* Implementation of GiNaC's indexed expressions. */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2003 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
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
+#include <iostream>
+#include <sstream>
#include <stdexcept>
-#include <algorithm>
#include "indexed.h"
#include "idx.h"
#include "mul.h"
#include "ncmul.h"
#include "power.h"
+#include "relational.h"
#include "symmetry.h"
+#include "operators.h"
#include "lst.h"
-#include "print.h"
#include "archive.h"
#include "utils.h"
-#include "debugmsg.h"
namespace GiNaC {
-GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
+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, destructor, copy constructor assignment operator and helpers
+// default constructor
//////////
indexed::indexed() : symtree(sy_none())
{
- debugmsg("indexed default constructor", LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_indexed;
}
-void indexed::copy(const indexed & other)
-{
- inherited::copy(other);
- symtree = other.symtree;
-}
-
-DEFAULT_DESTROY(indexed)
-
//////////
// other constructors
//////////
indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
{
- debugmsg("indexed constructor from ex", LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
{
- debugmsg("indexed constructor from ex,ex", LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
{
- debugmsg("indexed constructor from ex,ex,ex", LOGLEVEL_CONSTRUCT);
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(sy_none())
{
- debugmsg("indexed constructor from ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
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(sy_none())
{
- debugmsg("indexed constructor from ex,ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
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)
{
- debugmsg("indexed constructor from ex,symmetry,ex,ex", LOGLEVEL_CONSTRUCT);
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)
{
- debugmsg("indexed constructor from ex,symmetry,ex,ex,ex", LOGLEVEL_CONSTRUCT);
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)
{
- debugmsg("indexed constructor from ex,symmetry,ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_indexed;
validate();
}
indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
{
- debugmsg("indexed constructor from ex,exvector", LOGLEVEL_CONSTRUCT);
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)
{
- debugmsg("indexed constructor from ex,symmetry,exvector", LOGLEVEL_CONSTRUCT);
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)
{
- debugmsg("indexed constructor from symmetry,exprseq", LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_indexed;
}
indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
{
- debugmsg("indexed constructor from symmetry,exvector", LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_indexed;
}
-indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
+indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
{
- debugmsg("indexed constructor from symmetry,exvector *", LOGLEVEL_CONSTRUCT);
tinfo_key = TINFO_indexed;
}
// archiving
//////////
-indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
{
- debugmsg("indexed constructor from archive_node", LOGLEVEL_CONSTRUCT);
if (!n.find_ex("symmetry", symtree, sym_lst)) {
// GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
unsigned symm = 0;
symtree = sy_none();
break;
}
- ex_to_nonconst_symmetry(symtree).validate(seq.size() - 1);
+ const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
}
}
// functions overriding virtual functions from base classes
//////////
-void indexed::print(const print_context & c, unsigned level) const
+void indexed::printindices(const print_context & c, unsigned level) const
{
- debugmsg("indexed print", LOGLEVEL_PRINT);
- GINAC_ASSERT(seq.size() > 0);
+ if (seq.size() > 1) {
- if (is_of_type(c, print_tree)) {
+ exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
- c.s << std::string(level, ' ') << class_name()
- << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
- << ", " << seq.size()-1 << " indices"
- << ", symmetry=" << symtree << std::endl;
- unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
- seq[0].print(c, level + delta_indent);
- printindices(c, level + delta_indent);
+ if (is_a<print_latex>(c)) {
- } else {
+ // TeX output: group by variance
+ bool first = true;
+ bool covariant = true;
- bool is_tex = is_of_type(c, print_latex);
- const ex & base = seq[0];
- bool need_parens = is_ex_exactly_of_type(base, add) || is_ex_exactly_of_type(base, mul)
- || is_ex_exactly_of_type(base, ncmul) || is_ex_exactly_of_type(base, power)
- || is_ex_of_type(base, indexed);
- if (is_tex)
- c.s << "{";
- if (need_parens)
- c.s << "(";
- base.print(c);
- if (need_parens)
- c.s << ")";
- if (is_tex)
+ 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 << "}";
- printindices(c, level);
+
+ } 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;
int indexed::compare_same_type(const basic & other) const
{
- GINAC_ASSERT(is_of_type(other, indexed));
+ GINAC_ASSERT(is_a<indexed>(other));
return inherited::compare_same_type(other);
}
// If the base object is 0, the whole object is 0
if (base.is_zero())
- return _ex0();
+ return _ex0;
// If the base object is a product, pull out the numeric factor
- if (is_ex_exactly_of_type(base, mul) && is_ex_exactly_of_type(base.op(base.nops() - 1), numeric)) {
+ 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 * thisexprseq(v);
+ return f * thiscontainer(v);
}
// Canonicalize indices according to the symmetry properties
if (seq.size() > 2) {
exvector v = seq;
- GINAC_ASSERT(is_ex_exactly_of_type(symtree, symmetry));
+ GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
if (sig != INT_MAX) {
// Something has changed while sorting indices, more evaluations later
if (sig == 0)
- return _ex0();
- return ex(sig) * thisexprseq(v);
+ return _ex0;
+ return ex(sig) * thiscontainer(v);
}
}
// Let the class of the base object perform additional evaluations
- return base.bp->eval_indexed(*this);
+ return ex_to<basic>(base).eval_indexed(*this);
}
-int indexed::degree(const ex & s) const
-{
- return is_equal(*s.bp) ? 1 : 0;
-}
-
-int indexed::ldegree(const ex & s) const
-{
- return is_equal(*s.bp) ? 1 : 0;
-}
-
-ex indexed::coeff(const ex & s, int n) const
-{
- if (is_equal(*s.bp))
- return n==1 ? _ex1() : _ex0();
- else
- return n==0 ? ex(*this) : _ex0();
-}
-
-ex indexed::thisexprseq(const exvector & v) const
+ex indexed::thiscontainer(const exvector & v) const
{
return indexed(ex_to<symmetry>(symtree), v);
}
-ex indexed::thisexprseq(exvector * vp) const
+ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
{
return indexed(ex_to<symmetry>(symtree), vp);
}
{
GINAC_ASSERT(seq.size() > 0);
- if ((options & expand_options::expand_indexed) && is_ex_exactly_of_type(seq[0], add)) {
+ if ((options & expand_options::expand_indexed) && is_exactly_a<add>(seq[0])) {
// expand_indexed expands (a+b).i -> a.i + b.i
const ex & base = seq[0];
- ex sum = _ex0();
- for (unsigned i=0; i<base.nops(); i++) {
+ ex sum = _ex0;
+ for (size_t i=0; i<base.nops(); i++) {
exvector s = seq;
s[0] = base.op(i);
- sum += thisexprseq(s).expand();
+ sum += thiscontainer(s).expand();
}
return sum;
// non-virtual functions in this class
//////////
-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_of_type(c, print_latex)) {
-
- // TeX output: group by variance
- bool first = true;
- bool covariant = true;
-
- while (it != itend) {
- bool cur_covariant = (is_ex_of_type(*it, varidx) ? ex_to<varidx>(*it).is_covariant() : true);
- if (first || cur_covariant != covariant) {
- 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++;
- }
- }
- }
-}
-
/** 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(void) const
+void indexed::validate() const
{
GINAC_ASSERT(seq.size() > 0);
exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
while (it != itend) {
- if (!is_ex_of_type(*it, idx))
+ 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_ex_exactly_of_type(symtree, symmetry))
+ if (!is_exactly_a<symmetry>(symtree))
throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
- ex_to_nonconst_symmetry(symtree).validate(seq.size() - 1);
+ const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
}
}
* @see ex::diff */
ex indexed::derivative(const symbol & s) const
{
- return _ex0();
+ 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)
{
if (v1.size() != v2.size())
return false;
- return equal(v1.begin(), v1.end(), v2.begin(), ex_is_equal());
+ return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
}
-exvector indexed::get_indices(void) const
+exvector indexed::get_indices() const
{
GINAC_ASSERT(seq.size() >= 1);
return exvector(seq.begin() + 1, seq.end());
}
-exvector indexed::get_dummy_indices(void) const
+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 false;
}
-exvector indexed::get_free_indices(void) const
+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(void) const
+exvector add::get_free_indices() const
{
exvector free_indices;
- for (unsigned i=0; i<nops(); i++) {
+ for (size_t i=0; i<nops(); i++) {
if (i == 0)
free_indices = op(i).get_free_indices();
else {
return free_indices;
}
-exvector mul::get_free_indices(void) const
+exvector mul::get_free_indices() const
{
// Concatenate free indices of all factors
exvector un;
- for (unsigned i=0; i<nops(); i++) {
+ 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());
}
return free_indices;
}
-exvector ncmul::get_free_indices(void) const
+exvector ncmul::get_free_indices() const
{
// Concatenate free indices of all factors
exvector un;
- for (unsigned i=0; i<nops(); i++) {
+ 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());
}
return free_indices;
}
-exvector power::get_free_indices(void) const
+exvector power::get_free_indices() const
{
// Return free indices of basis
return basis.get_free_indices();
/** Rename dummy indices in an expression.
*
- * @param e Expression to be worked on
+ * @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
* by the function */
static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
{
- unsigned global_size = global_dummy_indices.size(),
- local_size = local_dummy_indices.size();
+ size_t global_size = global_dummy_indices.size(),
+ local_size = local_dummy_indices.size();
// Any local dummy indices at all?
if (local_size == 0)
// 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 (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(ex_is_equal(), *it)) == global_dummy_indices.end()) {
+ if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(op0_is_equal(), *it)) == global_dummy_indices.end()) {
global_dummy_indices.push_back(*it);
global_size++;
remaining--;
}
it++;
}
- }
- // Replace index symbols in expression
+ // If this is the first set of local indices, do nothing
+ if (old_global_size == 0)
+ return e;
+ }
GINAC_ASSERT(local_size <= global_size);
- bool all_equal = true;
- lst local_syms, global_syms;
- for (unsigned i=0; i<local_size; i++) {
- ex loc_sym = local_dummy_indices[i].op(0);
- ex glob_sym = global_dummy_indices[i].op(0);
- if (!loc_sym.is_equal(glob_sym)
- && ex_to<idx>(local_dummy_indices[i]).get_dim().is_equal(ex_to<idx>(global_dummy_indices[i]).get_dim())) {
- all_equal = false;
- local_syms.append(loc_sym);
- global_syms.append(glob_sym);
+
+ // 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; i<local_size; 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; i<local_size; i++) // don't use more global symbols than necessary
+ 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;
+
+ // If a dummy index is encountered for the first time in the
+ // product, pull it up, otherwise, pull it down
+ exvector::const_iterator it2, it2start, it2end;
+ for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; 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()) {
+ e = e.subs(lst(
+ *it2 == ex_to<varidx>(*it2).toggle_variance(),
+ ex_to<varidx>(*it2).toggle_variance() == *it2
+ ), subs_options::no_pattern);
+ something_changed = true;
+ it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
+ it2start = ex_to<indexed>(e).seq.begin();
+ it2end = ex_to<indexed>(e).seq.end();
+ }
+ 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()) {
+ e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
+ something_changed = true;
+ it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
+ it2start = ex_to<indexed>(e).seq.begin();
+ it2end = ex_to<indexed>(e).seq.end();
+ }
+ goto next_index;
+ }
+ }
+
+next_index: ;
}
- if (all_equal)
- return e;
- else
- return e.subs(local_syms, global_syms);
+
+ 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;
+ }
+};
+
/** 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)
{
// Remember whether the product was commutative or noncommutative
// (because we chop it into factors and need to reassemble later)
- bool non_commutative = is_ex_exactly_of_type(e, ncmul);
+ bool non_commutative = is_exactly_a<ncmul>(e);
// Collect factors in an exvector, store squares twice
exvector v;
v.reserve(e.nops() * 2);
- if (is_ex_exactly_of_type(e, power)) {
+ 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()));
+ GINAC_ASSERT(e.op(1).is_equal(_ex2));
v.push_back(e.op(0));
v.push_back(e.op(0));
} else {
- for (unsigned i=0; i<e.nops(); i++) {
+ for (size_t i=0; i<e.nops(); i++) {
ex f = e.op(i);
- if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2())) {
+ 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_ex_exactly_of_type(f, ncmul)) {
+ } 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 (unsigned j=0; j<f.nops(); j++)
+ for (size_t j=0; j<f.nops(); j++)
v.push_back(f.op(j));
} else
v.push_back(f);
for (it1 = v.begin(); it1 != next_to_last; it1++) {
try_again:
- if (!is_ex_of_type(*it1, indexed))
+ if (!is_a<indexed>(*it1))
continue;
bool first_noncommutative = (it1->return_type() != return_types::commutative);
exvector::iterator it2;
for (it2 = it1 + 1; it2 != itend; it2++) {
- if (!is_ex_of_type(*it2, indexed))
+ if (!is_a<indexed>(*it2))
continue;
bool second_noncommutative = (it2->return_type() != return_types::commutative);
// Check whether the two factors share dummy indices
exvector free, dummy;
find_free_and_dummy(un, free, dummy);
- unsigned num_dummies = dummy.size();
+ 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()) {
- if (sp.is_defined(*it1, *it2)) {
- *it1 = sp.evaluate(*it1, *it2);
- *it2 = _ex1();
- goto contraction_done;
+
+ // Find minimal dimension of all indices of both factors
+ exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
+ ex dim = ex_to<idx>(*dit).get_dim();
+ ++dit;
+ for (; dit != ditend; ++dit) {
+ dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
+ }
+ dit = ex_to<indexed>(*it2).seq.begin() + 1;
+ ditend = ex_to<indexed>(*it2).seq.end();
+ for (; dit != ditend; ++dit) {
+ dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
}
- }
- // Contraction of symmetric with antisymmetric object is zero
- if (num_dummies > 1
- && ex_to<symmetry>(ex_to<indexed>(*it1).symtree).has_symmetry()
- && ex_to<symmetry>(ex_to<indexed>(*it2).symtree).has_symmetry()) {
-
- // Check all pairs of dummy indices
- for (unsigned idx1=0; idx1<num_dummies-1; idx1++) {
- for (unsigned idx2=idx1+1; idx2<num_dummies; idx2++) {
-
- // Try and swap the index pair and check whether the
- // relative sign changed
- lst subs_lst(dummy[idx1].op(0), dummy[idx2].op(0)), repl_lst(dummy[idx2].op(0), dummy[idx1].op(0));
- ex swapped1 = it1->subs(subs_lst, repl_lst);
- ex swapped2 = it2->subs(subs_lst, repl_lst);
- if (it1->is_equal(swapped1) && it2->is_equal(-swapped2)
- || it1->is_equal(-swapped1) && it2->is_equal(swapped2)) {
- free_indices.clear();
- return _ex0();
- }
- }
+ // 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 = it1->op(0).bp->contract_with(it1, it2, v);
+ 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 = it2->op(0).bp->contract_with(it2, it1, v);
+ contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
}
if (contracted) {
contraction_done:
if (first_noncommutative || second_noncommutative
- || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
- || is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
- || is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
+ || 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
- // simplify_ncmul() the chance to re-order and canonicalize
+ // 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);
// Find free indices (concatenate them all and call find_free_and_dummy())
// and all dummy indices that appear
exvector un, individual_dummy_indices;
- it1 = v.begin(); itend = v.end();
- while (it1 != itend) {
+ for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
exvector free_indices_of_factor;
- if (is_ex_of_type(*it1, indexed)) {
+ 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());
- it1++;
}
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.
+ if (local_dummy_indices.size() >= 2) {
+ exvector dummy_syms;
+ dummy_syms.reserve(local_dummy_indices.size());
+ for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
+ dummy_syms.push_back(it->op(0));
+ if (symmetrize(r, dummy_syms).is_zero()) {
+ free_indices.clear();
+ return _ex0;
+ }
+ }
+
// Dummy index renaming
r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
// Product of indexed object with a scalar?
- if (is_ex_exactly_of_type(r, mul) && r.nops() == 2
- && is_ex_exactly_of_type(r.op(1), numeric) && is_ex_of_type(r.op(0), indexed))
- return r.op(0).op(0).bp->scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
+ 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);
+ }
+};
+
/** Simplify indexed expression, return list of free indices. */
ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
{
ex e_expanded = e.expand();
// Simplification of single indexed object: just find the free indices
- // and perform dummy index renaming
- if (is_ex_of_type(e_expanded, indexed)) {
+ // 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
return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
}
// Simplification of sum = sum of simplifications, check consistency of
// free indices in each term
- if (is_ex_exactly_of_type(e_expanded, add)) {
+ if (is_exactly_a<add>(e_expanded)) {
bool first = true;
- ex sum = _ex0();
+ ex sum;
free_indices.clear();
- for (unsigned i=0; i<e_expanded.nops(); i++) {
+ 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()) {
sum = term;
first = false;
} else {
- if (!indices_consistent(free_indices, free_indices_of_term))
- throw (std::runtime_error("simplify_indexed: inconsistent indices in sum"));
- if (is_ex_of_type(sum, indexed) && is_ex_of_type(term, indexed))
- sum = sum.op(0).bp->add_indexed(sum, term);
+ 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;
}
}
}
- return sum;
+ // 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;
+
+ // Yes, construct vector of all dummy index symbols
+ exvector dummy_syms;
+ dummy_syms.reserve(dummy_indices.size());
+ for (exvector::const_iterator it = dummy_indices.begin(); it != dummy_indices.end(); ++it)
+ dummy_syms.push_back(it->op(0));
+
+ // 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);
+ ex term_symm = symmetrize(term, dummy_syms);
+ 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_ex_exactly_of_type(e_expanded, mul)
- || is_ex_exactly_of_type(e_expanded, ncmul)
- || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2())))
+ 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
* the free indices in sums are consistent.
*
* @return simplified expression */
-ex ex::simplify_indexed(void) const
+ex ex::simplify_indexed(unsigned options) const
{
exvector free_indices, dummy_indices;
scalar_products sp;
*
* @param sp Scalar products to be replaced automatically
* @return simplified expression */
-ex ex::simplify_indexed(const scalar_products & sp) const
+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(void) const
+ex ex::symmetrize() const
{
return GiNaC::symmetrize(*this, get_free_indices());
}
/** Antisymmetrize expression over its free indices. */
-ex ex::antisymmetrize(void) const
+ex ex::antisymmetrize() const
{
return GiNaC::antisymmetrize(*this, get_free_indices());
}
/** Symmetrize expression by cyclic permutation over its free indices. */
-ex ex::symmetrize_cyclic(void) const
+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[make_key(v1, v2)] = 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)
+void scalar_products::add_vectors(const lst & l, const ex & dim)
{
// Add all possible pairs of products
- unsigned num = l.nops();
- for (unsigned i=0; i<num; i++) {
- ex a = l.op(i);
- for (unsigned j=0; j<num; j++) {
- ex b = l.op(j);
- add(a, b, a*b);
- }
- }
+ 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(void)
+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
+bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
{
- return spm.find(make_key(v1, v2)) != spm.end();
+ 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 scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
{
- return spm.find(make_key(v1, v2))->second;
+ return spm.find(spmapkey(v1, v2, dim))->second;
}
-void scalar_products::debugprint(void) const
+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.first << "," << k.second;
- std::cerr << "), value=" << i->second << std::endl;
+ std::cerr << "item key=";
+ k.debugprint();
+ std::cerr << ", value=" << i->second << std::endl;
++i;
}
}
-/** Make key from object pair. */
-spmapkey scalar_products::make_key(const ex & v1, const ex & v2)
-{
- // If indexed, extract base objects
- ex s1 = is_ex_of_type(v1, indexed) ? v1.op(0) : v1;
- ex s2 = is_ex_of_type(v2, indexed) ? v2.op(0) : v2;
-
- // Enforce canonical order in pair
- if (s1.compare(s2) > 0)
- return spmapkey(s2, s1);
- else
- return spmapkey(s1, s2);
-}
-
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff