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/* Tensor package. This code intends to simplify tensorial expressions.
*/
/* functions internal to tensors */
RuleBase("Delta",{ind1,ind2});
RuleBase("TList",{head,tail});
RuleBase("TSum",{indices,body});
RuleBase("TD",{ind});
RuleBase("X",{ind});
/* And the simplificaiton rules for X, addition, subtraction
and multiplication */
10 # (TD(_i)X(_j)) <-- Delta(i,j);
10 # (TD(_i) ( (_f) + (_g) ) ) <-- (TD(i)f) + (TD(i)g);
10 # (TD(_i) ( (_f) - (_g) ) ) <-- (TD(i)f) - (TD(i)g);
10 # (TD(_i) ( - (_g) ) ) <-- - TD(i)g;
10 # (TD(_i) ( (_f) * (_g) ) ) <-- (TD(i)f)*g + f*(TD(i)g);
10 # (TD(_i) ( (_f) ^ (n_IsPositiveInteger) ) ) <-- n*(TD(i)f)*f^(n-1);
10 # (TD(_i)Delta(_j,_k)) <-- 0;
10 # (TD(_i)f_IsNumber) <-- 0;
/* The only TSum summation simplification: summing over no indices
means no summation. */
10 # (TSum({})(_body)) <-- body;
/* Explicit summation when Ndim is defined. This summation will
be invoked when using TExplicitSum. */
20 # (TSum(_indices)(_body))_(IsInteger(Ndim)) <--
LocalSymbols(index,i,sum)
[
Local(index,i,sum);
index:=indices[1];
sum:=0;
MacroLocal(index);
For(i:=1,i<=Ndim,i++)
[
MacroSet(index,i);
sum:=sum+Eval(TSum(Tail(indices))body);
];
sum;
];
/* TExplicitSum sets the dimension of the space under consideration,
so summation can proceed */
(TExplicitSum(Ndim_IsInteger)(_body)) <-- Eval(body);
/* Move the delta factors to the front, so they can be simplified
away. It uses ApplyDelta to move a factor either to the front
or to the back of the list. Input is a list of factors, as
returned by Flatten(expressions,"*")
*/
MoveDeltas(_list) <--
[
Local(result,i,nr);
result:={};
nr:=Length(list);
For(i:=1,i<=nr,i++)
[
ApplyDelta(result,list[i]);
];
result;
];
10 # ApplyDelta(_result,Delta(_i,_j)) <--
DestructiveInsert(result,1,Delta(i,j));
20 # ApplyDelta(_result,(_x) ^ (n_IsInteger))_(n>0) <--
[
Local(i);
For(i:=1,i<=n,i++)
[
ApplyDelta(result,x);
];
];
100 # ApplyDelta(_result,_term) <--
DestructiveAppend(result,term);
/* TSimplify : expand brackets, and send the expression of addition
of terms to TSimplifyAux */
TSimplify(TSum(_indices)(_f)) <--
[
TSimplifyAux(TSum(indices)ExpandBrackets(f));
];
/* TSimplifyAux : simplify each term independently */
10 # TSimplifyAux(TSum(_indices)((_f) + (_g))) <--
TSimplifyAux(TSum(FlatCopy(indices))(f)) +
TSimplifyAux(TSum(FlatCopy(indices))(g));
10 # TSimplifyAux(TSum(_indices)((_f) - (_g))) <--
TSimplifyAux(TSum(FlatCopy(indices))(f)) -
TSimplifyAux(TSum(FlatCopy(indices))(g));
10 # TSimplifyAux(TSum(_indices)( - (_g))) <--
- TSimplifyAux(TSum(indices)(g));
40 # TSimplifyAux(TSum(_indices)_body) <--
[
Local(flat);
/* Convert expressions of the form (a*b*c) to {a,b,c} */
flat:=Flatten(body,"*");
/* Move the deltas to the front. */
flat:=MoveDeltas(flat);
/* Simplify the deltas away (removing the required indices) */
flat:=TSumRest(flat);
/* Determine if there are indices the summand still depends on */
Local(varlist,independ,nrdims);
varlist:=VarList(flat);
independ:=Intersection(indices,varlist);
nrdims:=Length(indices)-Length(independ);
/* Return result, still summing over the indices not removed by deltas */
Ndim^nrdims*TSum(independ)flat;
];
/* Terminating condition for the tensorial simplification */
10 # TSumSimplify(TList(Delta(_ind,_ind),_list))_Contains(indices,ind) <--
[
/* Remove the index from the list of indices to sum over, since
it is now implicitly summed over by simplifying the delta */
DestructiveDelete(indices,Find(indices,ind));
/* Return result simplified for this delta */
Ndim*TSumRest(list);
];
11 # TSumSimplify(TList(Delta(_ind1,_ind2),_list))_
Contains(indices,ind2) <--
[
/* Remove the index from the list of indices to sum over, since
it is now implicitly summed over by simplifying the delta */
DestructiveDelete(indices,Find(indices,ind2));
/* Return result simplified for this delta */
TSumRest( Subst(ind2,ind1)list );
];
11 # TSumSimplify(TList(Delta(_ind1,_ind2),_list))_
Contains(indices,ind1) <--
[
/* Remove the index from the list of indices to sum over, since
it is now implicitly summed over by simplifying the delta */
DestructiveDelete(indices,Find(indices,ind1));
/* Return result simplified for this delta */
TSumRest( Subst(ind1,ind2)list );
];
1010 # TSumSimplify(TList(_term,_list)) <--
[
term*TSumRest(list);
];
10 # TSumRest({}) <-- 1;
20 # TSumRest(_list) <--
[
TSumSimplify(TList(Head(list),Tail(list)));
];
UnFence("TSumSimplify",1);
UnFence("TSumRest",1);
UnFence("TSum",2);
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