X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fpseries.cpp;h=bed134ed10abfab7506c7f8cbecd8ede4662191b;hp=87d0e4377a6ccbb3023723759b5c6be3773b45d8;hb=81efb5392132944d3649351b1efc5dac1a2625b1;hpb=6564c3b961f7e0b69c20187b56f90d86d4fdcb9a

diff --git a/ginac/pseries.cpp b/ginac/pseries.cpp
index 87d0e437..bed134ed 100644
--- a/ginac/pseries.cpp
+++ b/ginac/pseries.cpp
@@ -47,43 +47,43 @@ GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
 
 pseries::pseries() : basic(TINFO_pseries)
 {
-    debugmsg("pseries default constructor", LOGLEVEL_CONSTRUCT);
+	debugmsg("pseries default constructor", LOGLEVEL_CONSTRUCT);
 }
 
 pseries::~pseries()
 {
-    debugmsg("pseries destructor", LOGLEVEL_DESTRUCT);
-    destroy(false);
+	debugmsg("pseries destructor", LOGLEVEL_DESTRUCT);
+	destroy(false);
 }
 
 pseries::pseries(const pseries &other)
 {
-    debugmsg("pseries copy constructor", LOGLEVEL_CONSTRUCT);
-    copy(other);
+	debugmsg("pseries copy constructor", LOGLEVEL_CONSTRUCT);
+	copy(other);
 }
 
 const pseries &pseries::operator=(const pseries & other)
 {
-    debugmsg("pseries operator=", LOGLEVEL_ASSIGNMENT);
-    if (this != &other) {
-        destroy(true);
-        copy(other);
-    }
-    return *this;
+	debugmsg("pseries operator=", LOGLEVEL_ASSIGNMENT);
+	if (this != &other) {
+		destroy(true);
+		copy(other);
+	}
+	return *this;
 }
 
 void pseries::copy(const pseries &other)
 {
-    inherited::copy(other);
-    seq = other.seq;
-    var = other.var;
-    point = other.point;
+	inherited::copy(other);
+	seq = other.seq;
+	var = other.var;
+	point = other.point;
 }
 
 void pseries::destroy(bool call_parent)
 {
-    if (call_parent)
-        inherited::destroy(call_parent);
+	if (call_parent)
+		inherited::destroy(call_parent);
 }
 
 
@@ -97,15 +97,16 @@ void pseries::destroy(bool call_parent)
  *  the last coefficient can be Order(_ex1()) to represent a truncated,
  *  non-terminating series.
  *
- *  @param var_  series variable (must hold a symbol)
- *  @param point_  expansion point
+ *  @param rel_  expansion variable and point (must hold a relational)
  *  @param ops_  vector of {coefficient, power} pairs (coefficient must not be zero)
  *  @return newly constructed pseries */
-pseries::pseries(const ex &var_, const ex &point_, const epvector &ops_)
-    : basic(TINFO_pseries), seq(ops_), var(var_), point(point_)
+pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
 {
-    debugmsg("pseries constructor from ex,ex,epvector", LOGLEVEL_CONSTRUCT);
-    GINAC_ASSERT(is_ex_exactly_of_type(var_, symbol));
+	debugmsg("pseries constructor from ex,epvector", LOGLEVEL_CONSTRUCT);
+	GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
+	GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
+	point = rel_.rhs();
+	var = *static_cast<symbol *>(rel_.lhs().bp);
 }
 
 
@@ -116,135 +117,200 @@ pseries::pseries(const ex &var_, const ex &point_, const epvector &ops_)
 /** Construct object from archive_node. */
 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 {
-    debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
-    for (unsigned int i=0; true; i++) {
-        ex rest;
-        ex coeff;
-        if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
-            seq.push_back(expair(rest, coeff));
-        else
-            break;
-    }
-    n.find_ex("var", var, sym_lst);
-    n.find_ex("point", point, sym_lst);
+	debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
+	for (unsigned int i=0; true; ++i) {
+		ex rest;
+		ex coeff;
+		if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
+			seq.push_back(expair(rest, coeff));
+		else
+			break;
+	}
+	n.find_ex("var", var, sym_lst);
+	n.find_ex("point", point, sym_lst);
 }
 
 /** Unarchive the object. */
 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
 {
-    return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
+	return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
 }
 
 /** Archive the object. */
 void pseries::archive(archive_node &n) const
 {
-    inherited::archive(n);
-    epvector::const_iterator i = seq.begin(), iend = seq.end();
-    while (i != iend) {
-        n.add_ex("coeff", i->rest);
-        n.add_ex("power", i->coeff);
-        i++;
-    }
-    n.add_ex("var", var);
-    n.add_ex("point", point);
+	inherited::archive(n);
+	epvector::const_iterator i = seq.begin(), iend = seq.end();
+	while (i != iend) {
+		n.add_ex("coeff", i->rest);
+		n.add_ex("power", i->coeff);
+		++i;
+	}
+	n.add_ex("var", var);
+	n.add_ex("point", point);
 }
 
-
-/*
- *  Functions overriding virtual functions from base classes
- */
+//////////
+// functions overriding virtual functions from bases classes
+//////////
 
 basic *pseries::duplicate() const
 {
-    debugmsg("pseries duplicate", LOGLEVEL_DUPLICATE);
-    return new pseries(*this);
+	debugmsg("pseries duplicate", LOGLEVEL_DUPLICATE);
+	return new pseries(*this);
+}
+
+void pseries::print(std::ostream &os, unsigned upper_precedence) const
+{
+	debugmsg("pseries print", LOGLEVEL_PRINT);
+	if (precedence<=upper_precedence) os << "(";
+	for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
+		// omit zero terms
+		if (i->rest.is_zero())
+			continue;
+		// print a sign, if needed
+		if (i!=seq.begin())
+			os << '+';
+		if (!is_order_function(i->rest)) {
+			// print 'rest', i.e. the expansion coefficient
+			if (i->rest.info(info_flags::numeric) &&
+				i->rest.info(info_flags::positive)) {
+				os << i->rest;
+			} else
+				os << "(" << i->rest << ')';
+			// print 'coeff', something like (x-1)^42
+			if (!i->coeff.is_zero()) {
+				os << '*';
+				if (!point.is_zero())
+					os << '(' << var-point << ')';
+				else
+					os << var;
+				if (i->coeff.compare(_ex1())) {
+					os << '^';
+					if (i->coeff.info(info_flags::negative))
+						os << '(' << i->coeff << ')';
+					else
+						os << i->coeff;
+				}
+			}
+		} else {
+			os << Order(power(var-point,i->coeff));
+		}
+	}
+	if (precedence<=upper_precedence) os << ")";
 }
 
-void pseries::print(ostream &os, unsigned upper_precedence) const
-{
-    debugmsg("pseries print", LOGLEVEL_PRINT);
-    convert_to_poly().print(os, upper_precedence);
-}
 
-void pseries::printraw(ostream &os) const
+void pseries::printraw(std::ostream &os) const
 {
 	debugmsg("pseries printraw", LOGLEVEL_PRINT);
 	os << "pseries(" << var << ";" << point << ";";
-	for (epvector::const_iterator i=seq.begin(); i!=seq.end(); i++) {
+	for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
 		os << "(" << (*i).rest << "," << (*i).coeff << "),";
 	}
 	os << ")";
 }
 
+
+void pseries::printtree(std::ostream & os, unsigned indent) const
+{
+	debugmsg("pseries printtree",LOGLEVEL_PRINT);
+	os << std::string(indent,' ') << "pseries " 
+	   << ", hash=" << hashvalue
+	   << " (0x" << std::hex << hashvalue << std::dec << ")"
+	   << ", flags=" << flags << std::endl;
+	for (unsigned i=0; i<seq.size(); ++i) {
+		seq[i].rest.printtree(os,indent+delta_indent);
+		seq[i].coeff.printtree(os,indent+delta_indent);
+		if (i!=seq.size()-1)
+			os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
+	}
+	var.printtree(os, indent+delta_indent);
+	point.printtree(os, indent+delta_indent);
+}
+
+/** Return the number of operands including a possible order term. */
 unsigned pseries::nops(void) const
 {
-    return seq.size();
+	return seq.size();
 }
 
+
+/** Return the ith term in the series when represented as a sum. */
 ex pseries::op(int i) const
 {
-    if (i < 0 || unsigned(i) >= seq.size())
-        throw (std::out_of_range("op() out of range"));
-    return seq[i].rest * power(var - point, seq[i].coeff);
+	if (i < 0 || unsigned(i) >= seq.size())
+		throw (std::out_of_range("op() out of range"));
+	return seq[i].rest * power(var - point, seq[i].coeff);
 }
 
+
 ex &pseries::let_op(int i)
 {
-    throw (std::logic_error("let_op not defined for pseries"));
+	throw (std::logic_error("let_op not defined for pseries"));
 }
 
+
+/** Return degree of highest power of the series.  This is usually the exponent
+ *  of the Order term.  If s is not the expansion variable of the series, the
+ *  series is examined termwise. */
 int pseries::degree(const symbol &s) const
 {
-    if (var.is_equal(s)) {
-        // Return last exponent
-        if (seq.size())
-            return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
-        else
-            return 0;
-    } else {
-        epvector::const_iterator it = seq.begin(), itend = seq.end();
-        if (it == itend)
-            return 0;
-        int max_pow = INT_MIN;
-        while (it != itend) {
-            int pow = it->rest.degree(s);
-            if (pow > max_pow)
-                max_pow = pow;
-            it++;
-        }
-        return max_pow;
-    }
+	if (var.is_equal(s)) {
+		// Return last exponent
+		if (seq.size())
+			return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
+		else
+			return 0;
+	} else {
+		epvector::const_iterator it = seq.begin(), itend = seq.end();
+		if (it == itend)
+			return 0;
+		int max_pow = INT_MIN;
+		while (it != itend) {
+			int pow = it->rest.degree(s);
+			if (pow > max_pow)
+				max_pow = pow;
+			++it;
+		}
+		return max_pow;
+	}
 }
 
+/** Return degree of lowest power of the series.  This is usually the exponent
+ *  of the leading term.  If s is not the expansion variable of the series, the
+ *  series is examined termwise.  If s is the expansion variable but the
+ *  expansion point is not zero the series is not expanded to find the degree.
+ *  I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
 int pseries::ldegree(const symbol &s) const
 {
-    if (var.is_equal(s)) {
-        // Return first exponent
-        if (seq.size())
-            return ex_to_numeric((*(seq.begin())).coeff).to_int();
-        else
-            return 0;
-    } else {
-        epvector::const_iterator it = seq.begin(), itend = seq.end();
-        if (it == itend)
-            return 0;
-        int min_pow = INT_MAX;
-        while (it != itend) {
-            int pow = it->rest.ldegree(s);
-            if (pow < min_pow)
-                min_pow = pow;
-            it++;
-        }
-        return min_pow;
-    }
+	if (var.is_equal(s)) {
+		// Return first exponent
+		if (seq.size())
+			return ex_to_numeric((*(seq.begin())).coeff).to_int();
+		else
+			return 0;
+	} else {
+		epvector::const_iterator it = seq.begin(), itend = seq.end();
+		if (it == itend)
+			return 0;
+		int min_pow = INT_MAX;
+		while (it != itend) {
+			int pow = it->rest.ldegree(s);
+			if (pow < min_pow)
+				min_pow = pow;
+			++it;
+		}
+		return min_pow;
+	}
 }
 
 ex pseries::coeff(const symbol &s, int n) const
 {
-    if (var.is_equal(s)) {
+	if (var.is_equal(s)) {
 		if (seq.size() == 0)
 			return _ex0();
-
+		
 		// Binary search in sequence for given power
 		numeric looking_for = numeric(n);
 		int lo = 0, hi = seq.size() - 1;
@@ -266,40 +332,59 @@ ex pseries::coeff(const symbol &s, int n) const
 			}
 		}
 		return _ex0();
-    } else
-        return convert_to_poly().coeff(s, n);
+	} else
+		return convert_to_poly().coeff(s, n);
 }
 
+
 ex pseries::collect(const symbol &s) const
 {
-	if (var.is_equal(s))
-		return convert_to_poly();
-	else
-		return inherited::collect(s);
+	return *this;
 }
 
+
+/** Evaluate coefficients. */
 ex pseries::eval(int level) const
 {
-    if (level == 1)
-        return this->hold();
-    
-    // Construct a new series with evaluated coefficients
-    epvector new_seq;
-    new_seq.reserve(seq.size());
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    while (it != itend) {
-        new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
-        it++;
-    }
-    return (new pseries(var, point, new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
-}
-
-/** Evaluate numerically.  The order term is dropped. */
+	if (level == 1)
+		return this->hold();
+	
+	if (level == -max_recursion_level)
+		throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
+	
+	// Construct a new series with evaluated coefficients
+	epvector new_seq;
+	new_seq.reserve(seq.size());
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
+	while (it != itend) {
+		new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
+		++it;
+	}
+	return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
+}
+
+
+/** Evaluate coefficients numerically. */
 ex pseries::evalf(int level) const
 {
-    return convert_to_poly().evalf(level);
+	if (level == 1)
+		return *this;
+	
+	if (level == -max_recursion_level)
+		throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
+	
+	// Construct a new series with evaluated coefficients
+	epvector new_seq;
+	new_seq.reserve(seq.size());
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
+	while (it != itend) {
+		new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
+		++it;
+	}
+	return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
 }
 
+
 ex pseries::subs(const lst & ls, const lst & lr) const
 {
 	// If expansion variable is being substituted, convert the series to a
@@ -307,43 +392,58 @@ ex pseries::subs(const lst & ls, const lst & lr) const
 	// no longer be a power series
 	if (ls.has(var))
 		return convert_to_poly(true).subs(ls, lr);
-
+	
 	// Otherwise construct a new series with substituted coefficients and
 	// expansion point
-	epvector new_seq;
-	new_seq.reserve(seq.size());
+	epvector newseq;
+	newseq.reserve(seq.size());
 	epvector::const_iterator it = seq.begin(), itend = seq.end();
 	while (it != itend) {
-		new_seq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
-		it++;
+		newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
+		++it;
 	}
-    return (new pseries(var, point.subs(ls, lr), new_seq))->setflag(status_flags::dynallocated);
+	return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
 }
 
+
+/** Implementation of ex::expand() for a power series.  It expands all the
+ *  terms individually and returns the resulting series as a new pseries.
+ *  @see ex::diff */
+ex pseries::expand(unsigned options) const
+{
+	epvector newseq;
+	newseq.reserve(seq.size());
+	for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
+		newseq.push_back(expair(i->rest.expand(), i->coeff));
+	return (new pseries(relational(var,point), newseq))
+	        ->setflag(status_flags::dynallocated | status_flags::expanded);
+}
+
+
 /** Implementation of ex::diff() for a power series.  It treats the series as a
  *  polynomial.
  *  @see ex::diff */
 ex pseries::derivative(const symbol & s) const
 {
-    if (s == var) {
-        epvector new_seq;
-        epvector::const_iterator it = seq.begin(), itend = seq.end();
-        
-        // FIXME: coeff might depend on var
-        while (it != itend) {
-            if (is_order_function(it->rest)) {
-                new_seq.push_back(expair(it->rest, it->coeff - 1));
-            } else {
-                ex c = it->rest * it->coeff;
-                if (!c.is_zero())
-                    new_seq.push_back(expair(c, it->coeff - 1));
-            }
-            it++;
-        }
-        return pseries(var, point, new_seq);
-    } else {
-        return *this;
-    }
+	if (s == var) {
+		epvector new_seq;
+		epvector::const_iterator it = seq.begin(), itend = seq.end();
+		
+		// FIXME: coeff might depend on var
+		while (it != itend) {
+			if (is_order_function(it->rest)) {
+				new_seq.push_back(expair(it->rest, it->coeff - 1));
+			} else {
+				ex c = it->rest * it->coeff;
+				if (!c.is_zero())
+					new_seq.push_back(expair(c, it->coeff - 1));
+			}
+			++it;
+		}
+		return pseries(relational(var,point), new_seq);
+	} else {
+		return *this;
+	}
 }
 
 
@@ -356,18 +456,25 @@ ex pseries::derivative(const symbol & s) const
  *  @param no_order flag: discard higher order terms */
 ex pseries::convert_to_poly(bool no_order) const
 {
-    ex e;
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    
-    while (it != itend) {
-        if (is_order_function(it->rest)) {
-            if (!no_order)
-                e += Order(power(var - point, it->coeff));
-        } else
-            e += it->rest * power(var - point, it->coeff);
-        it++;
-    }
-    return e;
+	ex e;
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
+	
+	while (it != itend) {
+		if (is_order_function(it->rest)) {
+			if (!no_order)
+				e += Order(power(var - point, it->coeff));
+		} else
+			e += it->rest * power(var - point, it->coeff);
+		++it;
+	}
+	return e;
+}
+
+/** Returns true if there is no order term, i.e. the series terminates and
+ *  false otherwise. */
+bool pseries::is_terminating(void) const
+{
+	return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
 }
 
 
@@ -377,42 +484,48 @@ ex pseries::convert_to_poly(bool no_order) const
 
 /** Default implementation of ex::series(). This performs Taylor expansion.
  *  @see ex::series */
-ex basic::series(const symbol & s, const ex & point, int order) const
-{
-    epvector seq;
-    numeric fac(1);
-    ex deriv = *this;
-    ex coeff = deriv.subs(s == point);
-    if (!coeff.is_zero())
-        seq.push_back(expair(coeff, numeric(0)));
-    
-    int n;
-    for (n=1; n<order; n++) {
-        fac = fac.mul(numeric(n));
-        deriv = deriv.diff(s).expand();
-        if (deriv.is_zero()) {
-            // Series terminates
-            return pseries(s, point, seq);
-        }
-        coeff = fac.inverse() * deriv.subs(s == point);
-        if (!coeff.is_zero())
-            seq.push_back(expair(coeff, numeric(n)));
-    }
-    
-    // Higher-order terms, if present
-    deriv = deriv.diff(s);
-    if (!deriv.is_zero())
-        seq.push_back(expair(Order(_ex1()), numeric(n)));
-    return pseries(s, point, seq);
+ex basic::series(const relational & r, int order, unsigned options) const
+{
+	epvector seq;
+	numeric fac(1);
+	ex deriv = *this;
+	ex coeff = deriv.subs(r);
+	const symbol *s = static_cast<symbol *>(r.lhs().bp);
+	
+	if (!coeff.is_zero())
+		seq.push_back(expair(coeff, numeric(0)));
+	
+	int n;
+	for (n=1; n<order; ++n) {
+		fac = fac.mul(numeric(n));
+		deriv = deriv.diff(*s).expand();
+		if (deriv.is_zero()) {
+			// Series terminates
+			return pseries(r, seq);
+		}
+		coeff = deriv.subs(r);
+		if (!coeff.is_zero())
+			seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
+	}
+	
+	// Higher-order terms, if present
+	deriv = deriv.diff(*s);
+	if (!deriv.expand().is_zero())
+		seq.push_back(expair(Order(_ex1()), numeric(n)));
+	return pseries(r, seq);
 }
 
 
 /** Implementation of ex::series() for symbols.
  *  @see ex::series */
-ex symbol::series(const symbol & s, const ex & point, int order) const
+ex symbol::series(const relational & r, int order, unsigned options) const
 {
 	epvector seq;
-	if (is_equal(s)) {
+	const ex point = r.rhs();
+	GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
+	const symbol *s = static_cast<symbol *>(r.lhs().bp);
+	
+	if (this->is_equal(*s)) {
 		if (order > 0 && !point.is_zero())
 			seq.push_back(expair(point, _ex0()));
 		if (order > 1)
@@ -421,7 +534,7 @@ ex symbol::series(const symbol & s, const ex & point, int order) const
 			seq.push_back(expair(Order(_ex1()), numeric(order)));
 	} else
 		seq.push_back(expair(*this, _ex0()));
-	return pseries(s, point, seq);
+	return pseries(r, seq);
 }
 
 
@@ -432,99 +545,99 @@ ex symbol::series(const symbol & s, const ex & point, int order) const
  *  @return the sum as a pseries */
 ex pseries::add_series(const pseries &other) const
 {
-    // Adding two series with different variables or expansion points
-    // results in an empty (constant) series 
-    if (!is_compatible_to(other)) {
-        epvector nul;
-        nul.push_back(expair(Order(_ex1()), _ex0()));
-        return pseries(var, point, nul);
-    }
-    
-    // Series addition
-    epvector new_seq;
-    epvector::const_iterator a = seq.begin();
-    epvector::const_iterator b = other.seq.begin();
-    epvector::const_iterator a_end = seq.end();
-    epvector::const_iterator b_end = other.seq.end();
-    int pow_a = INT_MAX, pow_b = INT_MAX;
-    for (;;) {
-        // If a is empty, fill up with elements from b and stop
-        if (a == a_end) {
-            while (b != b_end) {
-                new_seq.push_back(*b);
-                b++;
-            }
-            break;
-        } else
-            pow_a = ex_to_numeric((*a).coeff).to_int();
-        
-        // If b is empty, fill up with elements from a and stop
-        if (b == b_end) {
-            while (a != a_end) {
-                new_seq.push_back(*a);
-                a++;
-            }
-            break;
-        } else
-            pow_b = ex_to_numeric((*b).coeff).to_int();
-        
-        // a and b are non-empty, compare powers
-        if (pow_a < pow_b) {
-            // a has lesser power, get coefficient from a
-            new_seq.push_back(*a);
-            if (is_order_function((*a).rest))
-                break;
-            a++;
-        } else if (pow_b < pow_a) {
-            // b has lesser power, get coefficient from b
-            new_seq.push_back(*b);
-            if (is_order_function((*b).rest))
-                break;
-            b++;
-        } else {
-            // Add coefficient of a and b
-            if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
-                new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
-                break;  // Order term ends the sequence
-            } else {
-                ex sum = (*a).rest + (*b).rest;
-                if (!(sum.is_zero()))
-                    new_seq.push_back(expair(sum, numeric(pow_a)));
-                a++;
-                b++;
-            }
-        }
-    }
-    return pseries(var, point, new_seq);
+	// Adding two series with different variables or expansion points
+	// results in an empty (constant) series 
+	if (!is_compatible_to(other)) {
+		epvector nul;
+		nul.push_back(expair(Order(_ex1()), _ex0()));
+		return pseries(relational(var,point), nul);
+	}
+	
+	// Series addition
+	epvector new_seq;
+	epvector::const_iterator a = seq.begin();
+	epvector::const_iterator b = other.seq.begin();
+	epvector::const_iterator a_end = seq.end();
+	epvector::const_iterator b_end = other.seq.end();
+	int pow_a = INT_MAX, pow_b = INT_MAX;
+	for (;;) {
+		// If a is empty, fill up with elements from b and stop
+		if (a == a_end) {
+			while (b != b_end) {
+				new_seq.push_back(*b);
+				++b;
+			}
+			break;
+		} else
+			pow_a = ex_to_numeric((*a).coeff).to_int();
+		
+		// If b is empty, fill up with elements from a and stop
+		if (b == b_end) {
+			while (a != a_end) {
+				new_seq.push_back(*a);
+				++a;
+			}
+			break;
+		} else
+			pow_b = ex_to_numeric((*b).coeff).to_int();
+		
+		// a and b are non-empty, compare powers
+		if (pow_a < pow_b) {
+			// a has lesser power, get coefficient from a
+			new_seq.push_back(*a);
+			if (is_order_function((*a).rest))
+				break;
+			++a;
+		} else if (pow_b < pow_a) {
+			// b has lesser power, get coefficient from b
+			new_seq.push_back(*b);
+			if (is_order_function((*b).rest))
+				break;
+			++b;
+		} else {
+			// Add coefficient of a and b
+			if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
+				new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
+				break;  // Order term ends the sequence
+			} else {
+				ex sum = (*a).rest + (*b).rest;
+				if (!(sum.is_zero()))
+					new_seq.push_back(expair(sum, numeric(pow_a)));
+				++a;
+				++b;
+			}
+		}
+	}
+	return pseries(relational(var,point), new_seq);
 }
 
 
 /** Implementation of ex::series() for sums. This performs series addition when
  *  adding pseries objects.
  *  @see ex::series */
-ex add::series(const symbol & s, const ex & point, int order) const
-{
-    ex acc; // Series accumulator
-    
-    // Get first term from overall_coeff
-    acc = overall_coeff.series(s, point, order);
-
-    // Add remaining terms
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    for (; it!=itend; it++) {
-        ex op;
-        if (is_ex_exactly_of_type(it->rest, pseries))
-            op = it->rest;
-        else
-            op = it->rest.series(s, point, order);
-        if (!it->coeff.is_equal(_ex1()))
-            op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
-        
-        // Series addition
-        acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
-    }
-    return acc;
+ex add::series(const relational & r, int order, unsigned options) const
+{
+	ex acc; // Series accumulator
+	
+	// Get first term from overall_coeff
+	acc = overall_coeff.series(r, order, options);
+	
+	// Add remaining terms
+	epvector::const_iterator it = seq.begin();
+	epvector::const_iterator itend = seq.end();
+	for (; it!=itend; ++it) {
+		ex op;
+		if (is_ex_exactly_of_type(it->rest, pseries))
+			op = it->rest;
+		else
+			op = it->rest.series(r, order, options);
+		if (!it->coeff.is_equal(_ex1()))
+			op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
+		
+		// Series addition
+		acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
+	}
+	return acc;
 }
 
 
@@ -535,18 +648,18 @@ ex add::series(const symbol & s, const ex & point, int order) const
  *  @return the product as a pseries */
 ex pseries::mul_const(const numeric &other) const
 {
-    epvector new_seq;
-    new_seq.reserve(seq.size());
-    
-    epvector::const_iterator it = seq.begin(), itend = seq.end();
-    while (it != itend) {
-        if (!is_order_function(it->rest))
-            new_seq.push_back(expair(it->rest * other, it->coeff));
-        else
-            new_seq.push_back(*it);
-        it++;
-    }
-    return pseries(var, point, new_seq);
+	epvector new_seq;
+	new_seq.reserve(seq.size());
+	
+	epvector::const_iterator it = seq.begin(), itend = seq.end();
+	while (it != itend) {
+		if (!is_order_function(it->rest))
+			new_seq.push_back(expair(it->rest * other, it->coeff));
+		else
+			new_seq.push_back(*it);
+		++it;
+	}
+	return pseries(relational(var,point), new_seq);
 }
 
 
@@ -557,82 +670,82 @@ ex pseries::mul_const(const numeric &other) const
  *  @return the product as a pseries */
 ex pseries::mul_series(const pseries &other) const
 {
-    // Multiplying two series with different variables or expansion points
-    // results in an empty (constant) series 
-    if (!is_compatible_to(other)) {
-        epvector nul;
-        nul.push_back(expair(Order(_ex1()), _ex0()));
-        return pseries(var, point, nul);
-    }
-
-    // Series multiplication
-    epvector new_seq;
-    
-    const symbol *s = static_cast<symbol *>(var.bp);
-    int a_max = degree(*s);
-    int b_max = other.degree(*s);
-    int a_min = ldegree(*s);
-    int b_min = other.ldegree(*s);
-    int cdeg_min = a_min + b_min;
-    int cdeg_max = a_max + b_max;
-    
-    int higher_order_a = INT_MAX;
-    int higher_order_b = INT_MAX;
-    if (is_order_function(coeff(*s, a_max)))
-        higher_order_a = a_max + b_min;
-    if (is_order_function(other.coeff(*s, b_max)))
-        higher_order_b = b_max + a_min;
-    int higher_order_c = min(higher_order_a, higher_order_b);
-    if (cdeg_max >= higher_order_c)
-        cdeg_max = higher_order_c - 1;
-    
-    for (int cdeg=cdeg_min; cdeg<=cdeg_max; cdeg++) {
-        ex co = _ex0();
-        // c(i)=a(0)b(i)+...+a(i)b(0)
-        for (int i=a_min; cdeg-i>=b_min; i++) {
-            ex a_coeff = coeff(*s, i);
-            ex b_coeff = other.coeff(*s, cdeg-i);
-            if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
-                co += coeff(*s, i) * other.coeff(*s, cdeg-i);
-        }
-        if (!co.is_zero())
-            new_seq.push_back(expair(co, numeric(cdeg)));
-    }
-    if (higher_order_c < INT_MAX)
-        new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
-    return pseries(var, point, new_seq);
+	// Multiplying two series with different variables or expansion points
+	// results in an empty (constant) series 
+	if (!is_compatible_to(other)) {
+		epvector nul;
+		nul.push_back(expair(Order(_ex1()), _ex0()));
+		return pseries(relational(var,point), nul);
+	}
+	
+	// Series multiplication
+	epvector new_seq;
+	
+	const symbol *s = static_cast<symbol *>(var.bp);
+	int a_max = degree(*s);
+	int b_max = other.degree(*s);
+	int a_min = ldegree(*s);
+	int b_min = other.ldegree(*s);
+	int cdeg_min = a_min + b_min;
+	int cdeg_max = a_max + b_max;
+	
+	int higher_order_a = INT_MAX;
+	int higher_order_b = INT_MAX;
+	if (is_order_function(coeff(*s, a_max)))
+		higher_order_a = a_max + b_min;
+	if (is_order_function(other.coeff(*s, b_max)))
+		higher_order_b = b_max + a_min;
+	int higher_order_c = std::min(higher_order_a, higher_order_b);
+	if (cdeg_max >= higher_order_c)
+		cdeg_max = higher_order_c - 1;
+	
+	for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
+		ex co = _ex0();
+		// c(i)=a(0)b(i)+...+a(i)b(0)
+		for (int i=a_min; cdeg-i>=b_min; ++i) {
+			ex a_coeff = coeff(*s, i);
+			ex b_coeff = other.coeff(*s, cdeg-i);
+			if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
+				co += a_coeff * b_coeff;
+		}
+		if (!co.is_zero())
+			new_seq.push_back(expair(co, numeric(cdeg)));
+	}
+	if (higher_order_c < INT_MAX)
+		new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
+	return pseries(relational(var,point), new_seq);
 }
 
 
 /** Implementation of ex::series() for product. This performs series
  *  multiplication when multiplying series.
  *  @see ex::series */
-ex mul::series(const symbol & s, const ex & point, int order) const
-{
-    ex acc; // Series accumulator
-    
-    // Get first term from overall_coeff
-    acc = overall_coeff.series(s, point, order);
-    
-    // Multiply with remaining terms
-    epvector::const_iterator it = seq.begin();
-    epvector::const_iterator itend = seq.end();
-    for (; it!=itend; it++) {
-        ex op = it->rest;
-        if (op.info(info_flags::numeric)) {
-            // series * const (special case, faster)
-            ex f = power(op, it->coeff);
-            acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
-            continue;
-        } else if (!is_ex_exactly_of_type(op, pseries))
-            op = op.series(s, point, order);
-        if (!it->coeff.is_equal(_ex1()))
-            op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
-
-        // Series multiplication
-        acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
-    }
-    return acc;
+ex mul::series(const relational & r, int order, unsigned options) const
+{
+	ex acc; // Series accumulator
+	
+	// Get first term from overall_coeff
+	acc = overall_coeff.series(r, order, options);
+	
+	// Multiply with remaining terms
+	epvector::const_iterator it = seq.begin();
+	epvector::const_iterator itend = seq.end();
+	for (; it!=itend; ++it) {
+		ex op = it->rest;
+		if (op.info(info_flags::numeric)) {
+			// series * const (special case, faster)
+			ex f = power(op, it->coeff);
+			acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
+			continue;
+		} else if (!is_ex_exactly_of_type(op, pseries))
+			op = op.series(r, order, options);
+		if (!it->coeff.is_equal(_ex1()))
+			op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
+
+		// Series multiplication
+		acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
+	}
+	return acc;
 }
 
 
@@ -642,84 +755,98 @@ ex mul::series(const symbol & s, const ex & point, int order) const
  *  @param deg  truncation order of series calculation */
 ex pseries::power_const(const numeric &p, int deg) const
 {
-    int i;
-    const symbol *s = static_cast<symbol *>(var.bp);
-    int ldeg = ldegree(*s);
-    
-    // Calculate coefficients of powered series
-    exvector co;
-    co.reserve(deg);
-    ex co0;
-    co.push_back(co0 = power(coeff(*s, ldeg), p));
-    bool all_sums_zero = true;
-    for (i=1; i<deg; i++) {
-        ex sum = _ex0();
-        for (int j=1; j<=i; j++) {
-            ex c = coeff(*s, j + ldeg);
-            if (is_order_function(c)) {
-                co.push_back(Order(_ex1()));
-                break;
-            } else
-                sum += (p * j - (i - j)) * co[i - j] * c;
-        }
-        if (!sum.is_zero())
-            all_sums_zero = false;
-        co.push_back(co0 * sum / numeric(i));
-    }
-    
-    // Construct new series (of non-zero coefficients)
-    epvector new_seq;
-    bool higher_order = false;
-    for (i=0; i<deg; i++) {
-        if (!co[i].is_zero())
-            new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
-        if (is_order_function(co[i])) {
-            higher_order = true;
-            break;
-        }
-    }
-    if (!higher_order && !all_sums_zero)
-        new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
-    return pseries(var, point, new_seq);
+	int i;
+	const symbol *s = static_cast<symbol *>(var.bp);
+	int ldeg = ldegree(*s);
+	
+	// Calculate coefficients of powered series
+	exvector co;
+	co.reserve(deg);
+	ex co0;
+	co.push_back(co0 = power(coeff(*s, ldeg), p));
+	bool all_sums_zero = true;
+	for (i=1; i<deg; ++i) {
+		ex sum = _ex0();
+		for (int j=1; j<=i; ++j) {
+			ex c = coeff(*s, j + ldeg);
+			if (is_order_function(c)) {
+				co.push_back(Order(_ex1()));
+				break;
+			} else
+				sum += (p * j - (i - j)) * co[i - j] * c;
+		}
+		if (!sum.is_zero())
+			all_sums_zero = false;
+		co.push_back(co0 * sum / numeric(i));
+	}
+	
+	// Construct new series (of non-zero coefficients)
+	epvector new_seq;
+	bool higher_order = false;
+	for (i=0; i<deg; ++i) {
+		if (!co[i].is_zero())
+			new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
+		if (is_order_function(co[i])) {
+			higher_order = true;
+			break;
+		}
+	}
+	if (!higher_order && !all_sums_zero)
+		new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
+	return pseries(relational(var,point), new_seq);
+}
+
+
+/** Return a new pseries object with the powers shifted by deg. */
+pseries pseries::shift_exponents(int deg) const
+{
+	epvector newseq(seq);
+	for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
+		i->coeff = i->coeff + deg;
+	return pseries(relational(var, point), newseq);
 }
 
 
 /** Implementation of ex::series() for powers. This performs Laurent expansion
  *  of reciprocals of series at singularities.
  *  @see ex::series */
-ex power::series(const symbol & s, const ex & point, int order) const
-{
-    ex e;
-    if (!is_ex_exactly_of_type(basis, pseries)) {
-        // Basis is not a series, may there be a singulary?
-        if (!exponent.info(info_flags::negint))
-            return basic::series(s, point, order);
-        
-        // Expression is of type something^(-int), check for singularity
-        if (!basis.subs(s == point).is_zero())
-            return basic::series(s, point, order);
-        
-        // Singularity encountered, expand basis into series
-        e = basis.series(s, point, order);
-    } else {
-        // Basis is a series
-        e = basis;
-    }
-    
-    // Power e
-    return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
+ex power::series(const relational & r, int order, unsigned options) const
+{
+	ex e;
+	if (!is_ex_exactly_of_type(basis, pseries)) {
+		// Basis is not a series, may there be a singulary?
+		if (!exponent.info(info_flags::negint))
+			return basic::series(r, order, options);
+		
+		// Expression is of type something^(-int), check for singularity
+		if (!basis.subs(r).is_zero())
+			return basic::series(r, order, options);
+		
+		// Singularity encountered, expand basis into series
+		e = basis.series(r, order, options);
+	} else {
+		// Basis is a series
+		e = basis;
+	}
+	
+	// Power e
+	return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
 }
 
 
 /** Re-expansion of a pseries object. */
-ex pseries::series(const symbol & s, const ex & p, int order) const
-{
-	if (var.is_equal(s) && point.is_equal(p)) {
-		if (order > degree(s))
+ex pseries::series(const relational & r, int order, unsigned options) const
+{
+	const ex p = r.rhs();
+	GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
+	const symbol *s = static_cast<symbol *>(r.lhs().bp);
+	
+	if (var.is_equal(*s) && point.is_equal(p)) {
+		if (order > degree(*s))
 			return *this;
 		else {
-	        epvector new_seq;
-	        epvector::const_iterator it = seq.begin(), itend = seq.end();
+			epvector new_seq;
+			epvector::const_iterator it = seq.begin(), itend = seq.end();
 			while (it != itend) {
 				int o = ex_to_numeric(it->coeff).to_int();
 				if (o >= order) {
@@ -727,34 +854,59 @@ ex pseries::series(const symbol & s, const ex & p, int order) const
 					break;
 				}
 				new_seq.push_back(*it);
-				it++;
+				++it;
 			}
-			return pseries(var, point, new_seq);
+			return pseries(r, new_seq);
 		}
 	} else
-		return convert_to_poly().series(s, p, order);
+		return convert_to_poly().series(r, order, options);
 }
 
 
 /** Compute the truncated series expansion of an expression.
- *  This function returns an expression containing an object of class pseries to
- *  represent the series. If the series does not terminate within the given
+ *  This function returns an expression containing an object of class pseries 
+ *  to represent the series. If the series does not terminate within the given
  *  truncation order, the last term of the series will be an order term.
  *
- *  @param s  expansion variable
- *  @param point  expansion point
+ *  @param r  expansion relation, lhs holds variable and rhs holds point
  *  @param order  truncation order of series calculations
+ *  @param options  of class series_options
  *  @return an expression holding a pseries object */
-ex ex::series(const symbol &s, const ex &point, int order) const
-{
-    GINAC_ASSERT(bp!=0);
-    return bp->series(s, point, order);
+ex ex::series(const ex & r, int order, unsigned options) const
+{
+	GINAC_ASSERT(bp!=0);
+	ex e;
+	relational rel_;
+	
+	if (is_ex_exactly_of_type(r,relational))
+		rel_ = ex_to_relational(r);
+	else if (is_ex_exactly_of_type(r,symbol))
+		rel_ = relational(r,_ex0());
+	else
+		throw (std::logic_error("ex::series(): expansion point has unknown type"));
+	
+	try {
+		e = bp->series(rel_, order, options);
+	} catch (std::exception &x) {
+		throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
+	}
+	return e;
 }
 
+//////////
+// static member variables
+//////////
+
+// protected
+
+unsigned pseries::precedence = 38;  // for clarity just below add::precedence
+
+//////////
+// global constants
+//////////
 
-// Global constants
 const pseries some_pseries;
-const type_info & typeid_pseries = typeid(some_pseries);
+const std::type_info & typeid_pseries = typeid(some_pseries);
 
 #ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC