/** @file indexed.cpp
*
* Implementation of GiNaC's indexed expressions. */
/*
* GiNaC Copyright (C) 1999-2007 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
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include <iostream>
#include <sstream>
#include <stdexcept>
#include <limits>
#include "indexed.h"
#include "idx.h"
#include "add.h"
#include "mul.h"
#include "ncmul.h"
#include "power.h"
#include "relational.h"
#include "symmetry.h"
#include "operators.h"
#include "lst.h"
#include "archive.h"
#include "symbol.h"
#include "utils.h"
#include "integral.h"
#include "matrix.h"
namespace GiNaC {
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
print_func<print_context>(&indexed::do_print).
print_func<print_latex>(&indexed::do_print_latex).
print_func<print_tree>(&indexed::do_print_tree))
//////////
// default constructor
//////////
indexed::indexed() : symtree(not_symmetric())
{
tinfo_key = TINFO_indexed;
}
//////////
// other constructors
//////////
indexed::indexed(const ex & b) : inherited(b), symtree(not_symmetric())
{
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(not_symmetric())
{
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(not_symmetric())
{
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(not_symmetric())
{
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(not_symmetric())
{
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
{
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
{
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(symm)
{
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(not_symmetric())
{
seq.insert(seq.end(), v.begin(), v.end());
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
{
seq.insert(seq.end(), v.begin(), v.end());
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
{
tinfo_key = TINFO_indexed;
}
indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
{
tinfo_key = TINFO_indexed;
}
indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
{
tinfo_key = TINFO_indexed;
}
//////////
// archiving
//////////
indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
{
if (!n.find_ex("symmetry", symtree, sym_lst)) {
// GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
unsigned symm = 0;
n.find_unsigned("symmetry", symm);
switch (symm) {
case 1:
symtree = sy_symm();
break;
case 2:
symtree = sy_anti();
break;
default:
symtree = not_symmetric();
break;
}
const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
}
}
void indexed::archive(archive_node &n) const
{
inherited::archive(n);
n.add_ex("symmetry", symtree);
}
DEFAULT_UNARCHIVE(indexed)
//////////
// functions overriding virtual functions from base classes
//////////
void indexed::printindices(const print_context & c, unsigned level) const
{
if (seq.size() > 1) {
exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
if (is_a<print_latex>(c)) {
// TeX output: group by variance
bool first = true;
bool covariant = true;
while (it != itend) {
bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
if (first || cur_covariant != covariant) { // Variance changed
// The empty {} prevents indices from ending up on top of each other
if (!first)
c.s << "}{}";
covariant = cur_covariant;
if (covariant)
c.s << "_{";
else
c.s << "^{";
}
it->print(c, level);
c.s << " ";
first = false;
it++;
}
c.s << "}";
} else {
// Ordinary output
while (it != itend) {
it->print(c, level);
it++;
}
}
}
}
void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
{
if (precedence() <= level)
c.s << openbrace << '(';
c.s << openbrace;
seq[0].print(c, precedence());
c.s << closebrace;
printindices(c, level);
if (precedence() <= level)
c.s << ')' << closebrace;
}
void indexed::do_print(const print_context & c, unsigned level) const
{
print_indexed(c, "", "", level);
}
void indexed::do_print_latex(const print_latex & c, unsigned level) const
{
print_indexed(c, "{", "}", level);
}
void indexed::do_print_tree(const print_tree & c, unsigned level) const
{
c.s << std::string(level, ' ') << class_name() << " @" << this
<< std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
<< ", " << seq.size()-1 << " indices"
<< ", symmetry=" << symtree << std::endl;
seq[0].print(c, level + c.delta_indent);
printindices(c, level + c.delta_indent);
}
bool indexed::info(unsigned inf) const
{
if (inf == info_flags::indexed) return true;
if (inf == info_flags::has_indices) return seq.size() > 1;
return inherited::info(inf);
}
struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
bool operator() (const ex & e, unsigned inf) const {
return !(ex_to<idx>(e).get_value().info(inf));
}
};
bool indexed::all_index_values_are(unsigned inf) const
{
// No indices? Then no property can be fulfilled
if (seq.size() < 2)
return false;
// Check all indices
return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
}
int indexed::compare_same_type(const basic & other) const
{
GINAC_ASSERT(is_a<indexed>(other));
return inherited::compare_same_type(other);
}
ex indexed::eval(int level) const
{
// First evaluate children, then we will end up here again
if (level > 1)
return indexed(ex_to<symmetry>(symtree), evalchildren(level));
const ex &base = seq[0];
// If the base object is 0, the whole object is 0
if (base.is_zero())
return _ex0;
// If the base object is a product, pull out the numeric factor
if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
exvector v(seq);
ex f = ex_to<numeric>(base.op(base.nops() - 1));
v[0] = seq[0] / f;
return f * thiscontainer(v);
}
if(this->tinfo()==TINFO_indexed && seq.size()==1)
return base;
// Canonicalize indices according to the symmetry properties
if (seq.size() > 2) {
exvector v = seq;
GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
if (sig != std::numeric_limits<int>::max()) {
// Something has changed while sorting indices, more evaluations later
if (sig == 0)
return _ex0;
return ex(sig) * thiscontainer(v);
}
}
// Let the class of the base object perform additional evaluations
return ex_to<basic>(base).eval_indexed(*this);
}
ex indexed::thiscontainer(const exvector & v) const
{
return indexed(ex_to<symmetry>(symtree), v);
}
ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
{
return indexed(ex_to<symmetry>(symtree), vp);
}
ex indexed::expand(unsigned options) const
{
GINAC_ASSERT(seq.size() > 0);
if (options & expand_options::expand_indexed) {
ex newbase = seq[0].expand(options);
if (is_exactly_a<add>(newbase)) {
ex sum = _ex0;
for (size_t i=0; i<newbase.nops(); i++) {
exvector s = seq;
s[0] = newbase.op(i);
sum += thiscontainer(s).expand(options);
}
return sum;
}
if (!are_ex_trivially_equal(newbase, seq[0])) {
exvector s = seq;
s[0] = newbase;
return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
}
}
return inherited::expand(options);
}
//////////
// virtual functions which can be overridden by derived classes
//////////
// none
//////////
// non-virtual functions in this class
//////////
/** Check whether all indices are of class idx and validate the symmetry
* tree. This function is used internally to make sure that all constructed
* indexed objects really carry indices and not some other classes. */
void indexed::validate() const
{
GINAC_ASSERT(seq.size() > 0);
exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
while (it != itend) {
if (!is_a<idx>(*it))
throw(std::invalid_argument("indices of indexed object must be of type idx"));
it++;
}
if (!symtree.is_zero()) {
if (!is_exactly_a<symmetry>(symtree))
throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
}
}
/** Implementation of ex::diff() for an indexed object always returns 0.
*
* @see ex::diff */
ex indexed::derivative(const symbol & s) const
{
return _ex0;
}
//////////
// global functions
//////////
struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
bool operator() (const ex &lh, const ex &rh) const
{
if (lh.is_equal(rh))
return true;
else
try {
// Replacing the dimension might cause an error (e.g. with
// index classes that only work in a fixed number of dimensions)
return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
} catch (...) {
return false;
}
}
};
/** Check whether two sorted index vectors are consistent (i.e. equal). */
static bool indices_consistent(const exvector & v1, const exvector & v2)
{
// Number of indices must be the same
if (v1.size() != v2.size())
return false;
return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
}
exvector indexed::get_indices() const
{
GINAC_ASSERT(seq.size() >= 1);
return exvector(seq.begin() + 1, seq.end());
}
exvector indexed::get_dummy_indices() const
{
exvector free_indices, dummy_indices;
find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
return dummy_indices;
}
exvector indexed::get_dummy_indices(const indexed & other) const
{
exvector indices = get_free_indices();
exvector other_indices = other.get_free_indices();
indices.insert(indices.end(), other_indices.begin(), other_indices.end());
exvector dummy_indices;
find_dummy_indices(indices, dummy_indices);
return dummy_indices;
}
bool indexed::has_dummy_index_for(const ex & i) const
{
exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
while (it != itend) {
if (is_dummy_pair(*it, i))
return true;
it++;
}
return false;
}
exvector indexed::get_free_indices() const
{
exvector free_indices, dummy_indices;
find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
return free_indices;
}
exvector add::get_free_indices() const
{
exvector free_indices;
for (size_t i=0; i<nops(); i++) {
if (i == 0)
free_indices = op(i).get_free_indices();
else {
exvector free_indices_of_term = op(i).get_free_indices();
if (!indices_consistent(free_indices, free_indices_of_term))
throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
}
}
return free_indices;
}
exvector mul::get_free_indices() const
{
// Concatenate free indices of all factors
exvector un;
for (size_t i=0; i<nops(); i++) {
exvector free_indices_of_factor = op(i).get_free_indices();
un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
}
// And remove the dummy indices
exvector free_indices, dummy_indices;
find_free_and_dummy(un, free_indices, dummy_indices);
return free_indices;
}
exvector ncmul::get_free_indices() const
{
// Concatenate free indices of all factors
exvector un;
for (size_t i=0; i<nops(); i++) {
exvector free_indices_of_factor = op(i).get_free_indices();
un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
}
// And remove the dummy indices
exvector free_indices, dummy_indices;
find_free_and_dummy(un, free_indices, dummy_indices);
return free_indices;
}
struct is_summation_idx : public std::unary_function<ex, bool> {
bool operator()(const ex & e)
{
return is_dummy_pair(e, e);
}
};
exvector power::get_free_indices() const
{
// Get free indices of basis
exvector basis_indices = basis.get_free_indices();
if (exponent.info(info_flags::even)) {
// If the exponent is an even number, then any "free" index that
// forms a dummy pair with itself is actually a summation index
exvector really_free;
std::remove_copy_if(basis_indices.begin(), basis_indices.end(),
std::back_inserter(really_free), is_summation_idx());
return really_free;
} else
return basis_indices;
}
exvector integral::get_free_indices() const
{
if (a.get_free_indices().size() || b.get_free_indices().size())
throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
return f.get_free_indices();
}
template<class T> size_t number_of_type(const exvector&v)
{
size_t number = 0;
for(exvector::const_iterator i=v.begin(); i!=v.end(); ++i)
if(is_exactly_a<T>(*i))
++number;
return number;
}
/** Rename dummy indices in an expression.
*
* @param e Expression to work on
* @param local_dummy_indices The set of dummy indices that appear in the
* expression "e"
* @param global_dummy_indices The set of dummy indices that have appeared
* before and which we would like to use in "e", too. This gets updated
* by the function */
template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
{
size_t global_size = number_of_type<T>(global_dummy_indices),
local_size = number_of_type<T>(local_dummy_indices);
// Any local dummy indices at all?
if (local_size == 0)
return e;
if (global_size < local_size) {
// More local indices than we encountered before, add the new ones
// to the global set
size_t old_global_size = global_size;
int remaining = local_size - global_size;
exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
while (it != itend && remaining > 0) {
if (is_exactly_a<T>(*it) && find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(idx_is_equal_ignore_dim(), *it)) == global_dummy_indices.end()) {
global_dummy_indices.push_back(*it);
global_size++;
remaining--;
}
it++;
}
// If this is the first set of local indices, do nothing
if (old_global_size == 0)
return e;
}
GINAC_ASSERT(local_size <= global_size);
// Construct vectors of index symbols
exvector local_syms, global_syms;
local_syms.reserve(local_size);
global_syms.reserve(local_size);
for (size_t i=0; local_syms.size()!=local_size; i++)
if(is_exactly_a<T>(local_dummy_indices[i]))
local_syms.push_back(local_dummy_indices[i].op(0));
shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
for (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
if(is_exactly_a<T>(global_dummy_indices[i]))
global_syms.push_back(global_dummy_indices[i].op(0));
shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
// Remove common indices
exvector local_uniq, global_uniq;
set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
// Replace remaining non-common local index symbols by global ones
if (local_uniq.empty())
return e;
else {
while (global_uniq.size() > local_uniq.size())
global_uniq.pop_back();
return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
}
}
/** Given a set of indices, extract those of class varidx. */
static void find_variant_indices(const exvector & v, exvector & variant_indices)
{
exvector::const_iterator it1, itend;
for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
if (is_exactly_a<varidx>(*it1))
variant_indices.push_back(*it1);
}
}
/** Raise/lower dummy indices in a single indexed objects to canonicalize their
* variance.
*
* @param e Object to work on
* @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
* @param moved_indices The set of indices that have been repositioned (will be changed by this function)
* @return true if 'e' was changed */
bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
{
bool something_changed = false;
// Find dummy symbols that occur twice in the same indexed object.
exvector local_var_dummies;
local_var_dummies.reserve(e.nops()/2);
for (size_t i=1; i<e.nops(); ++i) {
if (!is_a<varidx>(e.op(i)))
continue;
for (size_t j=i+1; j<e.nops(); ++j) {
if (is_dummy_pair(e.op(i), e.op(j))) {
local_var_dummies.push_back(e.op(i));
for (exvector::iterator k = variant_dummy_indices.begin();
k!=variant_dummy_indices.end(); ++k) {
if (e.op(i).op(0) == k->op(0)) {
variant_dummy_indices.erase(k);
break;
}
}
break;
}
}
}
// In the case where a dummy symbol occurs twice in the same indexed object
// we try all posibilities of raising/lowering and keep the least one in
// the sense of ex_is_less.
ex optimal_e = e;
size_t numpossibs = 1 << local_var_dummies.size();
for (size_t i=0; i<numpossibs; ++i) {
ex try_e = e;
for (size_t j=0; j<local_var_dummies.size(); ++j) {
exmap m;
if (1<<j & i) {
ex curr_idx = local_var_dummies[j];
ex curr_toggle = ex_to<varidx>(curr_idx).toggle_variance();
m[curr_idx] = curr_toggle;
m[curr_toggle] = curr_idx;
}
try_e = e.subs(m, subs_options::no_pattern);
}
if(ex_is_less()(try_e, optimal_e))
{ optimal_e = try_e;
something_changed = true;
}
}
e = optimal_e;
if (!is_a<indexed>(e))
return true;
exvector seq = ex_to<indexed>(e).seq;
// If a dummy index is encountered for the first time in the
// product, pull it up, otherwise, pull it down
for (exvector::iterator it2 = seq.begin()+1, it2end = seq.end();
it2 != it2end; ++it2) {
if (!is_exactly_a<varidx>(*it2))
continue;
exvector::iterator vit, vitend;
for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
if (it2->op(0).is_equal(vit->op(0))) {
if (ex_to<varidx>(*it2).is_covariant()) {
/*
* N.B. we don't want to use
*
* e = e.subs(lst(
* *it2 == ex_to<varidx>(*it2).toggle_variance(),
* ex_to<varidx>(*it2).toggle_variance() == *it2
* ), subs_options::no_pattern);
*
* since this can trigger non-trivial repositioning of indices,
* e.g. due to non-trivial symmetry properties of e, thus
* invalidating iterators
*/
*it2 = ex_to<varidx>(*it2).toggle_variance();
something_changed = true;
}
moved_indices.push_back(*vit);
variant_dummy_indices.erase(vit);
goto next_index;
}
}
for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
if (it2->op(0).is_equal(vit->op(0))) {
if (ex_to<varidx>(*it2).is_contravariant()) {
*it2 = ex_to<varidx>(*it2).toggle_variance();
something_changed = true;
}
goto next_index;
}
}
next_index: ;
}
if (something_changed)
e = ex_to<indexed>(e).thiscontainer(seq);
return something_changed;
}
/* Ordering that only compares the base expressions of indexed objects. */
struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
bool operator() (const ex &lh, const ex &rh) const
{
return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
}
};
/* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
* It returns an exvector of factors from the supplied product */
static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
{
// Remember whether the product was commutative or noncommutative
// (because we chop it into factors and need to reassemble later)
non_commutative = is_exactly_a<ncmul>(e);
// Collect factors in an exvector, store squares twice
v.reserve(e.nops() * 2);
if (is_exactly_a<power>(e)) {
// We only get called for simple squares, split a^2 -> a*a
GINAC_ASSERT(e.op(1).is_equal(_ex2));
v.push_back(e.op(0));
v.push_back(e.op(0));
} else {
for (size_t i=0; i<e.nops(); i++) {
ex f = e.op(i);
if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
v.push_back(f.op(0));
v.push_back(f.op(0));
} else if (is_exactly_a<ncmul>(f)) {
// Noncommutative factor found, split it as well
non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
for (size_t j=0; j<f.nops(); j++)
v.push_back(f.op(j));
} else
v.push_back(f);
}
}
}
// Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
{ exvector dummy_syms;
dummy_syms.reserve(r.nops());
for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
if(is_exactly_a<T>(*it))
dummy_syms.push_back(it->op(0));
if(dummy_syms.size() < 2)
return r;
ex q=symmetrize(r, dummy_syms);
return q;
}
/** Simplify product of indexed expressions (commutative, noncommutative and
* simple squares), return list of free indices. */
ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
{
// Collect factors in an exvector
exvector v;
// Remember whether the product was commutative or noncommutative
// (because we chop it into factors and need to reassemble later)
bool non_commutative;
product_to_exvector(e, v, non_commutative);
// Perform contractions
bool something_changed = false;
GINAC_ASSERT(v.size() > 1);
exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
for (it1 = v.begin(); it1 != next_to_last; it1++) {
try_again:
if (!is_a<indexed>(*it1))
continue;
bool first_noncommutative = (it1->return_type() != return_types::commutative);
// Indexed factor found, get free indices and look for contraction
// candidates
exvector free1, dummy1;
find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
exvector::iterator it2;
for (it2 = it1 + 1; it2 != itend; it2++) {
if (!is_a<indexed>(*it2))
continue;
bool second_noncommutative = (it2->return_type() != return_types::commutative);
// Find free indices of second factor and merge them with free
// indices of first factor
exvector un;
find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
un.insert(un.end(), free1.begin(), free1.end());
// Check whether the two factors share dummy indices
exvector free, dummy;
find_free_and_dummy(un, free, dummy);
size_t num_dummies = dummy.size();
if (num_dummies == 0)
continue;
// At least one dummy index, is it a defined scalar product?
bool contracted = false;
if (free.empty() && it1->nops()==2 && it2->nops()==2) {
ex dim = minimal_dim(
ex_to<idx>(it1->op(1)).get_dim(),
ex_to<idx>(it2->op(1)).get_dim()
);
// User-defined scalar product?
if (sp.is_defined(*it1, *it2, dim)) {
// Yes, substitute it
*it1 = sp.evaluate(*it1, *it2, dim);
*it2 = _ex1;
goto contraction_done;
}
}
// Try to contract the first one with the second one
contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
if (!contracted) {
// That didn't work; maybe the second object knows how to
// contract itself with the first one
contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
}
if (contracted) {
contraction_done:
if (first_noncommutative || second_noncommutative
|| is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
|| is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
|| is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
// One of the factors became a sum or product:
// re-expand expression and run again
// Non-commutative products are always re-expanded to give
// eval_ncmul() the chance to re-order and canonicalize
// the product
ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
return simplify_indexed(r, free_indices, dummy_indices, sp);
}
// Both objects may have new indices now or they might
// even not be indexed objects any more, so we have to
// start over
something_changed = true;
goto try_again;
}
}
}
// Find free indices (concatenate them all and call find_free_and_dummy())
// and all dummy indices that appear
exvector un, individual_dummy_indices;
for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
exvector free_indices_of_factor;
if (is_a<indexed>(*it1)) {
exvector dummy_indices_of_factor;
find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free_indices_of_factor, dummy_indices_of_factor);
individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
} else
free_indices_of_factor = it1->get_free_indices();
un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
}
exvector local_dummy_indices;
find_free_and_dummy(un, free_indices, local_dummy_indices);
local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
// Filter out the dummy indices with variance
exvector variant_dummy_indices;
find_variant_indices(local_dummy_indices, variant_dummy_indices);
// Any indices with variance present at all?
if (!variant_dummy_indices.empty()) {
// Yes, bring the product into a canonical order that only depends on
// the base expressions of indexed objects
if (!non_commutative)
std::sort(v.begin(), v.end(), ex_base_is_less());
exvector moved_indices;
// Iterate over all indexed objects in the product
for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
if (!is_a<indexed>(*it1))
continue;
if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
something_changed = true;
}
}
ex r;
if (something_changed)
r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
else
r = e;
// The result should be symmetric with respect to exchange of dummy
// indices, so if the symmetrization vanishes, the whole expression is
// zero. This detects things like eps.i.j.k * p.j * p.k = 0.
ex q = idx_symmetrization<idx>(r, local_dummy_indices);
if (q.is_zero()) {
free_indices.clear();
return _ex0;
}
q = idx_symmetrization<varidx>(q, local_dummy_indices);
if (q.is_zero()) {
free_indices.clear();
return _ex0;
}
q = idx_symmetrization<spinidx>(q, local_dummy_indices);
if (q.is_zero()) {
free_indices.clear();
return _ex0;
}
// Dummy index renaming
r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
// Product of indexed object with a scalar?
if (is_exactly_a<mul>(r) && r.nops() == 2
&& is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
else
return r;
}
/** This structure stores the original and symmetrized versions of terms
* obtained during the simplification of sums. */
class terminfo {
public:
terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
ex orig; /**< original term */
ex symm; /**< symmtrized term */
};
class terminfo_is_less {
public:
bool operator() (const terminfo & ti1, const terminfo & ti2) const
{
return (ti1.symm.compare(ti2.symm) < 0);
}
};
/** This structure stores the individual symmetrized terms obtained during
* the simplification of sums. */
class symminfo {
public:
symminfo() : num(0) {}
symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
{
if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
coeff = symmterm_.op(symmterm_.nops()-1);
symmterm = symmterm_ / coeff;
} else {
coeff = 1;
symmterm = symmterm_;
}
}
ex symmterm; /**< symmetrized term */
ex coeff; /**< coefficient of symmetrized term */
ex orig; /**< original term */
size_t num; /**< how many symmetrized terms resulted from the original term */
};
class symminfo_is_less_by_symmterm {
public:
bool operator() (const symminfo & si1, const symminfo & si2) const
{
return (si1.symmterm.compare(si2.symmterm) < 0);
}
};
class symminfo_is_less_by_orig {
public:
bool operator() (const symminfo & si1, const symminfo & si2) const
{
return (si1.orig.compare(si2.orig) < 0);
}
};
bool hasindex(const ex &x, const ex &sym)
{
if(is_a<idx>(x) && x.op(0)==sym)
return true;
else
for(size_t i=0; i<x.nops(); ++i)
if(hasindex(x.op(i), sym))
return true;
return false;
}
/** Simplify indexed expression, return list of free indices. */
ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
{
// Expand the expression
ex e_expanded = e.expand();
// Simplification of single indexed object: just find the free indices
// and perform dummy index renaming/repositioning
if (is_a<indexed>(e_expanded)) {
// Find the dummy indices
const indexed &i = ex_to<indexed>(e_expanded);
exvector local_dummy_indices;
find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
// Filter out the dummy indices with variance
exvector variant_dummy_indices;
find_variant_indices(local_dummy_indices, variant_dummy_indices);
// Any indices with variance present at all?
if (!variant_dummy_indices.empty()) {
// Yes, reposition them
exvector moved_indices;
reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
}
// Rename the dummy indices
e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
return e_expanded;
}
// Simplification of sum = sum of simplifications, check consistency of
// free indices in each term
if (is_exactly_a<add>(e_expanded)) {
bool first = true;
ex sum;
free_indices.clear();
for (size_t i=0; i<e_expanded.nops(); i++) {
exvector free_indices_of_term;
ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
if (!term.is_zero()) {
if (first) {
free_indices = free_indices_of_term;
sum = term;
first = false;
} else {
if (!indices_consistent(free_indices, free_indices_of_term)) {
std::ostringstream s;
s << "simplify_indexed: inconsistent indices in sum: ";
s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
throw (std::runtime_error(s.str()));
}
if (is_a<indexed>(sum) && is_a<indexed>(term))
sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
else
sum += term;
}
}
}
// If the sum turns out to be zero, we are finished
if (sum.is_zero()) {
free_indices.clear();
return sum;
}
// More than one term and more than one dummy index?
size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
if (num_terms_orig < 2 || dummy_indices.size() < 2)
return sum;
// Chop the sum into terms and symmetrize each one over the dummy
// indices
std::vector<terminfo> terms;
for (size_t i=0; i<sum.nops(); i++) {
const ex & term = sum.op(i);
exvector dummy_indices_of_term;
dummy_indices_of_term.reserve(dummy_indices.size());
for(exvector::iterator i=dummy_indices.begin(); i!=dummy_indices.end(); ++i)
if(hasindex(term,i->op(0)))
dummy_indices_of_term.push_back(*i);
ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
if (term_symm.is_zero())
continue;
terms.push_back(terminfo(term, term_symm));
}
// Sort by symmetrized terms
std::sort(terms.begin(), terms.end(), terminfo_is_less());
// Combine equal symmetrized terms
std::vector<terminfo> terms_pass2;
for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
size_t num = 1;
std::vector<terminfo>::const_iterator j = i + 1;
while (j != terms.end() && j->symm == i->symm) {
num++;
j++;
}
terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
i = j;
}
// If there is only one term left, we are finished
if (terms_pass2.size() == 1)
return terms_pass2[0].orig;
// Chop the symmetrized terms into subterms
std::vector<symminfo> sy;
for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
if (is_exactly_a<add>(i->symm)) {
size_t num = i->symm.nops();
for (size_t j=0; j<num; j++)
sy.push_back(symminfo(i->symm.op(j), i->orig, num));
} else
sy.push_back(symminfo(i->symm, i->orig, 1));
}
// Sort by symmetrized subterms
std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
// Combine equal symmetrized subterms
std::vector<symminfo> sy_pass2;
exvector result;
for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
// Combine equal terms
std::vector<symminfo>::const_iterator j = i + 1;
if (j != sy.end() && j->symmterm == i->symmterm) {
// More than one term, collect the coefficients
ex coeff = i->coeff;
while (j != sy.end() && j->symmterm == i->symmterm) {
coeff += j->coeff;
j++;
}
// Add combined term to result
if (!coeff.is_zero())
result.push_back(coeff * i->symmterm);
} else {
// Single term, store for second pass
sy_pass2.push_back(*i);
}
i = j;
}
// Were there any remaining terms that didn't get combined?
if (sy_pass2.size() > 0) {
// Yes, sort by their original terms
std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
// How many symmetrized terms of this original term are left?
size_t num = 1;
std::vector<symminfo>::const_iterator j = i + 1;
while (j != sy_pass2.end() && j->orig == i->orig) {
num++;
j++;
}
if (num == i->num) {
// All terms left, then add the original term to the result
result.push_back(i->orig);
} else {
// Some terms were combined with others, add up the remaining symmetrized terms
std::vector<symminfo>::const_iterator k;
for (k=i; k!=j; k++)
result.push_back(k->coeff * k->symmterm);
}
i = j;
}
}
// Add all resulting terms
ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
if (sum_symm.is_zero())
free_indices.clear();
return sum_symm;
}
// Simplification of products
if (is_exactly_a<mul>(e_expanded)
|| is_exactly_a<ncmul>(e_expanded)
|| (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
// Cannot do anything
free_indices.clear();
return e_expanded;
}
/** Simplify/canonicalize expression containing indexed objects. This
* performs contraction of dummy indices where possible and checks whether
* the free indices in sums are consistent.
*
* @param options Simplification options (currently unused)
* @return simplified expression */
ex ex::simplify_indexed(unsigned options) const
{
exvector free_indices, dummy_indices;
scalar_products sp;
return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
}
/** Simplify/canonicalize expression containing indexed objects. This
* performs contraction of dummy indices where possible, checks whether
* the free indices in sums are consistent, and automatically replaces
* scalar products by known values if desired.
*
* @param sp Scalar products to be replaced automatically
* @param options Simplification options (currently unused)
* @return simplified expression */
ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
{
exvector free_indices, dummy_indices;
return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
}
/** Symmetrize expression over its free indices. */
ex ex::symmetrize() const
{
return GiNaC::symmetrize(*this, get_free_indices());
}
/** Antisymmetrize expression over its free indices. */
ex ex::antisymmetrize() const
{
return GiNaC::antisymmetrize(*this, get_free_indices());
}
/** Symmetrize expression by cyclic permutation over its free indices. */
ex ex::symmetrize_cyclic() const
{
return GiNaC::symmetrize_cyclic(*this, get_free_indices());
}
//////////
// helper classes
//////////
spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
{
// If indexed, extract base objects
ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
// Enforce canonical order in pair
if (s1.compare(s2) > 0) {
v1 = s2;
v2 = s1;
} else {
v1 = s1;
v2 = s2;
}
}
bool spmapkey::operator==(const spmapkey &other) const
{
if (!v1.is_equal(other.v1))
return false;
if (!v2.is_equal(other.v2))
return false;
if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
return true;
else
return dim.is_equal(other.dim);
}
bool spmapkey::operator<(const spmapkey &other) const
{
int cmp = v1.compare(other.v1);
if (cmp)
return cmp < 0;
cmp = v2.compare(other.v2);
if (cmp)
return cmp < 0;
// Objects are equal, now check dimensions
if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
return false;
else
return dim.compare(other.dim) < 0;
}
void spmapkey::debugprint() const
{
std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
}
void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
{
spm[spmapkey(v1, v2)] = sp;
}
void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
{
spm[spmapkey(v1, v2, dim)] = sp;
}
void scalar_products::add_vectors(const lst & l, const ex & dim)
{
// Add all possible pairs of products
for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
add(*it1, *it2, *it1 * *it2);
}
void scalar_products::clear()
{
spm.clear();
}
/** Check whether scalar product pair is defined. */
bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
{
return spm.find(spmapkey(v1, v2, dim)) != spm.end();
}
/** Return value of defined scalar product pair. */
ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
{
return spm.find(spmapkey(v1, v2, dim))->second;
}
void scalar_products::debugprint() const
{
std::cerr << "map size=" << spm.size() << std::endl;
spmap::const_iterator i = spm.begin(), end = spm.end();
while (i != end) {
const spmapkey & k = i->first;
std::cerr << "item key=";
k.debugprint();
std::cerr << ", value=" << i->second << std::endl;
++i;
}
}
/** Returns all dummy indices from the exvector */
exvector get_all_dummy_indices(const ex & e)
{
exvector p;
bool nc;
product_to_exvector(e, p, nc);
exvector::const_iterator ip = p.begin(), ipend = p.end();
exvector v, v1;
while (ip != ipend) {
if (is_a<indexed>(*ip)) {
v1 = ex_to<indexed>(*ip).get_dummy_indices();
v.insert(v.end(), v1.begin(), v1.end());
exvector::const_iterator ip1 = ip+1;
while (ip1 != ipend) {
if (is_a<indexed>(*ip1)) {
v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
v.insert(v.end(), v1.begin(), v1.end());
}
++ip1;
}
}
++ip;
}
return v;
}
ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
{
exvector va = get_all_dummy_indices(a), vb = get_all_dummy_indices(b), common_indices;
set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
if (common_indices.empty()) {
return b;
} else {
exvector new_indices, old_indices;
old_indices.reserve(2*common_indices.size());
new_indices.reserve(2*common_indices.size());
exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
while (ip != ipend) {
if (is_a<varidx>(*ip)) {
varidx mu((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(*ip).get_dim(), ex_to<varidx>(*ip).is_covariant());
old_indices.push_back(*ip);
new_indices.push_back(mu);
old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
new_indices.push_back(mu.toggle_variance());
} else {
old_indices.push_back(*ip);
new_indices.push_back(idx((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(*ip).get_dim()));
}
++ip;
}
return b.subs(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()), subs_options::no_pattern);
}
}
ex expand_dummy_sum(const ex & e, bool subs_idx)
{
ex e_expanded = e.expand();
pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
return e_expanded.map(fcn);
} else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded) || is_a<indexed>(e_expanded)) {
exvector v;
if (is_a<indexed>(e_expanded))
v = ex_to<indexed>(e_expanded).get_dummy_indices();
else
v = get_all_dummy_indices(e_expanded);
ex result = e_expanded;
for(exvector::const_iterator it=v.begin(); it!=v.end(); ++it) {
ex nu = *it;
if (ex_to<idx>(nu).get_dim().info(info_flags::nonnegint)) {
int idim = ex_to<numeric>(ex_to<idx>(nu).get_dim()).to_int();
ex en = 0;
for (int i=0; i < idim; i++) {
if (subs_idx && is_a<varidx>(nu)) {
ex other = ex_to<varidx>(nu).toggle_variance();
en += result.subs(lst(
nu == idx(i, idim),
other == idx(i, idim)
));
} else {
en += result.subs( nu.op(0) == i );
}
}
result = en;
}
}
return result;
} else {
return e;
}
}
} // namespace GiNaC
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