Sun Aug 18 16:14:14 2002 run on Windows
% Test of Assist Package version 2.31.
% DATE : 30 August 1996
% Author: H. Caprasse <hubert.caprasse@ulg.ac.be>
%load_package assist$
Comment 2. HELP for ASSIST:;
;
assist();
Argument of ASSISTHELP must be an integer between 3 and 14.
Each integer corresponds to a section number in the documentation:
3: switches 4: lists 5: bags 6: sets
7: utilities 8: properties and flags 9: control functions
10: handling of polynomials
11: handling of transcendental functions
12: handling of n-dimensional vectors
13: grassmann variables 14: matrices
;
assisthelp(7);
{{mkidnew,list_to_ids,oddp,followline,detidnum,dellastdigit,==},
{randomlist,mkrandtabl},
{permutations,perm_to_num,num_to_perm,combnum,combinations,cyclicpermlist,
symmetrize,remsym},
{extremum,sortnumlist,sortlist,algsort},
{funcvar,implicit,depatom,explicit,simplify,korderlist,remcom},
{checkproplist,extractlist,array_to_list,list_to_array},
{remvector,remindex,mkgam}}
;
Comment 3. CONTROL OF SWITCHES:;
;
switches;
**** exp:=t .................... allfac:= t ****
**** ezgcd:=nil ................. gcd:= nil ****
**** mcd:=t ....................... lcm:= t ****
**** div:=nil ................... rat:= nil ****
**** intstr:=nil ........... rational:= nil ****
**** precise:=t ............. reduced:= nil ****
**** complex:=nil ....... rationalize:= nil ****
**** factor:= nil ....... combineexpt:= nil ****
**** revpri:= nil ........ distribute:= nil ****
off exp;
on gcd;
off precise;
switches;
**** exp:=nil .................... allfac:= t ****
**** ezgcd:=nil ................. gcd:= t ****
**** mcd:=t ....................... lcm:= t ****
**** div:=nil ................... rat:= nil ****
**** intstr:=nil ........... rational:= nil ****
**** precise:=nil ............. reduced:= nil ****
**** complex:=nil ....... rationalize:= nil ****
**** factor:= nil ....... combineexpt:= nil ****
**** revpri:= nil ........ distribute:= nil ****
switchorg;
switches;
**** exp:=t .................... allfac:= t ****
**** ezgcd:=nil ................. gcd:= nil ****
**** mcd:=t ....................... lcm:= t ****
**** div:=nil ................... rat:= nil ****
**** intstr:=nil ........... rational:= nil ****
**** precise:=t ............. reduced:= nil ****
**** complex:=nil ....... rationalize:= nil ****
**** factor:= nil ....... combineexpt:= nil ****
**** revpri:= nil ........ distribute:= nil ****
;
if !*mcd then "the switch mcd is on";
the switch mcd is on
if !*gcd then "the switch gcd is on";
;
Comment 4. MANIPULATION OF THE LIST STRUCTURE:;
;
t1:=mklist(5);
t1 := {0,0,0,0,0}
Comment MKLIST does NEVER destroy anything ;
mklist(t1,10);
{0,0,0,0,0,0,0,0,0,0}
mklist(t1,3);
{0,0,0,0,0}
;
sequences 3;
{{0,0,0},
{1,0,0},
{0,1,0},
{1,1,0},
{0,0,1},
{1,0,1},
{0,1,1},
{1,1,1}}
lisp;
nil
sequences 3;
((0 0 0) (1 0 0) (0 1 0) (1 1 0) (0 0 1) (1 0 1) (0 1 1) (1 1 1))
algebraic;
;
for i:=1:5 do t1:= (t1.i:=mkid(a,i));
t1;
{a1,
a2,
a3,
a4,
a5}
;
t1.5;
a5
;
t1:=(t1.3).t1;
t1 := {a3,a1,a2,a3,a4,a5}
;
% Notice the blank spaces ! in the following illustration:
1 . t1;
{1,a3,a1,a2,a3,a4,a5}
;
% Splitting of a list:
split(t1,{1,2,3});
{{a3},
{a1,a2},
{a3,a4,a5}}
;
% It truncates the list :
split(t1,{3});
{{a3,a1,a2}}
;
% A KERNEL may be coerced to a list:
kernlist sin x;
{x}
;
% algnlist constructs a list which contains n-times a given list
algnlist(t1,2);
{{a3,
a1,
a2,
a3,
a4,
a5},
{a3,
a1,
a2,
a3,
a4,
a5}}
;
% Delete :
delete(x, {a,b,x,f,x});
{a,b,f,x}
;
% delete_all eliminates ALL occurences of x:
delete_all(x,{a,b,x,f,x});
{a,b,f}
;
remove(t1,4);
{a3,a1,a2,a4,a5}
;
% delpair deletes a pair if it is possible.
delpair(a1,pair(t1,t1));
{{a3,a3},
{a2,a2},
{a3,a3},
{a4,a4},
{a5,a5}}
;
elmult(a1,t1);
1
;
frequency append(t1,t1);
{{a3,4},
{a1,2},
{a2,2},
{a4,2},
{a5,2}}
;
insert(a1,t1,3);
{a3,a1,a1,a2,a3,a4,a5}
;
li:=list(1,2,5);
li := {1,2,5}
;
% Not to destroy an already ordered list during insertion:
insert_keep_order(4,li,lessp);
{1,2,4,5}
insert_keep_order(bb,t1,ordp);
{a3,
a1,
a2,
a3,
a4,
a5,
bb}
;
% the same function when appending two correctly ORDERED lists:
merge_list(li,li,<);
{1,1,2,2,5,5}
;
merge_list({5,2,1},{5,2,1},geq);
{5,5,2,2,1,1}
;
depth list t1;
2
;
depth a1;
0
% Any list can be flattened into a list of depth 1:
mkdepth_one {1,{{a,b,c}},{c,{{d,e}}}};
{1,
a,
b,
c,
c,
d,
e}
position(a2,t1);
3
appendn(li,li,li);
{1,2,5,1,2,5,1,2,5}
;
clear t1,li;
comment 5. THE BAG STRUCTURE AND OTHER FUNCTION FOR LISTS AND BAGS.
;
aa:=bag(x,1,"A");
aa := bag(x,1,A)
putbag bg1,bg2;
t
on errcont;
putbag list;
***** list invalid as BAG
off errcont;
aa:=bg1(x,y**2);
2
aa := bg1(x,y )
;
if bagp aa then "this is a bag";
this is a bag
;
% A bag is a composite object:
clearbag bg2;
;
depth bg2(x);
0
;
depth bg1(x);
1
;
if baglistp aa then "this is a bag or list";
this is a bag or list
if baglistp {x} then "this is a bag or list";
this is a bag or list
if bagp {x} then "this is a bag";
if bagp aa then "this is a bag";
this is a bag
;
ab:=bag(x1,x2,x3);
ab := bag(x1,x2,x3)
al:=list(y1,y2,y3);
al := {y1,y2,y3}
% The basic lisp functions are also active for bags:
first ab;
bag(x1)
third ab;
bag(x3)
first al;
y1
last ab;
bag(x3)
last al;
y3
belast ab;
bag(x1,x2)
belast al;
{y1,y2}
belast {a,b,a,b,a};
{a,b,a,b}
rest ab;
bag(x2,x3)
rest al;
{y2,y3}
;
% The "dot" plays the role of the function "part":
ab.1;
x1
al.3;
y3
on errcont;
ab.4;
***** Expression bag(x1,x2,x3) does not have part 4
off errcont;
a.ab;
bag(a,x1,x2,x3)
% ... but notice
1 . ab;
bag(1,x1,x2,x3)
% Coercion from bag to list and list to bag:
kernlist(aa);
2
{x,y }
;
listbag(list x,bg1);
bg1(x)
;
length ab;
3
;
remove(ab,3);
bag(x1,x2)
;
delete(y2,al);
{y1,y3}
;
reverse al;
{y3,y2,y1}
;
member(x3,ab);
bag(x3)
;
al:=list(x**2,x**2,y1,y2,y3);
2
al := {x ,
2
x ,
y1,
y2,
y3}
;
elmult(x**2,al);
2
;
position(y3,al);
5
;
repfirst(xx,al);
2
{xx,x ,y1,y2,y3}
;
represt(xx,ab);
bag(x1,xx)
;
insert(x,al,3);
2 2
{x ,x ,x,y1,y2,y3}
insert( b,ab,2);
bag(x1,b,xx)
insert(ab,ab,1);
bag(bag(x1,xx),x1,xx)
;
substitute (new,y1,al);
2 2
{x ,x ,new,y2,y3}
;
appendn(ab,ab,ab);
{x1,xx,x1,xx,x1,xx}
;
append(ab,al);
2 2
bag(x1,xx,x ,x ,y1,y2,y3)
append(al,ab);
2 2
{x ,x ,y1,y2,y3,x1,xx}
clear ab;
a1;
a1
;
comment Association list or bag may be constructed and thoroughly used;
;
l:=list(a1,a2,a3,a4);
l := {a1,a2,a3,a4}
b:=bg1(x1,x2,x3);
b := bg1(x1,x2,x3)
al:=pair(list(1,2,3,4),l);
al := {{1,a1},{2,a2},{3,a3},{4,a4}}
ab:=pair(bg1(1,2,3),b);
ab := bg1(bg1(1,x1),bg1(2,x2),bg1(3,x3))
;
clear b;
comment : A BOOLEAN function abaglistp to test if it is an association;
;
if abaglistp bag(bag(1,2)) then "it is an associated bag";
it is an associated bag
;
% Values associated to the keys can be extracted
% first occurence ONLY.
;
asfirst(1,al);
{1,a1}
asfirst(3,ab);
bg1(3,x3)
;
assecond(a1,al);
{1,a1}
assecond(x3,ab);
bg1(3,x3)
;
aslast(z,list(list(x1,x2,x3),list(y1,y2,z)));
{y1,y2,z}
asrest(list(x2,x3),list(list(x1,x2,x3),list(y1,y2,z)));
{x1,x2,x3}
;
clear a1;
;
% All occurences.
asflist(x,bg1(bg1(x,a1,a2),bg1(x,b1,b2)));
bg1(bg1(x,a1,a2),bg1(x,b1,b2))
asslist(a1,list(list(x,a1),list(y,a1),list(x,y)));
{{x,a1},{y,a1}}
restaslist(bag(a1,x),bg1(bag(x,a1,a2),bag(a1,x,b2),bag(x,y,z)));
bg1(bg1(x,b2),bg1(a1,a2))
restaslist(list(a1,x),bag(bag(x,a1,a2),bag(a1,x,b2),bag(x,y,z)));
bag(bag(x,b2),bag(a1,a2))
;
Comment 6. SETS AND THEIR MANIPULATION FUNCTIONS
;
ts:=mkset list(a1,a1,a,2,2);
ts := {a1,a,2}
if setp ts then "this is a SET";
this is a SET
;
union(ts,ts);
{a1,a,2}
;
diffset(ts,list(a1,a));
{2}
diffset(list(a1,a),ts);
{}
;
symdiff(ts,ts);
{}
;
intersect(listbag(ts,set1),listbag(ts,set2));
set1(a1,a,2)
Comment 7. GENERAL PURPOSE UTILITY FUNCTIONS :;
;
clear a1,a2,a3,a,x,y,z,x1,x2,op$
;
% DETECTION OF A GIVEN VARIABLE IN A GIVEN SET
;
mkidnew();
g0
mkidnew(a);
ag1
;
dellastdigit 23;
2
;
detidnum aa;
detidnum a10;
10
detidnum a1b2z34;
34
;
list_to_ids list(a,1,rr,22);
a1rr22
;
if oddp 3 then "this is an odd integer";
this is an odd integer
;
<<prin2 1; followline 7; prin2 8;>>;
1
8
;
operator foo;
foo(x):=x;
foo(x) := x
foo(x)==value;
value
x;
value
% it is equal to value
clear x;
;
randomlist(10,20);
{8,1,8,0,5,7,3,8,0,5,5,9,0,5,2,0,7,5,5,1}
% Generation of tables of random numbers:
% One dimensional:
mkrandtabl({4},10,ar);
{4}
array_to_list ar;
{5,4,4,7}
;
% Two dimensional:
mkrandtabl({3,4},10,ar);
*** array ar redefined
{3,4}
array_to_list ar;
{{9,5,2,8},{7,3,5,2},{8,1,6,0}}
;
% With a base which is a decimal number:
on rounded;
mkrandtabl({5},3.5,ar);
*** array ar redefined
{5}
array_to_list ar;
{2.77546499305,1.79693268486,3.43100115041,2.11636272025,3.45447023392}
off rounded;
;
% Combinatorial functions :
permutations(bag(a1,a2,a3));
bag(bag(a1,a2,a3),bag(a1,a3,a2),bag(a2,a1,a3),bag(a2,a3,a1),bag(a3,a1,a2),
bag(a3,a2,a1))
permutations {1,2,3};
{{1,2,3},{1,3,2},{2,1,3},{2,3,1},{3,1,2},{3,2,1}}
;
cyclicpermlist{1,2,3};
{{1,2,3},{2,3,1},{3,1,2}}
;
combnum(8,3);
56
;
combinations({1,2,3},2);
{{2,3},{1,3},{1,2}}
;
perm_to_num({3,2,1,4},{1,2,3,4});
5
num_to_perm(5,{1,2,3,4});
{3,2,1,4}
;
operator op;
symmetric op;
op(x,y)-op(y,x);
0
remsym op;
op(x,y)-op(y,x);
op(x,y) - op(y,x)
;
labc:={a,b,c};
labc := {a,b,c}
symmetrize(labc,foo,cyclicpermlist);
foo(a,b,c) + foo(b,c,a) + foo(c,a,b)
symmetrize(labc,list,permutations);
{a,b,c} + {a,c,b} + {b,a,c} + {b,c,a} + {c,a,b} + {c,b,a}
symmetrize({labc},foo,cyclicpermlist);
foo({a,b,c}) + foo({b,c,a}) + foo({c,a,b})
;
extremum({1,2,3},lessp);
1
extremum({1,2,3},geq);
3
extremum({a,b,c},nordp);
c
;
funcvar(x+y);
{x,y}
funcvar(sin log(x+y));
{x,y}
funcvar(sin pi);
funcvar(x+e+i);
{x}
funcvar sin(x+i*y);
{y,x}
;
operator op;
*** op already defined as operator
noncom op;
op(0)*op(x)-op(x)*op(0);
- op(x)*op(0) + op(0)*op(x)
remnoncom op;
t
op(0)*op(x)-op(x)*op(0);
0
clear op;
;
depatom a;
a
depend a,x,y;
depatom a;
{x,y}
;
depend op,x,y,z;
;
implicit op;
op
explicit op;
op(x,y,z)
depend y,zz;
explicit op;
op(x,y(zz),z)
aa:=implicit op;
aa := op
clear op;
;
korder x,z,y;
korderlist;
(x z y)
;
if checkproplist({1,2,3},fixp) then "it is a list of integers";
it is a list of integers
;
if checkproplist({a,b1,c},idp) then "it is a list of identifiers";
it is a list of identifiers
;
if checkproplist({1,b1,c},idp) then "it is a list of identifiers";
;
lmix:={1,1/2,a,"st"};
1
lmix := {1,---,a,st}
2
;
extractlist(lmix,fixp);
{1}
extractlist(lmix,numberp);
1
{1,---}
2
extractlist(lmix,idp);
{a}
extractlist(lmix,stringp);
{st}
;
% From a list to an array:
list_to_array({a,b,c,d},1,ar);
*** array ar redefined
array_to_list ar;
{a,b,c,d}
list_to_array({{a},{b},{c},{d}},2,ar);
*** array ar redefined
;
comment 8. PROPERTIES AND FLAGS:;
;
putflag(list(a1,a2),fl1,t);
t
putflag(list(a1,a2),fl2,t);
t
displayflag a1;
{fl1,fl2}
;
clearflag a1,a2;
displayflag a2;
{}
putprop(x1,propname,value,t);
x1
displayprop(x1,prop);
{}
displayprop(x1,propname);
{propname,value}
;
putprop(x1,propname,value,0);
displayprop(x1,propname);
{}
;
Comment 9. CONTROL FUNCTIONS:;
;
alatomp z;
t
z:=s1;
z := s1
alatomp z;
t
;
alkernp z;
t
alkernp log sin r;
t
;
precp(difference,plus);
t
precp(plus,difference);
precp(times,.);
precp(.,times);
t
;
if stringp x then "this is a string";
if stringp "this is a string" then "this is a string";
this is a string
;
if nordp(b,a) then "a is ordered before b";
a is ordered before b
operator op;
for all x,y such that nordp(x,y) let op(x,y)=x+y;
op(a,a);
op(a,a)
op(b,a);
a + b
op(a,b);
op(a,b)
clear op;
;
depvarp(log(sin(x+cos(1/acos rr))),rr);
t
;
clear y,x,u,v;
clear op;
;
% DISPLAY and CLEARING of user's objects of various types entered
% to the console. Only TOP LEVEL assignments are considered up to now.
% The following statements must be made INTERACTIVELY. We put them
% as COMMENTS for the user to experiment with them. We do this because
% in a fresh environment all outputs are nil.
;
% THIS PART OF THE TEST SHOULD BE REALIZED INTERACTIVELY.
% SEE THE ** ASSIST LOG ** FILE .
%v1:=v2:=1;
%show scalars;
%aa:=list(a);
%show lists;
%array ar(2);
%show arrays;
%load matr$
%matrix mm;
%show matrices;
%x**2;
%saveas res;
%show saveids;
%suppress scalars;
%show scalars;
%show lists;
%suppress all;
%show arrays;
%show matrices;
;
comment end of the interactive part;
;
clear op;
operator op;
op(x,y,z);
op(x,y,s1)
clearop op;
t
;
clearfunctions abs,tan;
*** abs is unprotected : Cleared ***
*** tan is a protected function: NOT cleared ***
"Clearing is complete"
;
comment THIS FUNCTION MUST BE USED WITH CARE !!!!!;
;
Comment 10. HANDLING OF POLYNOMIALS
clear x,y,z;
COMMENT To see the internal representation :;
;
off pri;
;
pol:=(x-2*y+3*z**2-1)**3;
3 2 2 2 2 4
pol := x + x *( - 6*y + 9*s1 - 3) + x*(12*y + y*( - 36*s1 + 12) + 27*s1 -
2 3 2 2 4 2
18*s1 + 3) - 8*y + y *(36*s1 - 12) + y*( - 54*s1 + 36*s1 - 6) + 27*
6 4 2
s1 - 27*s1 + 9*s1 - 1
;
pold:=distribute pol;
6 4 2 3 2 2 2 2 2
pold := 27*s1 - 27*s1 + 9*s1 + x - 6*x *y + 9*x *s1 - 3*x + 12*x*y + 27*x
4 2 2 3 2 2 2
*s1 - 18*x*s1 - 36*x*y*s1 + 12*x*y + 3*x - 8*y + 36*y *s1 - 12*y -
4 2
54*y*s1 + 36*y*s1 - 6*y - 1
;
on distribute;
leadterm (pold);
6
27*s1
pold:=redexpr pold;
4 2 3 2 2 2 2 2 4
pold := - 27*s1 + 9*s1 + x - 6*x *y + 9*x *s1 - 3*x + 12*x*y + 27*x*s1 -
2 2 3 2 2 2
18*x*s1 - 36*x*y*s1 + 12*x*y + 3*x - 8*y + 36*y *s1 - 12*y - 54*y*
4 2
s1 + 36*y*s1 - 6*y - 1
leadterm pold;
4
- 27*s1
;
off distribute;
polp:=pol$
leadterm polp;
3
x
polp:=redexpr polp;
2 2 2 2 4
polp := x *( - 6*y + 9*s1 - 3) + x*(12*y + y*( - 36*s1 + 12) + 27*s1 - 18*s1
2 3 2 2 4 2 6
+ 3) - 8*y + y *(36*s1 - 12) + y*( - 54*s1 + 36*s1 - 6) + 27*s1 -
4 2
27*s1 + 9*s1 - 1
leadterm polp;
2 2
x *( - 6*y + 9*s1 - 3)
;
monom polp;
6
{27*s1 ,
4
- 27*s1 ,
2
9*s1 ,
2
- 6*x *y,
2 2
9*x *s1 ,
2
- 3*x ,
2
12*x*y ,
4
27*x*s1 ,
2
- 18*x*s1 ,
2
- 36*x*y*s1 ,
12*x*y,
3*x,
3
- 8*y ,
2 2
36*y *s1 ,
2
- 12*y ,
4
- 54*y*s1 ,
2
36*y*s1 ,
- 6*y,
-1}
;
on pri;
;
splitterms polp;
2 2
{{9*s1 *x ,
2
12*x*y ,
12*x*y,
4
27*s1 *x,
3*x,
2 2
36*s1 *y ,
2
36*s1 *y,
6
27*s1 ,
2
9*s1 },
2
{6*x *y,
2
3*x ,
2
36*s1 *x*y,
2
18*s1 *x,
3
8*y ,
2
12*y ,
4
54*s1 *y,
6*y,
4
27*s1 ,
1}}
;
splitplusminus polp;
6 4 2 2 2 2 2 2 2
{3*(9*s1 + 9*s1 *x + 3*s1 *x + 12*s1 *y + 12*s1 *y + 3*s1 + 4*x*y + 4*x*y
+ x),
4 4 2 2 2 2 3 2
- 54*s1 *y - 27*s1 - 36*s1 *x*y - 18*s1 *x - 6*x *y - 3*x - 8*y - 12*y
- 6*y - 1}
;
divpol(pol,x+2*y+3*z**2);
4 2 2 2 2 2
{9*s1 + 6*s1 *x - 24*s1 *y - 9*s1 + x - 8*x*y - 3*x + 28*y + 18*y + 3,
3 2
- 64*y - 48*y - 12*y - 1}
;
lowestdeg(pol,y);
0
;
Comment 11. HANDLING OF SOME TRANSCENDENTAL FUNCTIONS:;
;
trig:=((sin x)**2+(cos x)**2)**4;
trig :=
8 6 2 4 4 2 6 8
cos(x) + 4*cos(x) *sin(x) + 6*cos(x) *sin(x) + 4*cos(x) *sin(x) + sin(x)
trigreduce trig;
1
trig:=sin (5x);
trig := sin(5*x)
trigexpand trig;
4 2 2 4
sin(x)*(5*cos(x) - 10*cos(x) *sin(x) + sin(x) )
trigreduce ws;
sin(5*x)
trigexpand sin(x+y+z);
cos(s1)*cos(x)*sin(y) + cos(s1)*cos(y)*sin(x) + cos(x)*cos(y)*sin(s1)
- sin(s1)*sin(x)*sin(y)
;
;
hypreduce (sinh x **2 -cosh x **2);
-1
;
;
clear a,b,c,d;
;
Comment 13. HANDLING OF N-DIMENSIONAL VECTORS:;
;
clear u1,u2,v1,v2,v3,v4,w3,w4;
u1:=list(v1,v2,v3,v4);
u1 := {v1,v2,v3,v4}
u2:=bag(w1,w2,w3,w4);
u2 := bag(w1,w2,w3,w4)
%
sumvect(u1,u2);
{v1 + w1,
v2 + w2,
v3 + w3,
v4 + w4}
minvect(u2,u1);
bag( - v1 + w1, - v2 + w2, - v3 + w3, - v4 + w4)
scalvect(u1,u2);
v1*w1 + v2*w2 + v3*w3 + v4*w4
crossvect(rest u1,rest u2);
{v3*w4 - v4*w3,
- v2*w4 + v4*w2,
v2*w3 - v3*w2}
mpvect(rest u1,rest u2, minvect(rest u1,rest u2));
0
scalvect(crossvect(rest u1,rest u2),minvect(rest u1,rest u2));
0
;
Comment 14. HANDLING OF GRASSMANN OPERATORS:;
;
putgrass eta,eta1;
grasskernel:=
{eta(~x)*eta(~y) => -eta y * eta x when nordp(x,y),
(~x)*(~x) => 0 when grassp x};
grasskernel := {eta(~x)*eta(~y) => - eta(y)*eta(x) when nordp(x,y),
~x*~x => 0 when grassp(x)}
;
eta(y)*eta(x);
eta(y)*eta(x)
eta(y)*eta(x) where grasskernel;
- eta(x)*eta(y)
let grasskernel;
eta(x)^2;
0
eta(y)*eta(x);
- eta(x)*eta(y)
operator zz;
grassparity (eta(x)*zz(y));
1
grassparity (eta(x)*eta(y));
0
grassparity(eta(x)+zz(y));
parity undefined
clearrules grasskernel;
grasskernel:=
{eta(~x)*eta(~y) => -eta y * eta x when nordp(x,y),
eta1(~x)*eta(~y) => -eta x * eta1 y,
eta1(~x)*eta1(~y) => -eta1 y * eta1 x when nordp(x,y),
(~x)*(~x) => 0 when grassp x};
grasskernel := {eta(~x)*eta(~y) => - eta(y)*eta(x) when nordp(x,y),
eta1(~x)*eta(~y) => - eta(x)*eta1(y),
eta1(~x)*eta1(~y) => - eta1(y)*eta1(x) when nordp(x,y),
~x*~x => 0 when grassp(x)}
;
let grasskernel;
eta1(x)*eta(x)*eta1(z)*eta1(w);
- eta(x)*eta1(s1)*eta1(w)*eta1(x)
clearrules grasskernel;
remgrass eta,eta1;
clearop zz;
t
;
Comment 15. HANDLING OF MATRICES:;
;
clear m,mm,b,b1,bb,cc,a,b,c,d,a1,a2;
load_package matrix;
baglmat(bag(bag(a1,a2)),m);
t
m;
[a1 a2]
on errcont;
;
baglmat(bag(bag(a1),bag(a2)),m);
***** (mat ((*sq ((((a1 . 1) . 1)) . 1) t) (*sq ((((a2 . 1) . 1)) . 1) t)))
should be an identifier
off errcont;
% **** i.e. it cannot redefine the matrix! in order
% to avoid accidental redefinition of an already given matrix;
clear m;
baglmat(bag(bag(a1),bag(a2)),m);
t
m;
[a1]
[ ]
[a2]
on errcont;
baglmat(bag(bag(a1),bag(a2)),bag);
***** operator bag invalid as matrix
off errcont;
comment Right since a bag-like object cannot become a matrix.;
;
coercemat(m,op);
op(op(a1),op(a2))
coercemat(m,list);
{{a1},{a2}}
;
on nero;
unitmat b1(2);
matrix b(2,2);
b:=mat((r1,r2),(s1,s2));
[r1 r2]
b := [ ]
[s1 s2]
b1;
[1 0]
[ ]
[0 1]
b;
[r1 r2]
[ ]
[s1 s2]
mkidm(b,1);
[1 0]
[ ]
[0 1]
;
seteltmat(b,newelt,2,2);
[r1 r2 ]
[ ]
[s1 newelt]
geteltmat(b,2,1);
s1
%
b:=matsubr(b,bag(1,2),2);
[r1 r2]
b := [ ]
[1 2 ]
;
submat(b,1,2);
[1]
;
bb:=mat((1+i,-i),(-1+i,-i));
[i + 1 - i]
bb := [ ]
[i - 1 - i]
cc:=matsubc(bb,bag(1,2),2);
[i + 1 1]
cc := [ ]
[i - 1 2]
;
cc:=tp matsubc(bb,bag(1,2),2);
[i + 1 i - 1]
cc := [ ]
[ 1 2 ]
matextr(bb, bag,1);
bag(i + 1, - i)
;
matextc(bb,list,2);
{ - i, - i}
;
hconcmat(bb,cc);
[i + 1 - i i + 1 i - 1]
[ ]
[i - 1 - i 1 2 ]
vconcmat(bb,cc);
[i + 1 - i ]
[ ]
[i - 1 - i ]
[ ]
[i + 1 i - 1]
[ ]
[ 1 2 ]
;
tpmat(bb,bb);
[ 2*i - i + 1 - i + 1 -1]
[ ]
[ -2 - i + 1 i + 1 -1]
[ ]
[ -2 i + 1 - i + 1 -1]
[ ]
[ - 2*i i + 1 i + 1 -1]
bb tpmat bb;
[ 2*i - i + 1 - i + 1 -1]
[ ]
[ -2 - i + 1 i + 1 -1]
[ ]
[ -2 i + 1 - i + 1 -1]
[ ]
[ - 2*i i + 1 i + 1 -1]
;
clear hbb;
hermat(bb,hbb);
[ - i + 1 - (i + 1)]
[ ]
[ i i ]
% id hbb changed to a matrix id and assigned to the hermitian matrix
% of bb.
;
load_package HEPHYS;
% Use of remvector.
;
vector v1,v2;
v1.v2;
v1.v2
remvector v1,v2;
on errcont;
v1.v2;
***** v1 v2 invalid as list or bag
off errcont;
% To see the compatibility with ASSIST:
v1.{v2};
{v1,v2}
;
index u;
vector v;
(v.u)^2;
v.v
remindex u;
t
(v.u)^2;
2
u.v
;
% Gamma matrices properties may be translated to any identifier:
clear l,v;
vector v;
g(l,v,v);
v.v
mkgam(op,t);
t
op(l,v,v);
v.v
mkgam(g,0);
operator g;
g(l,v,v);
g(l,v,v)
;
clear g,op;
;
% showtime;
end;
Time for test: 1561 ms, plus GC time: 90 ms