X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns.cpp;h=677d92f61925a47a10599375a72ae16e0f7be817;hp=28cadfb174ecf16680dd18d37d9af8ad0f1af1d2;hb=db5765dc91202851739e196ba11bfccb0b3fe7bc;hpb=4afb5bbe2c0b0a60928120a042997ba7d89e8f5c

diff --git a/ginac/inifcns.cpp b/ginac/inifcns.cpp
index 28cadfb1..677d92f6 100644
--- a/ginac/inifcns.cpp
+++ b/ginac/inifcns.cpp
@@ -3,7 +3,7 @@
  *  Implementation of GiNaC's initially known functions. */
 
 /*
- *  GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2000 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
@@ -33,54 +33,314 @@
 #include "numeric.h"
 #include "power.h"
 #include "relational.h"
-#include "series.h"
+#include "pseries.h"
 #include "symbol.h"
 #include "utils.h"
 
-#ifndef NO_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC
+
+//////////
+// absolute value
+//////////
+
+static ex abs_evalf(const ex & arg)
+{
+    BEGIN_TYPECHECK
+        TYPECHECK(arg,numeric)
+    END_TYPECHECK(abs(arg))
+    
+    return abs(ex_to_numeric(arg));
+}
+
+static ex abs_eval(const ex & arg)
+{
+    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_exactly_of_type(arg, mul)) {
+        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())
+        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));
+
 
 //////////
 // dilogarithm
 //////////
 
-static ex Li2_eval(ex const & x)
+static ex Li2_evalf(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();
+    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.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, Li2_eval, NULL, NULL, NULL);
+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));
 
 //////////
 // trilogarithm
 //////////
 
-static ex Li3_eval(ex const & x)
+static ex Li3_eval(const ex & x)
 {
     if (x.is_zero())
         return x;
     return Li3(x).hold();
 }
 
-REGISTER_FUNCTION(Li3, Li3_eval, NULL, NULL, NULL);
+REGISTER_FUNCTION(Li3, eval_func(Li3_eval));
 
 //////////
 // factorial
 //////////
 
-static ex factorial_evalf(ex const & x)
+static ex factorial_evalf(const ex & x)
 {
     return factorial(x).hold();
 }
 
-static ex factorial_eval(ex const & x)
+static ex factorial_eval(const ex & x)
 {
     if (is_ex_exactly_of_type(x, numeric))
         return factorial(ex_to_numeric(x));
@@ -88,18 +348,19 @@ static ex factorial_eval(ex const & x)
         return factorial(x).hold();
 }
 
-REGISTER_FUNCTION(factorial, factorial_eval, factorial_evalf, NULL, NULL);
+REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
+                             evalf_func(factorial_evalf));
 
 //////////
 // binomial
 //////////
 
-static ex binomial_evalf(ex const & x, ex const & y)
+static ex binomial_evalf(const ex & x, const ex & y)
 {
     return binomial(x, y).hold();
 }
 
-static ex binomial_eval(ex const & x, ex const &y)
+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));
@@ -107,52 +368,72 @@ static ex binomial_eval(ex const & x, ex const &y)
         return binomial(x, y).hold();
 }
 
-REGISTER_FUNCTION(binomial, binomial_eval, binomial_evalf, NULL, NULL);
+REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
+                            evalf_func(binomial_evalf));
 
 //////////
 // Order term function (for truncated power series)
 //////////
 
-static ex Order_eval(ex const & x)
+static ex Order_eval(const ex & x)
 {
-	if (is_ex_exactly_of_type(x, numeric)) {
+    if (is_ex_exactly_of_type(x, numeric)) {
+        // 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);
+        // 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();
+}
 
-		// O(c)=O(1)
-		return Order(_ex1()).hold();
+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;
+    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);
+}
 
-	} else if (is_ex_exactly_of_type(x, mul)) {
+// Differentiation is handled in function::derivative because of its special requirements
 
-	 	mul *m = static_cast<mul *>(x.bp);
-		if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric)) {
+REGISTER_FUNCTION(Order, eval_func(Order_eval).
+                         series_func(Order_series));
 
-			// O(c*expr)=O(expr)
-			return Order(x / m->op(m->nops() - 1)).hold();
-		}
-	}
-	return Order(x).hold();
-}
+//////////
+// Inert partial differentiation operator
+//////////
 
-static ex Order_series(ex const & x, symbol const & s, ex const & point, int order)
+static ex Derivative_eval(const ex & f, const ex & l)
 {
-	// Just wrap the function into a series object
-	epvector new_seq;
-	new_seq.push_back(expair(Order(_ex1()), numeric(min(x.ldegree(s), order))));
-	return series(s, point, new_seq);
+	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(Order, Order_eval, NULL, NULL, Order_series);
+REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
 
 //////////
 // Solve linear system
 //////////
 
-ex lsolve(ex const &eqns, ex const &symbols)
+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"));
-        }
+        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);
@@ -163,19 +444,19 @@ ex lsolve(ex const &eqns, ex const &symbols)
     
     // syntax checks
     if (!eqns.info(info_flags::list)) {
-        throw(std::invalid_argument("lsolve: 1st argument must be a list"));
+        throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
     }
-    for (int i=0; i<eqns.nops(); i++) {
+    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"));
+            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"));
+        throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
     }
-    for (int i=0; i<symbols.nops(); i++) {
+    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"));
+            throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
         }
     }
     
@@ -184,61 +465,48 @@ ex lsolve(ex const &eqns, ex const &symbols)
     matrix rhs(eqns.nops(),1);
     matrix vars(symbols.nops(),1);
     
-    for (int 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 (int c=0; c<symbols.nops(); c++) {
-            ex co=eq.coeff(ex_to_symbol(symbols.op(c)),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();
+        linpart = linpart.expand();
         rhs.set(r,0,-linpart);
     }
     
     // test if system is linear and fill vars matrix
-    for (int i=0; i<symbols.nops(); i++) {
+    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);
-    } catch (runtime_error const & e) {
-        // probably singular matrix (or other error)
-        // return empty solution list
-        // cerr << e.what() << endl;
+        solution = sys.solve(vars,rhs);
+    } catch (const 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 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,...)
+    // return list of equations of the form lst(var1==sol1,var2==sol2,...)
     lst sollist;
-    for (int i=0; i<symbols.nops(); i++) {
+    for (unsigned i=0; i<symbols.nops(); i++)
         sollist.append(symbols.op(i)==solution(i,0));
-    }
     
     return sollist;
 }
 
 /** non-commutative power. */
-ex ncpower(ex const &basis, unsigned exponent)
+ex ncpower(const ex &basis, unsigned exponent)
 {
     if (exponent==0) {
         return _ex1();
@@ -255,9 +523,9 @@ ex ncpower(ex const &basis, unsigned exponent)
 
 /** 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_GINAC_NAMESPACE
+#ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC
-#endif // ndef NO_GINAC_NAMESPACE
+#endif // ndef NO_NAMESPACE_GINAC