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module solvealg; % Solution of equations and systems which can % be lifted to algebraic (polynomial) systems. % Author: Herbert Melenk. % Copyright (c) 1992 The RAND Corporation and Konrad-Zuse-Zentrum. % All rights reserved. % August 1992: added material for % rule set for reduction of trig. polynomial terms to % elementary expressions in sin and cos, % constant expressions in sin, cos and constant roots, % closed form results for trigonometric systems. % general exponentials. % avoiding false solutions with surds. % % May 1993: better handling of products of exponentials % with common base, % additional computation branch for linear parts of % nonlinear systems. % July 1996: safe handling of twice (or more) the same input % (not handling the case, that one equation is a multiple % of an other one) fluid '(!*expandexpt); % from simp.red fluid '( system!* % system to be solved osystem!* % original system on input uv!* % user supplied variables iv!* % internal variables fv!* % restricted variables kl!* % kernels to be investigated sub!* % global substitutions inv!* % global inverse substitutions depl!* % REDUCE dependency list !*solvealgp % true if using this module solvealgdb!* % collecting some data last!-vars!* % collection of innermost aux variables const!-vars!* % variables representing constants root!-vars!* % variables representing root expressions !*expli % local switch: explicit solution groebroots!* % predefined roots from input surds !*test_solvealg % debugging support !*arbvars !*varopt solve!-gensymcounter ); fluid '(!*trnonlnr); % If set on, the modified system and the Groebner result % or the reason for the failure are printed. global '(loaded!-packages!* !!arbint); switch trnonlnr; !*solvealgp := t; % Solvenonlnrsys receives a system of standard forms and % a list of variables from SOLVE. The system is lifted to % a polynomial system (if possible) in substituting the % non-atomic kernels by new variables and appending additonal % relations, e.g. % replace add % sin u,cos u -> su,cu su^2+cu^2-1 % u^(1/3) -> v v^3 - u % ... % in a recursive style. If completely successful, the % system definitely can be treated by Groebner or any % other polynomial system solver. % % Return value is a pair % (tag . res) % where "res" is nil or a structure for !*solvelist2solveeqlist % and "tag" is one of the following: % % T a satisfactory solution was generated, % % FAILED the algorithm cannot be applied (res=nil) % % INCONSISTENT the algorithm could prove that the % the system has no solution (res=nil) % % NIL the complexity of the system could % be reduced, but some (or all) relations % remain still implicit. % rules to be applied locally for converting composite transcendental % function forms into simpler ones algebraic << solvealg!-rules1:= { sin(~alpha + ~beta) => sin(alpha)*cos(beta) + cos(alpha)*sin(beta), cos(~alpha + ~beta) => cos(alpha)*cos(beta) - sin(alpha)*sin(beta), sin(~n*~alpha) => sin(alpha)*cos((n-1)*alpha) + cos(alpha)*sin((n-1)*alpha) when fixp n, cos(~n*~alpha) => cos(alpha)*cos((n-1)*alpha) - sin(alpha)*sin((n-1)*alpha) when fixp n, sin(~alpha)**2 => 1 - cos(alpha)**2, sinh(~alpha+~beta) => sinh(alpha)*cosh(beta) + cosh(alpha)*sinh(beta), cosh(~alpha+~beta) => cosh(alpha)*cosh(beta) + sinh(alpha)*sinh(beta), sinh(~n*~alpha) => sinh(alpha)*cosh((n-1)*alpha) + cosh(alpha)*sinh((n-1)*alpha) when fixp n, cosh(~n*~alpha) => cosh(alpha)*cosh((n-1)*alpha) + sinh(alpha)*sinh((n-1)*alpha) when fixp n, sinh(~alpha)**2 => cosh(alpha)**2 - 1}; solvealg!-rules2:= { tan(~alpha) => sin(alpha)/cos(alpha), cot(~alpha) => cos(alpha)/sin(alpha), tanh(~alpha) => sinh(alpha)/cosh(alpha), coth(~alpha) => cosh(alpha)/sinh(alpha) } ; solvealg!-rules3:= { sin(~alpha)**2 => 1 - cos(alpha)**2, sinh(~alpha)**2 => cosh(alpha)**2 - 1}; % Artificial operator for matching powers in a % product. operator my!-expt; solvealg!-rules4:= { my!-expt(~a,~b)*my!-expt(a,~c) => my!-expt(a,b+c), my!-expt(~a,~b)*a => my!-expt(a,b+1) % my!-expt(~a,~b)/my!-expt(a,~c) => my!-expt(a,b-c) }; >>; symbolic procedure solvenonlnrsys(sys,uv); % interface to algebraic system solver. % factorize the system and collect solutions. % After factoring we resimplify with *expandexpt off % in order to have exponentials to one basis % collected. begin scalar q,r,s,tag,!*expandexpt; s := sys; sys := nil; for each x in s do sys := union(sys,{x}); s := '(nil); if solve!-psysp(sys,uv) then s := {sys} else for each p in sys do <<r := nil; for each q in cdr fctrf p do if topkernlis(car q,uv) then for each u in s do r := (car q . u) . r; s := r>>; tag := 'failed; r := nil; for each u in s do <<% collect exponentials with same base. u := solvenonlnrcollectexpt u; q := solvenonlnrsys1(u,uv); if eqcar(q,'failed) then q := solvenonlnrsyssep(u,uv); if eqcar(q,'failed) then q := solvenonlnrsyslin(u,uv,nil); if eqcar(q,'not) then q := solvenonlnrsyslin(u,uv,t); if eqcar(q,'not) then q := '(failed); if car q and car q neq 'failed then tag := car q; q := if car q neq 'failed then cdr q else for each j in u collect {{j ./ 1},nil,1}; r := union(q,r)>>; return if tag eq 'inconsistent or tag eq 'failed then {tag} else tag . r end; symbolic procedure topkernlis(u,v); v and (topkern(u,car v) or topkernlis(u,cdr v)); symbolic procedure solvenonlnrcollectexpt u; % u is a list of standard forms. Reform these % such that products of exponentials with same basis % are collected. if not smemq('expt,u) then u else <<eval'(let0 '(solvealg!-rules4)); u:=for each q in u collect numr simp subst('expt,'my!-expt, reval prepf subst('my!-expt,'expt,q)); eval'(clearrules '(solvealg!-rules4)); u>>; symbolic procedure solvenonlnrsyslin(eqs,vars,mode); % Eqs is a system of equations (standard forms, % implicitly equated to zero); this routine tries % to reduce the system recursively by separation, % if one variable occurs in one equation only linearly. % Mode=NIL: simple version: only pure linear variables % are substituted. % T: extended version: replacing variables with % degree 1 and potentially complicated % coefficients. % Returns solution or % '(not) if not applicable % '(failed) if applicable but solution failed. begin scalar d,e,e1,n,s,q,x,v,w,w1,neqs,nvars; v:=vars; var_loop: if null v then return '(not); x:=car v; v:=cdr v; w:=eqs; eqn_loop: if null w then goto var_loop; e:=car w; w:=cdr w; if null e then goto eqn_loop; if domainp e then return '(inconsistent); e1:= reorder e where kord!*={x}; if not(mvar e1 =x) or ldeg e1>1 or smemq(x,d:=lc e1) or smemq(x,n:=red e1) then goto eqn_loop; if not mode then <<w:=nil; for each y in vars do w:=w or smemq(y,d); if w then return '(not); >>; % linear form found: n*x+d=0. This is basis for a solution % x=-n/d. In a second branch the case {n=0,d=0} has to % be considered if n and d are not constants. n := reorder n; d:=reorder d; % Step 1: substitute in remaining equations, solve % and add linear formula to result. s:= quotsq(negf n ./ 1, d ./ 1); neqs := for each eqn in delete(e,eqs) join <<q:=numr subf(eqn,{x.prepsq s}); if q then {q}>>; nvars:=for each y in delete(x,vars) join if smemq(y,neqs) then {y}; w:= if null neqs then '(t (nil nil 1)) else if null nvars then '(inconsistent) else if cdr neqs then solvenonlnrsys(neqs,nvars) else solvenonlnrsysone(car neqs,car nvars); if car w eq 'failed then return w; w:=add!-variable!-to!-tagged!-solutions(x,s,w); % Step 2: add an eventual solution for n=0,d=0; if domainp d or not mode then return w; w1:=solvenonlnrsys(n.d.eqs,vars); return merge!-two!-tagged!-solutions(w,w1); end; symbolic procedure solvenonlnrsysone(f,x); % equation system has been reduced to one. Using solvesq. begin scalar w; w:=solvesq(f ./ 1,x,1); if null w then return '(inconsistent) else if null cadr car w then return '(failed); % if not smemq('root_of,w) then goto ret; % % here we try to find out whether a root_of % % expression is a useful information or whether % % it is simply an echo of the input. % if cdr w then goto ret; % multiple branches: good. % q := prepsq caar car w; % if not eqcar(q,'root_of) % not on top level: good. % then goto ret; % q:=subst(x,caddr q,cadr q); % if f = numr simp q then return '(failed); %ret: return t.w; end; symbolic procedure add!-variable!-to!-tagged!-solutions(x,s,y); % Y is a tagged solution. Add equation x=s to all members. if eqcar(y,'inconsistent) then y else if null y or null cdr y then {t,{{s},{x},1}} else car y . for each q in cdr y collect % Put new solution into the last position. {append(car q,{s}),append(cadr q,{x}),caddr q}; symbolic procedure merge!-two!-tagged!-solutions(w1,w2); % w1 and w2 are tagged solution sets. Merge these and % eliminated inconsistent cases. if car w1='failed or car w2='failed then '(failed) else if car w1='inconsistent then w2 else if car w2='inconsistent then w1 else car w1 . append(cdr w1,cdr w2); symbolic procedure solvenonlnrsyssep(eqs,vars); % Eqs is a system of equations (standard forms, % implicitly equated to zero); this routine tries % to reduce the system recursively by separation, % if one variable occurs only in one equation. begin scalar y,r,s,r0,u,w,tag; if null vars then return '(failed) else if null cdr eqs then <<if not smember(car vars,car eqs) then return solvenonlnrsyssep(eqs,cdr vars); r:=solvesq(!*f2q car eqs,car vars,1); return if r and cadr car r then 't.r else '(failed); >>; for each x in vars do if null y then <<r:=nil; for each u in eqs do if smember(x,u) then r:=u.r; if r and null cdr r then y:=x; >>; if null y then return '(failed); r:=car r; s:=solvenonlnrsys(delete(r,eqs),delete(y,vars)); if car s='failed then return s else s:=cdr s; tag := t; u:=for each s0 in s join << w:=for each q in pair(cadr s0,car s0) join if not smemq('root_of,cdr q) then {car q.prepsq cdr q}; r0:=subf(r,w); r0:=solvesq(r0,y,caddr s0); if null r0 or null cadr car r0 then tag:='failed; for each r1 in r0 collect {caar r1. car s0,y.cadr s0,caddr r1} >>; return tag.u; end; symbolic procedure solve!-psysp(s,uv); % T if s is a pure polynomial system. null s or (solve!-psysp1(car s,uv) and solve!-psysp(cdr s,uv)); symbolic procedure solve!-psysp1(f,uv); domainp f or ((member(mvar f,uv) or solve!-psysp2(mvar f,uv)) and solve!-psysp1(lc f,uv) and solve!-psysp1(red f,uv)); symbolic procedure solve!-psysp2(v,uv); % t if there is no interaction between v and uv. null uv or (not smember(car uv,v) and solve!-psysp2(v,cdr uv)); symbolic procedure solvenonlnrsys1(system!*,uv!*); % solve one system. begin scalar r,rules; osystem!* := system!*; if solvealgtrig0 system!* then rules:='(solvealg!-rules1); if smemq('tan,system!*) or smemq('cot,system!*) or smemq('tanh,system!*) or smemq('coth,system!*) then rules:='solvealg!-rules2.rules; r := evalletsub2({rules,'(solvenonlnrsyspre)},nil); if errorp r then return '(failed) else system!* := car r; r := solvenonlnrsys2(); return r; end; symbolic procedure solvenonlnrsyspre(); (for each p in system!* collect numr simp prepf p) where dmode!* = nil; symbolic procedure solvenlnrsimp(u); % a prepsq including resimplification with additional rules. % begin scalar r; % r := evalletsub2({'(solvealg!-rules3), % {'simp!* ,mkquote u}},nil); % if errorp r then error(99,list("error during postprocessing simp")); % return car r; % end; simp!* u; symbolic procedure solvenonlnrsys2(); % Main driver. We need non-local exits here % because of possibly hidden non algebraic variable % dependencies. if null !*solvealgp then system!*:='(FAILED) else % against recursion. (begin scalar iv!*,kl!*,inv!*,fv!*,r,w,!*solvealgp,solvealgdb!*,sub!*; scalar last!-vars!*,groebroots!*,const!-vars!*,root!-vars!*; % preserving the variable sequence if *varopt is off if not !*varopt then depl!* := append(pair(uv!*,append(cdr uv!*,{gensym()})),depl!*); % hiding dmode because exponentials need integers. for each f in system!* do solvealgk0 (if dmode!* then numr subf(f,nil) where dmode!*=nil else f); if !*trnonlnr then print list("original kernels:",kl!*); if null cdr system!* then if (smemq('sin,system!*)or smemq('cos,system!*)) and (r:=solvenonlnrtansub(prepf(w:=car system!*),car uv!*)) and car r then return solvenonlnrtansolve(r,car uv!*,w) else if (smemq('sinh,system!*)or smemq('cosh,system!*)) and (r:=solvenonlnrtanhsub(prepf(w:=car system!*),car uv!*)) and car r then return solvenonlnrtanhsolve(r,car uv!*,w); if atom (errorset('(solvealgk1),!*trnonlnr,nil)) where dmode!*=nil then return (system!*:='(FAILED)); system!*:='LIST.for each p in system!* collect prepf p; if not('groebner memq loaded!-packages!*) then load!-package 'groebner; for each x in iv!* do if not member(x,last!-vars!*) then for each y in last!-vars!* do depend1(x,y,t); iv!* := sort(iv!*,function (lambda(a,b);depends(a,b))); if !*trnonlnr then << prin2t "Entering Groebner for system"; writepri(mkquote system!*,'only); writepri(mkquote('LIST.iv!*), 'only); >>; r := list(system!*,'LIST.iv!*); r := groesolveeval r; if !*trnonlnr then << prin2t "leaving Groebner with intermediate result"; writepri(mkquote r,'only); terpri(); terpri(); >>; if 'sin memq solvealgdb!* then r:=solvealgtrig2 r; if 'sinh memq solvealgdb!* then r:=solvealghyp2 r; r:= if r='(LIST) then '(INCONSISTENT) else solvealginv r; system!* := r; % set value aside return r; end) where depl!*=depl!* ; symbolic procedure solvealgk0(p); % Extract new top level kernels from form p. if domainp p then nil else <<if not member(mvar p,kl!*) and not member(mvar p,iv!*) then kl!*:=mvar p.kl!*; solvealgk0(lc p); solvealgk0(red p)>>; symbolic procedure solvealgk1(); % Process all kernels in kl!*. Note that kl!* might % change during processing. begin scalar k,kl0,kl1; k := car kl!*; while k do <<kl0 := k.kl0; solvealgk2(k); kl1 := kl!*; k:= nil; while kl1 and null k do if not member(car kl1,kl0) then k:=car kl1 else kl1:=cdr kl1; >>; end; symbolic procedure solvealgk2(k); % Process one kernel. (if member(k,uv!*) then solvealgvb0 k and (iv!*:= k.iv!*) else if atom k then t else if eq(car k,'EXPT) then solvealgexpt(k,x) else if memq(car k,'(sin cos tan cot)) then solvealgtrig(k,x) else if memq(car k,'(sinh cosh tanh coth)) then solvealghyp(k,x) else if null x then t else solvealggen(k,x) ) where x=solvealgtest(k); symbolic procedure solvealgtest(k); % Test if the arguments of a composite kernel interact with % the variables known so far. if atom k then nil else solvealgtest0(k); symbolic procedure solvealgtest0(k); % Test if kernel k interacts with the known variables. solvealgtest1(k,iv!*) or solvealgtest1(k,uv!*); symbolic procedure solvealgtest1(k,kl); % list of those kernels in list kl, which occur somewhere % in the composite kernel k. if null kl then nil else if member(k,kl) then list k else if atom k then nil else union(if smember(car kl,cdr k) then list car kl else nil, solvealgtest1(k,cdr kl)); symbolic procedure solvealgvb k; % Restricted variables are those which might establish % non-algebraic relations like e.g. x + e**x. Test k % and add it to the list. fv!* := append(solvealgvb0 k,fv!*); symbolic procedure solvealgvb0 k; % test for restricted variables. begin scalar ak; ak := allkernels(k,nil); if intersection(ak,iv!*) or intersection(ak,fv!*) then error(99,list("transcendental variable dependency from",k)); return ak; end; symbolic procedure allkernels(a,kl); % a is an algebraic expression. Extract all possible inner % kernels of a and collect them in kl. if numberp a then kl else if atom a or a member uv!* then if not member(a,kl) then a.kl else kl else <<for each x in cdr a do kl := allkernels1(numr s,allkernels1(denr s,kl)) where s=simp x; kl>>; symbolic procedure allkernels1(f,kl); if domainp f then kl else <<if not member(mvar f,kl) then kl:=allkernels(mvar f,mvar f . kl); allkernels1(lc f, allkernels1(red f,kl)) >>; symbolic procedure solvealgexpt(k,x); % kernel k is an exponential form. ( if eqcar(m,'quotient) and fixp caddr m then if cadr m=1 then solvealgrad(cadr k,caddr m,x) else solvealgradx(cadr k,cadr m,caddr m,x) else if null x then solvealgid k else if ((null intersection(w,uv!*) and null intersection(w,iv!*) and null intersection(w,fv!*)) where w=allkernels(m,nil)) then solvealggen(k,x) else solvealgexptgen(k,x) ) where m = caddr k; symbolic procedure solvealgexptgen(k,x); % Kernel k is a general exponentiation u ** v. begin scalar bas,xp,nv; bas := cadr k; xp := caddr k; if solvealgtest1(xp,uv!*) then return solvealgexptgen1(k,x) else if solvealgtest1(bas,uv!*) then return solvealggen(k,x); % remaining case: "constant" exponential expression to % replaced by an id for syntatical reasons nv := '( % old kernel ( (expt !&alpha n)) % new variable ( !&beta) % substitution ( ((expt !&alpha n) . !&beta) ) % inverse ( (!&beta (expt !&alpha n) !& )) % new equations nil ); nv:=subst(bas,'!&alpha,nv); nv:=subst(solve!-gensym(),'!&beta,nv); nv:=subst(xp,'n,nv); return solvealgupd(nv,nil); end; symbolic procedure solve!-gensym(); begin scalar w; w := explode solve!-gensymcounter; solve!-gensymcounter := solve!-gensymcounter+1; while length w < 4 do w := '!0 . w; % If users have things to solve with names like G0001 in them, there % could be confusion. return compress ('g . w) end; symbolic procedure solvealgexptgen1(k,x); % Kernel k is a general exponentiation u ** v. % where v is an expression in a solution variable, u % is constant. Transform all kernels with same basis % and compatible exponent to common exponent denominator % form. begin scalar bas,xp,xpl,q,r,nk,sub; bas := cadr k; xp := caddr k; % collect all exponentials with this basis. xpl:={(1 ./ 1).xp}; for each k in kl!* do if eqcar(k,'expt) and cadr k=bas and <<q:=simp{'quotient,r:=caddr k,xp}; fixp numr q and fixp denr q>> then <<kl!*:=delete(k,kl!*); xpl:=(q.r).xpl>>; % compute common denominator. q:=1; for each e in xpl do q:=lcm(q,denr car e); % the new artificial kernel. nk:=reval{'expt,bas,{'quotient,xp,q}}; sub := for each e in xpl collect {'expt,bas,cdr e}. {'expt,nk,numr car e * q/denr car e}; system!*:=sublis(sub,system!*); return solvealggen(nk,x); end; symbolic procedure solvealgradx(x,m,n,y); % error(99,"forms e**(x/2) not yet implemented"); solvealgexptgen1({'expt,x,{'quotient,m,n}},y); symbolic procedure solvealgrad(x,n,y); % k is a radical exponentiation expression x**1/n. begin scalar nv,m,!β !&beta := solve!-gensym(); nv:= '( % old kernel ( (expt !&alpha (quotient 1 !&n))) % new variable ( !&beta) % substitution ( ((expt !&alpha (quotient 1 !&n)) . !&beta) ) % inverse % ( (!&beta !&alpha (expt !& !&n)) ) nil % new equation ( (difference (expt !&beta !&n) !&alpha) ) ); m := list('!&alpha.x,'!&beta.!&beta,'!&n.n); nv := subla(m,nv); root!-vars!* := !&beta . root!-vars!*; % prepare roots for simple surds. if null y or y={x} then groebroots!* := ({'plus,{'expt,!&beta,n},reval{'minus,x}} .{{{'equal,!&beta,{'expt,x,{'quotient,1,n}}}}}).groebroots!*; if null y then last!-vars!* := !&beta . last!-vars!*; return solvealgupd(nv,y); end; symbolic procedure solvealgtrig0(f); % examine if sin/cos identies must be applied. begin scalar args,r,c; args :=for each a in solvealgtrig01(f,nil) collect (union(kernels numr q,kernels denr q) where q=simp a); while args do <<c:=car args;args:=cdr args; for each q in args do r:=r or intersection(c,q)>>; return r; end; symbolic procedure solvealgtrig01(f,args); if atom f then args else if memq(car f,'(sin cos tan cot sinh cosh tanh coth)) then if constant_exprp cadr f then args else union({cadr f},args) else solvealgtrig01(cdr f,solvealgtrig01(car f,args)); algebraic << operator p_sign,the_1; let p_sign(~x) => if sign(x)=0 then 1 else sign(x); let the_1(~x) =>1; >>; symbolic procedure solvealgtrig(k,x); % k is a trigonometric function call. begin scalar nv,m,s,!&alpha,!β solvealgdb!* := union('(sin),solvealgdb!*); if x then if cdr x then error(99,"too many variables in trig. function") else x := car x; solvealgvb k; nv := '( % old kernels ( (sin !&alpha)(cos !&alpha)(tan !&alpha)(cot !&alpha) ) % new variables ( (sin !&beta) (cos !&beta) ) % substitutions ( ((sin !&alpha) . (sin !&beta)) ((cos !&alpha) . (cos !&beta)) %%% these should be handled now by the ruleset. %%% ((tan !&alpha) . (quotient (sin !&beta) (cos !&beta))) %%% ((cot !&alpha) . (quotient (cos !&beta) (sin !&beta))) ) % inverses ( ((sin !&beta) (cond ((and !*expli (test_trig)) '(!&loc (p_sign (!&!& !&)))) (t '(!&x (!&!& (root_of (equal (sin !&alpha) !&) !&x)))))) ((cos !&beta) (cond ((and !*expli (test_trig)) '(!&x (plus (!&!& (times !&loc (acos !&))) (times 2 pi !&arb)))) (t '(!&x (!&!& (root_of (equal (cos !&alpha) !&) !&x)))))) ) % new equation ( (plus (expt (sin !&beta) 2)(expt (cos !&beta) 2) -1) ) ); % invert the inner expression. s := if x then solvealginner(cadr k,x) else 'the_1; !&beta := solve!-gensym(); m := list('!&alpha . (!&alpha:=cadr k), '!&beta . !&beta, '!&loc . solve!-gensym(), '!&arb . {'arbint,!!arbint:=!!arbint+1}, '!&x . x, '!&!& . s); nv:=sublis!-pat(m , nv); if x then last!-vars!*:= append(last!-vars!*,{{'sin,!&beta},{'cos,!&beta}}) else const!-vars!* := append(const!-vars!*,{{'sin,!&beta}.{'sin,!&alpha}, {'cos,!&beta}.{'cos,!&alpha}}); return solvealgupd(nv,nil); end; symbolic procedure solvealghyp(k,x); % k is a hyperbolic function call. begin scalar nv,m,s,!&alpha,!β solvealgdb!* := union('(sinh),solvealgdb!*); if x then if cdr x then error(99,"too many variables in hyp. function") else x := car x; solvealgvb k; nv := '( % old kernels ( (sinh !&alpha)(cosh !&alpha)(tanh !&alpha)(coth !&alpha) ) % new variables ( (sinh !&beta) (cosh !&beta) ) % substitutions ( ((sinh !&alpha) . (sinh !&beta)) ((cosh !&alpha) . (cosh !&beta)) ) % inverses ( ((sinh !&beta) (cond ((and !*expli (test_hyp)) '(!&loc (p_sign (!&!& !&)))) (t '(!&x (!&!& (root_of (equal (sinh !&alpha) !&) !&x)))))) ((cosh !&beta) (cond ((and !*expli (test_hyp)) '(!&x (plus (!&!& (times !&loc (acosh !&))) (times 2 pi i !&arb)))) (t '(!&x (!&!& (root_of (equal (cosh !&alpha) !&) !&x)))))) ) % new equation ( (plus (minus (expt (sinh !&beta) 2))(expt (cosh !&beta) 2) -1) ) ); % invert the inner expression. s := if x then solvealginner(cadr k,x) else 'the_1; !&beta := solve!-gensym(); m := list('!&alpha . (!&alpha:=cadr k), '!&beta . !&beta, '!&loc . solve!-gensym(), '!&arb . {'arbint,!!arbint:=!!arbint+1}, '!&x . x, '!&!& . s); nv:=sublis!-pat(m , nv); if x then last!-vars!*:= append(last!-vars!*,{{'sinh,!&beta},{'cosh,!&beta}}) else const!-vars!* := append(const!-vars!*,{{'sinh,!&beta}.{'sinh,!&alpha}, {'cosh,!&beta}.{'cosh,!&alpha}}); return solvealgupd(nv,nil); end; symbolic procedure solvealgtrig2 u; % r is a result from goesolve; remove trivial relations % like sin^2 + cos^2 = 1. begin scalar r,w,op,v,rh; for each s in cdr u do <<w := nil; for each e in s do % delete "sin u = sqrt(-cos u^2+1)" etc if eqcar(e,'equal) and (eqcar(cadr e,'sin) or eqcar(cadr e,'cos)) and (op := caadr e) and (v := cadr cadr e) and member(if eqcar(rh:=caddr e,'!*sq!*) then cadr rh else rh, subst({if op='sin then 'cos else 'sin,v},'!-form!-, '((MINUS (SQRT (PLUS (MINUS (EXPT !-form!- 2)) 1))) (SQRT (PLUS (MINUS (EXPT !-form!- 2)) 1))))) then nil else w:=e.w; w := reverse w; if not member(w,r) then r:=w.r; >>; return 'list . reverse r; end; symbolic procedure solvealghyp2 u; % r is a result from goesolve; remove trivial relations % like cosh^2 - sinh^2 = 1. begin scalar r,w,op,v,rh; for each s in cdr u do <<w := nil; for each e in s do % delete "sinh u = sqrt(cosh u^2-1)","cosh u = sqrt(sinh u^2+1)" if eqcar(e,'equal) and (eqcar(cadr e,'sinh) or eqcar(cadr e,'cosh)) and (op := caadr e) and (v := cadr cadr e) and member(if eqcar(rh:=caddr e,'!*sq!*) then cadr rh else rh, if op='sinh then subst({'cosh,v},'!-form!-, '((MINUS (SQRT (PLUS (EXPT !-form!- 2) 1))) (SQRT (PLUS (EXPT !-form!- 2) 1)))) else subst({'sinh,v},'!-form!-, '((MINUS (SQRT (PLUS (EXPT !-form!- 2) (MINUS 1)))) (SQRT (PLUS (EXPT !-form!- 2) (MINUS 1)))))) then nil else w:=e.w; w := reverse w; if not member(w,r) then r:=w.r; >>; return 'list . reverse r; end; symbolic procedure solvealggen(k,x); % k is a general function call; processable if SOLVE % can invert the function. begin scalar nv,m,s; if cdr x then error(99,"too many variables in function expression"); x := car x; solvealgvb k; nv := '( % old kernels ( !&alpha ) % new variables ( !&beta ) % substitutions ( ( !&alpha . !&beta) ) % inverses (( !&beta '(!&x (!&!& !&)))) % new equation nil); % invert the kernel expression. s := solvealginner(k,x); m := list('!&alpha . k, '!&beta . solve!-gensym(), '!&x . x, '!&!& . s); nv:=sublis!-pat(m , nv); return solvealgupd(nv,nil); end; symbolic procedure solvealgid k; % k is a "constant" kernel, however in a syntax unprocessable % for Groebner (e.g. expt(a/2)); replace temporarily begin scalar nv,m; nv := '( % old kernels ( !&alpha ) % new variables ( ) % substitutions ( ( !&alpha . !&beta) ) % inverses (( !&beta nil . !&alpha)) % new equation nil); % invert the kernel expression. m := list('!&alpha . k, '!&beta . solve!-gensym()); nv:=sublis(m , nv); return solvealgupd(nv,nil); end; symbolic procedure solvealginner(s,x); <<s := solveeval1 {{'equal,s,'!#},{'LIST,x}}; s := reval cadr s; if not eqcar(s,'EQUAL) or not equal(cadr s,x) then error (99,"inner expression cannot be inverted"); {'lambda,'(!#),caddr s}>>; symbolic procedure solvealgupd(u,innervars); % Update the system and the structures. begin scalar ov,nv,sub,inv,neqs; ov := car u; u := cdr u; nv := car u; u := cdr u; sub:= car u; u := cdr u; inv:= car u; u := cdr u; neqs:=car u; u := cdr u; for each x in ov do kl!*:=delete(x,kl!*); for each x in innervars do for each y in nv do depend1(y,x,t); sub!* := append(sub,sub!*); iv!* := append(nv,iv!*); inv!* := append(inv,inv!*); system!* := append( for each u in neqs collect <<u:= numr simp u; solvealgk0 u; u>>, for each u in system!* collect numr subf(u,sub) ); return t; end; symbolic procedure solvealginv u; % Reestablish the original variables, produce inverse % mapping and do complete value propagation. begin scalar v,r,s,m,lh,rh,y,z,tag,sub0,sub,!*expli,noarb,arbs; scalar abort; integer n; sub0 := for each p in sub!* collect (cdr p.car p); tag := t; r := for each sol in cdr u join <<sub := sub0; abort := v:= r:= s:= noarb :=arbs :=nil; if !*test_solvealg then <<prin2t "================================"; prin2t const!-vars!*; prin2t " next basis:"; writepri(mkquote sol,'only); >>; for each eqn in reverse cdr sol do <<lh := cadr eqn; rh := subsq(simp!* caddr eqn,s); if !*test_solvealg then writepri(mkquote {'equal,lh,prepsq rh},'only); !*expli:=member(lh,iv!*); % look for violated constant relations. if (y:=assoc(lh,const!-vars!*)) and constant_exprp prepsq rh and numr subtrsq(rh,simp cdr y) then abort := t; % look for a "negative" root. if memq(lh,root!-vars!*) and numberp(y:=reval{'sign,prepsq rh}) and y<0 then abort := t; if not !*expli then noarb := t; if !*expli and not noarb then << % assign value to free variables; for each x in uv!* do if !*arbvars and solvealgdepends(rh,x) and not member(x,fv!*) and not member(x,arbs) then <<z := mvar makearbcomplex(); y := z; v := x . v; r := simp y . r; % rh := subsq(rh,list(x.y)); % s := (x . y) . s; arbs:=x.arbs; >>; if not smemq('root_of,rh) then s:=(lh.prepsq rh).s else fv!*:=lh.fv!*; >>; if (m:=assoc(lh,inv!*))then <<m:=cdr m; lh :=car m; kl!* := eqn; if eqcar(lh,'cond) or eqcar(lh,'quote) then lh:=car(m:=eval lh); rh:=solvenlnrsimp subst(prepsq rh,'!&,cadr m)>>; % if local variable, append to substitution. if not member(lh,uv!*) and !*expli then << sub:=append(sub,{lh .(z:=prepsq subsq(rh,sub))}); if smember(lh,r) then r:=subst(z,lh,r) >>; % append to the final output. if (member(lh,uv!*) or not !*expli) % inhibit repeated same values. and not<< z:=subsq(rh,sub); n:=length member(z,r); n>0 and lh=nth(v,length v + 1 - n)>> then <<r:=z.r; v:=lh.v;>>; >>; % Classify result. % for each x in uv!* do % if tag and not member(x,v) and smember(x,r) then tag:=nil; if !*test_solvealg then if abort then yesp "ABORTED" else <<prin2t " --------> "; writepri(mkquote ('list .for each u in pair(v,r) collect {'equal,car u,prepsq cdr u}) ,'only); prin2t "================================"; yesp "continue?"; >>; if not abort then {reverse r . reverse v} >>; return solvealg!-verify(tag,r); end; symbolic procedure solvealgdepends(u,x); % inspect u for explicit dependency of x, being careful for % root_of subexpressions. if u=x then t else if atom u then nil else if eqcar(u,'root_of) then if x=caddr u then nil else solvealgdepends(cadr u,x) else solvealgdepends(car u,x) or solvealgdepends(cdr u,x); symbolic procedure test_trig(); begin scalar lh,rh,r; lh := cadr kl!*; rh:= caddr kl!*; if member(lh . nil, solvealgdb!*) then return nil; r := not !*complex and not smemq('i,kl!*) and not smemq('!:gi!:,kl!*) and not smemq('!:cr!:,kl!*) and not smemq('root_of,kl!*); if not r then solvealgdb!* := append(solvealgdb!*,{('sin.cdr lh).nil,('cos.cdr lh).nil}); return r; end; symbolic procedure test_hyp(); begin scalar lh,rh,r; lh := cadr kl!*; rh:= caddr kl!*; if member(lh . nil, solvealgdb!*) then return nil; r := not !*complex and not smemq('i,kl!*) and not smemq('!:gi!:,kl!*) and not smemq('!:cr!:,kl!*) and not smemq('root_of,kl!*); if not r then solvealgdb!* := append(solvealgdb!*,{('sinh.cdr lh).nil,('cosh.cdr lh).nil}); return r; end; fluid '(!*solvealg_verify); % the idea of the following procedure is to exclude isolated % solutions which give a substantial residue when subsituted % into the equation system under "on rounded"; as long as no % good criterion for a residue to be small has been found, this % step is disabled. symbolic procedure solvealg!-verify(tag,r); <<if !*rounded and !*solvealg_verify then begin scalar min,s,cmpl,!*msg; % exclude solutions with a residue substantially % above the minimum of all nonzero residues. cmpl:=!*complex; if not cmpl then setdmode('complex,!*complex:=t); s := for each u in r collect solvealg!-verify1 u.u; min:=simp'(quotient 1 100); r:= for each u in s join if null car u or minusf numr subtrsq(car u,min) then {cdr u}; if not cmpl then <<setdmode('complex,nil); !*complex:=nil>>; end; tag . for each q in r collect car q . cdr q . list 1 >>; symbolic procedure solvealg!-verify1 s; % verify solution s for the current equation system. begin scalar sub,nexpli,x,y,sum,fail; sub:= for each u in pair(cdr s,car s) collect if not nexpli then <<y:=prepsq cdr u; if not (domainp y or constant_exprp y) then nexpli:=t; car u.y>>; % a non explicit solution cannot be tested. if nexpli then return nil; sum := nil ./ 1; for each u in osystem!* do if not fail then <<x:=subf(u,sub); if domainp numr x then sum:=addsq(sum,absf numr x ./ denr x) else fail := t; >>; return if fail then nil else sum; end; symbolic procedure sublis!-pat(a,u); % like sublis, but replace lambda expressions by matching their % actual arguments. begin scalar v; if atom u then return <<v:=assoc(u,a); if v then sublis!-pat(a,cdr v) else u>>; v:=assoc(car u,a); if v and (v:=cdr v) and eqcar(v,'lambda) then return sublis!-pat((caadr v.cadr u).a,caddr v); return sublis!-pat1(a,u); end; symbolic procedure sublis!-pat1(a,l); if null l then nil else if atom l then sublis!-pat(a,l) else sublis!-pat(a,car l) . sublis!-pat1(a,cdr l); %---------------------------------------------------------------- % section for single trigonometric polynomials %---------------------------------------------------------------- symbolic procedure solvenonlnrtansub(p,x); % Perform tangent substitution. if not smemq('sin,p) and not smemq('cos,p) then if smemq(x,p) then nil else nil.p else if car p='cos then if smemq(x,cdr p) then (cdr p). '(quotient (difference 1(expt tg!- 2)) (plus 1(expt tg!- 2))) else nil.p else if car p='sin then if smemq(x,cdr p) then (cdr p). '(quotient (times 2 tg!-) (plus 1(expt tg!- 2))) else nil.p else (if ca and cd and (car ca = car cd or null car ca or null car cd) then (car ca or car cd).(cdr ca.cdr cd)) where ca=solvenonlnrtansub(car p,x), cd=solvenonlnrtansub(cdr p,x); symbolic procedure solvenonlnrtansolve(u,x,w); begin scalar v,s,z,r,y; integer ar; % We reset arbint for each solve call such that equal forms can % be recognized by the function union. ar := !!arbint; v:=caar u; u:= prepf numr simp cdr u; s:=solveeval {u,'tg!-}; !!arbint:=ar; for each q in cdr s do <<z:=reval caddr q; z:=reval sublis(solvenonlnrtansolve1 z,z); !!arbint:=ar; y:=solve0({'equal,{'tan,{'quotient,V,2}},z},x); r:=union(y,r); >>; % test for the special cases x=pi(not covered % by tangent substitution). if null numr subf(w,{x.'pi}) then <<!!arbint:=ar; r:=union(solve0({'equal,{'cos,x},-1},x),r)>>; return t.r; end; symbolic procedure solvenonlnrtansolve1 u; % Find all cos**2. if atom u then nil else if car u='expt and eqcar(cadr u,'cos) and caddr u=2 then {u . {'difference,1,{'expt,{'sin,cadr cadr u},2}}} else union(solvenonlnrtansolve1 car u,solvenonlnrtansolve1 cdr u); %---------------------------------------------------------------- % section for single hyperbolic polynomials %---------------------------------------------------------------- symbolic procedure solvenonlnrtanhsub(p,x); % Perform hyperbolic tangent substitution. if not smemq('sinh,p) and not smemq('cosh,p) then if smemq(x,p) then nil else nil.p else if car p='cosh then if smemq(x,cdr p) then (cdr p). '(quotient (plus 1(expt tgh!- 2)) (difference 1(expt tgh!- 2))) else nil.p else if car p='sinh then if smemq(x,cdr p) then (cdr p). '(quotient (times 2 tgh!-) (difference 1(expt tgh!- 2))) else nil.p else (if ca and cd and (car ca = car cd or null car ca or null car cd) then (car ca or car cd).(cdr ca.cdr cd)) where ca=solvenonlnrtanhsub(car p,x), cd=solvenonlnrtanhsub(cdr p,x); symbolic procedure solvenonlnrtanhsolve(u,x,w); begin scalar v,s,z,r,y,ar; ar := !!arbint; v:=caar u; u:= prepf numr simp cdr u; s:=solveeval {u,'tgh!-}; ar := !!arbint; for each q in cdr s do <<z:=reval caddr q; z:=reval sublis(solvenonlnrtanhsolve1 z,z); !!arbint:=ar; y:=solve0({'equal,{'tanh,{'quotient,v,2}},z},x); r:=union(y,r); >>; if !*complex and null numr subf(w,{x.'(times pi i)}) then <<!!arbint:=ar; r:=union(solve0({'equal,{'cosh,x},-1},x),r)>>; return t.r; end; symbolic procedure solvenonlnrtanhsolve1 u; % Find all cosh**2. if atom u then nil else if car u='expt and eqcar(cadr u,'cosh) and caddr u=2 then {u . {'plus,1,{'expt,{'sinh,cadr cadr u},2}}} else union(solvenonlnrtanhsolve1 car u,solvenonlnrtanhsolve1 cdr u); endmodule; end;