* Implementation of GiNaC's initially known functions. */
/*
- * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2004 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 "operators.h"
#include "relational.h"
#include "pseries.h"
#include "symbol.h"
+#include "symmetry.h"
#include "utils.h"
-#ifndef NO_GINAC_NAMESPACE
namespace GiNaC {
-#endif // ndef NO_GINAC_NAMESPACE
+
+//////////
+// complex conjugate
+//////////
+
+static ex conjugate_evalf(const ex & arg)
+{
+ if (is_exactly_a<numeric>(arg)) {
+ return ex_to<numeric>(arg).conjugate();
+ }
+ return conjugate_function(arg).hold();
+}
+
+static ex conjugate_eval(const ex & arg)
+{
+ return arg.conjugate();
+}
+
+static void conjugate_print_latex(const ex & arg, const print_context & c)
+{
+ c.s << "\\bar{"; arg.print(c); c.s << "}";
+}
+
+static ex conjugate_conjugate(const ex & arg)
+{
+ return arg;
+}
+
+REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
+ evalf_func(conjugate_evalf).
+ print_func<print_latex>(conjugate_print_latex).
+ conjugate_func(conjugate_conjugate).
+ set_name("conjugate","conjugate"));
//////////
// absolute value
//////////
-static ex abs_evalf(const ex & x)
+static ex abs_evalf(const ex & arg)
+{
+ if (is_exactly_a<numeric>(arg))
+ return abs(ex_to<numeric>(arg));
+
+ return abs(arg).hold();
+}
+
+static ex abs_eval(const ex & arg)
+{
+ if (is_exactly_a<numeric>(arg))
+ return abs(ex_to<numeric>(arg));
+ else
+ return abs(arg).hold();
+}
+
+static void abs_print_latex(const ex & arg, const print_context & c)
+{
+ c.s << "{|"; arg.print(c); c.s << "|}";
+}
+
+static void abs_print_csrc_float(const ex & arg, const print_context & c)
+{
+ c.s << "fabs("; arg.print(c); c.s << ")";
+}
+
+static ex abs_conjugate(const ex & arg)
+{
+ return abs(arg);
+}
+
+REGISTER_FUNCTION(abs, eval_func(abs_eval).
+ evalf_func(abs_evalf).
+ print_func<print_latex>(abs_print_latex).
+ print_func<print_csrc_float>(abs_print_csrc_float).
+ print_func<print_csrc_double>(abs_print_csrc_float).
+ conjugate_func(abs_conjugate));
+
+
+//////////
+// Complex sign
+//////////
+
+static ex csgn_evalf(const ex & arg)
+{
+ if (is_exactly_a<numeric>(arg))
+ return csgn(ex_to<numeric>(arg));
+
+ return csgn(arg).hold();
+}
+
+static ex csgn_eval(const ex & arg)
+{
+ if (is_exactly_a<numeric>(arg))
+ return csgn(ex_to<numeric>(arg));
+
+ else if (is_exactly_a<mul>(arg) &&
+ is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
+ 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, subs_options::no_pattern);
+ 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);
+}
+
+static ex csgn_conjugate(const ex& arg)
+{
+ return csgn(arg);
+}
+
+REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
+ evalf_func(csgn_evalf).
+ series_func(csgn_series).
+ conjugate_func(csgn_conjugate));
+
+
+//////////
+// Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
+// This function is closely related to the unwinding number K, sometimes found
+// in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
+//////////
+
+static ex eta_evalf(const ex &x, const ex &y)
+{
+ // It seems like we basically have to replicate the eval function here,
+ // since the expression might not be fully evaluated yet.
+ if (x.info(info_flags::positive) || y.info(info_flags::positive))
+ return _ex0;
+
+ if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
+ const numeric nx = ex_to<numeric>(x);
+ const numeric ny = ex_to<numeric>(y);
+ const numeric nxy = ex_to<numeric>(x*y);
+ int cut = 0;
+ if (nx.is_real() && nx.is_negative())
+ cut -= 4;
+ if (ny.is_real() && ny.is_negative())
+ cut -= 4;
+ if (nxy.is_real() && nxy.is_negative())
+ cut += 4;
+ return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
+ (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
+ }
+
+ return eta(x,y).hold();
+}
+
+static ex eta_eval(const ex &x, const ex &y)
+{
+ // trivial: eta(x,c) -> 0 if c is real and positive
+ if (x.info(info_flags::positive) || y.info(info_flags::positive))
+ return _ex0;
+
+ if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
+ // don't call eta_evalf here because it would call Pi.evalf()!
+ const numeric nx = ex_to<numeric>(x);
+ const numeric ny = ex_to<numeric>(y);
+ const numeric nxy = ex_to<numeric>(x*y);
+ int cut = 0;
+ if (nx.is_real() && nx.is_negative())
+ cut -= 4;
+ if (ny.is_real() && ny.is_negative())
+ cut -= 4;
+ if (nxy.is_real() && nxy.is_negative())
+ cut += 4;
+ return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
+ (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
+ }
+
+ return eta(x,y).hold();
+}
+
+static ex eta_series(const ex & x, const ex & y,
+ const relational & rel,
+ int order,
+ unsigned options)
{
- BEGIN_TYPECHECK
- TYPECHECK(x,numeric)
- END_TYPECHECK(abs(x))
-
- return abs(ex_to_numeric(x));
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ const ex y_pt = y.subs(rel, subs_options::no_pattern);
+ if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
+ (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
+ ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
+ throw (std::domain_error("eta_series(): on discontinuity"));
+ epvector seq;
+ seq.push_back(expair(eta(x_pt,y_pt), _ex0));
+ return pseries(rel,seq);
}
-static ex abs_eval(const ex & x)
+static ex eta_conjugate(const ex & x, const ex & y)
{
- if (is_ex_exactly_of_type(x, numeric))
- return abs(ex_to_numeric(x));
- else
- return abs(x).hold();
+ return -eta(x,y);
}
-REGISTER_FUNCTION(abs, abs_eval, abs_evalf, NULL, NULL);
+REGISTER_FUNCTION(eta, eval_func(eta_eval).
+ evalf_func(eta_evalf).
+ series_func(eta_series).
+ latex_name("\\eta").
+ set_symmetry(sy_symm(0, 1)).
+ conjugate_func(eta_conjugate));
+
//////////
// dilogarithm
//////////
+static ex Li2_evalf(const ex & x)
+{
+ if (is_exactly_a<numeric>(x))
+ return Li2(ex_to<numeric>(x));
+
+ return Li2(x).hold();
+}
+
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(ex_to<numeric>(x));
+ }
+
+ return Li2(x).hold();
+}
+
+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(_ex1-x)/x;
+}
+
+static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
+{
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ 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), subs_options::no_pattern);
+ // 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.is_equal(_ex1)) {
+ // method:
+ // construct series manually in a dummy symbol s
+ const symbol s;
+ ex ser = zeta(_ex2);
+ // 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), subs_options::no_pattern);
+ // 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 = ex_to<symbol>(rel.lhs());
+ 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 (size_t 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, subs_options::no_pattern),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, Li2_eval, NULL, NULL, NULL);
+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).
+ latex_name("\\mbox{Li}_3"));
+
+//////////
+// Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
+//////////
+
+static ex zetaderiv_eval(const ex & n, const ex & x)
+{
+ if (n.info(info_flags::numeric)) {
+ // zetaderiv(0,x) -> zeta(x)
+ if (n.is_zero())
+ return zeta(x);
+ }
+
+ return zetaderiv(n, x).hold();
+}
+
+static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param<2);
+
+ if (deriv_param==0) {
+ // d/dn zeta(n,x)
+ throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
+ }
+ // d/dx psi(n,x)
+ return zetaderiv(n+1,x);
}
-REGISTER_FUNCTION(Li3, Li3_eval, NULL, NULL, NULL);
+REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
+ derivative_func(zetaderiv_deriv).
+ latex_name("\\zeta^\\prime"));
//////////
// 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_exactly_a<numeric>(x))
+ return factorial(ex_to<numeric>(x));
+ else
+ return factorial(x).hold();
+}
+
+static ex factorial_conjugate(const ex & x)
+{
+ return factorial(x);
}
-REGISTER_FUNCTION(factorial, factorial_eval, factorial_evalf, NULL, NULL);
+REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
+ evalf_func(factorial_evalf).
+ conjugate_func(factorial_conjugate));
//////////
// binomial
static ex binomial_evalf(const ex & x, const ex & y)
{
- return binomial(x, y).hold();
+ return binomial(x, y).hold();
+}
+
+static ex binomial_sym(const ex & x, const numeric & y)
+{
+ if (y.is_integer()) {
+ if (y.is_nonneg_integer()) {
+ const unsigned N = y.to_int();
+ if (N == 0) return _num0;
+ if (N == 1) return x;
+ ex t = x.expand();
+ for (unsigned i = 2; i <= N; ++i)
+ t = (t * (x + i - y - 1)).expand() / i;
+ return t;
+ } else
+ return _num0;
+ }
+
+ 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_exactly_a<numeric>(y)) {
+ if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
+ return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
+ else
+ return binomial_sym(x, ex_to<numeric>(y));
+ } else
+ return binomial(x, y).hold();
+}
+
+// At the moment the numeric evaluation of a binomail function always
+// gives a real number, but if this would be implemented using the gamma
+// function, also complex conjugation should be changed (or rather, deleted).
+static ex binomial_conjugate(const ex & x, const ex & y)
+{
+ return binomial(x,y);
}
-REGISTER_FUNCTION(binomial, binomial_eval, binomial_evalf, NULL, NULL);
+REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
+ evalf_func(binomial_evalf).
+ conjugate_func(binomial_conjugate));
//////////
// Order term function (for truncated power series)
static ex Order_eval(const ex & x)
{
- if (is_ex_exactly_of_type(x, numeric)) {
-
- // O(c)=O(1)
- return Order(_ex1()).hold();
-
- } 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)
- return Order(x / m->op(m->nops() - 1)).hold();
- }
+ if (is_exactly_a<numeric>(x)) {
+ // O(c) -> O(1) or 0
+ if (!x.is_zero())
+ return Order(_ex1).hold();
+ else
+ return _ex0;
+ } else if (is_exactly_a<mul>(x)) {
+ const mul &m = ex_to<mul>(x);
+ // O(c*expr) -> O(expr)
+ if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
+ 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_a<symbol>(r.lhs()));
+ const symbol &s = ex_to<symbol>(r.lhs());
+ new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
+ return pseries(r, new_seq);
+}
+
+static ex Order_conjugate(const ex & x)
+{
+ return Order(x);
}
-REGISTER_FUNCTION(Order, Order_eval, NULL, NULL, Order_series);
+// Differentiation is handled in function::derivative because of its special requirements
+
+REGISTER_FUNCTION(Order, eval_func(Order_eval).
+ series_func(Order_series).
+ latex_name("\\mathcal{O}").
+ conjugate_func(Order_conjugate));
//////////
// Solve linear system
//////////
-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);
-}
-
-/** 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_zeta1 = function_index_zeta1;
-
-#ifndef NO_GINAC_NAMESPACE
+ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
+{
+ // 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"));
+ const ex sol = lsolve(lst(eqns),lst(symbols));
+
+ GINAC_ASSERT(sol.nops()==1);
+ GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
+
+ 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 (size_t 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 (size_t 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 (size_t r=0; r<eqns.nops(); r++) {
+ const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
+ ex linpart = eq;
+ for (size_t c=0; c<symbols.nops(); c++) {
+ const 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 (size_t 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,options);
+ } 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 (size_t i=0; i<symbols.nops(); i++)
+ sollist.append(symbols.op(i)==solution(i,0));
+
+ return sollist;
+}
+
+/* Force inclusion of functions from inifcns_gamma and inifcns_zeta
+ * for static lib (so ginsh will see them). */
+unsigned force_include_tgamma = tgamma_SERIAL::serial;
+unsigned force_include_zeta1 = zeta1_SERIAL::serial;
+
} // namespace GiNaC
-#endif // ndef NO_GINAC_NAMESPACE
git://www.ginac.de / ginac.git / blobdiff