 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;
+unsigned force_include_tgamma = function_index_tgamma;
unsigned force_include_zeta1 = function_index_zeta1;
#ifndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff