%% dr.sm1 (Define Ring) 1994/9/25, 26 (dr.sm1 Version 10/8/1994. ) message %% n evenQ bool /evenQ { /arg1 set arg1 2 idiv 2 mul arg1 sub 0 eq { true } { false } ifelse } def %% (x,y,z) polynomial_ring [x-list, d-list , paramList] /ring_of_polynomials { /arg1 set [/vars /n /i /xList /dList /param] pushVariables %dup print (-----) message [ (mmLarger) (matrix) switch_function (mpMult) (poly) switch_function (red@) (module1) switch_function (groebner) (standard) switch_function [arg1 to_records pop] /vars set vars length evenQ { } { vars [(PAD)] join /vars set } ifelse vars length 2 idiv /n set [ << n 1 sub >> -1 0 { /i set vars i get } for ] /xList set [ << n 1 sub >> -1 0 { /i set vars << i n add >> get } for ] /dList set [(H)] xList join [(e)] join /xList set [(h)] dList join [(E)] join /dList set [0 %% dummy characteristic << xList length >> << xList length >> << xList length >> << xList length >> << xList length 1 sub >> << xList length >> << xList length >> << xList length >> ] /param set [xList dList param] /arg1 set ] pop popVariables arg1 } def %% (x,y,z) polynomial_ring [x-list, d-list , paramList] %% with no graduation and homogenization variables. /ring_of_polynomials2 { /arg1 set [/vars /n /i /xList /dList /param] pushVariables %dup print (-----) message [ (mmLarger) (matrix) switch_function (mpMult) (poly) switch_function (red@) (module1) switch_function (groebner) (standard) switch_function [arg1 to_records pop] /vars set vars length evenQ { } { vars [(PAD)] join /vars set } ifelse vars length 2 idiv /n set [ << n 1 sub >> -1 0 { /i set vars i get } for ] /xList set [ << n 1 sub >> -1 0 { /i set vars << i n add >> get } for ] /dList set [0 %% dummy characteristic << xList length >> << xList length >> << xList length >> << xList length >> << xList length >> << xList length >> << xList length >> << xList length >> ] /param set [xList dList param] /arg1 set ] pop popVariables arg1 } def /ring_of_differential_operators { /arg1 set [/vars /n /i /xList /dList /param] pushVariables [ (mmLarger) (matrix) switch_function (mpMult) (diff) switch_function (red@) (module1) switch_function (groebner) (standard) switch_function [arg1 to_records pop] /vars set %[x y z] vars reverse /xList set %[z y x] vars {(D) 2 1 roll 2 cat_n} map reverse /dList set %[Dz Dy Dx] [(H)] xList join [(e)] join /xList set [(h)] dList join [(E)] join /dList set [0 1 1 1 << xList length >> 1 1 1 << xList length 1 sub >> ] /param set [ xList dList param ] /arg1 set ] pop popVariables arg1 } def /ring_of_differential_operators3 { %% with no homogenization variables. /arg1 set [/vars /n /i /xList /dList /param] pushVariables [ (mmLarger) (matrix) switch_function (mpMult) (diff) switch_function (red@) (module1) switch_function (groebner) (standard) switch_function [arg1 to_records pop] /vars set %[x y z] vars reverse /xList set %[z y x] vars {(D) 2 1 roll 2 cat_n} map reverse /dList set %[Dz Dy Dx] xList [(e)] join /xList set dList [(E)] join /dList set [0 0 0 0 << xList length >> 0 0 0 << xList length 1 sub >> ] /param set [ xList dList param ] /arg1 set ] pop popVariables arg1 } def /ring_of_q_difference_operators { /arg1 set [/vars /n /i /xList /dList /param] pushVariables [ (mmLarger) (qmatrix) switch_function (mpMult) (diff) switch_function (red@) (qmodule1) switch_function (groebner) (standard) switch_function [arg1 to_records pop] /vars set %[x y z] vars reverse /xList set %[z y x] vars {(Q) 2 1 roll 2 cat_n} map reverse /dList set %[Dz Dy Dx] [(q)] xList join [(e)] join /xList set [(h)] dList join [(E)] join /dList set [0 1 << xList length >> << xList length >> << xList length >> 1 << xList length 1 sub >> << xList length >> << xList length >> ] /param set [ xList dList param ] /arg1 set ] pop popVariables arg1 } def /ring_of_q_difference_operators3 { %% with no homogenization and q variables. /arg1 set [/vars /n /i /xList /dList /param] pushVariables [ (mmLarger) (qmatrix) switch_function (mpMult) (diff) switch_function (red@) (qmodule1) switch_function (groebner) (standard) switch_function [arg1 to_records pop] /vars set %[x y z] vars reverse /xList set %[z y x] vars {(Q) 2 1 roll 2 cat_n} map reverse /dList set %[Dz Dy Dx] xList [(e)] join /xList set dList [(E)] join /dList set [0 0 << xList length >> << xList length >> << xList length >> 0 << xList length 1 sub >> << xList length >> << xList length >> ] /param set [ xList dList param ] /arg1 set ] pop popVariables arg1 } def /reverse { /arg1 set arg1 length 1 lt { [ ] } { [ << arg1 length 1 sub >> -1 0 { arg1 2 1 roll get } for ] } ifelse } def /memberQ { %% a set0 memberQ bool /arg2 set /arg1 set [/a /set0 /flag /i ] pushVariables [ /a arg1 def /set0 arg2 def /flag 0 def 0 1 << set0 length 1 sub >> { /i set << set0 i get >> a eq { /flag 1 def } { } ifelse } for ] pop /arg1 flag def popVariables arg1 } def /transpose { %% mat transpose mat2 /arg1 set [/i /j /m /n /flat /mat] pushVariables [ /mat arg1 def /n mat length def /m mat 0 get length def [ 0 1 << n 1 sub >> { /i set mat i get aload pop } for ] /flat set %% [[1 2] [3 4]] ---> flat == [1 2 3 4] [ 0 1 << m 1 sub >> { /i set [ 0 1 << n 1 sub >> { /j set flat << j m mul >> i add get } for ] } for ] /arg1 set ] pop popVariables arg1 } def /getPerm { %% old new getPerm perm /arg2 set /arg1 set [/old /new /i /j /p] pushVariables [ /old arg1 def /new arg2 def [ /p old length def 0 1 << p 1 sub >> { /i set 0 1 << p 1 sub >> { /j set old i get new j get eq { j } { } ifelse } for } for ] /arg1 set ] pop popVariables arg1 } def /permuteOrderMatrix { %% order perm puermuteOrderMatrix newOrder /arg2 set /arg1 set [/order /perm /newOrder /k ] pushVariables [ /order arg1 def /perm arg2 def order transpose /order set order 1 copy /newOrder set pop 0 1 << perm length 1 sub >> { /k set newOrder << perm k get >> << order k get >> put } for newOrder transpose /newOrder set ] pop /arg1 newOrder def popVariables arg1 } def /complement { %% set0 universe complement compl /arg2 set /arg1 set [/set0 /universe /compl /i] pushVariables /set0 arg1 def /universe arg2 def [ 0 1 << universe length 1 sub >> { /i set << universe i get >> set0 memberQ { } { universe i get } ifelse } for ] /arg1 set popVariables arg1 } def %%% from order.sm1 %% size i evec [0 0 ... 0 1 0 ... 0] /evec { /arg2 set /arg1 set [/size /iii] pushVariables /size arg1 def /iii arg2 def [ 0 1 << size 1 sub >> { iii eq { 1 } { 0 } ifelse } for ] /arg1 set popVariables arg1 } def %% size i evec_neg [0 0 ... 0 -1 0 ... 0] /evec_neg { /arg2 set /arg1 set [/size /iii] pushVariables /size arg1 def /iii arg2 def [ 0 1 << size 1 sub >> { iii eq { -1 } { 0 } ifelse } for ] /arg1 set popVariables arg1 } def %% size i j e_ij << matrix e(i,j) >> /e_ij { /arg3 set /arg2 set /arg1 set [/size /k /i /j] pushVariables [ /size arg1 def /i arg2 def /j arg3 def [ 0 1 << size 1 sub >> { /k set k i eq { size j evec } { k j eq { size i evec } { size k evec } ifelse } ifelse } for ] /arg1 set ] pop popVariables arg1 } def %% m1 m2 oplus /oplus { /arg2 set /arg1 set [/m1 /m2 /n /m /k ] pushVariables [ /m1 arg1 def /m2 arg2 def m1 length /n set m2 length /m set [ 0 1 << n m add 1 sub >> { /k set k n lt { << m1 k get >> << m -1 evec >> join } { << n -1 evec >> << m2 << k n sub >> get >> join } ifelse } for ] /arg1 set ] pop popVariables arg1 } def %%%%%%%%%%%%%%%%%%%%%%% /eliminationOrderTemplate { %% esize >= 1 %% if esize == 0, it returns reverse lexicographic order. %% m esize eliminationOrderTemplate mat /arg2 set /arg1 set [/m /esize /m1 /m2 /k ] pushVariables [ /m arg1 def /esize arg2 def /m1 m esize sub 1 sub def /m2 esize 1 sub def [esize 0 gt { [1 1 esize { pop 1 } for esize 1 << m 1 sub >> { pop 0 } for ] %% 1st vector } { } ifelse m esize gt { [1 1 esize { pop 0 } for esize 1 << m 1 sub >> { pop 1 } for ] %% 2nd vector } { } ifelse m1 0 gt { m 1 sub -1 << m m1 sub >> { /k set m k evec_neg } for } { } ifelse m2 0 gt { << esize 1 sub >> -1 1 { /k set m k evec_neg } for } { } ifelse ] /arg1 set ] pop popVariables arg1 } def /elimination_order { %% [x-list d-list params] (x,y,z) elimination_order %% vars evars %% [x-list d-list params order] /arg2 set /arg1 set [/vars /evars /univ /order /perm /univ0 /compl] pushVariables /vars arg1 def /evars [arg2 to_records pop] def [ /univ vars 0 get reverse vars 1 get reverse join def << univ length 2 sub >> << evars length >> eliminationOrderTemplate /order set [[1]] order oplus [[1]] oplus /order set /univ0 [univ reverse aload pop pop] reverse def %% [e,x,y,h] --> [x,y,h] /compl [univ 0 get] evars join evars univ0 complement join def compl univ getPerm /perm set %%perm :: univ :: compl :: order perm permuteOrderMatrix /order set vars [order] join /arg1 set ] pop popVariables arg1 } def /elimination_order2 { %% [x-list d-list params] (x,y,z) elimination_order %% vars evars %% [x-list d-list params order] %% with no graduation and homogenization variables. /arg2 set /arg1 set [/vars /evars /univ /order /perm /compl] pushVariables /vars arg1 def /evars [arg2 to_records pop] def [ /univ vars 0 get reverse vars 1 get reverse join def << univ length >> << evars length >> eliminationOrderTemplate /order set /compl evars << evars univ complement >> join def compl univ getPerm /perm set %%perm :: univ :: compl :: order perm permuteOrderMatrix /order set vars [order] join /arg1 set ] pop popVariables arg1 } def /elimination_order3 { %% [x-list d-list params] (x,y,z) elimination_order %% vars evars %% [x-list d-list params order] /arg2 set /arg1 set [/vars /evars /univ /order /perm /univ0 /compl] pushVariables /vars arg1 def /evars [arg2 to_records pop] def [ /univ vars 0 get reverse vars 1 get reverse join def << univ length 1 sub >> << evars length >> eliminationOrderTemplate /order set [[1]] order oplus /order set /univ0 [univ reverse aload pop pop] reverse def %% [e,x,y] --> [x,y] /compl [univ 0 get] evars join evars univ0 complement join def compl univ getPerm /perm set %%perm :: univ :: compl :: order perm permuteOrderMatrix /order set vars [order] join /arg1 set ] pop popVariables arg1 } def /define_ring { %[ (x,y,z) ring_of_polynominals % (x,y) elimination_order % 17 %] define_ring /arg1 set [/rp /param /foo] pushVariables [/rp arg1 def [ rp 0 get 0 get rp 0 get 1 get rp 0 get 2 get /param set param 0 << rp 1 get >> put param rp 0 get 3 get ] /foo set foo aload pop set_up_ring@ ] pop popVariables } def /defineTests1 { /test { [[1 2 3] [0 1 0] [0 1 2]] [0 2 1] permuteOrderMatrix :: } def /test2 { (x,y,z) ring_of_polynomials (z,y) elimination_order /ans set } def /test3 { [ (x,y,z) ring_of_polynomials (x,y) elimination_order 17 ] define_ring } def /test4 { [ (x,y,z) ring_of_polynomials ( ) elimination_order 17 ] define_ring } def } def %% misterious bug (x,y) miss /miss { /arg1 set %[/vars /n /i /xList /dList /param] pushVariables [/vars /i] pushVariables [ arg1 print [arg1 to_records pop] /vars set ] pop dup print popVariables arg1 } def /lexicographicOrderTemplate { % size lexicographicOrderTemplate matrix /arg1 set [/k /size] pushVariables [ /size arg1 def [ 0 1 << size 1 sub >> { /k set size k evec } for ] /arg1 set ] pop popVariables arg1 } def /lexicographic_order { %% [x-list d-list params] (x,y,z) lexicograhic_order %% vars evars %% [x-list d-list params order] /arg2 set /arg1 set [/vars /evars /univ /order /perm /univ0 /compl] pushVariables /vars arg1 def /evars [arg2 to_records pop] def [ /univ vars 0 get reverse vars 1 get reverse join def << univ length 2 sub >> lexicographicOrderTemplate /order set [[1]] order oplus [[1]] oplus /order set /univ0 [univ reverse aload pop pop] reverse def %% [e,x,y,h] --> [x,y,h] /compl [univ 0 get] evars join evars univ0 complement join def compl univ getPerm /perm set %%perm :: univ :: compl :: order perm permuteOrderMatrix /order set vars [order] join /arg1 set ] pop popVariables arg1 } def /lexicographic_order2 { %% [x-list d-list params] (x,y,z) lexicograhic_order %% vars evars %% [x-list d-list params order] %% with no graduation and homogenization variables /arg2 set /arg1 set [/vars /evars /univ /order /perm /compl] pushVariables /vars arg1 def /evars [arg2 to_records pop] def [ /univ vars 0 get reverse vars 1 get reverse join def << univ length >> lexicographicOrderTemplate /order set /compl evars << evars univ complement >> join def compl univ getPerm /perm set order perm permuteOrderMatrix /order set vars [order] join /arg1 set ] pop popVariables arg1 } def /lexicographic_order3 { %% [x-list d-list params] (x,y,z) lexicograhic_order %% vars evars %% [x-list d-list params order] %% with no homogenization variable. /arg2 set /arg1 set [/vars /evars /univ /order /perm /univ0 /compl] pushVariables /vars arg1 def /evars [arg2 to_records pop] def [ /univ vars 0 get reverse vars 1 get reverse join def << univ length 1 sub >> lexicographicOrderTemplate /order set [[1]] order oplus /order set /univ0 [univ reverse aload pop pop] reverse def %% [e,x,y] --> [x,y] /compl [univ 0 get] evars join evars univ0 complement join def compl univ getPerm /perm set %%perm :: univ :: compl :: order perm permuteOrderMatrix /order set vars [order] join /arg1 set ] pop popVariables arg1 } def %%%%%% add_rings %%%%%%%%%%%%%% 10/5 /getX { %% param [1|2|3|4] getX [var-lists] ; 1->c,2->l,3->m,4->n /arg2 set /arg1 set [/k /param /func /low /top] pushVariables [ /param arg1 def /func arg2 def func 1 eq { /low 0 def } { /low << param 2 get >> << func 1 sub >> get def } ifelse /top << param 2 get >> << func 4 add >> get 1 sub def [ low 1 top { /k set param 0 get k get } for ] /arg1 set ] pop popVariables arg1 } def /getD { %% param [1|2|3|4] getD [var-lists] ; 1->c,2->l,3->m,4->n /arg2 set /arg1 set [/k /param /func /low /top] pushVariables [ /param arg1 def /func arg2 def func 1 eq { /low 0 def } { /low << param 2 get >> << func 1 sub >> get def } ifelse /top << param 2 get >> << func 4 add >> get 1 sub def [ low 1 top { /k set param 1 get k get } for ] /arg1 set ] pop popVariables arg1 } def /getXV { %% param [1|2|3|4] getXV [var-lists] ; 1->c,2->l,3->m,4->n /arg2 set /arg1 set [/k /param /func /low /top] pushVariables [ /param arg1 def /func arg2 def /low << param 2 get >> << func 4 add >> get def /top << param 2 get >> func get 1 sub def [ low 1 top { /k set param 0 get k get } for ] /arg1 set ] pop popVariables arg1 } def /getDV { %% param [1|2|3|4] getDV [var-lists] ; 1->c,2->l,3->m,4->n /arg2 set /arg1 set [/k /param /func /low /top] pushVariables [ /param arg1 def /func arg2 def /low << param 2 get >> << func 4 add >> get def /top << param 2 get >> func get 1 sub def [ low 1 top { /k set param 1 get k get } for ] /arg1 set ] pop popVariables arg1 } def /reNaming { %% It also changes oldx2 and oldd2, which are globals. /arg1 set [/i /j /new /count /ostr /k] pushVariables [ /new arg1 def /count 0 def 0 1 << new length 1 sub >> { /i set << i 1 add >> 1 << new length 1 sub >> { /j set << new i get >> << new j get >> eq { new j get /ostr set (The two rings have the same name :) messagen new i get messagen (.) message (The name ) messagen new i get messagen ( is changed into ) messagen new j << new i get << 48 count add $string$ data_conversion >> 2 cat_n >> put new j get messagen (.) message /oldx2 ostr << new j get >> reNaming2 /oldd2 ostr << new j get >> reNaming2 /count count 1 add def } { } ifelse } for } for /arg1 new def ] pop popVariables arg1 } def /reNaming2 { %% array oldString newString reNaming2 %% /aa (x) (y) reNaming2 /arg3 set /arg2 set /arg1 set [/array /oldString /newString /k] pushVariables [ /array arg1 def /oldString arg2 def /newString arg3 def 0 1 << array load length 1 sub >> { /k set << array load k get >> oldString eq { array load k newString put } { } ifelse } for ] pop popVariables } def /add_rings { /arg2 set /arg1 set [/param1 /param2 /newx /newd /newv /k /const /od1 /od2 /od /oldx2 /oldd2 % these will be changed in reNaming. /oldv ] pushVariables [ /param1 arg1 def /param2 arg2 def %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /newx [ ] param2 1 getX join param1 1 getX join param2 1 getXV join param1 1 getXV join param2 2 getX join param1 2 getX join param2 2 getXV join param1 2 getXV join param2 3 getX join param1 3 getX join param2 3 getXV join param1 3 getXV join param2 4 getX join param1 4 getX join param2 4 getXV join param1 4 getXV join def /newd [ ] param2 1 getD join param1 1 getD join param2 1 getDV join param1 1 getDV join param2 2 getD join param1 2 getD join param2 2 getDV join param1 2 getDV join param2 3 getD join param1 3 getD join param2 3 getDV join param1 3 getDV join param2 4 getD join param1 4 getD join param2 4 getDV join param1 4 getDV join def /newv newx newd join def /oldx2 param2 0 get def /oldd2 param2 1 get def /oldx2 oldx2 {1 copy 2 1 roll pop} map def /oldd2 oldd2 {1 copy 2 1 roll pop} map def /newv newv reNaming def /newx [ 0 1 << newv length 2 idiv 1 sub >> { /k set newv k get } for ] def /newd [ 0 1 << newv length 2 idiv 1 sub >> { /k set newv << newv length 2 idiv k add >> get } for ] def /const [ << param1 2 get 0 get >> << param1 2 get 1 get param2 2 get 1 get add >> << param1 2 get 2 get param2 2 get 2 get add >> << param1 2 get 3 get param2 2 get 3 get add >> << param1 2 get 4 get param2 2 get 4 get add >> << param1 2 get 5 get param2 2 get 5 get add >> << param1 2 get 6 get param2 2 get 6 get add >> << param1 2 get 7 get param2 2 get 7 get add >> << param1 2 get 8 get param2 2 get 8 get add >> ] def /od1 param1 3 get def /od2 param2 3 get def od1 od2 oplus /od set %%oldx2 :: oldd2 :: << param1 0 get reverse >> << param1 1 get reverse >> join << oldx2 reverse >> << oldd2 reverse >> join join /oldv set od << oldv << newx reverse newd reverse join >> getPerm >> permuteOrderMatrix /od set /arg1 [newx newd const od] def ] pop popVariables arg1 } def /test5 { (t) ring_of_polynomials ( ) elimination_order /r1 set (x) ring_of_differential_operators (Dx) elimination_order /r2 set r2 r1 add_rings } def /test6 { (H,h) ring_of_polynomials2 (H,h) lexicographic_order2 /r0 set (x,y,z) ring_of_polynomials2 (x,y) elimination_order2 /r1 set (t) ring_of_differential_operators3 (Dt) elimination_order3 /r2 set [r2 r1 add_rings r0 add_rings 0] define_ring } def /test7 { (H,h) ring_of_polynomials2 (H,h) lexicographic_order2 /r0 set (a,b,c,cp) ring_of_polynomials2 ( ) elimination_order2 /r1 set (x,y) ring_of_differential_operators3 (Dx,Dy) elimination_order3 /r2 set [r2 r1 add_rings r0 add_rings 0] define_ring [(Dx (x Dx + c-1) - (x Dx + y Dy + a) (x Dx + y Dy + b)). (Dy (y Dy + cp-1) - (x Dx + y Dy + a) (x Dx + y Dy + b)).] /ff set ff {[[$h$. $1$.]] replace} map homogenize /ff set } def %%%% end of add_rings ;