module vdp2dip;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% interface for Virtual Distributive Polynomials(VDP)
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% "Distributive representation" with respect to a given set of
% variables(" vdpvars ")means for a polynomial, that the polynomial
% is regarded as a sequence of monomials, each of which is a
% product of a " coefficient " and of some powers of the variables.
% This internal representation is very closely connected to the
% standard external(printed)representation of a polynomial in
% REDUCE if nothing is factored out. The monomials are logically
% ordered by a term order mode based on the ordering which is
% given bye the sequence " vdpvars ";with respect to this ordering
% the representation of a polynomial is unique. The " highest " term
% is the car one. Monomials are represented by their coefficient
%(" vbc ")and by a vector of the exponents(" vev ")(in the order
% corresponding to the vector vars). The distributive representation
% is good for those algorithms,which base their decisions on the
% complete ledading monomial: this representation guarantees a
% fast and uniform access to the car monomial and to the reductum
%(the cdr of the polynomial beginning with the cadr monomial).
% The algorithms of the Groebner package are of this type. The
% interface defines the distributive polynomials as abstract data
% objects via their acess functions. These functions map the
% distributive operations to an arbitrary real data structure
%(" virtual "). The mapping of the access functions to an actual
% data structure is restricted only by the demand,that the typical
% " distributive operations " be efficient. Additionally to the
% algebraic value a VDP object has a property list. So the algorithms
% using the VDP interface can assign name - value - pairs to individual
% polynomials. The interface is defined by a set of routines which
% create and handle the distributive polynomials. In general the
% first letters of the routine name classifies the data its works on:
%
% vdp... complete virtual polynomial objects
% vbc... virtual base coefficients
% vev... virtual exponent vectors
%
% 0. general control
%
% vdpinit(dv)initialises the vdp package for the variables
% given in the list 'dv'. vdpinit modifies the
% torder and returns the prvevious torder as its
% result. 'vdpinit' sets the global variable
% 'vdpvars!*'.
%
% 1. Conversion
%
% a2vdp Algebraic(prefix)to vdp.
% f2vdp Standard form to vdp.
% a2vbc Algebraic(prefix)to vbc.
% vdp2a Vdp to algebraic(prefix).
% vdp2f Vdp to standard form.
% vbc2a Vbc to algebraic(prefix).
%
% 2. Composing/decomposing
%
% vdpfmon Make a vdp from a vbc and an vev.
% vdpmoncomp Add a monomial(vbc and vev)to the front of a vdp.
% vdpappendmon Add a monomial(vbc and vev)to the bottom of a vdp.
% vdpmonadd Add a monomial(vbc and vev)to a vdp,not yet
% knowing the place of the insertiona.
% vdpappendvdp Concat two vdps.
%
% vdplbc Extract leading vbc.
% vdpevlmon Extract leading vev.
% vdpred Reductum of vdp.
% vdplastmon Last monomial of polynomial.
% vevnth Nth element from exponent vector.
%
% 3. Testing
%
% vdpzero? Test vdp = 0.
% vdpredzero!? Test rductum of vdp = 0.
% vdpone? Test vdp = 1.
% vevzero? Test vev =(0 0 ... 0).
% vbczero? Test vbc = 0.
% vbcminus? Test vbc <= 0(not decidable for algebraic vbcs).
% vbcplus? Test vbc >= 0(not decidable for algebraic vbcs).
% vbcone!? Test vbc = 1.
% vbcnumberp Test vbc is a numeric value.
% vevdivides? Test if vev1 < vev2 elementwise.
% vevlcompless? Test ordering vev1 < vev2.
% vdpvevlcomp Calculate ordering vev1 / vev1 : -1, 0 or +1.
% vdpequal Test vdp1 = vdp2.
% vdpmember Member based on " vdpequal ".
% vevequal Test vev1 = vev2.
%
% 4. Arithmetic
%
% 4.1 Vdp arithmetic
%
% vdpsum vdp + vdp
% Special routines for monomials : see above(2.).
% vdpdif vdp - vdp.
% vdpprod vdp * vdp.
% vdpvbcprod vbc * vdp.
% vdpdivmon vdp /(vbc,vev) divisability presumed.
% vdpcancelvev Substitute all multiples of monomial(1,vev)in vdp by 0.
% vdlLcomb1 vdp1 *(vbc1,vev1)+ vdp2 *(vbc2,vev2).
% vdpcontent Calculate gcd over all vbcs.
%
% 4.2 Vbc arithmetic
%
% vbcsum vbc1 + vbc2.
% vbcdif vbc1 - vbc2.
% vbcneg - vbc.
% vbcprod vbc1 * vbc2.
% vbcquot vbc1 / vbc2 Divisability assumed if domain = ring.
% vbcinv 1 / vbc Only usable in field.
% vbcgcd gcd(vbc1,vbc2) Only usable in Euclidean field.
%
% 4.2 Vev arithmetic
%
% vevsum vev1 + vev2 Elementwise.
% vevdif vev1 - vev2 Elementwise.
% vevtdeg Sum over all exponents.
% vevzero Generate a zero vev.
%
% 5. Auxiliary
%
% vdpputprop Assign indicator - value - pair to vdp.
% The property " number " is used for printing.
% vdpgetprop Read value of indicator from vdp.
% vdplsort Sort list of polynomials with respect to ordering.
% vdplsortin Sort a vdp into a sorted list of vdps.
% vdpprint Print a vdp together with its number.
% vdpprin2t Print a vdp " naked ".
% vdpprin3t Print a vdp with closing ";".
% vdpcondense Replace exponent vectors by equal objects from
% global list dipevlist!* in order to save memory.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% RECORD STRUCTURE
%
% A virtual polynomial here is a record(list) with the entries
% ('vdp < vdpevlmon > < vdplbc > < form > < plist >)
%
% ´ vdp A type tag;
% < vdpevlmon > the exponents of the variables in the
% leading monomial;the positions correspond to
% the positions in vdpvars!*. Trailing zeroes
% can be omitted.
%
% < lcoeff > The " coefficient " of the leading monomial,which
% in general is a standard form.
%
% < form > The complete polynomial,e.g. as REDUCE standard form.
%
% < plist > An asso list for the properties of the polynomial.
%
% The components should not be manipulated only via the interface
% functions and macros,so that application programs remain
% independent from the internal representation.
% The only general assumption made on < form > is,that the zero
% polynomial is represented as NIL. That is the case e. g. for both,
% REDUCE standard forms and DIPOLYs.
%
% Conventions for the usage:
% -------------------------
%
% vdpint has to be called prveviously to all vdp calls. The list of
% vdp paraemters is passed to vdpinit. The value of vdpvars!*
% and the current torder must remain unmodfied afterwards.
% usual are simple id's,e.g.
%
% Modifications to vdpvars!* during calculations
% ----------------------------------------------
%
% This mapping of vdp operations to standard forms offers the
% ability to enlarge vdpvars during the calculation in order
% to add new(intermediate)variables. Basis is the convention,
% that exponent vectors logically have an arbitrary number
% of trailing zeros. All routines processing exponent vectors
% are able to handle varying length of exponent vectors.
% A new call to vdpinit is necessary.
%
% During calculation vdpvars may be enlarged(new variables
% suffixed)without needs to modify existing polynomials;only
% korder has to be set to the new variable sequence.
% modifications to the sequence in vdpvars requires a
% new call to vdpinit and a reordering of exisiting
% polynomials,e.g. by
% vdpint newvdpvars;
% f2vdp vdp2f p1;f2vdp vdp2f p2;.....
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% DECLARATION SECTION
%
% This module must be present during code generation for modules
% using the vdp - sf interface.
global '(vdpprintmax groebmonfac);
flag('(vdpprintmax),'share);
% Basic internal constructor of vdp-record:
smacro procedure makevdp(vbc,vev,form);
{'vdp,vev,vbc,form,nil};
% Basic selectors(conversions):
smacro procedure vdppoly u;cadr cddr u;
smacro procedure vdplbc u;caddr u;
smacro procedure vdpevlmon u;cadr u;
% Basic tests:
smacro procedure vdpzero!? u;null u or null vdppoly u;
smacro procedure vevzero!? u;
null u or(car u=0 and vevzero!?1 cdr u);
smacro procedure vdpone!? p;
not vdpzero!? p and vevzero!? vdpevlmon p;
% Manipulating of exponent vectors.
smacro procedure vevdivides!?(vev1,vev2);vevmtest!?(vev2,vev1);
smacro procedure vevzero();vevmaptozero1(vdpvars!*,nil);
smacro procedure vdpnumber f;vdpgetprop(f,'number);
% The code for checkpointing is factored out.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Interface for DIPOLY polynomials as records(objects).
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
flag('(vdpprintmax),'share);
symbolic procedure dip2vdp u;
% Is used when u can be empty.
(if dipzero!? uu then makevdp(a2bc 0,nil,nil)
else makevdp(diplbc uu,dipevlmon uu,uu))
where uu=if !*groebsubs then dipsubs2 u else u;
% Some simple mappings:
smacro procedure makedipzero();nil;
symbolic procedure vdpredzero!? u;dipzero!? dipmred vdppoly u;
symbolic procedure vdplastmon u;
% Return bc. ev of last monomial of u.
begin u:=vdppoly u;
if dipzero!? u then return nil;
while not dipzero!? u and not dipzero!? dipmred u do u:=dipmred u;
return diplbc u.dipevlmon u end;
symbolic procedure vbczero!? u;bczero!? u;
symbolic procedure vbcnumber u;
if pairp u and numberp car u and 1=cdr u then cdr u else nil;
symbolic procedure vbcfi u;bcfd u;
symbolic procedure a2vbc u;a2bc u;
symbolic procedure vbcquot(u,v);bcquot(u,v);
symbolic procedure vbcneg u;bcneg u;
symbolic procedure vbcabs u;if vbcminus!? u then bcneg u else u;
symbolic procedure vbcone!? u;bcone!? u;
symbolic procedure vbcprod(u,v);bcprod(u,v);
% Initializing vdp - dip polynomial package.
symbolic procedure vdpinit2 vars;
begin scalar oldorder;vdpcleanup();
oldorder:=kord!*;
if null vars then rerror(dipoly,8,"vdpinit: vdpvars not set");
vdpvars!*:=dipvars!*:=vars;torder2 vdpsortmode!*;
return oldorder end;
symbolic procedure vdpcleanup();dipevlist!*:={nil};
symbolic procedure vdpred u;
begin scalar r,s;r:=dipmred vdppoly u;
if dipzero!? r then return makevdp(nil ./ nil,nil,makedipzero());
r:=makevdp(diplbc r,dipevlmon r,r);
if !*gsugar and(s:=vdpgetprop(u,'sugar))then gsetsugar(r,s);
return r end;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Coefficient handling;here we assume that coefficients are
% standard quotients.
%
symbolic procedure vbcgcd(u,v);
begin scalar x;
if not vbcsize(u,-100)or not vbcsize(v,-100)
then return '(1 . 1);
x:=if denr u=1 and denr v=1 then
if fixp numr u and fixp numr v then gcdn(numr u,numr v) ./ 1
else gcdf!*(numr u,numr v)./ 1
else 1 ./ 1;
return x end;
symbolic procedure vbcsize(u,n);
if n #> -1 then nil
else if atom u then n
else begin n:=vbcsize(car u,n #+ 1);
if null n then return nil;return vbcsize(cdr u,n)end;
% Cofactors: compute(q,v)such that q*a=v*b.
symbolic procedure vbc!-cofac(bc1,bc2);
% Compute base coefficient cofactors.
<<if vbcminus!? bc1 and vbcminus!? bc2 then gcd:=vbcneg gcd;
vbcquot(bc2,gcd). vbcquot(bc1,gcd)>>
where gcd=vbcgcd(bc1,bc2);
symbolic procedure vev!-cofac(ev1,ev2);
% Compute exponent vector cofactors.
(vevdif(lcm,ev1).vevdif(lcm,ev2))
where lcm=vevlcm(ev1,ev2);
% The following functions must be redefinable.
symbolic procedure vbcplus!? u;(numberp v and v > 0)where v=numr u;
symbolic procedure bcplus!? u;(numberp v and v > 0)where v=numr u;
symbolic procedure vbcminus!? u;(numberp v and v < 0)where v=numr u;
symbolic procedure vbcinv u;bcinv u;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Conversion between forms, vdps and prefix expressions.
%
% Prefix to vdp.
symbolic procedure a2vdp u;
if u=0 or null u then makevdp(nil./1,nil,makedipzero())
else(makevdp(diplbc r,dipevlmon r,r)where r=a2dip u);
% Vdp to prefix.
symbolic procedure vdp2a u;dip2a vdppoly u;
symbolic procedure vbc2a u;bc2a u;
% Form to vdp.
symbolic procedure f2vdp u;
if u=0 or null u then makevdp(nil./1,nil,makedipzero())
else(makevdp(diplbc r,dipevlmon r,r)where r=f2dip u);
% Vdp to form.
symbolic procedure vdp2f u;dip2f vdppoly u;
% Vdp from monomial.
symbolic procedure vdpfmon(coef,vev);
begin scalar r;r:=makevdp(coef,vev,dipfmon(coef,vev));
if !*gsugar then gsetsugar(r,vevtdeg vev);return r end;
% Add a monomial to a vdp in front(new vev and coeff).
symbolic procedure vdpmoncomp(coef,vev,vdp);
if vdpzero!? vdp then vdpfmon(coef,vev)
else if vbczero!? coef then vdp
else makevdp(coef,vev,dipmoncomp(coef,vev,vdppoly vdp));
% Add a monomial to the end of a vdp(vev remains unchanged).
symbolic procedure vdpappendmon(vdp,coef,vev);
if vdpzero!? vdp then vdpfmon(coef,vev)
else if vbczero!? coef then vdp
else makevdp(vdplbc vdp,vdpevlmon vdp,dipsum(vdppoly vdp,dipfmon(coef,vev)));
% Add monomial to vdp;place of new monomial still unknown.
symbolic procedure vdpmonadd(coef,vev,vdp);
if vdpzero!? vdp then vdpfmon(coef,vev)else
(if c=1 then vdpmoncomp(coef,vev,vdp)else
if c=-1 then makevdp(vdplbc vdp,vdpevlmon vdp,
dipsum(vdppoly vdp,dipfmon(coef,vev)))
else vdpsum(vdp,vdpfmon(coef,vev))
)where c=vevcomp(vev,vdpevlmon vdp);
symbolic procedure vdpzero();a2vdp 0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Comparing of exponent vectors:
%
symbolic procedure vdpvevlcomp(p1,p2);dipevlcomp(vdppoly p1,vdppoly p2);
symbolic procedure vevilcompless!?(e1,e2);1=evilcomp(e2,e1);
symbolic procedure vevilcomp(e1,e2);evilcomp(e1,e2);
symbolic procedure vevcompless!?(e1,e2);1=evcomp(e2,e1);
symbolic procedure vevcomp(e1,e2);evcomp(e1,e2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Routines traversing the " coefficients ";
%
% CONTENT of a vdp:
% The content is the gcd of all coefficients.
symbolic procedure vdpcontent d;
if vdpzero!? d then a2bc 0 else
<<d:=vdppoly d;dipnumcontent(dipmred d,diplbc d)>>;
symbolic procedure vdpcontent1(d,c);dipnumcontent(vdppoly d,c);
symbolic procedure dipnumcontent(d,c);
if bcone!? c or dipzero!? d then c
else dipnumcontent(dipmred d,vbcgcd(c,diplbc d));
symbolic procedure dipcontenti p;
% The content is a pair of the lcm of the coefficients and the
% exponent list of the common monomial factor.
if dipzero!? p then 1 else
(if dipzero!? rp then diplbc p.
(if !*groebrm then dipevlmon p else nil)
else dipcontenti1(diplbc p, if !*groebrm then dipevlmon p else nil,rp))
where rp=dipmred p;
symbolic procedure dipcontenti1(n,ev,p1);
if dipzero!? p1 then n.ev
else begin scalar nn;nn:=vbcgcd(n,diplbc p1);
if ev then ev:=dipcontevmin(dipevlmon p1,ev);
if bcone!? nn and null ev then return nn.nil
else return dipcontenti1(nn,ev,dipmred p1)end;
% CONTENT and MONFAC(if groebrm is on).
symbolic procedure vdpcontenti d;
vdpcontent d.if !*groebrm then vdpmonfac d else nil;
symbolic procedure vdpmonfac d;dipmonfac vdppoly d;
symbolic procedure dipmonfac p;
% Exponent list of the common monomial factor.
if dipzero!? p or not !*groebrm then evzero()
else(if dipzero!? rp then dipevlmon p
else dipmonfac1(dipevlmon p,rp))where rp=dipmred p;
symbolic procedure dipmonfac1(ev,p1);
if dipzero!? p1 or evzero!? ev then ev
else dipmonfac1(dipcontevmin(ev,dipevlmon p1),dipmred p1);
% vdpcoeffcientsfromdomain?
symbolic procedure vdpcoeffcientsfromdomain!? w;
dipcoeffcientsfromdomain!? vdppoly w;
symbolic procedure dipcoeffcientsfromdomain!? w;
if dipzero!? w then t else
(if bcdomain!? v then dipcoeffcientsfromdomain!? dipmred w
else nil)where v=diplbc w;
symbolic procedure vdplength f;diplength vdppoly f;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Polynomial operations:
% coefficient normalization and reduction of monomial factors.
%
symbolic procedure vdpequal(p1,p2);
p1 eq p2
or(n1 and n1=n2 % number comparison is faster most times
or dipequal(vdppoly p1,vdppoly p2)
where n1=vdpgetprop(p1,'number),n2=vdpgetprop(p2,'number));
symbolic procedure dipequal(p1,p2);
if dipzero!? p1 then dipzero!? p2 else if dipzero!? p2 then nil
else diplbc p1=diplbc p2 and evequal(dipevlmon p1,dipevlmon p2)
and dipequal(dipmred p1,dipmred p2);
symbolic procedure evequal(e1,e2);
% Test equality with variable length exponent vectors.
if null e1 and null e2 then t
else if null e1 then evequal('(0),e2)
else if null e2 then evequal(e1,'(0))
else 0=(car e1 #- car e2)and evequal(cdr e1,cdr e2);
symbolic procedure vdplcm p;diplcm vdppoly p;
symbolic procedure vdprectoint(p,q);dip2vdp diprectoint(vdppoly p,q);
symbolic procedure vdpsimpcont(p);
begin scalar r,q;q:=vdppoly p;
if dipzero!? q then return p;r:=dipsimpcont q;
p:=dip2vdp cdr r;% the polynomial
r:=car r; % the monomial factor if any
if not evzero!? r and(dipmred q or evtdeg r>1)
then vdpputprop(p,'monfac,r);return p end;
symbolic procedure dipsimpcont(p);
if !*vdpinteger or not !*groebdivide then dipsimpconti p else dipsimpcontr p;
% Routines for integer coefficient case:
% calculation of contents and dividing all coefficients by it.
symbolic procedure dipsimpconti p;
% Calculate the contents of p and divide all coefficients by it.
begin scalar co,lco,res,num;
if dipzero!? p then return nil.p;co:=bcfd 1;
co:=if !*groebdivide then dipcontenti p
else if !*groebrm then co.dipmonfac p else co.nil;
num:=car co;
if not bcplus!? num then num:=bcneg num;
if not bcplus!? diplbc p then num:=bcneg num;
if bcone!? num and cdr co=nil then return nil.p;
lco:=cdr co;
if groebmonfac neq 0 then lco:=dipcontlowerev cdr co;
res:=p;
if not(bcone!? num and lco=nil)then res:=dipreduceconti(p,num,lco);
if null cdr co then return nil.res;
lco:=evdif(cdr co,lco);
return(if lco and not evzero!? evdif(dipevlmon res,lco)
then lco else nil).res end;
symbolic procedure vdpreduceconti(p,co,vev);
% Divide polynomial p by monomial from co and vev.
vdpdivmon(p,co,vev);
% Divide all coefficients of p by cont.
symbolic procedure dipreduceconti(p,co,ev);
if dipzero!? p then makedipzero()
else dipmoncomp(bcquot(diplbc p,co),
if ev then evdif(dipevlmon p,ev)
else dipevlmon p,dipreduceconti(dipmred p,co,ev));
% Routines for rational coefficient case:
% calculation of contents and dividing all coefficients by it
symbolic procedure dipsimpcontr p;
% Calculate the contents of p and divide all coefficients by it.
begin scalar co,lco,res;
if dipzero!? p then return nil.p;
co:=dipcontentr p;
if bcone!? diplbc p and co=nil then return nil.p;
lco:=dipcontlowerev co;res:=p;
if not(bcone!? diplbc p and lco=nil)then
res:=dipreducecontr(p,bcinv diplbc p,lco);
return(if co then evdif(co,lco)else nil).res end;
symbolic procedure dipcontentr p;
% The content is the exponent list of the common monomial factor.
(if dipzero!? rp then (if !*groebrm then dipevlmon p else nil)
else dipcontentr1(if !*groebrm then dipevlmon p else nil,rp))
where rp=dipmred p;
symbolic procedure dipcontentr1(ev,p1);
if dipzero!? p1 then ev
else begin
if ev then ev:=dipcontevmin(dipevlmon p1,ev);
if null ev then return nil
else return dipcontentr1(ev,dipmred p1)end;
% Divide all coefficients of p by cont.
symbolic procedure dipreducecontr(p,co,ev);
if dipzero!? p then makedipzero()
else dipmoncomp(bcprod(diplbc p,co),if ev then evdif(dipevlmon p,ev)
else dipevlmon p,dipreducecontr(dipmred p,co,ev));
symbolic procedure dipcontevmin(e1,e2);
% Calculates the minimum of two exponents;if one is shorter, trailing
% zeroes are assumed.
% e1 is an exponent vector.e2 is a list of exponents
begin scalar res;
while e1 and e2 do
<<res:=(if ilessp(car e1,car e2)then car e1 else car e2).res;
e1:=cdr e1;e2:=cdr e2>>;
while res and 0=car res do res:=cdr res;
return reversip res end;
symbolic procedure dipcontlowerev e1;
% Subtract a 1 from those elements of an exponent vector which
% are greater than 1.
% e1 is a list of exponents,the result is an exponent vector.
begin scalar res;
while e1 do
<<res:=(if igreaterp(car e1,0)then car e1 - 1 else 0).res;e1:=cdr e1>>;
while res and 0=car res do res:=cdr res;
if res and !*trgroebs then
<<prin2 " ***** exponent reduction : ";prin2t reverse res>>;
return reversip res end;
symbolic procedure dipappendmon(dip,bc,ev);append(dip,dipfmon(bc,ev));
smacro procedure dipnconcmon(dip,bc,ev);nconc(dip,dipfmon(bc,ev));
smacro procedure dipappenddip(dip1,dip2);append(dip1,dip2);
smacro procedure dipnconcdip(dip1,dip2);nconc(dip1,dip2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Basic polynomial arithmetic:
%
symbolic procedure vdpsum(d1,d2);
begin scalar r;
r:=dip2vdp dipsum(vdppoly d1,vdppoly d2);
if !*gsugar then gsetsugar(r,max(gsugar d1,gsugar d2));return r end;
symbolic procedure vdpdif(d1,d2);
begin scalar r;
r:=dip2vdp dipdif(vdppoly d1,vdppoly d2);
if !*gsugar then gsetsugar(r,max(gsugar d1,gsugar d2));return r end;
symbolic procedure vdpprod(d1,d2);
begin scalar r;
r:= dip2vdp dipprod(vdppoly d1,vdppoly d2);
if !*gsugar then gsetsugar(r,gsugar d1 + gsugar d2);return r end;
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%
% Linear combination: the Buchberger workhorse.
%
% LCOMB1: calculate mon1 * vdp1 + mon2 * vdp2.
symbolic procedure vdpilcomb1(d1,vbc1,vev1,d2,vbc2,vev2);
begin scalar r;
r:=
dip2vdp dipilcomb1(vdppoly d1,vbc1,vev1,vdppoly d2,vbc2,vev2);
if !*gsugar then gsetsugar(r,max(gsugar d1 + vevtdeg vev1,
gsugar d2 + vevtdeg vev2));return r end;
symbolic procedure dipilcomb1(p1,bc1,ev1,p2,bc2,ev2);
% Same as dipILcomb, exponent vectors multiplied in already.
begin scalar gcd;
gcd:=!*gcd;
return
begin scalar ep1,ep2,sl,res,sum,z1,z2,p1new,p2new,
lptr,bptr,c,!*gcd;
!*gcd:=if vbcsize(bc1,-100)and vbcsize(bc2,-100)then gcd;
z1:=not evzero!? ev1;z2:=not evzero!? ev2;
p1new:=p2new:=t;
lptr:=bptr:=res:=makedipzero();
loop:
if p1new then
<<if dipzero!? p1 then return if dipzero!? p2 then res else
dipnconcdip(res,dipprod(p2,dipfmon(bc2,ev2)));
ep1:=dipevlmon p1;
if z1 then ep1:=evsum(ep1,ev1);
p1new:=nil>>;
if p2new then
<<if dipzero!? p2 then
return dipnconcdip(res,dipprod(p1,dipfmon(bc1,ev1)));
ep2:=dipevlmon p2;
if z2 then ep2:=evsum(ep2,ev2);
p2new:=nil>>;
sl:=evcomp(ep1,ep2);
if sl=1 then
<<if !*gcd and not vbcsize(diplbc p1,-100)then !*gcd:=nil;
c:=bcprod(diplbc p1,bc1);
if not bczero!? c then
<<lptr:=dipnconcmon(bptr,c,ep1);
bptr:=dipmred lptr>>;
p1:=dipmred p1;p1new:=t;
>> else if sl=-1 then
<<if !*gcd and not vbcsize(diplbc p2,-100)then !*gcd:=nil;
c:=bcprod(diplbc p2,bc2);
if not bczero!? c then
<<lptr:=dipnconcmon(bptr,c,ep2);bptr:=dipmred lptr>>;
p2:=dipmred p2;p2new:=t>>
else
<<if !*gcd and(not vbcsize(diplbc p1,-100)or
not vbcsize(diplbc p2,-100)) then !*gcd:=nil;
sum:=bcsum(bcprod(diplbc p1,bc1),
bcprod(diplbc p2,bc2));
if not bczero!? sum then
<<lptr:=dipnconcmon(bptr,sum,ep1);
bptr:=dipmred lptr>>;
p1:=dipmred p1;p2:=dipmred p2;p1new:=p2new:=t>>;
if dipzero!? res then <<res:=bptr:=lptr>>;% initial
goto loop end;end;
symbolic procedure vdpvbcprod(p,a);
(if !*gsugar then gsetsugar(q,gsugar p)else q)
where q=dip2vdp dipbcprod(vdppoly p,a);
symbolic procedure vdpdivmon(p,c,vev);
(if !*gsugar then gsetsugar(q,gsugar p)else q)
where q=dip2vdp dipdivmon(vdppoly p,c,vev);
symbolic procedure dipdivmon(p,bc,ev);
% Divides a polynomial by a monomial;
% we are sure that the monomial ev is a factor of p.
if dipzero!? p then makedipzero()
else dipmoncomp(bcquot(diplbc p,bc),evdif(dipevlmon p,ev),
dipdivmon(dipmred p,bc,ev));
symbolic procedure vdpcancelmvev(p,vev);
(if !*gsugar then gsetsugar(q,gsugar p)else q)
where q=dip2vdp dipcancelmev(vdppoly p,vev);
symbolic procedure dipcancelmev(f,ev);
% Cancels all monomials in f which are multiples of ev
dipcancelmev1(f,ev,makedipzero());
symbolic procedure dipcancelmev1(f,ev,res);
if dipzero!? f then res
else if evmtest!?(dipevlmon f,ev)then dipcancelmev1(dipmred f,ev,res)
else dipcancelmev1(dipmred f,ev,
% dipappendmon(res,diplbc f,dipevlmon f));
dipnconcmon(res,diplbc f,dipevlmon f));
% Some prehistoric routines needed in resultant operation
symbolic procedure vevsum0(n,p);
% Exponent vector sum version 0 . n is the length of vdpvars!*.
% p is a distributive polynomial.
if vdpzero!? p then vevzero1 n else vevsum(vdpevlmon p,vevsum0(n,vdpred p));
symbolic procedure vevzero1 n;
% Returns the exponent vector power representation
% of length n for a zero power.
begin scalar x;for i:=1:n do x:=0 . x;return x end;
symbolic procedure vdpresimp u;
% if domain changes,the coefficients have to be resimped
dip2vdp dipresimp vdppoly u;
symbolic procedure dipresimp u;
if null u then nil else
(for each x in u collect
<<toggle:=not toggle;
if toggle then simp prepsq x else x>>)where toggle = t;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% printing of polynomials
%
symbolic procedure vdpprin2t u;<<vdpprint1(u,nil,9999);terpri()>>;
symbolic procedure vdpprin3t u;<<vdpprint1(u,nil,9999);prin2t ";">>;
symbolic procedure vdpprint u;<<vdpprin2 u;terpri()>>;
symbolic procedure vdpprin2 u;
<<(if x then <<prin2 " P(";prin2 x;
if s then <<prin2 " / ";prin2 s>>;prin2 "): ">>)
where x=vdpgetprop(u,'number),s= vdpgetprop(u,'sugar);
vdpprint1(u,nil,vdpprintmax)>>;
symbolic procedure vdpprint1(u,v,max);vdpprint1x(vdppoly u,v,max);
symbolic procedure vdpprint1x(u,v,max);
% Prints a distributive polynomial in infix form.
% U is a distributive form. V is a flag which is true if a term
% has preceded current form
% max limits the number of terms to be printed
if dipzero!? u then if null v then dipprin2 0 else nil
else if max=0 then % maximum of terms reached
<<terpri();prin2 " ### etc(";
prin2 diplength u;prin2 " terms)### ";terpri()>>
else begin scalar bool,w;
w:=diplbc u;
if bcminus!? w then<<bool:=t;w:=bcneg w>>;
if bool then dipprin2 " - " else if v then dipprin2 " + ";
(if not bcone!? w or evzero!? x then<<bcprin w;dipevlpri(x,t)>>
else dipevlpri(x,nil))
where x=dipevlmon u;
vdpprint1x(dipmred u,t,max - 1)end;
symbolic procedure dipprin2 u;<<if posn()>69 then terprit 2;prin2 u>>;
symbolic procedure vdpsave u;u;
% switching between term order modes
symbolic procedure torder2 u;dipsortingmode u;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% additional conversion utilities
% conversion dip to standard form / standard quotient
symbolic procedure dip2f u;
(if denr v neq 1 then
<<print u;
rerror(dipoly,9,
" Distrib . poly . with rat coeff cannot be converted ")>>
else numr v) where v=dip2sq u;
symbolic procedure dip2sq u;
% Convert a dip into a standard quotient.
if dipzero!? u then nil ./ 1
else addsq(diplmon2sq(diplbc u,dipevlmon u), dip2sq dipmred u);
symbolic procedure diplmon2sq(bc,ev);
% Convert a monomial into a standard quotient.
multsq(bc,dipev2f(ev,dipvars!*)./ 1);
symbolic procedure dipev2f(ev,vars);
if null ev then 1
else if car ev=0 then dipev2f(cdr ev,cdr vars)
else multf(car vars .** car ev .* 1 .+ nil,dipev2f(cdr ev,cdr vars));
% evaluate SUBS2 for the coefficients of a dip
symbolic procedure dipsubs2 u;
begin scalar v,secondvalue!*;
secondvalue!*:=1 ./ 1;v:=dipsubs21 u;
return diprectoint(v,secondvalue!*)end;
symbolic procedure dipsubs21 u;
if dipzero!? u then u else
begin scalar c;c:=groebsubs2 diplbc u;
if null numr c then return dipsubs21 dipmred u;
if not(denr c=1)then secondvalue!*:=bclcmd(c,secondvalue!*);
return dipmoncomp(c,dipevlmon u,dipsubs21 dipmred u)end;
% conversion standard form to dip
symbolic procedure f2dip u;f2dip1(u,evzero(),bcfd 1);
symbolic procedure f2dip1(u,ev,bc);
% f to dip conversion : scan the standard form. ev
% and bc are the exponent and coefficient parts collected
% so far from higher parts.
if null u then nil
else if domainp u then dipfmon(bcprod(bc,bcfd u), ev)
else dipsum(f2dip2(mvar u,ldeg u,lc u,ev,bc), f2dip1(red u,ev,bc));
symbolic procedure f2dip2(var,dg,c,ev,bc);
% f to dip conversion:
% multiply leading power either into exponent vector
% or into the base coefficient.
<<if ev1 then ev:=ev1
else bc:=multsq(bc,var .** dg .* 1 .+ nil ./ 1);
f2dip1(c,ev,bc)>>
where ev1=if memq(var,dipvars!*)then
evinsert(ev,var,dg,dipvars!*)else nil;
symbolic procedure evinsert(ev,v,dg,vars);
% f to dip conversion:
% Insert the "dg" into the ev in the place of variable v.
if null ev or null vars then nil
else if car vars eq v then dg.cdr ev
else car ev.evinsert(cdr ev,v,dg,cdr vars);
symbolic procedure vdpcondense f;dipcondense car cdddr f;
endmodule;;end;