module dipoly; % Header module for dipoly package.
%/*Authors: R. Gebauer, A. C. Hearn, H. Kredel*/
% For the time being, this contains the smacros that used to be in
% consel, and repeats those in bcoeff.
%/*Constructors and selectors for a distributed polynomial form*/
%/*A distributive polynomial has the following informal syntax:
%
% <dipoly> ::= dipzero
% | <exponent vector> . <base coefficient> . <dipoly>*/
% Vdp2dip modules included. They could be in a separate package.
create!-package('(dipoly a2dip bcoeff dip2a dipoly1 dipvars
expvec torder vdp2dip vdpcom),
'(contrib dipoly));
%define dipzero = 'nil;
fluid '(dipzero);
%/*Until we understand how to define something to nil*/
smacro procedure dipzero!? u; null u;
smacro procedure diplbc p;
% /* Distributive polynomial leading base coefficient.
% p is a distributive polynomial. diplbc(p) returns
% the leading base coefficient of p. */
cadr p;
smacro procedure dipmoncomp (a,e,p);
% /* Distributive polynomial monomial composition. a is a base
% coefficient, e is an exponent vector and p is a
% distributive polynomial. dipmoncomp(a,e,p) returns a dis-
% tributive polynomial with p as monomial reductum, e as
% exponent vector of the leading monomial and a as leading
% base coefficient. */
e . a . p;
smacro procedure dipevlmon p;
% /* Distributive polynomial exponent vector leading monomial.
% p is a distributive polynomial. dipevlmon(p) returns the
% exponent vector of the leading monomial of p. */
car p;
smacro procedure dipfmon (a,e);
% /* Distributive polynomial from monomial. a is a base coefficient
% and e is an exponent vector. dipfmon(a,e) returns a
% distributive polynomial with e as exponent vector and
% a as base coefficient. */
e . a . dipzero;
smacro procedure dipnov p;
% /* Distributive polynomial number of variables. p is a distributive
% polynomial. dipnov(p) returns a digit, the number of variables
% of the distributive polynomial p. */
length car p;
smacro procedure dipmred p;
% /* Distributive polynomial reductum. p is a distributive polynomial
% dipmred(p) returns the reductum of the distributive polynomial p,
% a distributive polynomial. */
cddr p;
% These smacros are also in bcoeff.
smacro procedure bcminus!? u;
% /* Boolean function. Returns true if u is a negative base coeff*/
minusf numr u;
smacro procedure bczero!? u;
% /* Returns a boolean expression, true if base coefficient u is zero*/
null numr u;
endmodule;
module a2dip;
%/*Convert an algebraic (prefix) form to distributive polynomial*/
%/*Authors: R. Gebauer, A. C. Hearn, H. Kredel*/
% Modified by: H. Melenk.
fluid '(dipvars!* dipzero);
symbolic procedure a2dip u;
% /*Converts the algebraic (prefix) form u to a distributive poly.
% We assume that all variables used have been previously
% defined in dipvars!*, but a check is also made for this*/
if atom u then a2dipatom u
else if not atom car u or not idp car u
then typerr(car u,"dipoly operator")
% Handling expt separately because the exponents should
% not be simplified as domain elements.
else if car u = 'expt then dipfnpow(a2dip cadr u,caddr u)
else (if x then apply(x,list for each y in cdr u collect a2dip y)
else a2dipatom u)
where x = get(car u,'dipfn);
expr procedure a2dipatom u;
% /*Converts the atom (or kernel) u into a distributive polynomial*/
if u=0 then dipzero
else if numberp u or not(u member dipvars!*)
then dipfmon(a2bc u,evzero())
else dipfmon(a2bc 1,mkexpvec u);
expr procedure dipfnsum u;
% /*U is a list of dip expressions. Result is the distributive poly
% representation for the sum*/
(<<for each y in cdr u do x := dipsum(x,y); x>>) where x = car u;
put('plus,'dipfn,'dipfnsum);
put('plus2,'dipfn,'dipfnsum);
expr procedure dipfnprod u;
% /*U is a list of dip expressions. Result is the distributive poly
% representation for the product*/
% /*Maybe we should check for a zero*/
(<<for each y in cdr u do x := dipprod(x,y); x>>) where x = car u;
put('times,'dipfn,'dipfnprod);
put('times2,'dipfn,'dipfnprod);
expr procedure dipfndif u;
% /*U is a list of two dip expressions. Result is the distributive
% polynomial representation for the difference*/
dipsum(car u,dipneg cadr u);
put('difference,'dipfn,'dipfndif);
symbolic procedure dipfnpow(v,n);
% V is a dip. Result is the distributive poly v**n.
(if not fixp n or n<0
then typerr(n,"distributive polynomial exponent")
else if n=0 then if dipzero!? v then rerror(dipoly,1,"0**0 invalid")
else w
else if dipzero!? v or n=1 then v
else if dipzero!? dipmred v
then dipfmon(bcpow(diplbc v,n),intevprod(n,dipevlmon v))
else <<while n>0 do
<<if not evenp n then w := dipprod(w,v);
n := n/2;
if n>0 then v := dipprod(v,v)>>;
w>>)
where w := dipfmon(a2bc 1,evzero());
% put('expt,'dipfn,'dipfnpow);
expr procedure dipfnneg u;
% /*U is a list of one dip expression. Result is the distributive
% polynomial representation for the negative*/
(if dipzero!? v then v
else dipmoncomp(bcneg diplbc v,dipevlmon v,dipmred v))
where v = car u;
put('minus,'dipfn,'dipfnneg);
expr procedure dipfnquot u;
% /*U is a list of two dip expressions. Result is the distributive
% polynomial representation for the quotient*/
if dipzero!? cadr u or not dipzero!? dipmred cadr u
or not evzero!? dipevlmon cadr u
then typerr(dip2a cadr u,"distributive polynomial denominator")
else dipfnquot1(car u,diplbc cadr u);
expr procedure dipfnquot1(u,v);
if dipzero!? u then u
else dipmoncomp(bcquot(diplbc u,v),
dipevlmon u,
dipfnquot1(dipmred u,v));
put('quotient,'dipfn,'dipfnquot);
endmodule;
module bcoeff; % Computation of base coefficients.
%/*Definitions of base coefficient operations for distributive
% polynomial package. We assume that only field elements are used, but
% no check is made for this. In this module, a standard quotient
% coefficient is assumed*/
%/*Authors: R. Gebauer, A. C. Hearn, H. Kredel*/
% H. Melenk: added routines for faster handling of standard case
% of standard quotients representing integers.
fluid '(dmode!*);
symbolic procedure bcint2op(a1,a2,op);
null dmode!* and
1=denr a1 and numberp (a1:=numr a1) and
1=denr a2 and numberp (a2:=numr a2) and
(a1 := apply(op,list(a1,a2))) and
((if a1=0 then nil else a1) ./ 1);
fluid '(!*nat);
symbolic procedure bcless!? (a1,a2);
% /* Base coefficient less. a1 and a2 are base coefficients.
% bcless!?(a1,a2) returns a boolean expression, true if
% a1 is less than a2 else false. */
minusf numr addsq(a1,negsq a2);
smacro procedure bcminus!? u;
% /* Boolean function. Returns true if u is a negative base coeff*/
minusf numr u;
smacro procedure bczero!? u;
% /* Returns a boolean expression, true if the base coefficient u is
% zero*/
null numr u;
symbolic procedure bccomp (a1,a2);
% /* Base coefficient compare a1 and a2 are base coefficients.
% bccomp(a1,a2) compares the base coefficients a1 and a2 and returns
% a digit 1 if a1 greater than a2, a digit 0 if a1 equals a2 else a
% digit -1. */
(if bczero!? sl then 0
else if bcminus!? sl then -1
else 1)
where sl = bcdif(a1, a2);
symbolic procedure bcfi a;
% /* Base coefficient from integer. a is an integer. bcfi(a) returns
% the base coefficient a. */
mkbc(a,1);
symbolic procedure bclcmd(u,v);
% Base coefficient least common multiple of denominators.
% u and v are two base coefficients. bclcmd(u,v) calculates the
% least common multiple of the denominator of u and the
% denominator of v and returns a base coefficient of the form
% 1/lcm(denom u,denom v).
if bczero!? u then mkbc(1,denr v)
else if bczero!? v then mkbc(1,denr u)
else mkbc(1,multf(quotf(denr u,gcdf(denr u,denr v)),denr v));
symbolic procedure bclcmdprod(u,v);
% Base coefficient least common multiple denominator product.
% u is a basecoefficient of the form 1/integer. v is a base
% coefficient. bclcmdprod(u,v) calculates (denom u/denom v)*nom v/1
% and returns a base coefficient.
mkbc(multf(quotf(denr u,denr v),numr v),1);
symbolic procedure bcquod(u,v);
% Base coefficient quotient. u and v are base coefficients.
% bcquod(u,v) calculates u/v and returns a base coefficient.
bcprod(u,bcinv v);
symbolic procedure bcone!? u;
% /* Base coefficient one. u is a base coefficient.
% bcone!?(u) returns a boolean expression, true if the
% base coefficient u is equal 1. */
denr u = 1 and numr u = 1;
symbolic procedure bcinv u;
% /* Base coefficient inverse. u is a base coefficient.
% bcinv(u) calculates 1/u and returns a base coefficient. */
invsq u;
symbolic procedure bcneg u;
% /* Base coefficient negative. u is a base coefficient.
% bcneg(u) returns the negative of the base coefficient
% u, a base coefficient. */
negsq u;
symbolic procedure bcprod (u,v);
% /* Base coefficient product. u and v are base coefficients.
% bcprod(u,v) calculates u*v and returns a base coefficient.
bcint2op(u,v,'bcnumtimes) or multsq(u,v);
symbolic procedure bcnumtimes(u,v); u*v;
symbolic procedure mkbc(u,v);
% /* Convert u and v into u/v in lowest terms*/
if v = 1 then u ./ v
else if minusf v then mkbc(negf u,negf v)
else quotf(u,m) ./ quotf(v,m) where m = gcdf(u,v);
symbolic procedure bcquot (u,v);
% /* Base coefficient quotient. u and v are base coefficients.
% bcquot(u,v) calculates u/v and returns a base coefficient. */
quotsq(u,v);
symbolic procedure bcsum (u,v);
% /* Base coefficient sum. u and v are base coefficients.
% bcsum(u,v) calculates u+v and returns a base coefficient. */
bcint2op(u,v,'bcnumplus) or addsq(u,v);
symbolic procedure bcnumplus(u,v); u+v;
symbolic procedure bcdif(u,v);
% /* Base coefficient difference. u and v are base coefficients.
% bcdif(u,v) calculates u-v and returns a base coefficient. */
bcint2op(u,v,'bcnumdifference) or bcsum(u,bcneg v);
symbolic procedure bcnumdifference(u,v); u-v;
symbolic procedure bcpow(u,n);
% /*Returns the base coefficient u raised to the nth power, where
% n is an integer*/
exptsq(u,n);
symbolic procedure a2bc u;
% /*Converts the algebraic (kernel) u into a base coefficient.
simp!* u;
symbolic procedure bc2a u;
% /* Returns the prefix equivalent of the base coefficient u*/
prepsq u;
fluid '(!*groebigpos !*groebigneg !*groescale);
!*groescale := 20;
!*groebigpos:= 10** !*groescale; !*groebigneg := - 10** !*groescale;
symbolic procedure bcprin u;
% /* Prints a base coefficient in infix form*/
begin scalar nat;
nat := !*nat;
!*nat := nil;
if cdr u = 1 and
numberp car u and
(car u>!*groebigpos or car u<!*groebigneg)
then bcprin2big car u
else
sqprint u;
!*nat := nat
end;
symbolic procedure bcprin2big u;
<< if u<0 then<< prin2 "-"; u:= -u>>;
bcprin2big1(u,0)>>;
symbolic procedure bcprin2big1 (u,n);
if u>!*groebigpos then
bcprin2big1 (u/!*groebigpos,n#+!*groescale)
else <<prin2 u; prin2 "e"; prin2 n>>;
endmodule;
module dip2a;
%/* Functions for converting distributive forms into prefix forms*/
%/*Authors: R. Gebauer, A. C. Hearn, H. Kredel*/
expr procedure dip2a u;
% /* Returns prefix equivalent of distributive polynomial u*/
if dipzero!? u then 0 else dipreplus dip2a1 u;
expr procedure dip2a1 u;
if dipzero!? u then nil
else ((if bcminus!? x then list('minus,dipretimes(bc2a bcneg x . y))
else dipretimes(bc2a x . y))
where x = diplbc u, y = expvec2a dipevlmon u)
. dip2a1 dipmred u;
expr procedure dipreplus u;
if atom u then u else if null cdr u then car u else 'plus . u;
expr procedure dipretimes u;
% /* U is a list of prefix expressions the first of which is a number.
% Result is prefix representation for their product*/
if car u = 1 then if cdr u then dipretimes cdr u else 1
else if null cdr u then car u
else 'times . u;
endmodule;
module dipoly; % /*Distributive polnomial algorithms*/
%/*Authors: R. Gebauer, A. C. Hearn, H. Kredel*/
% modification for REDUCE 3.4: H. Melenk.
fluid '(dipvars!* dipzero);
symbolic procedure dipconst!? p;
not dipzero!? p
and dipzero!? dipmred p
and evzero!? dipevlmon p;
symbolic procedure dfcprint pl;
% h polynomial factor list of distributive polynomials print.
for each p in pl do dfcprintin p;
symbolic procedure dfcprintin p;
% factor with exponent print.
( if cdr p neq 1 then << prin2 " ( "; dipprint1(p1,nil); prin2 " )** ";
prin2 cdr p; terprit 2 >> else << prin2 " "; dipprint p1>> )
where p1:= dipmonic a2dip prepf car p;
symbolic procedure dfcprin p;
% print content, factors and exponents of factorized polynomial p.
<< terpri(); prin2 " content of factorized polynomials = ";
prin2 car p;
terprit 2; dfcprint cdr p >>;
symbolic procedure diplcm p;
% Distributive polynomial least common multiple of denomiators.
% p is a distributive rational polynomial. diplcm(p) calculates
% the least common multiple of the denominators and returns a
% base coefficient of the form 1/lcm(denom bc1,.....,denom bci).
if dipzero!? p then mkbc(1,1)
else bclcmd(diplbc p, diplcm dipmred p);
symbolic procedure diprectoint(p,u);
% Distributive polynomial conversion rational to integral.
% p is a distributive rational polynomial, u is a base coefficient
% ( 1/lcm denom p ). diprectoint(p,u) returns a primitive
% associate pseudo integral ( denominators are 1 ) distributive
% polynomial.
if bczero!? u then dipzero
else if bcone!? u then p
else diprectoint1(p,u);
symbolic procedure diprectoint1(p,u);
% Distributive polynomial conversion rational to integral internall 1.
% diprectoint1 is used in diprectoint.
if dipzero!? p then dipzero
else dipmoncomp(bclcmdprod(u,diplbc p),dipevlmon p,
diprectoint1(dipmred p,u));
symbolic procedure dipresul(p1,p2);
% test for fast downwards calculation
% p1 and p2 are two bivariate distributive polynomials.
% dipresul(p1,p2) returns the resultant of p1 and p2 with respect
% respect to the first variable of the variable list (car dipvars!*).
begin scalar q1,q2,q,ct;
q1:=dip2a diprectoint(p1,diplcm p1);
q2:=dip2a diprectoint(p2,diplcm p2);
ct := time();
q:= a2dip prepsq simpresultant list(q1,q2,car dipvars!*);
ct := time() - ct;
prin2 " resultant : "; dipprint dipmonic q; terpri();
prin2 " time resultant : "; prin2 ct; terpri();
end;
symbolic procedure dipbcprod (p,a);
% /* Distributive polynomial base coefficient product.
% p is a distributive polynomial, a is a base coefficient.
% dipbcprod(p,a) computes p*a, a distributive polynomial. */
if bczero!? a then dipzero
else if bcone!? a then p
else dipbcprodin(p,a);
symbolic procedure dipbcprodin (p,a);
% /* Distributive polynomial base coefficient product internal.
% p is a distributive polynomial, a is a base coefficient,
% where a is not equal 0 and not equal 1.
% dipbcprodin(p,a) computes p*a, a distributive polynomial. */
if dipzero!? p then dipzero
else dipmoncomp(bcprod(a, diplbc p),
dipevlmon p,
dipbcprodin(dipmred p, a));
symbolic procedure dipdif (p1,p2);
% /* Distributive polynomial difference. p1 and p2 are distributive
% polynomials. dipdif(p1,p2) calculates the difference of the
% two distributive polynomials p1 and p2, a distributive polynomial*/
if dipzero!? p1 then dipneg p2
else if dipzero!? p2 then p1
else ( if sl = 1 then dipmoncomp(diplbc p1,
ep1,
dipdif(dipmred p1, p2) )
else if sl = -1 then dipmoncomp(bcneg diplbc p2,
ep2,
dipdif(p1,dipmred p2))
else ( if bczero!? al
then dipdif(dipmred p1, dipmred p2)
else dipmoncomp(al,
ep1,
dipdif(dipmred p1,
dipmred p2) )
) where al = bcdif(diplbc p1, diplbc p2)
) where sl = evcomp(ep1, ep2)
where ep1 = dipevlmon p1, ep2 = dipevlmon p2;
symbolic procedure diplength p;
% /* Distributive polynomial length. p is a distributive
% polynomial. diplength(p) returns the number of terms
% of the distributive polynomial p, a digit.*/
if dipzero!? p then 0 else 1 + diplength dipmred p;
symbolic procedure diplistsum pl;
% /* Distributive polynomial list sum. pl is a list of distributive
% polynomials. diplistsum(pl) calculates the sum of all polynomials
% and returns a list of one distributive polynomial. */
if null pl or null cdr pl then pl
else diplistsum(dipsum(car pl, cadr pl) . diplistsum cddr pl);
symbolic procedure diplmerge (pl1,pl2);
% /* Distributive polynomial list merge. pl1 and pl2 are lists
% of distributive polynomials where pl1 and pl2 are in non
% decreasing order. diplmerge(pl1,pl2) returns the merged
% distributive polynomial list of pl1 and pl2. */
if null pl1 then pl2
else if null pl2 then pl1
else ( if sl >= 0 then cpl1 . diplmerge(cdr pl1, pl2)
else cpl2 . diplmerge(cdr pl2, pl1)
) where sl = evcomp(ep1, ep2)
where ep1 = dipevlmon cpl1, ep2 = dipevlmon cpl2
where cpl1 = car pl1, cpl2 = car pl2;
symbolic procedure diplsort pl;
% /* Distributive polynomial list sort. pl is a list of
% distributive polynomials. diplsort(pl) returns the
% sorted distributive polynomial list of pl.
sort(pl, function dipevlcomp);
symbolic procedure dipevlcomp (p1,p2);
% /* Distributive polynomial exponent vector leading monomial
% compare. p1 and p2 are distributive polynomials.
% dipevlcomp(p1,p2) returns a boolean expression true if the
% distributive polynomial p1 is smaller or equal the distributive
% polynomial p2 else false. */
not evcompless!?(dipevlmon p1, dipevlmon p2);
symbolic procedure dipmonic p;
% /* Distributive polynomial monic. p is a distributive
% polynomial. dipmonic(p) computes p/lbc(p) if p is
% not equal dipzero and returns a distributive
% polynomial, else dipmonic(p) returns dipzero. */
if dipzero!? p then p
else dipbcprod(p, bcinv diplbc p);
symbolic procedure dipneg p;
% /* Distributive polynomial negative. p is a distributive
% polynomial. dipneg(p) returns the negative of the distributive
% polynomial p, a distributive polynomial. */
if dipzero!? p then p
else dipmoncomp ( bcneg diplbc p,
dipevlmon p,
dipneg dipmred p );
symbolic procedure dipone!? p;
% /* Distributive polynomial one. p is a distributive polynomial.
% dipone!?(p) returns a boolean value. If p is the distributive
% polynomial one then true else false. */
not dipzero!? p
and dipzero!? dipmred p
and evzero!? dipevlmon p
and bcone!? diplbc p;
symbolic procedure dippairsort pl;
% /* Distributive polynomial list pair merge sort. pl is a list
% of distributive polynomials. dippairsort(pl) returns the
% list of merged and in non decreasing order sorted
% distributive polynomials. */
if null pl or null cdr pl then pl
else diplmerge(diplmerge( car(pl) . nil, cadr(pl) . nil ),
dippairsort cddr pl);
symbolic procedure dipprod (p1,p2);
% /* Distributive polynomial product. p1 and p2 are distributive
% polynomials. dipprod(p1,p2) calculates the product of the
% two distributive polynomials p1 and p2, a distributive polynomial*/
if diplength p1 <= diplength p2 then dipprodin(p1, p2)
else dipprodin(p2, p1);
symbolic procedure dipprodin (p1,p2);
% /* Distributive polynomial product internal. p1 and p2 are distrib
% polynomials. dipprod(p1,p2) calculates the product of the
% two distributive polynomials p1 and p2, a distributive polynomial*/
if dipzero!? p1 or dipzero!? p2 then dipzero
else ( dipmoncomp(bcprod(bp1, diplbc p2),
evsum(ep1, dipevlmon p2),
dipsum(dipprodin(dipfmon(bp1, ep1),
dipmred p2),
dipprodin(dipmred p1, p2) ) )
) where bp1 = diplbc p1,
ep1 = dipevlmon p1;
symbolic procedure dipprodls (p1,p2);
% /* Distributive polynomial product. p1 and p2 are distributive
% polynomials. dipprod(p1,p2) calculates the product of the
% two distributive polynomials p1 and p2, a distributive polynomial
% using distributive polynomials list sum (diplistsum). */
if dipzero!? p1 or dipzero!? p2 then dipzero
else car diplistsum if diplength p1 <= diplength p2
then dipprodlsin(p1, p2)
else dipprodlsin(p2, p1);
symbolic procedure dipprodlsin (p1,p2);
% /* Distributive polynomial product. p1 and p2 are distributive
% polynomials. dipprod(p1,p2) calculates the product of the
% two distributive polynomials p1 and p2, a distributive polynomial
% using distributive polynomials list sum (diplistsum). */
if dipzero!? p1 or dipzero!? p2 then nil
else ( dipmoncomp(bcprod(bp1, diplbc p2),
evsum(ep1, dipevlmon p2),
car dipprodlsin(dipfmon(bp1, ep1),
dipmred p2))
. dipprodlsin(dipmred p1, p2)
) where bp1 = diplbc p1,
ep1 = dipevlmon p1;
symbolic procedure dipsum (p1,p2);
% /* Distributive polynomial sum. p1 and p2 are distributive
% polynomials. dipsum(p1,p2) calculates the sum of the
% two distributive polynomials p1 and p2, a distributive polynomial*/
if dipzero!? p1 then p2
else if dipzero!? p2 then p1
else ( if sl = 1 then dipmoncomp(diplbc p1,
ep1,
dipsum(dipmred p1, p2) )
else if sl = -1 then dipmoncomp(diplbc p2,
ep2,
dipsum(p1,dipmred p2))
else ( if bczero!? al then dipsum(dipmred p1,
dipmred p2)
else dipmoncomp(al,
ep1,
dipsum(dipmred p1,
dipmred p2) )
) where al = bcsum(diplbc p1, diplbc p2)
) where sl = evcomp(ep1, ep2)
where ep1 = dipevlmon p1, ep2 = dipevlmon p2;
endmodule;
module dipvars;
%/* Determine distributive polynomial variables in a prefix form*/
%/*Authors: R. Gebauer, A. C. Hearn, H. Kredel*/
expr procedure dipvars u;
% /* Returns list of variables in prefix form u*/
dipvars1(u,nil);
expr procedure dipvars1(u,v);
if atom u then if constantp u or u memq v then v else u . v
else if idp car u and get(car u,'dipfn)
then dipvarslist(cdr u,v)
else if u memq v then v
else u . v;
expr procedure dipvarslist(u,v);
if null u then v
else dipvarslist(cdr u,union(dipvars car u,v));
endmodule;
module expvec;
% /*Specific support for distributive polynomial exponent vectors*/
% /* Authors: R. Gebauer, A. C. Hearn, H. Kredel */
% We assume here that an exponent vector is a list of integers. This
% version uses small integer arithmetic on the individual exponents
% and assumes that a compiled function can be dynamically redefined*/
% Modification H. Melenk (August 1988)
% 1. Most ev-routines handle exponent vectors with variable length:
% the convention is, that trailing zeros may be omitted.
% 2. evcompless!? is mapped to evcomp such that each term order mode
% is supported by exactly one procedure entry.
% 3. complete exponent vector compare collected in separate module
% TORDER (TORD33)
fluid '(dipsortmode!* dipvars!*);
expr procedure evperm (e1,n);
% /* Exponent vector permutation. e1 is an exponent vector, n is a
% index list , a list of digits. evperm(e1,n) returns a list e1
% permuted in respect to n. */
if null n then nil
else evnth(e1, car n) . evperm(e1, cdr n);
expr procedure evcons (e1,e2);
% /* Exponent vector construct. e1 and e2 are exponents. evcons(e1,e2)
% constructs an exponent vector. */
e1 . e2;
expr procedure evnth (e1,n);
% /* Exponent vector n-th element. e1 is an exponent vector, n is a
% digit. evnth(e1,n) returns the n-th element of e1, an exponent. */
if null e1 then 0 else
if n = 1 then evfirst e1 else evnth(evred e1, n - 1);
expr procedure evred e1;
% /* Exponent vector reductum. e1 is an exponent vector. evred(e1)
% returns the reductum of the exponent vector e1. */
if e1 then cdr e1 else nil;
expr procedure evfirst e1;
% /* Exponent vector first. e1 is an exponent vector. evfirst(e1)
% returns the first element of the exponent vector e1, an exponent. */
if e1 then car e1 else 0;
expr procedure evsum0(n,p);
% exponent vector sum version 0. n is the length of dipvars!*.
% p is a distributive polynomial.
if dipzero!? p then evzero1 n else
evsum(dipevlmon p, evsum0(n,dipmred p));
expr procedure evzero1 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;
expr procedure indexcpl(ev,n);
% returns a list of indexes of non zero exponents.
if null ev then ev else ( if car ev = 0 then
indexcpl(cdr ev,n + 1) else
( n . indexcpl(cdr ev,n + 1)) );
expr procedure evzer1!? e;
% returns a boolean expression. true if e is null else false.
null e;
expr procedure evzero!? e;
% /* Returns a boolean expression. True if all exponents are zero*/
null e or car e = 0 and evzero!? cdr e;
expr procedure evzero;
% /* Returns the exponent vector representation for a zero power*/
% for i := 1:length dipvars!* collect 0;
begin scalar x;
for i := 1:length dipvars!* do <<x := 0 . x>>;
return x
end;
expr procedure mkexpvec u;
% /* Returns an exponent vector with a 1 in the u place*/
if not(u member dipvars!*) then typerr(u,"dipoly variable")
else for each x in dipvars!* collect if x eq u then 1 else 0;
expr procedure evlcm (e1,e2);
% /* Exponent vector least common multiple. e1 and e2 are
% exponent vectors. evlcm(e1,e2) computes the least common
% multiple of the exponent vectors e1 and e2, and returns
% an exponent vector. */
% for each lpart in e1 each rpart in e2 collect
% if lpart #> rpart then lpart else rpart;
begin scalar x;
while e1 and e2 do
<<x := (if car e1 #> car e2 then car e1 else car e2) . x;
e1 := cdr e1; e2 := cdr e2>>;
return reversip x
end;
symbolic procedure evmtest!? (e1,e2);
% /* Exponent vector multiple test. e1 and e2 are compatible exponent
% vectors. evmtest!?(e1,e2) returns a boolean expression.
% True if exponent vector e1 is a multiple of exponent
% vector e2, else false. */
if e1 and e2 then not(car e1 #< car e2) and evmtest!?(cdr e1,cdr e2)
else evzero!? e2 ;
expr procedure evsum (e1,e2);
% /* Exponent vector sum. e1 and e2 are exponent vectors.
% evsum(e1,e2) calculates the sum of the exponent vectors.
% e1 and e2 componentwise and returns an exponent vector. */
% for each lpart in e1 each rpart in e2 collect lpart #+ rpart;
begin scalar x;
while e1 and e2 do
<<x := (car e1 #+ car e2) . x; e1 := cdr e1; e2 := cdr e2>>;
x := reversip x;
return if e1 then nconc(x,e1) else
if e2 then nconc(x,e2) else x;
end;
expr procedure evdif (e1,e2);
% /* Exponent vector difference. e1 and e2 are exponent
% vectors. evdif(e1,e2) calculates the difference of the
% exponent vectors e1 and e2 componentwise and returns an
% exponent vector. */
% for each lpart in e1 each rpart in e2 collect lpart #- rpart;
begin scalar x;
while e2 do
<<if null e1 then e1 := '(0);
x := (car e1 #- car e2) . x; e1 := cdr e1; e2 := cdr e2>>;
return nconc (reversip x,e1);
end;
expr procedure intevprod(n,e);
% /* Multiplies each element of the exponent vector u by the integer n*/
for each x in e collect n #* x;
expr procedure expvec2a e;
% /* Returns list of prefix equivalents of exponent vector e*/
expvec2a1(e,dipvars!*);
expr procedure expvec2a1(u,v);
% /* Sub function of expvec2a */
if null u then nil
else if car u = 0 then expvec2a1(cdr u,cdr v)
else if car u = 1 then car v . expvec2a1(cdr u,cdr v)
else list('expt,car v,car u) . expvec2a1(cdr u,cdr v);
expr procedure dipevlpri(e,v);
% /* Print exponent vector e in infix form. V is a boolean variable
% which is true if an element in a product has preceded this one*/
dipevlpri1(e,dipvars!*,v);
expr procedure dipevlpri1(e,u,v);
% /* Sub function of dipevlpri */
if null e then nil
else if car e = 0 then dipevlpri1(cdr e,cdr u,v)
else <<if v then dipprin2 "*";
dipprin2 car u;
if car e #> 1 then <<dipprin2 "**"; dipprin2 car e>>;
dipevlpri1(cdr e,cdr u,t)>>;
endmodule;
module torder; % Term order modes for distributive polynomials.
% New interface, based on one routine per order
% mode.
% H. Melenk, ZIB Berlin
fluid '(dipsortmode!* dipsortevcomp!* olddipsortmode!*);
fluid '(vdpsortmode!* vdpsortextension!*);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% switching between term order modes: TORDER statement.
%
symbolic procedure torder u;
begin scalar oldmode,oldex;
oldmode := vdpsortmode!*; oldex := vdpsortextension!*;
vdpsortmode!* := reval car u; u:=cdr u;
if eqcar(vdpsortmode!*,'list) then
<<u:=cddr vdpsortmode!*; vdpsortmode!* := cadr vdpsortmode!*>>;
vdpsortextension!* := for each x in u join
(if eqcar(y,'list) then cdr y else {y}) where y=reval x;
return if null oldex then oldmode
else 'list . oldmode . oldex;
end;
put('torder,'stat,'rlis);
symbolic procedure dipsortingmode u;
% /* Sets the exponent vector sorting mode. Returns the previous mode*/
begin scalar x,y,z;
if not idp u or
(not flagp(u,'dipsortmode)
and null (y:= assoc(u,olddipsortmode!*)))
then return typerr(u,"term ordering mode");
if y then
<<prin2 "**** warning: TORDER ";
prin2 u;
prin2 " no longer supported; using ";
prin2 cdr y;
prin2t " instead";
u := cdr y>>;
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);
return x
end;
olddipsortmode!*:= '((invlex . lex)(invtotaldegree . totaldegree)
(totaldegree . revgradlex));
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 =0 then evlexcomp(cdr e1,cdr e2)
else if car e1 #> car e2 then 1
else -1;
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 =0 then evgradlexcomp(cdr e1, cdr e2)
else (if te1#-te2=0 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 evlexcomp(e2,e1) % here lex reversed
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 secion 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=evweightedcomp1(e1,vdpsortextension!*),
dg2=evweightedcomp1(e2,vdpsortextension!*);
symbolic procedure evweightedcomp1 (e,w);
% scalar product of exponent and weight vector
if null e then 0 else
if null w then evweightedcomp1 (e,'(1 1 1 1 1)) else
car e #* car w #+ evweightedcomp1(cdr e,cdr w);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% PRIVATE
%
%
fluid '(dipvars!*);
flag ('(private),'dipsortmode);
flag ('(lexgradlex),'dipsortextension);
put('private,'evcomp,'evprivatecomp);
put('private,'evcompinit,'evprivateinit);
fluid '(evprivate1!*,evprivate2!*,evprivatel!*,evprivatefn!*);
symbolic procedure evprivateinit();
begin integer n; scalar m,v1,v2,v3;
n:=length dipvars!*;
evprivatefn!* := car vdpsortextension!*;
if null getd evprivatefn!* then
rerror(dipoly,3,
"Second parameter for private torder is not a function");
evprivatel!* := n;
evprivate1!* := mkvect n;
evprivate2!* := mkvect n;
% compatibility test
v1 := for i:=1:n collect 1+random(50);
while (null v2 or v1=v2) do
v2 := for i:=1:n collect 1+random(50);
v3 := ((car v1) + 1 ) . cdr v1;
m := list(
evprivatecomp (v1,v1),
evprivatecomp (v2,v2),
evprivatecomp (v1,v3),
evprivatecomp (v3,v1),
evprivatecomp (reverse v1,reverse v3),
evprivatecomp (reverse v3,reverse v1),
evprivatecomp (v1,v2)*evprivatecomp (v2,v1));
if not(m='(0 0 -1 1 -1 1 -1))then
rerror(dipoly,4,"Private order not admissible")
end;
symbolic procedure evprivatecomp (e1,e2);
<<evprivatespread(e1,e2,1);
apply(evprivatefn!*,list(evprivate1!*,evprivate2!*))>>;
symbolic procedure evprivatespread (e1,e2,n);
if n #> evprivatel!* then nil
else (<<putv(evprivate1!*,n,x1);putv(evprivate2!*,n,x2);
evprivatespread (y1,y2,n#+1)>>)
where x1 = if e1 then car e1 else 0,
x2 = if e2 then car e2 else 0,
y1 = if e1 then cdr e1 else nil,
y2 = if e2 then cdr e2 else nil;
% symbolic procedure specimen(v1,v2);
% % simulating a 2 dim lex ordering.
% if getv(v1,1) < getv(v2,1) then -1 else
% if getv(v1,1) > getv(v2,1) then 1 else
% if getv(v1,2) < getv(v2,2) then -1 else
% if getv(v1,2) > getv(v2,2) then 1 else 0;
%
% torder private,specimen;
endmodule;
module vdp2dip;
imports dipoly;
% create!-package('(vdp2dip vdpcom vdp2dip1),
% '(contrib dipoly));
% load!-package 'dipoly;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% 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 locically
% 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 cdrricted 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
% car letters of the routine name classifies the data its works on:
% vdp... complete virt. polynomial objects
% vbc... virt. base coefficients
% vev... virt. 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 insertion
% vdpAppendVdp concat two vdps
%
% vdpLbc extract leading vbc
% vdpevlmon extract leading vev
% vdpred reductum of vbc
% 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
% vdpLcomb1 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 ";"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% RECCORD 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
% leading monomial; the positions correspond to
% the positions in vdpvars!*. Trailing zeroes
% can be omitted.
%
% <lcoeff> the "coeffcient" 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
fluid '(vdpvars!* intvdpvars!* secondvalue!* vdpsortmode!* !*groebrm
!*vdpinteger !*trgroeb !*trgroebs !*groebdivide pcount!*
!*gsugar);
global '(vdpprintmax groebmonfac);
flag('(vdpprintmax),'share);
% basic internal constructor of vdp-record
smacro procedure makevdp (vbc,vev,form); list('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;
% base coefficients
% 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
% This version: NO CHECKPOINT FACILITY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% interface for DIPOLY polynomials as records (objects).
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
fluid '(intvdpvars!* vdpvars!* secondvalue!* vdpsfsortmode!* !*groebrm
!*vdpinteger !*trgroeb !*trgroebs !*groebdivide pcount!*
!*groebsubs);
fluid '(vdpsortmode!*);
global '(vdpprintmax groebmonfac);
flag('(vdpprintmax),'share);
fluid '(dipvars!* !*vdpinteger);
symbolic procedure dip2vdp u;
% is unsed 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 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; bcfi 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;
oldorder := kord!*;
if null vars then rerror(dipoly,8,"Vdpinit: vdpvars not set");
vdpvars!* := dipvars!* := vars;
torder2 vdpsortmode!*;
return oldorder;
end;
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);
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;
% 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 ./ nil,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 ./ nil,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 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 denr v = 1 and domainp numr 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 := bcfi 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 s1,s2,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 asl dipILcomb, exponent vectors multiplied in already
begin scalar ep1,ep2,sl,res,sum,z1,z2,p1new,p2new,lptr,bptr;
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
<< lptr := dipnconcmon (bptr,
bcprod(diplbc p1,bc1),
ep1);
bptr := dipmred lptr;
p1 := dipmred p1; p1new := t;
>>
else if sl = -1 then
<< lptr := dipnconcmon (bptr,
bcprod(diplbc p2,bc2),
ep2);
bptr := dipmred lptr;
p2 := dipmred p2; p2new := t;
>>
else
<< 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;
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;
% fi 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;
begin scalar c;
if dipzero!? u then return u;
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(),1 ./ 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(multsq(bc,u ./ 1),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);
endmodule;
module vdpcom;
% common routines to all vdp mappings
fluid '(intvdpvars!* vdpvars!* secondvalue!* vdpsortmode!* !*groebrm
!*vdpinteger !*trgroeb !*groebdivide pcount!*
vdpsortextension!* );
global '(vdpprintmax);
flag('(vdpprintmax),'share);
vdpprintmax := 5;
% Repeat of smacros defined in vdp2dip.
smacro procedure vdppoly u; cadr cddr u;
smacro procedure vdpzero!? u;
null u or null vdppoly u;
smacro procedure vdpevlmon u; cadr u;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% manipulating of exponent vectors
%
symbolic procedure vevnth (a,n);
% extract nth element from a
if null a then 0 else if n=1 then car a else vevnth(cdr a,n #- 1);
% unrolled code for zero test (very often called)
smacro procedure vevzero!? u;
null u or (car u = 0 and vevzero!?1 cdr u);
symbolic procedure vevzero!?1 u;
null u or (car u = 0 and vevzero!? cdr u);
symbolic procedure veveq(vev1,vev2);
if null vev1 then vevzero!? vev2
else if null vev2 then vevzero!? vev1
else (car vev1 = car vev2 and vevequal(cdr vev1,vev2));
symbolic procedure vevmaptozero e;
% generate an exponent vector with same length as e and zeros only
vevmaptozero1(e,nil);
symbolic procedure vevmaptozero1(e,vev);
if null e then vev else vevmaptozero1(cdr e, 0 . vev);
symbolic procedure vevmtest!? (e1,e2);
% /* Exponent vector multiple test. e1 and e2 are compatible exponent
% vectors. vevmtest?(e1,e2) returns a boolean expression.
% True if exponent vector e1 is a multiple of exponent
% vector e2, else false. */
if null e2 then t
else if null e1 then if vevzero!? e2 then t else nil
else not(car e1 #<car e2)and vevmtest!?(cdr e1,cdr e2);
symbolic procedure vevlcm (e1,e2);
% /* Exponent vector least common multiple. e1 and e2 are
% exponent vectors. vevlcm(e1,e2) computes the least common
% multiple of the exponent vectors e1 and e2, and returns
% an exponent vector. */
begin scalar x;
while e1 and e2 do
<<x := (if car e1 #> car e2 then car e1 else car e2) . x;
e1 := cdr e1; e2 := cdr e2>>;
x := reversip x;
if e1 then x := nconc(x,e1)
else if e2 then x := nconc(x,e2);
return x;
end;
symbolic procedure vevmin (e1,e2);
% Exponent vector minima
begin scalar x;
while e1 and e2 do
<<x := (if car e1 #< car e2 then car e1 else car e2) . x;
e1 := cdr e1; e2 := cdr e2>>;
while x and 0=car x do x := cdr x; % cut trailing zeros
return reversip x;
end;
symbolic procedure vevsum (e1,e2);
% /* Exponent vector sum. e1 and e2 are exponent vectors.
% vevsum(e1,e2) calculates the sum of the exponent vectors.
% e1 and e2 componentwise and returns an exponent vector. */
begin scalar x;
while e1 and e2 do
<<x := (car e1 #+ car e2) . x;e1 := cdr e1; e2 := cdr e2>>;
x := reversip x;
if e1 then x := nconc(x,e1)
else if e2 then x := nconc(x,e2);
return x;
end;
symbolic procedure vevtdeg u;
% calculate the total degree of u
if null u then 0 else car u #+ vevtdeg cdr u;
symbolic procedure vdptdeg u;
if vdpzero!? u then 0 else
max(vevtdeg vdpevlmon u,vdptdeg vdpred u);
symbolic procedure vevdif (ee1,ee2);
% Exponent vector difference. e1 and e2 are exponent
% vectors. vevdif(e1,e2) calculates the difference of the
% exponent vectors e1 and e2 componentwise and returns an
% exponent vector.
begin scalar x,y,break,e1,e2;
e1 := ee1; e2 := ee2;
while e1 and e2 and not break do
<<y := (car e1 #- car e2); x := y . x;
break := y #< 0;
e1 := cdr e1; e2 := cdr e2>>;
if break or (e2 and not vevzero!? e2) then
<<print ee1; print ee2;
if getd 'backtrace then backtrace();
return rerror(dipoly,5,"Vevdif, difference would be < 0")>>;
return nconc(reversip x,e1);
end;
symbolic procedure vevdivides!?(e1,e2);
% test if e2 is a multiple of e1
null e1
or (null e2 and vevzero!? e1)
or (car e1 leq car e2 and vevdivides!?(cdr e1,cdr e2));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% numbering of polynomials
%
symbolic procedure vdpenumerate f;
% f is a temporary result. Prepare it for medium range storage
% and ssign a number
if vdpzero!? f then f else
<< f := vdpsave f;
if not vdpgetprop(f,'number) then
f := vdpputprop(f,'number,(pcount!* := pcount!* #+ 1));
f>>;
%smacro procedure vdpNumber f;
% vdpGetProp(f,'NUMBER) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% SUGAR of polynomials
%
symbolic procedure gsugar p;
if !*gsugar then
((s or
<< print list("*** missing sugar",p);
backtrace();
s:=vdptdeg p;
gsetsugar(p,s);
s>>
) where s= if vdpzero!? p then 0 else vdpgetprop(p,'sugar));
symbolic procedure gsetsugar(p,s);
!*gsugar and vdpputprop(p,'sugar,s or vdptdeg p) or p;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% operations on sets of polynomials
%
symbolic procedure vdpmember(p1,l);
% test membership of a polynomial in a list of polys
if null l then nil
else
if vdpequal(p1,car l) then l
else
vdpmember(p1,cdr l);
symbolic procedure vdpunion (s1,s2);
% s1 and s2 are two sets of polynomials.
% union of the sets using vdpMember as crit
if null s1 then s2
else
if vdpmember(car s1,s2) then vdpunion(cdr s1,s2)
else car s1 . vdpunion(cdr s1,s2);
symbolic procedure vdpintersection (s1,s2);
% s1 and s2 are two sets of polynomials.
% intersection of the sets using vdpMember as crit
if null s1 then nil
else
if vdpmember(car s1,s2) then car s1 . vdpunion(cdr s1,s2)
else vdpunion(cdr s1,s2);
symbolic procedure vdpsetequal!?(s1,s2);
% tests if s1 and s2 have the same polynomials as members
if not (length s1 = length s2) then nil
else vdpsetequal!?1(s1,append(s2,nil));
symbolic procedure vdpsetequal!?1(s1,s2);
% destroys its second parameter (is therefor copied when called)
if null s1 and null s2 then t
else
if null s1 or null s2 then nil
else
(if hugo then vdpsetequal!?1(cdr s1,groedeletip(car hugo,s2))
else nil) where hugo = vdpmember(car s1,s2);
symbolic procedure vdpsortedsetequal!?(s1,s2);
% tests if s1 and s2 have the same polynomials as members
% here assuming, that both sets are sorted by the same
% principles
if null s1 and null s2 then t
else
if null s1 or null s2 then nil
else
if vdpequal(car s1,car s2) then
vdpsortedsetequal!?(cdr s1,cdr s2)
else nil;
symbolic procedure vdpdisjoint!? (s1,s2);
% s1 and s2 are two sets of polynomials.
% test that there are no common members
if null s1 then t
else
if vdpmember(car s1,s2) then nil
else vdpdisjoint!?(cdr s1,s2);
symbolic procedure vdpsubset!? (s1,s2);
not length s1 > length s2 and vdpsubset!?1(s1,s2);
symbolic procedure vdpsubset!?1 (s1,s2);
% s1 and s2 are two sets of polynomials.
% test if s1 is subset of s2
if null s1 then t
else
if vdpmember(car s1,s2) then vdpsubset!?1 (cdr s1,s2)
else nil;
symbolic procedure vdpdeletemember(p,l);
% delete polynomial p from list l
if null l then nil
else
if vdpequal(p,car l) then vdpdeletemember(p,cdr l)
else car l . vdpdeletemember(p,cdr l);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% sorting of polynomials
%
symbolic procedure vdplsort pl;
% /* Distributive polynomial list sort. pl is a list of
% distributive polynomials. vdplsort(pl) returns the
% sorted distributive polynomial list of pl.
sort(pl, function vdpvevlcomp);
symbolic procedure vdplsortin (po,pl);
% po is a polynomial, pl is a list of polynomials.
% po is inserted into pl at its place determined by vevlcompless?.
% the result is the updated pl;
if null pl then list po
else if vevcompless!? (vdpevlmon po, vdpevlmon car pl)
then car pl . vdplsortin (po, cdr pl)
else po . pl;
symbolic procedure vdplsortinreplacing (po,pl);
% po is a polynomial, pl is a list of polynomials.
% po is inserted into pl at its place determined by vevlcompless?.
% if there is a multiple of the exponent of pl, this is deleted
% the result is the updated pl;
if null pl then list po
else if vevdivides!? (vdpevlmon po, vdpevlmon car pl)
then vdplsortinreplacing (po, cdr pl)
else if vevcompless!? (vdpevlmon po, vdpevlmon car pl)
then car pl . vdplsortinreplacing (po, cdr pl)
else po . pl;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% property lists for polynomials
%
symbolic procedure vdpputprop (poly,prop,val);
begin scalar c,p;
if not pairp poly or not pairp (c := cdr poly)
or not pairp (c := cdr c)
or not pairp (c := cdr c)
or not pairp (c := cdr c )
then rerror(dipoly,6,
list("vdpPutProp given a non-vdp as 1st parameter",
poly,prop,val));
p := assoc(prop,car c);
if p then rplacd(p,val)
else rplaca(c,(prop . val) . car c);
return poly;
end;
symbolic procedure vdpgetprop (poly,prop);
if null poly then nil % nil is a legal variant of vdp=0
else
if not eqcar(poly,'vdp)
then rerror(dipoly,7,
list("vdpGetProp given a non-vdp as 1st parameter",
poly,prop))
else
(if p then cdr p else nil)
where p=assoc(prop,cadr cdddr poly);
symbolic procedure vdpremallprops u;
begin scalar c;
if not pairp u or not pairp (c := cdr u)
or not pairp (c := cdr c)
or not pairp (c := cdr c)
or not pairp (c := cdr c)
then return u;
rplaca(c,nil); return u;
end;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Groebner interface to power substitution
fluid '(!*sub2);
symbolic procedure groebsubs2 q;
(subs2 q) where !*sub2=t;
% and a special print
symbolic procedure vdpprintshort u;
begin scalar m;
m := vdpprintmax;
vdpprintmax:= 2;
vdpprint u;
vdpprintmax:=m;
end;
endmodule;
end;