module torder; % Term order modes for distributive polynomials.
% H. Melenk, ZIB Berlin.
% The routines of this module should be coded as efficiently as
% possible.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% switching between term order modes: TORDER statement.
%
global!-dipvars!*:='(list);
symbolic procedure torder u;
begin scalar oldmode,oldex,oldvars,w;
oldmode:=vdpsortmode!*; oldex:=vdpsortextension!*;
oldvars:=global!-dipvars!*;
global!-dipvars!*:='(list);
a:
w:=reval car u; u:=cdr u;
if eqcar(w,'list) and null u then<<u:=cdr w; goto a>>;
if eqcar(w,'list) then
<<global!-dipvars!*:=w; w:=reval car u; u:=cdr u>>;
vdpsortmode!*:=w;
% dipsortevcomp!*:=get(w, 'evcomp);
vdpsortextension!*:=for each x in u join
(if eqcar(x:=reval x,'list) then cdr x else {x});
if flagp(vdpsortmode!*,'dipsortextension) and null vdpsortextension!*
then rederr "term order needs additional parameter(s)";
return 'list . oldvars . oldmode . oldex end ;
remprop('torder,'number!-of!-args);
put('torder,'psopfn,'torder);
symbolic procedure dipsortingmode u;
% Sets the exponent vector sorting mode. Returns the previous mode
begin scalar x,z;
if not idp u or not flagp(u,'dipsortmode)
then return typerr(u,"term ordering mode");
x:=dipsortmode!*; dipsortmode!*:=u;
% saves thousands of calls to GET;
dipsortevcomp!*:=get(dipsortmode!*,'evcomp);
if not getd dipsortevcomp!* then
rerror(dipoly,2,
"No compare routine for term order mode found");
if (z:=get(dipsortmode!*,'evcompinit)) then apply(z,nil);
if (z:=get(dipsortmode!*,'evlength)) and z neq length dipvars!*
then rederr
"wrong variable number for fixed length term order";
vdplastvar!*:=length dipvars!* ;
return x end ;
flag('(lex gradlex revgradlex),'dipsortmode);
put('lex,'evcomp,'evlexcomp);
put('gradlex,'evcomp,'evgradlexcomp);
put('revgradlex,'evcomp,'evrevgradlexcomp);
symbolic procedure evcompless!?(e1,e2);
% Exponent vector compare less. e1, e2 are exponent vectors
% in some order. Evcompless? is a boolean function which returns
% true if e1 is ordered less than e2 .
% Mapped to evcomp .
1=evcomp(e2,e1) ;
symbolic procedure evcomp (e1,e2);
% Exponent vector compare. e1, e2 are exponent vectors in some
% order. Evcomp(e1,e2) returns the digit 0 if exponent vector e1 is
% equal exponent vector e2, the digit 1 if e1 is greater than e2,
% else the digit -1. This function is assigned a value by the
% ordering mechanism, so is dummy for now .
% IDapply would be better here, but is not within standard LISP!
apply(dipsortevcomp!*,list(e1,e2)) ;
symbolic procedure evlexcomp (e1,e2);
% Exponent vector lexicographical compare. The
% exponent vectors e1 and e2 are in lexicographical
% ordering. evLexComp(e1,e2) returns the digit 0 if exponent
% vector e1 is equal exponent vector e2, the digit 1 if e1 is
% greater than e2, else the digit -1.
if null e1 then 0
else if null e2 then evlexcomp(e1,'(0))
else if car e1 #= car e2 then evlexcomp(cdr e1,cdr e2)
else if car e1 #> car e2 then 1
else -1 ;
symbolic procedure evinvlexcomp (e1,e2);
% Exponent vector inverse lexicographical compare .
if null e1 then
if null e2 then 0 else evinvlexcomp('(0),e2)
else if null e2 then evlexcomp(e1,'(0))
else if car e1 #= car e2 then evinvlexcomp(cdr e1,cdr e2)
else (if not(n#=0) then n
% else if car e2 #= car e1 then 0
else if car e2 #> car e1 then 1
else -1) where n=evinvlexcomp(cdr e1,cdr e2);
symbolic procedure evgradlexcomp (e1,e2);
% Exponent vector graduated lex compare.
% The exponent vectors e1 and e2 are in graduated lex
% ordering. evGradLexComp(e1,e2) returns the digit 0 if exponent
% vector e1 is equal exponent vector e2, the digit 1 if e1 is
% greater than e2, else the digit -1.
if null e1 then 0
else if null e2 then evgradlexcomp(e1,'(0))
else if car e1 #= car e2 then evgradlexcomp(cdr e1, cdr e2)
else (if te1#=te2 then if car e1 #> car e2 then 1 else -1
else if te1 #> te2 then 1 else -1)
where te1=evtdeg e1, te2=evtdeg e2;
symbolic procedure evrevgradlexcomp (e1,e2);
% Exponent vector reverse graduated lex compare.
% The exponent vectors e1 and e2 are in reverse graduated lex
% ordering. evRevGradLexcomp(e1,e2) returns the digit 0 if exponent
% vector e1 is equal exponent vector e2, the digit 1 if e1 is
% greater than e2, else the digit -1.
if null e1 then 0
else if null e2 then evrevgradlexcomp(e1,'(0))
else if car e1 #= car e2 then evrevgradlexcomp(cdr e1, cdr e2)
else (if te1 #= te2 then evinvlexcomp(e1,e2)
else if te1 #> te2 then 1 else -1)
where te1=evtdeg e1, te2=evtdeg e2;
symbolic procedure evtdeg e1;
% Exponent vector total degree. e1 is an exponent vector.
% evtdeg(e1) calculates the total degree of the exponent
% e1 and returns an integer.
(<<while e1 do <<x:=car e1 #+ x; e1:=cdr e1>>; x>>) where x=0;
% The following section contains additional term order modes.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% gradlexgradlex
%
% this order can have several steps
% torder gradlexgradlex,3,2,4;
%
flag ('(gradlexgradlex),'dipsortmode);
flag ('(gradlexgradlex),'dipsortextension);
put('gradlexgradlex,'evcomp,'evgradgradcomp);
symbolic procedure evgradgradcomp (e1,e2);
evgradgradcomp1 (e1,e2,car vdpsortextension!*,
cdr vdpsortextension!*);
symbolic procedure evgradgradcomp1 (e1,e2,n,nl);
if null e1 then 0
else if null e2 then evgradgradcomp1(e1,'(0),n,nl)
else if n#=0 then if null nl then evgradlexcomp(e1,e2)
else evgradgradcomp1 (e1,e2,car nl,cdr nl)
else if car e1 #= car e2 then
evgradgradcomp1(cdr e1,cdr e2,n#-1,nl)
else (if te1 #= te2 then if car e1 #> car e2 then 1 else -1
else if te1 #> te2 then 1 else -1)
where te1=evpartdeg(e1,n), te2=evpartdeg(e2,n);
symbolic procedure evpartdeg(e1,n); evpartdeg1(e1,n,0);
symbolic procedure evpartdeg1(e1,n,sum);
if n #= 0 or null e1 then sum
else evpartdeg1(cdr e1,n #-1, car e1 #+ sum);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% gradlexrevgradlex
%
%
flag ('(gradlexrevgradlex),'dipsortmode);
flag ('(gradlexrevgradlex),'dipsortextension);
put('gradlexrevgradlex,'evcomp,'evgradrevgradcomp);
symbolic procedure evgradrevgradcomp (e1,e2);
evgradrevgradcomp1 (e1,e2,car vdpsortextension!*);
symbolic procedure evgradrevgradcomp1 (e1,e2,n);
if null e1 then 0
else if null e2 then evgradrevgradcomp1(e1,'(0),n)
else if n#=0 then evrevgradlexcomp(e1,e2)
else if car e1 #= car e2 then evgradrevgradcomp1(cdr e1,cdr e2,n#-1)
else (if te1 #= te2 then if car e1 #< car e2 then 1 else -1
else if te1 #> te2 then 1 else -1)
where te1=evpartdeg(e1,n), te2=evpartdeg(e2,n);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% LEXGRADLEX
%
%
flag ('(lexgradlex),'dipsortmode);
flag ('(lexgradlex),'dipsortextension);
put('lexgradlex,'evcomp,'evlexgradlexcomp);
symbolic procedure evlexgradlexcomp (e1,e2);
evlexgradlexcomp1 (e1,e2,car vdpsortextension!*);
symbolic procedure evlexgradlexcomp1 (e1,e2,n);
if null e1 then (if evzero!? e2 then 0 else -1)
else if null e2 then evlexgradlexcomp1(e1,'(0),n)
else if n#=0 then evgradlexcomp(e1,e2)
else if car e1 #= car e2 then evlexgradlexcomp1(cdr e1,cdr e2,n#-1)
else if car e1 #> car e2 then 1 else -1;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% LEXREVGRADLEX
%
%
flag ('(lexrevgradlex),'dipsortmode);
flag ('(lexrevgradlex),'dipsortextension);
put('lexrevgradlex,'evcomp,'evlexrevgradlexcomp);
symbolic procedure evlexrevgradlexcomp (e1,e2);
evlexrevgradlexcomp1 (e1,e2,car vdpsortextension!*);
symbolic procedure evlexrevgradlexcomp1 (e1,e2,n);
if null e1 then (if evzero!? e2 then 0 else -1)
else if null e2 then evlexrevgradlexcomp1(e1,'(0),n)
else if n#=0 then evrevgradlexcomp(e1,e2)
else if car e1 #= car e2 then
evlexrevgradlexcomp1(cdr e1,cdr e2,n#-1)
else if car e1 #> car e2 then 1 else -1;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% WEIGHTED
%
%
flag ('(weighted),'dipsortmode);
flag ('(weighted),'dipsortextension);
put('weighted,'evcomp,'evweightedcomp);
symbolic procedure evweightedcomp (e1,e2);
(if dg1 #= dg2 then evlexcomp(e1,e2) else
if dg1 #> dg2 then 1 else -1
) where dg1=evweightedcomp2(0,e1,vdpsortextension!*),
dg2=evweightedcomp2(0,e2,vdpsortextension!*);
symbolic procedure evweightedcomp1 (e,w); evweightedcomp2(0, e, w);
symbolic procedure evweightedcomp2 (n,e,w);
% scalar product of exponent and weight vector
if null e then n else
if null w then evweightedcomp2(n, e, '(1 1 1 1 1)) else
if car w=0 then evweightedcomp2(n, cdr e, cdr w) else
if car w=1 then evweightedcomp2(n #+ car e, cdr e, cdr w) else
evweightedcomp2(car e #* car w #+ n, cdr e, cdr w);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% GRADED term order
% cascading a graded sorting with another term order.
%
% The grade of a term is defined as a scalar product of the exponent
% vector and a grade vector which contains non-negative integers.
% In contrast to a weight vector the grade vector may contain also
% zeros. A vector of ones is used if no vector is given explicitly.
%
fluid '(gradedrec!*);
flag ('(graded),'dipsortmode);
flag ('(graded),'dipsortextension);
put('graded,'evcomp,'evgradedcomp);
put('graded,'evcompinit,'evgradedinit);
symbolic procedure evgradedinit();
begin scalar w,gvect,vse;
vse:=vdpsortextension!*;
while pairp vdpsortextension!* and numberp car vdpsortextension!*
do <<gvect:=car vdpsortextension!* . gvect;
vdpsortextension!*:=cdr vdpsortextension!*>>;
if vdpsortextension!* then
<<w:=car vdpsortextension!*;
vdpsortextension!*:=cdr vdpsortextension!*>>
else w:='lex;
dipsortingmode w;
gradedrec!*:={reversip gvect,dipsortevcomp!*,vdpsortextension!*};
dipsortevcomp!*:='evgradedcomp;
dipsortmode!*:='graded;
vdpsortextension!*:=vse end;
symbolic procedure evgradedcomp (e1,e2);
(if dg1 #= dg2 then
apply(cadr gradedrec!*,{e1,e2})
where vdpsortextension!*=caddr gradedrec!*
else
if dg1 #> dg2 then 1 else -1
) where dg1=ev!-gamma e1, dg2=ev!-gamma e2;
symbolic procedure ev!-gamma(ev);
% compute the grade of an exponent vector;
evweightedcomp1 (ev,
if dipsortmode!*='graded then car gradedrec!* else
if dipsortmode!*='weighted then vdpsortextension!* else nil);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% MATRIX
%
%
% In the following routines I assume that 99 percent of the matrix
% entries will be 0 or 1 such that the special branches for these
% numbers makes sense. We save lots of memory read and
% multiplication is needed only entries other than 0 and 1.
%
% I could do the same optimization step for -1, but I don't
% expect that many people will use term orders with negative
% numbers.
%
% This package includes a compilation mode for matrix term orders
% for fixed length variable lists. Compilation is done implicilty
% when *comp is on, or explicitly by callint torder_compile.
flag ('(matrix),'dipsortmode);
flag ('(matrix),'dipsortextension);
put('matrix,'evcomp,'evmatrixcomp);
put('matrix,'evcompinit,'evmatrixinit);
symbolic procedure evmatrixcomp(e1,e2); evmatrixcomp1(e1,e2,vdpmatrix!*);
symbolic procedure evmatrixcomp1(e1,e2,m);
if null m then 0 else
(if w1 #= w2 then evmatrixcomp1(e1,e2,cdr m) else % #=
if w1 #> w2 then 1 else -1)
where
w1= evmatrixcomp2 (e1,car m,0),
w2= evmatrixcomp2 (e2,car m,0);
symbolic procedure evmatrixcomp2(e,l,w);
if null e or null l then w else
(if l1 #= 0 then
evmatrixcomp2(cdr e,cdr l,w) else
if l1 #= 1 then
evmatrixcomp2(cdr e,cdr l,w #+ car e)
else evmatrixcomp3(e,l1,l,w)) where l1=car l;
symbolic procedure evmatrixcomp3(e,l1,l,w);
evmatrixcomp2(cdr e,cdr l,w #+ car e #* l1);
symbolic procedure evmatrixinit1(w,mode);
begin scalar m,mm;
if not eqcar(w,'mat) or
mode and length cadr w neq length dipvars!* then
typerr(w,"term order matrix for". dipvars!*);
for each row in cdr w do
<<row:=for each c in row collect ieval c;
mm:=row . mm;
row:=reverse row;
while eqcar(row,0) do row:=cdr row;
m:=reversip row . m>>;
m:=reversip m; mm:=reversip mm;
if m neq vdpmatrix!* then
<<if length cadr w > length cdr w then
lprim "Warning: non-square matrix used in torder"
else if 0=reval{'det,w} then
typerr(w,"term order (singular matrix)");
if not evmatrixcheck mm then
typerr(w,"term order (non admissible)")
>>;
return m end;
symbolic procedure evmatrixinit();
begin scalar c,m,w;
w:=reval car vdpsortextension!*;
m:=evmatrixinit1(w,t);
if (c:=assoc(m,compiled!-orders!*)) then
dipsortevcomp!*:=cdr c else
if !*comp then dipsortevcomp!*:=evmatrixcompile m;
vdpmatrix!*:=m end;
symbolic procedure evmatrixcheck m;
% Check the usability of the term order matrix: the
% top elements of each column must be positive. This
% approach goes back to a recommendation of J. Apel.
begin scalar bad,c,w;
integer i,j,r;
r:=length m;
for i:=1:length car m do
<<c:=nil;
for j:=1:r do
if (w:=nth(nth(m,j),i)) neq 0 and null c then
<<c:=t; bad:=w < 0>>
>>;
return not bad end;
symbolic procedure evmatrixcompile m;
begin scalar w;
w:= evmatrixcompile1 m;
putd(car w,'expr,caddr w);
compiled!-orders!*:=(m.car w).compiled!-orders!*;
return car w end;
symbolic procedure evmatrixcompile1 m;
begin scalar c,n,x,w,lvars,code;
integer ld,p,k;
for each row in m do k:=max(k,length row);
lvars:=for i:=1:k collect gensym();
code:={{'setq,car lvars,
'(idifference (car e1) (car e2))}};
ld:=1;
for each row in m do
<<p:=0; w:=nil;
while row do
<<c:=car row; row:=cdr row; p:=p+1;
if c neq 0 then
<< % load the differences up to the current point.
for i:=ld+1:p do
<< code:= append(code,'((setq e1(cdr e1))(setq e2(cdr e2))));
code:=append(code,
{{'setq,nth(lvars,i),
'(idifference (car e1) (car e2))}});
ld:=i; >>;
% collect the terms of the row sum
x:=nth(lvars,p);
if c=-1 then x:={'iminus,x} else
if c neq 1 then x:={'itimes,c,x};
w:=if w then {'iplus2,x,w} else x >>;
>>;
if not atom w then
<<code:=append(code,{{'setq,'w,w}});w:='w>>;
code:=append(code,
{{'cond,{{'iequal,w,0},t},
{{'igreaterp,w,0},'(return 1)},
'(t (return -1))}}); >>;
% common trailor
code:=append(code,'((return 0)));
n:=gensym();
return {n,k,evform {'lambda,'(e1 e2), 'prog.('w.lvars). code}} end;
symbolic procedure evform(u);
% Let form play on the generated code.
form1(u,nil,'symbolic);
symbolic procedure torder_compile_form(w,c,m);
begin scalar n;
if length w < 3 then rederr "illegal arguments";
m:=evmatrixinit1(eval form caddr w,nil);
c:=evmatrixcompile1 m;
n:=eval form cadr w;
return
{'progn,
{'putd,mkquote n,mkquote 'expr,mkquote caddr c},
{'setq,'compiled!-orders!*,
{'cons,{'cons,mkquote m,mkquote n}, 'compiled!-orders!*}},
{'put,mkquote n,''evcomp,mkquote n},
{'put,mkquote n,''evlength,cadr c},
{'flag,mkquote{n},''dipsortmode}, mkquote n} end;
put('torder_compile,'formfn,'torder_compile_form);
endmodule;;end;