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module crack; % Top level CRACK module. %********************************************************************** Comment Title: CRACK Authors: Andreas Brand Institut fuer Informatik Friedrich Schiller Universitaet Jena 07740 Jena, Germany email: maa@hpux.rz.uni-jena.de tel.: + 49 3641 630698, Thomas Wolf School of Mathematical Sciences Queen Mary and Westfield College University of London London E1 4NS email: T.Wolf@maths.qmw.ac.uk tel.: + 44 71 975 5493 Date of last change: 1995-03-20 Abstract: CRACK is a package for solving overdetermined systems of partial or ordinary differential equations (PDEs, ODEs). Examples of programs which make use of CRACK for investigating ODEs (finding symmetries, first integrals, an equivalent Lagrangian or a "differential factorization") are added. The manual CRACK.TEX gives further details. REDUCE version: 3.4. CRACK uses the package ODESOLVE of Malcolm MacCallum which is included in REDUCE 3.4. T. Wolf, An Analytic Algorithm for Decoupling and Integrating systems of Nonlinear Partial Differential Equations, J. Comp. Phys., no. 3, 60 (1985) 437-446. T. Wolf, The Symbolic Integration of Exact PDEs, preprint. M.A.H. MacCallum, An Ordinary Differential Equation Solver for REDUCE, Proc. ISAAC'88, Springer Lect. Notes in Comp Sci. 358, 196--205. Keywords: partial differential equations, computer analytic $ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % CRACK Version 1995-03-20 % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% create!-package('(crack crackstr decouple separate integrat genint intfactr odeint simplf utilts simpsols intfix dfint), '(contrib crack)); endmodule; module crackstr; % Author: Andreas Brand % 1991,1993 symbolic fluid '(reducefunctions_ cont_ odesolve_ fcteval_ print_ facint_ potint_ safeint_ solvealg_ logoprint_ independence_ tr_gensep decouple_ tr_genint contradiction_ poly_only fname_ factorize_ nfct_ fnew_ genint_ gensep_ ineq_ time_ adjust_fnc)$ symbolic operator setcrackflags$ symbolic procedure setcrackflags$ <<reducefunctions_:= '(abs acos acosd acosh acot acotd acoth acsc acscd acsch asec asecd asech asin asind asinh atan atand atanh atan2 atan2d cbrt cos cosd cosh cot cotd coth csc cscd csch exp factorial hypot ln log logb log10 sec secd sech sin sind sinh sqrt tan tand tanh plus difference df minus times quotient expt int)$ if not fixp nfct_ then nfct_:=1$ % index of new functions and constants initialized cont_:=nil$ % if t then the user will be asked if a de % is too long for integration or substitution independence_:=nil$% if t then the user will be asked whether or not % expr. are considered to be linear independent genint_:=t$ % generalized integration disabled/enabled facint_:=nil$ % allowing the search of integrating factors potint_:=nil$ % allowing `potential integration' safeint_:=t$ % uses only solns of ODEs with non-vanishing denom. gensep_:=8$ % max. size of expressions to separate in a % generalized way factorize_:=5$ % recursion depth for factorizing equations odesolve_:=100$ % maximal length of a de (number of terms) to be % integrated as ode fcteval_:=100$ % maximal length of an expression to be substituted % for a function decouple_:=10$ % max. number of decoupling processes for a function tr_gensep:=nil$ % Trace generalized separation tr_genint:=nil$ % Trace generalized integration solvealg_:=nil$ % Use SOLVE for algebraic equations fname_:='c$ % name of new functions and constants (integration) logoprint_:=t$ % print logo for crack call poly_only:=t$ % all equations are polynomials only time_:=t$ % print the time needed for running crack print_:=8$ % maximal length of an expression to be printed adjust_fnc:=nil>>$ % if t then free constants/functions are scaled and % redundant ones are droped to simplify the result symbolic operator crackhelp$ symbolic procedure crackhelp$ << terpri()$ write "The call: CRACK({ex1,..,exn},{ineq1,..,ineql},", " {fct1,..,fctm},{var1,..,vark})"$ terpri()$terpri()$ write "ex1,..,exn are the equations ex1=0,..,exn=0."$ write "ineq1,..,ineql are the inequalities inequ1<>0,..,ineql<>0"$ write "fct1,..,fctm are the unknown functions depending on ", "arguments"$ terpri()$ write " declared by DEPEND fcti,...$ ."$terpri()$ write "var1,..,vark are further independent free variables which"$ terpri()$ write " none of the fcti depends on."$terpri()$terpri()$ write "The Return: {res1,...,resr}, each resi representing a", " solution of the form"$terpri()$ write "resi = {list_remain_ness_and_suff_conditions, ", "list_solved_functions,"$ terpri()$ write " list_free_or_unsolved_functions, list_inequalities}"$ terpri()$ write "i.e. CRACK returns {} if it found that no solution exists"$ terpri()$terpri()$ write "The performance of CRACK can be controlled with special flags."$ terpri()$ write "The procedure SETCRACKFLAGS() sets the defaults for all flags"$ terpri()$ write "and the procedure LISTCRACKFLAGS() shows the flags and their ", "defaults."$ terpri()$ write "All flags are GLOBAL or FLUID symbolic mode variables."$ terpri()$ write "They can be changed with: SYMBOLIC (flagname:=value)$"$ terpri()$terpri()$ write "This helptext can be recalled by: CRACKHELP()"$terpri()$ write "For more detailed information see the manual." >>$ symbolic operator listcrackflags$ symbolic procedure listcrackflags$ << write "The values in parenthesis are the defaults. ", "For more details see the manual."$ terpri()$terpri()$write "CONT_ (nil): ", "if t then the user will be asked if a de is too long"$ terpri()$write " (wrt. odesolve_, fcteval_) ", "for integration or substitution"$ terpri()$write "DECOUPLE_ (10): maximal number of decoupling attempts ", "for each function"$ terpri()$write "FACTORIZE_ (5): recursion depth for factorizing equations"$ terpri()$write "FCTEVAL_ (100): maximal length of an expression to be ", "substituted"$ terpri()$write " for a function"$ terpri()$write "FNAME_ ('c): name of new functions and constants (integr.)"$ terpri()$write "GENINT_ (T): generalized integration disabled/enabled"$ terpri()$write "GENSEP_ (8): maximal length of an expression for doing a", "generalized"$ terpri()$write " separation"$ terpri()$write "INDEPENDENCE_ (nil): if t then the user will be asked whether or not"$ terpri()$write " expressions are considered to be linear ", "independent"$ terpri()$write "ODESOLVE_ (100): maximal length of a de (number of terms) to be"$ terpri()$write " integrated as ode"$ terpri()$write "FACINT_ (nil): allowing the search for integrating factors"$ terpri()$write "PRINT_ (8): maximal length of an expression to be printed"$ terpri()$write " which contain unknown functions or constants ", "which are"$ terpri()$write "TIME_ (T): print the time needed for running crack"$ terpri()$write " parametric and not to calculate"$ terpri()$write "TR_GENINT (nil): Trace generalized integration"$ terpri()$write "TR_GENSEP (nil): Trace generalized separation"$ terpri()$write "ADJUST_FNC (nil): simplification of result by droping redundant ", "free "$ terpri()$write " functions and constants or rescaling them"$ terpri()$ >>$ symbolic operator crack$ symbolic procedure crack(el,il,fl,vl)$ % Main procedure % el - list of PDEs , il - list of nonvanishing expr. % fl - list of functions, vl - list of variables begin scalar l,l1,l2,n,m,ezgcdold,mcdold,gcdold,expold,ratold, ratargold,failure$ scalar !*precise; if print_ and logoprint_ then << "**************************************************************", "************"$ terpri()$ write "This is CRACK - a solver for overdetermined partial differential ", "equations"$ terpri()$ write "Version 1995-03-20"$terpri()$ write "***************************************************************", "***********"$ terpri()$ >>$ if el then if pairp el and (car el='list) then el:=cdr el else el:=list el$ if fl then if pairp fl and (car fl='list) then fl:=cdr fl else fl:=list fl$ if il then if pairp il and (car il='list) then il:=cdr il else il:=list il$ if vl then if pairp vl and (car vl='list) then vl:=cdr vl else vl:=list vl$ ezgcdold:=!*ezgcd$ gcdold:=!*gcd$ expold:=!*exp$ mcdold:=!*mcd$ ratold:=!*rational$ ratargold:=!*ratarg$ !*ezgcd:=t$ !*gcd:=nil$ !*exp:=t$ !*mcd:=t$ %!*rational:=nil$ !*rational:=t$ !*ratarg:=t$ fnew_:=nil$ vl:=union(reverse argset fl,vl)$ % if print_ then % <<terpri()$write "CRACK is called with :"$terpri()$ % terpri()$write "equations : "$deprint el$terpri()$ % terpri()$write "functions : "$fctprint fl$terpri()$ % terpri()$write "variables : "$fctprint vl$terpri()>>$ n:=time()$ m:=gctime()$ l1:=el$ if poly_only then while l1 do if rationaltest(car l1,fl) then l1:=cdr l1 else <<failure:=t$l1:=nil>>$ if not failure then <<l:=union(crack1(el,nil,il,fl,vl,factorize_,nil,nil),nil)$ if !*time or time_ then <<terpri()$write "CRACK needed : ",time()-n," ms GC time : ", gctime()-m," ms">>$ l:=for each a in l collect <<l1:=nil$ l2:=caddr a$ for each b in cadr a do if (pairp b) and (car b = 'equal) then l1:=cons(b,l1) else l2:=cons(b,l2)$ list(car a,l1,l2,cadddr a)>>$ if adjust_fnc then l:=for each a in l collect if l1:=dropredund(a,vl) then cdr l1 else a>> else l:=nil$ !*ezgcd:=ezgcdold$ !*gcd:=gcdold$ !*exp:=expold$ !*mcd:=mcdold$ !*rational:=ratold$ !*ratarg:=ratargold$ return if l then cons('list,for each a in l collect list('list,cons('list,car a), cons('list,cadr a), cons('list,caddr a), cons('list,cadddr a))) else list('list) end$ symbolic procedure crack1(ss,sf,ineq,forg,vl,fact,level,ass)$ % Main program begin scalar q,p,s,ftem,fges,ls,ls1,lf,l,l1,se,decfl$ contradiction_:=nil$ ineq_:=ineq$ ss:=desort ss$ % sort the de's w.r.t. its length sf:=desort sf$ fges:=union(smemberl(fnew_,append(forg,append(ss,sf))),forg)$ % add new functions fnew_:=nil$ rep: ftem:=smemberl(fges,union(ss,sf))$ % only occuring functions if print_ and (ss or sf) then <<terpri()$write "equations: "$ deprint(ss)$ deprint(sf)>>$ if print_ and ineq_ then <<terpri()$write "non-vanishing expressions: "$ for each aa in ineq_ do eqprint aa>>$ if print_ then <<terpri()$write "functions: "$ fctprint(forg)>>$ if print_ then fctprint(setdiff(ftem,forg))$ for each p in union(sf,ss) do if member(p,ftem) then l:=cons(cons(0,p),l)$ while not contradiction_ and (l or ss) do <<for each p in l do <<l1:=ls:=lf:=nil$ if print_ then <<terpri()$write "substitution : "$ substprint(cdr p,car p)>>$ if ineq_ and null (ineq_:=ineqsubst(p,ftem)) then l:=ss:=nil$ while not contradiction_ and ss do <<s:=car ss$ ss:=cdr ss$ ls:=cons(reval subst(car p,cdr p,s),ls)$ contradiction(car ls,fges)>>$ while not contradiction_ and sf do <<s:=car sf$ sf:=cdr sf$ if my_freeof(s,cdr p) then lf:=cons(s,lf) else <<ls:=cons(reval subst(car p,cdr p,s),ls)$ contradiction(car ls,fges)>> >>$ if not contradiction_ then <<for each s in se do if my_freeof(s,cdr p) then l1:=cons(s,l1)$ se:=reverse l1$ ss:=desort delete(0,union(ls,nil))$ sf:=desort delete(0,union(lf,nil))$ forg:=fctsubst(car p,cdr p,forg)$ if print_ then fctprint(forg); ftem:=delete(cdr p,ftem)$ fges:=delete(cdr p,fges)>>$ >>$ % alle Fkt. subst. ls:=ss$ l:=nil$ while not contradiction_ and ls do <<q:=car ls$ ls:=cdr ls$ ss:=delete(q,ss)$ if pairp q then <<l1:=desort gensep(q,smemberl(union(fnew_,ftem),q),vl)$ if not contradiction_ then <<sf:=union(reverse l1,sf)$ ftem:=union(fnew_,smemberl(ftem,append(sf,ss)))$ fges:=union(fnew_,smemberl(fges,append(sf,ss)))$ for each p in l1 do if member(p,ftem) then l:=union(list cons(0,p),l) else se:=cons(p,se)$ if l then ls:=nil>> >> else se:=cons(q,se)$ se:=desort union(se,nil)$ ls1:=se$ if not contradiction_ then <<while ls1 and not l do <<if q:=fctevalstrict(car ls1,ftem,vl) then <<l:=union(list q,l)$ ls:=nil>>$ ls1:=cdr ls1>>$ while se and not l do <<if q:=fcteval(car se,ftem,vl) then <<l:=union(list q,l)$ ls:=nil>>$ se:=cdr se>> >> >>$ if not contradiction_ then <<ls1:=se$ while ls1 and not l do <<if q:=fctevalstrict(car ls1,ftem,vl) then l:=union(list q,l)$ ls1:=cdr ls1>>$ while se and not l do <<if q:=fcteval(car se,ftem,vl) then l:=union(list q,l)$ se:=cdr se>> >> >>$ if contradiction_ then return nil$ if fact and (fact>0) then q:=splitandcrack(sf,ineq_,forg,ftem,vl,fact,level,ass) else q:=nil$ if contradiction_ then return nil$ decfl:=makedecfl(fges,decfl)$ if q then p:=nil else % decoupling process starts if (p:=decouple(sf,forg,ftem,fges,vl,decfl)) then if setdiff(car p,sf) or setdiff(sf,car p) or (cadr p) then <<sf:=car p$ ss:=cadr p$ forg:=caddr p$ fges:=cadddr p$ decfl:=car cddddr p$ goto rep>>$ if contradiction_ then return nil$ if not q then <<q:=list list( sf, forg, smemberl(setdiff(fges,forg), append(sf,for each l in equallist forg collect cddr l)), ineq_)$ if print_ then resout(car q)>>$ ineq_:=ineq$ return q end$ % of crack1 symbolic procedure ineqsubst(p,ftem)$ % tests all q's in ineq_ for subst(car p, cdr p,q)=0 % result: nil, if 0 occurs % otherwise list of the subst(car p,...) begin scalar l,a$ while ineq_ do if not my_freeof(car ineq_,cdr p) then <<a:=reval subst(car p, cdr p,car ineq_)$ if zerop a then <<if print_ then <<terpri()$write "contradiction :"$ eqprint list('equal,cdr p,car p)$ write "because of non-vanishing expression:"$ deprint list car ineq_>>$ contradiction_:=t$ l:=nil$ ineq_:=nil>> else <<if pairp a and (car a='quotient) then a:=cadr a$ a:=cdr reval list('factorize,a)$ for each s in a do if smemberl(ftem,s) and not member(s,l) then l:=cons(s,l)$ ineq_:=cdr ineq_>> >> else <<l:=cons(car ineq_,l)$ ineq_:=cdr ineq_>>$ return reverse l$ end$ symbolic procedure contradiction(s,fges)$ % tests expression s for contradiction an non-zero expressions if (s=0) or smemberl(fges,s) then if member(s,ineq_) then <<contradiction_:=t$ if print_ then <<terpri()$write "non-zero expression vanishes : "$ deprint list s>>$ t>> else nil else <<contradiction_:=t$ if print_ then <<terpri()$write "Contradiction found : "$ deprint list s>>$ t>>$ symbolic procedure makedecfl(fges,decfl)$ % makes decouple flag list of functions in fges begin scalar s$ while decfl do <<if member(caar decfl,fges) then s:=cons(car decfl,s)$ decfl:=cdr decfl>>$ for each f in fges do if not assoc(f,s) then s:=cons(cons(f,decouple_),s)$ return s$ end$ symbolic procedure resout(reslist)$ % print results % reslist=list(sorg,forg,fnew,ineq) begin scalar l,fnew$ fnew:=caddr reslist$ if print_ then <<for each a in cadr reslist do if (pairp a) and (car a='equal) then l:=cons(a,l) else fnew:=cons(a,fnew)$ terpri()$write "End of this CRACK run"$terpri()$terpri()$ write "The solution : "$ if l then fctprint l$ if car reslist then <<write "Remaining conditions : "$ deprint car reslist$ if fnew then <<write "for the functions : "$fctprint fnew>> >> else if fnew then <<write "Free functions or constants : "$fctprint fnew>>$ if cadddr reslist then <<terpri()$write "Non-vanishing expressions: "$ for each a in cadddr reslist do eqprint a>>$ terpri()$ write "*************************************************************", "*************"$terpri()$terpri()>>$ end$ symbolic procedure splitandcrack(sorg,ineq,forg,ftem,vl,fact,level,ass)$ % If possible one de form sorg is factoirized and % crack1 is called with each factor instead begin scalar l,p,q,a,n,contrad$ l:=sorg$ while l do if (p:=splitde(car l,ftem)) then % factorizing a de <<a:=car l$ contrad:=t$ if print_ then <<terpri()$ write "factorizing "$deprint(list a)$ if length p>1 then write "we get the alternative equations: " else write "we get the equation : "$ deprint(p)>>$ l:=nil>> else l:=cdr l$ if p then <<n:=0$ for each d in p do % for each factor if not member(d,ineq_) then <<ass:=cons(d,ass)$ n:=n+1$ level:=cons(n,level)$ if print_ then <<terpri()$ write "Level : "$ for each m in reverse level do write m,"."$terpri()$ write "CRACK is now called with the assumtion(s) : "$ deprint(reverse ass)>>$ l:=crack1(list(d),delete(a,sorg),ineq, union(forg,smemberl(ftem,sorg)), vl, if fact<2 then nil else fact-1,level,ass)$ ineq:=union(list d,ineq)$ if not contradiction_ then contrad:=nil$ ass:=cdr ass$ level:=cdr level$ if l then q:=union(clearphantomfct(l,forg),q)>> >>$ % crack1 is called contradiction_:=contrad$ return q end$ symbolic procedure clearphantomfct(l,forg)$ % functions from higher level ftem beeing removed from forg for each a in l collect clrphfct(a,forg)$ symbolic procedure clrphfct(a,forg)$ begin scalar l,l1,fl$ if a then <<for each f in forg do if (pairp f) and (car f='equal) then fl:=cons(cadr f,fl) else fl:=cons(f,fl)$ % List of functions in forg for each f in cadr a do if (pairp f) and (car f='equal) then (if member(cadr f,fl) then l:=cons(f,l)) % Member of forg else l1:=cons(f,l1)$ % free or unknown function l1:=smemberl(l1,append(car a,append(fl,l)))$ % occuring in forg or sorg return list(car a,l,union(l1,caddr a),cadddr a)>> end$ endmodule; module decouple; % Routines for decoupling de's. % Author: Andreas Brand % August 1991 symbolic procedure eqderiv(p,f,vl)$ % yields a list of de's by differentiating p w.r.t all those % variables from vl on which f not depends begin scalar l$ vl:=setdiff(vl,fctargs f)$ % variables on which f not depends for each v in vl do if not my_freeof(p,v) then l:=cons(reval list('df,p,v),l)$ return l end$ symbolic procedure decouplelist(liste,f)$ % produces a list of de's for decoupling process w.r.t function f % the elements are of the type: ((expr (deriv.pot)) flag), % f Funktion begin scalar p,q$ p:=nexteq(liste,t)$ % find first de marked with t q:=nexteq(liste,nil)$ % find first de marked with nil return if minausdp(p,q) then append(derivlist(list(list(q,nil),list(p,t)),f, difdiff(cadr q,cadr p),fctargs f,t),liste) % simplier de is marked with t % the other with nil % than the de's are diff. else append(derivlist(list(list(p,t),list(q,nil)),f, difdiff(cadr p,cadr q),fctargs f,nil),liste) end$ symbolic procedure nexteq(liste,flag)$ % finds the first element in liste marked with flag begin scalar stop$ while (liste and (not stop)) do if flag=cadar liste then stop:=t else liste:=cdr liste$ return if stop then caar liste else nil$ end$ symbolic procedure derivlist(dl,f,l,v,flag)$ % differentiates de's from dl, dl:list of de's % f:function, l:list of numbers, v:list of variables, % flag:de to be substituted by a simplier one begin scalar m,p,q,li$ for each x in v do <<m:=car l$ l:=cdr l$ if m<0 then % second de to be diff. <<if not q then q:=car nexteq(dl,not flag)$ q:=reval list('df,q,x,list('minus,m))$ li:=cons(list(cons(q,ldiff(q,f)),not flag),li)>> else if 0<m then % first de to be differentiated <<if not p then p:=car nexteq(dl,flag)$ p:=reval list('df,p,x,m)$ li:=cons(list(cons(p,ldiff(p,f)),flag),li)>> >>$ return li$ end$ symbolic procedure fctchoose(stem,fl)$ % finds function from fl with the maximal number of arguments if null (fl:=smemberl(fl,stem)) then nil else begin scalar s,n,m,ma$ ma:=length argset fl$ s:=car fl$ n:=fctlength s$ fl:=cdr fl$ while fl and (n<ma) do <<m:=fctlength car fl$ if m>n then <<s:=car fl$n:=m>>$ fl:=cdr fl>>$ return s end$ symbolic procedure dechoose(gl,f)$ % chooses two de's from list gl witch are optimal to be decoupled % w.r.t function f begin scalar d,c$ d:=shortdesearch(gl,f)$ c:=secondde(delete(d,gl),d,f)$ return if null c then list(list(d,nil),list(nil,t)) else if minausdp(d,c) then list(list(c,nil),list(d,t)) else list(list(d,nil),list(c,t)) end$ symbolic procedure shortdesearch(gl,f)$ % finds shortest de from list gl in which function f occurs if null gl then nil else if my_freeof(caar gl,f) then shortdesearch(cdr gl,f) else shortde(car gl,shortdesearch(cdr gl,f))$ symbolic procedure shortde(g1,g2)$ % finds the shorter of two de's if null g1 then g2 else if null g2 then g1 else begin scalar n,m$ n:=delength(car g1)$ m:=delength(car g2)$ return if n<m then g1 else if n=m then minausd(g1,g2) % the simplier de else g2$ end$ symbolic procedure secondde(gl,g,f)$ % Auswahl der 2. Dgl. aus gl,g 1.Dgl.,f Funktion begin scalar c$ c:=secondde1(gl,g,f)$ if null c then c:=shortdesearch(gl,f)$ return c end$ symbolic procedure secondde1(gl,g,f)$ % Auswahl der kuerzesten Gl. aus gl, die die Funktion f enthaelt und % einfacher als die Gleichung g ist if null gl then nil else if my_freeof(caar gl,f) or minausdp(car gl,g) then secondde1(cdr gl,g,f) % Gl., die f nicht enthalten oder % schwieriger als g sind, nicht else shortde(car gl,secondde1(cdr gl,g,f))$ % die kuerzere aus der 1. Gl. und % der kuerzesten des Rests symbolic procedure decouple(sorg,forg,ftem,fges,vl,decfl)$ begin scalar repe,a,f,fl,l,l1,s,p,q,r,ss,s13,stem,ftem1,ftem2,stop2,h, fctflag$ fl:=t$ repe:=1$ ftem1:=ftem$ stem:=sorg$ rep: stop2:=nil$ ftem1:=smemberl(ftem1,stem)$ ftem2:=nil$ for each f in ftem1 do if (a:=assoc(f,decfl)) and not zerop cdr a then ftem2:=cons(f,ftem2)$ if null ftem2 then <<l:=union(setdiff(stem,sorg),setdiff(s13,sorg))$ % new equations stop2:=t$ if null l or repe<=0 then <<for each a in l do sorg:=union(list reval aeval a,sorg)$ sorg:=desort sorg$ fl:=nil>> else <<repe:=sub1 repe$ for each a in l do sorg:=union(list reval aeval a,sorg)$ sorg:=desort sorg>> >> else if (f:=fctchoose (stem,ftem2)) then <<fctflag:=t$ f:=assoc(f,decfl)$ decfl:=cons(cons(car f,sub1 cdr f),delete(f,decfl))$ f:=car f$ if print_ then <<terpri()$write "decoupling: "$eqprint(f)>>$ stem:=powappend(stem,f)$ if print_ then <<terpri()$write "new equations: ">>$ begin scalar stop3$ rep1: stop3:=nil$ l1:=dechoose(stem,f)$ % optimale Gl. suchen p:=caar l1$ % p schwierigere Gl. if (null caadr l1) and fctflag then if length varslist(p,smemberl(ftem,p),vl)>fctlength f then % f kommt nur in einer % Gl. vor und haengt nicht % von allen Variablen ab <<fctflag:=nil$ stem:=union(powappend(eqderiv(car p,f,vl),f),stem)$ % Hinzufuegen neuer Gl. l1:=dechoose(stem,f)$ p:=caar l1>>$ if not null caadr l1 then <<repeat <<l1:=decouplelist(l1,f)$ %list of de's to be decoupled h:=mkldiff(f,cadaar l1)$ % h leading derivative s:=car nexteq(l1,t)$ if pairp s and (car s='quotient) then s:=cadr s$ r:=car nexteq(l1,nil)$ if pairp r and (car r='quotient) then r:=cadr r$ s:=reval !*q2a simpresultant list(s,r,h)$ % alg. decoupl. s:=car simplifyde(s,ftem1,vl,nil)$ ftem1:=union(fnew_,ftem1)$ if contradiction_ then <<stem:=nil$ stop2:=t$ stop3:=t$ fl:=nil>> else <<fges:=union(fnew_,fges)$ if my_freeof(s,f) and stardep(s,smemberl(ftem1,s),varslist(s,ftem1,vl)) then <<ss:=list s$ sorg:=desort sorg$ fges:=union(fnew_,fges)$ stem:=nil$ stop2:=t$ stop3:=t>> >>$ if not stop3 then if (q:=fcteval(s,ftem1,vl)) then <<if print_ then <<terpri()$write "substitution : "$ substprint(cdr q,car q)>>$ if member(cdr q,ftem) then <<l:=nil$ while sorg do <<a:=car sorg$ sorg:=cdr sorg$ if my_freeof(a,cdr q) then l:=cons(a,l) else <<ss:=cons(reval subst(car q,cdr q,a),ss)$ if contradiction(car ss,ftem1) then <<sorg:=nil$ fl:=nil>> >> >>$ if not contradiction_ then <<sorg:=desort reverse l$ ss:=desort delete(0,union(ss,nil))$ forg:=fctsubst(car q,cdr q,forg); if print_ then fctprint(forg)>>$ stop3:=t$ stop2:=t >> else <<stem:=powdelete(stem)$ stem:=substandsep(q,stem,ftem1,vl)$ if not contradiction_ then <<stem:=powappend(stem,f)$ s13:=substandsep(q,s13,ftem1,vl)>> else <<stop2:=t$ fl:=nil>>$ stop3:=t$ >>$ ftem1:=delete(cdr q,ftem1)>>$ if not stop3 then <<s:=cons(s,ldiff(s,f))$ % fuehr. Abl. + Potenz l1:=reverse l1$ % Reihenfolge umkehren l:=list(car l1)$ % einfach. Ausgangsgl. l1:=cdr l1$ % Glgn., von denen eine % ersetzt wird while not null l1 and minausdsp(caar l1,s) do <<l:=cons(car l1,l)$ % erste einfachere Gl. l1:=cdr l1>>$ l1:=cons(list(s,not cadar l),l)$ % durch s ersetzen l:=nil$ stop3:=nil$ if print_ and (length l1=2) then <<eqprint(list('equal,0,car s))$ write "with leading derivative "$ write reval list('expt,mkldiff(f,cadr s),cddr s)$ write " replaces a de with "$ write reval list('expt,mkldiff(f,cadr p),cddr p)$ terpri()>> >> >> until ((length(l1)=2) and not q) or stop3$ if not stop3 then if null car s or zerop car s then <<stem:=delete(p,stem)$ p:=nil$ stop2:=nil>> else << stem:=cons(s,delete(p,stem))$ p:=s:=nil$ stop2:=nil>> >> else <<s13:=cons(car p,s13)$ stem:=delete(p,stem)$ p:=nil$ stop3:=t>>$ if not stop3 then goto rep1$ end$ stem:=powdelete(stem)>>$ if not stop2 then goto rep$ return if fl then list(sorg,ss,forg,fges,decfl) else nil end$ endmodule; module separate; % Routines for separation of de's. % Author: Andreas Brand % June 1990 fluid '(fnew_ genint_ gensep_ independence_ tr_gensep); symbolic procedure termsep(a,x)$ % Separieren eines Produktes % a Ausdr. in LISP-Notation, x Var. % Ergebnis: nil, falls keine Sep. moegl.,sonst Liste von Produktpaaren % ((abh. von x)(unabh. von x)) begin scalar l,p,q,sig,l1,l2$ if atom a then if x=a then l:=list(a,1) % Variable else l:=list(1,a) % Konstante else if my_freeof(a,x) then l:=list(1,a) % a unabh. von x else <<if car a='minus then <<a:=cadr a$ sig:=not sig>>$ if pairp a and (car a='times) then l:=cdr a else l:=list a$ % l Liste der Faktoren p:=nil$ % Liste der Faktoren, % die x enthalten q:=nil$ % Liste der Faktoren, % die x nicht enth. while l do <<if my_freeof(car l,x) then q:=cons(car l,q) % Faktor enth. x nicht else if pairp car l and (caar l='expt) and my_freeof(cadar l,x) and (pairp caddar l) and (car caddar l='plus) then <<for each s in cdr caddar l do if my_freeof(s,x) then l1:=cons(s,l1) else l2:=cons(s,l2)$ if l1 then <<if cdr l1 then l1:=cons('plus,l1) else l1:=car l1$ q:=cons(list('expt,cadar l,l1),q)>>$ if l2 and cdr l2 then l2:=cons('plus,l2) else l2:=car l2$ p:=cons(list('expt,cadar l,l2),p)>> else p:=cons(car l,p)$ l:=cdr l>>$ if p then if length p>1 then p:=cons('times,p) else p:=car p$ if q then if length q>1 then q:=cons('times,q) else q:=car q$ if p or q then % war Sep. moegl.? if null p then if sig then l:=list(1,list('minus,q)) else l:=list(1,q) else if q then if sig then l:=list(p,list('minus,q)) else l:=list(p,q) else if sig then l:=list(p,list('minus,1)) else l:=list(p,1)>>$ return l end$ symbolic procedure sumsep(l,x)$ % Separieren einer Liste von Summanden % l Liste von Ausdr. in LISP-Notation, x Var. % Ergebnis: nil, falls keine Sep. moegl., % sonst Liste von Listen von Summanden begin scalar cl,p,q,s$ while l do if pairp car l and (caar l='quotient) then <<p:=termsep(cadar l,x)$ if not q then q:=termsep(caddar l,x)$ % Quotient immer gleich if p and q then <<l:=cdr l$ if car q=1 then s:=car p else s:=list('quotient,car p,car q)$ if cadr q=1 then p:=list(s,cadr p) else p:=list(s,list('quotient,cadr p,cadr q))$ cl:=termsort(cl,p)>> else <<l:=nil$ cl:=nil>> >> else <<p:=termsep(car l,x)$ % Separieren eines Summanden if p then % erfolgreich <<l:=cdr l$ cl:=termsort(cl,p)>> % Terme einordnen else <<l:=nil$ % Abbruch cl:=nil>> >>$ % keine Separ. moegl. if cl and print_ and (length cl>1) then % output for test <<terpri()$write "separation w.r.t. "$fctprint list x$ % of linear independence if pairp caar cl and (caaar cl='quotient) then l:=for each s in cl collect cadar s else l:=for each s in cl collect car s$ if not linearindeptest(l,list x) then cl:=nil>>$ return cl % Liste der Terme, die von x % unabh. sind end$ symbolic procedure linearindeptest(l,vl)$ begin scalar l1,flag$ l1:=l$flag:=t$ while flag and pairp l1 do if freeoflist(car l1,vl) then l1:=cdr l1 else if member(car l1,vl) then l1:=cdr l1 else if (pairp car l1) and (caar l1='expt) and (numberp caddar l1) and member(cadar l1,vl) then l1:=cdr l1 else flag:=nil$ if not flag then <<terpri()$write "linear independent expressions : "$ for each x in l do eqprint(x)$ if independence_ then if yesp "Are the expressions linear independent?" then flag:=t else flag:=nil else flag:=t>>$ return flag$ end$ symbolic procedure termsort(cl,p)$ % Zusammenfassen der Terme % cl Liste von Paaren,p Paar % Sind bei einem Element von cl und bei p die ersten Teile gleich, % wird der zweite Teil von p an den des Elements von cl angehaengt if null cl then list p else if caar cl=car p then cons(cons(car p,cons(cadr p,cdar cl)),cdr cl) else cons(car cl,termsort(cdr cl,p))$ symbolic procedure eqsep(eql,ftem)$ % Separieren einer Liste von: Gl. als Liste von Koeffizient + Summ. % + Liste der Var. nach denen schon erfolglos sep. wurde % + Liste der Var. nach denen noch nicht separiert wurde begin scalar vlist1,vlist2,a,x,l,eql1$ while eql do <<a:=car eql$ % erste Gl. +Listen vlist1:=cadr a$ % Var. nach d. erfolglos sep. wurde vlist2:=caddr a$ % Var. nach denen nicht sep. wurde eql:=cdr eql$ if vlist2 then % noch Var. zu sep. <<x:=car vlist2$ vlist2:=cdr vlist2$ if my_freeof(cdar a,x) then eql:=cons(list(car a,vlist1,vlist2),eql) else if fctdepend(list cdar a,x,ftem) then eql:=cons(list(car a,cons(x,vlist1),vlist2),eql) else <<l:=sumsep(cdar a,x)$ % Liste der Gl. die sich durch Sep. % nach x ergeben if l then eql:=append(varapp(l,caar a,nil,append(vlist1,vlist2)),eql) % nach erfolgr. Sep. wird nach bisher % erfolglosen Var. sep. else eql:=cons(list(car a,cons(x,vlist1),vlist2),eql)>> >> % erfolgloses Sep.,x wird als % erfolglose Var. registriert else eql1:=cons(a,eql1)>>$ return eql1$ end$ symbolic procedure sep(p,ftem,varl)$ % Die Gl. p (in LISP-Notation) wird nach den Var. aus varl separiert % varl Liste der Variabl. begin scalar eql,eqlist,a,q$ if pairp p and (car p='quotient) then <<q:=cdr reval list('factorize,caddr p)$ if length q>1 then q:=cons('times,q) else q:=car q$ p:=cadr p >>$ if pairp p and (car p='plus) then a:=cons(nil,if not q then cdr p else for each b in cdr p collect list('quotient,b,q)) else if not q then a:=list(nil,p) else a:=list(nil,list('quotient,p,q))$ % Gl. als Liste von Summanden eql:=list(list(a,nil,varl))$ % Listen der Var. anhaengen eql:=eqsep(eql,ftem)$ while eql do <<a:=caar eql$ % Listen der Var. streichen if cddr a then a:=cons(car a,cons('plus,cdr a)) else a:=cons(car a,cadr a)$ % PLUS eintragen if car a then if cdar a then a:=cons(cons('times, car a),cdr a) else a:=cons(caar a,cdr a) else a:=cons(1,cdr a)$ eqlist:=cons(a,eqlist)$ eql:=cdr eql >>$ return eqlist end$ symbolic procedure separ2(p,ftem,varl)$ % Die Gl. p (in LISP-Notation) wird nach den Var. aus varl separiert % varl Liste der Variabl. begin scalar eqlist$ if p and not zerop p then if not (pairp p and (car p='quotient) and intersection(argset smemberl(ftem,cadr p),varl)) then <<eqlist:=sep(p,ftem,varl)$ if eqlist then eqlist:=union(cdr eqlist,list car eqlist)$ >>; % else eqlist is nil return eqlist end$ symbolic procedure separ(p,ftem,varl)$ % Die Gl. p (in LISP-Notation) wird nach den Var. aus varl separiert % varl Liste der Variabl. begin scalar eql,eqlist,a,b,l,s$ if p and not zerop p then if not (pairp p and (car p='quotient) and intersection(argset smemberl(ftem,cadr p),varl)) then <<eqlist:=sep(p,ftem,varl)$ if eqlist then eql:=union(cdr eqlist,list car eqlist)$ eqlist:=nil$ while eql do <<a:=car eql$ l:=eql:=cdr eql$ for each b in l do <<s:=reval list('quotient,cdr b,cdr a)$ if not smemberl(append(varl,ftem),s) then <<eql:=delete(b,eql)$ a:=cons(reval list('plus,car a,list('times,s,car b)),cdr a)>> >>$ eqlist:=cons(a,eqlist) >> >> else eqlist:=list cons(1,p) % FTEM functions in the DENR of p else eqlist:=list cons(0,0)$ return eqlist end$ symbolic procedure separate(p,ftem,vl)$ % Separieren der Gleichung p % ftem Liste der Fkt., vl Liste der Variablen begin scalar l$ l:=separ(p,ftem,vl)$ return union(for each a in l collect cdr a,nil)$ end$ symbolic procedure starp(ft,vl)$ % vl are all variables in an expression, yields a boolean value % whether a function of ft contains all variables of vl begin scalar b,n$ n:=length vl$ % number of all variables while not b and ft do % searching a fct of all vars if fctlength car ft=n then b:=t else ft:=cdr ft$ return not b end$ symbolic procedure stardep(p,ftem,vl)$ % yields: nil, if a function (from ftem) in p depends % on all variables (from vl) % cons(v,n) otherweise, with v beeing the list of variables % on which depend a minimal number n of functions begin scalar b,v,n$ vl:=varslist(p,ftem,vl)$ if starp(ftem,vl) then <<n:=sub1 length ftem$ while vl do % searching var.s on which depend a % minimal number of functions <<if n> (b:=for each h in ftem sum if member(car vl,fctargs h) then 1 else 0) then <<n:=b$ % a new minimum v:=list car vl>> else if b=n then v:=cons(car vl,v)$ vl:=cdr vl>> >>$ return if v then cons(v,n) % on each varible from v depend n % functions else nil end$ symbolic procedure gensep(p,ftem,vl)$ begin scalar a,aa,q,l1,l2,ft,pl,pl1,x,vl1,deno$ if pairp p and (car p = 'quotient) then p:=cadr p$ ftem:=smemberl(ftem,p)$ vl:=varslist(p,ftem,vl)$ if not (a:=stardep(p,ftem,vl)) then pl:=list p else if zerop cdr a then pl:=separate(p,ftem,vl) else % e.g. a=((x y z).2) if delength(p) leq gensep_ then <<pl:=list p$ if print_ then <<terpri()$write "generalized separation ">>$ if tr_gensep then <<terpri()$write "de to be separated : "$ deprint pl$ terpri()$write "variable list for separation : ",car a$ terpri()$write "for each of these variables ",cdr a, " function(s) do(es) not depend on it ">>$ a:=car a$ while pairp a do % i.e. being able to make a separation for each x <<x:=car a$ q:=p$ ft:=l2:=nil$ for each f in ftem do if member(x,fctargs f) and not my_freeof(q,f) then ft:=cons(f,ft)$ if tr_gensep then <<terpri()$write "to separate directly w.r.t. ",x$ write " the expression : "$deprint list q$ write "will be differentiated">>$ ft:=reverse fctsort ft$ % sorting w.r.t. number of args while ft do % throwing out all functions ft if (l1:=felim(q,car ft,ftem,vl,l2)) then <<q:=car l1$ l2:=cadr l1$ aa:=stardep(q,smemberl(ftem,q),vl)$ if not aa or zerop cdr aa then ft:=nil else ft:=smemberl(cdr ft,q)>> else ft:=nil$ if l1 then <<if (pairp q) and (car q='quotient) then <<deno:=caddr q$ q:=cadr q$ ft:=smemberl(ftem,q)>> else ft:=nil$ if ft then <<vl1:=nil$ for each y in vl do if my_freeof(ft,y) then vl1:=cons(y,vl1) >> else vl1:=vl$ if aa and zerop cdr aa and not (q=0) then <<if tr_gensep then <<terpri()$write "trying direct separation of "$ deprint list q$ write "Remaining variables: ",vl1>>$ l1:=separate(q,ftem,vl1)$ if tr_gensep then <<write "The result of direct separation: "$deprint l1>>$ if l1 and cdr l1 and tr_gensep then <<terpri()$ write "direct separation yields ",length l1," equations">> >> else l1:=list q$ l1:=desort for each s in l1 collect if deno then backint(list('quotient,s,deno),l2,union(fnew_,ftem),vl) else backint(s,l2,union(fnew_,ftem),vl)$ ftem:=union(fnew_,ftem)$ pl:=union(l1,pl)$ % while l1 do % <<if fctevalstrict(car l1,ftem,vl) then % <<pl1:=cons(car l1,pl1)$ % a:=list nil$ % ftem:=union(fnew_,ftem)>>$ % l1:=cdr l1 % >> l1:=nil >>$ a:=cdr a$ >>$ >>$ if pl1 then pl:=union(simplifyde(reval p,union(fnew_,ftem),vl,genint_),pl1) else <<if null pl then pl:=list p$ for each p in pl do pl1:=union(simplifyde(reval p,union(fnew_,ftem),vl,genint_),pl1)$ pl:=pl1$ if cdr pl and print_ then <<terpri()$write "separation yields ",length pl," equations"$ if tr_gensep then deprint pl>> >>$ return pl$ end$ symbolic procedure felim(q,f,ftem,vl,l2)$ begin scalar a,b,l,l1,ft1,v,prflag$ v:=car setdiff(vl,fctargs f)$ % getting rid of f through diff. wrt. v ft1:=nil$ for each f in ftem do if my_freeof(f,v) then ft1:=cons(f,ft1)$ if not (pairp q and (car q='quotient) and smemberl(ft1,caddr q)) then <<prflag:=print_$print_:=nil$ l:=desort separ(q,ft1,list v)$ % det. all lin. ind. factors with v if tr_gensep then <<terpri()$write "To get rid of ",f, " we differentiate w.r.t. : ",v>>$ print_:=prflag$ l1:=nil$ while l do <<a:=car l$ b:=nil$ if not my_freeof(cdr a,f) and (not zerop car a) then if (pairp cdr a) and (cadr a='plus) then <<for each c in cddr a do if not my_freeof(c,f) then b:=cons(c,b)$ if b then b:=cons('plus,b)>> else b:=cdr a$ if b then <<l1:=cons(car a,l1)$ % q:=reval list('DIFFERENCE,q,list('times,b,car a)) >>$ l:=cdr l >>$ if tr_gensep then <<terpri()$ write v," - depending coefficients of terms containing ",f," : "$ for each ss in l1 do eqprint ss>>$ while l1 do <<b:=reval aeval car l1$ l1:=cdr l1$ if not zerop b then <<a:=reval aeval list('quotient,q,b)$ % if not my_freeof(a,v) then <<l:=cons(b,l)$ % for later backward integrations l1:=for each c in l1 collect reval list('df,list('quotient,c,b),v)$ % casecheck(b,ftem,vl)$ % if not freeofzero(b,ftm,vl) then ..... q:=reval list('df,a,v)$ if tr_gensep then <<write "The new equation: "$ eqprint q$ write "The remaining factors:"$ for each ss in l1 do eqprint ss$>> >> % else <<q:=b$ l1:=nil>> >> >>$ if tr_gensep then <<terpri()$write "new expression (should not depend on ",f,") : "$ eqprint q$>>$ if l then l2:=cons(list(v,l),l2)$ if tr_gensep and l2 then <<write "The list of multiplications and integrations ", "to go backwards after direct separation:"$ for each ss in l2 do <<write "integr. wrt. ",car ss$ terpri()$ write "multiply with "$ for each aa in cadr ss do eqprint aa>> >>$ l1:=list(q,l2) >>$ return l1$ end$ symbolic procedure backint(s,l2,ftem,vl)$ begin scalar fl,ft,q,l,v,v1,vf,s2,p,f2,fnew1$ fnew1:=fnew_$ fl:=q:=t$ p:=s$ while l2 and fl do <<l:=car l2$ l2:=cdr l2$ v:=car l$ if tr_gensep then <<terpri()$ write "backward integration w.r.t. ",v," of the expression : "$ eqprint p>>$ l:=cadr l$ while l and q do <<ft:=smemberl(ftem,p)$ %terpri()$write "vor int: p= "$eqprint p$ fnew_:=nil$ q:=integratepde(p,ft,v,nil,nil)$ % genflag:=nil, potflag=nil fnew1:=append(fnew_,fnew1)$ if q then <<fl:=t$ p:=reval list('times,car l,car q)$ % Substituting the new functions of integration by derivatives % of them such that back-integration can be made % hat fnew_ nur ein element, d.h. wird nur eine Integration gemacht % oder mehrere? for each f1 in fnew_ do <<f2:=f1$ vf:=setdiff(vl,fctargs f1)$ for each s1 in reverse l2 do <<v1:=car s1$ if not my_freeof(f1,v1) then % only then integration makes difficulties <<s2:=reverse cadr s1$ while s2 do <<if not smemberl(vf,car s2) then f2:=reval list('df,list('quotient,f2,car s2),v1)$ % actually only dividing through those factors of (car s2) % which depend on v1 and which do not contain variables % which f2 does not depent on. s2:=cdr s2 >> >> >>$ if f1 neq f2 then <<if tr_gensep then <<write f1," is replaced by "$ eqprint f2>>$ p:=subst(f2,f1,p)$ >> >>$ ftem:=union(fnew_,ftem) >> else fl:=nil$ l:=cdr l >> >>$ if tr_gensep then if fl then <<terpri()$write "yields : "$ eqprint p$>> else <<terpri()$ write "was not successful.">>$ fnew_:=union(fnew1,fnew_)$ return if fl then p else s end$ endmodule; module integrat; % Routines for integration of pde's. % Author: Thomas Wolf, Andreas Brand % June 1993 fluid '(facint_ fnew_ poly_only potint_ tr_genint); %symbolic procedure intpoly(p,x)$ %% increases all powers of x (also the zeroth) of the polynomial p of x %% with x-independent coefficients, copying terms with x-dependent %% functions %if not pairp p then if my_freeof(p,x) then list('TIMES,x,p) % else p % else %if car p='MINUS then cons('MINUS,intpoly(cadr p,x)) else %if car p='TIMES then %<<p:=cdr p; % if my_freeof(car p,x) then % reval list('TIMES, car p, % intpoly(if (1<length cdr p) then % cons('TIMES,cdr p) else cadr p,x)) else % if car p=x then % reval cons('TIMES, cons(list('TIMES,x,car p), cdr p)) else % if (caar p='EXPT) and (cadar p=x) then % reval cons('TIMES, cons(list('TIMES,x,car p), cdr p)) else % cons('TIMES, cdr p) %>> else %if car p='PLUS then %list('PLUS,intpoly(cadr p , x), % intpoly(if (1<length cddr p) then cons('PLUS,cddr p) % else caddr p , x) ) % else if my_freeof(p,x) then list('TIMES,x,p) % else p$ %----------------- symbolic procedure ldlist(p,f,vl)$ % provides a reverse list of leading derivatives of f in p, vl is list % of variables begin scalar a$ if not atom p then if member(car p,list('expt,'plus,'minus,'times,'quotient,'df,'equal)) then << if (car p='plus) or (car p='times) or (car p='quotient) or (car p='equal) then <<p:=cdr p$ while p do <<a:=diffincl(a,ldlist(car p,f,vl))$ p:=cdr p>> >> else if car p='minus then a:=ldlist(cadr p,f,vl) else if car p='expt then % if numberp caddr p then a:=ldlist(cadr p,f,vl) else % fuehrende Abl. aus der Basis % else a:=nil if car p='df then if cadr p=f then <<p:=cddr p; while vl do <<a:=cons(diffdeg(p,car vl),a); vl:=cdr vl>>; a:=list a >> >>$ return a end$ symbolic procedure diffincl(a,b)$ % a,b are lists of leading derivatives which are to be united begin scalar p; while a and b do <<a:=ddroplow(a,car b); if car a then p:=cons(car a,p); a:=cdr a; b:=cdr b>>; return if null a then if p then nconc(p,b) else b else if p then a:=nconc(p,a) else a end$ symbolic procedure ddroplow(a,cb)$ % loescht Elemente von a, von denen cb eine Ableitung ist, loescht cb, % wenn ein Element von a eine Ableitung von cb ist begin scalar h; return if null a then list(cb) else <<h:=compdiffer(car a,cb); if numberp(h) then if h>0 then cons(nil,a) % car a=df(cb,.. else ddroplow(cdr a,cb) % cb=df(car a,.. else <<h:=ddroplow(cdr a,cb); % neither cons(car h,cons(car a,cdr h))>> >> end$ symbolic procedure compdiffer(p,q)$ % finds whether p is a derivative of q or q of p or neither begin scalar p!>q,q!>p; while p and ((null p!>q) or (null q!>p)) do <<if car p>car q then p!>q:=t else % compare orders of derivatives if car p<car q then q!>p:=t; p:=cdr p;q:=cdr q >>; return if q!>p then if p!>q then nil % neither else 0 % q is derivative of p else if p!>q then 2 % p is derivative of q else 1 % p equal q end$ symbolic procedure ldintersec(a)$ % determines the intersection of derivatives in the list a begin scalar b; return if null a then nil else <<b:=car a;a:=cdr a; while a do <<b:=isec(b,car a)$ a:=cdr a >>; b >> end$ symbolic procedure isec(ca,b)$ % determines the minimum derivatives between ca and b begin scalar newb; while ca do <<newb:=cons(if car b<car ca then car b else car ca, newb); ca:=cdr ca;b:=cdr b >>; return reverse newb end$ symbolic procedure disjun(a,b)$ <<while a do if (car a neq 0) and (car b neq 0) then a:=nil else <<a:=cdr a;b:=cdr b>>; if b then nil else t >>$ symbolic procedure ddrophigh(a,cb)$ % loescht Elemente von a, die Ableitung von cb sind, % loescht cb, wenn cb Ableitung eines Elements von a ist oder =a ist, % haengt cb ansonsten an begin scalar h; return if null a then list(cb) else <<h:=compdiffer(car a,cb); if numberp(h) then if h<2 then a % cb is deriv or = car a else ddrophigh(cdr a,cb) % car a=df(cb,.. else cons(car a,ddrophigh(cdr a,cb)) % neither >> end$ symbolic procedure elibar(l1,l2,lds)$ begin scalar found1,found2,h; % found1=t if an LD=l1 is found, found2=t if contradiction found while lds and (not found2) do <<if car lds=l1 then found1:=t else <<h:=compdiffer(l2,car lds); if (null h) or (h eq 2) then found2:=t >>; lds:=cdr lds >>; return found1 and (not found2) end$ symbolic procedure intminderiv(p,ftem,vlrev,maxvanz,nfsub)$ % yields a list {nv1,nv2, ... } such that nvi is the minimum % of the highest derivatives w.r.t. vi of all the leading derivatives % of all functions of ftem which are functions of all maxvanz variables. % Only those are kept for which nvi>0. % further a list ((f1 ld's of f1) (f2 ld's of f2)...), begin scalar l,a,listoflds$ while ftem do <<if (maxvanz=(fctlength car ftem)) or (nfsub=0) then <<l:=ldlist(p,car ftem,vlrev); listoflds:=cons(cons(car ftem,l),listoflds)$ a:=if a then ldintersec(cons(a,l)) else ldintersec(l) >>$ ftem:=cdr ftem >>$ return list(a,listoflds) end$ symbolic procedure potintegrable(listoflds)$ begin scalar h,l1,l2,f,n,lds,f1,f2$ if tr_genint then write "Does a potential exist?"$ %------- Determining 2 minimal sets of integration variables % There must be two disjunct LDs such that all others are in their % ideal. This is necessary if we want to eliminate 2 functions. h:=listoflds; l1:=nil; while h do <<l2:=cdar h; % the list of LDs for the function caar h while l2 do <<l1:=ddrophigh(l1,car l2)$ l2:=cdr l2>>; h:=cdr h >>; return if length l1 neq 2 then nil else if not disjun(car l1,cadr l1) then nil else % if there would be an overlap between l1 and l2 then it would have % to be integrated for elimination but it cannot --> no overlap % possible <<% selecting interesting functions for which one LD is = l1 and all % others are derivatives of l2 or equal l2 and vice versa. Two % necessary (one with an LD=l1 and one with an LD=l2) from which at % functions are least one (f) has no further LD. % Exception: 2 functions with each 2 LDs equal to (l1,l2) (these % functions are counted by n) l2:=cadr l1;l1:=car l1; f:=nil;f1:=nil;f2:=nil;n:=0; h:=listoflds; while h and ((not f1) or (not f2) or ((not f) and (n neq 2))) do <<lds:=cdar h; if (not f1) or (not f) then if elibar(l1,l2,lds) then <<f1:=cons(caar h,f1); if length lds eq 1 then f:=caar h else if length lds eq 2 then if (car lds = l2) or (cadr lds = l2) then n:=n+1 >>; if (not f2) or (not f) then if elibar(l2,l1,lds) then <<f2:=cons(caar h,f2); if length lds eq 1 then f:=caar h >>; h:=cdr h >>; if f1 and ((n>1) or (f2 and f)) then list(l1,l2) else nil >> end$ % of potintegrable symbolic procedure vlofintlist(vl,intlist)$ % provides a list of elements of vl for which the corresponding % elements of intlist are non-zero begin scalar a; while intlist do <<if (car intlist) and (not zerop car intlist) then a:=cons(car vl,a); vl:=cdr vl; intlist:=cdr intlist >>; return a end$ symbolic procedure vlofintfaclist(vl,intfaclist)$ % determining the list of all variables of vl in intfaclist begin scalar e1,a; for each e1 in vl do if not my_freeof(intfaclist,e1) then a:=cons(e1,a); return a end$ symbolic procedure multipleint(intvar,ftem,q,vari,vl,genflag, potflag,partial,doneintvar)$ % multiple integration of q wrt. variables in vl, max. number of % integrations specified by intvar % integrating factors must not depend on doneintvar, doneintvar is % a list of integration variables so far % partial=t then as much as possible of an expression is integrated % even if there is a remainder begin scalar pri,vlcop,v,nmax,geni,intlist,iflag,n,nges,newcond, intfaclist,ph,pih,qh,qih,intfacdepnew,intfacdep$ % intfacdep is a list of variables on which factors of integration % depend so far, other than the integration variable in their % integration --> no integration wrt. these variables by potint % because there the diff. operators wrt. to different variables % need not commute because the integrations are not done % pri:=t$ if (not vari) and (zerop q) then return nil; nges:=0; vlcop:=vl; pih:=t; % Splitting of the equation into the homogeneous and inhomogeneous % part which is of advantage for finding integrating factors q:=splitinhom(q,ftem,vl)$ qh:=car q; qih:=cdr q; q:=nil; while (vari or vlcop) and (pih or (not potflag)) do %------- if for potflag=t one variable can not be integrated the %------- maximal number of times (nmax) then immediate stop because %------- then no elimination of two functions will be possible << %-- The next integration variable: x, no of integrations: nmax if vari then <<v:=vari;nmax:=1>> else <<v:=car vlcop; vlcop:=cdr vlcop; nmax:=car intvar; intvar:=cdr intvar>>; if zerop nmax then intlist:=cons(nil,intlist) else <<if pri then write"anf: intvar=",intvar," vari=",vari," q=",q$ if vari and (not member(v,vl)) then <<iflag:='success$ intlist:=cons(list(1),intlist)$ q:=reval list('int,q,v)$ >> else <<n:=0$ if pri then write"333"$ intfaclist:=nil; %-- the list of integr. factors in v-integr. if potflag or my_freeof(intfacdep,v) then % otherwise v-integration not allowed because one integrating % factor already depends on v % for potflag=t this `commutativity'-demand plays no role repeat << %--- max nmax integrations of qh and qih wrt. v if pri then <<write"444 vor intpde:"$eqprint q$terpri()$ write"potflag=",potflag," v=",v, " ftem=",ftem>>$ % At first trying a direct integration of the homog. part qh ph:=intpde(qh,ftem,vl,v,potflag)$ % faster if potflag=nil if pri then <<write"nach intpde(qh):"$deprint ph>>$ %------ At first the integration of the homogeneous part intfacdepnew:=intfacdep; if ph and (partial or (zerop cadr ph)) then << %---- For the homogen. part cadr ph must be zero intfaclist:=cons(1,intfaclist); ph:=car ph >> else if vari then ph:=nil else if facint_ then << ph:=findintfac(list(qh),ftem,vl,v,doneintvar,intfacdep, not zerop n); % factorize before ivestig. if ph then << %--- Complete integr. of qh was possible %--- update the list of variables on which all integr. %--- factors depend apart from the integration variable intfacdepnew:=caddr ph; %--- extend the list of integrating factors, cadr ph %--- is a list of integr. factors, here only one intfaclist:=cons(caadr ph,intfaclist); %--- multiply the inhomogeneous part with integ. factor qih:=reval list('times,car intfaclist,qih); ph:=car ph % the integral of qh >> >>; %------ Now the integration of the inhomogeneous part if not ph then pih:=nil %--- no integration possible else << if zerop qih then pih:=list(0,0) else pih:=intpde(qih,ftem,vl,v,potflag)$ if null pih then write"error2: inhom. expr. can not be integrated"$ if pri then <<write"nach intpde(qih):"$deprint pih>>$ if pih then if zerop cadr pih then <<qih:=car pih$n:=add1 n$iflag:='success>> else if not genflag then pih:=nil else <<if pri then write"555"$ if (geni:=partint(cadr pih,smemberl(ftem,cadr pih), vl,v)) then <<qih:=reval list('plus,car pih,car geni)$ n:=add1 n$ ftem:=union(fnew_,ftem)$ newcond:=append(cdr geni,newcond)$ % additional de's if pri then <<terpri()$write"nach gen newcond:"$deprint newcond>>$ iflag:='genint >> else if partial then qih:=car pih else pih:=nil >>; if pih then << qh:=ph; if (not potflag) and (n neq 1) then intfacdep:=intfacdepnew %-The first integr. factor of all v-integrations does not % depend on any earlier integration variables and can % therefore be taken out of all integrals --> no incease % of intfacdep for n=1. %-For potential integration the integration variables and % extra-dependencies-variables of integr. factors need not % be disjunct therefore the intfacdep-update only for % not-potflag >> else << % inhomogeneous part can not be integrated therefore % reversing the succesful integration of the hom. part if car intfaclist neq 1 then qih:=reval list('quotient,qih,car intfaclist); intfaclist:=cdr intfaclist >>; >> %-- end of the integration of the inhomog. part >> until (n=nmax) or (not pih)$ %---- end of v-integration %------- variables of done integrations are collected for %------- determining integrating factors in later integr. if not zerop n then doneintvar:=cons(v,doneintvar)$ nges:=nges+n; intlist:=cons(intfaclist,intlist) >>$ % of not ( vari and (not member(v,vl))) if vari then <<vari:=nil;vlcop:=nil>>; if pri then write"ende: intvar=",intvar," vari=",vari, " qh=",qh," qih=",qih$ >> % of (nmax neq zero) >>$ % of ( while (vari or vlcop) and (pih or (not potflag)) ) % intlist and its elements intfaclist are in the inverse order % to vl and the sequence of integrations done q:=reval list('plus,qh,qih)$ %--- adding homog. and inhomog. part if pri then <<terpri()$write"\\\ newcond:"$deprint newcond; write"multiple result:",if null iflag then nil else list(q,intlist,newcond,nges) >>; return if null iflag then nil else list(q,intlist,newcond,nges) end$ % of multipleint symbolic procedure uplistoflds(intlist,listoflds)$ begin scalar f,h1,h2,h3,h4,lds,itl; while listoflds do <<f:=caar listoflds; lds:=cdar listoflds; listoflds:=cdr listoflds; h2:=nil; % h2 becomes the new list of lds of f while lds do <<h3:=car lds; lds:=cdr lds; itl:=intlist; h4:=nil; % h4 becomes one new ld of f while h3 do <<h4:=cons(car h3 - if null car itl then 0 else length car itl, h4); h3:=cdr h3;itl:=cdr itl >>; h2:=cons(reverse h4,h2) >>; h1:=cons(cons(f,h2),h1) >>; return h1 % updated listoflds end$ % of uplistoflds symbolic procedure addintco(q, ftem, ifac, vl, vari)$ begin scalar v,f,l,vl1; % multi.ing factors to the constants/functions of integration if zerop q then l:=1 else <<ftem:=reverse fctsort ftem$ while ftem do if fctlength car ftem<length vl then ftem:=nil else if fctlinear(q,f) then <<f:=car ftem$ ftem:=nil>> else ftem:=cdr ftem$ if f then <<l:=lderiv2(q,f,fctargs f)$ l:=reval coeffn(q,reval car l,cdr l) >> else l:=1 >>; % the constants and functions of integration if vari then q:=list('plus,q,intconst(l,vl,vari,list(1))) else <<vl1:=vl; while vl1 do <<v:=car vl1; vl1:=cdr vl1; if car ifac then q:=list('plus,q,intconst(l,vl,v,car ifac))$ % l..product of factors in the coefficient of the function to be % eliminated, car ifac .. list of integrating factors ifac:=cdr ifac; >> >>$ return reval q end$ % of addintco symbolic procedure integratepde(p,ftem,vari,genflag,potflag)$ % Generalized integration of the expression p % if not genflag then "normal integration" % Equation p must not be directly separable, otherwise the depen- % dencies of functions of integration on their variables is wrong, % i.e. no dependence on explicit variables % ftem are all functions from the `global' ftem which occur in p, i.e. % ftem:=smemberl(ftem,p)$ % if vari=nil then integration w.r.t. all possible variables within % the equation % else only w.r.t. vari one time begin scalar vl,vlrev,v,intlist, ili1a,ili2a,maxvanz,fsub,h,hh,nfsub,iflag,newcond, n1,n2,pot1,pot2,p1,p2,listoflds,secnd,ifac0, ifac1a,ifac1b,ifac2a,ifac2b,cop,v1a,v2a,revvl,pri$ % pri:=t; if pri then <<terpri()$write"Start Integratepde">>$ vl:=argset ftem$ vlrev:=reverse vl; if vari then <<potflag:=nil; if zerop p then iflag:='success>> else <<%------- determining fsub=list of functions of all variables maxvanz:=length vl$ fsub:=nil; h:=ftem; while h do <<if fctlength car h=maxvanz then fsub:=cons(car h,fsub)$ h:=cdr h >>$ nfsub:=length fsub$ % must be >1 for potential-integration h:=intminderiv(p,ftem,vlrev,maxvanz,nfsub)$ % fsub is also for below intlist:=car h$ %-- list of necessary integrations of the whole equation to solve %-- for a function of all variables listoflds:=cadr h$ %-- list of leading derivatives >>$ if pri then <<terpri()$ write"complete integrations:",intlist," for:",vl>>; %-- n1 is the number of integrations which must be done to try %-- potential integration which must enable to eliminate 2 functions %-- n2 is the number of integrations actually done n1:=for each h in intlist sum h; if (not vari) and (zerop n1) then <<n2:=0; if potflag then % else not necessary for h:=1:(length vl) do ifac0:=cons(nil,ifac0) >> else <<if tr_genint then <<terpri()$write "integration of the expression : "$ eqprint p>>$ if pri then <<terpri()$write"at first all multiple complete integration">>; %-- At first if possible n2 integrations of the whole equation h:=multipleint(intlist,ftem,p,vari,vl,genflag,nil,nil,nil)$ % potflag=nil, partial=nil, doneintvar=nil if h then <<p:=car h; ifac0:=cadr h; % ifac0 is the list of lists of integr. factors newcond:=caddr h; n2:=cadddr h; % number of done integrations % doneintvar & intfacdep for the two halfs of integrations % from the two parts of ifac0 h:=nil; iflag:='success; >> else n2:=0; ftem:=union(fnew_,ftem)$ >>; %------------ Existence of a potential ? if (n1=n2) and potflag and (nfsub>1) then %---- at least 2 functions to solve for <<if not zerop n2 then %---- update listoflds listoflds:=uplistoflds(reverse ifac0,listoflds)$ if pri then <<terpri()$write"uplistoflds:",listoflds>>$ if h:=potintegrable(listoflds) then <<ili1a:=car h; ili2a:=cadr h; % The necess. differentiations of the potential if pri then <<terpri()$write"potintegrable:",ili1a," ",ili2a>>$ if pri then <<write"+++ intlist=",intlist, " ili1a=",ili1a, " ili2a=",ili2a>>$ %-- distributing the integrating factors of ifac0 among %-- the two lists ifac1b and ifac2b which are so far nil %-- such that (ifac1b and ili1a are disjunct) and %-- (ifac2b and ili2a are disjunct) v1a:=vlofintlist(vl,ili1a); v2a:=vlofintlist(vl,ili2a); hh:=t; cop:=ifac0; while hh and cop do << % cop is a list of lists of integr. factors if car cop then h:=vlofintfaclist(vl,cdar cop) else h:=nil; if freeoflist(h,v2a) then << ifac1b:=cons( nil, ifac1b); ifac2b:=cons( reverse car cop, ifac2b) >> else if freeoflist(h,v1a) then << ifac2b:=cons( nil, ifac2b); ifac1b:=cons( reverse car cop, ifac1b) >> else hh:=nil; cop:=cdr cop; >>; % the elements of ifac1b,ifac2b are in reverse order to % ifac1a,ifac2a and are in the same order as vl, also % the elements in the infac-lists are in inverse order, % i.e. in the order the integrations have been done if pri then <<terpri()$ write "ifac1a=",ifac1a," ifac1b=",ifac1b, " ifac2a=",ifac2a," ifac2b=",ifac2b >>$ %-- lists of integrations to be done to both parts if hh then repeat % possibly a second try with part2 integrated first <<n1:=for each n1 in ili1a sum n1; % n1 .. number of remaining integrations of the first half p1:=multipleint(ili1a,ftem,p,nil,vl,genflag,t,t, % potflag=t, partial=t union(vlofintlist(vl,ili2a), vlofintlist(vl,ifac1b)))$ % that the variables of integration are not in ifac1b % was already checked. Only restriction: the integrating % factors must not depend on the last argument. ftem:=union(fnew_,ftem)$ if p1 then << ifac1a:=cadr p1; % ifac1a is now the list of integrating factors if newcond then newcond:=nconc(newcond,caddr p1) else newcond:=caddr p1; if pri then <<terpri()$write"mul2: newcond=",newcond>>$ n2:=cadddr p1; p1:=car p1 >>; if p1 and (n1=n2) then %--- if the first half has been integrated suff. often <<%--- integrating the second half sufficiently often revvl:=reverse vl; n1:=for each n1 in ili2a sum n1; % calculation of the 2. part which is not contained in p1 p2:=p1; cop:=ifac1a; hh:=revvl; % because ifac1a is reversed while cop do << h:=car cop;cop:=cdr cop; v:=car hh;hh:=cdr hh; % h is the list of integrating factors of the v-integr. while h do << p2:=reval list('quotient,list('df,p2,v),car h); h:=cdr h >> >>; p2:=reval reval list('plus,p,list('minus,p2)); p2:=multipleint(ili2a,ftem,p2,nil,vl,genflag,t,nil, % potflag=t, partial=nil union(vlofintlist(vl,ili1a), vlofintlist(vl,ifac2b)))$ ftem:=union(fnew_,ftem)$ if p2 then << ifac2a:=cadr p2; % ifac2a is now list of integrating factors if newcond then newcond:=nconc(newcond,caddr p2) else newcond:=caddr p2; if pri then <<terpri()$write"mul3: newcond=",newcond>>$ n2:=cadddr p2; p2:=car p2 >>; if p2 and (n1=n2) then % if the second half has been integrated sufficiently often <<iflag:='potint; % ifac1b,ifac2b are in reverse order to ifac1a,ifac2a! pot1:=newfct(fname_,vl,nfct_)$ % the new potential fct. pot2:=pot1; nfct_:=add1 nfct_$ fnew_:=cons(pot1,fnew_); v:=revvl; while v do <<cop:=car ifac1a; ifac1a:=cdr ifac1a; while cop do << pot1 := reval list('quotient,list('df,pot1,car v),car cop); cop:=cdr cop >>; cop:=car ifac2a; ifac2a:=cdr ifac2a; while cop do << pot2 := reval list('quotient,list('df,pot2,car v),car cop); cop:=cdr cop >>; v:=cdr v; >>; p:=addintco(list('plus,p1,reval pot2), ftem,ifac1b,vlrev,nil)$ newcond:=cons(addintco(list('plus,p2, list('minus,reval pot1)), ftem,ifac2b,vlrev,nil), newcond) % vari=nil >> ;if pri then write":::"$ >>; secnd:=not secnd; % retry in different order of integration, p is still the same if (iflag neq 'potint) and secnd then <<cop:=ili1a;ili1a:=ili2a;ili2a:=cop>> >> until (iflag eq 'potint) or (not secnd) >>; >>$ %--------- returning the result return if not iflag then nil else <<if iflag neq 'potint then % constants of integration p:=addintco(p, ftem, % the completely reversed ifac0 <<h:=nil; while ifac0 do <<h:=cons(reverse car ifac0,h);ifac0:=cdr ifac0>>; h >>, vl, vari)$ if pri then <<terpri()$write"ENDE INTEGRATEPDE"$deprint(cons(p,newcond))>>$ cons(p,newcond) >> end$ % of integratepde symbolic procedure intpde(p,ftem,vl,x,potint)$ % integration of an polynomial expr. p w.r.t. x begin scalar f,ft,l,l1,vl,k,s,a,iflag,flag$ ft:=ftem$ vl:=cons(x,delete(x,vl))$ while ftem and not flag do <<f:=car ftem$ % integrating all terms including car ftem if member(x,fctargs f) then <<l1:=l:=lderiv2(p,f,vl)$ while not (flag or (iflag:=intlintest(l,x))) do <<k:=reval coeffn(p,car l,cdr l)$ if intcoefftest(lderiv1(k,f,vl),car l,vl) then <<a:=decderiv(car l,x)$ k:=reval list('int,subst('v_a_r_,a,k),'v_a_r_)$ k:=reval subst(a,'v_a_r_,k)$ s:=cons(k,s)$ p:=reval aeval list('difference,p,list('df,k,x))$ if (l:=lderiv2(p,f,vl))=l1 then flag:='neverending else l1:=l >> else flag:='coeffld >>$ % iflag='nofct is the so far integrable case % the non-integrable cases are: flag neq nil, % (iflag neq nil) and (iflag neq 'nofct), an exception to the % second case is potint where non-integrable rests are allowed if iflag='nofct then ftem:=smemberl(ftem,p) else if potint or (fctlength f<length vl) then <<ftem:=cdr ftem$flag:=nil>> else flag:=(iflag or flag) >> else ftem:=cdr ftem >>$ return if not flag then <<l:=explicitpart(p,ft,x)$ l:=list('int,l,x)$ s:=reval aeval cons('plus,cons(l,s))$ p:=list('difference,p,list('df,l,x))$ if poly_only then if ratexp(s,ft) then list(s,reval reval aeval p) else nil else list(s,reval reval aeval p) >> else nil$ end$ % of intpde symbolic procedure explicitpart(p,ft,x)$ % part of a sum p which only explicitly depends on a variable x begin scalar l$ if not member(x,argset smemberl(ft,p)) then l:=p else if pairp p then <<if car p='minus then l:=reval list('minus,explicitpart(cadr p,ft,x))$ if (car p='quotient) and not member(x,argset smemberl(ft,caddr p)) then l:=reval list('quotient,explicitpart(cadr p,ft,x),caddr p)$ if car p='plus then <<for each a in cdr p do if not member(x,argset smemberl(ft,a)) then l:=cons(a,l)$ if pairp l then l:=reval cons('plus,l)>> >>$ if not l then l:=0$ return l$ end$ symbolic procedure intconst(co,vl,x,ifalist)$ % The factors in ifalist must be in the order of integrations done if null ifalist then 0 else begin scalar l,l2,f,coli,cotmp$ while ifalist do << cotmp:=coli; coli:=nil; while cotmp do << coli:=cons(list('int,list('times,car ifalist,car cotmp),x),coli); cotmp:=cdr cotmp >>; coli:=cons(1,coli); ifalist:=cdr ifalist >>; while coli do <<f:=newfct(fname_,delete(x,vl),nfct_)$ nfct_:=add1 nfct_$ fnew_:=cons(f,fnew_)$ l:=cons(list('times,f,car coli),l)$ coli:=cdr coli >>$ if length l>1 then l:=cons('plus,l) else if pairp l then l:=car l else l:=0$ if co and co neq 1 then if pairp co then <<if car co='times then co:=cdr co else co:=list co$ while co do <<if my_freeof(car co,x) then l2:=cons(car co,l2)$ co:=cdr co>> >> else if co neq x then l2:=list co$ return reval if l2 then cons('times,cons(l,l2)) else l end$ symbolic procedure intcoefftest(l,l1,vl)$ begin scalar s$ return if not pairp l then t else if car l='df then <<s:=decderiv(l1,car vl)$ if pairp s and pairp cdr s then s:=cddr s else s:=nil$ if diffrelp(cons(cddr l,1),cons(s,1),vl) then t else nil>> else t$ end$ symbolic procedure fctlinear(p,f)$ <<p:=ldiffp(p,f)$ (null car p) and (cdr p=1)>>$ symbolic procedure intlintest(l,x)$ % Test,ob in einem Paar (fuehrende Ableitung.Potenz) % eine x-Ableitung linear auftritt if not car l then if zerop cdr l then 'nofct else 'nonlin else % Fkt. tritt auf if (car l) and (cdr l=1) then % fuehr. Abl. tritt linear auf if pairp car l then % Abl. der Fkt. tritt auf if caar l='df then if member(x,cddar l) then nil % x-Abl. tritt auf else if member(x,fctargs cadar l) then 'noxdrv else 'noxdep else 'nodrv else 'nodrv else 'nonlin$ symbolic procedure decderiv(l,x)$ % in Diff.ausdr. l wird Ordn. d. Abl. nach x um 1 gesenkt begin scalar l1$ return if l then if car l='df then <<l1:=decderiv1(cddr l,x)$ if l1 then cons('df,cons(cadr l,l1)) else cadr l>> else l else nil$ end$ symbolic procedure decderiv1(l,x)$ if null l then nil else if car l=x then if cdr l then if numberp cadr l then if cadr l>2 then cons(car l,cons(sub1 cadr l,cddr l)) else cons(car l,cddr l) else cdr l else nil else cons(car l,decderiv1(cdr l,x))$ symbolic procedure integratede(q,ftem,genflag)$ % Integration of a de % result: newde if successfull, nil otherwise begin scalar l,l1,l2,fl$ ftem:=smemberl(ftem,q)$ if l1:=integrableode(q,ftem) then % looking for an integrable ode if l1:=integrateode(q,car l1,cadr l1,caddr l1,ftem) then % trying to integrate it <<q:=algsimple(l1,ftem,argset ftem)$ ftem:=smemberl(union(fnew_,ftem),q)$ fl:=t>>$ while l1:=integratepde(q,ftem,nil,genflag,potint_) do % trying to integrate a pde <<q:=car l1$ for each a in cdr l1 do <<ftem:=union(fnew_,ftem)$ if (l2:=integratede(a,ftem,nil)) then l:=append(l2,l) else l:=cons(a,l)>>$ fl:=t$ if null genflag then l1:=nil$ ftem:=smemberl(union(fnew_,ftem),q)>>$ if fl then <<l:=cons(q,l)$ l:=for each a in l collect reval aeval a$ l:=for each a in l collect if pairp a and (car a='quotient) then cadr a else a>>$ return l$ end$ endmodule; module genint; %Generalized integration of pde's containing unknown fns. % Author: Andreas Brand % December 1991 fluid '(fname_ fnew_ nfct_ tr_genint tr_gensep); symbolic procedure gintorder(p,vl,x)$ % reorder equation p begin scalar l,l1,q,m,b,c$ if pairp(p) and (car p='quotient) then <<q:=caddr p$p:=cadr p>>$ if pairp(p) and (car p='plus) then p:=cdr p % list of summands else p:=list p$ while p do <<l1:=gintorder1(car p,x)$ if deponallvars(car l1,x,vl) then l:=p:=nil else <<l:=termsort(l,l1)$p:=cdr p>> >>$ if l then <<l:=for each a in l collect if cddr a then <<b:=car a$ c:=cdr reval coeff1(cons('plus,cdr a),x,nil)$ m:=0$ while c and (car c=0) do <<c:=cdr c$m:=add1 m>>$ if m>0 then b:=list('times,list('expt,x,m),b)$ cons(reval b,c)>> else cons(reval car a,cdr reval coeff1(cadr a,x,nil))$ if q then if my_freeof(q,x) then l:=for each a in l collect cons(car a,for each s in cdr a collect reval list('quotient,s,q)) else l:=for each a in l collect cons(reval list('quotient,car a,q),cdr a)>>$ return l$ end$ symbolic procedure deponallvars(c,x,vl)$ % tests for occurence off vars from vl in factors of c depending on x begin scalar l$ if pairp c and (car c='times) then c:=cdr c else c:=list c$ while c and vl do <<if not my_freeof(car c,x) then for each v in vl do if not my_freeof(car c,v) then l:=cons(v,l)$ vl:=setdiff(vl,l)$ c:=cdr c >>$ return (null vl)$ end$ symbolic procedure gintorder1(p,x)$ % reorder a term p begin scalar l1,l2,sig$ % l2:list of factors of p not depending % on x or beeing a power of x % l1:all other factors if pairp p and (car p='minus) then <<sig:=t$p:=cadr p>>$ if pairp p and (car p='times) then p:=cdr p else p:=list p$ for each a in p do <<if my_freeof(a,x) then l2:=cons(a,l2) else if a=x then l2:=cons(a,l2) else if pairp a and (car a='expt) and (cadr a=x) and fixp caddr a then l2:=cons(a,l2) else l1:=cons(a,l1)>>$ if pairp l1 then if cdr l1 then l1:=cons('times,l1) else l1:=car l1$ if pairp l2 then if cdr l2 then l2:=cons('times,l2) else l2:=car l2$ if sig then if l2 then l2:=list('minus,l2) else l2:=list('minus,1)$ return list(if l1 then l1 else 1,if l2 then l2 else 1)$ end$ symbolic procedure partint(p,ftem,vl,x)$ begin scalar f,neg,l1,l2,n,k,l$ if tr_genint then <<terpri()$write "generalized integration of the unintegrated rest : "$ eqprint p >>$ l:=gintorder(p,vl,x)$ l:=for each s in l collect <<f:=newfct(fname_,varslist(car s,ftem,vl),nfct_)$ nfct_:=add1 nfct_$ fnew_:=cons(f,fnew_)$ neg:=t$ n:=sub1 length cdr s$ k:=-1$ if (pairp car s) and (caar s='df) then <<l1 := reval list('difference,cadar s,list('df,f,x,add1 n)) . l1; l2:=cddar s>> else <<l1:=cons(signchange reval list('difference,car s, list('df,f,x,add1 n)),l1)$ l2:=nil>>$ reval cons('plus, for each sk on cdr s collect <<neg:=not neg$ k:=add1 k$ reval list('times,if neg then -1 else 1, append(list('df,f,x,n-k),l2), tailsum(sk,k,x))>> )>>$ if l then l:=cons(reval cons('plus,l),l1)$ if tr_gensep then <<terpri()$ write "result (without constant or function of integration): "$ eqprint car l$ write "additional equations : "$ deprint cdr l$ >>$ return l$ end$ symbolic procedure tailsum(sk,k,x)$ begin scalar j$ j:=-1$ return reval cons('plus, for each a in sk collect <<j:=j+1$ list('times,a,prod(j+1,j+k),list('expt,x,j)) >> ) end$ symbolic procedure prod(m,n)$ if m>n then 1 else for i:=m:n product i$ endmodule; module intfactr; % Routines for finding integrating factors of PDEs. % Author: Thomas Wolf % July 1994 % The following without factorization --> faster but less general %symbolic procedure fctrs(p,vl,v)$ %begin scalar fl1,fl2,neg; % %write"p=",p; % % if car p='MINUS then <<neg:=t;p:=cdr p>>$ % return % if not pairp p then if my_freeof(p,v) and (not freeoflist(p,vl)) then % list(p,1,neg) else % list(1,p,neg) % else if car p='PLUS then list(1,p,neg) % else % if car p='TIMES then % <<for each el in cdr p do % if my_freeof(el,v) and (not freeoflist(p,vl)) then % fl1:=cons(el,fl1) else % fl2:=cons(el,fl2); % if pairp fl1 then fl1:=cons('TIMES,fl1); % if pairp fl2 then fl2:=cons('TIMES,fl2); % if not fl1 then fl1:=1; % if not fl2 then fl2:=1; % list(fl1,fl2,neg) % >> else if my_freeof(p,v) and (not freeoflist(p,vl)) then % list(p,1,neg) else % list(1,p,neg) %end$ % of fctrs % symbolic procedure fctrs(p,indep,v)$ begin scalar fl1,fl2; p:=cdr reval factorize p; for each el in p do if freeoflist(el,indep) and (not my_freeof(el,v)) then fl1:=cons(el,fl1) else fl2:=cons(el,fl2); if null fl1 then fl1:=1; if null fl2 then fl2:=1; if pairp fl1 then fl1:=cons('times,fl1); if pairp fl2 then fl2:=cons('times,fl2); return list(fl1,fl2) end$ % of fctrs symbolic procedure extractfac(p,indep,v)$ % Looks for factors of p dependent of v and independent of indep and % returns a list of the numerator factors and a list of the denominator % factors. begin scalar nu,de$ return if car p='quotient then <<nu:=fctrs( cadr p,indep,v); de:=fctrs(caddr p,indep,v); list( reval list('quotient, car nu, car de), list('quotient, cadr nu, cadr de) ) >> else fctrs(p,indep,v) end$ % of extractfac %---------------------------- symbolic procedure get_kernels(ex)$ % gets the list of all kernels in ex begin scalar res,pri$ % pri:=t; ex:=reval ex$ if pri then <<terpri()$write"ex=",ex>>; if pairp ex then if (car ex='quotient) or (car ex='plus) or (car ex='times) then for each s in cdr ex do res:=union(get_kernels s,res) else if (car ex='minus) or ((car ex='expt) and % (numberp caddr ex)) then % not for e.g. (quotient,2,3) (cadr ex neq 'e) and (cadr ex neq 'e) and (not fixp cadr ex) ) then res:=get_kernels cadr ex else res:=list ex else if idp ex then res:=list ex$ if pri then <<terpri()$write"res=",res>>; return res$ end$ %------------------ symbolic procedure specialsol(p,vl,fl,x,indep,gk)$ % tries a power ansatz for the functions in fl in the kernels % of p to make p to zero % indep is a list of kernels, on which the special solution should % not depend. Is useful, to reduce the search-space, e.g. when an % integrating factor for a linear equation for, say f, is to be % found then f itself can not turn up in the integrating factor fl % gk are kernels which occur in p and possibly extra ones which % e.g. are not longer in p because of factorizing p but which are % likely to play a role, if nil then determined below begin scalar e1,e2,n,nl,h,sublist,eqs,startval,pri,printold; % pri:=t; p:=num p; if pri then << terpri()$write"The equation for the integrating factor:"; terpri()$eqprint p; >>; if null gk then gk:=get_kernels(p); for each e1 in fl do << h:=nil; %---- h is the power ansatz if pri then for each e2 in gk do << terpri()$write"e2=",e2; if my_freeof(e2,x) then write" freeof1"; if not freeoflist(e2,fl) then write" not freeoflist"$ if not freeoflist(e2,indep) then write" dependent on indep" >>; %----- nl is the list of constants to be found for each e2 in gk do if (not my_freeof(e2,x)) and % integ. fac. should depend on x freeoflist(e2,fl) and % if the equation is linear in fl % e.g. if an integr. factor is to % be found even for a non-lin. de freeoflist(e2,indep) then << n:=gensym();nl:=cons(n,nl); h:=cons(list('expt,e2,n),h); >>; if h then << if length h > 1 then h:=cons('times,h) else h:=car h; %-- the list of substitutions for the special ansatz sublist:=cons((e1 . h),sublist); if pri then <<terpri()$write"Ansatz: ",e1," = ",h>>; p:= reval num reval subst(h,e1,p); if pri then <<terpri()$write"p=";eqprint p>> >> >>; if null h then return nil; %------- An numerical approach to solve for the constants if nil then << % numerical approach %--- Substituting all kernels in p by numbers on rounded; precision 20; terpri()$terpri()$write"Before substituting numerical values:"; eqprint p; terpri()$terpri()$write"Constants to be calculated: "; for each n in nl do write n," "; for each e1 in nl do << h:=p; for each e2 in gk do if freeoflist(e2,fl) then if pairp e2 and ((car e2 = 'df) or (car e2 = 'int)) then << n:=list('quotient,1+random 30028,30029); terpri();write"substitution done: ",e2," = ",n; h:=subst(list('quotient,1+random 30028,30029),e2,h) >>; for each e2 in gk do if freeoflist(e2,fl) then if not(pairp e2 and ((car e2 = 'df) or (car e2 = 'int))) then << n:=list('quotient,1+random 30028,30029); terpri();write"substitution done: ",e2," = ",n; h:=subst(list('quotient,1+random 30028,30029),e2,h) >>; terpri()$terpri()$write"The equation after all substitutions: "; terpri()$ eqprint h; eqs:=cons(reval h,eqs); startval:=cons(list('equal,e1,1),startval) >>; if length eqs = 1 then eqs:=cdr eqs else eqs:=cons('list,eqs); if length startval = 1 then startval:=cdr startval else startval:=cons('list,startval); terpri()$write"start rdsolveeval!";terpri()$terpri()$ h:=rdsolveeval list(eqs,startval); eqs:=nil; off rounded; >>; %----- An analytical approach to solve for the constants if null pri then <<printold:=print_;print_:=nil>>; if p and not zerop p then % uebernommen aus SEPAR if not (pairp p and (car p='quotient) and % " " " intersection(argset smemberl(fl,cadr p),vl)) then p:=separ2(p,fl,vl) else p:=nil; if null pri then print_:=printold; if p then << % possibly investigating linear dependencies of different caar p % solve(lasse x-abhaengige und nl-unabhaengige faktoren fallen von % factorize df(reval list('QUOTIENT, caar p1, caar p2),x),nl). while p do if freeoflist(cdar p,nl) then <<eqs:=nil;p:=nil>> % singular system --> no solution else << eqs:=cons(cdar p,eqs); p:=cdr p >>; >>; if pri then <<terpri()$write"eqs1=",eqs>>; if null eqs then return nil else << if pri then << terpri()$write"The algebraic system to solve for ",nl," is:"; if length eqs > 1 then deprint eqs else eqprint car eqs >>; if length eqs > 1 then eqs:=cons('list,eqs) else eqs:=car eqs; if pri then <<terpri()$write"eqs2=",eqs;terpri();write"nl=",nl>>$ % for catching the error message `singular equations' if pri then << terpri()$write"The algebraic system to solve for ",nl," is:"; eqprint eqs >>; h:=cons('list,nl); eqs:=<< h:=errorset({'solveeval,mkquote{eqs, h}},nil,nil) where !*protfg=t; erfg!*:=nil; if errorp h then nil else cdar h % cdr for deleting 'LIST >>; if pri then <<terpri()$write"eqs3=",eqs;terpri()>>$ %--- eqs is the list of solutions for the constant exponents of the %--- integrating factor if null eqs then return nil; if length nl=1 then eqs:=list eqs; if pri then <<write"nl=",nl," eqs4=",eqs;terpri()>>; for each e1 in eqs do << % each e1 is a list of substitutions if pri then <<terpri()$write"e2=",e1;terpri()>>$ if car e1='list then e1:=cdr e1; if pri then <<terpri()$write"e3=",e1;terpri()>>$ for each e2 in e1 do << if pri then algebraic write"solution:",symbolic e2; sublist:=subst(caddr e2,cadr e2,sublist) >>; if pri then <<terpri()$write"The sublist is:",sublist>> >>; >>; return for each e1 in sublist collect (car e1 . reval cdr e1) end$ % of specialsol %------------------ symbolic operator findintfac$ symbolic procedure findintfac(pl,ftem,vl,x,doneintvar,intfacdep, factr)$ % - pl is a list of equations from which the *-part (inhomogeneous % terms) have been dropped. % - each equation of pl gets an integrating factor h % - doneintvar is a list of variables, on which the integrating factor % should not depend. The chances to find an integrating factor % increase if the inhomogeneous part of pl is dropped and % separately integrated with generalized integration later. % - if factr is not nil then the equation(s) pl is(are) at first % factorized, e.g. if integration(s) have already been done % and there is a chance that the equation can factorize, thereby % simplify and giving a higher chance for integrability. begin scalar h,newequ,tozero,fl,e1,pri,factr,exfactors,ftr,gk; % exfactors is the list of factors extracted at the beginning % pri:=t; factr:=t; % whether tozero should be factorized if pri then <<terpri()$write"START VON FINDINTFAC">>; %--- Generation of the condition for the integrating factor(s) in fl for each e1 in pl do << %--- extracting factors dependend on x and independent of %--- doneintvar but only if integrations have already been done, %--- i.e. (doneintvar neq nil) gk:=union(gk,get_kernels(e1)); if factr then <<ftr:=extractfac(e1,append(doneintvar,ftem),x); if not evalnumberp car ftr then gk:=union(gk,car ftr); if (cadr ftr) and (not evalnumberp cadr ftr) then gk:=union(gk,cadr ftr)>> else ftr:=list(1,nil); exfactors:=cons(car ftr,exfactors); if car ftr neq 1 then << e1:=cadr ftr; if pri then <<terpri()$write"extracted factor:",eqprint car ftr>>; >>; %--- making an ansatz for the integrating factor h:=gensym(); depl!*:=cons(list(h,x),depl!*)$ depend h,x; fl:=h . fl; e1:=intpde(reval list('times,h,e1),ftem,vl,x,t); if null car e1 then << pl:=nil; tozero:=nil >> else << newequ:=car e1 . newequ; tozero:=cadr e1 . tozero; if pri then << terpri()$write" the main part of integration:"$ eqprint(car e1); terpri()$write"car e1=",car e1; terpri()$write" the remainder of integration:"$ eqprint(cadr e1); terpri()$write"cadr e1=",cadr e1; >> >>; >>; if null tozero then return nil; %-------- newequ is the integral newequ:=if length pl > 1 then cons('plus,newequ) else car newequ; %-------- tozero is the PDE for the integrating factor tozero:=reval if length pl > 1 then cons('plus,tozero) else car tozero; if pairp tozero and (car tozero='quotient) then tozero:=cadr tozero$ if factr then << h:=cdr reval list('factorize,tozero)$ if pri then <<terpri()$write"The factors of tozero:",h>>; tozero:=nil; for each e1 in h do if smemberl(fl,e1) then tozero:=cons(e1,tozero)$ tozero:= reval if length tozero > 1 then cons('times,tozero) else car tozero; >>; if nil and pri then <<write"tozero =";eqprint tozero >>; h:=nil; % actually only those f in ftem, in which pl is nonlinear, but also % then only integrating factors with a leading derivative low enough h:=specialsol(tozero,vl,fl,x,append(ftem,doneintvar),gk); % h:=specialsol(tozero,vl,fl,x,doneintvar,gk); if pri then <<write"h=",h;terpri()>>; if h then << if print_ then << if length pl = 1 then write"An integrating factor has been found:" else write"Integrating factors have been found: "$ for each e1 in h do algebraic write lisp cdr e1," "; >>; for each e1 in h do << % each e1 is one integrating factor determined if pri then <<terpri()$write"e1=",e1; terpri()$write"newequ=",newequ;terpri()>>; newequ:=reval subst(cdr e1,car e1,newequ); if pri then <<terpri()$write"newequ=",newequ>>; >> >> else if pri then write"no integrating factor"; %--- delete all dependencies of the functions in fl %--- must come before the following update for each e1 in fl do depl!*:=delete(assoc(e1,depl!*),depl!*)$ %--- update intfacdep for each e1 in vl do if (e1 neq x) and my_freeof(intfacdep,e1) and ((not my_freeof(h,e1)) or (not my_freeof(exfactors,e1))) then intfacdep:=cons(e1,intfacdep); %--- returns nil if no integrating factor else a list of the %--- factors and the integral if h and print_ then << terpri()$write"The integrated equation:"; eqprint newequ >>; return if (null h) or (zerop newequ) then nil else list(newequ, for each e1 in h collect << ftr:=car exfactors; exfactors:=cdr exfactors; if ftr=1 then cdr e1 else reval list('quotient,cdr e1,ftr) >>, intfacdep) end$ endmodule; module odeint; % Integration of ode's containing unknown functions. % Author: Thomas Wolf % August 1991 fluid '(cont_ fnew_ odesolve_ poly_only safeint_); symbolic procedure integrateode(de,fold,xnew,ordr,ftem)$ begin scalar newde,newnewde,l,factrs,fc,changd$ return if not xnew then begin % Integr. einer alg. Gl. fuer eine Abl. if pairp de and (car de='quotient) then de:=cadr de$ factrs:=cdr reval list('factorize,de); l:=for each fc in factrs collect if smember(fold,fc) then << newde:=cadr solveeval list(fc,fold)$ if not freeof(newde,'root_of) then fc else << newde:=list('plus,cadr newde,list('minus,caddr newde))$ if (l:=integratepde(newde,ftem,nil,t,nil)) then <<changd:=t;car l>> else fc >> >> else fc; return if not changd then nil else if length l > 1 then cons('times,l) else cdr l % newde:=cadr solveeval list(de,fold)$ % l:='ROOT_OF; % if not freeof(newde,l) then return nil; % newde:=list('PLUS,cadr newde,list('MINUS,caddr newde))$ % return if (l:=integratepde(newde,ftem,nil,t,nil)) then car l % %genflag=t,potflag=t else nil % if newde=de then nil % else if (l:=integratede(newde,ftem,nil)) then car l % else nil$ end else % eine ode fuer ein f? if not pairp fold then % i.e. not df(...,...), i.e. fold=f odeconvert(de,fold,xnew,ordr,ftem) % --> ode fuer eine Abl. von f else begin newde:=odeconvert(de,fold,xnew,ordr,ftem)$ % if print_ then % if newde then <<terpri()$write "One ode has been integrated.">>$ return if not newde then nil else << newnewde:=cadr solveeval list(newde,fold)$ newnewde:=reval list('plus,cadr newnewde,list('minus, caddr newnewde))$ ftem:=union(fnew_,ftem)$ newnewde:=integratede(newnewde,ftem,nil)$ if newnewde then car newnewde else newde >> end end$ % of integrateode symbolic operator polyans$ symbolic procedure polyans(ordr,dgr,x,y,d_y,fn)$ % generates a polynom % for i:=0:dgr sum fn"i"(x,y,d_y(1),..,d_y(ordr-1))*d_y(ordr)**i % with fn as the function names and d_y as names or derivatives of y % w.r.t. x begin scalar ll,fl,a,i,f$ i:=sub1 ordr$ while i>0 do <<ll:=cons(list(d_y,i),ll)$ i:=sub1 i>>$ ll:=cons(y,ll)$ ll:=reverse cons(x,ll)$ fl:=nil$ i:=0$ while i<=dgr do <<f:=newfct(fn,ll,i)$ fl:=(f . fl)$ a:=list('plus,list('times,f,list('expt,list(d_y,ordr),i)),a)$ i:=add1 i>>$ return list('list,reval a,cons('list,fl)) end$ % of polyans symbolic operator sepans$ symbolic procedure sepans(kind,v1,v2,fn)$ % Generates a separation ansatz % v1,v2 = lists of variables, fn = new function name + index added % The first variable of v1 occurs only in one sort of the two sorts of % functions and the remaining variables of v1 in the other sort of % functios. % The variables of v2 occur in all functions. % Returned is a sum of products of each one function of both sorts. % form: fn1(v11;v21,v22,v23,..)*fn2(v12,..,v1n;v21,v22,v23,..)+... % the higher "kind", the more general and difficult the ansatz is % kind = 0 is the full case begin scalar n,vl1,vl2,h1,h2,h3,h4,fl$ if cdr v1 = nil then <<vl1:=cdr v2$vl2:=cdr v2>> else <<vl1:=cons(cadr v1,cdr v2)$ vl2:=append(cddr v1,cdr v2)>>$ return if kind = 0 then <<vl1:=append(cdr v1,cdr v2)$ h1:=newfct(fn,vl1,'_)$ list('list,h1,list('list,h1))>> else if kind = 1 then <<h1:=newfct(fn,vl1,1)$ list('list,h1,list('list,h1))>> else if kind = 2 then <<h1:=newfct(fn,vl2,1)$ list('list,h1,list('list,h1))>> else if kind = 3 then <<h1:=newfct(fn,vl1,1)$ h2:=newfct(fn,vl2,2)$ list('list,reval list('plus,h1,h2), list('list,h1,h2))>> else if kind = 4 then <<h1:=newfct(fn,vl1,1)$ h2:=newfct(fn,vl2,2)$ list('list,reval list('times,h1,h2), list('list,h1,h2))>> else if kind = 5 then <<h1:=newfct(fn,vl1,1)$ h2:=newfct(fn,vl2,2)$ h3:=newfct(fn,vl1,3)$ list('list,reval list('plus,list('times,h1,h2),h3), list('list,h1,h2,h3))>> else if kind = 6 then <<h1:=newfct(fn,vl1,1)$ h2:=newfct(fn,vl2,2)$ h3:=newfct(fn,vl2,3)$ list('list,reval list('plus,list('times,h1,h2),h3), list('list,h1,h2,h3))>> else if kind = 7 then <<h1:=newfct(fn,vl1,1)$ h2:=newfct(fn,vl2,2)$ h3:=newfct(fn,vl1,3)$ h4:=newfct(fn,vl2,4)$ list('list,reval list('plus, list('times,h1,h2),h3,h4), list('list,h1,h2,h3,h4))>> else % ansatz of the form FN = FN1(v11,v2) + FN2(v12,v2) + ... + FNi(v1i,v2) if kind = 8 then <<n:=1$ vl1:=cdr v1$ vl2:=cdr v2$ fl:=()$ while vl1 neq () do << h1:=newfct(fn,cons(car vl1,vl2),n)$ vl1:=cdr vl1$ fl:=cons(h1, fl)$ n:=n+1 >>$ list('list, cons('plus,fl), cons('list,fl))>> else <<h1:=newfct(fn,vl1,1)$ h2:=newfct(fn,vl2,2)$ h3:=newfct(fn,vl1,3)$ h4:=newfct(fn,vl2,4)$ list('list,reval list('plus,list('times,h1,h2), list('times,h3,h4)), list('list,h1,h2,h3,h4))>> end$ % of sepans symbolic procedure odecheck(ex,fint,ftem)$ % assumes an revaled expression ex % Does wrong if car ex is a list! begin scalar a,b,op,ex1$ %***** ex is a ftem-function ***** if ex=fint then % list(ex,0,0,..) <<a:=list ex$ ex:=fctargs ex$ while ex do <<a:=append(list(0,0),a)$ ex:=cdr ex>>$ % not checked if it is a function of an expression of x op:=reverse a>> else if pairp ex then %***** car ex is 'df ***** if (car ex)='df then <<a:=odecheck(cadr ex,fint,ftem)$ if not pairp a then op:=a else % a is list(fctname,0,0,..,0,0) <<op:=list(car a)$ a:=fctargs car a$ % a is list(variables), not checked ex:=cddr ex$ % ex is list(derivatives) while a do <<ex1:=ex$ while ex1 and ((car a) neq (car ex1)) do ex1:=cdr ex1$ if null ex1 then op:=cons(0,cons(0,op)) else <<if not cdr ex1 then b:=1 % b is number of deriv. of that var. else <<b:=cadr ex1$ if not numberp b then b:=1>>$ op:=cons(b,cons(b,op))>>$ a:=cdr a>>$ op:=reverse op>> >> else %***** car ex is a standard or other function ***** <<a:=car ex$ % for linearity check ex:=cdr ex$ if a='int then ex:=list reval car ex$ while (op neq '!_abb) and ex do <<b:=odecheck(car ex,fint,ftem)$ if b then % function found if b eq '!_abb then op:='!_abb % occures properly else op:=odetest(op,b)$ ex:=cdr ex>> >>$ return op end$ symbolic procedure integrableode(p,ftem)$ if delength p>(if odesolve_ then odesolve_ else 0) then (if cont_ then if yesp("expression to be integrated ? ") then integrableode1(p,ftem)) else integrableode1(p,ftem)$ symbolic procedure integrableode1(p,ftem)$ begin scalar a,b,u,vl,le,uvar, fint,fivar,% the function to be integrated and its variables fold, % the new function of the ode xnew, % the independ. variable of the ode ordr1, % order of the ode ordr2, % order of the derivative for which it is an ode intlist$ % list of ode's ftem:=smemberl(ftem,p)$ vl:=argset ftem$ % p muss genau eine Funktion aus ftem von allen Variablen enthalten. % Die Integrationsvariable darf nicht Argument anderer in p enthaltener % ftem-Funktionen sein. a:=ftem$ b:=nil$ le:=length vl$ while a and vl do <<u:=car a$ uvar:=fctargs u$ if (length uvar) = le then if b then <<vl:=nil$a:=list(nil)>> else <<b:=t$ fint:=u$ fivar:=uvar>> else vl:=setdiff(vl,uvar)$ a:=cdr a>>$ if not b then vl:=nil$ le:=length p$ if ((1<le) and vl) then <<a:=odecheck(p,fint,ftem)$ if not atom a then % The equation is an ode <<ordr1:=0$ ordr2:=0$ xnew:=nil$ a:=cdr a$ b:=fivar$ while b do <<if (car a) neq 0 then <<fold:=cons(car b,fold)$ if (car a) > 1 then fold:=cons(car a,fold)>>$ ordr2:=ordr2+car a$ if (car a) neq (cadr a) then <<xnew:=car b$ if not member(xnew,vl) then <<b:=list(nil)$vl:=nil>>$ ordr1:=(cadr a) - (car a)>>$ b:=cdr b$ a:=cddr a>>$ fold:=reverse fold$ %fold is the list of diff.variables + number of diff. if fold then fold:=cons('df,cons(fint,fold)) else fold:=fint$ if vl and ((ordr1 neq 0) or (ordr2 neq 0)) then intlist:=list(fold,xnew,ordr1,ordr2) >> % of variable found >>$ % of if return intlist end$ % of integrableode1 symbolic procedure odetest(op,b)$ if not op then b else % op=nil --> first function found if (car op) neq (car b) then '!_abb else % f occurs in differ. fct.s begin scalar dif,a$ dif:=nil$ % dif=t --> different derivatives a:=list(car op)$ % in one variable already found op:=cdr op$ b:=cdr b$ while op do <<a:=cons(max(cadr op,cadr b),cons(min(car op,car b),a))$ if (car a) neq ( cadr a) then if dif then <<a:='!_abb$ op:=list(1,1)>> else dif:=t$ op:=cddr op$ b:=cddr b>>$ if not atom a then a:=reverse a$ return a % i.e. '!_abb or (fctname,min x1-der.,max x1-der.,...) end$ symbolic procedure odeconvert(de,ford,xnew,ordr,ftem)$ begin scalar j,ford_,newco,oldde,newde,newvl,null_$ ford_:='y_$%gensym$ depl!*:=delete(assoc(ford_,depl!*),depl!*)$ depend1(ford_,xnew,t)$ oldde:=reval subst(ford_,reval ford,de)$ if pairp ford then % i.e. (car ford) eq 'DF then << for j:=1:ordr do oldde:= subst( reval list('df,ford_,xnew,j), reval list('df,ford,xnew,j), oldde)>>$ algebraic !!arbconst:=0$ algebraic ( explog:= { cot(~x) => 1/tan(x), e**(~x+~y) => e**x*e**y, sqrt(e)**(~x+~y) => sqrt(e)**x*sqrt(e)**y, e**((~x+~y)/~z) => e**(x/z)*e**(y/z), sqrt(e)**((~x+~y)/~z) => sqrt(e)**(x/z)*sqrt(e)**(y/z), sqrt(e)**(log(~y)/~x) => y**(1/x/2), sqrt(e)**(-log(~y)/~x) => y**(-1/x/2), sqrt(e)**(~x*log(~y)/~z) => y**(x/z/2), sqrt(e)**(-~x*log(~y)/~z) => y**(-x/z/2), sqrt(e)**((~x*log(~y))/~z) => y**(x/z/2), e**(log(~y)/~x) => y**(1/x), e**(~x*log(~y)/~z) => y**(x/z), e**((~x*log(~y))/~z) => y**(x/z), int(df(~y,~x)/~y,~x) => log(y) })$ algebraic( let explog )$ newde:=reval algebraic first odesolve(symbolic oldde,symbolic ford_,symbolic xnew)$ % Instead of the following test one should return several cases if safeint_ and zero_den(newde,ftem,argset ftem) then newde:=nil; if newde and (cadr newde neq oldde) then begin % solution found % Test der Loesung klappt nur, wenn Loesung explizit gegeben if cadr newde neq ford_ then << if print_ then <<write "Solution of the ode is not explicitly given."$ algebraic write "Equation is: ",algebraic symbolic oldde$ algebraic write "Solution is: ",algebraic symbolic newde >>; if poly_only then % The solution must be rational in the % function and constants of integration if not ratexp1(newde,ford_) then newde:=nil else << j:=1; while (j leq ordr) and ratexp1(subst(ford_,list('arbconst,j),newde),ford_) do j:=j+1; if j leq ordr then newde:=nil >>; >> else begin null_:=reval reval aeval subst(caddr newde, ford_, oldde)$ % reval reval because of a REDUCE bug for special data, % to be droped as soon as possible if (null_ neq 0) then begin % newde:=nil$ if print_ then begin write "odesolve solves : "$ deprint list oldde$ write "to"$ deprint list newde$ write "which inserted in the equation yields"$ deprint list null_$ end end end end$ algebraic ( clear explog )$ depl!*:=delete(assoc(ford_,depl!*),depl!*)$ if newde then <<newde:=list('plus,cadr newde,list('minus,caddr newde))$ if 0 = reval list('plus,newde,list('minus,oldde)) then newde:=nil>>$ return if not newde then nil else <<newde:=subst(ford,ford_,newde)$ newvl:=delete(xnew,if (pairp ford and (car ford='df)) then fctargs cadr ford else fctargs ford)$ % if pairp ford then newvl:=delete(xnew,cdr assoc(cadr ford,depl!*)) % else newvl:=delete(xnew,cdr assoc(ford,depl!*))$ for j:=1:ordr do begin newco:=newfct(fname_,newvl,nfct_)$ nfct_:=add1 nfct_$ fnew_:=cons(newco,fnew_)$ newde:=subst(newco,list('arbconst,j),newde) % newde:=subst(newco, prepf !*kk2f list('arbconst,j),newde) % newde:=reval subst(newco,list('arbconst,j),newde) % newde:=reval subst(newco, prepf !*kk2f list('arbconst,j),newde) end$ newde>> end$ endmodule; module simplf; % Simplifications and substitution of functions. % Author: Andreas Brand, Thomas Wolf % August 1991, May 1995 fluid '(cont_ contradiction_ fcteval_ fnew_ ineq_ one_argument_functions_ print_ solvealg_); symbolic procedure clearfactors(sorg,ftem,vl)$ % liefert Liste der Gl. die entstehen, wenn in allen Gl. aus sorg % gemeinsame Faktoren, die keine der Funktionen aus ftem sowie deren % Ableitungen enthalten, gekuerzt werden if sorg and ftem then begin scalar l,ft$ while sorg do <<if not zerop car sorg then if not(ft:=smemberl(ftem,car sorg)) then l:=cons(car sorg,l) else l:=cons(reval algsimple(reval car sorg,ft,vl),l)$ sorg:=cdr sorg>>$ return l end else sorg$ symbolic procedure simplifyterm(p,ftem)$ % simplify a single factor p of g=p*q*r*...=0 if (ftem:=smemberl(ftem,p)) then if pairp p and (member(car p,'(minus sqrt quotient)) or ((car p='expt) and numberp caddr p)) then simplifyterm(cadr p,ftem) else p else 1$ symbolic procedure signchange(g)$ % ensure, that the first term is positive if pairp g and (car g='minus) then cadr g else if pairp g and (car g='plus) and (pairp cadr g) and (caadr g='minus) then reval list('minus,g) else g$ symbolic procedure algsimple(g,ftem,vl)$ % Gl. die entsteht, wenn in g gemeinsame Faktoren, die keine der Fkt. % aus ftem sowie deren Ableitungen enthalten, gekuerzt werden if pairp g then begin scalar h,l$ g:=reval aeval g$ if (car g='expt) and freeoflist(cadr g,vl) then g:=1 else if car g='minus then g:=algsimple(cadr g,ftem,vl) else if (car g='quotient) or (car g='times) then g:=reval aeval cons(car g, for each a in cdr g collect algsimple(a,ftem,vl)) else <<h:=ftem$ g:=signchange(g)$ l:=cons(nil,0)$ while h do <<l:=lderiv(g,car h,fctargs car h)$ if l then h:=nil else h:=cdr h>>$ if l then <<h:=car reverse cdr reval list('coeff,g,l)$ %highest coefficient if lowpow!*>1 then g:=reval list('quotient,g,list('expt,l,lowpow!*-1))$ if not zerop h then if pairp h then <<h:=independpart(reval list('gcd,g,h),ftem,vl)$ if not zerop h then g:=reval list('quotient,g,h)>> else g:=reval list('quotient,g,list('gcd,g,h))>> >>$ return g end else if zerop g then 0 else if my_freeof(ftem,g) then 1 else g$ symbolic procedure intsimplify_drop(factlist,xl,ex)$ % Find factors in factlist which can't be droped and % the GCD of the exponents begin scalar fl,ex1$ for each s in factlist do if not freeoflist(s,xl) then <<fl:=cons(s,fl)$ if ex and pairp s and (car s='expt) then if zerop ex then ex:=caddr s else ex:=reval list('gcd,ex,caddr s) else ex:=nil>>$ if zerop ex or (ex=1) then ex:=nil$ if ex and not freeoflist(ex,xl) then if pairp ex then <<if car ex='times then ex1:=cdr ex else ex1:=cdr reval list('factorize,ex)$ ex:=nil$ for each s in ex1 do if freeoflist(s,xl) then ex:=cons(s,ex)$ if pairp ex then if pairp cdr ex then ex:=cons('times,reverse ex) else ex:=car ex else ex:=nil>> else ex:=nil$ return cons(reverse fl,ex)$ end$ symbolic procedure facsum(ex)$ begin scalar newex,fa,ep; return if (pairp ex) and (car ex = 'plus) then << ex:=cdr reval factorize ex; while ex do << fa:=car ex;ex:=cdr ex; ep:=1; while (pairp ex) and (car ex=fa) do <<ep:=ep+1;ex:=cdr ex>>; if ep=1 then newex:=cons(fa,newex) else newex:=cons(list('expt,fa,ep),newex) >>; if length newex=1 then car newex else cons('times,newex) >> else ex end$ symbolic procedure intsimplify_divide(fact_ex,xl)$ % fact_ex is (list_of_factors . common_expt_of_the_factors) % if rekur then rekursive call of intsimplify if pairp car fact_ex then <<if cdr fact_ex then fact_ex:= % there exists a common exponent for each s in car fact_ex collect % devide each factors exponent reval list('expt,cadr s,list('quotient,caddr s,cdr fact_ex)) else fact_ex:=car fact_ex$ % fact_ex is now the list of factors !!! if pairp cdr fact_ex then cons('times,fact_ex) else car fact_ex >> else 0$ symbolic procedure simpli(p,vl); begin scalar cp,k,kernl,diffr; if p then if atom p then if not freeof(vl,p) then kernl:=p else else << if car p = 'plus then cp:=facsum p else cp:=p; repeat << k:=simpli(car cp,vl); cp:=cdr cp; if k then if not kernl then kernl:=k else if kernl neq k then diffr:=t; >> until (not cp) or diffr; >>; if diffr then kernl:=p; return kernl end$ symbolic procedure intsimplify(q,xl)$ % Simplification of the equation q=c, where c is arbitrarily % adjustable, but independent of the variable and function list xl begin scalar q1,q2,p$ if pairp q then << q:=simpli(q,xl); if pairp q then if car q='expt then p:=reval intsimplify_divide(intsimplify_drop(list q,xl,0),xl) else if car q='times then p:=intsimplify_divide(intsimplify_drop(cdr q,xl,0),xl) else if car q='quotient then <<p:=list('quotient,facsum cadr q,facsum caddr q); if pairp cadr p and (caadr p='times) then q1:=cdadr p else q1:=list cadr p$ if pairp caddr p and (caaddr p='times) then q2:=cdaddr p else q2:=list caddr p$ q1:=intsimplify_drop(q1,xl,0)$ q2:=intsimplify_drop(q2,xl,cdr q1)$ q1:=cons(car q1,cdr q2)$ p:=list('quotient,intsimplify_divide(q1,xl), intsimplify_divide(q2,xl)) >> else if car q='plus then p:=facsum q; if p and (p neq q) then q:=intsimplify(p,xl); >>; return q end$ symbolic procedure independpart(g,ftem,vl)$ % determines factors which can be get rid of begin scalar l$ g:=cdr reval list('factorize,g)$ while g do <<if (pairp car g and (caar g='expt) and freeoflist(cadar g,vl)) or member(car g,cdr g) or member(signchange(car g),ineq_) or not smemberl(ftem,car g) then l:=cons(car g,l)$ g:=cdr g>>$ return if l then if cdr l then cons('times,l) else car l else 1 end$ symbolic procedure solvesplit(q,ftem)$ begin scalar l$ q:=list q$ if length(ftem:=alldfs(car q,ftem))=1 then begin l:=cdr reval solveeval(list(car q,car ftem))$ if length l>1 then q:=for each a in l collect reval if pairp caddr a and (caaddr a='root_of) then subst(cadr a,cadd caddr a,cadr caddr a) else list('plus,cadr a,list('minus,caddr a))$ end$ return q$ end$ symbolic procedure splitde(p,ftem)$ begin scalar l$ if pairp p and (car p='quotient) then p:=cadr p$ for each x in cdr reval list('factorize,p) do if smemberl(ftem,x) then l:=cons(reval x,l)$ l:=union(l,nil)$ if solvealg_ then begin p:=nil$ for each q in l do p:=union(solvesplit(q,ftem),p)$ l:=p$ end$ return if length l>1 then l else nil end$ symbolic procedure simplifyde(a,ftem,vl,genflag)$ begin scalar l,f,p$ ftem:=smemberl(ftem,a)$ if not contradiction(a,ftem) and (f:=fctchoose(list a,ftem)) then <<p:=reval algsimple(reval a,cons(f,delete(f,ftem)),vl)$ if not p or zerop p then <<write "***** Error in ALGSIMPLE : "$ mathprint a>> else a:=p$ if not stardep(a,ftem,vl) then if (l:=integratede(a,ftem,genflag)) and print_ then if null cdr l then <<terpri()$write "integrated equation : "$deprint(l)>> else <<terpri()$write "generalized integration yields ", length l," equations : "$deprint(l)>> >>$ return if l then l else list a end$ symbolic procedure fctsubst(ex,fo,forg)$ % substitution of a function fo in the list forg by an expression ex <<%if not member(fo,forg) then depl!*:=delete(assoc(fo,depl!*),depl!*)$ for each f in forg collect if f=fo then list('equal,fo,ex) else if my_freeof(f,fo) then f else reval subst(ex,fo,f)>>$ symbolic procedure substandsep(g,sorg,ftem,vlist)$ % in allen Gl. aus sorg wird die Fkt. car g durch cdr g ersetzt begin scalar l,p,q,a$ while sorg do <<p:=car sorg$ sorg:=cdr sorg$ if my_freeof(p,cdr g) then l:=union(list p,l) else <<q:=reval subst(car g,cdr g,p)$ contradiction(q,ftem)$ if (ineq_:=ineqsubst(g,ftem)) then if pairp q then <<a:=gensep(q,smemberl(union(fnew_,ftem),q),vlist)$ if contradiction_ then l:=sorg:=nil else l:=union(list car a,l)>> else if q neq 0 then l:=union(list q,l)>> >>$ return l end$ symbolic procedure fcteval(p,ftem,vl)$ begin if pairp p and (car p='quotient) then p:=cadr p$ return if delength p>(if fcteval_ then fcteval_ else 0) then (if cont_ then if yesp("function to be evaluated ? ") then fcteval1(p,ftem,vl)) else fcteval1(p,ftem,vl) end$ symbolic procedure fcteval1(p,ftem,vl)$ begin scalar l,f,n,ft$ ft:=smemberl(ftem,p)$ n:=length varslist(p,ft,vl)$ for each x in ft do if fctlength x>=n then <<l:=ldiffp(p,x)$ if (null car l) and (cdr l=1) then if freeofzero(coeffn(p,x,1),delete(x,ft),vl) then f:=x>>$ if f then p:=reval list('difference,f,list('quotient,p,coeffn(p,f,1)))$ return if f then cons(p,f) else nil$ end$ symbolic procedure freeofzero(p,ft,vl)$ % gets p (factorized), if p not vanishes identically if null ft then p else begin scalar a,b,fr,pri$ pri:=print_$ print_:=nil$ % terpri()$write "FREEOFZERO called with ineq_= ",ineq_$ for each s in cdr reval list('factorize,p) do a:=union(list simplifyterm(s,ft),a)$ % terpri()$write "a= ",a$ if length a>1 then p:=cons('times,a)$ while a do if null smemberl(ft,car a) or member(car a,ineq_) then a:=cdr a else if pairp cdr (b:=separate(car a,ft,vl)) then <<fr:=nil$ while b do if freeofzero(car b,ft,vl) then <<b:=nil$fr:=t>> else b:=cdr b$ if fr then a:=cdr a else <<a:=nil$p:=nil>> >> else <<a:=nil$p:=nil>>$ print_:=pri$ return p end$ symbolic procedure fctevalstrict(a,ftem,vl)$ % fcteval with additional condition % substituted expr. contains only functions of less arguments % than the function begin scalar l,n,ft$ l:=fcteval(a,ftem,vl)$ if l then <<ft:=smemberl(ftem,car l)$ n:=for each f in ft collect length fctargs f$ if not (reval cons('max,n)<length fctargs cdr l) then l:=nil>>$ return l$ end$ endmodule; module utilts; % Routines for finding leading derivatives and others. % Author: Andreas Brand % June 1990 symbolic procedure diffrel(p,q,v)$ % liefert komplizierteren Differentialausdruck$ if diffrelp(p,q,v) then q else p$ symbolic procedure diffrelp(p,q,v)$ % liefert t, falls p einfacherer Differentialausdruck, sonst nil % p, q Paare (liste.power), v Liste der Variablen % liste Liste aus Var. und Ordn. der Ableit. in Diff.ausdr., % power Potenz des Differentialausdrucks if cdr p='infinity then nil else if cdr q='infinity then t else diffrelp1(p,q,v)$ symbolic procedure diffrelp1(p,q,v)$ if null v then % alle Variable untersucht ? if cdr p>cdr q then nil else t else begin scalar a,b$ a:=diffdeg(car p, car v)$ % Ordnung der Ableitung nach b:=diffdeg(car q,car v)$ % der ersten Variablen return if a<b then t else if b<a then nil else diffrelp1(p,q,cdr v) % falls Ableitungen % erste Variable gleich, dann % restliche Variablen end$ symbolic procedure diffdeg(p,v)$ % liefert Ordnung der Ableitung von p nach v$ % p Liste Varible+Ordnung der Ableitung, v Variable (Atom) if null p then 0 % alle Variable bearbeitet ? else if car p=v then % v naechste Variable ? if cdr p then if numberp(cadr p) then cadr p % folgt eine Zahl ? else 1 else 1 else diffdeg(cdr p,v)$ % weitersuchen symbolic procedure ldiff(p,f)$ % Suchen der fuehrenden Ableitung der Fkt. f(arg1.,...,argn) in p % p Ausdruck in Listenform,f Funktion in listenform % Ergebnis: (p,(liste.power))$ liste: Liste der Ordn. der Ableitungen begin scalar l$ l:=ldiffp(p,f)$ % fuerende Liste mit Potenz return cons(ldiff1(car l,fctargs f),cdr l)$ % aus Liste Variablen + Ordnung % wird Liste der Ordnungen end$ symbolic procedure highdiff(p,f,x)$ % hoechste Ableitung einer Funktion f nach der Variable x % in einem Ausdruck p % p,f in LISP-Notation, x Atom diffdeg(car ldiffp1(p,f,list x),x)$ symbolic operator fargs$ symbolic procedure fargs f$ cons('list,fctargs f)$ symbolic procedure fctargs f$ % arguments of a function %if pairp f then cdr f$ if (f:=assoc(f,depl!*)) then cdr f$ symbolic procedure fctlength f$ % number of arguments length fctargs f$ symbolic procedure argset(ftem)$ % List of arguments of all functions in ftem if ftem then union(reverse fctargs car ftem,argset(cdr ftem)) else nil$ symbolic procedure alldfs(p,ftem)$ if not pairp p then smemberl(ftem,p) else if (car p='df) and member(cadr p,ftem) then list p else union(alldfs(car p,ftem),alldfs(cdr p,ftem))$ symbolic procedure ldiff1(l,v)$ % liefert Liste der Ordnungen der Ableitungen nach den Variablen aus v % l Liste (Variable + Ordnung)$ v Liste der Variablen if null v then nil % alle Variable abgearbeitet ? else cons(diffdeg(l,car v),ldiff1(l,cdr v))$ % Ordnung der Ableitung nach % erster Variable anhaengen symbolic procedure ldiffp(p,f)$ % liefert Liste der Variablen + Ordnungen mit Potenz % p Ausdruck in LISP - Notation, f Funktion ldiffp1(p,f,fctargs f)$ symbolic procedure ldiffp1(p,f,vl)$ % liefert Liste der Variablen + Ordnungen mit Potenz % p Ausdruck in LISP - Notation, f Funktion, lv Variablenliste begin scalar a$ if p=f then a:=cons(nil,1) else <<a:=cons(nil,0)$ if pairp p then if member(car p,'(plus times quotient equal)) then <<p:=cdr p$ while p do <<a:=diffrel(ldiffp1(car p,f,vl),a,vl)$ if cdr a='infinity then p:=nil else p:=cdr p >> >> else if car p='minus then a:=ldiffp1(cadr p,f,vl) else if car p='expt then <<a:=ldiffp1(cadr p,f,vl)$ % fuehrende Abl. aus der Basis if numberp cdr a then if numberp caddr p then a:=cons(car a,times(caddr p,cdr a)) else if not zerop cdr a then a:=cons(nil,'infinity) else if not my_freeof(caddr p,f) then a:=cons(nil,'infinity) >> else if car p='df then if cadr p=f then a:=cons(cddr p,1) % f wird differenziert? else if my_freeof(cadr p,f) then a:=cons(nil,0) % sonst Konstante bzgl. f else a:=cons(nil,'infinity) else if my_freeof(p,f) then a:=cons(nil,0) else a:=cons(nil,'infinity) >>$ return a end$ symbolic procedure lderiv(p,f,vl)$ % fuehrende Ableitung in LISP-Notation ohne Potenz %lderiv1(p,f,fctargs f)$ lderiv1(p,f,vl)$ symbolic procedure lderiv1(p,f,vl)$ % fuerende Ableitung in LISP-Notation ohne Potenz % mit Angabe der Rangfolge der Variablen begin scalar l$ l:=ldiffp1(p,f,vl)$ return if car l then cons('df,cons(f,car l)) else if zerop cdr l then nil else f end$ symbolic procedure lderiv2(p,f,vl)$ % fuehrende Ableitung in LISP-Notation mit Potenz (als dotted pair) begin scalar l$ l:=ldiffp1(p,f,vl)$ return cons(if car l then cons('df,cons(f,car l)) else if zerop cdr l then nil else f ,cdr l) end$ symbolic procedure minausd(p,q)$ % minimalen Differentialausdruck suchen$ % p,q, Ausdr. in Listenform mit fuehrender Ableitung + Potenz % zur Berechnung wird nur fuehrender Ableitung + Potenz benutzt if minausdp1(cadr p,cddr p,cadr q,cddr q) then p else q$ symbolic procedure minausdsp(p,q)$ % falls p minimaler Diff.ausdr. und p nicht laenger als q, p sonst nil % p,q, Ausdr. in Listenform mit fuerender Ableitung und Potenz % benutzt wird nur fuerende Ableitung und Potenz begin scalar s$ s:=minausdp1(cadr p,cddr p,cadr q,cddr q)$ return if s then if s='equal!= then % falls die Ausdruecke gleich % schwierig sind, if shorter(car p,car q) then p else nil % geht auch die Laenge ein else p else nil end$ symbolic procedure minausdp(p,q)$ % falls p minimaler Differentialausdruck, p sonst nil % p,q, Ausdr. in Listenform mit fuehrender Ableitung und Potenz % benutzt wird nur fuerende Ableitung und Potenz if minausdp1(cadr p,cddr p,cadr q,cddr q) then p else nil$ symbolic procedure minausdp1(l1,p1,l2,p2)$ % liefert t, wenn der erste von zwei Differentialausdr. echt einfacher, % 'EQUAL!=, wenn sie gleich schwierig sind, sonst nil % l1,l2 Listen der Ordnungen der Ableit., p1,p2 Potenzen if null l1 then % Ordnungen gleich ? if p1=p2 then 'equal!= % Potenzen gleich ? else p1<p2 else if null l2 then nil % darf eigentlich nicht sein else if car l1<car l2 then t % Ordn. der 1. Var. kleiner ? else if car l1>car l2 then nil % Ordn. der 1. Var. groeaer ? else minausdp1(cdr l1,p1,cdr l2,p2)$ % restliche Variable testen symbolic procedure difdiff(r,s)$ % liefert Liste der Differenzen der Elemente zweier Listen if null r then nil else cons (car r-car s,difdiff(cdr r,cdr s))$ symbolic procedure mkldiff(f,l)$ % erzeugt die Ableitung der Funktion f nach % den Variablen (einschliesslich Ordnungen) aus der Liste l cons('df,cons(f,cr_difflist(fctargs f,l)))$ symbolic procedure cr_difflist(v,l)$ % erzeugt aus der Liste der Variablen und der Liste der Ordnungen der % Ableitungen eine Liste von Variablen + Ordnungen if null v then nil else if car l=0 then cr_difflist(cdr v,cdr l) % Ordnung 0 else if car l=1 then cons(car v,cr_difflist(cdr v,cdr l)) % bei Ordnung 1 wird Variable % ohne Ordnung angehaengt else cons(car v,cons(car l,cr_difflist(cdr v,cdr l)))$ symbolic procedure delength(d)$ % Laenge eines Polynoms in LISP - Notation if not pairp d then if d then 1 else 0 else if car d='plus then length(d)-1 % Laenge in LISP ohne PLUS else if car d='equal then delength(cadr d)+delength(caddr d) else if car d='quotient then delength(cadr d) else 1$ symbolic procedure shorter(a,b)$ delength a<=delength b$ symbolic procedure desort(l)$ % list sorting begin scalar l1,l2,l3,m,n$ return if null l then nil else <<n:=delength car l$ l2:=list car l$ l:=cdr l$ while l do <<m:=delength car l$ if m<n then l1:=cons(car l,l1) else if m>n then l3:=cons(car l,l3) else l2:=cons(car l,l2)$ l:=cdr l>>$ append(desort(l1),append(l2,desort(l3)))>> end$ symbolic procedure smemberl(fl,ex)$ if fl and ex then if smember(car fl,ex) then cons(car fl,smemberl(cdr fl,ex)) else smemberl(cdr fl,ex)$ symbolic procedure fctsort(l)$ % list sorting begin scalar l1,l2,l3,m,n$ return if null l then nil else <<n:=fctlength car l$ l2:=list car l$ l:=cdr l$ while l do <<m:=fctlength car l$ if m<n then l1:=cons(car l,l1) else if m>n then l3:=cons(car l,l3) else l2:=cons(car l,l2)$ l:=cdr l>>$ append(fctsort(l1),append(l2,fctsort(l3)))>> end$ symbolic procedure fctprint(fl)$ % Ausdrucken der Funktionen aus fl begin scalar n,l$ n:=0$ for each f in fl do if pairp f then if car f='equal then <<n:=delength reval f$ if n>print_ then <<terpri()$write cadr f,"= expr. with ",n," terms"$terpri()>> else mathprint f$ n:=0>> else <<if n eq 5 then <<terpri()$n:=0>>$ write car f$ if pairp cdr f then <<write "("$write cadr f$ for each v in cddr f do <<write ","$write v>>$ write ") ">> else <<write car f$ write " ">>$ n:=add1 n>> else <<if n eq 5 then <<terpri()$n:=0>>$ write f$ if (l:=fctargs f) then <<write "("$write car l$ for each v in cdr l do write ",",v$ write ")">>$ write " "$ n:=add1 n>>$ %if n neq 0 then terpri() end$ symbolic procedure eqprint(e)$ % Ausdrucken der Gl. e if print_ then begin scalar n$ n:=delength reval e$ if n>print_ then <<write "expr. with "$write n$write " terms"$terpri()>> else mathprint e$ end$ symbolic procedure substprint(a,b)$ if print_ then begin scalar n$ n:=delength reval b$ if n>print_ then <<fctprint list a$write "= expr. with ",n," terms"$ terpri()>> else mathprint list('equal,a,b) end$ symbolic procedure deprint(l)$ % Ausdrucken der Gl. aus der Liste l if l and print_ then for each x in l do eqprint(list('equal,0,x))$ symbolic procedure powappend(l,f)$ % Anhaengen der fuehrenden Ableitung if null l then nil else cons(cons(car l,ldiff(car l,f)),powappend(cdr l,f))$ symbolic procedure powdelete(l)$ % Loeschen der fuehrenden Ableitung if null l then nil else cons(caar l,powdelete(cdr l))$ symbolic procedure fctdepend(e,v,fctset)$ % Test,ob variable v im Ausdruck e nicht in Abhaeng. von % Fkt. aus fctset vorkommt if member(v,argset smemberl(fctset,e)) then t else my_freeof(e,v) $ symbolic procedure subset(a,b)$ % test,ob a Teilmenge von b ist if null a then t else if member(car a,b) then subset(cdr a,b) else nil$ symbolic procedure idlist(l)$ if pairp l then union(idlist car l,idlist cdr l) else if l and idp l then list l else nil$ symbolic procedure varapp(l,a,v1,v2)$ % an jede Gl. aus l werden v1 und v2 angehaengt if null l then nil else cons(list(cons(cons(caar l,a),cdar l),v1,v2),varapp(cdr l,a,v1,v2))$ symbolic procedure varslist(p,ftem,vl)$ begin scalar l$ ftem:=argset smemberl(ftem,p)$ for each v in vl do if not my_freeof(p,v) or member(v,ftem) then l:=cons(v,l)$ return reverse l$ end$ symbolic procedure newfct(id,l,nfct)$ begin scalar f$ f:=mkid(id,nfct)$ depl!*:=delete(assoc(f,depl!*),depl!*)$ %put(f,'simpfn,'simpiden)$ %if pairp l then f:=cons(f,l)$ if pairp l then depl!*:=cons(cons(f,l),depl!*)$ if print_ then <<terpri()$ if pairp l then <<write "new function: "$ fctprint list f>> else write "new constant: ",f>>$ return f$ end$ symbolic procedure equallist(forg)$ % List of all members of forg which are pairs if forg then if pairp car forg then cons(car forg,equallist cdr forg) else equallist cdr forg$ symbolic procedure spmin(a,b)$ if null a then b else if null b then a else if cadr a<cadr b then a else b$ symbolic procedure specons(a,l)$ if my_freeof(l,car a) then cons(car a,cons(cadr a,l)) else specons1(a,l)$ symbolic procedure specons1(a,l)$ if null l then nil else if car l=car a then if cadr l<cadr a then cons(car a,cons(cadr a,cddr l)) else l else cons(car l,cons(cadr l,specons1(a,cddr l)))$ symbolic procedure polyp(p,ftem)$ begin scalar a$ a:=t$ while ftem and a do if polyp1(p,car ftem) then ftem:=cdr ftem else a:=nil$ return a end$ symbolic procedure polynomtest(p,ftem)$ begin scalar a$ a:=t$ while ftem do if polyp1(p,car ftem) then ftem:=cdr ftem else <<terpri()$ write "***** This equation is not a polynomial w.r.t. "$ write car ftem$write " and its derivatives : "$ deprint(list p)$ ftem:=nil$ a:=nil>>$ return a end$ symbolic procedure polyp1(p,f)$ % prueft, ob p f nur in polynomialer Form enthaelt if my_freeof(p,f) then t else begin scalar a$ if atom p then a:=t else if member(car p,list('expt,'plus,'minus,'times,'quotient,'df)) then % erlaubte Funktionen <<if (car p='plus) or (car p='times) then <<p:=cdr p$ while p do if a:=polyp1(car p,f) then p:=cdr p else p:=nil>> else if (car p='minus) then a:=polyp1(cadr p,f) else if (car p='quotient) then <<if my_freeof(caddr p,f) then a:=polyp1(cadr p,f)>> else if car p='expt then % Exponent <<if (fixp caddr p) then if caddr p>0 then a:=polyp1(cadr p,f)>> else if car p='df then % Ableitung if (cadr p=f) or my_freeof(cadr p,f) then a:=t>> else a:=(p=f)$ return a end$ symbolic operator rationalexp$ symbolic procedure rationalexp(p,ftem)$ ratexp(p,cdr ftem)$ lisp flag('(rationalexp),'boolean)$ symbolic procedure rationaltest(p,ftem)$ begin scalar a$ a:=t$ while ftem do if ratexp1(p,car ftem) then ftem:=cdr ftem else <<terpri()$ write "***** This equation is not a rational expr. w.r.t. "$ write car ftem$write " and its derivatives : "$ deprint(list p)$ ftem:=nil$ a:=nil>>$ return a end$ symbolic procedure ratexp(p,ftem)$ if null ftem then t else if ratexp1(p,car ftem) then ratexp(p,cdr ftem) else nil$ symbolic procedure ratexp1(p,f)$ % prueft, ob p f nur in rationaler Form enthaelt if not pairp p or my_freeof(p,f) then t else if car p='equal then ratexp1(cadr p,f) and ratexp1(caddr p,f) else if car p='quotient then polyp1(cadr p,f) and polyp1(caddr p,f) else polyp1(p,f)$ symbolic operator my_freeof$ symbolic procedure my_freeof(u,v)$ not(smember(v,u)) and freeofdepl(depl!*,u,v)$ lisp flag('(my_freeof),'boolean)$ symbolic procedure freeofdepl(de,u,v)$ if null de then t else if smember(v,cdar de) and smember(caar de,u) then nil else freeofdepl(cdr de,u,v)$ symbolic procedure freeoflist(l,m)$ % liefert t, falls kein Element aus m in l auftritt if null m then t else if my_freeof(l,car m) then freeoflist(l,cdr m) else nil$ symbolic procedure splitinhom(q,ftem,vl)$ % Splitting the equation q into the homogeneous and inhom. part % returns dotted pair qhom . qinhom begin scalar qhom,qinhom,denm; vl:=varslist(q,ftem,vl)$ if pairp q and (car q = 'quotient) then if starp(smemberl(ftem,caddr q),vl) then <<denm:=caddr q; q:=cadr q>> else return (q . 0) else denm:=1; if pairp q and (car q = 'plus) then q:=cdr q else q:=list q; while q do << if starp(smemberl(ftem,car q),vl) then qinhom:=cons(car q,qinhom) else qhom :=cons(car q,qhom); q:=cdr q >>; if null qinhom then qinhom:=0 else if length qinhom > 1 then qinhom:=cons('plus,qinhom) else qinhom:=car qinhom; if null qhom then qhom:=0 else if length qhom > 1 then qhom:=cons('plus,qhom) else qhom:=car qhom; if denm neq 1 then <<qhom :=list('quotient, qhom,denm); qinhom:=list('quotient,qinhom,denm)>>; return qhom . qinhom end$ symbolic procedure search_den(l)$ % get all denominators anywhere in a list l if pairp l then if car l='quotient then union(cddr l,search_den(cdr l)) else if member(car l,'(log ln logb log10)) then union(cdr l,search_den(cdr l)) else union(search_den(car l),search_den(cdr l))$ symbolic procedure zero_den(l,ftem,vl)$ begin scalar l1$ for each s in search_den(l) do if not freeofzero(s,ftem,vl) then l1:=cons(s,l1)$ return l1 end$ endmodule; module simpsols; % Routines for simplifying expressions by changing % free functions and constants of integration. % Author: Thomas Wolf % Nov 1993 fluid '(factorize_ nfct_ print_); symbolic operator dropredundant$ symbolic procedure dropredundant(ex,fl,vl)$ Comment All arguments are algebraic, ex is the list of expressions or equations from which the right side is taken, fl is the list of functions to be sorted out, vl the list of all extra independent variables, not already included in fl. returns algebraic list of redundant functions/const.=0, new EX, new FL; begin scalar a; vl:=union(reverse argset cdr fl,cdr vl)$ a:=dropredund(list(list(nil),cdr ex,cdr fl),vl); return if a then list('list,cons('list,car a), cons('list,caddr a), cons('list,cadddr a) ) else nil end$ symbolic procedure dropredund(a,vl); begin scalar sol,arbit,fl,el1,el2,corres,b,condi,oldcon,printold,redund, flstart,fldrop$ % a has the structure of one solution of CRACK in symbolic mode, % makes a copy of the free constants and functions in the solution % sets to zero the difference between the old solution and the new % with replaced constants and functions and finds the non-essential % returns cons(list of redundant fnc./cons=0,new solution) if cadr a then if length cadr a > 0 then if caddr a then if length caddr a > 0 then << printold:=print_; print_:=nil; sol:=for each el1 in cadr a collect if pairp el1 then if car el1 = 'equal then caddr el1 else el1 else el1; b:=sol; arbit:=caddr a; for each el1 in arbit do if (not my_freeof(b,el1)) or (not my_freeof(car a,el1)) then << flstart:=cons(el1,flstart); el2:=newfct(fname_,fctargs(el1),nfct_)$ nfct_:=add1 nfct_$ sol:=subst(el2,el1,sol); fl:=cons(el1,cons(el2,fl)); corres:=cons((el1 . el2),corres); >> else fldrop:=cons(el1,fldrop); while b do << condi:=cons(reval list('plus,car b,list('minus,car sol)),condi); b:=cdr b; sol:=cdr sol >>; b:=car crack1(condi,nil,nil,fl,vl,factorize_,nil,nil); print_:=printold; if not car b then << arbit:=caddr b; for each el1 in cadr b do if not((pairp el1) and (car el1 = 'equal)) then arbit:=cons(el1,arbit)$ oldcon:=car a; sol:=cadr a; % the remaining essential free functions : flstart for each el1 in corres do if member(car el1,arbit) and member(cdr el1,arbit) then << redund:=cons(list('equal,car el1,0),redund); fldrop:=cons(car el1,fldrop); % the function and its copy are both not essential oldcon:=for each el2 in oldcon collect reval subst(0,car el1,el2); sol:=for each el2 in sol collect reval subst(0,car el1,el2); flstart:=delete(car el1,flstart) >>; >>; if fldrop and printold then << write"non-essential dropped constant(s) or function(s): "; fctprint fldrop >>$ return if null fldrop then nil else list(redund,oldcon,sol,flstart) >> end$ algebraic procedure absorbconst(exlist,flist)$ % absorbing numerical factors into free constants/functions of flist % if the list of expressions in exlist is known to be linear in flist % returns an algebraic list of substitutions to be done begin scalar e1,e2,n,n1,n2,sb; sb:={}; for each e1 in flist do << n1:=nil; for each e2 in exlist do << n:=coeffn(e2,e1,1); if n neq 0 then << n:=ncontent(n); if n1=nil then <<n1:=num n; n2:=den n>> else << n1:=gcd(n1,num(n)); n2:=n2*den(n)/gcd(n2,den(n)) >> >> >>; if n1 and ((n1 neq 1) or (n2 neq 1)) then sb:=cons(e1=e1*n2/n1 , sb) >>; return if sb={} then nil else sb end$ % of absorbconst algebraic procedure drop_const(oldsoln, varlist, additive)$ Comment oldsoln is the output of a CRACK call. In all solutions functions which are independent of all elements of varlist are dropped from the list of free functions/constants and - set to zero if additive=t and they are additive or - set to 1 if additive=nil and they are multiplicative; begin scalar soln, sl, fncn, h1, h2, newfl, vcopy, constnt, v, fcopy,f1,co, mcdold; soln := {}; mcdold:=lisp !*mcd$ off mcd; while oldsoln neq {} do << sl := first oldsoln; oldsoln := rest oldsoln; fncn := second sl; h1 := third sl; newfl:={}; for each h2 in h1 do << % is h2 constant ? vcopy := varlist; constnt := t; while constnt and (vcopy neq {}) do << v := first vcopy; vcopy := rest vcopy; if not my_freeof(co,v) then constnt := nil >>; if constnt then if (not my_freeof(first sl, h2)) or my_freeof(fncn, h2) then constnt := nil; if constnt then << % is the coefficient of h2 constant in all solved expressions % and occurs h2 only linear ? fcopy := fncn; while constnt and (fcopy neq {}) do << f1 := rhs first fcopy; fcopy := rest fcopy; on mcd; co:=coeffn(f1,h2,1); if (not my_freeof(co,h2)) or ( additive and (not my_freeof(f1 - co*h2, h2))) or ((not additive) and ((f1 - co*h2) neq 0)) then constnt := nil; off mcd; if constnt and additive then << vcopy := varlist; while constnt and (vcopy neq {}) do << v := first vcopy; vcopy := rest vcopy; if not my_freeof(co,v) then constnt := nil >> >> >> >>; if constnt then if additive then fncn := sub(h2=0, fncn) else fncn := sub(h2=1, fncn) else newfl := cons(h2, newfl) >>; soln := cons({first sl, fncn, newfl}, soln) >>; if mcdold then on mcd; return soln end$ % of drop_const endmodule; module intfix; % Further fixes to the integration package. load_package int$ % die folgende Aenderung verhindert das Erzeugen von int* ... remd('simpint!*)$ symbolic procedure simpint!* u$ begin scalar x$ return if (x := opmtch('int . u)) then simp x else simpiden('int . u) % statt else simpiden('int!* . u) end$ setcrackflags()$ !*fullroots:=t$ % ein Patch fuer das REDUCE 3.5 EZGCD %symbolic procedure simpexpt u$ % % We suppress reordering during exponent evaluation, otherwise % % internal parts (as in e^(a*b)) can have wrong order. % begin scalar expon; % expon := simpexpon carx(cdr u,'expt) where kord!*=nil; % expon := resimp expon; % We still need right order. <--- change. % return simpexpt1(car u,expon,nil) % end$ % Zum Integrieren % put('int, 'simpfn, 'SimpIntPatch)$ algebraic << let { int(1/~x^(~n),~x) => -x/(x^n*(n-1)) when numberp n, ~x^(~m/~n)*~x => x**((m+n)/n) when (numberp n and numberp m), int(~z/~y,~x) => log(y) when z = df(y,x)}$ if sin(!%x)**2+cos(!%x)**2 neq 1 then let {sin(~x)**2 => 1-cos(x)**2}$ if cosh(!%x)**2 neq (sinh(!%x)**2 + 1) then let {cosh(~x)**2 => (sinh(x)**2 + 1)}$ if sin(!%x)*tan(!%x/2)+cos(!%x) neq 1 then let {tan(!%x/2) => (1-cos(!%x))/sin(!%x)}$ if sin(!%x)*cot(!%x/2)-cos(!%x) neq 1 then let {cot(!%x/2) => (1+cos(!%x))/sin(!%x)}$ if sqrt(!%x**2-!%y**2)-sqrt(!%x-!%y)*sqrt(!%x+!%y) neq 0 then let {sqrt(~x)*sqrt(~y) => sqrt(x*y)} >>$ endmodule; module dfint; % Patch to improve differentiation, mainly of integrals. % This version specifically for use by the crack package. % Author: Francis J. Wright <F.J.Wright@QMW.ac.uk> % QMW, London and CBPF, Rio. % Date: Thu Nov 19 21:11:12 1992 % Revised 12 April 1995 for REDUCE 3.6 switch allowdfint, dfint; deflist('((allowdfint ((t (rmsubs)))) (dfint ((t (rmsubs))))), 'simpfg); !*allowdfint := !*dfint := t; rmsubs(); put('int, 'dfform, 'dfform_int); symbolic procedure dfform_int(u, v, n); % Simplify a SINGLE derivative of an integral. % u = '(int y x) [as main variable of SQ form] % v = kernel % n = integer power % Return SQ form of df(u**n, v) = n*u**(n-1)*df(u, v) % This routine is called by diffp via the hook % if x := get(car u,'dfform) then return apply3(x,u,v,n) begin scalar result, x, y; y := simp!* cadr u; % SQ form integrand x := caddr u; % kernel result := if v eq x then y % df(int(y,x), x) -> y replacing the let rule in INT.RED else if (!*allowdfint or !*dfint) and << y := diffsq(y, v); !*dfint or not_df_p y % it has simplified >> then simp{'int, mk!*sq y, x} % MUST re-simplify it!!! % i.e. differentiate under the integral sign % df(int(y, x), v) -> int(df(y, v), x). % (Perhaps I should use prepsq - kernels are normally % true prefix?) else !*kk2q{'df, u, v}; % remain unchanged if not(n eq 1) then result := multsq( (((u .** (n-1)) .* n) .+ nil) ./ 1, result); return result end; symbolic procedure not_df_p y; % True if the SQ form y is not a df kernel. not(denr y eq 1 and not domainp (y := numr y) and eqcar(mvar y, 'df)); % The remainder of this patch is included in the procedure diffp in % module diff in the poly package in REDUCE 3.6. But it currently % still needs to be explicitly turned on: fluid '(!*fjwflag); !*fjwflag := t; endmodule; end;